# The trace amplitude method and its application to the NLO QCD calculation

Zi-Qiang Chen1222<EMAIL_ADDRESS>and Cong-Feng
<EMAIL_ADDRESS>corresponding author
1 School of Physics, University of Chinese Academy of Sciences, Yuquan Road
19A, Beijing 100049
2 CAS Key Laboratory of Vacuum Physics, Beijing 100049, China

###### Abstract

The trace amplitude method (TAM) provides us a straightforward way to
calculate the helicity amplitudes with massive fermions analytically. In this
work, we review the basic idea of this method, and then discuss how it can be
applied to next-to-leading order (NLO) quantum chromodynamics (QCD)
calculations, which has not been explored before. By analyzing the singularity
structures of both virtual and real corrections, we show that the TAM can be
generalized to NLO QCD calculations straightforwardly, the only caution is
that the unitarity should be guaranteed. We also present a simple example to
demonstrate the application of this method.

PACS number(s): 12.38.–t, 12.38.Bx

##  I Introduction

In high energy colliders like the Large Hadron Collider (LHC), processes with
multi-particle final states are of great important in signal and background
analysis. To describe these processes at the same precision level as the
experimental measurements, one has to calculate the cross sections to at least
next-to-leading order (NLO). However, the NLO calculations for multi-particle
processes are very challenging: as the number of external particles increases,
both the number of Feynman diagrams and the computational difficulty of each
diagram grows rapidly. The conventional amplitude squaring approach (CAS),
i.e. squaring the Feynman amplitude, summing over the spins of external
states, and taking the trace of each possible fermion string loop, is proved
to be tedious and time consuming when the number of external particles is more
than 5. Another drawback of this approach is that it loses the spin
information of final state particles, which is attainable in present
experimental measurements.

An alternative approach is to compute the helicity amplitude explicitly, then
the amplitude squaring and polarization summation can be performed easily
during the numerical evaluation. The development of this approach has
experienced a long history. Many techniques have been developed for
calculating the tree- DeCausmaecker:1981jtq ; Berends:1981uq ; Nam:1983gt ;
Kleiss:1985yh ; Xu:1986xb ; Parke:1986gb ; Berends:1987me ; Chang:1992bb ;
Yehudai:1992rt ; Ballestrero:1994jn ; Vega:1995cc ; Bondarev:1997kf ;
Andreev:2001se ; Qiao:2003ue ; Cachazo:2004kj ; Britto:2004ap ; Schwinn:2005pi
and loop- Bern:1991aq ; Bern:1994zx ; Bern:1994cg ; Brandhuber:2004yw ;
Luo:2004ss ; Bena:2004xu ; Quigley:2004pw ; Bedford:2004py ; Britto:2004nc ;
Roiban:2004ix ; Bern:2005hs ; Bidder:2005ri  level helicity amplitudes. It
should be noted that literatures on this subject are vast, and to give a
complete survey of them is beyond the scope of this paper. For reviews, see
for instance Refs. Mangano:1990by ; Dixon:1996wi ; Bern:2008ef ;
Elvang:2013cua ; Dixon:2013uaa .

For processes of fermion production or decays, Feynman amplitudes incorporate
one or more open fermion line, which can be expressed as

$\bar{U}(p_{1},\lambda_{1})\Gamma U(p_{2},\lambda_{2})={\rm tr}[\Gamma
U(p_{2},\lambda_{2})\otimes\bar{U}(p_{1},\lambda_{1})],$ (1)

where $p_{1}$ and $p_{2}$ are the momenta of the external fermions,
$\lambda_{1}$ and $\lambda_{2}$ denote their polarization states; $\Gamma$
stands for the string of Dirac gamma matrices between the spinors;
$U(p,\lambda)$ stands for either fermion spinor $u(p,\lambda)$ or anti-fermion
spinor $v(p,\lambda)$. In $4$-dimensional spinor space, the spinor product
$U(p_{2},\lambda_{2})\otimes\bar{U}(p_{1},\lambda_{1})$ can be re-expressed by
basic Dirac gamma matrices through different ways Nam:1983gt ; Kleiss:1985yh ;
Chang:1992bb ; Yehudai:1992rt ; Ballestrero:1994jn ; Vega:1995cc ;
Bondarev:1997kf ; Andreev:2001se ; Qiao:2003ue , then the trace in Eq. (1) can
be evaluated straightforwardly. For convenience, we call this method the trace
amplitude method (TAM) hereafter. The TAM is different from the so called
helicity amplitude method (HAM), which has been proposed in Refs.
Kleiss:1985yh ; Xu:1986xb  and generalized to NLO in Refs. Bern:1991aq ;
Bern:1994zx . These two methods are complementary to each other: the results
obtained from the HAM is more compact, while the TAM is more transparent to
beginners and more convenient for a realization on a computer algebra system.
Although the TAM has been proposed for a long time, its validity in higher-
order calculations has not been discussed before. Considering the fact that
the NLO corrections are usually important in phenomenological study, in this
work we discuss the generalization of TAM to NLO QCD calculation.

The rest of the paper is organized as follows. In Sec. II, we review some
basic formulas used to derive the TAM; In Sec. III, we analyze different types
of singularities encountered in NLO QCD calculations; In Sec. III, we present
a scheme that enable the application of TAM in NLO QCD calculation; In Sec.
IV, illustrative examples, the NLO QCD corrections to $g+g\to t+\bar{t}$ and
$q+\bar{q}\to t+\bar{t}$ processes, are presented. The last section is
reserved for a summary.

##  II Spinor product

The key ingredient of the TAM is to re-express the spinor product
$U(p_{2},\lambda_{2})\otimes\bar{U}(p_{1},\lambda_{1})$ by basic Dirac gamma
matrices. This re-expression can be done through various means, such as
constructing the transformation matrix between spinors with different momenta
and polarization states Nam:1983gt , introducing auxiliary vectors
Kleiss:1985yh ; Chang:1992bb ; Ballestrero:1994jn ; Andreev:2001se ;
Qiao:2003ue , making the use of orthogonal basis of the $4$-dimensional spinor
space Yehudai:1992rt , making the use of the Bouchiat-Michel identity
Vega:1995cc ; B-MIdentity , etc. In fact, as revealed in Ref. Bondarev:1997kf
, in fact all these approaches may attribute to the same mathematical scheme.
In this section, we follow the lines of auxiliary vector approach, and present
some basic formulas of TAM.

Consider a (anti)fermion with momentum $p$, polarization vector $s$, and mass
$m$, the on-shell and polarization conditions require that

$p^{2}=m^{2},\quad s^{2}=-1,\quad p\cdot s=0.$ (2)

The corresponding spinor can be defined as the common eigenstate of the two
commuting operators $\not{p}$ and $\gamma_{5}\not{s}$:

$\displaystyle\not{p}U_{s}(p,\lambda)=MU_{s}(p,\lambda),$ (3)
$\displaystyle\gamma_{5}\not{s}U_{s}(p,\lambda)=\lambda U_{s}(p,\lambda).$ (4)

Here, for fermion $U_{s}(p,\lambda)=u_{s}(p,\lambda)$, $M=m$; for antifermion
$U_{s}(p,\lambda)=v_{s}(p,\lambda)$, $M=-m$; $\lambda=\pm 1$ denote the two
different polarization states.

The massive spinor $U_{s}(p,\lambda)$ can be constructed with massless spinor.
By introducing two auxiliary vectors that fulfil the conditions

$k_{0}^{2}=0,\quad k_{1}^{2}=-1,\quad k_{0}\cdot k_{1}=0,$ (5)

one can construct a massless spinor $w(k_{0},\lambda)$ in light of the light-
like vector $k_{0}$, satisfying

$\displaystyle\not{k}_{0}w(k_{0},\lambda)=0,$ (6)
$\displaystyle\gamma_{5}w(k_{0},\lambda)=\lambda w(k_{0},\lambda).$ (7)

From above two equations we have

$\displaystyle
w(k_{0},\lambda)\bar{w}(k_{0},\lambda)=\frac{1+\lambda\gamma_{5}}{2}\not{k}_{0},$
$\displaystyle\not{k}_{1}w(k_{0},\lambda)=\lambda w(k_{0},-\lambda).$ (8)

Here the relative phase between $w(k_{0},+)$ and $w(k_{0},-)$ may be fixed by
$k_{1}$.

With the massless spinor $w(k_{0},\lambda)$, the massive spinor
$U_{s}(p,\lambda)$ can be expressed as

$U_{s}(p,\lambda)=\frac{(\not{p}+M)(1+\not{s})}{2\sqrt{k_{0}\cdot(p+Ms)}}w(k_{0},-\lambda),$
(9)

which satisfies (3) and (4). The normalization factor here is fixed by the
fermion spin sum relation

$\sum_{\lambda}U_{s}(p,\lambda)\bar{U}_{s}(p,\lambda)=\not{p}+M.$ (10)

Combine Eqs. (8) and (9), one can readily get the desired spinor product:

$U_{s_{1}}(p_{1},\lambda_{1})\otimes\bar{U}_{s_{2}}(p_{2},\lambda_{2})=\frac{({\not{p}}_{1}+M_{1})(1+{\not{s}}_{1})\Lambda(\lambda_{1},\lambda_{2}){\not{k}}_{0}(1+{\not{s}}_{2})({\not{p}}_{2}+M_{2})}{8\sqrt{k_{0}\cdot(p_{1}+M_{1}s_{1})}\sqrt{k_{0}\cdot(p_{2}+M_{2}s_{2})}},$
(11)

with

$\displaystyle\Lambda(\lambda,\lambda)=1-\lambda\gamma_{5},$
$\displaystyle\Lambda(\lambda,-\lambda)=\not{k}_{1}(\lambda+\gamma_{5}).$ (12)

The polarization vector of a fermion can be expressed through the momentum of
the fermion as

$s=\frac{(p\cdot q)p-m^{2}q}{m\sqrt{(p\cdot q)^{2}-m^{2}q^{2}}},$ (13)

where $q$ can be an arbitrary vector except for those paralleling to momentum
$p$. For $q=(1,\vec{0})$, the polarization vector is found to be
$s=(\frac{|\vec{p}|}{m},\frac{E}{m}\frac{\vec{p}}{|\vec{p}|})$, which
indicates that the corresponding spinor is in helicity eigenstate. In
computation, it is more convenient to take $q=\frac{M}{m}k_{0}$, the so called
Kleiss-Stirling (KS) Kleiss:1985yh  polarization basis. In this basis, Eq.
(11) can be simplified to

$U_{\rm KS}(p_{1},\lambda_{1})\otimes\bar{U}_{\rm
KS}(p_{2},\lambda_{2})=\frac{({\not{p}}_{1}+M_{1})\Lambda(\lambda_{1},\lambda_{2}){\not{k}}_{0}({\not{p}}_{2}+M_{2})}{4\sqrt{k_{0}\cdot
p_{1}}\sqrt{k_{0}\cdot p_{2}}}.$ (14)

Note, in phenomenological study, other choices of polarization basis may lead
to certain convenience. The transformation rule between spinors in different
polarization basis can be obtained by taking an explicit representation for
Dirac matrices.

