#  Forward-backward asymmetries of the heavy quark pair production in
$e^{+}e^{-}$ collisions at $\mathcal{O}(\alpha_{s}^{2})$

Long Chen111E-mail<EMAIL_ADDRESS>
Institut für Theoretische Teilchenphysik und Kosmologie,
RWTH Aachen University, 52056 Aachen, Germany

###### Abstract

The talk is on a computational set-up for calculating the production of a
massive quark-antiquark pair in electron positron collisions to order
$\alpha_{s}^{2}$ in the coupling of quantum chromodynamics (QCD) at the
differential level using the antenna subtraction method. Theoretical
predictions on the production of top quark pairs in the continuum, and the
bottom quark pairs at the $Z$ resonance, will be discussed. In particular, we
would be focusing on the order $\alpha_{s}^{2}$ QCD corrections to the heavy
quark forward-backward asymmetry (AFB) in electron positron collisions. In the
case of the AFB of bottom quarks at the $Z$ resonance, the QCD corrections are
determined with respect to both the bottom quark axis and the thrust axis. We
will also briefly discuss improvements on these QCD corrections brought by
applying the optimization procedure based on the Principle of Maximum
Conformality.

Talk presented at the International Workshop on Future Linear Colliders
(LCWS2021)

15-18 March 2021. C21-03-15.1

##  1 Introduction

The exploration of the physics of heavy quarks, that is, bottom and top
quarks, is among the core physics issues at both the past and the future
linear or circular electron-positron colliders [1, 2, 3, 4, 5], such as the
International Linear Collider (ILC) [4], the Compact Linear Collider (CLIC)
[6], the Future Circular electron-positron Collider (FCC-ee) [7] and the
Circular Electron Positron Collider (CEPC) [8, 9]. Among the set precision
observables associated with the heavy quark production at these lepton
colliders include the forward-backward asymmetries, the key observables for
the determination of the neutral current couplings of leptons and quarks in
the reactions $e^{+}e^{-}\to Q{\bar{Q}}$. As far as quarks are concerned, the
most precisely known asymmetry is that of the $b$ quark at the $Z$ resonance
$A_{\rm FB}^{b}$, which was measured at SLAC and LEP with an accuracy of 1.7
percent [1, 2]. The measured $A_{\rm FB}^{b}$ at the $Z$ resonance shows a
relatively large deviation, about 2.9 $\sigma$, from the respective Standard
Model (SM) fit 222It is noted that the deviation of this experimental
measurement from the SM fit result given in a later ref. [10] reduces to 2.5
$\sigma$, and to 2.4 $\sigma$ compared to the updated SM prediction in ref.
[11].. So far, it has not been clarified whether this deviation is due to
underestimated experimental and/or theoretical uncertainties or whether it is
a hint of new physics.

At a future linear or circular $e^{+}e^{-}$ collider, precision determinations
of electroweak parameters will again involve forward-backward asymmetries. If
such a collider will be operated at the $Z$ resonance, an accuracy of about
0.1 percent may be reached for these observables [12, 13, 14]. At the
International Linear Collider (ILC), the top-quark forward-backward asymmetry
can be measured to a precision of below one percent in relative [15, 16, 14].
Needless to say, precise predictions are required, too, on the theoretical
side.

As far as the production of $t{\bar{t}}$, or more general, the production of a
heavy quark-antiquark pair $(Q{\bar{Q}})$ in $e^{+}e^{-}$ collisions in the
continuum is concerned, differential predictions at next-to-leading order
(NLO) QCD (corresponding to $\mathcal{O}(\alpha_{s})$ for this process) have
been known for a long time for $Q{\bar{Q}}$ [17] and $Q{\bar{Q}}$ \+ jet [18,
19, 20, 21, 22, 23] final states. Also the NLO electroweak corrections are
known [24, 25, 26, 27, 28, 29, 30]. Recently, the QED initial state
corrections to the forward-backward asymmetry for
$e^{+}e^{-}\to\gamma^{*}/Z^{*}$ are calculated in the leading logarithmic
approximation to high orders in ref. [31]. The total $Q{\bar{Q}}$ cross
section $\sigma_{Q{\bar{Q}}}$ was computed to order $\alpha_{s}^{2}$ and order
$\alpha_{s}^{3}$ in [32, 33, 34, 35] and [36], respectively, using
approximations as far as the dependence of $\sigma_{Q{\bar{Q}}}$ on the mass
of $Q$ is concerned. A computation of the cross section and of differential
distributions for $t{\bar{t}}$ production at order $\alpha_{s}^{2}$ with full
top-mass dependence was reported in [37, 38] using a hybrid approach combining
phase-space slicing and the dipole subtraction, and also in [39] using the
antenna subtraction method [40, 41]. A large effort has been made to
investigate $t{\bar{t}}$ production at threshold, presently known at next-to-
next-to-next-to-leading order QCD [42]. For $b$ quarks, the next-to-next-to-
leading order (NNLO) corrections (corresponding to
$\mathcal{O}(\alpha^{2}_{s})$ for this process) were calculated previously in
the limit of vanishing $b$-quark mass [43, 44, 45], and later with full $b$
quark mass dependence accounted for in ref. [46].

