#  DESY 20–214 ISSN 0418-9833
Dezember 2020

Angular analysis of bottom-flavored hadron production in semileptonic decays
of polarized top quarks

Bernd A. Kniehl111E-mail<EMAIL_ADDRESS>
II. Institut für Theoretische Physik, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany

S. Mohammad Moosavi Nejad222E-mail<EMAIL_ADDRESS>
Faculty of Physics, Yazd University, P.O. Box 89195–741, Yazd, Iran

###### Abstract

We study the inclusive production of bottom-flavored hadrons from semileptonic
decays of polarized top quarks at next-to-leading order in QCD using
fragmentation functions recently determined from a global fit to $e^{+}e^{-}$
data. We provide the relevant differential decay widths at parton level in
analytic form. These results fill an important gap in the theoretical
interpretation of recent measurements of the top-quark polarization and the
$t\bar{t}$ spin correlations using dilepton final states in proton-proton
collisions at the CERN Large Hadron Collider. As an illustration, we study the
distributions in the scaled bottom-hadron energy of the polarized-top-quark
decay widths for different $W$-boson helicities.

PACS numbers: 12.38.Bx, 13.85.Ni, 14.40.Nd, 14.65.Ha

##  1 Introduction

The top quark $t$ of the standard model (SM) is the heaviest known elementary
particle. Due to its high mass, it plays a crucial role in testing the
electroweak symmetry breaking mechanism and in searching for new physics
beyond the SM. The precise determination of its properties, including its mass
$m_{t}$ and total decay width $\Gamma_{t}$, and its process-dependent
features, like its polarization or the correlation of its spin with that of a
co-produced antitop quark, is of prime importance. The latter quantities are
particularly sensitive probes of deviations from the SM and allow us to
constrain, e.g., the anomalous chromoelectric and chromomagnetic dipole
moments of the top quark. The top-quark lifetime
$\tau_{t}=\hbar/\Gamma_{t}\approx 5\times 10^{-25}$ s [1] is shorter than the
typical time scale of quantum chromodynamics (QCD)
$\hbar/\Lambda_{\mathrm{QCD}}\approx 10^{-24}$ s and much shorter than the
spin correlation time scale $\hbar m_{t}/\Lambda_{\mathrm{QCD}}^{2}\approx
10^{-21}$ s, where $\Lambda_{\mathrm{QCD}}$ is the asymptotic scale parameter
of QCD. Therefore, the top quark decays before it can hadronize, and its full
spin information is preserved during its decay process and fully encoded in
the angular distribution of its decay products.

The top-quark polarization and the $t\bar{t}$ spin correlations have recently
been measured using dilepton final states in Run 2 at the CERN Large Hadron
Collider (LHC) with center-of-mass energy $\sqrt{s}=13$ TeV, by the ATLAS [2]
and CMS [3] Collaborations. Intriguingly, ATLAS found a deviation of 3.8
standard deviations ($\sigma$) from the SM prediction of the asymmetry
$A_{|\Delta\phi_{\ell\ell}|}$ of the distribution in the azimuthal angular
difference $\Delta\phi_{\ell\ell}$ of the decay leptons. This deviation was
confirmed by CMS, albeit with a smaller significance of about $2\sigma$. These
analyses are relying on the factorization of the squared matrix element of the
full process, $|\mathcal{M}(q\bar{q}/gg\to t\bar{t}\to
b\ell^{+}\nu\bar{b}\ell^{-}\bar{\nu})|^{2}\propto\rho\times
R\times\bar{\rho}$, into the spin density matrices $R$, $\rho$, and
$\bar{\rho}$ for on-shell $t\bar{t}$ hadroproduction and semileptonic $t$ and
$\bar{t}$ decays, respectively, via the narrow-width approximation. While $R$
is treated in Refs. [2, 3] at next-to-leading order (NLO) in QCD [4, 5],
$\rho$ and $\bar{\rho}$ are only modeled at leading order (LO) using the
program package madspin [6]. Moreover, the formation of bottom-flavored
hadrons is not taken into account within the rigorous framework of the QCD
parton model with fragmentation functions (FFs) whose universality is
guaranteed by the factorization theorem [7]. It is an urgent matter to clarify
in how far the observed deviations may be related to a lack of precision in
the theoretical treatment of the semileptonic $t$ and $\bar{t}$ decays.

It is the purpose of the present paper to provide theoretical input needed to
fill this gap. Specifically, we calculate the partial width of the inclusive
decay$t(\uparrow)\to bW^{+}(\uparrow)\to Bl^{+}\nu_{l}+X$, where $B$
generically denotes a bottom-flavored hadron, at NLO in QCD allowing for top-
quark polarization and definite $W$-boson helicity and properly accounting for
parton-to-hadron fragmentation and finite-hadron-mass effects. By doing so, we
generalize our previous work [8], where the top-quark spin was averaged over.

The general-mass variable-flavor-number scheme (GM-VFNS), which has been
elaborated for inclusive heavy-flavored-hadron production in $e^{+}e^{-}$
annihilation [9], two-photon collisions [10], photoproduction [11], and
hadroproduction [12, 13, 14, 15], provides an ideal theoretical framework also
here. However, owing to the large mass hierarchy $m_{b}\ll m_{t}$,
finite-$m_{b}$ corrections are expected to be negligible in the case at hand.
This expectation was actually confirmed in Ref. [8], by a comparative analysis
of the partial width of the decay $t\to B+W^{+}$ in the GM-VFNS and the zero-
mass variable-flavor-number scheme (ZM-VFNS), where bottom is included among
the massless quark flavors. In fact, the finite-$m_{b}$ corrections were found
to be much smaller than the contribution from gluon fragmentation. Therefore,
we will adopt the ZM-VFNS in the following. However, we will include
finite-$m_{B}$ effects, which modify the relations between partonic and
hadronic variables and reduce the available phase space, as explained in Sec.
2 of Ref. [8].