In general, arbitrary vectors $k_{0}$ and $k_{1}$ will cause the result of the
amplitude extra complication. To avoid this, in actual computation, one may
either specify $k_{0}$ and $k_{1}$ explicitly, or construct $k_{0}$ and
$k_{1}$ with external momenta, as demonstrated in Ref. Chang:1992bb .

##  III Singularity structure of NLO QCD calculation

In this section, we analyze the singularity structure of NLO QCD calculation.
The dimensional regularization with space-time dimension $D=4-2\epsilon$ is
used to regularize both ultraviolet (UV) and infrared (IR) singularities.
Although the results are well known Kunszt:1994np ; Catani:2000ef , we discuss
them in detail for a twofold reason: (i) they are essential to the NLO
generalization of TAM, and (ii) we provide a new perspective on this subject.
Specifically, in Ref. Catani:2000ef , the singular terms of the virtual loop
corrections are derived from that of the real corrections, by exploiting the
fact that the IR singularities of the virtual and real corrections cancel each
other. While here, we derive these singular terms through direct loop integral
analysis, and show that they are exactly canceled by their counterparts in
real corrections.

###  III.1 Singular terms in virtual corrections

The one-loop virtual corrections contain UV and IR singularities, which appear
as $\frac{1}{\epsilon^{n}}-$pole under the dimensional regularization. In a
renormalizable theory like QCD, UV singularities are contained in the diagrams
or subdiagrams with a small number of external legs, and can be removed by
renormalization procedure. In renormalized perturbation theory, the
renormalized UV-finite one-loop amplitude $\tilde{\mathcal{M}}^{\rm loop}$ is
defined as

$\tilde{\mathcal{M}}^{\rm loop}=\mathcal{M}^{\rm loop}+\mathcal{M}^{\rm CT},$
(15)

where $\mathcal{M}^{\rm CT}$ denotes the amplitudes of counterterms.

To study the IR singularity structure of $\tilde{\mathcal{M}}^{\rm loop}$, we
use lightcone gauge, where the gluon propagator is

$D_{\mu\nu}^{ab}(p)=\delta^{ab}\frac{i\Pi_{\mu\nu}(p)}{p^{2}+i\varepsilon},$
(16)

with

$\Pi_{\mu\nu}(p)=-g_{\mu\nu}+\frac{r_{\mu}p_{\nu}+r_{\nu}p_{\mu}}{r\cdot p},$
(17)

where $r$ is a light-like vector. Lightcone gauge is a physical gauge, means a
sum over physical transverse polarization states while the gluon is on its
mass shell:

$\Pi_{\mu\nu}(p)\overset{p^{2}=0}{\longrightarrow}\sum_{i=1,2}\epsilon_{\mu}^{(i)}(p,r)\epsilon_{\nu}^{(i)*}(p,r).$
(18)

For one-loop amplitude without soft or mutually collinear external lines, soft
singularities originate from the exchange of soft gluon between two on-shell
legs. To isolate the soft singularities, we impose a cutoff $\delta_{0}$ to
all components of loop momentum $k$, that is

$|k^{\mu}|<\delta_{0}\ll\text{particle masses or other kinematic scales}.$
(19)

This region will be referred as soft region.

The structure of external leg attached by a soft gluon can be approximated as

$\vbox{\hbox{\includegraphics[scale={0.5}]{eikonal.eps}}}\simeq-
g_{s}T^{a}_{c_{i}c_{j}}\frac{1}{k^{2}+2p\cdot
k}\left(2p^{\mu}\vbox{\hbox{\includegraphics[scale={0.5}]{eikonalB.eps}}}+\mathcal{O}(|k|)\right),$
(20)

with

$T^{a}_{c_{i}c_{j}}=\begin{cases}t^{a}_{c_{i}c_{j}},i=\text{outgoing quark or
incoming antiquark}\\\ -t^{a}_{c_{j}c_{i}},i=\text{outgoing antiquark or
incoming quark}\\\ -if^{ac_{i}c_{j}},i=\text{gluon}\end{cases}.$ (21)

Here the dashed line denotes either quark (massive or massless) or gluon,
$c_{i}$ denotes the color index of parton $i$, $t^{a}_{c_{i}c_{j}}$ and
$f^{ac_{i}c_{j}}$ are the generators of $SU(3)$ fundamental and adjoint
representation respectively. By default, the momentum of parton $i$ is defined
as outgoing. For incoming case, one should take the replacement $p\to-p$.

For the structure of two external legs connected by one soft gluon, we have

$\displaystyle\vbox{\hbox{\includegraphics[scale={0.5}]{softloop.eps}}}\simeq$
$\displaystyle
ig_{s}^{2}T^{a}_{c_{i}c_{i^{\prime}}}T^{a}_{c_{j}c_{j^{\prime}}}\mu^{4-D}\int\limits_{|k^{\mu}|<\delta_{0}}\frac{d^{D}k}{(2\pi)^{D}}\frac{1}{k^{2}(k^{2}+2k\cdot
p_{i})(k^{2}-2k\cdot p_{j})}$ (22)
$\displaystyle\times\left(4p_{i}\cdot\Pi(k)\cdot
p_{j}\vbox{\hbox{\includegraphics[scale={0.5}]{softloopB.eps}}}+\mathcal{O}(|k|)\right).$

Here, all the three propagators have poles in the region
$|k^{\mu}|<\delta_{0}$. However, at NLO, only the poles of $1/k^{2}$ are
concerned. For the case where both partons are incoming or outgoing, the poles
of $1/(k^{2}+2k\cdot p_{i})$ and $1/(k^{2}-2k\cdot p_{j})$ lead to pure
imaginary singular terms111These terms can be obtained through the Cutkosky
rules Cutkosky:1960sp ., which eventually canceled each other between
$\tilde{\mathcal{M}}^{\rm loop}(\mathcal{M}^{\rm tree})^{*}$ and
$(\tilde{\mathcal{M}}^{\rm loop})^{*}\mathcal{M}^{\rm tree}$. For the case
where one parton is outgoing while another is incoming, no imaginary
singularities arise. In fact, the poles of $1/(k^{2}+2k\cdot p_{i})$ and
$1/(k^{2}-2k\cdot p_{j})$ are located in the region where
$k^{+}k^{-}\ll|\vec{k}_{T}|^{2}$ 222Here we work in the $p_{i}+p_{j}$ center-
of-mass frame. We take the lightcone coordinates so that the momenta are
$p_{i}=(p_{i}^{+},m_{i}^{2}/(2p_{i}^{+}),\vec{0}_{T})$,
$p_{j}=(m_{j}^{2}/(2p_{j}^{-}),p_{j}^{-},\vec{0}_{T})$,
$k=(k^{+},k^{-},\vec{k}_{T})$., and we can perform contour deformations on
both $k^{+}$ and $k^{-}$ to get out of this region Collins:2011zzd . Note,
since we work in lightcone guage, the singularities at $r\cdot k=0$ may
obstruct the contour deformations. In our simple case, this can be overcome by
appropriate choice of $r$. For example, we may choose a generic $r$ that do
not parallel to any $p_{i}$.

When both partons $i$ and $j$ are massive, we can deform the integral to a
contour where all components of $k$ are comparable. Then the asymptotic
behavior of loop momentum is $|k^{\mu}|\sim\kappa^{2}$ as $\kappa\to 0$. Thus
we can neglect the $k^{2}$ term compared to $k\cdot p_{i}$ or $k\cdot p_{j}$
(eikonal approximation). It can also be seen that the $\mathcal{O}(|k|)$ term
does not contribute to soft singularity, as it leads to terms scaling like
$\kappa^{2}$ or higher. When either or both of $i$ and $j$ are massless, there
is an overlapping soft-collinear region, where the scaling behavior of $k$ is
$|k^{+}|\sim\kappa$, $|k^{-}|\sim\kappa^{3}$ and $|\vec{k}_{T}|\sim\kappa^{2}$
(or $|k^{+}|\sim\kappa^{3}$, $|k^{-}|\sim\kappa$ and
$|\vec{k}_{T}|\sim\kappa^{2}$). It can be seen that the eikonal approximation
still hold in this region. Thus for both massive and massless cases, we have

$\displaystyle\vbox{\hbox{\includegraphics[scale={0.5}]{softLoop.eps}}}\overset{\rm
soft}{\sim}-ig_{s}^{2}T^{a}_{c_{i}c_{i^{\prime}}}T^{a}_{c_{j}c_{j^{\prime}}}\mu^{4-D}\int\limits_{|k^{\mu}|<\delta_{0}}\frac{d^{D}k}{(2\pi)^{D}}\frac{p_{i}\cdot\Pi(k)\cdot
p_{j}}{k^{2}(k\cdot p_{i})(k\cdot
p_{j})}\vbox{\hbox{\includegraphics[scale={0.5}]{softLoopB.eps}}}$
$\displaystyle\overset{\rm
soft}{\sim}-g_{s}^{2}T^{a}_{c_{i}c_{i^{\prime}}}T^{a}_{c_{j}c_{j^{\prime}}}\mu^{4-D}\int\limits_{|\vec{k}|<\delta_{0}}\frac{d^{D-1}k}{2k_{0}(2\pi)^{D-1}}\frac{p_{i}\cdot\Pi(k)\cdot
p_{j}}{(k\cdot p_{i})(k\cdot
p_{j})}\bigg{|}_{k_{0}=|\vec{k}|}\vbox{\hbox{\includegraphics[scale={0.5}]{softLoopB.eps}}}$
$\displaystyle\ .$ (23)

Here, the symbol “$\overset{\rm soft}{\sim}$” denotes that the real part of
soft singular terms on each side are equal.