The talk is on the QCD corrections to the production of top and bottom quarks
at $e^{+}e^{-}$ colliders in the continuum, based on the work reported in
refs.[39, 46, 47]. Namely, we analyze heavy $Q{\bar{Q}}$ production in
$e^{+}e^{-}$ collisions,

$e^{+}e^{-}\to\gamma^{*},Z^{*}\to Q~{}{\bar{Q}}+X\,,$ (1.1)

at the differential level to $\alpha_{s}^{2}$ in pertrubative QCD and apply it
to $t{\bar{t}}$ production above the pair-production threshold and to
$b{\bar{b}}$ production at the $Z$ resonance.

##  2 Ingredients for computing $e^{+}e^{-}\rightarrow Q\bar{Q}~{}+~{}X$ at
${\cal O}(\alpha_{s}^{2})$

### Matrix elements to ${\cal O}(\alpha_{s}^{2})$

To order $\alpha_{s}^{2}$, the (differential) cross section of the reaction
(1.1) receives contributions from

  * i)

the two-parton $Q\bar{Q}$ state (at Born level, to order $\alpha_{s}$, and to
order $\alpha_{s}^{2}$),

  * ii)

the three-parton state $Q{\bar{Q}}g$ (to order $\alpha_{s}$ and to order
$\alpha_{s}^{2}$),

  * iii)

and the four-parton states $Q{\bar{Q}}gg$, $Q{\bar{Q}}q\bar{q}$, and above the
$4Q$-threshold from $Q{\bar{Q}}Q{\bar{Q}}$ (to order $\alpha_{s}^{2}$).

Contributions from the $Q{\bar{Q}}Q{\bar{Q}}$ final state should be treated
with care, in view of the fact that for these states the particle-multiplicity
$n_{Q}\neq 1$ if one considers so-called inclusive heavy-quark distributions
$d\sigma(e^{+}e^{-}\to Q+X)/dO_{Q}$, where $O_{Q}$ is some observable
associated with $Q$.

We start with the two-parton $Q\bar{Q}$ contributions. Fig. 1 shows
representative Feynman diagrams for this two-parton final state up to
$\alpha_{s}^{2}$.

Figure 1: Examples of diagrams that contribute to the $Q{\bar{Q}}$ final state
to order $\alpha_{s}^{2}$. The dashed line represents the electroweak neutral
current, that is, the virtual photon or $Z$ boson, the thick line the massive
$Q$ quark, and the thin line any of the six quarks. The triangle diagrams (d)
are summed over the six quark flavors.

According to the classification used in [48, 49, 50], the two-loop diagram
fig. 1c belongs to the type-A two-loop contributions where the external
current couples to to $Q{\bar{Q}}$. This diagram and the one-loop diagram fig.
1c are also examples of so-called universal QCD corrections because the same
electroweak couplings as in the lowest order diagram fig. 1a are involved.
Fig. 1d is an example of the so-called type-B two-loop contributions where a
fermion triangle loop is involved. Since in fig. 1d it is not necessarily the
heavy quark $Q$ that is coupled to the electroweak current, it is an example
for the so-called non-universal QCD corrections to the lowest order
$Q{\bar{Q}}$ production amplitude.

Examples of diagrams associated with the $Q{\bar{Q}}g$ final state that lead
to contributions of order $\alpha_{s}$ and $\alpha_{s}^{2}$ to the
differential cross section of (1.1) are displayed in fig. 2.

Figure 2: Examples of diagrams that contribute to the $Q{\bar{Q}}g$ final
state to order $\alpha_{s}^{2}$. The assignment of the lines is as in fig. 1.

The diagrams fig. 2a,b and fig. 2c belong to the universal and non-universal
QCD corrections, respectively. The four-parton final-state diagrams of fig. 3
contribute at order $\alpha_{s}^{2}$ to the differential cross section. An
example of a diagram corresponding to the final state $Q{\bar{Q}}gg$ is shown
in fig. 3a, while two of the four diagrams associated with
$Q{\bar{Q}}q{\bar{q}}$ $(q\neq Q)$ are exhibited in fig. 3b,c. The square of
the $Q{\bar{Q}}gg$ diagrams and the square of fig. 3a lead to universal QCD
corrections while the square of fig. 3c and the interference of figs. 3b and
3c belong to the non-universal QCD corrections. The calculations of these
tree-level and one-loop Feynman diagrams are nowadays rather straightforward.

