††institutetext: Max-Planck-Institut für Physik, Föhringer Ring 6, 80805
München, Germany

# A prescription for projectors to compute helicity amplitudes in D dimensions

Long Chen<EMAIL_ADDRESS>
###### Abstract

This article discusses a prescription to get polarized dimensionally
regularized amplitudes, providing a recipe for constructing simple and general
polarized amplitude projectors in D dimensions that avoids conventional
Lorentz tensor decomposition and avoids also dimensional splitting. Because of
the latter property commutation between Lorentz index contraction and loop
integration is preserved within this prescription, which entails many
technical advantages. The usage of these D-dimensional polarized amplitude
projectors results in helicity amplitudes that can be expressed solely in
terms of external momenta. Furthermore, we argue that despite being different
from the conventional dimensional regularization scheme (CDR), owing to the
amplitude-level factorization of ultraviolet and infrared singularities, our
prescription can be used, within an infrared subtraction framework, in a
hybrid way without re-calculating the (process-independent) integrated
subtraction coefficients, many of which are available in CDR. This hybrid CDR-
compatible prescription is shown to be unitary. We include two simple examples
to demonstrate this explicitly and also to illustrate its usage in practice.

###### Keywords:

Perturbative QCD, helicity amplitudes, regularization schemes, projection
method

††preprint: MPP-2019-39

##  1 Introduction

Helicity scattering amplitudes in Quantum Field Theory (QFT) encode the full
dependence on the spin degrees of freedom of the particles involved in the
scattering, and are the building blocks for computing various kinds of
physical observables through which we try to understand the interactions among
particles observed in nature. The incorporation of spin degrees of freedom, or
polarization effects, in terms of spin- respectively polarization-dependent
physical observables, leads to a richer phenomenology. Such observables offer
valuable means to discriminate different dynamical models, in particular for
discovering potential Beyond-Standard-Model effects. For a review of the role
of particle polarizations in testing the Standard Model and searching for new
physics, we refer to refs. Lampe:1998eu ; Leader:2001gr ; Accomando:1997wt ;
MoortgatPick:2005cw  and references therein.

Unlike physical observables, individual scattering amplitudes in QFT generally
possess infrared111We use the term “infrared” (IR) to denote both soft and
collinear divergences. (IR) and ultraviolet (UV) divergences, and thus a
regularization scheme (RS) for handling these intermediate divergences needs
to be introduced. Dimensional regularization tHooft:1972tcz ; Bollini:1972ui
is by far the most convenient one to use in gauge theories as it respects
gauge and Lorentz invariance222We ignore the treatment of $\gamma_{5}$ in
dimensional regularization for the moment. and allows one to handle both UV
and IR divergences in the same manner. The key ingredient of dimensional
regularization is the analytic continuation of loop momenta to $D=4-2\epsilon$
space-time dimensions with indefinite $\epsilon$. Having done this, one is
still left with some freedom regarding the dimensionality of the momenta of
the external particles, of algebraic objects like the space-time metric tensor
and Dirac matrices, as well as the number of polarizations of both external
and internal particles. This gives rise to different dimensional
regularization variants (for a review see e.g. ref. Gnendiger:2017pys  and
references therein), which in general leads to different expressions for
singular amplitudes. Apparently the RS dependence is intimately connected to
the singularity structures of amplitudes, which fortunately obey a nice
factorization form at the amplitude level Sen:1982bt ; Collins:1989bt ;
Catani:1998bh ; Sterman:2002qn ; Dixon:2008gr ; Gardi:2009qi ; Gardi:2009zv ;
Becher:2009cu ; Becher:2009kw ; Becher:2009qa ; Feige:2014wja . The result for
a physical quantity, such as a physical cross section which is free of any
such divergence, must not depend on the RS that has been used. However, in
practice, such a result is obtained as a sum of several partial contributions,
which usually are individually divergent and computed separately before being
combined. Therefore, these intermediate results can depend on the RS, and have
to be computed consistently to ensure the cancellation of the spurious RS-
dependence.

The conventional dimensional regularization (CDR)333By the acronym “CDR” we
refer in this article to the usual CDR Collins:1984xc  where, in addition,
$\gamma_{5}$ is treated by Larin’s prescription Larin:1991tj ; Larin:1993tq .
scheme Collins:1984xc  is a very popular RS, where all vector bosons are
treated as D-dimensional objects. It is conceptually the simplest one and does
guarantee a consistent treatment. It is typically employed in calculating
(unpolarized) amplitude interferences where the sum over the polarizations of
an external particle is conveniently made by using the respective unpolarized
Landau density matrix. For computing helicity amplitudes at the loop level,
the two commonly used RS are the 't Hooft-Veltman (HV) scheme tHooft:1972tcz
and the Four-Dimensional-Helicity (FDH) scheme Bern:1991aq ; Bern:2002zk . In
the FDH, the usage of spinor-helicity representations DeCausmaecker:1981wzb ;
DeCausmaecker:1981jtq ; Gunion:1985vca ; Kleiss:1985yh ; Xu:1986xb ;
Kleiss:1986qc ; Dittmaier:1998nn ; Schwinn:2005pi ; Arkani-Hamed:2017jhn  and
unitarity-cut based methods Bern:1994zx ; Bern:1994cg ; Bern:1997sc ;
Britto:2004nc ; Bern:2007dw  lead to compact expressions for helicity
amplitudes, which are computationally very advantageous, while the proper
renormalization procedure for non-supersymmetric theories beyond one loop
order requires some expertise Kilgore:2011ta ; Kilgore:2012tb ;
Gnendiger:2016cpg ; Gnendiger:2017rfh . Another widely used dimensional
regularization variant, the Dimensional-Reduction (DRED) scheme Siegel:1979wq
, was initially devised for application to supersymmetric theories and was
later shown to be applicable also to non-supersymmetric theories Capper:1979ns
; Jack:1993ws . The DRED and FDH have much in common, while there are also
subtle differences between the two Bern:2002zk ; Kilgore:2012tb ;
Broggio:2015dga ; Gnendiger:2017pys .

For computing D-dimensional helicity amplitudes, especially for amplitudes at
the loop level, one typically uses the projection method Karplus:1950zza ;
Kniehl:1990iva ; Binoth:2002xg  which is based on Lorentz covariant tensor
decomposition of scattering amplitudes (with external state vectors being
stripped off). The entire dependence of loop amplitudes on loop integrals is
encoded in the Lorentz invariant decomposition coefficients which multiply the
relevant Lorentz tensor structures. Lorentz tensor decomposition is employed,
for instance, at one loop in the Passarino-Veltman reduction procedure
Passarino:1978jh , and also in the systematic constructions of dimensionally
regularized QCD helicity amplitudes Abreu:2018jgq ; Boels:2018nrr . During the
last decades, there have been many computations done for high order QCD
corrections to scattering amplitudes using the projection method, for example,
Glover:2003cm ; Glover:2004si ; Binoth:2006hk ; Gehrmann:2011aa ;
Gehrmann:2013vga ; Gehrmann:2015ora ; vonManteuffel:2015msa ;
Bernreuther:2004ih ; Bernreuther:2004th ; Bernreuther:2005gw ;
Bernreuther:2005rw ; Aglietti:2004nj ; Moch:2005id ; Gehrmann:2005pd ;
Bonciani:2008wf ; Bonciani:2008az ; Asatrian:2008uk ; Beneke:2008ei ;
Bell:2008ws ; Assadsolimani:2014oga ; Ablinger:2017hst ; Liu:2017axv ;
Henn:2016tyf ; Lee:2018nxa ; Lee:2018rgs ; Ablinger:2018yae ; Ahmed:2015qpa ;
Borowka:2016ehy ; Heinrich:2017bvg ; Chen:2017jvi ; Jones:2018hbb ;
Anastasiou:2018fjr ; Maltoni:2018zvp ; vonManteuffel:2019wbj .

Despite being very generic, versatile, and widely used in many high-order
applications, there are a few aspects of the Lorentz tensor decomposition
approach that makes the traditional projection method not so easy to be
carried out in certain cases, as will be discussed in the next section. For
example, besides facing complexities in deriving D-dimensional projectors for
tensor decomposition coefficients in some multiple-parton, multiple-scale
scattering processes, evanescent Lorentz structures444The evanescent Lorentz
structures appearing in a Lorentz tensor decomposition should not be confused
with operator mixings in the renormalization of composite operators in
effective field theories Collins:1984xc ; Buras:1989xd , nor with evanescent
terms in the DRED or FDH regularized Lagrangian vanDamme:1984ig ; Jack:1993ws
; Jack:1994bn ; Stockinger:2005gx ; Harlander:2006rj ; Bern:2002zk ;
Kilgore:2011ta ; Kilgore:2012tb ; Gnendiger:2016cpg ; Gnendiger:2017rfh . can
appear in the D-dimensional basis for the loop amplitudes in question. Their
presence can lead to intermediate spurious poles in the resulting
D-dimensional projectors Binoth:2002xg ; Glover:2004si ; Gehrmann:2015ora .
Furthermore, when there are several external fermions involved in the
scattering Glover:2004si ; Abreu:2018jgq , the complete and linearly
independent set of basis structures in D dimensions will generally increase
with the perturbative order at which the virtual amplitude is computed (as the
Dirac algebra is formally infinite-dimensional in non-integer D dimensions).

Given the impressive long list of high-order QCD calculations of important
phenomenological consequences done in CDR and, moreover, having in mind the
aforementioned critical features of D-dimensional Lorentz tensor
decomposition, it should be justified to think of possible add-ons in order to
facilitate the computations of polarized amplitudes in a way fully compatible
with CDR. In this article we propose an alternative regularization
prescription of external states (for both bosons and fermions) in order to
avoid Lorentz tensor decomposition in the conventional projection method for
extracting helicity amplitudes. The prescription outlined below is devised to
be fully compatible with CDR so that certain results known in CDR can be
directly recycled.

As will become clear in following sections, the idea is based on the following
simple observation. In 4 dimensions, there are only four linearly independent
Lorentz 4-vectors, and hence any Lorentz 4-vector can be expressed linearly
using just three linearly independent Lorentz 4-vectors with the aid of the
Levi-Civita tensor. Therefore all polarization vectors can be built up by just
using three linearly independent external momenta in a Lorentz covariant way,
provided that there are enough linearly independent momenta involved in the
process. This basic mathematical fact is of course well known, and without
surprise it was already exploited about forty years ago in calculating (tree-
level) multiple photon bremsstrahlung processes in massless QED
DeCausmaecker:1981wzb ; DeCausmaecker:1981jtq . It was initially used for
simplifying the massless QED vertex by rewriting the slashed photon
polarization vector in terms of the slashed momenta of external charged
fermions (from which the photon was radiated), a trick that preluded the
introduction of the 4-dimensional massless spinor-helicity formalism
Gunion:1985vca ; Kleiss:1985yh ; Xu:1986xb ; Kleiss:1986qc . In this article,
instead of seeking simplifications of the gauge interaction vertices of
fermions in 4-dimensional massless theories, this mathematical fact is
employed for finding a CDR-compatible way to directly project out polarized
loop amplitudes, circumventing Lorentz tensor decomposition. Furthermore,
despite being different from CDR, we would like to argue that thanks to the
amplitude-level factorization of UV and IR singularities, such a prescription
can be used in a hybrid way together with results known in CDR to obtain RS-
independent finite remainders of loop amplitudes, without the need to
recalculate the integrated subtraction coefficients involved in an IR
subtraction framework. In other words, we will show that such a hybrid CDR-
compatible prescription is unitary in the sense defined in refs.
vanDamme:1984ig ; Catani:1996pk .

The article is organized as follows. In the next section, the conventional
projection method for computing polarized amplitudes is reviewed with comments
on a few aspects which motivated the work presented in this article. In
section 3 the proposed prescription to obtain polarized dimensionally
regularized scattering amplitudes is outlined in detail. Section 4 is devoted
to the discussion of the unitarity of the hybrid regularization prescription
of section 3. In particular we show that pole-subtracted RS-independent finite
remainders are always obtained and furthermore demonstrate this feature in the
context of an IR subtraction method. In section 5, we provide two simple
examples of calculating finite remainders of one-loop virtual amplitudes in
order to illustrate the usage of the prescription and to comment on a few
practical points worthy of attention. We conclude in section 6.

##  2 A Recap of the Projection Method

In this section, we review the projection method for computing polarized
amplitudes, and discuss a few aspects that motivated the work in this article.

The projection method Karplus:1950zza ; Kniehl:1990iva ; Binoth:2002xg , based
on Lorentz covariant tensor decomposition, can be used to obtain helicity
amplitudes for a generic scattering process at any loop order. The entire
dependence of scattering amplitudes on loop integrals is encoded in their
Lorentz-invariant decomposition coefficients that multiply the corresponding
Lorentz tensor structures and are independent of the external particles’
polarization vectors. These Lorentz-invariant decomposition coefficients are
sometimes called form factors of the amplitudes, a relativistic generalization
of the concept of charge distributions. In order to extract these form factors
containing dimensionally regularized loop integrals, projectors defined in D
dimensions should be constructed and subsequently applied directly to the
Feynman-diagrammatic expression of the amplitude, which can proceed diagram by
diagram.

###  2.1 Gram matrix and projectors

Scattering amplitudes in QFT with Poincaré symmetry are multi-linear in the
state vectors of the external particles, i.e., proportional to the tensor
product of all external polarization vectors, to all loop orders in
perturbative calculations, as manifestly shown by the Feynman diagram
representations. The color structure of QCD amplitudes can be conveniently
described using the color-decomposition Berends:1987cv ; Mangano:1987xk ;
Mangano:1987kp ; Mangano:1988kk ; Bern:1990ux  or the color-space formalism of
ref. Catani:1996vz . QCD amplitudes are thus viewed as abstract vectors in the
color space of external colored particles. Since projecting QCD amplitudes
onto the factorized color space and spin (Lorentz) structures can be done
independently of each other, we suppress for ease of notation possible color
indices of scattering amplitudes in the following discussions.

As nicely summarized and exploited in Boels:2017gyc ; Boels:2018nrr , every
scattering amplitude is a vector in a linear space spanned by a finite set of
Lorentz covariant structures, in dimensional regularization at any given
perturbative order. These structures are constrained by physical requirements
such as on-shell kinematics and symmetries of the dynamics. Scattering
amplitudes can thus be written as a linear combination of a set of chosen
Lorentz basis structures, where the decomposition coefficients are functions
of Lorentz invariants of external kinematics. All non-rational dependence of
the decomposition coefficients on external kinematics appear via loop
integrals. This implies the following linear ansatz for a scattering amplitude
$\hat{\mathcal{M}}$ at a fixed perturbative order,

$\displaystyle\hat{\mathcal{M}}=\sum_{n=1}^{N_{P}}c_{n}~{}\hat{T}_{n}\,,$ (1)

where each form factor $c_{n}$ is a function of Lorentz invariants of external
momenta, and each Lorentz structure $\hat{T}_{n}$ is multi-linear in the
external polarization state vectors. In general, $\hat{T}_{n}$ contains
contractions of external gauge bosons’ polarization state vectors with either
the space-time metric tensor connecting two different polarizations or with
external momenta, and contains also products of Dirac matrices sandwiched
between external on-shell spinors. The Levi-Civita tensor can also occur if
the scattering process involves parity-violating vertices. The complete and
linearly independent set of Lorentz structures for $\hat{\mathcal{M}}$ at any
given perturbative order depends on its symmetry properties as well as the
Lorentz and Dirac algebra in use.

Note that, as discussed in detail for the four-quark scattering amplitude
$q\bar{q}\rightarrow Q\bar{Q}$ in Glover:2004si ; Abreu:2018jgq , the complete
and linearly independent set of D-dimensional basis structures must in general
be enlarged according to the perturbative order at which $q\bar{q}\rightarrow
Q\bar{Q}$ is computed, because the Dirac algebra is infinite-dimensional for
non-integer dimensions. At each perturbative order only a finite number of
linearly independent Lorentz structures can appear in an amplitude, as is
evident from inspecting the corresponding Feynman diagrams which is a set of
finite elements.

To be specific, we consider in the following the Lorentz tensor decomposition
of scattering amplitudes in CDR at fixed order in perturbation theory. In the
following discussion of the projection method, we investigate also how to
uncover linear dependent relations among a set of (preliminarily chosen)
Lorentz tensor structures arising from on-shell constraints, without making
explicit reference to the origin of these linear dependencies.

Let us assume that by construction the set of the $N_{P}$ Lorentz structures
$\hat{T}_{n}$ in eq. (1), denoted by
$\mathbf{T}_{P}\equiv\\{\hat{T}_{1},\cdots,\hat{T}_{N_{P}}\\}$, is linearly
complete for the $\hat{\mathcal{M}}$ in question, but the $\hat{T}_{n}$ may
not be linearly independent of each other. For an analogy we recall the
representation of QCD amplitudes in terms of a set of color structures in
color space without demanding linear independence of these color structures.
Let us thus call eq. (1) a primitive Lorentz covariant decomposition of
$\hat{\mathcal{M}}$. Possible linear relations among the $N_{P}$ Lorentz
structures $\hat{T}_{n}$ due to Lorentz and/or Dirac algebra and also on-shell
constraints, such as equations of motion as well as transversality satisfied
by external state vectors, can be uncovered by computing their $N_{P}\times
N_{P}$ Gram matrix $\hat{\mathrm{\mathbf{G}}}$, whose matrix elements are
defined as

$\displaystyle\hat{\mathrm{\mathbf{G}}}_{ij}=\langle\hat{T}^{\dagger}_{i},\hat{T}_{j}\rangle.$
(2)

The symbol $\langle\hat{T}^{\dagger}_{i},\hat{T}_{j}\rangle$ denotes the
Lorentz invariant inner product between these two linear Lorentz structures.
It is typically defined as the trace of the matrix product of $\hat{T}_{i}$’s
Hermitian conjugate, i.e. $\hat{T}^{\dagger}_{i}$, and $\hat{T}_{j}$ with
tensor products of external state vectors (spinors) being substituted by the
corresponding unpolarized Landau density matrices. In other words, this
Lorentz invariant quantity can be viewed as the unpolarized interference
between two linear Lorentz structures $\hat{T}_{i}$ and $\hat{T}_{j}$ over all
helicity states of external particles in accordance with certain polarization
sum rules (encoded in the unpolarized Landau density matrices).