Besides soft gluon exchange between external legs, soft singularities also
come from on-shell renormalization constants. The corresponding terms can be
re-expressed as self-energy insertions to external lines. For this case, we
have (see Appendix A for detailed derivation):

$\displaystyle\frac{1}{2}\vbox{\hbox{\includegraphics[scale={0.5}]{softSE.eps}}}\overset{\rm
soft}{\sim}$
$\displaystyle-\frac{1}{2}g_{s}^{2}T^{a}_{c_{i}c_{i^{\prime\prime}}}T^{a}_{c_{i^{\prime\prime}}c_{i^{\prime}}}\mu^{4-D}\int\limits_{|\vec{k}|<\delta_{0}}\frac{d^{D-1}k}{2k_{0}(2\pi)^{D-1}}\frac{p_{i}\cdot\Pi(k)\cdot
p_{i}}{(k\cdot p_{i})^{2}}\bigg{|}_{k_{0}=|\vec{k}|}$
$\displaystyle\times\vbox{\hbox{\includegraphics[scale={0.5}]{softSEB.eps}}}.$
(24)

Summing up all configurations where a gluon connects two external legs and
self-energy corrections to each external line, we obtain the complete soft
singularities for one-loop amplitude:

$\displaystyle\tilde{\mathcal{M}}^{\rm loop}_{c_{1}\cdots c_{n}}\overset{\rm
soft}{\sim}$
$\displaystyle-\frac{1}{2}g_{s}^{2}\sum_{i,j}^{n}\mu^{4-D}\int\limits_{|\vec{k}|<\delta_{0}}\frac{d^{D-1}k}{2k_{0}(2\pi)^{D-1}}\frac{p_{i}\cdot\Pi(k)\cdot
p_{j}}{(k\cdot p_{i})(k\cdot p_{j})}\bigg{|}_{k_{0}=|\vec{k}|}$
$\displaystyle\times\left(\boldsymbol{T}\cdot\boldsymbol{T}\mathcal{M}^{\rm
tree}\right)_{c_{1}\cdots c_{i}\cdots c_{j}\cdots c_{n}},$ (25)

where

$\left(\boldsymbol{T}\cdot\boldsymbol{T}\mathcal{M}^{\rm
tree}\right)_{c_{1}\cdots c_{i}\cdots c_{j}\cdots
c_{n}}=\begin{cases}T^{a}_{c_{i}c_{i^{\prime}}}T^{a}_{c_{j}c_{j^{\prime}}}\mathcal{M}^{\rm
tree}_{c_{1}\cdots c_{i^{\prime}}\cdots c_{j^{\prime}}\cdots c_{n}},&i\neq
j\\\
T^{a}_{c_{i}c_{i^{\prime\prime}}}T^{a}_{c_{i^{\prime\prime}}c_{i^{\prime}}}\mathcal{M}^{\rm
tree}_{c_{1}\cdots c_{i^{\prime}}\cdots c_{n}},&i=j\end{cases}$ (26)

is the color connected Born amplitude.

The collinear singularities arise when the virtual gluon is collinear to any
massless external momentum. Here we work in lightcone coordinates where loop
momentum $k$ and external momentum $p$ can be expressed as

$\displaystyle k=(k^{+},k^{-},\vec{k}_{T}),$ $\displaystyle
p=(p^{+},0,\vec{0}_{T}).$ (27)

Since the soft-collinear singularities have been incorporated in Eq. (25), to
avoid double counting, we consider only the hard-collinear singularities. The
corresponding scaling behavior of $k$ is $|k^{+}|\sim\kappa^{0}$,
$|k^{-}|\sim\kappa^{2}$ and $|\vec{k}_{T}|\sim\kappa$. Then the involved
propagators $1/k^{2}$ and $1/(k\pm p)^{2}$ scale like $\kappa^{-2}$, other
propagators with non-collinear external momenta scale like $\kappa^{0}$. The
hard-collinear integral region for $k$ is defined as

$|k^{+}|>\frac{p^{+}}{p^{0}}\delta_{0},\quad\quad|\vec{k}_{T}|<\delta_{T},\quad\quad|k^{-}|\sim\frac{|\vec{k}_{T}|^{2}}{|k^{+}|},$
(28)

with the cut on $|k^{+}|$ exclude the soft-collinear region. The explicit
integration range of $|k^{-}|$ does not concern with the collinear singular
terms. Note, to validate the scaling behavior of $k$, the soft cutoff
parameter should be much larger than the collinear cutoff parameter:
$\delta_{T}\ll\delta_{0}$.

In lightcone gauge, the structures where virtual gluon $k$ connects the
external leg $p$ to hard part (or other external leg) scale like $\kappa$:

$\vbox{\hbox{\includegraphics[scale={0.5}]{colLoop.eps}}}\simeq\mathcal{O}(\kappa).$
(29)

Therefore, the only collinear singularites come from the self-energy
corrections to external legs. We have (see Appendix B for detailed
derivation):

$\displaystyle\frac{1}{2}\vbox{\hbox{\includegraphics[scale={0.5}]{colSEQuark.eps}}}\overset{\rm
coll}{\sim}$
$\displaystyle-\frac{g_{s}^{2}}{16\pi^{2}}\frac{1}{\Gamma(1-\epsilon)}\left(\frac{4\pi\mu^{2}}{\delta_{T}^{2}}\right)^{\epsilon}\frac{1}{\epsilon}C_{F}\left(\frac{3+\epsilon}{2}+2\ln\frac{\delta_{0}}{p_{0}}\right)$
$\displaystyle\times\vbox{\hbox{\includegraphics[scale={0.5}]{colSEQuarkB.eps}}},$
$\displaystyle\frac{n_{lf}}{2}\vbox{\hbox{\includegraphics[scale={0.5}]{colSEGluonF.eps}}}\overset{\rm
coll}{\sim}$
$\displaystyle-\frac{g_{s}^{2}}{16\pi^{2}}\frac{1}{\Gamma(1-\epsilon)}\left(\frac{4\pi\mu^{2}}{\delta_{T}^{2}}\right)^{\epsilon}\frac{1}{\epsilon}n_{lf}\left(-\frac{1-\epsilon}{3-2\epsilon}\right)$
$\displaystyle\times\vbox{\hbox{\includegraphics[scale={0.5}]{colSEGluonB.eps}}},$
$\displaystyle\frac{1}{2}\vbox{\hbox{\includegraphics[scale={0.5}]{colSEGluonG.eps}}}\overset{\rm
coll}{\sim}$
$\displaystyle-\frac{g_{s}^{2}}{16\pi^{2}}\frac{1}{\Gamma(1-\epsilon)}\left(\frac{4\pi\mu^{2}}{\delta_{T}^{2}}\right)^{\epsilon}\frac{1}{\epsilon}C_{A}\left(\frac{11}{6}+2\ln\frac{\delta_{0}}{p_{0}}\right)$
$\displaystyle\times\vbox{\hbox{\includegraphics[scale={0.5}]{colSEGluonB.eps}}}.$
(30)

Here, $n_{lf}=3$ denotes the number of light quark flavors, $C_{A}=3$,
$C_{F}=4/3$ are QCD color factors; the symbol “$\overset{\rm coll}{\sim}$”
means that the collinear singular terms on each side are equal. Then the
collinear singular terms for one-loop amplitude have the form

$\tilde{\mathcal{M}}^{\rm loop}_{c_{1}\cdots c_{n}}\overset{\rm
coll}{\sim}\frac{g_{s}^{2}}{16\pi^{2}}\frac{1}{\Gamma(1-\epsilon)}\left(\frac{4\pi\mu^{2}}{\delta_{T}^{2}}\right)^{\epsilon}\sum_{i}\left(\frac{\gamma(i)}{\epsilon}\right)\mathcal{M}^{\rm
tree}_{c_{1}\cdots c_{n}},$ (31)

where

$\displaystyle\gamma(q)=-C_{F}\left(\frac{3+\epsilon}{2}+2\ln\frac{\delta_{0}}{p_{0}}\right),$
$\displaystyle\gamma(g)=n_{lf}\frac{1-\epsilon}{3-2\epsilon}-C_{A}\left(\frac{11}{6}+2\ln\frac{\delta_{0}}{p_{0}}\right).$
(32)

###  III.2 Dimensional regularization prescriptions and singular terms in real
corrections

The analysis in the previous subsection is based on dimensional
regularization. The key ingredient of dimensional regularization is to
continue the dimensions of loop momentum from $4$ to $D=4-2\epsilon$. While
for the treatments of external momenta and gluons’ polarization, one is left
with some freedom, which result in different variants of dimensional
regularization. Two commonly used variants are333Another commonly used
regularization scheme is the dimensional reduction Siegel:1979wq ; Bern:1991aq
; Stockinger:2005gx , where a quasi-4-dimensional space should be introduced
Stockinger:2005gx . The transition rules between dimensional reduction and
dimensional regularization are discussed in Refs. Kunszt:1994np ;
Catani:1996pk ; Signer:2008va ; Catani:2000ef .:

  * •

‘t Hooft-Veltman (HV) tHooft:1972tcz  scheme: Loop momentum are treated as
$D$-dimensional, while external ones are treated as $4$-dimensional. The
gluons inside loop have $D-2$ polarization states, while other gluons have 2
polarizations.

  * •

Conventional dimensional regularization (CDR) scheme: All momenta are treated
as $D$-dimensional, and all gluons have $D-2$ polarization states.

The analysis in the previous subsection is legitimate in both HV and CDR
schemes. In Eqs. (25) and (31), the only quantity that concerning the choice
of HV or CDR is the tree-level amplitude $\mathcal{M}^{\rm tree}$.

In general, only a combination of virtual loop corrections and real emission
contributions lead to IR-finite results. Their dependence on regularization
prescriptions should also be canceled as the IR singularities. This
cancellation is only achieved if the regularization prescriptions employed in
the virtual and real corrections are consistent, which means unitarity. Hence,
in real corrections, the emitted soft or collinear particles should be treated
in the same way as the particles inside loop in virtual corrections. As an
example, Fig. 1 shows the case of a gluon splitting into soft or collinear
gluons under HV and CDR schemes separately.

Figure 1: Gluon splitting into soft or collinear gluons under HV and CDR
schemes. Here the label $D$ (4) indicates that the momentum of corresponding
gluon is $D$\- ($4$-) dimensional, and the number of polarization states is
$D-2$ (2).

There are essentially two types of approaches to evaluate the cross sections
of real emission processes: one based on the phase-space slicing method
Fabricius:1981sx ; Kramer:1986mc ; Harris:2001sx , and the other based on the
subtraction method Ellis:1980wv ; Catani:1996vz ; Phaf:2001gc . In both
approaches, IR singular terms are isolated, and the remaining finite parts can
be calculated numerically in 4-dimensional space-time. To match our analysis
on virtual corrections, here we take the two cutoff phase-space slicing
method. As the corresponding implementation is described in detail in Ref.
Harris:2001sx , we only introduce some main results here.

Considering the real emission process

$p_{a}+p_{b}\to p_{1}+\cdots+p_{n}+p_{n+1}\ ,$ (33)

where $(n+1)$ is the “additional” particle that may soft or collinear to
another massless external line. By introducing soft cut $\delta_{0}$ and
collinear cut $\delta_{T}$444The cutoff parameters used here is different from
that used in Ref. Harris:2001sx . They are related by the relations
$\delta_{0}=\frac{\sqrt{s_{12}}}{2}\delta_{s}^{\text{\tiny\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Harris:2001sx}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ and
$\delta_{T}^{2}=z(1-z)\delta_{c}^{\text{\tiny\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Harris:2001sx}{\@@citephrase{(}}{\@@citephrase{)}}}}}s_{12}$.
which satisfies $\delta_{0}\gg\delta_{T}$, the phase space can be separated
into three regions:

  * •

soft: $p_{n+1}^{0}<\delta_{0}$;

  * •

hard-collinear (HC): $p_{n+1}^{0}>\delta_{0}$ and $|\vec{p}_{n+1\
T}|<\delta_{T}$;

  * •

hard-non-collinear (HNC): $p_{n+1}^{0}>\delta_{0}$ and $|\vec{p}_{n+1\
T}|>\delta_{T}$.