In Ref. [8], we adopted the $B$ FFs from Ref. [16], which were determined at
NLO in the ZM-VFNS through a joint fit to $e^{+}e^{-}$ annihilation data taken
by ALEPH [17] and OPAL [18] at CERN LEP1 and by SLD [19] at SLAC SLC.
Specifically, the power ansatz
$D_{b}(z,\mu_{F}^{\text{ini}})=Nz^{\alpha}(1-z)^{\beta}$ was used as the
initial condition for the $b\to B$ FF at factorization scale
$\mu_{F}^{\text{ini}}=m_{b}=4.5$ GeV, while the gluon and light-quark FFs were
generated via the Dokshitzer-Gribov-Lipatov-Altatelli-Parisi (DGLAP) [20, 21,
22] evolution. In Ref. [23], the analysis of Ref. [16] was updated by
including the data taken by DELPHI [24] at CERN LEP1, which were published
after Ref. [16], working both at NLO and next-to-next-to-leading order (NNLO)
with the same theoretical assumptions. In our numerical analysis, we employ
the new $B$ FFs from Ref. [23].

A similar analysis, albeit without fragmentation and finite-$m_{B}$ effects,
was reported in Refs. [25, 26]. Our analysis provides an independent check of
analytic results presented therein. A related NLO analysis with a different
treatment of the final state, for bottom jets instead of bottom hadrons, was
performed in Ref. [27], leading to results that, unfortunately, cannot be
compared with ours in any straightforward way. Recently, the analysis of Refs.
[25, 26] has been extended to NNLO in QCD using the optical theorem [28]. Due
to the totally inclusive treatment of the hadronic part of the final state,
this result does not allow for the implementation of FFs.

This paper is organized as follows. In Sec. 2, we list our parton-level
results in analytic form, relegating lengthy formulas to the Appendix. In Sec.
3, we present our numerical analysis. In Sec. 4, we summarize our conclusions.

##  2 Analytic results

We work at NLO in the ZM-VFNS, implemented in the modified minimal-subtraction
($\overline{\mathrm{MS}}$) scheme, and consider the decay process

$t(p_{t},s)\to b(p_{b})+W^{+}(p_{W},\lambda)({}+g(p_{g}))\to
B(p_{B})+\ell^{+}(p_{\ell})+\nu_{\ell}(p_{\nu})+X,$ (1)

where $X$ collectively denotes the unobserved final-state hadrons and the
four-momentum, spin, and helicity assignments are indicated in parentheses. We
have $s=\pm 1/2$ and $\lambda=0,\pm 1$. The gluon in Eq. (1) contributes to
the real radiation at NLO. Both the $b$ quark and the gluon may hadronize into
the $B$ hadron. For simplicity, we employ the narrow-width approximation,
where $p_{W}^{2}=m_{W}^{2}$ and small terms of order
$\mathcal{O}(\Gamma_{W}^{2}/m_{W}^{2})$ are neglected. As mentioned in Sec. 1,
we put $m_{b}=0$, but keep $m_{B}$ finite. In the top-quark rest frame, the
$b$ quark, gluon, and $B$ hadron have energies $E_{i}=p_{t}\cdot p_{i}/m_{t}$
($i=b,g,B$), which nominally range from
$E_{b}^{\text{min}}=E_{g}^{\text{min}}=0$ and $E_{B}^{\text{min}}=m_{B}$ to
$E_{b}^{\text{max}}=E_{g}^{\text{max}}=(m_{t}^{2}-m_{W}^{2})/(2m_{t})$ and
$E_{B}^{\text{max}}=(m_{t}^{2}+m_{B}^{2}-m_{W}^{2})/(2m_{t})$, respectively.
As in Ref. [8], we choose the scaling variable $z$ by setting $E_{B}=zE_{a}$,
with $a=b,g$, in the range $0\leq z\leq 1$. Introducing the scaled energies
$x_{i}=E_{i}/E_{b}^{\text{max}}$ ($i=b,g,B$), we then have
$m_{B}/E_{b}^{\text{max}}\leq x_{B}\leq x_{a}\leq 1$. An alternative
definition of the scaling variable, in terms of light-cone variables, is
discussed in Sec. 3 of Ref. [8], to where we refer the interested reader.

We implement the polarization of the top quark by writing its (average) spin
four-vector in its rest frame as
$s_{t}^{\mu}=P(0,\sin\theta_{P}\cos\phi_{P},\sin\theta_{P}\sin\phi_{P},\cos\theta_{P})$,
where $P$ is the magnitude of the top-quark polarization, taking values in the
range $0\leq P\leq 1$. Here, it is understood that the $z$ axis is chosen to
point along the $W$-boson flight direction, so that $s_{t}\cdot
p_{W}=-P|\vec{p}_{W}|\cos\theta_{P}$. To describe the leptonic decay of the
$W$ boson, we boost into the rest frame of the latter, which leaves the $z$
axis invariant, and define the charged-lepton four-momentum to be
$p_{\ell}^{\mu}=E_{\ell}(1,\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$.
The polar angles $\theta_{P}$ and $\theta$, which appear in our final results,
are also illustrated in Fig. 1.

Figure 1: Definitions of the polar angle $\theta_{P}$ of the top-quark
polarization three-vector in the top-quark rest frame and of the polar angle
$\theta$ of the charged-lepton three-momentum in the $W$-boson rest frame. In
both cases, the $z$ axis is chosen to point along the $W$-boson three-momentum
in the top-quark rest frame.

We wish to calculate the triply differential partial decay width
$d^{3}\Gamma/(dx_{B}\,d\cos\theta\,d\cos\theta_{P})$ of process (1).
Analogously to Eq. (3) in Ref. [8], we have

$\frac{d^{3}\Gamma}{dx_{B}\,d\cos\theta\,d\cos\theta_{P}}=\sum_{a=b,g}\int_{x_{B}}^{1}\frac{dx_{a}}{x_{a}}\,\frac{d^{3}\hat{\Gamma}_{a}}{dx_{a}\,d\cos\theta\,d\cos\theta_{P}}(\mu_{R},\mu_{F})D_{a}\left(\frac{x_{B}}{x_{a}},\mu_{F}\right),$
(2)

where $\mu_{R}$ is the renormalization scale, $\mu_{R}$ is the factorization
scale, $D_{a}(z,\mu_{F})$ is the $a\to B$ FF, and