This $N_{P}\times N_{P}$ Gram matrix $\hat{\mathrm{\mathbf{G}}}$ in eq. (2)
can then be used to determine the linearly independent subset of
$\mathbf{T}_{P}$ spanning the vector space where the considered amplitude
$\hat{\mathcal{M}}$ lives. If the determinant of $\hat{\mathrm{\mathbf{G}}}$
is not identically zero, then the set $\mathbf{T}_{P}$ is both complete and
linearly independent, and thus qualifies as a basis of the vector space where
$\hat{\mathcal{M}}$ lives. Otherwise, $\hat{\mathrm{\mathbf{G}}}$ is not a
full-rank matrix, and its matrix rank
$N_{R}\equiv\mathrm{R}[\hat{\mathrm{\mathbf{G}}}]$ tells us the number of
linearly independent members of $\mathbf{T}_{P}$. Since $\mathbf{T}_{P}$ is
assumed to be linearly complete w.r.t. $\hat{\mathcal{M}}$ by construction,
$N_{R}$ is thus the number of basis elements of a linear basis of the vector
space that contains $\hat{\mathcal{M}}$.

The number $N_{P}-N_{R}$ of linear dependent relations in $\mathbf{T}_{P}$ can
be extracted from the null-space of this Gram matrix
$\hat{\mathrm{\mathbf{G}}}$. Technically, the null-space of a matrix (not
necessarily a square matrix) is the solution space of the corresponding
homogeneous system of linear algebraic equations defined by taking this matrix
as the system’s coefficient matrix. The null-space of
$\hat{\mathrm{\mathbf{G}}}$ can be conveniently represented as a list of
linearly independent $N_{P}$-dimensional basis vectors of the solution space
of the homogeneous linear algebraic system defined by
$\hat{\mathrm{\mathbf{G}}}$. The number of members of this list is equal to
the dimension of $\hat{\mathrm{\mathbf{G}}}$ minus its matrix rank, i.e.,
$N_{P}-N_{R}$. For the information we would like to extract555To just identify
the linearly dependent columns and/or rows of the multivariate Gram matrix,
numerical samples of this matrix at a few test points are usually enough.,
this null-space provides the complete set of linear combination coefficients
(being rational in the external kinematics) of the column vectors of
$\hat{\mathrm{\mathbf{G}}}$ that lead to vanishing $N_{P}$-dimensional
vectors. After having removed those linearly dependent columns (and their
corresponding transposed rows), we end up with a reduced full-rank Gram matrix
among the thus-selected linearly independent set of Lorentz structures,
denoted by $\mathbf{T}_{R}$. The set $\mathbf{T}_{R}$ can then be directly
taken as the basis of the vector space of $\hat{\mathcal{M}}$.

Elimination of redundancies in the set $\mathbf{T}_{P}$ for
$\hat{\mathcal{M}}$ involving external gauge bosons, due to Ward identities of
local gauge interactions, can be effectively accounted for by choosing
physical polarization sum rules for those external gauge bosons (with their
reference vectors chosen as momenta of other external particles). This point
can be easily seen once we realize that any unphysical structure, which may
happen to be just one specific $\hat{T}_{n}$ or a linear combination of some
of them (with rational coefficients in external kinematics), gets nullified by
the physical polarization sum rules of external gauge bosons. Notice, however,
reduction in the number of linearly independent basis structures of
$\hat{\mathcal{M}}$ due to additional process-specific symmetries such as
charge, parity, and/or Bose symmetry is not achieved by analyzing
$\hat{\mathrm{\mathbf{G}}}$ in this way. Instead they have to be accounted for
from the outset when determining the primitive set $\mathbf{T}_{P}$ in eq.
(1).

In terms of the thus-determined basis $\mathbf{T}_{R}$, the linear
decomposition of $\hat{\mathcal{M}}$ can be recast into

$\displaystyle\hat{\mathcal{M}}=\sum_{n=1}^{N_{R}}\tilde{c}_{n}~{}\hat{T}_{n}~{},~{}$
(3)

and the Gram matrix $\hat{\mathrm{\mathbf{G}}}_{R}$ of $\mathbf{T}_{R}$ with
matrix elements defined similarly as eq. (2) is now an invertible $N_{R}\times
N_{R}$ matrix.

Now we are ready to discuss projectors $\hat{P}_{n}$ for the Lorentz
decomposition coefficients (or form factors) $\tilde{c}_{n}$ of $\hat{T}_{n}$
in eq. (3). They are defined by

$\displaystyle\tilde{c}_{n}=\langle\hat{P}^{\dagger}_{n},\hat{\mathcal{M}}\rangle~{}~{}~{}\text{for
any $n~{}\in~{}\\{1,\cdots,N_{R}\\}$}\,,$ (4)

where the same Lorentz-invariant inner product operation as in eq. (2) is used
in the above projection. The defining equation (4) of $\hat{P}_{n}$ holds for
any linear object from the vector space spanned by the basis $\mathbf{T}_{R}$,
rather than just for a particular scattering amplitude $\hat{\mathcal{M}}$.
Inserting eq. (3) into eq. (4) then, taking the aforementioned property into
account, the defining equation for the projectors translates into

$\displaystyle\langle\hat{P}^{\dagger}_{n},\hat{T}_{m}\rangle=\delta_{nm}\quad\text{for
any $n,m~{}\in~{}\\{1,\cdots,N_{R}\\}$}\,.$ (5)

Each projector $\hat{P}^{\dagger}_{n}$ can be expressed in terms of a linear
combination of Hermitian conjugate members of $\mathbf{T}_{R}$ that span also
a vector space. We thus write

$\displaystyle\hat{P}^{\dagger}_{n}=\sum_{k=1}^{N_{R}}\left(\hat{\mathrm{\mathbf{H}}}\right)_{nk}~{}\hat{T}^{\dagger}_{k}\,,$
(6)

where the elements $\left(\hat{\mathrm{\mathbf{H}}}\right)_{nk}$ are to be
determined. Inserting eq. (6) into eq. (5) and using the definition of Gram
matrix elements we get

$\displaystyle\sum_{k=1}^{N_{R}}\left(\hat{\mathrm{\mathbf{H}}}\right)_{nk}~{}\left(\hat{\mathrm{\mathbf{G}}}_{R}\right)_{km}=\delta_{nm}~{},\quad\text{i.e.,}\;\hat{\mathrm{\mathbf{H}}}~{}\hat{\mathrm{\mathbf{G}}}_{R}=\hat{1}.$
(7)

Recall that $\hat{\mathrm{\mathbf{G}}}_{R}$ is invertible by the
aforementioned trimming procedure. This then answers the question of how to
construct the projectors $\hat{P}^{\dagger}_{n}$ from linear combinations of
the Hermitian conjugates of $\mathbf{T}_{R}$. In the special and ideal case of
a norm-orthogonal basis $\mathbf{T}_{R}$, its Gram matrix
$\hat{\mathrm{\mathbf{G}}}_{R}$ is equal to the identity matrix of dimension
$N_{R}$ and hence
$\hat{\mathrm{\mathbf{H}}}=\hat{\mathrm{\mathbf{G}}}_{R}^{-1}=\hat{1}$.
Subsequently, we have $\hat{P}^{\dagger}_{n}=\hat{T}^{\dagger}_{n}$, as is
well known for a norm-orthogonal basis.

By taking the Dirac traces and keeping all Lorentz indices in D dimensions in
the projection, these Lorentz-invariant tensor decomposition coefficients, or
form factors, are evaluated in D dimensions. These form factors are
independent of the external polarization vectors, and all their non-rational
dependence on external momenta is confined to loop integrals. Scalar loop
integrals appearing in these form factors can be reduced to a finite set of
master integrals with the aid of linear integration-by-parts identities
Tkachov:1981wb ; Chetyrkin:1981qh . Once these dimensionally regularized form
factors have been determined, external particles’ state vectors can be
conveniently chosen in 4 dimensions, leading to helicity amplitudes in
accordance with the HV scheme. In fact, once the (renormalized) virtual
amplitudes are available at hand in such a D-dimensional tensor-decomposed
form (with all (singular) Lorentz-invariant form factors computed in D
dimensions), then changing the regularization convention for the external
particles’ states consistently in both the virtual amplitude and the
corresponding IR-subtraction terms, should not alter the finite remainder that
is left after subtracting all poles, although the individual singular pieces
do change accordingly.

###  2.2 Comments on the D-dimensional projection

We now discuss a few delicate aspects of the Lorentz tensor decomposition in D
dimensions that motivated the work presented in this article.

In general, the Gram matrix $\hat{\mathrm{\mathbf{G}}}$ or
$\hat{\mathrm{\mathbf{G}}}_{R}$ computed using Lorentz and Dirac algebra in
CDR depends on the space-time dimension D. We can examine its 4-dimensional
limit by inserting $D=4-2\epsilon$ and check whether its determinant power-
expanded in $\epsilon$ is zero or not in the limit $\epsilon=0$. A determinant
vanishing at $\epsilon=0$ implies the presence of Lorentz structures in the
D-dimensional linearly independent basis set $\mathbf{T}_{R}$ that are
redundant in 4 dimensions.

To be more specific, we can compute the matrix rank of
$\hat{\mathrm{\mathbf{G}}}_{R}$ at $D=4$, denoted by
$\mathrm{R}[\hat{\mathrm{\mathbf{G}}}_{R}^{D=4}]$, and the difference
$N_{R}-\mathrm{R}[\hat{\mathrm{\mathbf{G}}}_{R}^{D=4}]$ tells us the number of
Lorentz structures appearing in $\mathbf{T}_{R}$ that are redundant in D=4.
Furthermore, if we compute the null-space of the 4-dimensional limit of
$\hat{\mathrm{\mathbf{G}}}$, then we can explicitly uncover all these special
linear relations among $\hat{T}_{n}$ due to the constraint of integer
dimensionality666Any potential non-linear relation among the $\hat{T}_{n}$ is
irrelevant here as we use a linear basis. in a similar way as one identifies
$\mathbf{T}_{R}$ out of $\mathbf{T}_{P}$. These special linear relations can
be used to construct exactly the number
$N_{R}-\mathrm{R}[\hat{\mathrm{\mathbf{G}}}_{R}^{D=4}]$ of evanescent Lorentz
structures out of $\mathbf{T}_{R}$ that are non-vanishing in D dimensions but
vanishing in 4 dimensions. In this way, the original basis set
$\mathbf{T}_{R}$ can be re-cast into a union of two subsets: one is linearly
independent and complete in 4 dimensions, and the other one only consists of
$N_{R}-\mathrm{R}[\hat{\mathrm{\mathbf{G}}}_{R}^{D=4}]$ evanescent Lorentz
structures. Such a reformulation of the Lorentz tensor decomposition basis in
D dimensions can be very useful in exhibiting the additional non-four-
dimensional structures involved in the virtual amplitude.

In case the number of structures in $\mathbf{T}_{R}$ is not very small (say,
not less than 10) and if there are several kinematic variables involved,
algebraically inverting $\hat{\mathrm{\mathbf{G}}}_{R}$ can be computationally
quite cumbersome Boels:2018nrr . Moreover, the resulting projectors
constructed in the above fashion may be hardly usable if the amplitudes
themselves are already quite complicated. This situation occurs naturally in
multiple-parton multiple-scale scattering processes. Possible simplifications
may be obtained by suitably recombining the linear basis structures in
$\mathbf{T}_{R}$ classified into several groups, such that they are mutually
orthogonal or decoupled from each other Boels:2018nrr . For example, we could
divide the set of tensor structures into symmetric and anti-symmetric sectors,
and also choose the anti-symmetrized product basis for strings of Dirac
matrices Dugan:1990df ; Gehrmann:2011aa . This amounts to choosing the basis
structures in $\mathbf{T}_{R}$ such that a partial triangularization of the
corresponding Gram matrix $\hat{\mathrm{\mathbf{G}}}_{R}$ is achieved already
by construction. This will facilitate the subsequent inversion operation, and
also make the results simpler. In addition, in case the set of tensor
structures all observe factorized forms in terms of products of a smaller set
of lower rank tensor structures, then this factorization can also be exploited
to greatly facilitate the construction of projectors Binoth:2002xg .
Alternatively, it is also a good practice to “compactify” the vector space as
much as possible, before the aforementioned construction procedure is applied,
by employing all possible physical constraints and symmetries, such as parity
and/or charge symmetry of the amplitudes in question, and also by fixing the
gauge of the external gauge bosons Gehrmann:2013vga ; vonManteuffel:2015msa ;
Boels:2017gyc ; Boels:2018nrr .

Other than the aforementioned technical complexity in inverting the Gram
matrix, there is another delicate point about the Lorentz tensor decomposition
approach in D dimensions, as already briefly mentioned above. In cases where
the external state consists only of bosons, a list of fixed number of Lorentz
tensor structures is indeed linearly complete in D dimensions to all orders in
perturbation theory Binoth:2002xg ; Gehrmann:2013vga . However, if there are
external fermions involved in the scattering, the complete and linearly
independent set of basis structures will generally increase with the
perturbative order at which the scattering amplitude is computed, because the
Dirac algebra is formally infinite dimensional in non-integer D dimensions, as
discussed for the four-quark scattering amplitude $q\bar{q}\rightarrow
Q\bar{Q}$ in Glover:2004si ; Abreu:2018jgq . Of course, at each given
perturbative order only a finite number of linearly independent Lorentz
structures can appear in an amplitude, because the corresponding Feynman
diagrams are just a set of finite elements. These additional D-dimensional
Lorentz structures are either evanescent by themselves or will lead to
additional evanescent structures of the same number computed by the procedure
discussed above.

The last comment we would like to make about the projection method in D
dimensions is the possible appearance of intermediate spurious poles in these
projectors Binoth:2002xg ; Glover:2004si ; Gehrmann:2015ora , which are
closely related to the presence of the aforementioned evanescent Lorentz
structures in the D-dimensional linearly independent basis. Since the presence
of evanescent Lorentz structures in the D-dimensional basis implies a Gram
matrix that vanishes in 4 dimensions, one expects that projectors resulting
from its inverse can contain poles in $D-4=-2\epsilon$, for instance in
Binoth:2002xg  for four-photon scattering. Of course, all intermediate
spurious poles generated this way in individual form factors projected out
should cancel in the physical amplitudes composed out of them, such as
helicity amplitudes or linearly polarized amplitudes.

All these sometimes cumbersome issues discussed above motivated the work that
will be presented in the following: the construction of simple and general
polarized amplitude projectors in D dimensions that avoids conventional
Lorentz tensor decomposition, yet is still fully compatible with CDR.

##  3 The Prescription

The idea behind the proposed prescription to obtain polarized dimensionally
regularized scattering amplitudes can be briefly outlined as follows, with
details to be exposed in the subsequent subsections.

For external gauge bosons of a scattering amplitude, massless and/or massive,
we decompose each external polarization vector in terms of external momenta.
We then keep the form of Lorentz covariant decomposition fixed while formally
promote all its open Lorentz indices, which are now all carried by external
momenta, from 4 dimensions to D dimensions, like every Lorentz vector in CDR.
If external fermions are present in the scattering amplitude, strings of Dirac
matrices sandwiched between external on-shell spinors will show up. For each
open fermion line, we first rewrite this quantity as a trace of products of
Dirac matrices with the aid of external spinors’ Landau density matrices, up
to an overall Lorentz-invariant normalization factor. The space-like
polarization vectors of a massive spinor can also be represented in terms of
external momenta. Again, once such a momentum basis representation is
established in 4 dimensions, the Lorentz covariant form will be kept fixed
while all open Lorentz and Dirac indices, carried by either external momenta
or Dirac matrices, will be respectively promoted in accordance with CDR.

As scattering amplitudes are multi-linear in the state vectors of the external
particles to all loop orders in perturbation theory, the tensor products of
momentum basis representations of all external gauge bosons and all properly
re-written external spinor products, with their open indices promoted
accordingly as in CDR, will be taken as the external projectors for polarized
amplitudes. Helicity amplitude projectors of a generic scattering process
defined in this way naturally obey a simple factorized pattern as the tensor
product of the respective polarization projector of each external gauge boson
and open fermion line. Features and subtleties worthy of attention during
these rewriting procedures will be discussed and explained below.