### The antenna subtraction terms

The antenna subtraction method is a systematic (process-independent) procedure
for the construction of infrared subtraction terms that was developed for NLO
QCD calculations in refs. [51, 40] and has been generalized to NNLO QCD in
ref. [41]. The antenna subtraction method is based on the use of color-ordered
amplitudes into which an $n$-point parton amplitude at $l$ loops,
$M_{n}^{(l)}$, can be decomposed[52, 53, 54]:
$M_{n}^{(l)}=\sum_{i}C_{i,n}^{(l)}{\cal M}^{(l)}_{i}$ where $i$ runs over the
$(n-1)!$ non-cyclic permutations of the external parton spin and momentum
labels. This means that the ordering of the external partons is fixed in the
color-stripped partial amplitudes ${\cal M}^{(l)}_{i}$. The coefficients
$C_{i,n}^{(l)}$ contain the color information. The development of the antenna
subtraction method relies on two important features. First, the construction
of subtraction terms for the color-ordered partial amplitudes is considerably
simpler than for the full amplitudes. It relies on the IR factorization
properties of the color-ordered partial amplitudes [52, 53, 55, 56, 57, 51,
58, 59]). The determination of the integrated subtraction terms in analytic
fashion in $D\neq 4$ dimensions takes advantage of the modern multi-loop
integration techniques. It has been accomplished at NNLO QCD for the massless
case [60, 41], but also for a number of cases involving massive quarks [61,
62, 63, 64, 65, 66, 67]. For the computation of the reaction (1.1) at the
differential level, we use the unintegrated and integrated NNLO real radiation
antenna subtraction terms and the NNLO real-virtual antenna functions worked
out in [68, 69, 70]. For a synopsis of these objects needed, we refer to ref.
[39].

Figure 3: (a): Examples of diagrams that contribute to the $Q{\bar{Q}}gg$
final state at order $\alpha_{s}^{2}$. (b,c): Two of the four diagrams that
contribute to the $Q{\bar{Q}}q{\bar{q}}$ $(q\neq Q)$ final state at order
$\alpha_{s}^{2}$.

##  3 Application to the top quark pair prodution in the continuum

We now apply our computational set-up to the study of the top quark pair
production for unpolarized $e^{+}$ and $e^{-}$ beams and $e^{+}e^{-}$ center-
of-mass (c.m.) energies $\sqrt{s}$ sufficiently away from the $t{\bar{t}}$
threshold, where the fixed order perturbation theory in $\alpha_{s}$ is
applicable. Fig. 4

Figure 4: The total $t{\bar{t}}$ cross section at LO, NLO, and NNLO QCD as a
function of the c.m. energy. The solid lines that refer to $\sigma_{\rm
NLO}^{t{\bar{t}}}$ and $\sigma_{\rm NNLO}^{t{\bar{t}}}$ correspond to the
choice $\mu=\sqrt{s}$, the dashed lines correspond to $\mu=\sqrt{s}/2$ and
$2\sqrt{s}$.

shows our result for the $e^{+}e^{-}\rightarrow t{\bar{t}}$ cross section at
LO, NLO, and NNLO QCD for c.m. energies $\sqrt{s}\lesssim 4m_{t}$ (with input
values of the SM parameters specified in ref. [39]). In the case of
$\sigma_{\rm NLO}^{t{\bar{t}}}$ and $\sigma_{\rm NNLO}^{t{\bar{t}}}$ the solid
lines represent the values computed with the choice $\mu=\sqrt{s}$ for the
renormalization scale. Uncertainties due to undetermined higher-order
corrections are conventionally estimated by varying the renormalization-scale
$\mu$ between $\sqrt{s}/2$ and $2\sqrt{s}$. The upper and lower dashed lines
correspond to these scale variations. In order to exhibit the size of the
higher order QCD corrections relative to the LO cross section, we represent
the $t{\bar{t}}$ cross section in the form

$\sigma_{\rm NNLO}^{t{\bar{t}}}=\sigma_{\rm
LO}^{t{\bar{t}}}\,(1+\Delta_{1}+\Delta_{2})\,.$ (3.1)

We list, for a reference, in table 1

Table 1:  The QCD corrections $\Delta_{1}$ and $\Delta_{2}$ to the LO ${t{\bar{t}}}$ cross section defined in Eq. (3.1) for several c.m. energies and $\mu=\sqrt{s}$. $\sqrt{s}$ [GeV] | 360 | 381.3 | 400 | 500
---|---|---|---|---
$\Delta_{1}$ | 0.627 | 0.352 | 0.266 | 0.127
$\Delta_{2}$ | 0.281 | 0.110 | 0.070 | 0.020

the QCD corrections $\Delta_{1}$ and $\Delta_{2}$ for selected c.m. energies
$\sqrt{s}$ for $\mu=\sqrt{s}$.

From figures 4 we can see that the higher order QCD corrections to the
$t{\bar{t}}$ cross sections decrease with increasing c.m. energies $\sqrt{s}$
after $\sim 380$ GeV. As clearly shown, in the region where the perturbative
calculation is reliable, the scale-variation band at NNLO is narrower than at
NLO, as expected in perturbative calculations. Notice also that the scale-
variation bands do not overlap here. In the near threshold region, the QCD
corrections become rather large. The blue thick line in fig. 4 representing
the total NNLO result already shows a hint of blowing up when $\sqrt{s}\to
2m_{t}.$ This is due to the well-known Coulomb singularities [71, 33] in this
process. The NLO cross section, i.e., the red line in fig. 4, does not diverge
but goes to a constant for $\beta\rightarrow 0$, because it involves only a
single ${1}/{\beta}$ Coulomb singularity, due to one virtual longitudinal
gluon exchange, which is compensated by the suppression factor $\beta$ due to
the shrinking phase-space. The presence of Coulomb singularities in fixed
order perturbative calculations near threshold clearly signifies the break-
down of such a calculation. Within perturbation theory, one way to improve in
the near threshold region is to sum up these Coulomb singularities through all
perturbative orders, which then leads to a finite result (see, for instance,
[71, 72, 42]).