$\frac{d^{3}\hat{\Gamma}_{a}}{dx_{a}\,d\cos\theta\,d\cos\theta_{P}}=\frac{1}{2}\left(\frac{d^{2}\hat{\Gamma}_{a}^{\text{unpol}}}{dx_{a}\,d\cos\theta}+P\frac{d^{2}\hat{\Gamma}_{a}^{\text{pol}}}{dx_{a}\,d\cos\theta}\cos\theta_{P}\right).$
(3)

The factor $1/2$ on the right-hand side of Eq. (3) ensures that

$\int_{-1}^{1}d\cos\theta_{P}\,\frac{d^{3}\hat{\Gamma}}{dx_{a}\,d\cos\theta\,d\cos\theta_{P}}=\frac{d^{2}\hat{\Gamma}_{a}^{\text{unpol}}}{dx_{a}\,d\cos\theta}.$
(4)

Each $W$-boson helicity is featured by a characteristic $\theta$ dependence,
which is encoded in the structure

$\frac{d^{2}\hat{\Gamma}_{a}}{dx_{a}\,d\cos\theta}=\frac{3}{8}(1+\cos\theta)^{2}\frac{d\hat{\Gamma}_{a}^{+}}{dx_{a}}+\frac{3}{8}(1-\cos\theta)^{2}\frac{d\hat{\Gamma}_{a}^{-}}{dx_{a}}+\frac{3}{4}\sin^{2}\theta\frac{d\hat{\Gamma}_{a}^{0}}{dx_{a}}.$
(5)

This holds for both the unpolarized and polarized terms in Eq. (3). Notice
that the $\theta$-dependent coefficients on the right-hand side of Eq. (5) are
normalized so that, upon integration over $\cos\theta$, we have

$\frac{d\hat{\Gamma}_{a}}{dx_{a}}=\sum_{\lambda=-1}^{1}\frac{d\hat{\Gamma}_{a}^{\lambda}}{dx_{a}}.$
(6)

At LO, we only have $a=b$, and $x_{b}=1$ is fixed, i.e. the $x_{b}$ dependence
comes as a delta-function peak. Specifically, we have

$\displaystyle\hat{\Gamma}_{b,\text{LO}}^{0,\text{unpol}}$ $\displaystyle=$
$\displaystyle\hat{\Gamma}_{b,\text{LO}}^{0,\text{pol}}=F(1-\omega)^{2},$
$\displaystyle\hat{\Gamma}_{b,\text{LO}}^{-,\text{unpol}}$ $\displaystyle=$
$\displaystyle-\hat{\Gamma}_{b,\text{LO}}^{-,\text{pol}}=F(2\omega)(1-\omega)^{2},$
$\displaystyle\hat{\Gamma}_{b,\text{LO}}^{+,\text{unpol}}$ $\displaystyle=$
$\displaystyle\hat{\Gamma}_{b,\text{LO}}^{+,\text{pol}}=0,$ (7)

where $\omega=m_{W}^{2}/m_{t}^{2}$ and
$F=G_{F}m_{t}^{3}|V_{tb}|^{2}B(W^{+}\to\ell^{+}\nu_{\ell})/(8\pi\sqrt{2})$.
Here, $G_{F}$ is Fermi’s constant, $V_{ij}$ is the $ij$ element of the
Cabibbo-Kobayashi-Matrix quark mixing matrix [29, 30], and
$B(W^{+}\to\ell^{+}\nu_{\ell})$ is the branching ratio of the leptonic
$W$-boson decay mode considered. Neglecting the masses of the charged leptons
and the first five quark flavors, we have

$\displaystyle B(W^{+}\to\ell^{+}\nu_{\ell})$ $\displaystyle=$
$\displaystyle\frac{1}{3+N_{c}\sum_{\begin{subarray}{c}i=u,c\\\
j=d,s,b\end{subarray}}|V_{ij}|^{2}\left[1+3C_{F}\alpha_{s}(\mu_{R})/(4\pi)\right]}$
(8) $\displaystyle\approx$
$\displaystyle\frac{1}{9}\left[1-\frac{2}{3}\,\frac{\alpha_{s}(\mu_{R})}{\pi}\right],$

where we have included the NLO QCD correction, with color factors $N_{c}=3$
and $C_{F}=(N_{c}^{2}-1)/(2N_{c})=4/3$. In the last equality in Eq. (8), we
have approximated $V_{ij}\approx\delta_{ij}$. At LO, the top-quark spin is
passed on to the bottom quark for $\lambda=0$, while it is flipped for
$\lambda=-1$; the case $\lambda=+1$ is forbidden by angular-momentum
conservation in the limit $m_{b}\to 0$, as is reflected in Eq. (7). Using Eq.
(6), we have

$\displaystyle\hat{\Gamma}_{b,\text{LO}}^{\text{unpol}}$ $\displaystyle=$
$\displaystyle F(1+2\omega)(1-\omega)^{2},$
$\displaystyle\hat{\Gamma}_{b,\text{LO}}^{\text{pol}}$ $\displaystyle=$
$\displaystyle F(1-2\omega)(1-\omega)^{2}.$ (9)

Inserting Eq. (7) in Eq. (2), we obtain a very simple formula for the final LO
result:

$\displaystyle\frac{d^{3}\Gamma_{\text{LO}}}{dx_{B}\,d\cos\theta\,d\cos\theta_{P}}$
$\displaystyle=$
$\displaystyle\frac{3}{8}F(1-\omega)^{2}D_{b}(x_{B},\mu_{F})[\sin\theta(1+P\cos\theta_{P})$
(10) $\displaystyle{}+\omega(1-\cos\theta)^{2}(1-P\cos\theta_{P})].$

Figure 2: Feynman diagrams contributing to the partial decay width of the
process in Eq. (1) at NLO: (a) initial-state radiation; (b) final-state
radiation; (c) vertex correction; and (d) combination of wave function
renormalizations and vertex counterterm. The leptonic $W$-boson decay is not
shown.

The Feynman diagrams contributing to the partial decay width of the process in
Eq. (1) at NLO are depicted in Fig. 2. The NLO coefficient functions of the
unpolarized case may be found in Appendix A of Ref. [8] and those of the
polarized case are presented in the Appendix of this paper. At NLO, the case
$\lambda=+1$ is enabled by the presence of the additional spin-one gluon even
for $m_{b}=0$.