Here, the transverse momentum $|\vec{p}_{n+1\ T}|$ is relative to the “parent”
particle $i^{\prime}$, which splitting into $i$ and $(n+1)$: $i^{\prime}\to
i+(n+1)$. The cross section of real emission process can be written as

$\sigma_{\rm real}=\sigma_{\rm real}^{\rm soft}+\sigma_{\rm real}^{\rm
HC}+\sigma_{\rm real}^{\rm HNC}.$ (34)

After neglecting terms of order $\delta_{0}$ and $\delta_{T}$, the soft and
hard-collinear pieces take the form555Here we present the results for
indistinguishable final state case only. The results for other cases, like
tagged final state or hadron in initial state, can be found in Ref.
Harris:2001sx .

$\displaystyle\sigma_{\rm real}^{\rm soft}=$
$\displaystyle\frac{1}{2\Phi}\int\limits_{\rm
S}d\Gamma_{n+1}\overline{\sum}|\mathcal{M}^{\rm real}_{c_{1}\cdots
c_{n+1}}|^{2}$ $\displaystyle=$ $\displaystyle
g_{s}^{2}\mu^{4-D}\int\limits_{|\vec{p}_{n+1}|<\delta_{0}}\frac{d^{D-1}p_{n+1}}{2p_{n+1}^{0}(2\pi)^{D-1}}\sum_{i,j}^{n}\bigg{\\{}\frac{p_{i}\cdot\Pi(p_{n+1})\cdot
p_{j}}{(p_{i}\cdot p_{n+1})(p_{j}\cdot p_{n+1})}$
$\displaystyle\frac{1}{2\Phi}\int
d\Gamma_{n}\overline{\sum}\left[\mathcal{M}^{\rm tree}_{c_{1}\cdots
c_{i^{\prime}}\cdots c_{j}\cdots
c_{n}}T^{a}_{c_{i}c_{i^{\prime}}}\right]\left[\mathcal{M}^{\rm
tree}_{c_{1}\cdots c_{i}\cdots c_{j^{\prime}}\cdots
c_{n}}T^{a}_{c_{j}c_{j^{\prime}}}\right]^{*}\bigg{\\}},$ (35)

and

$\displaystyle\sigma_{\rm real}^{\rm HC}=$
$\displaystyle\frac{1}{2\Phi}\int\limits_{\rm
HC}d\Gamma_{n+1}\overline{\sum}|\mathcal{M}^{\rm real}_{c_{1}\cdots
c_{n+1}}|^{2}$ $\displaystyle=$
$\displaystyle-\frac{g_{s}^{2}}{8\pi^{2}}\frac{1}{\Gamma(1-\epsilon)}\left(\frac{4\pi\mu^{2}}{\delta_{T}^{2}}\right)^{\epsilon}\left(\sum_{i}\frac{\gamma(i)}{\epsilon}\right)\frac{1}{2\Phi}\int
d\Gamma_{n}\overline{\sum}|\mathcal{M}^{\rm tree}_{c_{1}\cdots c_{n}}|^{2},$
(36)

where $\Phi$ is the flux factor, $d\Gamma_{n}$ and $d\Gamma_{n+1}$ stand for
the $n$\- and $(n+1)$\- body phase space respectively, $\gamma(i)$ has been
defined in Eq. (32).

Comparing Eqs. (35) and (36) with Eqs. (25) and (31), we can see that the IR
singularities in real corrections are canceled explicitly against their
counterparts in virtual corrections, as expected by the Kinoshita-Lee-
Nauenberg (KLN) theorem Kinoshita:1962ur ; Lee:1964is . Their dependence on
dimensional regularization prescription are also canceled, as long as the
corresponding $\gamma(i)$ and $\mathcal{M}^{\rm tree}$ are obtained with the
same scheme.

##  IV Generalize the TAM to NLO QCD calculation

In this section, we discuss how to apply TAM in NLO QCD calculations. We also
compare the result obtained from TAM with that from CAS.

###  IV.1 Calculation scheme

In fact, the application of TAM in the calculation of one-loop helicity
amplitude is transparent, and will not lead to any additional technical
difficulty. The only concern is that one should find a proper scheme for real
corrections to guarantee the unitarity.

In Section III.1, we analyze the singularity structure of one-loop amplitude.
The analysis there is performed at amplitude level, and is irrelevant with the
treatment of spinors. Therefore, for one-loop helicity amplitude which
calculated through TAM, the formulas (15), (25) and (31) still hold. While in
real corrections, the singularity formulas (35) and (36) are at cross section
(decay width) level. We can decompose $\sigma_{\rm real}^{\rm soft}$ and
$\sigma_{\rm real}^{\rm HC}$ according to the helicity states of the $1$st to
$n$-th particles, which leads to

$\displaystyle\sum_{\lambda_{n+1}}\sigma_{\rm real}^{\rm
soft}(\lambda_{1},\cdots,\lambda_{n},\lambda_{n+1})$ $\displaystyle=$
$\displaystyle
g_{s}^{2}\mu^{4-D}\int\limits_{|\vec{p}_{n+1}|<\delta_{0}}\frac{d^{D-1}p_{n+1}}{2p_{n+1}^{0}(2\pi)^{D-1}}\sum_{i,j}^{n}\bigg{\\{}\frac{p_{i}\cdot\Pi(p_{n+1})\cdot
p_{j}}{(p_{i}\cdot p_{n+1})(p_{j}\cdot p_{n+1})}$
$\displaystyle\frac{1}{2\Phi}\int d\Gamma_{n}\left[\mathcal{M}^{\rm
tree}_{c_{1}\cdots c_{i^{\prime}}\cdots c_{j}\cdots
c_{n}}(\lambda_{1},\cdots,\lambda_{n})T^{a}_{c_{i}c_{i^{\prime}}}\right]\left[\mathcal{M}^{\rm
tree}_{c_{1}\cdots c_{i}\cdots c_{j^{\prime}}\cdots
c_{n}}(\lambda_{1},\cdots,\lambda_{n})T^{a}_{c_{j}c_{j^{\prime}}}\right]^{*}\bigg{\\}},$
(37)

and

$\displaystyle\sum_{\lambda_{n+1}}\sigma_{\rm real}^{\rm
HC}(\lambda_{1},\cdots,\lambda_{n},\lambda_{n+1})$ $\displaystyle=$
$\displaystyle-\frac{g_{s}^{2}}{8\pi^{2}}\frac{1}{\Gamma(1-\epsilon)}\left(\frac{4\pi\mu^{2}}{\delta_{T}^{2}}\right)^{\epsilon}\left(\sum_{i}\frac{\gamma(i)}{\epsilon}\right)\frac{1}{2\Phi}\int
d\Gamma_{n}|\mathcal{M}^{\rm tree}_{c_{1}\cdots
c_{n}}(\lambda_{1},\cdots,\lambda_{n})|^{2}\ .$ (38)

Here, $\lambda_{i}$ denotes the helicity state of the $i$-th particle,
$\mathcal{M}^{\rm tree}(\lambda_{1},\cdots,\lambda_{n})$ denotes the tree-
level helicity amplitude, which should be calculated through the TAM to
guarantee the unitarity.

With the above preparation, we then obtain the scheme that can be used in NLO
QCD calculations: the one-loop helicity amplitude can be calculated
straightforwardly by using the TAM, and the soft and hard-collinear pieces of
real corrections should be calculated through Eqs. (37) and (38). There are
some remarkable points in practice as follows:

1) As we have discussed in Sec. III.2, this calculation scheme is legitimate
under both CDR and HV schemes.

2) In the TAM, additional $\gamma_{5}$ is introduced in Eq. (12). Although in
dimensional regularization, $\gamma_{5}$ is always a difficult object to deal
with, the $\gamma_{5}$ here will not cause additional trouble. Because this
$\gamma_{5}$ is only concerned with the treatment of spinor, which is
independent with singularity structure. Hence, our scheme is valid under
arbitrary self-consistent $\gamma_{5}$ prescription, like the ‘t Hooft-
Veltman-Breitenlohner-Maison (HVBM) prescription tHooft:1972tcz ;
Breitenlohner:1976te  or the Kreimer-Korner prescription Kreimer:1989ke ;
Korner:1991sx .

3) The integrand in Eq. (37) contains the factor

$\frac{p_{i}\cdot\Pi(p_{n+1})\cdot p_{j}}{(p_{i}\cdot p_{n+1})(p_{j}\cdot
p_{n+1})}=\frac{-p_{i}\cdot p_{j}}{(p_{i}\cdot p_{n+1})(p_{j}\cdot
p_{n+1})}+\frac{1}{r\cdot p_{n+1}}\left(\frac{r\cdot p_{i}}{p_{i}\cdot
p_{n+1}}+\frac{r\cdot p_{j}}{p_{j}\cdot p_{n+1}}\right).$ (39)

By using the color conservation relation
$\sum_{i=1}^{n}T^{a}_{c_{i}c_{i^{\prime}}}\mathcal{M}^{\rm tree}_{c_{1}\cdots
c_{i^{\prime}}\cdots c_{n}}=0$ Catani:2000ef , we can see that the second term
is vanished after summing over $i$ and $j$. Hence in actual calculation, we
can replace $p_{i}\cdot\Pi(p_{n+1})\cdot p_{j}$ by $-p_{i}\cdot p_{j}$.