The numbers in table 1 suggest that fixed order perturbation theory can be
applied for $\sqrt{s}>360$ GeV. In the computation of $\sigma_{\rm
NNLO}^{t{\bar{t}}}$, we have included also the non-universal contributions of
order $\mathcal{O}(\alpha_{s}^{2})$ that contain the electroweak couplings of
quarks $q\neq t$. These contributions are free of divergences and are very
small. For instance, at $\sqrt{s}$ = 500 GeV they amount to $-0.16\%$ of the
total second order correction $\Delta_{2}$ defined in Eq. (3.1), and this
fraction decreases further in magnitude for smaller c.m. energies.

We have numerically compared our exact NNLO result with the expanded results
in the near threshold region [73, 74, 75, 76] and in the asymptotic region
[32, 33, 34, 35] where $m_{t}^{2}/s\ll 1$, and found agreement. Furthermore,
our results shown in table 1 agree with the calculation of the $t{\bar{t}}$
cross section in ref.[38]. For a detailed discussion of our results on
differential distributions of the top quark pair prodution we refer to ref.
[39].

##  4 The forward-backward asymmetries in $e^{+}e^{-}\to Q{\bar{Q}}$

###  4.1 Definitions of the un-expanded and expanded $A_{\rm FB}$

The forward-backward asymmetry $A_{\rm FB}$ in the heavy $Q{\bar{Q}}$
production, is defined by

$\displaystyle A_{\rm FB}\equiv\frac{N_{F}-N_{B}}{N_{F}+N_{B}}\,,$ (4.1)

where $N_{F}$ $(N_{B})$ is the number of quarks $Q$ produced in the forward
(backward) direction. The asymmetry $A_{\rm FB}$ in the reaction
$e^{+}e^{-}\to Q{\bar{Q}}+X$ is generated by those terms in the squared
S-matrix elements which are odd under the interchange of $Q$ and $\bar{Q}$
while the initial state is kept fixed. The identification of the
forward/backward direction involves a choice of reference axis. The definition
of the reference axis must be such that the resulting forward-backward
asymmetry is an infrared safe (IR-safe) quantity so that it can be reliably
calculated and subsequently compared with experimental measurements. Common
choices include the direction of flight of the heavy quark and the oriented
thrust axis defined by a certain thrust-finding algorithm [77, 78].

The heavy quark forward-backward asymmetry $A_{\rm FB}$ can also be expressed
in terms of the symmetric cross section $\sigma_{S}$ and the antisymmetric
cross section $\sigma_{A}$ for the inclusive production of the heavy quark
$Q$, i.e.,

$\displaystyle A_{\rm
FB}=\frac{\sigma_{A}}{\sigma_{S}}=\frac{\sigma_{F}-\sigma_{B}}{\sigma_{F}+\sigma_{B}}.$
(4.2)

The $\sigma_{F}$ and $\sigma_{B}$ are the forward and backward cross sections,
respectively, which can be written in terms of differential cross sections as

$\displaystyle\sigma_{F}=\int_{0}^{1}d\cos\theta\int_{x_{0}}^{1}dx\frac{d\sigma}{dx\,d\cos\theta}\,,\,\,\sigma_{B}=\int_{-1}^{0}d\cos\theta\int_{x_{0}}^{1}dx\frac{d\sigma}{dx\,d\cos\theta},$
(4.3)

where $\theta$ is the angle between the electron three-momentum and the axis
defining the forward hemisphere. The energy ratio $x$ is defined as
$2\,E_{Q}/\sqrt{s}$ where $E_{Q}$ is the energy of the heavy quark $Q$.

To order $\alpha_{s}^{2}$ the symmetric and antisymmetric cross sections
receive the following perturbative contributions:

$\displaystyle\sigma_{A,S}=\sigma_{A,S}^{(2,0)}+\sigma_{A,S}^{(2,1)}+\sigma_{A,S}^{(3,1)}+\sigma_{A,S}^{(2,2)}+\sigma_{A,S}^{(3,2)}+\sigma_{A,S}^{(4,2)}+{\cal
O}(\alpha_{s}^{3})\,,$ (4.4)

where the first number in the superscripts $(i,j)$ denotes the number of
final-state partons associated with the respective term and the second one the
order of $\alpha_{s}$. Inserting (4.4) into (4.2) we get the unexpanded
$A^{b}_{\rm FB}$ to first and to second order in $\alpha_{s}$:

$\displaystyle A_{\rm FB}(\alpha_{s})=$
$\displaystyle\frac{\sigma_{A}^{(2,0)}+\sigma_{A}^{(2,1)}+\sigma_{A}^{(3,1)}}{{\sigma_{S}^{(2,0)}+\sigma_{S}^{(2,1)}+\sigma_{S}^{(3,1)}}}\equiv
A_{\rm FB}^{\rm LO}~{}C_{1}\,,$ (4.5) $\displaystyle A_{\rm
FB}(\alpha_{s}^{2})=$
$\displaystyle\frac{\sigma_{A}^{(2,0)}+\sigma_{A}^{(2,1)}+\sigma_{A}^{(3,1)}+\sigma_{A}^{(2,2)}+\sigma_{A}^{(3,2)}+\sigma_{A}^{(4,2)}}{\sigma_{S}^{(2,0)}+\sigma_{S}^{(2,1)}+\sigma_{S}^{(3,1)}+\sigma_{S}^{(2,2)}+\sigma_{S}^{(3,2)}+\sigma_{S}^{(4,2)}}\equiv
A_{\rm FB}^{\rm LO}~{}C_{2}\,,$ (4.6)

where

$A_{\rm FB}^{\rm LO}=\frac{\sigma_{A}^{(2,0)}}{\sigma_{S}^{(2,0)}}\,,$ (4.7)

is the forward-backward asymmetry at Born level. The factors $C_{1}$ and
$C_{2}$ defined by the respective ratio on the left of eq. (4.5) and (4.6) are
the unexpanded first-order and second-order QCD correction factors.

Taylor expanding eq. (4.5) to first order and of (4.6) to second order in
$\alpha_{s}$ gives

$\displaystyle A_{\rm FB}^{\rm NLO}=$ $\displaystyle A_{\rm FB}^{\rm
LO}\left[1+A_{1}\right]\;+\;{\cal O}(\alpha_{s}^{2})\,,$ (4.8) $\displaystyle
A_{\rm FB}^{\rm NNLO}=$ $\displaystyle A_{\rm FB}^{\rm
LO}\;\left[1\;+\;A_{1}\;+\;A_{2}\right]\;+\;{\cal O}(\alpha_{s}^{3})\,,$ (4.9)

where $A_{1}$ and $A_{2}$ are the QCD corrections of ${\cal O}(\alpha_{s})$
and ${\cal O}(\alpha_{s}^{2})$, respectively.

$\displaystyle A_{1}=$
$\displaystyle\sum\limits_{i=2,3}\big{[}\frac{\sigma_{A}^{(i,1)}}{\sigma_{A}^{(2,0)}}\;-\;\frac{\sigma_{S}^{(i,1)}}{\sigma_{S}^{(2,0)}}\big{]}\,,$
(4.10) $\displaystyle A_{2}=$
$\displaystyle\sum\limits_{i=2,3,4}\big{[}\frac{\sigma_{A}^{(i,2)}}{\sigma_{A}^{(2,0)}}\;-\;\frac{\sigma_{S}^{(i,2)}}{\sigma_{S}^{(2,0)}}\big{]}-\frac{\sigma_{S}^{(2,1)}+\sigma_{S}^{(3,1)}}{\sigma_{S}^{(2,0)}}\,A_{1}\,.$
(4.11)

Eqs. (4.8) and (4.9) are the expanded forms of the forward-backward asymmetry
at NLO and NNLO QCD. The unexpanded and expanded first and second-order
forward-backward asymmetries differ by terms of order $\alpha_{s}^{2}$ and
order $\alpha_{s}^{3}$, respectively. The differences between the two forms
may be considered as an estimate of the theory uncertainties.

###  4.2 Numerical results for the $A_{\rm FB}^{t}$ in the continuum

Table 2 contains our results for the expanded form of the top-quark forward-
backward asymmetry $A_{\rm FB}^{t}$ at NLO and NNLO QCD defined in eq. (4.8)
and (4.9), respectively, and the correction terms $A_{1}$ and $A_{2}$ given in
Eq. (4.10) and (4.11) for several c.m. energies. The central values and the
uncertainties refer to the scales $\mu=\sqrt{s}$ and $\mu=\sqrt{s}/2$ and
$2\sqrt{s}$, respectively. Obviously, here we vary $\mu$ simultaneously in the
$\sigma_{S}$ and $\sigma_{A}$ of different partonic channels between
$\sqrt{s}/2$ and $2\sqrt{s}$. We note that we have included in $A_{2}$ also
the non-universal contributions that contain the electroweak couplings of
quarks $q\neq t$.