At this point, we compare our new analytic results with the literature. In
Refs. [25, 26], process (1) was also considered at NLO using the narrow-width
approximation and putting $m_{b}=0$, but treating the final state in a less
differential fashion, which does not allow for the convolution with $a\to B$
FFs on the basis of the ZM-VFNS. We can compare our results for
$\hat{\Gamma}_{b}^{\lambda,\text{pol}}$ with Refs. [25, 26] upon integration
over $x_{b}$ in the range $0\leq x_{b}\leq 1$. Specifically, the quantities
$\hat{\Gamma}_{U}^{P}$, $\hat{\Gamma}_{L}^{P}$, and $\hat{\Gamma}_{F}^{P}$
listed in Eqs. (18)–(20) of Ref. [25] (see also Eqs. (42)–(44) in Ref. [26])
are related to $\hat{\Gamma}_{b}^{\lambda,\text{pol}}$ as
$\hat{\Gamma}_{b}^{0,\text{pol}}/\hat{\Gamma}_{b,\text{LO}}^{\text{unpol}}=\hat{\Gamma}_{L}^{P}$
and
$\hat{\Gamma}_{b}^{\pm,\text{pol}}/\hat{\Gamma}_{b,\text{LO}}^{\text{unpol}}=(\hat{\Gamma}_{U}^{P}\pm\hat{\Gamma}_{F}^{P})/2$.
With these identifications, we fully agree with Refs. [25, 26].

##  3 Numerical results

We are now in a position to explore the phenomenological consequences of our
results by performing a numerical analysis. We adopt from Ref. [1] the input
parameter values $G_{F}=1.1663787\times 10^{-5}$ GeV-2, $m_{W}=80.379$ GeV,
$m_{t}=172.4$ GeV, $m_{B}=5.279$ GeV, $|V_{tb}|=1$, and
$B(W^{+}\to\ell^{+}\nu_{\ell})=10.86\%$. We evaluate
$\alpha_{s}^{(n_{f})}(\mu_{R})$ at NLO (NNLO) in the $\overline{\text{MS}}$
scheme using Eq. (4) of Ref. [31], retaining only the first two (three) terms
on the right-hand side, with $n_{f}=5$ active quark flavors and asymptotic
scale parameter $\Lambda_{\overline{\text{MS}}}^{(5)}=225$ MeV (207 MeV)
adjusted such that $\alpha_{s}^{(5)}(m_{Z})=0.1179$ for $m_{Z}=91.1876$ GeV
[1]. This yields $\alpha_{s}^{(5)}(m_{t})=0.1076$ (0.1076). As already
mentioned in Sec. 1, we use the up-to-date $B$ FFs from Ref. [23], both at NLO
and NNLO. For definiteness, we identify $\mu_{R}=\mu_{F}=\xi m_{t}$ and vary
$\xi$ from 1/2 to 2 about the default value 1 to estimate the theoretical
uncertainty due to the lack of knowledge of higher-order corrections.

The angular dependencies at the parton level, in Eqs. (3) and (5), are passed
on to the hadron level via Eq. (2). The hadron level counterparts,
$d\Gamma^{\lambda}/dx_{B}$, of the coefficient functions
$d\hat{\Gamma}_{a}^{\lambda}/dx_{a}$ in Eq. (5) may be projected out from the
measured $\theta$ distribution $d\Gamma/(dx_{B}\,d\cos\theta)$ as explained in
Sec. 4 of Ref. [8] (see Eqs. (21)–(23) therein) and thus represent physical
observables by themselves. In Ref. [8], the top quarks were taken to be
unpolarized, i.e., according to our present notation,
$d\Gamma^{\lambda,\text{unpol}}/dx_{B}$ were considered. In the following, we
complement the study of Ref. [8] by presenting predictions for
$d\Gamma^{\lambda,\text{pol}}/dx_{B}$. For the sake of a coherent treatment,
we also provide the analogous predictions for
$d\Gamma^{\lambda,\text{unpol}}/dx_{B}$, thus updating the analysis of Ref.
[8].

Our central predictions are of NLO. To assess their significance, we compare
them with the respective LO results. Our LO predictions are slightly
inconsistent because they are evaluated with NLO FFs. Unfortunately, Ref. [23]
does not provide a LO FF set. By the way, the same is true for Ref. [16], to
which we could have resorted otherwise. On the other hand, Ref. [23] also
supplies a NNLO set. While consistent NNLO predictions are unfeasible in the
absence of NNLO parton-level results, this still offers us the opportunity to
get a first impression of the typical magnitude of the NNLO effects. In our
pseudo-NNLO analysis, besides using NNLO FFs, we also evaluate
$\alpha_{s}(\mu_{R})$ at NNLO as explained above.

Figure 3:  LO and NLO results for (a) $d\Gamma^{\lambda,\text{unpol}}/dx_{B}$
and (b) $d\Gamma^{\lambda,\text{pol}}/dx_{B}$ with $\lambda=0,\pm 1$ as
functions of $x_{B}$. $d\Gamma_{\text{NLO}}^{+,\text{unpol}}/dx_{B}$ and
$d\Gamma_{\text{NLO}}^{+,\text{pol}}/dx_{B}$ are rescaled by a factor of 10
for better visibility. The theoretical uncertainties are indicated by the
shaded bands.

Figure 4:  NLO results for (a) $d\Gamma^{\lambda,\text{unpol}}/dx_{B}$ and (b)
$d\Gamma^{\lambda,\text{pol}}/dx_{B}$ with $\lambda=0,-1$, normalized to the
respective default LO results, as functions of $x_{B}$. The theoretical
uncertainties of the NLO results are indicated by the shaded bands.