###  IV.2 Consistency between TAM and CAS

To compare the result obtained from TAM with that from CAS, we introduce a
quantity $\mathcal{F}(\Gamma_{A},\Gamma_{B})$:

$\mathcal{F}(\Gamma_{A},\Gamma_{B})=\sum_{\lambda_{1},\lambda_{2}}\bar{U}(p_{2},\lambda_{2})\Gamma_{A}U(p_{1},\lambda_{1})\bar{U}(p_{1},\lambda_{1})\Gamma_{B}U(p_{2},\lambda_{2})\
,$ (40)

where $\Gamma_{A}$ and $\Gamma_{B}$ denote series of Dirac gamma matrices.
This quantity can be calculated through either CAS or TAM:

$\displaystyle\mathcal{F}_{\rm CAS}(\Gamma_{A},\Gamma_{B})=$
$\displaystyle{\rm
tr}[\Gamma_{A}(\not{p}_{1}+M_{1})\Gamma_{B}(\not{p}_{2}+M_{2})]\ ,$
$\displaystyle\mathcal{F}_{\rm TAM}(\Gamma_{A},\Gamma_{B})=$
$\displaystyle\frac{1}{16(k_{0}\cdot p_{1})(k_{0}\cdot
p_{2})}\sum_{\lambda_{1},\lambda_{2}}\Big{\\{}{\rm
tr}[\Gamma_{A}(\not{p}_{1}+M_{1})\Lambda(\lambda_{1},\lambda_{2})(\not{p}_{2}+M_{2})]$
$\displaystyle{\rm
tr}[\Gamma_{B}(\not{p}_{2}+M_{2})\Lambda(\lambda_{2},\lambda_{1})(\not{p}_{1}+M_{1})]\Big{\\}}\
.$ (41)

In 4-dimensions, the gamma matrices can be represented explicitly by $4\times
4$ matrices, and the construction (9) is compatible with the fermion spin sum
relation (10), which means $\mathcal{F}_{\rm TAM}(\Gamma_{A},\Gamma_{B})$ is
equivalent with $\mathcal{F}_{\rm CAS}(\Gamma_{A},\Gamma_{B})$. While in
general $D$-dimensions, this equivalence will no longer hold666For example,
considering $\Gamma_{A}=\gamma^{\mu}$, $\Gamma_{B}=\gamma_{\mu}$. By using the
HVBM $\gamma_{5}$-scheme, we obtain $\mathcal{F}_{\rm
CAS}(\gamma^{\mu},\gamma_{\mu})=4DM_{1}M_{2}-(4D-8)p_{1}\cdot p_{2}$, and
$\mathcal{F}_{\rm TAM}(\gamma^{\mu},\gamma_{\mu})=16M_{1}M_{2}-8p_{1}\cdot
p_{2}$.. The subtle difficulty is that there is inconsistency between the
continuous space-time dimensions and the fixed spinor space dimensions, as
revealed in Ref. Siegel:1980qs . Therefore, we may write

$\mathcal{F}_{\rm CAS}(\Gamma_{A},\Gamma_{B})-\mathcal{F}_{\rm
TAM}(\Gamma_{A},\Gamma_{B})=\mathcal{O}(\epsilon).$ (42)

At NLO, one may worry that the $\mathcal{O}(\epsilon)$ term will interfere
with UV or IR singularities, which eventually lead to terms that are finite or
even divergent as $\epsilon\to 0$. In fact, the $\mathcal{O}(\epsilon)$ term
arises due to different treatment of spinor, whose effect will be eliminated
as we sum up all pieces of NLO corrections, as discussed in the preceding
subsection.

In HV dimensional regularization scheme, all momenta and gluons outside loop
are treated as 4-dimensional. The product of one-loop amplitude and Born
amplitude takes the form $\mathcal{F}(\Gamma_{A},\overline{\Gamma}_{B})$,
where $\overline{\Gamma}_{B}$ denotes a series of 4-dimensional Dirac gamma
matrices that comes from Born amplitude. In this case, we have

$\mathcal{F}_{\rm CAS}(\Gamma_{A},\overline{\Gamma}_{B})-\mathcal{F}_{\rm
TAM}(\Gamma_{A},\overline{\Gamma}_{B})=0,$ (43)

which indicates that the consistency can be obtained even without a
combination of virtual and real corrections.

##  V Example

To demonstrate the calculation scheme proposed in Sec. IV, in this section, we
apply it to the calculation of the NLO QCD corrections to $g+g\to t+\bar{t}$
and $q+\bar{q}\to t+\bar{t}$ processes. We find that the final results are
consistent with that from the CAS. Another successful application of the TAM
at one-loop level can be found in Ref. Chen:2012ju , where Higgs boson decays
to $l\bar{l}\gamma$ was studied.

The momenta and polarization states of incoming and outgoing particles are
denoted as:

$\displaystyle g(p_{1},\lambda_{1})+g(p_{2},\lambda_{2})\to
t(p_{3},\lambda_{3})+\bar{t}(p_{4},\lambda_{4}),$ $\displaystyle
q(p_{1},\lambda_{1})+\bar{q}(p_{2},\lambda_{2})\to
t(p_{3},\lambda_{3})+\bar{t}(p_{4},\lambda_{4}).$ (44)

Here, initial and final state particles are all on their mass shells:
$p_{1}^{2}=p_{2}^{2}=0$ and $p_{3}^{2}=p_{4}^{2}=m^{2}_{t}$. The gluons’
polarization vectors are denoted as $\epsilon_{1}^{(\lambda_{1})}$ and
$\epsilon_{2}^{(\lambda_{2})}$, which satisfy the constrains
$\epsilon_{1}^{(\lambda_{1})}\cdot\epsilon_{1}^{(\lambda_{1})*}=\epsilon_{2}^{(\lambda_{2})}\cdot\epsilon_{2}^{(\lambda_{2})*}=-1$
and
$p_{1}\cdot\epsilon_{1}^{(\lambda_{1})}=p_{2}\cdot\epsilon_{2}^{(\lambda_{2})}=0$.

In the center-of-mass system, the momenta and gluons’ polarization vectors are
chosen as:

$\displaystyle p_{1}=\frac{\sqrt{s}}{2}(1,0,0,1),\quad
p_{2}=\frac{\sqrt{s}}{2}(1,0,0,-1),$ $\displaystyle
p_{3}=\frac{\sqrt{s}}{2}(1,0,r_{y},r_{z}),\quad
p_{4}=\frac{\sqrt{s}}{2}(1,0,-r_{y},-r_{z}),$ (45)

and

$\epsilon_{1}^{(1)}=\epsilon_{2}^{(1)}=(0,1,0,0),\quad\epsilon_{1}^{(2)}=\epsilon_{2}^{(2)}=(0,0,1,0),$
(46)

where $s=(p_{1}+p_{2})^{2}$, and the on-shell condition constrains
$r_{y}^{2}+r_{z}^{2}=1-4m_{t}^{2}/s$. The helicity states of fermions are
defined under the KS basis with the auxiliary vectors chosen as:

$k_{0}=(1,1,0,0),\quad k_{1}=(0,0,0,1).$ (47)

Then the tree-level and one-loop helicity amplitudes can be calculated
straightforwardly by using the spinor product formula (14).

In the computation of one-loop amplitudes, the HV dimensional regularization
is adopted to regularize the UV and IR singularities. For the $\gamma_{5}$
introduced in Eq. (14), the HVBM prescription is adopted. The UV singularities
are removed by renormalization procedure. The renormalization constants
include $Z_{2}$, $Z_{m}$, $Z_{l}$, $Z_{3}$ and $Z_{g}$, corresponding to heavy
quark field, heavy quark mass, light quark field, gluon field and strong
coupling constant, respectively. We define $Z_{2}$, $Z_{m}$, $Z_{l}$ and
$Z_{3}$ in the on-shell (OS) scheme, $Z_{g}$ in the modified minimal-
subtraction ($\overline{\rm MS}$) scheme. The corresponding counterterms are

$\displaystyle\delta Z_{2}^{\rm OS}=$ $\displaystyle-
C_{F}\frac{\alpha_{s}}{4\pi}\left[\frac{1}{\epsilon_{\rm
UV}}+\frac{2}{\epsilon_{\rm
IR}}-3\gamma_{E}+3\ln\frac{4\pi\mu^{2}}{m_{t}^{2}}+4\right],$
$\displaystyle\delta Z_{m}^{\rm OS}=$
$\displaystyle-3C_{F}\frac{\alpha_{s}}{4\pi}\left[\frac{1}{\epsilon_{\rm
UV}}-\gamma_{E}+\ln\frac{4\pi\mu^{2}}{m_{t}^{2}}+\frac{4}{3}\right],$
$\displaystyle\delta Z_{l}^{\overline{\rm OS}}=$ $\displaystyle-
C_{F}\frac{\alpha_{s}}{4\pi}\left[\frac{1}{\epsilon_{\rm
UV}}-\frac{1}{\epsilon_{\rm IR}}\right],$ $\displaystyle\delta
Z_{3}^{\overline{\rm OS}}=$
$\displaystyle\frac{\alpha_{s}}{4\pi}\left[(\beta^{\prime}_{0}-2C_{A})\left(\frac{1}{\epsilon_{\rm
UV}}-\frac{1}{\epsilon_{\rm
IR}}\right)-\frac{4}{3}T_{F}\sum_{i=c,b,t}\left(\frac{1}{\epsilon_{\rm
UV}}-\gamma_{E}+\ln\frac{4\pi\mu^{2}}{m_{i}^{2}}\right)\right],$
$\displaystyle\delta Z_{g}^{\overline{\rm MS}}=$
$\displaystyle-\frac{\beta_{0}}{2}\frac{\alpha_{s}}{4\pi}\left[\frac{1}{\epsilon_{\rm
UV}}-\gamma_{E}+\ln(4\pi)\right].$ (48)

Here, $\mu$ is the renormalization scale, $\gamma_{E}$ is the Euler’s
constant; $C_{A}=3$, $C_{F}=4/3$ and $T_{F}=1/2$ are QCD color factors;
$\beta_{0}=(11/3)C_{A}-(4/3)T_{F}n_{f}$ is the one-loop coefficient of QCD
beta function, in which $n_{f}=6$ is the number of active quark flavors,
$\beta^{\prime}_{0}=(11/3)C_{A}-(4/3)T_{F}n_{lf}$ and $n_{lf}=3$ is the number
of light quark flavors.

In real corrections, IR singularities arise from the phase-space integration
of the additional emitted gluon, whose momentum is denoted by $p_{5}$
hereafter. To isolate the singularities, we follows the lines of Ref.
Harris:2001sx . By introducing two cutoff parameters $\delta_{s}$ and
$\delta_{c}$, the real correction phase space is split into three regions:

  * •

soft: $p_{5}^{0}\leq\delta_{s}\sqrt{s}/2$;

  * •

hard-collinear: including the collinear-to-$p_{1}$ region where
$p_{5}^{0}>\delta_{s}\sqrt{s}/2$ and $(p_{1}+p_{5})^{2}\leq\delta_{c}s$, and
the the collinear-to-$p_{2}$ region where $p_{5}^{0}>\delta_{s}\sqrt{s}/2$ and
$(p_{2}+p_{5})^{2}\leq\delta_{c}s$.

  * •

hard-non-collinear: $p_{5}^{0}>\delta_{s}\sqrt{s}/2$ and
$(p_{1}+p_{5})^{2}>\delta_{c}s$ and $(p_{2}+p_{5})^{2}>\delta_{c}s$.