Table 2:  The top-quark forward-backward asymmetry at LO, NLO, and NNLO QCD for several c.m. energies using the expansions in eq. (4.8) and (4.9). The numbers are given in percent. $\sqrt{s}$ [GeV] |  $A_{\rm FB}^{\rm LO}$ [%] |  $A_{\rm FB}^{\rm NLO}$ [%] |  $A_{\rm FB}^{\rm NNLO}$ [%] |  $A_{1}$ [%] |  $A_{2}$ [%] |  $\delta A_{\rm FB}^{\rm NNLO}$ [%]
---|---|---|---|---|---|---
360 | 14.94 | $15.54^{+0.05}_{-0.04}$ | $16.23^{+0.12}_{-0.10}$ | $4.01^{+0.35}_{-0.29}$ | $4.58^{+0.46}_{-0.38}$ | $\pm 0.59$
400 | 28.02 | $28.97^{+0.08}_{-0.07}$ | $29.63^{+0.11}_{-0.10}$ | $3.41^{+0.29}_{-0.25}$ | $2.36^{+0.11}_{-0.11}$ | $\pm 0.27$
500 | 41.48 | $42.42^{+0.08}_{-0.07}$ | $42.91^{+0.08}_{-0.07}$ | $2.28^{+0.19}_{-0.16}$ | $1.18^{+0.01}_{-0.01}$ | $\pm 0.13$
700 | 51.34 | $51.81^{+0.04}_{-0.03}$ | $52.05^{+0.04}_{-0.04}$ | $0.91^{+0.07}_{-0.06}$ | $0.47^{+0.01}_{-0.01}$ | $\pm 0.06$

Table 3 contains our results for the unexpanded form of the top-quark
$A^{t}_{\rm FB}$ at NLO and NNLO QCD and the associated QCD correction factors
$C_{1}$ and $C_{2}$ defined in eq. (4.5) and (4.6) for several c.m. energies.
This form of computing the asymmetry corresponds to the simulation using
Monte-Carlo event generators. The $A^{t}_{\rm FB}$ at NNLO QCD was computed
before in ref. [38] in the unexpanded form with values of $m_{t}$ and
$\alpha_{s}$ that differ slightly from the ones that we use here, while the
results basically agree.

Table 3:  The unexpanded form of the top-quark forward-backward asymmetry (4.2) for several c.m. energies. The numbers are given in percent. $\sqrt{s}$ [GeV] |  $A_{\rm FB}^{\rm LO}$ [%] |  $A_{\rm FB}^{\rm NLO}$ [%] |  $A_{\rm FB}^{\rm NNLO}$ [%]
---|---|---|---
360 | 14.94 | $15.31^{+0.02}_{-0.02}$ | $15.82^{+0.08}_{-0.06}$
400 | 28.02 | $28.77^{+0.05}_{-0.04}$ | $29.42^{+0.10}_{-0.09}$
500 | 41.48 | $42.32^{+0.06}_{-0.05}$ | $42.83^{+0.08}_{-0.07}$
700 | 51.34 | $51.78^{+0.03}_{-0.03}$ | $52.03^{+0.04}_{-0.04}$

One may take the spread between the values of the expanded and unexpanded
forms of $A_{FB}^{\rm NLO}$ and $A_{FB}^{\rm NNLO}$ given in tables 2, 3 as an
estimate of the uncalculated higher order corrections.

###  4.3 Numerical results for the $A_{\rm FB}^{b}$ at the $Z$ resonance

We now present our numerical results on the $b$-quark forward-backward
asymmetry at the $Z$ resonance to $\alpha_{s}^{2}$ in QCD and lowest order in
the electroweak couplings. The input values of the SM couplings and masses are
specified in detail in ref. [46]. In particular, we use $\alpha_{s}$ defined
in the 5-flavor QCD and the bottom-quark on-shell mass $m_{b}=4.89$ GeV. For
the mass of the top quark that appears in the triangle-loop diagrams
contributing to the non-universal corrections to $A_{\rm FB}^{b}$ we use
$m_{t}=173.34$ GeV. The $u,d,c,s$ quarks are taken to be massless. With these
input parameters, the Born-level value of the $b$-quark forward-backward
asymmetry at $\sqrt{s}=m_{Z}$ is $A_{\rm FB}^{\rm LO}=0.1512$. The value of
$A_{\rm FB}^{\rm LO}$ is very sensitive to the input value of $s^{2}_{W}$ but
insensitive to the value of $m_{b}$ within its uncertainty. Here, we discuss
only the results of the QCD corrections for the expanded $A_{\rm FB}^{b}$ with
respect to both the $b$-quark and the thrust direction, the later of which
enters the subsequent discussion of the phenomenological consequences. Table 4
lists the values of the NLO and NNLO correction factor, both for the quark
axis and the thrust axis definition, for the three renormalization scales
$\mu=m_{Z}/2,m_{Z},2m_{Z}$. The numbers given in this table show that the
order $\alpha_{s}^{2}$ corrections are significant. For $\mu=m_{Z}$ the ratio
$A_{2}/A_{1}$ is $43\%$ and $37\%$ for the quark and thrust axis definition,
respectively. The scale variations change both the first- and second-order QCD
correction factors by about $\pm 0.003$ with respect to their values at
$\mu=m_{Z}$. The fact that inclusion of the second-order correction term
$A_{2}$ does not reduce the scale uncertainty is not unusual for an observable
that is defined as a ratio. Contrary to the case of the top quark for the c.m.
energies below the $4m_{t}$ threshold, the order $\alpha_{s}$ and
$\alpha^{2}_{s}$ corrections $A_{1}$ and $A_{2}$ are dominated by the
contributions from the three-parton and three- and four-parton final states,
respectively. We have included in the computation of $A_{\rm FB}$ the non-
universal corrections of order $\alpha_{s}^{2}$ that contain the vector and
axial vector couplings of quarks $q\neq b$, which turn out to be quite
sizable. We note that the non-universal corrections depend on the value of
$\sin^{2}\theta_{W}$.