In Fig. 3, we study the $x_{B}$ dependencies of
$d\Gamma^{\lambda,\text{unpol}}/dx_{B}$ and
$d\Gamma^{\lambda,\text{pol}}/dx_{B}$ for $\lambda=0,\pm 1$ at NLO and compare
them with the respective LO results for $\lambda=0,-1$. As explained in Sec.
2, $d\Gamma^{+,\text{unpol}}/dx_{B}$ and $d\Gamma^{+,\text{pol}}/dx_{B}$
vanish at LO in our approximation. The theoretical uncertainties are indicated
by the shaded bands. The slight inconsistency in our LO analysis mentioned
above, not only affects the default predictions, but also the estimation of
the theoretical uncertainty, which is expected to be slightly larger if the
$\mu_{F}$ dependence is subject to LO DGLAP evolution. Comparing Figs. 3(a)
and (b), we observe that $d\Gamma^{\lambda,\text{unpol}}/dx_{B}$ and
$d\Gamma^{\lambda,\text{pol}}/dx_{B}$ are very similar as for normalization
and line shape. Each $x_{B}$ distribution exhibits a maximum close to
$x_{B}=0.8$. Longitudinal $W$-boson helicity is favored, with
$d\Gamma_{\text{LO}/\text{NLO}}^{0,\text{unpol}/\text{pol}}/dx_{B}$ being more
than twice as large as
$d\Gamma_{\text{LO}/\text{NLO}}^{-,\text{unpol}/\text{pol}}/dx_{B}$ for
negative $W$-boson helicity $\lambda=-1$. On the other hand, positive
$W$-boson helicity $\lambda=+1$ is perturbatively suppressed; rescaling
$d\Gamma_{\text{NLO}}^{+,\text{unpol}/\text{pol}}/dx_{B}$ by the inverse
couplant, $2\pi/\alpha_{s}^{(5)}(m_{t})$, brings it up to the level of
$d\Gamma_{\text{NLO}}^{-,\text{unpol}/\text{pol}}/dx_{B}$. The NLO corrections
have a significant effect on the $x_{B}$ distributions for $\lambda=0,-1$, by
raising their peaks and lowering their small-$x_{B}$ tails. To render these
features more visible, we present, in Fig. 4, the QCD correction ($K$) factors
for $\lambda=0,-1$, which we evaluate by normalizing the NLO predictions
including their theoretical-uncertainty bands relative to the default LO
predictions. From Fig. 4(a) and (b), we observe that, both for unpolarized and
polarized top quarks, the $K$ factors steadily increase by one order of
magnitude, typically from 0.2 to 2, as $x_{B}$ runs across its range of
values. We conclude from Figs. 3 and 4 that the NLO corrections are quite
significant and should be taken into account in theoretical interpretations of
future top-quark polarization measurements.

Figure 5:  NLO and pseudo-NNLO results for (a)
$d\Gamma^{\lambda,\text{unpol}}/dx_{B}$ and (b)
$d\Gamma^{\lambda,\text{pol}}/dx_{B}$ with $\lambda=0,\pm 1$ as functions of
$x_{B}$. $d\Gamma_{\text{NLO}/\text{NNLO}}^{+,\text{unpol}}/dx_{B}$ and
$d\Gamma_{\text{NLO}/\text{NNLO}}^{+,\text{pol}}/dx_{B}$ are rescaled by a
factor of 10 for better visibility. The theoretical uncertainties are
indicated by the shaded bands.

Figure 6:  Pseudo-NNLO results for (a) $d\Gamma^{\lambda,\text{unpol}}/dx_{B}$
and (b) $d\Gamma^{\lambda,\text{pol}}/dx_{B}$ with $\lambda=0,\pm 1$,
normalized to the respective default NLO results, as functions of $x_{B}$. The
theoretical uncertainties of the pseudo-NNLO results are indicated by the
shaded bands.

In Fig. 5, compare our pseudo-NNLO evaluations of
$d\Gamma^{\lambda,\text{unpol}}/dx_{B}$ and
$d\Gamma^{\lambda,\text{pol}}/dx_{B}$ for $\lambda=0,\pm 1$ with the
respective NLO results already presented in Fig. 3. For better visibility, we
also plot, in Fig. 6, the ratios of the pseudo-NNLO results including their
theoretical-uncertainty bands and the default NLO predictions. We conclude
from Figs. 5 and 6 that the NNLO effects are likely to be relatively modest,
of the order of 10% or less.

##  4 Summary and Conclusions

We studied the inclusive production of bottom-flavored hadrons from
semileptonic decays of polarized top quarks at NLO in the ZM-FVNS using the
narrow-width approximation for the intermediate $W$ bosons, whose helicities
we distinguished. Specifically, we considered the partial decay width
differential in the scaled bottom-hadron energy $x_{B}$, the azimuthal angle
$\theta$ of the charged lepton in the $W$-boson rest frame, and the azimuthal
angle $\theta_{P}$ of the top-quark polarization in the top-quark rest frame.
In our numerical analysis, we employed up-to-date $B$ FFs, recently determined
from a global fit to all available $e^{+}e^{-}$ data [23]. We thus extended
our previous study in Ref. [8], which was restricted to unpolarized top
quarks. In Ref. [8], we had convinced ourselves, by comparing evaluations in
the GM-VFNS and ZM-VFNS, that finite-$m_{b}$ corrections may be safely
neglected. On the other hand, we retained finite-$m_{B}$ corrections, which
reduce the available phase space and lead to visible effects at small values
of $x_{B}$. We provided full analytic results, ready to be used by the
interested reader.

We found the NLO QCD corrections to be quite significant, inducing a reduction
in the lower $x_{B}$ range and an enhancement in the upper $x_{B}$ range, with
$K$ factors ranging from 0.2 to 2. On the other hand, including partial
information from NNLO, contained in the evaluation of the $B$ FFs and
$\alpha_{s}(\mu_{R})$, turned out to yield only mild modifications, with the
due caveat that such results suffer from a violation of renormalization group
invariance already in the considered order.

Our combined results, from Ref. [8] and this paper, help us to fill an
important gap in the theoretical interpretation of recent measurements of the
top-quark polarization and the $t\bar{t}$ spin correlations in dilepton final
states at the LHC [2, 3]. In particular, it will be interesting to see if
these theoretical improvements will contribute to reconciling the ATLAS [2]
and CMS [3] measurements of $A_{|\Delta\phi_{\ell\ell}|}$ with the SM
expectations.

## Acknowledgments

We thank G. Kramer for useful discussions at the initial stage of this work
and C. Schwanenberger for detailed information on Ref. [3]. The work of B.A.K.
was supported in part by the German Federal Ministry for Education and
Research BMBF through Grant No. 05H18GUCC1. The work of S.M.M.N. was supported
in part by the Iran National Science Foundation INSF through Grant No.
97005414.