The soft and hard-collinear contributions can be obtained through Eqs. (2.22)
and (2.74) of Ref. Harris:2001sx , with the unpolarized Born cross section
$\sigma^{0}$ replaced by helicity cross section
$\sigma^{0}(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})$, which should be
calculated through the TAM. The remaining hard-non-collinear contribution is
IR finite, and can be calculated numerically in 4-dimensions. After summing up
these three pieces, their dependence on technical cuts are eliminated as
expected.

The total NLO corrections are obtained by summing up the virtual and real
corrections. We verify that all singularities are canceled exactly. By taking
the same renormalization scheme, we find that our results agree with Fig. 3
and Fig. 4 of Ref. Nason:1987xz  within 1% accuracy.

For illustrative purpose, we present the numerical values of $\mathcal{A}^{\rm
loop}$, $\mathcal{A}^{\rm CT}$, $\mathcal{A}^{\rm soft}_{\rm real}$ and
$\mathcal{A}^{\rm HC}_{\rm real}$, which are defined as:

$\displaystyle\mathcal{A}^{\rm loop}=2\sum_{\lambda}{\rm
Re}\left[\mathcal{M}^{\rm loop}(\mathcal{M}^{\rm tree})^{*}\right],$
$\displaystyle\mathcal{A}^{\rm CT}=2\sum_{\lambda}{\rm
Re}\left[\mathcal{M}^{\rm CT}(\mathcal{M}^{\rm tree})^{*}\right],$
$\displaystyle\mathcal{A}^{\rm soft}_{\rm
real}=g_{s}^{2}\mu^{2\epsilon}\int\limits_{p_{5}^{0}<\delta_{s}\sqrt{s}/2}\frac{d^{D-1}p_{5}}{2p_{5}^{0}(2\pi)^{D-1}}\sum_{i,j=1}^{4}\frac{p_{i}\cdot
p_{j}}{(p_{i}\cdot p_{5})(p_{j}\cdot p_{5})}$
$\displaystyle\quad\quad\quad\sum_{\lambda}\left[\mathcal{M}^{\rm
tree}_{c_{1}\cdots c_{i^{\prime}}\cdots c_{j}\cdots
c_{4}}T^{a}_{c_{i}c_{i^{\prime}}}\right]\left[\mathcal{M}^{\rm
tree}_{c_{1}\cdots c_{i}\cdots c_{j^{\prime}}\cdots
c_{4}}T^{a}_{c_{j}c_{j^{\prime}}}\right]^{*},$ $\displaystyle\mathcal{A}^{\rm
HC}_{\rm
real}=\frac{g_{s}^{2}}{4\pi^{2}}\frac{1}{\Gamma(1-\epsilon)}\left(\frac{4\pi\mu^{2}}{\mu_{f}^{2}}\right)\frac{A_{1}}{\epsilon}\sum_{\lambda}|\mathcal{M}^{\rm
tree}|^{2}\ .$ (49)

Here, $\mu_{f}$ is the initial state factorization scale, and for gluon-gluon
channel $A_{1}=-n_{lf}/3+C_{A}(11/6+\ln\delta_{s})$ , for quark-antiquark
channel $A_{1}=C_{F}(3/2+2\ln\delta_{s})$. Note, for comparison convenience,
the hard-collinear contribution $\mathcal{A}^{\rm HC}_{\rm real}$ here only
include singular or approach dependent (TAM or CAS) terms.

The mass of heavy quarks are taken as $m_{t}=173$, $m_{b}=4.9$, $m_{c}=1.5$,
and the strong coupling $g_{s}$ is set to one. The numerical results for
$g+g\to t+\bar{t}$ and $q+\bar{q}\to t+\bar{t}$ processes at the point
$s=500^{2}$, $r_{z}=0.6$, $\mu_{r}=\mu_{f}=500$ are given in Table 1 and Table
2, respectively. As a comparison, we also list the results obtained from the
CAS, where CDR scheme is employed. It can be seen that for each piece of NLO
corrections, the results from TAM and CAS are generally different. However,
after summing up all pieces, consistent results are obtained. The numerical
results confirm our statement in Sec. IV.2.

Table 1: Numerical results for $g+g\to t+\bar{t}$ process. |  TAM with HV |  CAS with CDR
---|---|---
$\mathcal{A}^{\rm loop}$ |  $\scriptstyle-10.08118054436082\epsilon^{-2}-3.437853927720160\epsilon^{-1}$ $\scriptstyle+75.75337372837709$ |  $\scriptstyle-10.08118054436082\epsilon^{-2}+17.83506953428062\epsilon^{-1}$ $\scriptstyle+69.74008746354726$
$\mathcal{A}^{\rm CT}$ |  $\scriptstyle-20.99049132259232\epsilon^{-1}-87.04718302407158$ |  $\scriptstyle-20.47530745414479\epsilon^{-1}-41.20686937309857$
$\mathcal{A}^{\rm soft}_{\rm real}$ |  $\scriptstyle 10.08118054436082\epsilon^{-2}+9.306574433771248\epsilon^{-1}$ $\scriptstyle-20.16236108872165\ln\delta_{s}\epsilon^{-1}-20.38707598960415$ $\scriptstyle+20.16236108872165\ln^{2}\delta_{s}-18.61314886754250\ln\delta_{s}$ |  $\scriptstyle 10.08118054436082\epsilon^{-2}-12.48153289667707\epsilon^{-1}$ $\scriptstyle-20.16236108872165\ln\delta_{s}\epsilon^{-1}-27.53194238007485$ $\scriptstyle+20.16236108872165\ln^{2}\delta_{s}+24.96306579335413\ln\delta_{s}$
$\mathcal{A}^{\rm HC}_{\rm real}$ |  $\scriptstyle 20.16236108872165\ln\delta_{s}\epsilon^{-1}+15.12177081654124\epsilon^{-1}$ $\scriptstyle-39.39339412989338\ln\delta_{s}+29.54504559742003$ |  $\scriptstyle 20.16236108872165\ln\delta_{s}\epsilon^{-1}+15.12177081654124\epsilon^{-1}$ $\scriptstyle-4.182820531003251\ln\delta_{s}-3.137115398252439$
sum |  $\scriptstyle 20.78024526235088\ln\delta_{s}+20.16236108872165\ln^{2}\delta_{s}$ $\scriptstyle-2.1358396878786$ |  $\scriptstyle 20.78024526235088\ln\delta_{s}+20.16236108872165\ln^{2}\delta_{s}$ $\scriptstyle-2.1358396878786$
Table 2: Numerical results for $q+\bar{q}\to t+\bar{t}$ process. |  TAM with HV |  CAS with CDR
---|---|---
$\mathcal{A}^{\rm loop}$ |  $\scriptstyle-0.3998042041986813\epsilon^{-2}+0.6577945202719476\epsilon^{-1}$ $\scriptstyle+6.474889692802957$ |  $\scriptstyle-0.3998042041986813\epsilon^{-2}+1.198174166364416\epsilon^{-1}$ $\scriptstyle+5.585807571271599$
$\mathcal{A}^{\rm CT}$ |  $\scriptstyle-1.649192342319560\epsilon^{-1}-5.294770961065026$ |  $\scriptstyle-1.649192342319560\epsilon^{-1}-3.065704920933595$
$\mathcal{A}^{\rm soft}_{\rm real}$ |  $\scriptstyle 0.399804204198681\epsilon^{-2}+0.3916915157495908\epsilon^{-1}$ $\scriptstyle-0.7996084083973625\ln\delta_{s}\epsilon^{-1}-1.043296995492568$ $\scriptstyle+0.7996084083973625\ln^{2}\delta_{s}-0.7833830314991816\ln\delta_{s}$ |  $\scriptstyle 0.399804204198681\epsilon^{-2}-0.1486881303428773\epsilon^{-1}$ $\scriptstyle-0.7996084083973625\ln\delta_{s}\epsilon^{-1}-1.572711444953939$ $\scriptstyle+0.7996084083973625\ln^{2}\delta_{s}+0.2973762606857546\ln\delta_{s}$
$\mathcal{A}^{\rm HC}_{\rm real}$ |  $\scriptstyle 0.7996084083973625\ln\delta_{s}\epsilon^{-1}+0.5997063062980219\epsilon^{-1}$ $\scriptstyle 1.562281770620308\ln\delta_{s}+1.171711327965231$ |  $\scriptstyle 0.7996084083973625\ln\delta_{s}\epsilon^{-1}+0.5997063062980219\epsilon^{-1}$ $\scriptstyle+0.4815224784353714\ln\delta_{s}+0.3611418588265285$
sum |  $\scriptstyle 0.7788987391211260\ln\delta_{s}+0.7996084083973625\ln^{2}\delta_{s}$ $\scriptstyle+1.30853306421059$ |  $\scriptstyle 0.7788987391211260\ln\delta_{s}+0.7996084083973625\ln^{2}\delta_{s}$ $\scriptstyle+1.30853306421059$

##  VI Summary

Helicity amplitude method is not only a technique to perform the perturbative
calculation of Feynman diagrams, it provides more information than
conventional amplitude squaring approach in phenomenological study. In this
paper, we reviewed the basic idea of TAM, and discussed how to generalize this
method to NLO QCD calculation. By analyzing the singularity structures of
virtual and real corrections, we proposed a scheme that can guarantee the
unitarity in the NLO QCD calculation. This scheme is legitimate under both CDR
and HV schemes, and is compatible with arbitrary self-consistent $\gamma_{5}$
prescription. We also provided an illustrative example by this scheme. Another
noteworthy aspect of this work is that, instead of relying on KLN theorem, we
shown the cancellations of soft and collinear singularities explicitly, at
least at NLO, by using the power counting technique.

Last, it should be mentioned that the IR divergence cancellation at next-to-
next-to-leading order (NNLO) is much more complicated than that at NLO. To
tackle this issue, the techniques developed in Refs. Grammer:1973db ;
Catani:1998bh ; Becher:2009kw ; Becher:2009qa ; Feige:2014wja  may be useful.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of
China(NSFC) under the Grants 11975236 ,11635009, and 12047553.

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## Appendix A: Soft singularities of external self-energy diagrams

In this appendix, we present the derivation of Eq. (24). As we work in
lightcone gauge, a proper prescription should be introduced to treat the
unphysical pole $(r\cdot k)^{-1}$. It has been shown that the usual principal-
value prescription is incompatible with Wick rotation, and will lead to a
violation of Ward identity Leibbrandt:1983pj . These difficulties can be
avoided by using the Mandelstam-Leibbrandt (ML) prescription Mandelstam:1982cb
; Leibbrandt:1983pj :

$\frac{1}{k\cdot r}\to\frac{k\cdot r^{*}}{(k\cdot r^{*})(k\cdot
r)+i\varepsilon}=\frac{1}{k\cdot r+i\varepsilon\ {\rm sign}(r^{*}\cdot k)}\ ,$
(50)

where $r^{*}=(r^{0},-\vec{r})$. Although our analysis here does not involve
any explicit computation of Feynman integral, we as well use the ML
prescription.