###  3.1 Momentum basis representations of polarization vectors

Let us start with the cases with only bosons in the external states. We recall
that the polarization vector $\varepsilon^{\mu}_{\lambda}(p)$ of a physical
vector-boson state of momentum $p^{\mu}$ has to satisfy
$\varepsilon^{\mu}_{\lambda}(p)~{}p_{\mu}=0$. Here the subscript $\lambda$
labels the number of physical spin degrees of freedom, i.e., $\lambda=1,2,3$
in D = 4. By convention the physical polarization state vectors are orthogonal
and normalized by
$\varepsilon^{*}_{\lambda}(p)\cdot\varepsilon_{\lambda^{\prime}}(p)=-\delta_{\lambda\lambda^{\prime}}$.
The polarization vectors of a massless gauge boson obey an additional
condition in order to encode the correct number of physical spin degrees of
freedom. In practice, this additional condition is usually implemented by
introducing an auxiliary reference vector $\hat{r}_{\mu}$ that is not aligned
with the boson’s momentum but otherwise arbitrary, to which the physical
polarization vectors have to be orthogonal,
$\varepsilon^{\mu}_{\lambda}(p)~{}\hat{r}_{\mu}=0$. Thus, the reference vector
$\hat{r}_{\mu}$ and the bosons’ momentum $p_{\mu}$ define a plane to which the
massless gauge boson’s physical polarization vectors are orthogonal. We also
recall that in CDR the number of physical polarizations of a massless gauge
boson in D dimension is taken to be D-2. This is in contrast to our
prescription, where the number of physical polarizations remains two in D
dimensions, see below.

####  3.1.1 The $2\to 2$ scatterings among massless gauge bosons

Let us first consider a prototype $2\to 2$ scattering among 4 external
massless gauge bosons:

$\displaystyle g_{1}(p_{1})+g_{2}(p_{2})\rightarrow
g_{3}(p_{3})+g_{4}(p_{4}),$ (8)

with on-shell conditions $p_{j}^{2}=0,~{}j=1,...,4.$ The Mandelstam variables
associated with (8)

$\displaystyle
s\equiv\left(p_{1}+p_{2}\right)^{2}=\left(p_{3}+p_{4}\right)^{2}~{},\qquad
t\equiv\left(p_{2}-p_{3}\right)^{2}=\left(p_{1}-p_{4}\right)^{2}$ (9)

encode the independent external kinematic invariants.

The representation of the gauge bosons’ polarization state vectors in terms of
three linearly independent external momenta, $p_{1},p_{2},p_{3}$ can be
determined in the following way. We first write down a Lorentz covariant
parameterization ansatz for the linear representation and then solve the
aforementioned orthogonality and normalization conditions for the linear
decomposition coefficients. Once we have established a definite Lorentz
covariant decomposition form in 4 dimensions solely in terms of external
momenta and kinematic invariants, this form will be used as the definition of
the corresponding polarization state vector in D dimensions.

While the decomposition of polarization state vectors in terms of external
momenta is Lorentz covariant, it is always helpful to have in mind a
particular reference frame so that a clear geometric picture can be used to
illustrate the choices of and constraints on polarization state vectors. To
this end, we consider in the following discussion the center-of-mass frame of
the two incoming particles, as illustrated in figure 1, where the beam axis is
taken as the Z-axis with its positive direction set along $p_{1}$.
Furthermore, the scattering plane determined by $p_{1}$ and $p_{3}$ is set as
the X-O-Z plane with $p_{3}$ having a non-negative X-component by definition.
The positive direction of the Y-axis of the coordinate system will be
determined according to the right-hand rule. The reference frame is shown in
figure 1.

Figure 1: The chosen coordinate system in the center-of-mass reference frame
of the two incoming particles.

Let’s now come to the momentum basis representations of the polarization
vectors in this reference frame. There are two common basis choices regarding
the transverse polarization states, the linear and the circular polarization
basis, the latter represents helicity eigenstates of gauge bosons. These two
bases can be related via a ${\pi}/{2}$ rotation in the complex plane. In the
following, we will first establish a Lorentz covariant decomposition of a set
of elementary linear polarization state vectors in terms of external 4-momenta
and then compose circular polarization states of all external gauge bosons,
i.e., their helicity eigenstates, out of these elementary ones. It is
beneficial to postpone such an explicit basis transformation until the very
last step of the calculation, as will become clear from the later discussions.

For the two initial-state massless gauge bosons $g_{1}(p_{1})$ and
$g_{2}(p_{2})$, whose momenta are taken as the reference momenta for each
other, we first introduce a common linear polarization state vector
$\varepsilon_{X}^{\mu}$ along the X-axis direction, i.e., transverse to the
beam axis but within the X-O-Z plane. The set of equations that determines
$\varepsilon_{X}^{\mu}$ reads:

$\displaystyle\varepsilon^{\mu}_{X}=c^{X}_{1}~{}p^{\mu}_{1}+c^{X}_{2}~{}p^{\mu}_{2}+c^{X}_{3}~{}p^{\mu}_{3}\,,$
$\displaystyle\varepsilon_{X}\cdot p_{1}=0\,,$
$\displaystyle\varepsilon_{X}\cdot p_{2}=0\,,$
$\displaystyle\varepsilon_{X}\cdot\varepsilon_{X}=-1\,.$ (10)

Solving eq. (3.1.1) for the coefficients
$c^{X}_{1},~{}c^{X}_{2},~{}c^{X}_{3}$, and subsequently inserting the solution
back to the first line of eq. (3.1.1), we obtain the following momentum basis
representation for $\varepsilon_{X}$:

$\displaystyle\varepsilon^{\mu}_{X}=\mathcal{N}_{X}\Big{(}t~{}p^{\mu}_{1}+(-s-t)~{}p^{\mu}_{2}+s~{}p^{\mu}_{3}\Big{)}\,,$
(11)

where $\mathcal{N}^{-2}_{X}=-ts(s+t)$. Notice that, as will be made clear
later, the overall Lorentz-invariant normalization factor $\mathcal{N}_{X}$
needs to be included only in the very last step of the computation of
polarized loop amplitudes, for instance after UV renormalization and IR
subtraction if an IR subtraction method is employed. Therefore, we never have
to deal with $\mathcal{N}_{X}$, i.e., with a square root explicitly in the
intermediate stages. If we choose to incorporate the overall normalization
factors only at the level of squared amplitudes (or interferences), then
square roots of kinematic invariants never appear. Furthermore, we can always
by convenience define this overall normalization factor such that the
coefficients exhibited in eq. (11) are polynomials in the external kinematic
invariants (rather than rational functions). This can be helpful as computer
algebra systems are typically more efficient when dealing with polynomials
only.

Concerning the two final-state massless gauge bosons $g_{3}(p_{3})$ and
$g_{4}(p_{4})$, whose momenta are also taken as reference momenta for each
other, we can introduce a common linear polarization state vector
$\varepsilon_{T}^{\mu}$ defined to be transverse to $p_{3}$ and $p_{4}$ but
still lying within the X-O-Z plane, in analogy to $\varepsilon_{X}^{\mu}$. The
definition of $\varepsilon_{T}^{\mu}$ then translates into the following set
of equations:

$\displaystyle\varepsilon^{\mu}_{T}=c^{T}_{1}~{}p^{\mu}_{1}+c^{T}_{2}~{}p^{\mu}_{2}+c^{T}_{3}~{}p^{\mu}_{3}\,,$
$\displaystyle\varepsilon_{T}\cdot p_{3}=0\,,$
$\displaystyle\varepsilon_{T}\cdot p_{4}=0\,,$
$\displaystyle\varepsilon_{T}\cdot\varepsilon_{T}=-1\,.$ (12)

Solving eq. (3.1.1) for the coefficients
$c^{T}_{1},~{}c^{T}_{2},~{}c^{T}_{3}$, one obtains

$\displaystyle\varepsilon^{\mu}_{T}=\mathcal{N}_{T}\Big{(}t~{}p^{\mu}_{1}+(s+t)~{}p^{\mu}_{2}+(-s-2t)~{}p^{\mu}_{3}\Big{)}\,,$
(13)

where $\mathcal{N}_{T}^{~{}-2}=-ts(s+t)$. The comments given above on
$\varepsilon_{X}^{\mu}$ apply here as well.

The last elementary polarization state vector needed for constructing helicity
eigenstates of all four external massless gauge bosons is the one orthogonal
to $p_{1}$, $p_{2}$, and $p_{3}$, denoted by $\varepsilon_{Y}$, which is thus
perpendicular to the X-O-Z plane. In 4 dimensions, we obtain it using the
Levi-Civita tensor:

$\displaystyle\varepsilon^{\mu}_{Y}=\mathcal{N}_{Y}~{}\epsilon^{\nu\rho\sigma\mu}p_{1\nu}p_{2\rho}p_{3\sigma}=\mathcal{N}_{Y}~{}\epsilon^{\mu}_{p_{1}p_{2}p_{3}}\,,$
(14)

where $\mathcal{N}_{Y}^{~{}-2}=-st(s+t)/4$, and in the last line we introduced
the notation
$\epsilon^{\mu}_{p_{1}p_{2}p_{3}}\equiv\epsilon^{\nu\rho\sigma\mu}p_{1\nu}p_{2\rho}p_{3\sigma}$.
We use the convention $\epsilon^{0123}=+1$ and
$\epsilon_{\mu\nu\rho\sigma}=-\epsilon^{\mu\nu\rho\sigma}$.

A comment concerning $\epsilon^{\mu\nu\rho\sigma}$ is appropriate here. The
above polarization state vectors will be eventually used in D-dimensional
calculations. To this end, following Larin:1991tj ; Larin:1993tq ;
Moch:2015usa , we will treat $\epsilon^{\mu\nu\rho\sigma}$ merely as a symbol
denoting an object whose algebraic manipulation rules consist of the following
two statements.

  * •

Antisymmetry: it is completely anti-symmetric regarding any odd permutation of
its arguments.

  * •

Contraction Rule777There is a subtle point concerning this when there are
multiple Levi-Civita tensors in the contraction, related to the choice of
pairing, as will be briefly commented on in section 5.2.: the product of two
$\epsilon^{\mu\nu\rho\sigma}$ is replaced by a combination of products of
space-time metric tensors $g^{\mu\nu}$ of the same tensor rank according to
the following fixed pattern:

$\displaystyle\epsilon^{\mu\nu\rho\sigma}\epsilon^{\mu^{\prime}\nu^{\prime}\rho^{\prime}\sigma^{\prime}}=\mathrm{Det}\Big{[}g^{\alpha\alpha^{\prime}}\Big{]}~{},\text{~{}
with $\alpha=\mu,~{}\nu,~{}\rho,~{}\sigma$ and
$\alpha^{\prime}=\mu^{\prime},~{}\nu^{\prime},~{}\rho^{\prime},~{}\sigma^{\prime}$,}$
(15)

which agrees with the well-known mathematical identity for Levi-Civita tensors
in 4 dimensions.

Using eq. (15) with the D-dimensional space-time metric-tensor in determining
$\mathcal{N}_{Y}$ in eq. (14), one gets
${\mathcal{N}_{Y}^{-2}}=(3-D)st(s+t)/4$ with $D=4-2\epsilon$. Because
$\mathcal{N}_{Y}$ is an overall normalization factor which must be used
consistently in computing both the (singular) virtual loop amplitudes, the UV-
renormalization counter-terms, as well as potential IR subtraction terms, it
is merely a normalization convention whether the explicit $D$ appearing in
$\mathcal{N}_{Y}$ is set to 4 or to $4-2\epsilon$, on which the final
4-dimensional finite remainder should not depend (albeit the individual
singular objects do of course differ). This point can be made even more
transparent if one chooses to incorporate this overall normalization factor
only in the very last stage of the consistent computation of finite remainders
where the 4-dimensional limit has already been explicitly taken.

The circular polarization state vectors of all four external massless gauge
bosons, namely their helicity eigenstates, can be easily constructed from the
three linear polarization states given above by a suitable ${\pi}/{2}$
rotation in the complex plane. The two helicity eigenstates of each gauge
boson are given by

$\displaystyle\varepsilon_{\pm}(p_{1};p_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\varepsilon_{X}\pm
i*\varepsilon_{Y}\right)\,,$ $\displaystyle\varepsilon_{\pm}(p_{2};p_{1})$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}\left(\varepsilon_{X}\mp
i*\varepsilon_{Y}\right)\,,$ $\displaystyle\varepsilon_{\pm}(p_{3};p_{4})$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}\left(\varepsilon_{T}\pm
i*\varepsilon_{Y}\right)\,,$ $\displaystyle\varepsilon_{\pm}(p_{4};p_{3})$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}\left(\varepsilon_{T}\mp
i*\varepsilon_{Y}\right)\,,$ (16)

where the first argument of $\varepsilon_{\pm}(p;r)$ is the particle’s
momentum while the second shows the reference momentum. Eq. (3.1.1) shows that
the helicity flips once the particle’s 3-momentum gets reversed or if the
polarization vector is subject to complex conjugation. Furthermore, owing to
the Ward identities fulfilled by the gauge amplitudes, the representations of
helicity state vectors in eq. (3.1.1) can be further reduced respectively for
each gauge boson by removing the component proportional to the gauge boson’s
own 4-momentum. For instance, for the gauge boson $g_{1}$ with 4-momentum
$p_{1}$, the component of $\varepsilon_{X}$ proportional to $p_{1}^{\mu}$ in
eq. (11) can be safely dropped when constructing
$\varepsilon_{\pm}(p_{1};p_{2})$, and similar reductions hold also for the
other gauge bosons. However, as will become clear in the following
discussions, it is beneficial to project polarized amplitudes first in the
linear polarization basis and then have the helicity amplitudes composed at
the last stage of the computation. Since these elementary linear polarization
state vectors will be used to construct helicity states of several scattered
particles, we should keep their complete momentum basis representation forms
as given by eqs. (11), (13), and (14).

We emphasize again that in our prescription the number of physical
polarizations in D dimensions of a massless gauge boson remains two, see eq.
(3.1.1). In order to illustrate resulting differences to CDR let us do a
simple exercise about polarization sums. In CDR the sum over the physical
polarizations of a massless gauge boson $g_{1}$ with 4-momentum $p_{1}^{\mu}$
and gauge reference vector $r^{\mu}=p^{\mu}_{2}$ (cf. eq. (8)) is

$\displaystyle\sum_{\bar{\lambda}=\pm,~{}D-4}\bar{\varepsilon}_{\bar{\lambda}}^{~{}\mu}(p_{1};p_{2})\bar{\varepsilon}_{\bar{\lambda}}^{~{}*\nu}(p_{1};p_{2})$
$\displaystyle=$
$\displaystyle-g^{\mu\nu}+\frac{p_{1}^{\mu}p_{2}^{\nu}+p_{2}^{\mu}p_{1}^{\nu}}{p_{1}\cdot
p_{2}}$ (17) $\displaystyle=$
$\displaystyle-g^{\mu\nu}+\frac{2}{s}\left(p_{1}^{\mu}p_{2}^{\nu}+p_{2}^{\mu}p_{1}^{\nu}\right)$

which is also the unpolarized Landau density matrix of the polarization states
of $g_{1}$. All Lorentz indices in (17) are D dimensional and the
$\bar{\lambda}$ labels the polarization states in D dimensions. On the other
hand, in our prescription we sum over just the two transverse polarization
states of $g_{1}$ that are defined by their respective momentum basis
representations in eqs. (11), (13). We get

$\displaystyle\sum_{\lambda=X,Y}\varepsilon^{~{}\mu}_{\lambda}(p_{1};p_{2})\varepsilon^{~{}*\nu}_{\lambda}(p_{1};p_{2})$
$\displaystyle=$
$\displaystyle\frac{1}{D-3}\left(-g^{\mu\nu}+\frac{D-2}{s}\left(p_{1}^{\mu}p_{2}^{\nu}+p_{2}^{\mu}p_{1}^{\nu}\right)\right)$
(18) $\displaystyle+$
$\displaystyle\frac{4-D}{D-3}\Bigg{(}\frac{t}{s(s+t)}p_{1}^{\mu}p_{1}^{\nu}+\frac{s+t}{st}p_{2}^{\mu}p_{2}^{\nu}+\frac{s}{t(s+t)}p_{3}^{\mu}p_{3}^{\nu}$
$\displaystyle~{}~{}~{}+\frac{1}{s+t}\left(p_{1}^{\mu}p_{3}^{\nu}+p_{3}^{\mu}p_{1}^{\nu}\right)-\frac{1}{t}\left(p_{2}^{\mu}p_{3}^{\nu}+p_{3}^{\mu}p_{2}^{\nu}\right)\Bigg{)}$

where, as part of the definition of this expression, we have rewritten the
product of two Levi-Civita tensors in
$\varepsilon^{\mu}_{Y}(p_{1};p_{2})\varepsilon^{*\nu}_{Y}(p_{1};p_{2})$ in
terms of space-time metric tensors. Apparently eq. (18) is not identical to
eq. (17),888Note that with our prescription unpolarized squared amplitudes are
supposed to be computed by incoherently summing over helicity amplitudes, and
not by using polarization sums like (18). but the two expressions agree of
course in D = 4 dimensions.

Before we move on to establish explicit momentum basis representations of
longitudinal polarization vectors for massive vector bosons and also for
massive fermions, let us emphasize that by construction these momentum basis
representations of polarization state vectors fulfill all the defining
physical constraints, i.e., orthogonality to momenta and reference vectors,
which are assured even if the open Lorentz indices (carried by either the
external momenta or the Levi-Civita symbol) are taken to be D-dimensional.

Mathematically, the procedure of determining norm-orthogonal polarization
vectors eqs. (3.1.1), (3.1.1) from a given set of linearly independent momenta
in 4 dimensions resembles the Gram-Schmidt orthogonalization procedure. Our
key insight here is that we establish these Lorentz covariant decomposition
representations in 4 dimensions in a form that facilitates the subsequent
promotion of their open Lorentz indices from 4 to D, resulting in expressions
which will be taken as their definitions in D dimensions.