Table 4:  The first- and second-order QCD correction factors defined in (4.8) - (4.11) to the LO $b$-quark forward-backward asymmetry at the $Z$ peak $\mu=m_{Z}$. The numbers in superscript (subscript) refer to the changes if $\mu=2m_{Z}$ ($\mu=m_{Z}/2$) is chosen. | $1+A_{1}$ | $1+A_{1}+A_{2}$ | $A_{1}$ | $A_{2}$
---|---|---|---|---
quark axis: | $0.9710^{+0.0028}_{-0.0034}$ | $0.9587^{+0.0026}_{-0.0028}$ | $-0.0290$ | $-0.0123$
thrust axis: | $0.9713^{+0.0027}_{-0.0026}$ | $0.9608^{+0.0022}_{-0.0025}$ | $-0.0287$ | $-0.0105$

###  4.4 Phenomenological consequences

The $A_{\rm FB}$ with QCD corrections calculated above cannot be compared
directly with experimental measurements. In the measurements of the $b$-quark
asymmetry by the experiments at the previous $e^{+}e^{-}$ colliders LEP and
SLAC, which are reviewed in [1, 79, 80], the forward and backward hemispheres
was defined using the thrust axis. In our computation the thrust axis is used
but identified only for partonic final states, while the hadronization of
partons causes a smearing of this axis (causing discrepancy between thrust at
the parton level and at the level of stable hadrons). Results from several
experimental measurements with different acceptance regions in phase space,
and different sensitivity to QCD-correction effects etc, are combined after
unfolded respectively. Bearing this in mind, we proceed with the analysis as
described in ref. [46], which, albeit being very preliminary and far from
full-fledged, can give a quick idea about what might be the possible
consequence of a revised QCD-correction factor 333See ref. [81] for an updated
analysis of the theoretical uncertainties.. In short, a pseudo-observable, the
bare $b$-quark asymmetry $A^{0,b}_{\rm FB}$, is extracted from the measured
asymmetry $A^{b,T}_{\rm FB,exp}$ [82, 83] in the following way. The the
measured asymmetry $A^{b,T}_{\rm FB,exp}$ was first corrected for QCD effects
as follows,

$\displaystyle A^{b,T}_{\rm FB,exp}$ $\displaystyle=$
$\displaystyle(A^{0,b}_{\rm FB})_{\rm
exp}\left[1+\delta^{(1)}_{T}a_{s}+\delta^{(2)}_{T}a_{s}^{2}\right]\,,$ (4.12)

and then the QCD corrected “experimental” asymmetry $(A^{0,b}_{\rm FB})_{\rm
exp}$ was further corrected for higher order electroweak corrections etc,
before a value of the bare asymmetry $A^{0,b}_{\rm FB}$ was deduced:

$A_{\rm FB}^{0,b}=(A_{\rm FB}^{0,b})_{\rm exp}+\delta A_{\rm FB}^{b}\,.$
(4.13)

The corrections $\delta A_{\rm FB}^{b}$ include the energy shift from
$\sqrt{s}=91.26$ GeV (all measured asymmetries were corrected, in a first
step, to this energy) to $\sqrt{s}=m_{Z}$, QED corrections, corrections due to
$\gamma$ exchange and $Z-\gamma$ interference, and due to imaginary parts of
effective weak couplings (cf. for instance, [27]). In this way, the
experimental value determined for this pseudo-observable reads as
$A^{0,b}_{\rm FB}=0.0992\pm 0.0016$ and shows a large deviation, about
$2.9\sigma$, from the SM fit $A^{0,b}_{\rm FB}=0.1038$ [1, 80] if the NNLO QCD
corrections for massless $b$ quarks are used.

The QCD correction factor determined in [82] is $(1-C^{T}_{\rm QCD})=0.9646\pm
0.0063$ where the error includes estimates of hadronization effects. Our
thrust axis correction factor $(1+A_{1}+A_{2})=0.9608\pm 0.0025$ where the
error is due to scale uncertainties only, agrees with that factor within the
uncertainties. Our central value is smaller than $0.9646$ by $0.4\%$, and
accordingly our correction changes the value of the pseudo-observable
$A_{FB}^{0,b}$ to $0.0996\pm 0.0016$. Thus the pull between $A_{FB}^{0,b}$ and
the SM fit cited above is slightly decreased, namely from $2.9\sigma$ to
$2.6\sigma$.