## Appendix

In this appendix, we list the coefficient functions
$d\hat{\Gamma}_{a}^{\lambda,\text{pol}}/dx_{a}$ appearing in Eq. (5) at NLO in
the ZM-VFNS with $m_{b}=0$. They possess the following structure:

$\displaystyle\frac{1}{\hat{\Gamma}_{b,\text{LO}}^{\text{pol}}}\,\frac{d\hat{\Gamma}_{a}^{0,\text{pol}}}{dx_{a}}$
$\displaystyle=$
$\displaystyle\frac{1}{1-2\omega}\left[\delta_{ab}\delta(1-x_{a})+\frac{\alpha_{s}(\mu_{R})}{2\pi}\left(P_{ab}(x_{a})\ln\frac{m_{t}^{2}}{\mu_{F}^{2}}+C_{F}C_{a}^{0,\text{pol}}(x_{a})\right)\right],$
$\displaystyle\frac{1}{\hat{\Gamma}_{b,\text{LO}}^{\text{pol}}}\,\frac{d\hat{\Gamma}_{a}^{-,\text{pol}}}{dx_{a}}$
$\displaystyle=$
$\displaystyle\frac{-2\omega}{1-2\omega}\left[\delta_{ab}\delta(1-x_{a})+\frac{\alpha_{s}(\mu_{R})}{2\pi}\left(P_{ab}(x_{a})\ln\frac{m_{t}^{2}}{\mu_{F}^{2}}+C_{F}C_{a}^{-,\text{pol}}(x_{a})\right)\right],$
$\displaystyle\frac{1}{\hat{\Gamma}_{b,\text{LO}}^{\text{pol}}}\,\frac{d\hat{\Gamma}_{a}^{+,\text{pol}}}{dx_{a}}$
$\displaystyle=$
$\displaystyle\frac{4\omega}{(1-2\omega)(1-\omega)^{2}}\frac{\alpha_{s}(\mu_{R})}{2\pi}C_{F}C_{a}^{+,\text{pol}}(x_{a}),$
(11)

where

$\displaystyle P_{qq}(x)$ $\displaystyle=$ $\displaystyle
C_{F}\left(\frac{1+x^{2}}{1-x}\right)_{+},$ $\displaystyle P_{gq}(x)$
$\displaystyle=$ $\displaystyle C_{F}\frac{1+(1-x)^{2}}{x}$ (12)

are the timelike $q\to q$ and $q\to g$ splitting functions at LO.

For $a=b$, we have

$\displaystyle C_{b}^{0,\text{pol}}(x)$ $\displaystyle=$
$\displaystyle-\delta(1-x)\big{[}2\ln\omega\ln(1-\omega)+4\mathop{\mathrm{Li}}\nolimits_{2}(\omega)+\frac{2\omega}{1-\omega}\ln\omega+6\big{]}+2(1+x^{2})\bigg{(}\frac{\ln(1-x)}{1-x}\bigg{)}_{+}$
(13)
$\displaystyle{}+2\frac{1+x^{2}}{(1-x)_{+}}\ln(x(1-\omega))+\frac{8(x-2)(1-x)^{2}}{x^{2}(Sx^{2}-2x+2)}+2\omega(1-x)-\frac{12}{x}+\frac{8}{x^{2}}+\frac{6}{1-x}-\frac{(1+x)^{2}}{1-x}-\frac{1+x^{2}}{1-x}R_{1}$
$\displaystyle{}+\bigg{(}1-|1-2x+2Sx^{2}|\bigg{)}\bigg{(}6\frac{1-x}{x^{2}}-\frac{1}{(1-x)(1-\omega)}+6\frac{(1-x)^{2}(x-2)^{2}}{x^{2}(Sx^{2}-2x+2)^{2}}+\frac{1}{2x^{2}(Sx^{2}-2x+2)}\big{[}9x^{3}$
$\displaystyle{}-57x^{2}+97x-\frac{1}{1-x}-47\big{]}\bigg{)}+\frac{R_{2}}{S^{3/2}(Sx^{2}-2x+2)^{5/2}}\bigg{(}x^{3}S^{4}(-x^{3}+7x^{2}-22x+10+\frac{2}{1-x})+xS^{3}(x^{4}$
$\displaystyle{}+4x^{3}+25x^{2}-32x+12)-2(1-x)^{2}+S^{2}(1-x)(5x^{3}+21x^{2}-12x+4)-S(1-x)(9x^{2}-4x-4)\bigg{)},$
$\displaystyle C_{b}^{-,\text{pol}}(x)$ $\displaystyle=$
$\displaystyle-\delta(1-x)\big{[}2\ln\omega\ln(1-\omega)+4\mathop{\mathrm{Li}}\nolimits_{2}(\omega)+\frac{2\omega}{1-\omega}\ln\omega+6+\frac{1-\omega}{\omega}\ln(1-\omega)\big{]}+2(1+x^{2})\bigg{(}\frac{\ln(1-x)}{1-x}\bigg{)}_{+}$
(14)
$\displaystyle{}+2\frac{1+x^{2}}{(1-x)_{+}}\ln(x(1-\omega))-\frac{1+x^{2}}{2(1-x)}R_{1}+\frac{(1-S)^{2}}{\omega
S}+\frac{\omega+Sx}{\omega}-\frac{\omega}{2S(1-x)}+\frac{(1-S-Sx)}{S}\ln(1-2Sx)$
$\displaystyle{}-\frac{3S(2x-1)+6(1-x)}{2S(Sx^{2}-2x+2)^{2}}+\frac{S^{2}(1+3x)+4x\omega-3S}{2\omega
S(Sx^{2}-2x+2)}+B_{3}|1-2x+2Sx^{2}|+B_{1}\ln|2Sx^{2}-2x+1|$
$\displaystyle{}+\frac{R_{3}}{S^{2}\sqrt{\omega}}B_{2}-\frac{R_{2}}{2\sqrt{Sx^{2}-2x+2}}\bigg{(}x^{2}\sqrt{S}+\frac{(S-1)x}{\sqrt{S}}-\frac{2\sqrt{S}}{1-x}+\frac{2S^{2}-5S+7}{S^{\frac{3}{2}}}$
$\displaystyle{}+\frac{Sx(4S^{2}-23S+16)-S(28-19S)+6}{S^{\frac{5}{2}}(Sx^{2}-2x+2)}-\frac{3Sx(12-9S+S^{2})+15S(S-2)+12(1-x)}{S^{\frac{5}{2}}(Sx^{2}-2x+2)^{2}}\bigg{)},$
$\displaystyle C_{b}^{+,\text{pol}}(x)$ $\displaystyle=$
$\displaystyle\frac{(1+x^{2})R_{1}}{1-x}S^{2}-\frac{2(1+x)S^{3}}{\omega}+2S\big{[}1+Sx+(1-S-Sx)\ln(1-2Sx)\big{]}$
(15)
$\displaystyle{}-\frac{(17-21x)S^{3}+(20x-27)S^{2}+4(2-x)S}{\omega(Sx^{2}-2x+2)}+\frac{6(1-x)S-3(1-2x)S^{2}}{(Sx^{2}-2x+2)^{2}}+\frac{\omega
S}{1-x}+\frac{2R_{3}}{\sqrt{\omega}}B_{2}$
$\displaystyle{}-2S^{2}B_{3}|1-2x+2Sx^{2}|+2S^{2}B_{1}\ln|2Sx^{2}-2x+1|-\frac{R_{2}(1-x)\sqrt{S}}{x^{3}\sqrt{Sx^{2}-2x+2}}\bigg{(}11+7x^{2}-16x$
$\displaystyle{}+\frac{6(2-x)(1-x)^{3}}{(2-2x+Sx^{2})^{2}}+\frac{16x^{3}-58x^{2}+69x-28}{2-2x+Sx^{2}}+\frac{Sx^{2}(x^{2}+5x-4)}{1-x}+\frac{S^{2}x^{4}(1+x^{2})}{(1-x)^{2}}\bigg{)},$