####  3.1.2 Massive particles in the final state

Next we consider the scattering process eq. (8) but with massive final-state
vector bosons, for instance W or Z bosons, with on-shell conditions

$\displaystyle p_{1}^{2}=p_{2}^{2}=0~{},~{}~{}p_{3}^{2}=p_{4}^{2}=m^{2}\,.$
(19)

Concerning the three elementary transverse polarization state vectors,
$\varepsilon^{\mu}_{X}~{},~{}\varepsilon^{\mu}_{T}~{},~{}\varepsilon^{\mu}_{Y}$,
the above constructions can be repeated but with slightly different
kinematics. It is straightforward to arrive at the following explicit
representations:

$\displaystyle\varepsilon^{\mu}_{X}$ $\displaystyle=$
$\displaystyle\mathcal{N}_{X}\,\Big{(}(t-m^{2})~{}p^{\mu}_{1}+(-s-t+m^{2})~{}p^{\mu}_{2}+s~{}p^{\mu}_{3}\Big{)}\,,$
$\displaystyle\varepsilon^{\mu}_{T}$ $\displaystyle=$
$\displaystyle\mathcal{N}_{T}\Big{(}(t+m^{2})~{}p^{\mu}_{1}+(s+t-3m^{2})~{}p^{\mu}_{2}+(-s-2t+2m^{2})~{}p^{\mu}_{3}\Big{)}\,,$
$\displaystyle\varepsilon^{\mu}_{Y}$ $\displaystyle=$
$\displaystyle\mathcal{N}_{Y}~{}\epsilon^{\mu}_{p_{1}p_{2}p_{3}}\,,$ (20)

with the normalization factors

$\displaystyle\mathcal{N}_{X}^{~{}-2}$ $\displaystyle=$ $\displaystyle
s\left(-t(s+t)+2m^{2}t-m^{4}\right)\,,$ $\displaystyle\mathcal{N}_{T}^{~{}-2}$
$\displaystyle=$ $\displaystyle-st(s+t)+2m^{2}t(3s+2t)-m^{4}(s+8t)+4m^{6}\,,$
$\displaystyle\mathcal{N}_{Y}^{~{}-2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}s\left(-t(s+t)+2m^{2}t-m^{4}\right)\,,$ (21)

which, as already emphasized above, can always be conveniently chosen to be
incorporated only at the very last stage of the computation.

Compared to the massless case, the helicity eigenstates of massive gauge
bosons are reference-frame dependent and their helicities are not Lorentz-
invariant. Helicity eigenstates constructed from the above elementary linear
polarization state vectors are defined in the center-of-mass reference frame
of the two colliding particles. The third physical polarization state of a
massive gauge boson is described by the longitudinal polarization vector
(defined in the same reference frame), which has its spatial part aligned with
the momentum of the boson. For the massive particle $g_{3}(p_{3})$ these
conditions translate into the following set of equations for its longitudinal
polarization vector $\varepsilon_{L3}^{\mu}$:

$\displaystyle\varepsilon_{L3}^{\mu}=c^{L3}_{1}\left(p^{\mu}_{1}+p^{\mu}_{2}-p^{\mu}_{3}\right)+c^{L3}_{2}p^{\mu}_{3}\,,$
$\displaystyle\varepsilon_{L3}^{\mu}\cdot p_{3}=0\,,$
$\displaystyle\varepsilon_{L3}\cdot\varepsilon_{L3}=-1\,.$ (22)

Solving eq. (3.1.2) for $c^{L3}_{1},~{}c^{L3}_{2}$, one obtains

$\displaystyle\varepsilon^{\mu}_{L3}=\mathcal{N}_{L3}\Big{(}-2m^{2}~{}\left(p^{\mu}_{1}+p^{\mu}_{2}\right)+s~{}p^{\mu}_{3}\Big{)}\,,$
(23)

where $\mathcal{N}_{L3}^{~{}-2}=sm^{2}(s-4m^{2})$. For the massive vector
boson $g_{4}(p_{4})$ one gets for its longitudinal polarization vector
$\varepsilon_{L4}^{\mu}$:

$\displaystyle\varepsilon^{\mu}_{L4}=\mathcal{N}_{L4}\Big{(}(s-2m^{2})~{}\left(p^{\mu}_{1}+p^{\mu}_{2}\right)-s~{}p^{\mu}_{3}\Big{)}\,,$
(24)

where $\mathcal{N}_{L4}=\mathcal{N}_{L3}$. By construction the defining
physical properties, such as orthogonality to the momenta, are fulfilled by
these momentum basis representations, even if their open Lorentz indices are
taken to be D-dimensional. We emaphasize that in our prescription the number
of physical polarizations of a massive vector boson remains three in D
dimensions.

There are also polarization vectors associated with massive fermions. The
helicity eigenstate of a massive fermion with 4-momentum $k$ can be described
by a Dirac spinor, e.g. $u(k,S_{k})$, characterized by the normalized space-
like polarization vector $S_{k}^{\mu}$. And component-wisely

$\displaystyle
S_{k}^{\mu}=\left(\frac{|\vec{k}|}{m},\frac{k^{0}}{m}\frac{\vec{k}}{|\vec{k}|}\right)\,,$
(25)

where $k^{0}$ and $m$ are, respectively, the energy and mass of the massive
fermion, while $\vec{k}$ represents its 3-momentum. Interestingly, this
polarization vector has the same momentum basis decomposition form as the
longitudinal polarization vector of a massive vector boson (of the same
momentum), provided the same external kinematic configuration applies. By
identifying $p_{3}^{\mu}=k^{\mu}$, eq. (23) can be viewed as the momentum
basis representation of $S_{k}^{\mu}$ for the same external kinematic
configuration as above. Namely,

$\displaystyle
S_{k}^{\mu}=\mathcal{N}_{S_{k}}\Big{(}-2m^{2}~{}\left(p^{\mu}_{1}+p^{\mu}_{2}\right)+s~{}k^{\mu}\Big{)}$
(26)

with $\mathcal{N}_{S_{k}}^{~{}-2}=sm^{2}(s-4m^{2})$. This is because the set
of norm-orthogonal conditions that $S^{\mu}$ has to fulfill, namely $k\cdot
S=0~{},~{}S\cdot S=-1~{},~{}\vec{S}\parallel\vec{k}$, which are sufficient to
determine it up to an overall phase, are exactly the same as those that the
longitudinal polarization vector in eq. (23) has to fulfill.

###  3.2 Normalized tensor products of external spinors

In cases where external fermions are involved in scattering amplitudes,
strings of Dirac matrices sandwiched between external on-shell spinors will
show up. In order to evaluate each open fermion line using trace techniques,
we employ the standard trick of multiplying and dividing this quantity by
appropriate auxiliary Lorentz-invariant spinor inner products, which can be
traced back to ref. Bjorken:1966kh . Pulling out the chosen overall Lorentz-
invariant normalization factor, the rest can be cast into a trace of products
of Dirac matrices with the aid of Landau density matrices of external spinors.
The momentum basis representations of (massive) fermion’s space-like
polarization vectors, such as eq. (26), can be used in these density matrices.
For massless fermions, the spin density matrices are reduced to left-
respectively right-chirality projectors, which thus spares us from introducing
any explicit polarization vector in this case. This is because helicity states
of massless fermions coincide with chiral spinors.

From a single open fermion line in a Feynman diagram, we get a contribution
which can be generically written as
$\langle\psi_{A}|\hat{\mathrm{M}}|\psi_{B}\rangle$. The symbol
$\hat{\mathrm{M}}$ denotes a product of Dirac matrices with their Lorentz
indices either contracted or left open, and
$|\psi_{A}\rangle,~{}|\psi_{B}\rangle$ stand for the two external on-shell
Dirac spinors, either of $u$-type or $v$-type. Viewed as a spinor inner
product, $\langle\psi_{A}|\hat{\mathrm{M}}|\psi_{B}\rangle$ can always be
rewritten as a trace of a product of Dirac-matrices in the Dirac-spinor space:

$\displaystyle\langle\psi_{A}|\hat{\mathrm{M}}|\psi_{B}\rangle=\mathrm{Tr}\Big{[}|\psi_{B}\rangle\langle\psi_{A}|\hat{\mathrm{M}}\Big{]}.$
(27)

This formal rewriting is not really useful unless we can further exploit the
matrix structure of the external spinors’ tensor product
$|\psi_{B}\rangle\langle\psi_{A}|$ in the spinor space (explicitly in terms of
elementary Dirac matrices). To this end, we rewrite
$|\psi_{B}\rangle\langle\psi_{A}|$ by introducing an auxiliary spinor inner
product along the following line:

$\displaystyle|\psi_{B}\rangle\langle\psi_{A}|$ $\displaystyle=$
$\displaystyle\frac{\langle\psi_{B}|\hat{\mathrm{N}}|\psi_{A}\rangle}{\langle\psi_{B}|\hat{\mathrm{N}}|\psi_{A}\rangle}|\psi_{B}\rangle\langle\psi_{A}|$
(28) $\displaystyle=$
$\displaystyle\frac{1}{\langle\psi_{B}|\hat{\mathrm{N}}|\psi_{A}\rangle}|\psi_{B}\rangle\langle\psi_{B}|\hat{\mathrm{N}}|\psi_{A}\rangle\langle\psi_{A}|$
$\displaystyle=$
$\displaystyle\mathcal{N}_{AB}~{}|\psi_{B}\rangle\langle\psi_{B}|\hat{\mathrm{N}}|\psi_{A}\rangle\langle\psi_{A}|\,,$

where
$\mathcal{N}_{AB}\equiv({\langle\psi_{B}|\hat{\mathrm{N}}|\psi_{A}\rangle})^{-1}$.
The auxiliary matrix $\hat{\mathrm{N}}$ is only required to have a non-
vanishing matrix element $\langle\psi_{B}|\hat{\mathrm{N}}|\psi_{A}\rangle$
and otherwise can be chosen to be as simple as desired. For instance, for
massive external spinors of some particular helicity configurations,
$\hat{\mathrm{N}}$ may be chosen to be the identity matrix in spinor space,
provided that the spinor inner products between those helicity spinors are not
vanishing. A generally valid and simple choice is
$\hat{\mathrm{N}}=\gamma_{\mu}p^{\mu}$ with a 4-momentum $p^{\mu}$ that is not
linearly dependent on the on-shell momenta $p_{A}$ and $p_{B}$ of
$\langle\psi_{A}|$ and $|\psi_{B}\rangle$, respectively.

We manipulate eq. (28) further by first substituting Landau density matrices
for $|\psi_{A}\rangle\langle\psi_{A}|$ and $|\psi_{B}\rangle\langle\psi_{B}|$,
conventionally given by

$\displaystyle u(p,S_{p})\otimes\bar{u}(p,S_{p})$ $\displaystyle=$
$\displaystyle\left(\not{p}+m\right)\frac{1+\gamma_{5}\not{S}_{p}}{2}\,,$
$\displaystyle v(p,S_{p})\otimes\bar{v}(p,S_{p})$ $\displaystyle=$
$\displaystyle\left(\not{p}-m\right)\frac{1+\gamma_{5}\not{S}_{p}}{2}\,.$ (29)

Then we simplify the resulting composite Dirac matrix object before finally
obtaining a form that is suitable for being unambiguously used in eq. (27)
with the trace to be done in D dimensions.

There are several equivalent forms of these on-shell Dirac-spinors’ projectors
in 4 dimensions. In particular, one may commute the on-shell projection
operator $\not{p}\pm m$ and the polarization projection operator
$({1+\gamma_{5}\not{S}_{p}})/{2}$ using $p\cdot S_{p}=0$. However, it is well
known that a fully anticommuting $\gamma_{5}$ can not be implemented in
dimensional regularization in an algebraically consistent way Collins:1984xc ,
if we still want this object to coincide with the usual $\gamma_{5}$ in 4
dimensions. In this article, we adopt the $\gamma_{5}$ prescription of ref.
Larin:1991tj ; Larin:1993tq , conventionally known as Larin’s scheme, whose
equivalent but more efficient implementations in high-order perturbative
calculations are discussed in ref. Moch:2015usa . In our work, all appearances
of $\gamma_{5}$ matrices, originating either from interaction vertices or
external polarization projectors, should be regarded just for bookkeeping
purposes and their interpretations shall be based on Larin:1991tj ;
Larin:1993tq ; Moch:2015usa . As a consequence of this prescription,
$\gamma_{5}$ no longer anticommutes with all Dirac $\gamma$ matrices, and
4-dimensional equivalent forms of eq. (3.2) are no longer necessarily
algebraically equivalent in D dimensions.

In order to eliminate potential ambiguities — after having simplified (27),
(3.2) using 4-dimensional Lorentz and Dirac algebra as much as possible —, we
should agree on one definite fixed form of (27), (3.2), solely in terms of a
string of Dirac $\gamma$ matrices with fixed product ordering, the Levi-Civita
tensor, and external momenta. We may call these their canonical forms in 4
dimensions. This allows an unambiguous interpretation999This is up to a
potential subtlety related to the contraction order among multiple Levi-Civita
tensors Moch:2015usa , as will be commented on in section 5.2. of the
expression in D dimensions where it will be manipulated according to the
$D$-dimensional algebra after being inserted back into eq. (27).

Let us now be more specific about this by working out a representative case, a
single open fermion line with two massive external $u$-type spinors,
$u(p_{A},S_{A})$ and $u(p_{B},S_{B})$. We choose $\hat{\mathrm{N}}=\not{q}$
where $q^{\mu}$ is a 4-momentum that is linearly independent of $p_{A}$ and
$p_{B}$. Pulling out the normalization factor
$\mathcal{N}_{AB}=\left({\bar{u}(p_{A},S_{A})\not{q}u(p_{B},S_{B})}\right)^{-1}$,
eq. (28) reads in this case:

$\displaystyle\frac{1}{\mathcal{N}_{AB}}~{}u(p_{B},S_{B})\otimes\bar{u}(p_{A},S_{A})$
$\displaystyle=$
$\displaystyle\left(\not{p}_{B}+m\right)\frac{1+\gamma_{5}\not{S}_{B}}{2}\not{q}\frac{1+\gamma_{5}\not{S}_{A}}{2}\left(\not{p}_{A}+m\right)\,,$
(30)

which can be brought into the form

$\displaystyle\frac{1}{\mathcal{N}_{AB}}~{}u(p_{B},S_{B})\otimes\bar{u}(p_{A},S_{A})$
$\displaystyle=$
$\displaystyle\left(\not{p}_{B}+m\right)\frac{1}{4}\not{q}\left(\not{p}_{A}+m\right)$
(31) $\displaystyle+$
$\displaystyle\left(\not{p}_{B}+m\right)\frac{1}{4}\left(\frac{-i}{3!}\epsilon_{\gamma\gamma\gamma
S_{B}}\right)\not{q}\left(\not{p}_{A}+m\right)$ $\displaystyle+$
$\displaystyle\left(\not{p}_{B}+m\right)\frac{1}{4}\not{q}\left(\frac{-i}{3!}\epsilon_{\gamma\gamma\gamma
S_{A}}\right)\left(\not{p}_{A}+m\right)$ $\displaystyle+$
$\displaystyle\left(\not{p}_{B}+m\right)\frac{1}{4}\not{S}_{B}\not{q}\not{S}_{A}\left(\not{p}_{A}+m\right)\,.$

Strictly speaking, eq. (31) is identical to (30) only in 4 dimensions. The
unambiguous eq. (31), which no longer contains any explicit $\gamma_{5}$, will
be taken as the definition of $u(p_{B},S_{B})\otimes\bar{u}(p_{A},S_{A})$ when
it is inserted into eq. (27) and manipulated in accordance with the
D-dimensional algebra.

Notice that in eq. (30) the auxiliary matrix $\not{q}$ and the polarization
projection operators were placed inside the on-shell projection operators
$\not{p}_{\tiny{A/B}}+m$, a point which will be explained and become clear in
section 4.1. We emphasize again that the momentum basis representations of the
massive fermions’ helicity polarization vectors $S^{\mu}_{A}$ and
$S^{\mu}_{B}$ will be eventually inserted, whose open Lorentz indices are
carried by external momenta that are assumed to be D dimensional. Similar
rewritings and definitions like eq. (31) can be made also for fermion lines
with external $v$-type spinors, whose Landau density matrices are given in eq.
(3.2).

In practice, it is very convenient to keep projections associated with each of
the four terms in eq. (31) separate from each other, for two reasons. First,
this organization is in accordance with the power of the Levi-Civita tensor
appearing in the terms, which is advantageous especially when Feynman diagram
expressions are also split into terms with even and odd products of
$\gamma_{5}$ (arising from axial vertices). Second, for a fermion with fixed
momentum, its polarization vector, e.g. $S_{A}$ or $S_{B}$ in eq.(31), changes
just by an overall minus sign when its helicity is flipped. Therefore, the
expressions of eq. (27) for the four different helicity configurations can all
be obtained by suitably combining the traces in eq. (27) of the product of
$\hat{\mathrm{M}}$ and each of the four terms in eq. (31).

Notice that in general the normalization factor $\mathcal{N}_{AB}$ in eq. (27)
depends on the helicities of the external fermions A and B, as will be
explicitly shown in the example given in section 5.2. Using (31) we need to
compute these four individual projections separately just once, out of which
all four different helicity configurations can be obtained.