The fixed-order results discussed above were determined at the conventional
scale choice $\mu=m_{z}$. Due to the missing higher order perturbative
corrections, the truncated fixed order results on $A_{\rm FB}^{b}$ still have
a remaining $\mu$ dependence. One way to improve the perturbative result
regarding this aspect is to resum all explicit $\ln\mu^{2}$ terms in $A_{\rm
FB}^{b}$ that are related the QCD
$\beta=\mathrm{d}\ln\alpha_{s}/\mathrm{d}\ln\mu^{2}$ function, absorbed into a
new effective coupling for this particular observable, by means of the well-
known optimization procedure based on the Principle of Maximum Conformality
(PMC). Following the PMC procedure we determined the PMC scale to be $\mu^{\rm
PMC}_{r}=9.7~{}\rm{GeV}$ for the $b$ quark forward-backward asymmetry with the
thrust axis definition in ref. [47]. After applying PMC scale setting, the QCD
correction factor with the thrust axis definition is
$(1+\delta^{(1)}_{T}a_{s}+\delta^{(2)}_{T}a_{s}^{2})=0.9529$, which is smaller
than $0.9646$ by $1.2\%$. Consequently, the QCD correction factor with
refinements from the PMC method changes $A^{0,b}_{\rm FB}=0.0992\pm 0.0016$ to

$\displaystyle A^{0,b}_{\rm FB}=0.1004\pm 0.0016,$ (4.14)

which shows a $2.1\sigma$ deviation from the SM fit $A^{0,b}_{\rm FB}=0.1038$.

It is noted that there are updated SM fit results reported in refs. [10, 11]
with improved electroweak corrections taken into account. To see more clearly
how the deviations are changed due to the incorporation of updated electroweak
and QCD corrections, in table (5), we list the deviations computed using
different central values of the SM fit $A^{b,\text{fit}}_{FB}$, and also the
$A^{0,b}_{FB}$ determined by making use of the different parton-level QCD
correction factors.

$A^{b,\text{fit}}_{FB}$ v.s. $A^{0,b}_{FB}$ | 0.0992 ref. [80] | 0.0996 ref. [46] | 1.004 ref. [47]
---|---|---|---
0.1038 ref. [1] | 2.9 $\sigma$ | 2.6 $\sigma$ | 2.1 $\sigma$
0.1032 ref. [10] | 2.5 $\sigma$ | 2.25 $\sigma$ | 1.8 $\sigma$
0.1030 ref. [11] | 2.4 $\sigma$ | 2.1 $\sigma$ | 1.6 $\sigma$
Table 5:  Deviations between different central values of the SM fit
$A^{b,\text{fit}}_{FB}$ and the $A^{0,b}_{FB}$ determined in literature

So far the tension between $A^{b,\text{fit}}_{FB}$ and $A^{0,b}_{FB}$ seems to
decrease progressively over the years. However, the experimental uncertainty
of $A_{\rm FB}^{b}$ measurement used in determining the numbers in this table
is still the one at the SLAC and LEP experiments, about $1.7\%$ [1]. On the
other hand, the experimental uncertainty, currently dominated by the
statistical uncertainties, is expected to be greatly reduced at future lepton
colliders [12, 13, 15, 16, 14]. Confirmation or resolution of this long-term
discrepancy at the future lepton colliders would, of course, make a great
impact on the development of our theories.

##  5 Conclusions

We have formulated a set-up for computing the production of a heavy quark-
antiquark pair $Q{\bar{Q}}$ in $e^{+}e^{-}$ collisions at
$\mathcal{O}(\alpha_{s}^{2})$ in the perturbative QCD at the differential
level, using the antenna subtraction method for handling the intermediate
infrared soft and collinear divergences of the individual contributions. This
set-up applies to the differential calculation of any infrared-safe observable
of this process to $\mathcal{O}(\alpha_{s}^{2})$ in QCD, and works in a
numerically stable way owing to the local subtraction of IR divergences in
each partonic phase space.

Besides the application to $t\bar{t}$ pair production at (future) $e^{+}e^{-}$
collisions, we have determined the $\mathcal{O}(\alpha_{s}^{2})$ QCD
corrections to the b-quark forward-backward asymmetry $A_{\rm{FB}}$ in
$e^{+}e^{-}$ collisions at the $Z$ resonance, aimed at analysing the long-
standing tension between its direct determination at the SLAC and LEP
experiments and its global SM fit. We found that after taking into account
exact b-quark mass effects, the pull between the direct determination of
$A_{\rm{FB}}$ and its SM fit is reduced from 2.9$\sigma$ to 2.6$\sigma$.
Furthermore, we performed an improved determination of this QCD correction
factor by means of the optimization procedure based on the Principle of
Maximum Conformality, with which the tension is eventually reduced down to
2.1$\sigma$. Once compared to the updated SM-fit results, the deviation gets
reduced below 2$\sigma$. As a future application of our computational set-up,
one may consider the determination of the $\mathcal{O}(\alpha_{s}^{2})$
corrections to the forward-backward asymmetry for two $b$-jet final states,
for which one expects a decrease of the magnitude of the QCD corrections.
Furthermore, this setup can be applied to the prodution of polarized top
quarks, with polarized $\mathit{e}^{+}\mathit{e}^{-}$ beams.

## Acknowledgments

The author is grateful to Werner Bernreuther, Oliver Dekkers, Thomas Gehrmann,
Dennis Heisler, Rui-qing Meng, Jian-ming Shen, Zong-guo Si, Sheng-quan Wang,
Xing-gang Wu for collaborating on the work presented in this proceeding.

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