where $S=(1-\omega)/2$,

$\displaystyle R_{1}$ $\displaystyle=$
$\displaystyle\ln\bigg{[}(1-S)x^{2}-x+\frac{1}{2}+\frac{1}{2}|2Sx^{2}-2x+1|\bigg{]},$
$\displaystyle R_{2}$ $\displaystyle=$
$\displaystyle\ln\bigg{[}(1-x)(1-3Sx)+Sx^{2}(1-2Sx)+\sqrt{S(Sx^{2}-2x+2)}|2Sx^{2}-2x+1|\bigg{]}-$
$\displaystyle\ln\bigg{[}1+(S-1)x+\sqrt{S(Sx^{2}-2x+2)}\bigg{]},$
$\displaystyle R_{3}$ $\displaystyle=$
$\displaystyle\ln\big{[}1-x(1-\sqrt{\omega})\big{]}-\ln\big{|}1-x(1+\sqrt{\omega})|,$
$\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle
x-\frac{1-S}{S}-\frac{1-x}{2S(2-2x+Sx^{2})^{2}}\big{[}5S^{2}x^{2}-2(1-x)-2S(x^{2}+3x-3)\big{]},$
$\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle-\omega
S+xS^{2}+\frac{Sx(-4S^{2}+7S-2)+S(8-10S)-2}{2(Sx^{2}-2x+2)}-\frac{\omega(S^{2}x-4Sx+3S+2x-2)}{(Sx^{2}-2x+2)^{2}},$
$\displaystyle B_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2S(1-x)}+\frac{1}{1-2Sx}+\frac{6(1-x)-3S(1-2x)}{2S(Sx^{2}-2x+2)^{2}}+\frac{S^{2}(3x-5)+S(11-2x)-4}{2\omega
S(Sx^{2}-2x+2)}.$ (16)

For $a=g$, we have

$\displaystyle C_{g}^{0,\text{pol}}(x)$ $\displaystyle=$
$\displaystyle\frac{1+(1-x)^{2}}{x}\bigg{(}-R_{1}+2\ln[x(1-\omega)(1-x)]\bigg{)}+\bigg{(}8S-x-26+\frac{13}{S}+\frac{26S^{2}-38S+13}{2S^{3}x^{2}}$
(17)
$\displaystyle{}+\frac{-44S^{2}+88S-35}{2xS^{2}}\bigg{)}R_{4}+\bigg{(}\frac{1-6S}{2(2Sx-1)}+9\frac{1-S}{2Sx}+\frac{-4S^{2}+25S-13}{2S^{2}x^{2}}+\frac{\omega}{2(1-2Sx)^{2}}\bigg{)}|2Sx^{2}-2x+1|$
$\displaystyle{}+2(7-6S)+(1-4S)x+7\frac{7S-5}{2Sx}+\frac{4S^{2}-25S+13}{2S^{2}x^{2}},$
$\displaystyle C_{g}^{-,\text{pol}}(x)$ $\displaystyle=$
$\displaystyle\frac{1+(1-x)^{2}}{2x}\bigg{(}4\ln[x(1-\omega)(1-x)]-R_{1}\bigg{)}+\frac{1-6S}{4\omega}x-\frac{\omega^{2}}{32S^{3}x^{2}(1-2Sx)^{2}}-\frac{(10S-7)\omega}{32S^{3}x^{2}(1-2Sx)}$
(18)
$\displaystyle{}+\frac{1+5S-8S^{2}}{2S\omega}-A_{3}R_{4}+\frac{|2Sx^{2}-2x+1|}{8\omega
S^{2}x^{2}}A_{5}+\frac{A_{2}}{2S^{3}x^{2}}\ln(1-2Sx)+\frac{R_{3}A_{4}}{2\sqrt{\omega}S^{3}x^{2}}-A_{1}\frac{\ln|2Sx^{2}-2x+1|}{2x^{2}S^{3}}$
$\displaystyle{}+\frac{1}{16\omega
xS^{2}}(48S^{3}+72S^{2}-50S-5)+\frac{1}{16\omega
S^{3}x^{2}}(32S^{3}-88S^{2}+44S-3),$ $\displaystyle C_{g}^{+,\text{pol}}(x)$
$\displaystyle=$
$\displaystyle\frac{1}{2Sx^{2}}\Big{[}\frac{2R_{3}}{\sqrt{\omega}}A_{4}+2x(1+(1-x)^{2})S^{3}R_{1}+4S^{3}x^{2}A_{3}R_{4}+2A_{2}\ln(1-2Sx)-2A_{1}\ln|2Sx^{2}-2x+1|$
(19)
$\displaystyle{}-\frac{SA_{5}}{2\omega}|2Sx^{2}-2x+1|+\frac{S}{4\omega}\bigg{(}4S^{2}(1+2S)x^{3}+8S(4S^{2}-7S+1)x^{2}+\frac{8\omega^{2}x}{S(1-4S)}-(96S^{2}-70S+5)x$
$\displaystyle{}+\frac{32S^{2}-12S-3}{S}-\frac{40S^{3}-68S^{2}+38S-7}{2S(1-2Sx)}+\frac{8S^{3}-12S^{2}+6S-1}{2S(1-2Sx)^{2}}\bigg{)}\Big{]},$