Once a definite unambiguous form of the right-hand side of eq. (31) has been
established in 4 dimensions, it will be kept fixed while all open Lorentz and
Dirac indices will be promoted in accordance with computations in CDR.
Additionally, just like the aforementioned normalization factors associated
with the gauge boson’s polarization vectors, the factor $\mathcal{N}_{AB}$ in
eq. (27) is an overall normalization factor which must be adopted consistently
in computing all amplitudes involved in the calculations of finite remainders.
If one chooses to incorporate this overall normalization factor only in the
very last stage of calculating finite remainders where the 4 dimensional limit
can already be taken, it is then evident that we can evaluate these Lorentz
invariant factors in 4 dimensions.

As already mentioned above, in the massless limit the spin density matrices in
eq. (3.2) are reduced to left- or right-chiral projectors. Thus no
polarization vectors are needed. For instance, the massless limit of eq. (31)
with $++$ helicity configuration reads:

$\displaystyle\frac{1}{\mathcal{N}_{AB}}~{}u(p_{B},+)\otimes\bar{u}(p_{A},+)$
$\displaystyle=$
$\displaystyle\not{p}_{B}\frac{1-\gamma_{5}}{2}\not{q}\frac{1+\gamma_{5}}{2}\not{p}_{A}$
(32) $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{(}\not{p}_{B}\not{q}\not{p}_{A}-\not{p}_{B}\left(\frac{-i}{3!}\epsilon_{\gamma\gamma\gamma
q}\right)\not{p}_{A}\Bigg{)}.$

The remarks below eq. (31) concerning the use of this equation in
D-dimensional calculations apply also to (32). The above reformulations of
tensor products of two external helicity spinors, such as eq. (31) and eq.
(32), can be applied to each single open fermion line, besides using for each
external boson a momentum basis representation of its polarization vector.

To summarize, the tensor product of momentum basis representations of all
external gauge bosons’ polarization vectors and all properly re-written
external spinor products, such as those given by eqs. (11) - (14), and eq.
(31), (32), with their open indices promoted in accordance with CDR, will be
taken as the external projectors for polarized amplitudes. Polarized
amplitudes are thus first projected in the linear polarization basis for
external gauge bosons and the basis indicated by eq. (31), (32) for each open
fermion line. It is a good practice to first combine Levi-Civita tensors that
appear in external projectors in order to reach an unambiguous canonical form
that is homogeneous in the Levi-Civita tensor whose power is at most
one101010See the end of section 5.2 for more discussions about this.. Helicity
amplitudes can be subsequently obtained from these these polarized amplitudes
by linear combinations, such as those implied in eq. (3.1.1). For instance,
the transformation matrix of polarized scattering amplitudes among four
massless gauge bosons from the linear to the circular polarization basis is a
$16\times 16$ constant matrix that can be extracted from eq. (3.1.1).
Likewise, constant transformation matrices can also be extracted from eq. (31)
for massive fermion lines and eq. (32) for massless fermion lines.

Eventually every helicity amplitude thus composed is manifestly given as a
function of Lorentz invariant variables solely made out external momenta. This
is owing to the fact that the momentum basis representations of polarization
vectors allow us to find a Lorentz covariant representation of the tensor
product of external particle states solely in terms of external momenta and
algebraic constants (such as the metric tensor, the Levi-Civita tensor and
Dirac matrices). Subsequently this makes it feasible to directly take these
objects as the external polarization projectors. From the point of view of the
projection method as outlined in section 2.1, the set of external polarization
projectors described above might be loosely viewed as a special choice of
Lorentz decomposition basis which by construction are orthogonal among each
other. Consequently, the corresponding Gram matrix is diagonal and its
inversion is trivial. Furthermore, each structure that arises from such a
decomposition is directly related to a physical quantity, and therefore its
singularity pattern is protected by physical constraints obeyed by these
physical quantities. In this way the issues related to the conventional form
factor decomposition as discussed in section 2.2 are circumvented.

###  3.3 Comments on other processes

In the preceding subsections we have discussed a prototype $2\to 2$ scattering
process where there are only three linearly independent external momenta and
consequently the Lorentz invariant scattering amplitude cannot contain a term
composed of one Levi-Civita tensor fully contracted with external momenta.
This fact can lead to a reduction of terms that are to be included in external
projectors. For instance, if the $2\to 2$ scattering process is parity-
invariant, then all terms in the external projectors that are linear in Levi-
Civita tensor can be dropped from the outset. This simplification does no
longer occur if there are more than four particles involved in the scattering,
e.g. in a $2\to 3$ process. We comment on a few technical aspects in handling
these cases.

####  $2\to 3$ scattering.

We have seen in section 3.1 and 3.2 that three linearly independent external
momenta are sufficient to build momentum basis representations of external
polarization vectors, and the concrete decomposition coefficients depend on
the particular kinematics. For constructing momentum basis representations of
polarization vectors for final-state particles, it is convenient to take a
group of three linearly independent external momenta of which two are always
chosen to be the momenta of the initial-state (massless) particles and the
third one is the one of the final state particle in question.

For convenience, let us document below the momentum basis representations of
linear polarization vectors introduced in section 3.1, but without
specializing the external kinematic configuration. We consider a generic
configuration with two massless initial state particles with momenta $p_{1}$
and $p_{2}$ (which is applicable to most of the phenomenologically interesting
high-energy scattering processes), while the mass of the particular final
state particle, with momentum labeled as $p_{3}$, in question is left
unspecified. These three external momenta are assumed to be linearly
independent. No specification is made of the kinematics of the other particles
in the final state.

We introduce the following symbols for kinematic invariants:

$\displaystyle s_{12}=2~{}p_{1}\cdot p_{2}~{},~{}~{}s_{13}=2~{}p_{1}\cdot
p_{3}~{},~{}~{}s_{23}=2~{}p_{2}\cdot p_{3}~{},~{}~{}m^{2}=p_{3}\cdot
p_{3}~{},$ (33)

which are assumed to be independent of each other. Repeating the construction
made in section 3.1, we obtain for this generic kinematic setting:

$\displaystyle\varepsilon^{\mu}_{X}$ $\displaystyle=$
$\displaystyle\mathcal{N}_{X}\Big{(}(-s_{23})~{}p^{\mu}_{1}+(-s_{13})~{}p^{\mu}_{2}+s_{12}~{}p^{\mu}_{3}\Big{)},$
$\displaystyle\varepsilon^{\mu}_{T}$ $\displaystyle=$
$\displaystyle\mathcal{N}_{T}\Big{(}(s_{23}(s_{13}+s_{23})-2m^{2}s_{12})~{}p^{\mu}_{1}+(-s_{13}(s_{13}+s_{23})+2m^{2}s_{12})~{}p^{\mu}_{2}+(s_{12}(s_{13}-s_{23}))~{}p^{\mu}_{3}\Big{)},$
$\displaystyle\varepsilon^{\mu}_{Y}$ $\displaystyle=$
$\displaystyle\mathcal{N}_{Y}~{}\epsilon^{\mu}_{p_{1}p_{2}p_{3}},$
$\displaystyle\varepsilon^{\mu}_{L3}$ $\displaystyle=$
$\displaystyle\mathcal{N}_{L3}\Big{(}-2m^{2}~{}\left(p^{\mu}_{1}+p^{\mu}_{2}\right)+(s_{13}+s_{23})~{}p^{\mu}_{3}\Big{)}\,,$
(34)

with normalization factors

$\displaystyle\mathcal{N}_{X}^{~{}-2}$ $\displaystyle=$ $\displaystyle
s_{12}\left(s_{13}s_{23}-m^{2}s_{12}\right)\,,$
$\displaystyle\mathcal{N}_{T}^{~{}-2}$ $\displaystyle=$ $\displaystyle
s_{12}\left(s_{13}s_{23}(s_{13}+s_{23})^{2}-m^{2}s_{12}(s_{13}^{2}+6s_{13}s_{23}+s_{23}^{2})+4m^{4}s_{12}^{2}\right)\,,$
$\displaystyle\mathcal{N}_{Y}^{~{}-2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}s_{12}\left(s_{13}s_{23}-m^{2}s_{12}\right)\,,$
$\displaystyle\mathcal{N}_{L3}^{~{}-2}$ $\displaystyle=$ $\displaystyle
m^{2}\left((s_{13}+s_{23})^{2}-4m^{2}s_{12}\right)\,.$ (35)

All previous comments on polarization vectors and normalization factors apply
here as well.

Due to the presence of four linearly independent external momenta in a $2\to
3$ process, the aforementioned reduction of terms in the external projectors
for $2\to 2$ processes no longer applies in general. However, this fact also
offers an opportunity to eliminate the explicit appearance of the Levi-Civita
tensor $\epsilon^{\mu\nu\rho\sigma}$ from external projectors (which may
already be pre-processed to be at most linear in
$\epsilon^{\mu\nu\rho\sigma}$), by applying the trick used in defining the van
Neerven-Vermaseren basis vanNeerven:1983vr . To be more specific, let us
consider a $2\to 3$ scattering process where the four linearly independent
4-momenta are denoted by $p_{1},p_{2},p_{3},p_{4}$. A single power
$\epsilon^{\mu\nu\rho\sigma}$ in external polarization projectors can be
rewritten as

$\displaystyle\epsilon^{\mu\nu\rho\sigma}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{p_{1}p_{2}p_{3}p_{4}}}{\epsilon_{p_{1}p_{2}p_{3}p_{4}}}~{}\epsilon^{\mu\nu\rho\sigma}$
$\displaystyle=$
$\displaystyle\Delta~{}\Big{(}p_{1}^{\rho}p_{2}^{\nu}p_{3}^{\mu}p_{4}^{\sigma}-p_{1}^{\nu}p_{2}^{\rho}p_{3}^{\mu}p_{4}^{\sigma}-p_{1}^{\rho}p_{2}^{\mu}p_{3}^{\nu}p_{4}^{\sigma}+p_{1}^{\mu}p_{2}^{\rho}p_{3}^{\nu}p_{4}^{\sigma}+p_{1}^{\nu}p_{2}^{\mu}p_{3}^{\rho}p_{4}^{\sigma}-p_{1}^{\mu}p_{2}^{\nu}p_{3}^{\rho}p_{4}^{\sigma}$
$\displaystyle~{}~{}~{}~{}~{}-p_{1}^{\rho}p_{2}^{\nu}p_{3}^{\sigma}p_{4}^{\mu}+p_{1}^{\nu}p_{2}^{\rho}p_{3}^{\sigma}p_{4}^{\mu}+p_{1}^{\rho}p_{2}^{\sigma}p_{3}^{\nu}p_{4}^{\mu}-p_{1}^{\sigma}p_{2}^{\rho}p_{3}^{\nu}p_{4}^{\mu}-p_{1}^{\nu}p_{2}^{\sigma}p_{3}^{\rho}p_{4}^{\mu}+p_{1}^{\sigma}p_{2}^{\nu}p_{3}^{\rho}p_{4}^{\mu}$
$\displaystyle~{}~{}~{}~{}~{}+p_{1}^{\rho}p_{2}^{\mu}p_{3}^{\sigma}p_{4}^{\nu}-p_{1}^{\mu}p_{2}^{\rho}p_{3}^{\sigma}p_{4}^{\nu}-p_{1}^{\rho}p_{2}^{\sigma}p_{3}^{\mu}p_{4}^{\nu}+p_{1}^{\sigma}p_{2}^{\rho}p_{3}^{\mu}p_{4}^{\nu}+p_{1}^{\mu}p_{2}^{\sigma}p_{3}^{\rho}p_{4}^{\nu}-p_{1}^{\sigma}p_{2}^{\mu}p_{3}^{\rho}p_{4}^{\nu}$
$\displaystyle~{}~{}~{}~{}~{}-p_{1}^{\nu}p_{2}^{\mu}p_{3}^{\sigma}p_{4}^{\rho}+p_{1}^{\mu}p_{2}^{\nu}p_{3}^{\sigma}p_{4}^{\rho}+p_{1}^{\nu}p_{2}^{\sigma}p_{3}^{\mu}p_{4}^{\rho}-p_{1}^{\sigma}p_{2}^{\nu}p_{3}^{\mu}p_{4}^{\rho}-p_{1}^{\mu}p_{2}^{\sigma}p_{3}^{\nu}p_{4}^{\rho}+p_{1}^{\sigma}p_{2}^{\mu}p_{3}^{\nu}p_{4}^{\rho}\Big{)},$

where the normalization factor
$\Delta\equiv\frac{1}{\epsilon_{p_{1}p_{2}p_{3}p_{4}}}$ can be conveniently
pulled out and grouped together with other normalization factors of external
projectors (and used consistently through out the whole calculation). The
treatment of the Levi-Civita tensor in eq. (3.3) complies with the two rules
listed in section 3.1. In this way, no Levi-Civita tensor appears in external
polarization projectors for $2\to 3$ scattering amplitudes any more, up to a
global normalization factor, and hence it is manifest that the form of
external projectors can be unambiguously constructed.

####  $1\to 2$ decay.

For a $1\to 2$ decay amplitude, the conventional Lorentz tensor decomposition
and projection method can be carried out quite simply (due to the limited
number of basis structures and scales). For instance, for the fermion’s gauge
interaction vertex a general form factor decomposition can be found in
Hollik:1998vz . Here we briefly comment on how one can compute polarized $1\to
2$ decay amplitudes if one wants to use the above prescription.

The computation requires the introduction of an intermediate auxiliary
reference-vector, denoted by $\hat{r}^{\mu}$, which will be formally treated
on the same footing as an external 4-momentum. The reference-vector
$\hat{r}^{\mu}$ may be associated with the polarization vector of the decaying
particle (in which case it has a physical meaning), or chosen to be an
auxiliary coordinate-system-specific vector merely for intermediate usage. The
important point we would like to emphasize here is that the definition of
$\hat{r}^{\mu}$ can be achieved by simply specifying the values of a complete
set of quadratic Lorentz invariant products between $\hat{r}^{\mu}$ and two
linearly independent external momenta, which we denote by $p_{1}$ and $p_{2}$.
For instance, the normalized space-like $\hat{r}^{\mu}$ can be implicitly
specified by

$\displaystyle\hat{r}\cdot p_{1}=0~{},~{}~{}\hat{r}\cdot
p_{2}=0~{},~{}~{}\hat{r}\cdot\hat{r}=-1,$ (37)

which guarantees that it lies in the plane transverse to $p_{1}$ and $p_{2}$.
This set of assignments (37) is sufficient to algebraically manipulate
$\hat{r}$ in the computation of polarized $1\to 2$ decay amplitudes. There is
no need for its explicit component-wise specification in a definite
coordinate-reference system. With the aid of the thus-defined $\hat{r}$, all
procedures outlined above for the $2\to 2$ scattering processes, discussed in
section 3.1, can be repeated here. To be a bit more specific, in this case the
set of three linearly independent 4-vectors $\\{p_{1},p_{2},\hat{r}\\}$ will
take over the roles that were played by the three linearly independent
external momenta $\\{p_{1},p_{2},p_{3}\\}$ in the $2\to 2$ scattering
processes. In fact the $\hat{r}$ defined in eq. (37) fulfills the same set of
conditions that $\varepsilon_{X}$ in eq. (3.1.1) satisfies. Moreover, it never
appears in Feynman propagators111111This means that the sectors of loop
integrals appearing in the projected amplitudes will not be enlarged by the
introduction of this external reference-vector $\hat{r}$., and the Lorentz
invariants appearing in the resulting projections are still just those made
out of $p_{1}$ and $p_{2}$ (as the right-hand side of eq. (37) are all
constants). In the end the physical decay rates are independent of the choice
of this auxiliary $\hat{r}$. In the case of a scalar decaying into a pair of
fermions, the introduction of such an auxiliary vector could be avoided
because the helicity polarization vector of a massive fermion, eq. (26), makes
no reference at all to any transverse direction w.r.t. its momentum.

##  4 Unitarity of the Prescription

The potential RS dependence of amplitudes is intimately connected to the
structure of their UV and IR singularities. Fortunately, in QCD they obey a
factorized form at the amplitude level Sen:1982bt ; Collins:1989bt ;
Catani:1998bh ; Sterman:2002qn ; Dixon:2008gr ; Gardi:2009qi ; Gardi:2009zv ;
Becher:2009cu ; Becher:2009kw ; Becher:2009qa ; Feige:2014wja . The final
result for a physical quantity, for instance a cross section, is of course
finite and must not depend on the RS used.

The usage of the polarization projectors defined in the previous sections
yields helicity amplitudes that differ in general from those defined in many
existing dimensional regularization variants, in particular the CDR. In this
section, we argue that our prescription of external state vectors will however
lead to the same RS-independent finite remainders as for instance in CDR, and
can therefore be used in a hybrid way with CDR to achieve a maximal
convenience owing to the amplitude-level factorization of UV and IR
singularities in QCD amplitudes.

###  4.1 Pole subtracted amplitudes

We recall that in the D-dimensional Lorentz decomposition representation of a
scattering amplitude, the Lorentz-invariant form factors encode all dependence
on dimensionally regularized loop integrals and are independent of the
external polarization vectors. Once the (renormalized) loop amplitudes are
available in such a tensor decomposed form, with all (singular) Lorentz-
invariant form factors computed in D dimensions, then merely changing the RS
for the external particles’ state vectors, consistently both for the loop
amplitudes and the corresponding IR subtraction terms, should not alter the
finite remainders resulting from subtracting all poles and subsequently taking
the 4-dimensional limit121212The equivalence between CDR and HV in leading to
the same RS-independent finite remainders with the identical set of
renormalization constants and anomalous dimensions Kilgore:2012tb ;
Broggio:2015dga  can be appreciated this way, and the same arguments apply
here as well.. Because in the form-factor representation of an amplitude the
loop-integral dependent part is separated from the part depending on the
external states, it is thus unambiguous to implement whatever non-CDR
convention for external state vectors in the computation of singular
amplitudes. The crucial question for our purpose is whether our non-CDR
prescription for external state vectors can still be unambiguously and
directly applied in the computation of amplitudes without performing the form
factor decomposition first.