The self-energy correction to external massive quark takes the form

$\frac{1}{2}\vbox{\hbox{\includegraphics[scale={0.5}]{softSEQuark.eps}}}=\frac{1}{2}\bar{u}(p)[-i\Sigma_{c_{i}c_{i^{\prime}}}(p)]\frac{i(\not{p}+m)}{p^{2}-m^{2}}\Gamma_{c_{i^{\prime}}}(p)\bigg{|}_{p_{0}\to\omega_{p}},$
(51)

where $\omega_{p}=\sqrt{|\vec{p}|^{2}+m^{2}}$, and

$-i\Sigma_{c_{i}c_{i^{\prime}}}(p)=g_{s}^{2}C_{F}\delta_{c_{i}c_{i^{\prime}}}\mu^{4-D}\int\frac{d^{D}k}{(2\pi)^{D}}\frac{\gamma^{\mu}(\not{p}-\not{k}+m)\gamma^{\nu}}{k^{2}[(p-k)^{2}-m^{2}]}\Pi_{\mu\nu}(k).$
(52)

By performing the Passarino-Veltman tensor reduction, the
$-i\Sigma_{c_{i}c_{i^{\prime}}}(p)$ can be reduced to the form777The vector
$r^{*}$ is introduced by the ML prescription. Since our derivation does not
involve explicit integral computation, the final expression should be
prescription independent. As a check, we performed an analysis without any
$r^{*}$ involved, and consistent result was obtained.

$-i\Sigma_{c_{i}c_{i^{\prime}}}(p)=g_{s}^{2}C_{F}\delta_{c_{i}c_{i^{\prime}}}[f_{1}m+f_{2}(\not{p}-m)+f_{3}\not{r}+f_{4}\not{r}^{*}],$
(53)

where $f_{i}$ should be expand near $p_{0}=\omega_{p}$:

$f_{i}=f_{i}^{(0)}+f_{i}^{(1)}(p_{0}-\omega_{p})+\mathcal{O}\left((p_{0}-\omega_{p})^{2}\right),$
(54)

with

$f_{i}^{(n)}=\frac{d^{n}f_{i}}{dp_{0}^{n}}\bigg{|}_{p_{0}=\omega_{\vec{p}}}.$
(55)

By rescaling the loop momentum as $k\to\kappa^{2}k$, we can perform power
counting to pick the terms that are potentially soft divergent:

$\displaystyle f_{1}^{(0)}\overset{\rm soft}{\sim}$ $\displaystyle 0,\quad
f_{2}^{(0)}\overset{\rm soft}{\sim}0\ ,\quad f_{3}^{(0)}\overset{\rm
soft}{\sim}0\ ,\quad f_{4}^{(0)}\overset{\rm soft}{\sim}0,$ $\displaystyle
f_{1}^{(1)}\overset{\rm soft}{\sim}$ $\displaystyle\frac{2(p\cdot
r)}{m^{2}(r\cdot r^{*})-2(p\cdot r)(p\cdot r^{*})}[\omega_{p}(r\cdot
r^{*})I_{1}-(p\cdot r^{*})I_{2}-(p\cdot r)I_{3}]-2I_{2},$ $\displaystyle
f_{3}^{(1)}\overset{\rm soft}{\sim}$ $\displaystyle\frac{2(p\cdot
r)}{m^{2}(r\cdot r^{*})-2(p\cdot r)(p\cdot
r^{*})}\bigg{\\{}\bigg{[}\frac{\omega_{p}m^{2}(r\cdot r^{*})}{p\cdot
r}-3\omega_{p}(p\cdot r^{*})\bigg{]}I_{1}+\frac{(p\cdot r^{*})^{2}}{r\cdot
r^{*}}I_{2}$ $\displaystyle+\bigg{[}m^{2}-\frac{(p\cdot r)(p\cdot
r^{*})}{r\cdot r^{*}}\bigg{]}I_{3}\bigg{\\}},$ $\displaystyle
f_{4}^{(1)}\overset{\rm soft}{\sim}$ $\displaystyle\frac{2(p\cdot
r)}{m^{2}(r\cdot r^{*})-2(p\cdot r)(p\cdot r^{*})}\bigg{\\{}-\omega_{p}(p\cdot
r)I_{1}+\bigg{[}m^{2}-\frac{(p\cdot r)(p\cdot r^{*})}{r\cdot
r^{*}}\bigg{]}I_{2}$ $\displaystyle+\frac{(p\cdot r)^{2}}{r\cdot
r^{*}}I_{3}\bigg{\\}},$ (56)

where

$\displaystyle
I_{1}=\int\frac{d^{D}k}{(2\pi)^{D}}\frac{1}{k^{2}[(p-k)^{2}-m^{2}](k\cdot
r)}\bigg{|}_{p_{0}=\omega_{p}}\overset{\rm
soft}{\sim}-\frac{1}{2}\int\frac{d^{D}k}{(2\pi)^{D}}\frac{1}{k^{2}(k\cdot
p)(k\cdot r)}\ ,$ $\displaystyle
I_{2}=\frac{d}{dp_{0}}\int\frac{d^{D}k}{(2\pi)^{D}}\frac{1}{k^{2}[(p-k)^{2}-m^{2}]}\bigg{|}_{p_{0}=\omega_{p}}\overset{\rm
soft}{\sim}-\frac{\omega_{p}}{2}\int\frac{d^{D}k}{(2\pi)^{D}}\frac{1}{k^{2}(k\cdot
p)^{2}}\ ,$ $\displaystyle
I_{3}=\frac{d}{dp_{0}}\int\frac{d^{D}k}{(2\pi)^{D}}\frac{k\cdot
r^{*}}{k^{2}[(p-k)^{2}-m^{2}](k\cdot
r)}\bigg{|}_{p_{0}=\omega_{p}}\overset{\rm
soft}{\sim}-\frac{\omega_{p}}{2}\int\frac{d^{D}k}{(2\pi)^{D}}\frac{k\cdot
r^{*}}{k^{2}(k\cdot p)^{2}(k\cdot r)}\ .$ (57)

Note, to regularize the soft divergence by dimensional regularization, the
derivative and integral operation should be performed in an order:

$\frac{d}{dp_{0}}\int\frac{d^{D}k}{(2\pi)^{D}}\frac{\cdots}{\cdots}\bigg{|}_{p_{0}=\omega_{p}}\to\int\frac{d^{D}k}{(2\pi)^{D}}\left(\frac{d}{dp_{0}}\frac{\cdots}{\cdots}\bigg{|}_{p_{0}=\omega_{p}}\right).$
(58)

Finally, we obtain the desired soft singular term:

$\displaystyle\frac{1}{2}\bar{u}(p)[-i\Sigma_{c_{i}c_{i^{\prime}}}(p)]\frac{i(\not{p}+m)}{p^{2}-m^{2}}\Gamma_{c_{i^{\prime}}}(p)\bigg{|}_{p_{0}\to\omega_{p}}$
$\displaystyle\overset{\rm soft}{\sim}$
$\displaystyle\frac{i}{2}g_{s}^{2}C_{F}\frac{m^{2}f_{1}^{(1)}+(p\cdot
r)f_{3}^{(1)}+(p\cdot
r^{*})f_{4}^{(1)}}{\omega_{p}}\bar{u}(p)\Gamma_{c_{i}}(p)$
$\displaystyle\overset{\rm soft}{\sim}$
$\displaystyle-\frac{i}{2}g_{s}^{2}C_{F}\mu^{4-D}\int\frac{d^{D}k}{(2\pi)^{D}}\frac{p\cdot\Pi(k)\cdot
p}{k^{2}(k\cdot p)^{2}}\bar{u}(p)\Gamma_{c_{i}}(p).$ (59)

The result for massless quark can be obtained by simply set $m=0$, as we have
checked that the soft-collinear region will not produce new terms.

The self-energy correction to external gluon is of the form

$\frac{1}{2}\vbox{\hbox{\includegraphics[scale={0.5}]{softSEGluon.eps}}}=\frac{1}{4}\epsilon^{*\mu}(p)[-i\Sigma^{c_{i}c_{i^{\prime}}}_{\mu\nu}(p)]\frac{i\Pi^{\nu\nu^{\prime}}(p)}{p^{2}}\Gamma^{c_{i^{\prime}}}_{\nu^{\prime}}(p)\bigg{|}_{p_{0}\to|\vec{p}|},$
(60)

where

$\displaystyle-i\Sigma_{\mu\nu}^{c_{i}c_{i^{\prime}}}(p)=-C_{A}\delta^{c_{i}c_{i^{\prime}}}g_{s}^{2}\mu^{4-D}\int\frac{d^{D}k}{(2\pi)^{D}}$
$\displaystyle\frac{i\Pi^{\alpha\rho}(k)i\Pi^{\beta\sigma}(p-k)}{k^{2}(p-k)^{2}}$
$\displaystyle[g_{\rho\sigma}(2k-p)_{\mu}+g_{\sigma\mu}(2p-k)_{\rho}+g_{\mu\rho}(-p-k)_{\sigma}]$
$\displaystyle[g_{\nu\beta}(2p-k)_{\alpha}+g_{\beta\alpha}(2k-p)_{\nu}+g_{\alpha\nu}(-k-p)_{\beta}]\
.$ (61)

By performing power counting, we obtain the terms that are potentially soft or
soft-collinear divergent

$\displaystyle\frac{1}{4}\epsilon^{*\mu}(p)[-i\Sigma^{c_{i}c_{i^{\prime}}}_{\mu\nu}(p)]\frac{i\Pi^{\nu\nu^{\prime}}(p)}{p^{2}}\Gamma^{c_{i^{\prime}}}_{\nu^{\prime}}(p)\bigg{|}_{p_{0}\to|\vec{p}|}$
$\displaystyle\overset{\rm soft}{\sim}$
$\displaystyle-\frac{i}{2}g_{s}^{2}C_{A}\mu^{4-D}\int\frac{d^{D}k}{(2\pi)^{D}}\frac{p\cdot\Pi(k)\cdot
p}{k^{2}(p\cdot k)^{2}}\Gamma^{c_{i}}(p)\cdot\epsilon^{*}(p).$ (62)

Note, a factor of $2$ is included to incorporate the contribution from
$(k-p)\to 0$ region.