where

$\displaystyle R_{4}$ $\displaystyle=$
$\displaystyle\ln\bigg{[}1-S(-2Sx^{2}+2x+1-|2Sx^{2}-2x+1|)\bigg{]}-\ln[1-2Sx],$
$\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle
2x^{2}(x-1)S^{3}+(-2x^{2}+5x+1)S^{2}-2(2+x)S+2,$ $\displaystyle A_{2}$
$\displaystyle=$ $\displaystyle
x(x^{2}-2)S^{3}+(-2x^{2}+5x+1)S^{2}-2(2+x)S+2,$ $\displaystyle A_{3}$
$\displaystyle=$
$\displaystyle\frac{2+x^{2}}{2x}-\frac{5x^{2}+5x+1}{2Sx^{2}}-\frac{7-S(8+15x)}{4x^{2}S^{3}},$
$\displaystyle A_{4}$ $\displaystyle=$ $\displaystyle
2x(2-x^{2})S^{3}+(2x^{2}-7x-4)S^{2}+2(x+3)S-2,$ $\displaystyle A_{5}$
$\displaystyle=$ $\displaystyle
20S-11-Sx(10S-7)-\frac{\omega(7-10S)}{2(1-2Sx)}+\frac{\omega^{2}}{2(1-2Sx)^{2}}.$
(20)

Adding Eqs. (13)–(15) for $a=b$ and Eqs. (17)–(19) for $a=g$ according to Eq.
(6), we obtain the respective coefficient functions pertaining to the case
where $\theta$ is integrated over. Specifically, we have

$\displaystyle\frac{1}{\hat{\Gamma}_{b,\text{LO}}^{\text{pol}}}\,\frac{d\hat{\Gamma}_{b}^{\text{pol}}}{dx_{b}}$
$\displaystyle=$
$\displaystyle\delta(1-x_{b})+\frac{\alpha_{s}(\mu_{R})}{2\pi}\left(P_{qq}(x_{b})\ln\frac{m_{t}^{2}}{\mu_{F}^{2}}+C_{F}C_{b}^{\text{pol}}(x_{b})\right),$
$\displaystyle\frac{1}{\hat{\Gamma}_{b,\text{LO}}^{\text{pol}}}\,\frac{d\hat{\Gamma}_{g}^{\text{pol}}}{dx_{g}}$
$\displaystyle=$
$\displaystyle\frac{\alpha_{s}(\mu_{R})}{2\pi}\left(P_{gq}(x_{g})\ln\frac{m_{t}^{2}}{\mu_{F}^{2}}+C_{F}C_{g}^{\text{pol}}(x_{g})\right),$
(21)

where

$\displaystyle C_{b}^{\text{pol}}(x)$ $\displaystyle=$
$\displaystyle\delta(1-x)\big{[}-2\ln\omega\ln(1-\omega)+\frac{2(1-\omega)}{1-2\omega}\ln(1-\omega)-4\mathop{\mathrm{Li}}\nolimits_{2}(\omega)-\frac{2\omega}{1-\omega}\ln\omega-6\big{]}$
$\displaystyle{}+2(1+x^{2})\bigg{(}\frac{\ln(1-x)}{1-x}\bigg{)}_{+}+2\frac{1+x^{2}}{(1-x)_{+}}\ln(x(1-\omega))+\frac{1-S}{S(Sx^{2}-2x+2)}-\frac{2}{1-4S}-\frac{\omega}{S(1-x)}-1-x$
$\displaystyle{}-\frac{1+x^{2}}{1-x}R_{1}+\bigg{(}\frac{1}{S(1-x)}+\frac{4\omega}{(1-4S)(1-2Sx)}-\frac{1-S}{S(Sx^{2}-2x+2)}\bigg{)}|2Sx^{2}-2x+1|$
$\displaystyle{}-\frac{R_{2}}{\sqrt{S(Sx^{2}-2x+2)}}\bigg{(}\frac{1}{S}+\frac{2}{1-4S}+\frac{(13S-5-4S^{2})x}{1-4S}-\frac{Sx(x^{2}-x+2)}{1-x}-\frac{(1-S)[1-(1-S)x]}{S(Sx^{2}-2x+2)}\bigg{)},$
$\displaystyle C_{g}^{\text{pol}}(x)$ $\displaystyle=$
$\displaystyle\frac{1+(1-x)^{2}}{x}\bigg{(}-R_{1}+2\ln[x(1-\omega)(1-x)]\bigg{)}-\bigg{(}x-\frac{1+\omega^{2}}{4S^{3}x^{2}}-\frac{8S^{2}-6S+3}{S(4S-1)}-\frac{2-\omega^{2}(2\omega-5)}{2S^{2}(1-4S)x}\bigg{)}R_{4}$
(22)
$\displaystyle{}+\bigg{(}\frac{S-1}{2S^{2}x^{2}}+\frac{\omega}{2(1-2Sx)^{2}}-\frac{12S^{2}-15S+7}{2S(1-4S)x}-\frac{24S^{2}-26S+9}{2(1-4S)(1-2Sx)}\bigg{)}|2Sx^{2}-2x+1|+x+\frac{1-S}{2S^{2}x^{2}}$
$\displaystyle{}-\frac{2}{1-4S}+\frac{12S^{2}-7S+5}{2S(1-4S)x}.$

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