In our prescription all open Lorentz indices of the polarization projectors
defined in section 3 are set to be D-dimensional and no dimensional splitting
is ever introduced, just like in CDR. Thus, commutation between Lorentz index
contraction and loop integration is preserved within our prescription. This
means that applying our polarization projectors directly to the original
Feynman-diagrammatic representation of a loop amplitude should lead to the
same polarized amplitudes as would be obtained by applying these projectors to
the D-dimensional form-factor decomposition representation of that amplitude.
No matter whether or not evanescent Lorentz structures appear explicitly or
implicitly in the form-factor decomposition of the loop amplitude, they are
taken into account exactly as they are in the original Feynman-diagrammatic
representation of this amplitude. From this perspective we could already
expect to end up with the same (4-dimensional) finite remainder as one would
obtain from a computation purely within CDR.

Below we demonstrate this crucial point more clearly via providing an
alternative formulation of finite remainders introduced in the proposed
prescription, which also helps to clarify a few points alluded in the
preceding section. Let us consider the finite remainders of amplitudes in CDR
as defined by the celebrated amplitude-level factorization formula.
Singularities in the dimensionally regularized QCD amplitudes are known to
factorize Sen:1982bt ; Collins:1989bt ; Catani:1998bh ; Sterman:2002qn ;
Dixon:2008gr ; Gardi:2009qi ; Gardi:2009zv ; Becher:2009cu ; Becher:2009kw ;
Becher:2009qa ; Feige:2014wja . For our purpose, we can sketch this
factorization property of a bare QCD scattering amplitude
$\hat{\mathcal{A}}(\epsilon)$ among several resolved external particles (with
fixed external kinematics) schematically as follows:

$\displaystyle\hat{\mathcal{A}}(\epsilon)=\hat{\mathcal{Z}}_{\mathrm{IR}}(\epsilon)~{}\mathcal{Z}_{\mathrm{UV}}(\epsilon)~{}\hat{\mathcal{F}}(\epsilon)\,,$
(38)

where131313The need of mass renormalizations in the case of massive quarks is
understood. we have suppressed the dependence of the quantities on external
kinematics and masses as well as on auxiliary dimensional scales except the
dimensional regulator $\epsilon$, (for a detailed exposition, see e.g.
Catani:1998bh ; Feige:2014wja ; Broggio:2015dga ; Magnea:2018ebr  and
references therein). The bare amplitude $\hat{\mathcal{A}}(\epsilon)$ and the
finite pole-subtracted amplitude $\hat{\mathcal{F}}(\epsilon)$ should be
viewed as vectors in the color space of the external particles, and the
multiplicative singular IR-factor $\hat{\mathcal{Z}}_{\mathrm{IR}}(\epsilon)$
as a matrix. The RS-dependent singular factors
$\mathcal{Z}_{\mathrm{UV}}(\epsilon)$ and
$\hat{\mathcal{Z}}_{\mathrm{IR}}(\epsilon)$ encode all UV and IR pole-
singularities of $\hat{\mathcal{A}}(\epsilon)$, and are independent of the
detailed kinematic configuration, such as polarization states, of the external
resolved particles. (This is the meaning of “factorization”.) By the very
meaning of pole factorization in eq. (38), $\hat{\mathcal{F}}(\epsilon)$ is
regular in $\epsilon$ and has a finite 4-dimensional limit,
$\hat{\mathcal{F}}(\epsilon=0)$. We call this quantity the (4-dimensional)
finite remainder of $\hat{\mathcal{A}}(\epsilon)$ defined by subtracting all
poles minimally by the multiplicative factors as sketched in eq. (38).

We may summarize this by the following expression for the finite remainder
$\hat{\mathcal{F}}_{4}\equiv\hat{\mathcal{F}}(\epsilon=0)$, namely

$\displaystyle\hat{\mathcal{F}}_{4}=\Bigg{(}\hat{\mathcal{Z}}^{-1}_{\mathrm{IR};\mathrm{CDR}}(\epsilon)~{}\mathcal{Z}^{-1}_{\mathrm{UV};\mathrm{CDR}}(\epsilon)~{}\hat{\mathcal{A}}_{\mathrm{CDR}}(\epsilon)\Bigg{)}_{\epsilon=0}$
(39)

where we added the subscript “CDR” to all singular RS-dependent quantities
given in CDR. For the point to be demonstrated here, the concrete expressions
of these singular multiplicative factors taken from CDR are irrelevant. The
claim is that replacing all CDR-regularized external states of the fixed-angle
bare scattering amplitude $\hat{\mathcal{A}}_{\mathrm{CDR}}(\epsilon)$ by
their respective counterparts given in terms of momentum basis representations
defined in section 3 will still result in the same finite remainder
$\hat{\mathcal{F}}_{4}$, where all poles have been subtracted in a minimal way
by the same untouched
$\hat{\mathcal{Z}}^{-1}_{\mathrm{IR};\mathrm{CDR}}(\epsilon)~{}\mathcal{Z}^{-1}_{\mathrm{UV};\mathrm{CDR}}(\epsilon)$,
without appealing to the Lorentz tensor decomposition representation of
$\hat{\mathcal{A}}_{\mathrm{CDR}}(\epsilon)$.

In order to facilitate the discussion, let us exhibit the dependence of
$\hat{\mathcal{A}}_{\mathrm{CDR}}(\epsilon)$ on the CDR-regularized
polarization state $\bar{\varepsilon}_{\bar{\lambda}}(p_{i},r_{i})$ of a
representative external massless gauge boson with momentum $p_{i}$ and
reference vector $r_{i}$. Because the bare scattering amplitude
$\hat{\mathcal{A}}_{\mathrm{CDR}}$ is linear in
$\bar{\varepsilon}_{\bar{\lambda}}(p_{i},r_{i})$, we write

$\hat{\mathcal{A}}_{\mathrm{CDR}}\Big{(}\epsilon;\bar{\varepsilon}_{\bar{\lambda}}(p_{i},r_{i})\Big{)}=g_{\mu\nu}~{}\Big{(}\hat{\mathcal{A}}_{\mathrm{CDR}}\Big{)}^{\mu}~{}\bar{\varepsilon}^{~{}\nu}_{\bar{\lambda}}(p_{i},r_{i})\,.$
(40)

where we have introduced a compact notation
$\Big{(}\hat{\mathcal{A}}_{\mathrm{CDR}}\Big{)}^{\mu}$. For the pole-subracted
amplitude we have

$\displaystyle\hat{\mathcal{F}}_{\mathrm{CDR}}\Big{(}\epsilon;\bar{\varepsilon}_{\bar{\lambda}}(p_{i},r_{i})\Big{)}$
$\displaystyle\equiv$
$\displaystyle\hat{\mathcal{Z}}^{-1}_{\mathrm{IR};\mathrm{CDR}}\left(\epsilon\right)~{}\mathcal{Z}^{-1}_{\mathrm{UV};\mathrm{CDR}}\left(\epsilon\right)~{}\hat{\mathcal{A}}_{\mathrm{CDR}}\Big{(}\epsilon;\bar{\varepsilon}_{\bar{\lambda}}(p_{i},r_{i})\Big{)}$
(41) $\displaystyle=$
$\displaystyle\hat{\mathcal{F}}_{4}\Big{(}\varepsilon_{\lambda}(p_{i},r_{i})\Big{)}+\mathcal{O}(\epsilon)\,,$

whose limit at $\epsilon=0$ is precisely the finite remainder
$\hat{\mathcal{F}}_{4}$ in eq. (39) with 4-dimensional external polarization
vector $\varepsilon_{\lambda}(p_{i},r_{i})$. Now we multiply this regular
finite quantity by a generalized D-dependent Lorentz-invariant norm-orthogonal
factor $\Delta_{\bar{\lambda}\lambda}$ defined by

$\displaystyle\Delta_{\bar{\lambda}\lambda}$ $\displaystyle\equiv$
$\displaystyle-\bar{\varepsilon}^{~{}*}_{\bar{\lambda}}(p_{i},r_{i})\cdot\varepsilon^{\text{MBR}}_{\lambda}(p_{i},r_{i})$
(42) $\displaystyle=$
$\displaystyle\delta_{\bar{\lambda}\lambda}+\mathcal{O}(\epsilon).$

Here $\varepsilon^{\text{MBR}}_{\lambda}$ refers to a polarization vector for
a massless gauge boson of our prescription141414The acronym “MBR” denotes
mometum basis representation. of section 3, and the dot product in (42) refers
to the D-dimensional Minkowski scalar product. We recall that the polarization
index $\bar{\lambda}$ labels the $D-2$ polarization states of CDR while in our
prescription the index $\lambda$ of $\varepsilon^{\text{MBR}}_{\lambda}$ takes
only two values for massless gauge bosons (and three for massive ones), which
are $\pm$ in helicity basis for polarizations. The 4-dimensional limits of
these simple Lorentz-invariant contractions $\Delta_{\bar{\lambda}\lambda}$
are the norm-orthogonal factors (i.e. the Kronecker deltas) among different
4-dimensional physical polarization/helicity states.

Next we consider the sum of products

$\sum_{\bar{\lambda}=\pm,~{}D-4}\hat{\mathcal{F}}_{\mathrm{CDR}}\Big{(}\epsilon;\bar{\varepsilon}_{\bar{\lambda}}(p_{i},r_{i})\Big{)}~{}\Delta_{\bar{\lambda}\lambda}.$
(43)

As exhibited in eqs. (41) and (42), both
$\hat{\mathcal{F}}_{\mathrm{CDR}}\Big{(}\epsilon;\bar{\varepsilon}_{\bar{\lambda}}(p_{i},r_{i})\Big{)}$
and $\Delta_{\bar{\lambda}\lambda}$ are regular in $\epsilon$. Thus they can
be power expanded in $\epsilon$, and their 4-dimensional limits can be taken
separately before being multiplied together and subsequently summed over
polarizations. Proceeding in this way, we first insert the $\epsilon$-expanded
expressions of these two factors given above, and the resulting quantity is
precisely the finite remainder
$\hat{\mathcal{F}}_{4}\Big{(}\varepsilon_{\lambda}(p_{i},r_{i})\Big{)}$ of eq.
(39):

$\sum_{\bar{\lambda}=\pm,~{}D-4}\hat{\mathcal{F}}_{\mathrm{CDR}}\Big{(}\epsilon;\bar{\varepsilon}_{\bar{\lambda}}(p_{i},r_{i})\Big{)}~{}\Delta_{\bar{\lambda}\lambda}=\hat{\mathcal{F}}_{4}\Big{(}\varepsilon_{\lambda}(p_{i},r_{i})\Big{)}+\mathcal{O}(\epsilon).$
(44)

On the other hand, we can first perform the polarization sum in (43) in D
dimensions and take the 4-dimensional limit afterwards. Proceeding this way,
we have

$\displaystyle\sum_{\bar{\lambda}=\pm,~{}D-4}\hat{\mathcal{F}}_{\mathrm{CDR}}\Big{(}\epsilon;\bar{\varepsilon}_{\bar{\lambda}}(p_{i},r_{i})\Big{)}~{}\Delta_{\bar{\lambda}\lambda}$
$\displaystyle=$
$\displaystyle-\sum_{\bar{\lambda}=\pm,~{}D-4}\hat{\mathcal{Z}}^{-1}_{\mathrm{IR};\mathrm{CDR}}\left(\epsilon\right)~{}\mathcal{Z}^{-1}_{\mathrm{UV};\mathrm{CDR}}\left(\epsilon\right)~{}\hat{\mathcal{A}}_{\mathrm{CDR}}\Big{(}\epsilon;~{}\bar{\varepsilon}_{\bar{\lambda}}(p_{i},r_{i})\Big{)}~{}\bar{\varepsilon}^{~{}*}_{\bar{\lambda}}(p_{i},r_{i})\cdot\varepsilon^{\text{MBR}}_{\lambda}(p_{i},r_{i})$
$\displaystyle=$
$\displaystyle-\hat{\mathcal{Z}}^{-1}_{\mathrm{IR};\mathrm{CDR}}\left(\epsilon\right)~{}\mathcal{Z}^{-1}_{\mathrm{UV};\mathrm{CDR}}\left(\epsilon\right)~{}\sum_{\bar{\lambda}=\pm,~{}D-4}\Big{(}\hat{\mathcal{A}}_{\mathrm{CDR}}\Big{)}_{\mu}~{}\bar{\varepsilon}_{\bar{\lambda}}^{~{}\mu}(p_{i},r_{i})~{}\bar{\varepsilon}^{~{}*\nu}_{\bar{\lambda}}(p_{i},r_{i})~{}\varepsilon^{\text{MBR}}_{\lambda,~{}\nu}(p_{i},r_{i})\,,$

where we have used the fact that
$\hat{\mathcal{A}}_{\mathrm{CDR}}\Big{(}\epsilon;~{}\bar{\varepsilon}_{\bar{\lambda}}(p_{i},r_{i})\Big{)}$
is linear in the external polarization vector
$\bar{\varepsilon}_{\bar{\lambda}}(p_{i},r_{i})$. Now we employ eq. (17) for
summing over the physical polarizations of the CDR-regularized external gauge
boson151515Note that here we should sum over physical polarizations only,
especially in the case of gluons, which ensures that unphysical components
such as scalar and longitudinal polarizations are absent from the outset. With
this choice there is no need to incorporate diagrams involving ghost fields in
the external states (when there are multiple external non-Abelian gauge
bosons). and obtain

$\displaystyle-\sum_{\bar{\lambda}=\pm,~{}D-4}\Big{(}\hat{\mathcal{A}}_{\mathrm{CDR}}\Big{)}_{\mu}~{}\bar{\varepsilon}_{\bar{\lambda}}^{~{}\mu}(p_{i},r_{i})~{}\bar{\varepsilon}^{~{}*\nu}_{\bar{\lambda}}(p_{i},r_{i})~{}\varepsilon^{\text{MBR}}_{\lambda,~{}\nu}(p_{i},r_{i})$
(46) $\displaystyle=$
$\displaystyle\Big{(}\hat{\mathcal{A}}_{\mathrm{CDR}}\Big{)}_{\mu}~{}\Bigg{(}g^{\mu\nu}-\frac{p_{i}^{\mu}r_{i}^{\nu}+r_{i}^{\mu}p_{i}^{\nu}}{p_{i}\cdot
r_{i}}\Bigg{)}~{}\varepsilon^{\text{MBR}}_{\lambda,~{}\nu}(p_{i},r_{i})$
$\displaystyle=$
$\displaystyle\Big{(}\hat{\mathcal{A}}_{\mathrm{CDR}}\Big{)}_{\mu}~{}\Bigg{(}g^{\mu\nu}\Bigg{)}~{}\varepsilon^{\text{MBR}}_{\lambda,~{}\nu}(p_{i},r_{i})$
$\displaystyle=$
$\displaystyle\hat{\mathcal{A}}_{\mathrm{CDR}}\Big{(}\epsilon;~{}\varepsilon^{\text{MBR}}_{\lambda}(p_{i},r_{i})\Big{)}$

where we have used the orthogonality of
$\varepsilon^{\text{MBR}}_{\lambda}(p_{i},r_{i})$ w.r.t. the particle’s
momentum $p_{i}$ and its reference vector $r_{i}$ in D dimensions, which
$\varepsilon^{\text{MBR}}_{\lambda}(p_{i},r_{i})$ has to satisfy by
construction. Inserting eq. (46) back into eq. (4.1) we end up with

$\displaystyle\sum_{\bar{\lambda}=\pm,~{}D-4}\hat{\mathcal{F}}_{\mathrm{CDR}}\Big{(}\epsilon;~{}\bar{\varepsilon}_{\bar{\lambda}}(p_{i},r_{i})\Big{)}~{}\Delta_{\bar{\lambda}\lambda}$
$\displaystyle=$
$\displaystyle\hat{\mathcal{F}}_{\mathrm{CDR}}\Big{(}\epsilon;~{}\varepsilon^{\text{MBR}}_{\lambda}(p_{i},r_{i})\Big{)}\,,$
(47)

whose left-hand side has, according to eq. (44), a 4-dimensional limit that is
equal to the finite remainder
$\hat{\mathcal{F}}_{4}\Big{(}\varepsilon_{\lambda}(p_{i},r_{i})\Big{)}$ given
in eq. (39). Notice that eq. (47) is an identity holding to all orders in
$\epsilon$. The right-hand side of (47), more explicitly,

$\hat{\mathcal{F}}_{\mathrm{CDR}}\Big{(}\epsilon;~{}\varepsilon^{\text{MBR}}_{\lambda}(p_{i},r_{i})\Big{)}=\hat{\mathcal{Z}}^{-1}_{\mathrm{IR};\mathrm{CDR}}\left(\epsilon\right)~{}\mathcal{Z}^{-1}_{\mathrm{UV};\mathrm{CDR}}\left(\epsilon\right)~{}\hat{\mathcal{A}}_{\mathrm{CDR}}\Big{(}\epsilon;~{}\varepsilon^{\text{MBR}}_{\lambda}(p_{i},r_{i})\Big{)}$
(48)

is exactly the quantity suggested by our prescription. In order to avoid
confusion we emphasize that the subscript “CDR” on
$\hat{\mathcal{F}}_{\mathrm{CDR}}$ at the right-hand side of (47), and on
$\hat{\mathcal{F}}_{\mathrm{CDR}}$ and $\hat{\mathcal{A}}_{\mathrm{CDR}}$ in
eq. (48) means that these are the respective CDR expressions with the
exception that the CDR polarization vector of the external gluon with momentum
$p_{i}$ is replaced by the polarization vector of our MBR prescription. If
there are more gluons in the external state then the procedure outlined by
eqs. (40) - (48) can be iterated.