## Appendix B: Hard-collinear singularities of external self-energy diagrams

In this appendix, we present the derivation of Eq. (30), the hard-collinear
singular terms of external self-energy diagrams. We decompose the loop
momentum as

$k^{\mu}=zp^{\mu}+k^{-}\bar{p}^{\mu}+k_{T}^{\mu},$ (63)

where $p$ is the external momentum and $\bar{p}=(p_{0},-\vec{p})$. Then the
propagator and the integral measures take the forms

$\displaystyle k^{2}=4p_{0}^{2}zk^{-}-\vec{k}_{T}^{2},$
$\displaystyle(p-k)^{2}=-4p_{0}^{2}(1-z)k^{-}-\vec{k}_{T}^{2},$ $\displaystyle
d^{D}k=2p_{0}^{2}\ dz\ dk^{-}\ d^{D-2}k_{T}.$ (64)

For the quantity $(r\cdot k)^{-1}$, we use the ML prescription as in Appendix
A. To reduce the number of independent vectors, we set $r$ to be parallel with
$\bar{p}$ (then $r^{*}$ is parallel with $p$). In fact, these treatments is
not much concern to the final result, since $(r\cdot k)^{-1}$ is nonvanishing
in hard-collinear region (except when choosing $r\varpropto p$).

The self-energy correction to external massless quark is of the form

$\frac{1}{2}\vbox{\hbox{\includegraphics[scale={0.5}]{softSEQuark.eps}}}=\frac{1}{2}\bar{u}(p)[-i\Sigma_{c_{i}c_{i^{\prime}}}(p)]\frac{i\not{p}}{p^{2}}\Gamma_{c_{i^{\prime}}}(p)\bigg{|}_{p_{0}\to|\vec{p}|},$
(65)

with

$-i\Sigma_{c_{i}c_{i^{\prime}}}(p)=g_{s}^{2}C_{F}\delta_{c_{i}c_{i^{\prime}}}\mu^{4-D}\int\frac{d^{D}k}{(2\pi)^{D}}\frac{\gamma^{\mu}(\not{p}-\not{k})\gamma^{\nu}}{k^{2}(p-k)^{2}}\Pi_{\mu\nu}(k).$
(66)

In hard-collinear region, the scaling behaviors of lightcone components are
$z\sim\kappa^{0}$, $k^{-}\sim\kappa^{2}$, $|\vec{k}_{T}|\sim\kappa$. By taking
the terms which scale like $\mathcal{O}(\kappa^{0})$, we have

$\displaystyle\frac{1}{2}\bar{u}(p)[-i\Sigma_{c_{i}c_{i^{\prime}}}(p)]\frac{i\not{p}}{p^{2}}\Gamma_{c_{i^{\prime}}}(p)\bigg{|}_{p_{0}\to|\vec{p}|}\overset{\rm
coll}{\sim}ig_{s}^{2}C_{F}\mu^{4-D}\int\frac{2p_{0}^{2}dzd^{D-2}k_{T}}{(2\pi)^{D-1}}\frac{(D-10)z+8}{4z}$
$\displaystyle\times\int\frac{dk^{-}}{2\pi}\frac{1}{[4p_{0}^{2}zk^{-}-\vec{k}_{T}^{2}+i\varepsilon][-4p_{0}^{2}(1-z)k^{-}-\vec{k}_{T}^{2}+i\varepsilon]}\bar{u}(p)\Gamma_{c_{i}}(p).$
(67)

It can be seen that there are two poles located at
$k^{-}=\frac{\vec{k}_{T}^{2}-i\varepsilon}{4p_{0}^{2}z}$ and
$k^{-}=\frac{\vec{k}_{T}^{2}-i\varepsilon}{-4p_{0}^{2}(1-z)}$. In the region
$z<0$ or $z>1$, both poles lie in the same half-plane, and no singular term
will arise. In the region $0<z<1$, we have

$\displaystyle\overset{\rm
coll}{\sim}-\frac{g_{s}^{2}}{4\pi}C_{F}\mu^{4-D}\int_{\frac{\delta_{0}}{p_{0}}}^{1}dz\frac{(D-10)z+8}{4z}\int_{|\vec{k}_{T}|<\delta_{T}}\frac{d^{D-2}k_{T}}{(2\pi)^{D-2}}\frac{1}{\vec{k}_{T}^{2}}\
\bar{u}(p)\Gamma_{c_{i}}(p)$ $\displaystyle\overset{\rm
coll}{\sim}-\frac{g_{s}^{2}}{8\pi^{2}}C_{F}\frac{1}{\Gamma(1-\epsilon)}\left(\frac{4\pi\mu^{2}}{\delta_{T}^{2}}\right)^{\epsilon}\frac{1}{2\epsilon}\left(\frac{3+\epsilon}{2}+2\ln\frac{\delta_{0}}{p_{0}}\right)\
\bar{u}(p)\Gamma_{c_{i}}(p).$ (68)

Here, the cutoff parameter $\delta_{0}$ is introduced to exclude the soft-
collinear region.

The fermion loop self-energy correction to external gluon takes the form:

$\frac{n_{lf}}{2}\vbox{\hbox{\includegraphics[scale={0.5}]{softSEGluonFermionLoop.eps}}}=\frac{n_{lf}}{2}\epsilon^{*\mu}(p)[-i\Sigma^{c_{i}c_{i^{\prime}}}_{\mu\nu}(p)]\frac{i\Pi^{\nu\nu^{\prime}}(p)}{p^{2}}\Gamma^{c_{i^{\prime}}}_{\nu^{\prime}}(p)\bigg{|}_{p_{0}\to|\vec{p}|}\
,$ (69)

where

$\displaystyle-i\Sigma_{\mu\nu}^{c_{i}c_{i^{\prime}}}(p)=-\frac{g_{s}^{2}}{2}\delta^{c_{i}c_{i^{\prime}}}\mu^{4-D}\frac{d^{D}k}{(2\pi)^{D}}\frac{{\rm
Tr}[\gamma^{\nu}\cdot(\not{k}-\not{p})\cdot\gamma^{\mu}\cdot\not{k}]}{k^{2}(p-k)^{2}}.$
(70)

Similarly, we have

$\displaystyle\frac{n_{lf}}{2}\epsilon^{*\mu}(p)[-i\Sigma^{c_{i}c_{i^{\prime}}}_{\mu\nu}(p)]\frac{i\Pi^{\nu\nu^{\prime}}(p)}{p^{2}}\Gamma^{c_{i}}_{\nu^{\prime}}(p)\bigg{|}_{p_{0}\to|\vec{p}|}$
$\displaystyle\overset{\rm coll}{\sim}$ $\displaystyle
i\frac{n_{lf}}{2}g_{s}^{2}\frac{D-2}{D-1}\mu^{4-D}\int\frac{d^{D}k}{(2\pi)^{D}}\frac{1}{k^{2}(p-k)^{2}}\
\Gamma_{b}(p)\cdot\epsilon^{*}(p)$ $\displaystyle\overset{\rm coll}{\sim}$
$\displaystyle-
n_{lf}\frac{g_{s}^{2}}{8\pi}\frac{D-2}{D-1}\mu^{4-D}\int_{|\vec{k}_{T}|<\delta_{T}}\frac{d^{D-2}k_{T}}{(2\pi)^{D-2}}\frac{1}{\vec{k}_{T}^{2}}\
\Gamma_{c_{i}}(p)\cdot\epsilon^{*}(p)$ $\displaystyle\overset{\rm coll}{\sim}$
$\displaystyle
n_{lf}\frac{g_{s}^{2}}{8\pi^{2}}\frac{1}{\Gamma(1-\epsilon)}\left(\frac{4\pi\mu^{2}}{\delta_{T}^{2}}\right)^{\epsilon}\frac{1}{2\epsilon}\frac{1-\epsilon}{3-2\epsilon}\
\Gamma_{c_{i}}(p)\cdot\epsilon^{*}(p).$ (71)

The fermion loop self-energy correction to external gluon takes the form:

$\frac{1}{2}\vbox{\hbox{\includegraphics[scale={0.5}]{softSEGluon.eps}}}=\frac{1}{4}\epsilon^{*\mu}(p)[-i\Sigma^{c_{i}c_{i^{\prime}}}_{\mu\nu}(p)]\frac{i\Pi^{\nu\nu^{\prime}}(p)}{p^{2}}\Gamma^{c_{i^{\prime}}}_{\nu^{\prime}}(p)\bigg{|}_{p_{0}\to|\vec{p}|},$
(72)

where

$\displaystyle-i\Sigma_{\mu\nu}^{c_{i}c_{i^{\prime}}}(p)=-C_{A}\delta^{c_{i}c_{i^{\prime}}}g_{s}^{2}\mu^{4-D}\int\frac{d^{D}k}{(2\pi)^{D}}$
$\displaystyle\frac{i\Pi^{\alpha\rho}(k)i\Pi^{\beta\sigma}(p-k)}{k^{2}(p-k)^{2}}$
$\displaystyle[g_{\rho\sigma}(2k-p)_{\mu}+g_{\sigma\mu}(2p-k)_{\rho}+g_{\mu\rho}(-p-k)_{\sigma}]$
$\displaystyle[g_{\nu\beta}(2p-k)_{\alpha}+g_{\beta\alpha}(2k-p)_{\nu}+g_{\alpha\nu}(-k-p)_{\beta}].$
(73)

We have

$\displaystyle\frac{1}{4}\epsilon^{*\mu}(p)[-i\Sigma^{c_{i}c_{i^{\prime}}}_{\mu\nu}(p)]\frac{i\Pi^{\nu\nu^{\prime}}(p)}{p^{2}}\Gamma^{c_{i^{\prime}}}_{\nu^{\prime}}(p)\bigg{|}_{p_{0}\to|\vec{p}|}$
$\displaystyle\overset{\rm coll}{\sim}$ $\displaystyle
ig_{s}^{2}C_{A}\mu^{4-D}\int\frac{d^{D}k}{(2\pi)^{D}}\frac{(z^{2}-z+1)^{2}}{z(1-z)}\frac{1}{k^{2}(p-k)^{2}}\
\Gamma_{c_{i}}(p)\cdot\epsilon^{*}(p)$ $\displaystyle\overset{\rm coll}{\sim}$
$\displaystyle-\frac{g_{s}^{2}}{4\pi}C_{A}\mu^{4-D}\int_{\frac{\delta_{0}}{p_{0}}}^{1-\frac{\delta_{0}}{p_{0}}}dz\frac{(z^{2}-z+1)^{2}}{z(1-z)}\int_{|\vec{k}_{T}|<\delta_{T}}\frac{d^{D-2}k_{T}}{(2\pi)^{D-2}}\frac{1}{\vec{k}_{T}^{2}}\
\Gamma_{c_{i}}(p)\cdot\epsilon^{*}(p)$ $\displaystyle\overset{\rm coll}{\sim}$
$\displaystyle-\frac{g_{s}^{2}}{8\pi^{2}}C_{A}\frac{1}{\Gamma(1-\epsilon)}\left(\frac{4\pi\mu^{2}}{\delta_{T}^{2}}\right)^{\epsilon}\frac{1}{2\epsilon}\left(\frac{11}{6}+2\ln\frac{\delta_{0}}{p_{0}}\right)\
\Gamma_{c_{i}}(p)\cdot\epsilon^{*}(p).$ (74)