What the above reformulations show is that, to all orders in $\epsilon$, the
$\hat{\mathcal{F}}_{\mathrm{CDR}}\Big{(}\epsilon;~{}\varepsilon^{\text{MBR}}_{\lambda}(p_{i},r_{i})\Big{)}$
can be formally viewed as an unpolarized interference between
$\hat{\mathcal{F}}_{\mathrm{CDR}}\Big{(}\epsilon;~{}\bar{\varepsilon}_{\bar{\lambda}}(p_{i},r_{i})\Big{)}$
and the Lorentz-invariant generalized norm-orthogonal factor defined in eq.
(42), using physical polarization sum rules for all CDR external states. The
unpolarized Landau density matrices of external gauge bosons reduce into the
unique spacetime metric tensor by the virtue of the built-in orthogonality
between $\varepsilon^{\text{MBR}}_{\lambda}(p_{i},r_{i})$ and $p_{i},r_{i}$.

An analogous reformulation can be made for external fermions in the scattering
amplitude. In fact, for each open fermion line such a reformulation is more
straightforward than in the above gauge boson case, because there is no
redundancy in the spinor representation of the Lorentz algebra and the number
of the polarization/helicity states of a fermion is two both in CDR and in our
prescription. The unpolarized Landau density matrix of an external fermion is
the well-known projection operator onto the space of on-shell Dirac-spinors.
After performing a similar reformulation of an open fermion line in the
scattering amplitude, denoted by
$\langle\psi_{A}|\hat{\mathrm{M}}|\psi_{B}\rangle$ as in eq. (27), we end up
with the following replacement:

$\displaystyle\langle\psi^{\mathrm{CDR}}_{A}|\hat{\mathrm{M}}|\psi^{\mathrm{CDR}}_{B}\rangle~{}\longrightarrow~{}\mathrm{Tr}\Big{[}\frac{\hat{P}_{on}\left(p_{B},m_{B}\right)}{2f_{B}~{}m_{B}}\Big{(}~{}|\psi^{\mathrm{MBR}}_{B}\rangle\langle\psi^{\mathrm{MBR}}_{A}|~{}\Big{)}\frac{\hat{P}_{on}\left(p_{A},m_{A}\right)}{2f_{A}~{}m_{A}}~{}\hat{\mathrm{M}}\Big{]}$

where $\hat{P}_{on}(p,m)=(\not{p}\pm m)$ denotes the aforementioned on-shell
projection operator for a $u$\- respectively $v$-type Dirac spinor with
momentum $p$ and mass $m$, and
$|\psi^{\mathrm{MBR}}_{B}\rangle\langle\psi^{\mathrm{MBR}}_{A}|$ is exactly
the matrix (28) that was further discussed in eqs. (3.2) - (31). The
appearance of ${1}/{(2f_{A}~{}m_{A})}$ and ${1}/{(2f_{B}~{}m_{B})}$ in eq.
(4.1) is due to the conventional choice of normalization factors of on-shell
Dirac spinors. Here the factors $f_{A},f_{B}=1~{}(-1)$ when the fermion $A$
respectively $B$ is associated with a $u$-type ($v$-type) spinor.

Quantities that are sandwiched between the pair of on-shell projection
operators, $\hat{P}_{on}(p_{A},m_{A})$ and $\hat{P}_{on}(p_{B},m_{B})$,
associated with the two external spinors of the open fermion line, can be
manipulated and simplified according to the 4-dimensional Lorentz/Dirac-
algebra. We just have to agree on one definite form that will be taken as its
canonical form (out of all its forms that are equivalent in 4 dimensions) and
used unambiguously in D-dimensional algebraic computations. This pair of on-
shell projection operators sets the domain where matrices related to external
fermions’ states, namely
$|\psi^{\mathrm{MBR}}_{B}\rangle\langle\psi^{\mathrm{MBR}}_{A}|$, can be
manipulated and moved around using just 4 dimensional Lorentz/Dirac-algebra.
While, in general, moving any of these matrices beyond this range must be done
in accordance with the D-dimensional Lorentz/Dirac-algebra in order to not
introduce artificial terms by mistake. For instance, the object
$\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}S_{\sigma}\epsilon^{\mu\nu\rho\sigma}$
commutes with $\not{P}$ in 4 dimensions because of the orthogonality condition
$S\cdot P=0$. However, this is no longer true w.r.t. the D-dimensional
algebra, and there is thus a non-vanishing evanescent commutator resulting
from interchanging the product order between the two. In section 5.2 we will
briefly comment on this subtle point again.

Finally, in order to bring the external projector in eq. (4.1) into a form
analogous to eq. (27) with the tensor product of external spinors given by eq.
(31), the following defining property of the on-shell projection operators,
valid for $p^{2}=m^{2}$ in D dimensions, can be used:

$\displaystyle\hat{P}_{on}(p,m)\frac{\hat{P}_{on}(p,m)}{2f~{}m}=\hat{P}_{on}(p,m)\,,$
(50)

where $f=\pm 1$ depending on whether ${\hat{P}}_{on}$ is associated with a
$u$-type or $v$-type spinor. Notice also that such an identity has a
continuous limit at $m\rightarrow 0$, despite the superficial appearance of
the singular $\frac{1}{m}$ factor which does prevent setting $m=0$ directly in
eq.(50). Such an alternative perspective thus helps to explain the choice made
in eq. (31) where the polarization projection operators were placed inside the
on-shell projection operators.

We thus achieved what we aimed at in this subsection. We found an alternative
formulation of pole-subtracted finite amplitudes which helps to prove the
following claim: despite the fact that usage of the polarization projectors
defined in section 3 results in helicity amplitudes different from those in
CDR, replacing all CDR-regularized external polarization states of
$\hat{\mathcal{A}}_{\mathrm{CDR}}(\epsilon)$ in eq. (38) by their counterparts
given in terms of momentum basis representations constructed in section 3
still results in the same RS-independent finite remainder, where all poles are
chosen to be subtracted by the same factorized (singular) coefficients given
in CDR, without appealing to Lorentz tensor decomposition representations of
$\hat{\mathcal{A}}_{\mathrm{CDR}}(\epsilon)$. The validity of this statement
is not confined to one-loop or next-to-leading order (NLO) corrections to a
Born-level scattering amplitude, but holds as long as the amplitude-level
factorization formula sketched in eq. (38) holds in CDR.

###  4.2 Finite remainders in an IR subtraction framework

In this subsection, we move on and analyze finite remainders defined in an IR-
subtraction method that are obtained with our MBR prescription for external
polarization vectors. We will then show that this hybrid CDR-compatible
prescription is unitary as defined in the sense of refs. vanDamme:1984ig ;
Catani:1996pk .

In practice the finite RS-independent physical observables at NLO and beyond
are usually computed as combinations of separate, in general UV and/or IR
divergent contributions living in different partonic phase spaces. (UV
renormalization is understood in what follows.) To render individual
contributions from each partonic phase space IR-finite and RS-independent
respectively, one can add and subtract properly defined auxiliary IR-
subtraction terms. The introduction of these auxiliary terms are designed to
ensure the cancellation of all intermediate IR-divergences of amplitudes in
each partonic phase space, while on the other hand they leave no trace in the
final properly combined physical observables. This is the idea of IR-
subtraction methods Ellis:1980wv ; Kunszt:1992tn , which are nowadays
available in many different versions (e.g., Frixione:1995ms ; Catani:1996vz ;
Nagy:2003qn ; Kosower:1997zr ; GehrmannDeRidder:2005cm ; Czakon:2010td ;
Czakon:2014oma ; Caola:2017dug ; Magnea:2018hab ; Herzog:2018ily ;
Catani:2007vq ; Somogyi:2006da ).

Let us now sketch an IR-subtraction method by only being explicit about
aspects that are relevant for showing that our MBR prescription of external
states is unitary.

Assume that the Born-level scattering amplitude $\mathcal{A}_{n}$ lives in a
n-particle phase space, and we consider an IR-safe observable defined by the
measurement function $F_{J}$. The leading-order (LO) observable $\sigma_{\rm
LO}$ is given by

$\displaystyle\sigma_{\rm LO}$ $\displaystyle=$
$\displaystyle\int_{d\Phi_{n}}|\mathcal{A}_{n}|^{2}~{}\mathit{F}_{\mathit{J}}^{~{}(n)},$
(51)

where we suppressed all pre-factors related to spin averaging for the initial
state and the incident flux. The NLO QCD correction $\sigma_{\rm NLO}$
consists of real radiations $\int_{d\Phi_{n+1}}d\sigma_{\rm
NLO}^{\mathcal{R}}$ in the (n+1)-particle phase space and the (renormalized)
virtual corrections $\int_{d\Phi_{n}}d\sigma_{\rm NLO}^{\mathcal{V}}$ in the
n-particle phase space. To render individual contributions in each of these
two phase spaces finite, one adds and subtracts an appropriate IR-subtraction
term $d\sigma^{\mathcal{S}}$. Subsequently $\sigma_{\rm NLO}$ can then be
rewritten in an IR subtraction method as follows161616 For the sake of
simplicity, we suppressed here an initial-state collinear subtraction term
related to the (re)definition of parton-distribution functions, which does not
add any additional conceptual complexity to what we want to show.

$\displaystyle\sigma_{\rm NLO}$ $\displaystyle=$
$\displaystyle\int_{d\Phi_{n+1}}d\sigma_{\rm
NLO}^{\mathcal{R}}+\int_{d\Phi_{n}}d\sigma_{\rm NLO}^{\mathcal{V}}$
$\displaystyle=$
$\displaystyle\int_{d\Phi_{n+1}}|\mathcal{A}_{n+1}^{\mathcal{R}}|^{2}~{}\mathit{F}_{\mathit{J}}^{~{}(n+1)}+\left(\int_{d\Phi_{n+1}}d\sigma^{\mathcal{S}}~{}\mathit{F}_{\mathit{J}}^{~{}(n)}-\int_{d\Phi_{n+1}}d\sigma^{\mathcal{S}}~{}\mathit{F}_{\mathit{J}}^{~{}(n)}\right)+\int_{d\Phi_{n}}2\mathrm{Re}\left[\mathcal{A}_{n}^{*}\mathcal{A}_{n}^{\mathcal{V}}\right]~{}\mathit{F}_{\mathit{J}}^{~{}(n)}$
$\displaystyle=$
$\displaystyle\int_{d\Phi_{n+1}}\Bigg{[}\left(|\mathcal{A}_{n+1}^{\mathcal{R}}|^{2}~{}\mathit{F}_{\mathit{J}}^{~{}(n+1)}\right)_{\epsilon=0}-\left(d\sigma^{\mathcal{S}}~{}\mathit{F}_{\mathit{J}}^{~{}(n)}\right)_{\epsilon=0}\Bigg{]}+\int_{d\Phi_{n}}\left[2~{}\mathrm{Re}\left[\mathcal{A}_{n}^{*}\mathcal{A}_{n}^{\mathcal{V}}\right]+\int_{1}d\sigma^{\mathcal{S}}\right]_{\epsilon=0}\mathit{F}_{\mathit{J}}^{~{}(n)}.$

By construction, the subtraction term $d\sigma^{\mathcal{S}}$ should have the
same local IR-singular behavior as the squared real-radiation matrix
$|\mathcal{A}_{n+1}^{\mathcal{R}}|^{2}$ everywhere in the (n+1)-particle phase
space (subject to the constraint implied by $\mathit{F}_{\mathit{J}}$).
Consequently, the resulting subtracted phase-space integrand
$\left[\left(|\mathcal{A}_{n+1}^{\mathcal{R}}|^{2}~{}\mathit{F}_{\mathit{J}}^{~{}(n+1)}\right)_{\epsilon=0}-\left(d\sigma^{\mathcal{S}}~{}\mathit{F}_{\mathit{J}}^{~{}(n)}\right)_{\epsilon=0}\right]$
can be numerically evaluated and integrated over the phase space in 4
dimensions, as indicated by $\epsilon=0$. Notice that it is
$\mathit{F}_{\mathit{J}}^{~{}(n)}$ that is associated with
$d\sigma^{\mathcal{S}}$, the same as for virtual corrections living in
n-particle phase space. The integration of $d\sigma^{\mathcal{S}}$ over the
unresolved phase space has to be done in D dimensions with the IR unresolved
partonic d.o.f. regularized in the same way as those in the virtual correction
$2~{}\mathrm{Re}\left[\mathcal{A}_{n}^{*}\mathcal{A}_{n}^{\mathcal{V}}\right]$,
following from the unitarity constraint. The resulting IR singularities that
appear as poles in $\epsilon$ must cancel those appearing in
$2~{}\mathrm{Re}\left[\mathcal{A}_{n}^{*}\mathcal{A}_{n}^{\mathcal{V}}\right]$,
which renders the quantity in the second square bracket of the last line of
eq. (4.2) finite in 4 dimensions as well.

In order that eq. (4.2) is useful in practice, one must be able to perform the
D-dimensional integration $\int_{1}d\sigma^{\mathcal{S}}$, either analytically
or numerically. Thanks to the IR factorization, $d\sigma^{\mathcal{S}}$ and
likewise its integrated counterpart $\int_{1}d\sigma^{\mathcal{S}}$ can be
constructed, schematically, as a convoluted product of certain universal
(process-independent) multiplicative coefficient and the (process-specific)
squared Born amplitude $|\mathcal{A}_{n}|^{2}$:

$\displaystyle d\sigma^{\mathcal{S}}$ $\displaystyle=$
$\displaystyle\left(d\hat{I}_{RS}\right)\otimes|\mathcal{A}_{n}|^{2}~{},$
$\displaystyle\int_{1}d\sigma^{\mathcal{S}}$ $\displaystyle=$
$\displaystyle\hat{I}_{RS}\otimes|\mathcal{A}_{n}|^{2}.$ (53)

The factor $\hat{I}_{RS}$ plays a similar role as the multiplicative factors
$\hat{\mathcal{Z}}_{\mathrm{IR}}(\epsilon)$ in eq. (38). At NLO it encodes all
IR pole-singularities and is to be viewed as an operator in the color space of
the external particles.

In fact each variant of an IR-subtraction method can be seen as providing a
concrete constructive prescription for the integral representations of
factorized IR-subtraction coefficients, like the factor $\hat{I}_{RS}$, that
contain all the explicit pole-singularities of the loop amplitudes (after
multiplication with certain relevant process-dependent hard-scattering
amplitudes). The crucial point relevant for the following discussion is that
these integral representations are based on the amplitude-level IR
factorization, and are manifestly independent of the polarization states of
external particles which appear in the (remaining) hard-scattering matrix
elements.171717The dependence of factorized collinear $\epsilon$-pole
singularities on the polarization of a parent parton drops once one sums over
the polarizations of all other particles and also integrates over all
unresolved degrees of freedom in the collinear limit, notably the transverse
plane of the radiated partons (which essentially eliminates any preference in
the transverse direction).

All quantities in eq. (4.2) that contain explicit IR-divergences, i.e. poles
in $\epsilon$, contain RS-dependent pieces in their truncated Laurent series
to order $\epsilon^{0}$, especially the integrated $\hat{I}_{RS}$. At NLO,
this concerns only $\int_{d\Phi_{n}}d\sigma_{\rm NLO}^{\mathcal{V}}$ and
$\int_{1}d\sigma^{\mathcal{S}}=\hat{I}_{RS}\otimes|\mathcal{A}_{n}|^{2}$ that
live in the same n-particle phase space. By appealing to an IR-subtraction
method the unitarity constraint, originally imposed between the calculations
of $\int_{d\Phi_{n+1}}d\sigma_{\rm NLO}^{\mathcal{R}}$ and
$\int_{d\Phi_{n}}d\sigma_{\rm NLO}^{\mathcal{V}}$ is translated into the
following “locally distributed” version: we just need to make sure that
contributions associated with the same partonic phase space are computed
consistently with a unitarity-respecting prescription, while pole-subtracted
4-dimensional remainders living in different partonic phase spaces can be
computed independently of each other (using different methods). Thus, as
argued in ref. Catani:1996pk , IR subtraction methods offer a convenient way
to isolate and investigate the RS-dependence of individual singular pieces and
subsequently ensure the unitarity of regularization prescriptions used in the
calculation.

With the skeleton of an IR-subtraction framework ready, we can discuss how
each of the two square brackets in the last line of eq. (4.2) should be
evaluated with our proposed prescription in order to ensure a correct NLO
observable $\sigma_{\rm NLO}$.

First, the subtraction of implicit IR-singularities in $d\sigma_{\rm
NLO}^{\mathcal{R}}$, i.e. terms in the first square bracket of the last line
of eq. (4.2), is to be done at the integrand level of phase-space integrals.
This results in a subtracted real-radiation contribution that is numerically
integrable in 4 dimensions. In the 4-dimensional limit ($\epsilon=0$) the
external polarization states defined by the momentum basis representations
given in section 3, all coincide with their respective standard 4-dimensional
expressions. Therefore the RS-independence of the finite remainders of real-
radiation contributions associated with the 4-dimensional (n+1)-particle phase
space is manifest as dimensional regularization can be avoided from the
outset. Thus we just have to make sure that in this hybrid prescription, the
integral-level subtraction of explicit $\epsilon$-pole singularities in
$2~{}\mathrm{Re}\left[\mathcal{A}_{n}^{*}\mathcal{A}_{n}^{\mathcal{V}}\right]$,
i.e. the second square bracket of the last line of eq. (4.2), is also done in
a unitarity-respecting way so as to lead to the correct RS-independent finite
remainder in the n-particle phase space.

To this end, we can proceed in two ways. We could devise a proof analogous to
the previous subsection, but now applied to the finite remainder
$\left[2\mathrm{Re}\left[\mathcal{A}_{n}^{*}\mathcal{A}_{n}^{\mathcal{V}}\right]+\hat{I}_{RS}\otimes|\mathcal{A}_{n}|^{2}\right]_{\epsilon=0}$,
where the integrated factor $\hat{I}_{RS}$ plays a similar role as the
perturbatively-expanded multiplicative factor
$\hat{\mathcal{Z}}_{\mathrm{IR}}(\epsilon)$ in eq. (38). Alternatively, we
argue in this subsection that the unitarization recipe of ref. Catani:1996pk
is indeed respected by our hybrid prescription. We examine this now one by
one.

  1. 1.

The external partons in the Born-level hard-scattering matrix element
$\mathcal{A}_{n}$ of the factorized IR-subtraction term
$\hat{I}_{RS}\otimes|\mathcal{A}_{n}|^{2}$ have to be treated like the
external partons in the virtual loop amplitude $\mathcal{A}_{n}^{\mathcal{V}}$
(of the same external kinematic configuration).

This is guaranteed by applying the same set of polarization projectors defined
in section 3 consistently to $\mathcal{A}_{n}$ at LO and
$\mathcal{A}^{\mathcal{V}}_{n}$ at NLO, computed respectively to the required
powers in $\epsilon$.

  2. 2.

The parent parton and its (soft and collinear) daughter partons involved in
the integral representation of the factorized process-independent (singular)
coefficient function $\hat{I}_{RS}$ have to be treated like the corresponding
partons inside the loop integrals of $\mathcal{A}_{n}^{\mathcal{V}}$.

This is guaranteed by performing integrals involving IR-unresolved d.o.f.
consistently regularized with CDR. In particular, the phase-space integrals in
$\hat{I}_{RS}$ are done in D dimensions like D-dimensional loop integrals
subject to Cutkosky cuts.

Concerning the first point, as long as there is an unambiguous and consistent
way of directly applying such a non-CDR regularization convention of external
states in the computation of the virtual loop amplitude
$\mathcal{A}_{n}^{\mathcal{V}}$ (without appealing to its Lorentz tensor
decomposition representation), then the demonstration is completed. Similar as
in section 4.1, this point is guaranteed in our projection prescription by the
fact that all open Lorentz indices of the polarization projectors defined in
section 3 are taken to be a D-dimensional and no dimensional splitting is ever
introduced, just like in CDR.

Thus we have argued that our hybrid prescription can be conveniently used in a
NLO IR subtraction framework to correctly obtain all RS-independent finite
remainders needed for computing physical observables, with the (process-
independent) integrated IR-subtraction coefficients directly taken from CDR.
In other words, we have argued that our hybrid CDR-compatible prescription is
unitary.

Although beyond the scope of this article, it is possible, by analogy to the
NLO case, to ensure unitarity of the prescription at NNLO and beyond, owing to
the following generic features of an IR subtraction method (which the above
NLO discussions essentially rely on).

  * •

In a typical IR subtraction framework, all explicit IR-singularities in loop
amplitudes, manifested as poles in $\epsilon$, are always subtracted by IR
subtraction terms whose constructions are based on amplitude-level singularity
factorization formulae, and the factorized IR-subtraction coefficients are
independent of all external polarization states;

  * •

Any potential implicit IR singularity of the ($\epsilon$-pole-free) finite
remainders will always be further subtracted at the integrand level of phase-
space integrals over the external kinematics, and will be directly evaluated
in 4 dimensions without employing dimensional regularization.

Thus concerning the 4-dimensional integrand level subtractions of implicit IR-
singularities in those finite remainders, their $\epsilon$-suppressed terms
are never needed because the phase-space integration over the external
kinematics is done (numerically) in 4 dimensions. We leave a detailed
exposition of this at NNLO for a future publication.

##  5 Examples at NLO QCD

The polarization projectors constructed in section 3 are independent of the
loop order of virtual amplitudes, regardless of possible evanescent Lorentz
structures that may be generated in D dimensions. To illustrate its usage
without being overwhelmed by irrelevant complications, we consider two
prototype examples of a NLO QCD virtual amplitude, the 1-loop QCD corrections
to $gg\rightarrow gg$ and to $e^{+}e^{-}\rightarrow Q\bar{Q}$, in order to
show that RS-independent finite remainders are indeed obtained as was
discussed in the preceding sections. We will comment along the way points
worthy of attention.

###  5.1 $gg\rightarrow gg$

Because singular amplitudes are RS-dependent, it is only meaningful for our
purpose to compare properly defined finite remainders between computations
done using different regularization schemes. As discussed in section 4.2, the
IR-subtracted real-radiation contribution at NLO is obviously RS-independent.
Thus we just need to show how the same finite remainders of the virtual
corrections are obtained in computations using our hybrid prescription and in
CDR.

We consider the scattering process among 4 massless gluons:

$g_{1}(p_{1})~{}+~{}g_{2}(p_{2})\to g_{3}(p_{3})~{}+~{}g_{4}(p_{4}),$ (54)

in QCD without fermions for simplicity. The Mandelstam variables are given in
eq. (9).

The corresponding scattering amplitude perturbatively expanded up to NLO reads

$\displaystyle\Big{|}\mathcal{A}_{gggg}\Big{\rangle}=\Big{|}\mathcal{A}^{[\text{tree}]}_{gggg}\Big{\rangle}+\Big{|}\mathcal{A}^{[\text{1-loop}]}_{gggg}\Big{\rangle}+\mathcal{O}(\alpha^{3}_{s})\,,$
(55)

which is a vector in the color space of the external gluons. The 1-loop
virtual amplitudes were computed in refs. Bern:1991aq ; Schubert:2001he ;
Binoth:2006hk . For representing color structures of multi-gluon scattering
amplitudes, like eq. (55), it is most convenient to perform a color
decomposition using the choice of basis of refs. Berends:1987cv ;
Mangano:1987xk ; Mangano:1987kp ; Mangano:1988kk ; Bern:1990ux . It is well
known that tree-level QCD amplitudes with n external gluons can be decomposed
into color-ordered partial amplitudes, multiplied by associated single color
traces (over all noncylic permutations of fundamental color generators).
Decomposition of color structures of one-loop QCD amplitudes can be done in a
similar way but with an extended color basis including products of two color
traces181818This can be easily understood by combing the statement about tree-
level color decomposition and the Fierz identities of SU(N) color algebra..

For the amplitude eq. (55) we generate symbolic expressions of all
contributing Feynman diagrams using QGRAF Nogueira:1991ex , and subsequently
decompose them as follows:

$\displaystyle\Big{|}\mathcal{A}^{[\text{tree}]}_{gggg}\Big{\rangle}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{6}\mathcal{A}^{[\text{tree}]}_{gggg}(i)~{}|c_{i}\rangle~{},~{}$
$\displaystyle\Big{|}\mathcal{A}^{[\text{1-loop}]}_{gggg}\Big{\rangle}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{9}\mathcal{A}^{[\text{1-loop}]}_{gggg}(i)~{}|c_{i}\rangle~{},~{}$
(56)

using the following basis of 9 color structures:

$\displaystyle|c_{1}\rangle$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\Big{[}T_{1}~{}T_{2}~{}T_{3}~{}T_{4}\Big{]}~{},~{}~{}|c_{2}\rangle=\mathrm{Tr}\Big{[}T_{1}~{}T_{2}~{}T_{4}~{}T_{3}\Big{]}~{},~{}~{}|c_{3}\rangle=\mathrm{Tr}\Big{[}T_{1}~{}T_{3}~{}T_{4}~{}T_{2}\Big{]}$
$\displaystyle|c_{4}\rangle$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\Big{[}T_{1}~{}T_{3}~{}T_{2}~{}T_{4}\Big{]}~{},~{}~{}|c_{5}\rangle=\mathrm{Tr}\Big{[}T_{1}~{}T_{4}~{}T_{3}~{}T_{2}\Big{]}~{},~{}~{}|c_{6}\rangle=\mathrm{Tr}\Big{[}T_{1}~{}T_{4}~{}T_{2}~{}T_{3}\Big{]}$
$\displaystyle|c_{7}\rangle$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\Big{[}T_{1}~{}T_{2}\Big{]}\mathrm{Tr}\Big{[}T_{3}~{}T_{4}\Big{]}~{},~{}|c_{8}\rangle=\mathrm{Tr}\Big{[}T_{1}~{}T_{3}\Big{]}\mathrm{Tr}\Big{[}T_{2}~{}T_{4}\Big{]}~{},~{}|c_{9}\rangle=\mathrm{Tr}\Big{[}T_{1}~{}T_{4}\Big{]}\mathrm{Tr}\Big{[}T_{2}~{}T_{3}\Big{]}~{},$

where the subscripts of the color generators label the associated gluons while
their explicit color indices are suppressed. These 9 color structures are
linearly independent, as can be checked by computing its Gram matrix. The
amplitude $\Big{|}\mathcal{A}^{[\text{tree}]}_{gggg}\Big{\rangle}$ involves
only the first 6 non-cylic single color traces given in eq. (5.1), which can
be further reduced to 4 structures by reflection symmetries. The color
structures $|c_{7}\rangle,~{}|c_{8}\rangle,~{}|c_{9}\rangle$ are needed in
addition to represent
$\Big{|}\mathcal{A}^{[\text{1-loop}]}_{gggg}\Big{\rangle}$.

Each of the color decomposition coefficients
$\mathcal{A}^{[\text{tree}]}_{gggg}(i),~{}\mathcal{A}^{[\text{1-loop}]}_{gggg}(i)$
is a function of external kinematics and polarization state vectors, to which
we now apply the polarization projectors prescribed in section 3. We extract
polarized amplitudes in the linear polarization basis for all four external
gluons (cf. section 3.1.1), from which helicity amplitudes can be easily
obtained. Because the reaction (54) is parity-invariant the scattering
amplitude does not contain terms involving $\gamma_{5}$ or an odd number of
Levi-Civita tensors. We thus need to consider only the following 8 linear
polarization projectors, which are even in $\varepsilon_{Y}$, respectively in
the number of Levi-Civita tensors:

$\displaystyle\varepsilon^{\mu_{1}}_{X}\varepsilon^{\mu_{2}}_{X}\varepsilon^{\mu_{3}}_{T}\varepsilon^{\mu_{4}}_{T}~{},~{}\varepsilon^{\mu_{1}}_{X}\varepsilon^{\mu_{2}}_{X}\varepsilon^{\mu_{3}}_{Y}\varepsilon^{\mu_{4}}_{Y}~{},~{}\varepsilon^{\mu_{1}}_{X}\varepsilon^{\mu_{2}}_{Y}\varepsilon^{\mu_{3}}_{T}\varepsilon^{\mu_{4}}_{Y}~{},~{}\varepsilon^{\mu_{1}}_{X}\varepsilon^{\mu_{2}}_{Y}\varepsilon^{\mu_{3}}_{Y}\varepsilon^{\mu_{4}}_{T}~{},~{}$
$\displaystyle\varepsilon^{\mu_{1}}_{Y}\varepsilon^{\mu_{2}}_{X}\varepsilon^{\mu_{3}}_{T}\varepsilon^{\mu_{4}}_{Y}~{},~{}\varepsilon^{\mu_{1}}_{Y}\varepsilon^{\mu_{2}}_{X}\varepsilon^{\mu_{3}}_{Y}\varepsilon^{\mu_{4}}_{T}~{},~{}\varepsilon^{\mu_{1}}_{Y}\varepsilon^{\mu_{2}}_{Y}\varepsilon^{\mu_{3}}_{T}\varepsilon^{\mu_{4}}_{T}~{},~{}\varepsilon^{\mu_{1}}_{Y}\varepsilon^{\mu_{2}}_{Y}\varepsilon^{\mu_{3}}_{Y}\varepsilon^{\mu_{4}}_{Y}.$
(58)

For the sake of simplicity of notation, the arguments of these polarization
vectors are suppressed while their subscripts at the open Lorentz indices
indicate the associated gluons.

The number of linear polarization projectors in eq. (5.1) equals the number of
independent helicity amplitudes, taking into account the parity symmetry of
the scattering amplitude. We do not consider additional relations among the
linear polarized amplitudes arising from Bose symmetry, which involve
kinematic crossings. The set of 8 linear polarization projectors in eq. (5.1)
are sufficient for any parity-even scattering amplitude among four external
massless bosons to any loop order, irrespective of any possible (evanescent)
Lorentz structures therein.191919In case a $2\to 2$ amplitude involves parity-
violating couplings, 8 linear polarization projectors containing an odd number
of $\varepsilon_{Y}$ (or Levi-Civita tensors) can be used in addition.

We insert the expressions (11), (13), and (14) for the polarization vectors in
(5.1). All the Lorentz algebra is carried out using FORM Vermaseren:2000nd .
Let us emphasize again that, in order to avoid possible ambiguities in the
definition and application of these external projectors, all pairs of Levi-
Civita tensors in eq. (5.1) are replaced according to the contraction rule eq.
(15) before being used in the projection. Then the projectors (5.1) are
expressed solely in terms of external momenta and space-time metric tensors.
After pulling out the normalization factors as prescribed in section (3), the
resulting tensor projectors (which have only a polynomial dependence on
external momenta and kinematics) will be applied to the color stripped
amplitudes
$\mathcal{A}^{[\text{tree}]}_{gggg}(i),~{}\mathcal{A}^{[\text{1-loop}]}_{gggg}(i)$.
We use the convention to set the variable $D=4$ in the projectors (5.1), in
particular in the normalization factors that are pulled out. Of course this
convention is used both for the amplitudes and the associated UV and/or IR
subtraction terms. Then the normalization factors pulled out from the
respective projectors (5.1) are

$\displaystyle\mathcal{N}_{XXTT}=\frac{1}{s^{2}t^{2}(s+t)^{2}}~{},~{}\mathcal{N}_{XXYY}=\frac{4}{s^{2}t^{2}(s+t)^{2}}~{},~{}$
$\displaystyle\mathcal{N}_{XYTY}=\frac{4}{s^{2}t^{2}(s+t)^{2}}~{},~{}\mathcal{N}_{XYYT}=\frac{4}{s^{2}t^{2}(s+t)^{2}}~{},~{}$
$\displaystyle\mathcal{N}_{YXTY}=\frac{4}{s^{2}t^{2}(s+t)^{2}}~{},~{}\mathcal{N}_{YXTX}=\frac{4}{s^{2}t^{2}(s+t)^{2}}~{},~{}$
$\displaystyle\mathcal{N}_{YYTT}=\frac{4}{s^{2}t^{2}(s+t)^{2}}~{},~{}\mathcal{N}_{YYYY}=\frac{16}{s^{2}t^{2}(s+t)^{2}}~{}.$
(59)

The linear polarized amplitudes projected out by applying eq. (5.1) to
$\mathcal{A}^{[\text{1-loop}]}_{gggg}(i)$ contain both UV and IR
singularities, manifested as poles in $\epsilon$. We are only interested in
the finite remainders defined by subtracting all these singularities in
accordance with a certain convention. For our purpose, there is no need to
stick to a specific IR-subtraction scheme. All we need to know is a
factorization formula providing us with a set of terms that capture all
singularities in $\mathcal{A}^{[\text{1-loop}]}_{gggg}(i)$ (with the process-
independent singular coefficients obtained in CDR). To be specific, we choose
to define the finite remainders of the virtual amplitude
$\Big{|}\mathcal{A}^{[\text{1-loop}]}_{gggg}\Big{\rangle}$ by the following
explicit on-shell UV and IR subtraction terms:

$\displaystyle\Big{|}\mathcal{A}^{[\text{UV}]}_{gggg}\Big{\rangle}$
$\displaystyle=$
$\displaystyle\Big{(}-\frac{11}{3}\mathrm{N}_{c}\frac{1}{\epsilon}+\mathcal{O}(\epsilon^{0})\Big{)}~{}\Big{|}\mathcal{A}^{[\text{tree}]}_{gggg}\Big{\rangle}\,,$
$\displaystyle\Big{|}\mathcal{A}^{[\text{IR}]}_{gggg}\Big{\rangle}$
$\displaystyle=$ $\displaystyle
2\mathrm{N}_{c}\left(\frac{-2}{\epsilon^{2}}+\frac{1}{\epsilon}\left(\frac{-11}{3}+\mathrm{log}\left(\frac{-s_{12}}{\mu^{2}_{DR}}\right)+\mathrm{log}\left(\frac{+s_{23}}{\mu^{2}_{DR}}\right)\right)+\mathcal{O}(\epsilon^{0})\right)2\mathcal{A}^{[\text{tree}]}_{gggg}(1;\epsilon)~{}\Big{|}c_{1}\Big{\rangle}\,$
$\displaystyle+$ $\displaystyle