# Top-Quark Decay at Next-to-Next-to-Next-to-Leading Order in QCD

Long Chen<EMAIL_ADDRESS>School of Physics, Shandong University, Jinan
250100, China  Xiang Chen<EMAIL_ADDRESS>School of Physics, Peking
University, Beijing 100871, China  Xin Guan<EMAIL_ADDRESS>School of
Physics, Peking University, Beijing 100871, China  Yan-Qing Ma
<EMAIL_ADDRESS>School of Physics, Peking University, Beijing 100871, China
Center for High Energy Physics, Peking University, Beijing 100871, China

###### Abstract

We present the first complete high-precision QCD corrections to the inclusive
decay width $\mathrm{\Gamma}_{t}$, the $W$-helicity fractions
$f_{\mathrm{L,R,0}}$ and semi-inclusive distributions for the top-quark decay
process $t\rightarrow b+W^{+}+X_{\mathrm{\tiny QCD}}$ at NNNLO in the strong
coupling constant $\alpha_{s}$. In particular, the pure NNNLO QCD correction
decreases the $\mathrm{\Gamma}_{t}$ by about $0.8\%$ of the previous NNLO
result at the top-quark pole mass scale, exceeding the error estimated by the
usual scale-variation prescription. After taking into account all sources of
errors, we get
$\mathrm{\Gamma}_{t}=1.3148^{+0.003}_{-0.005}+0.027\,(m_{t}-172.69)\,\text{GeV}$,
the error of which meets the request by future colliders. On the other hand,
the NNNLO QCD effects on $f_{\mathrm{L,R,0}}$ are found to be much smaller, at
the level of one per-mille for the dominating $f_{0}$, predestining them to
act as precision observables for the top-quark decay process.

Introduction. —  The top-quark $t$, to-date heaviest fundamental particle ever
discovered in experiments, plays a special role both in the precision test of
the Standard Model (SM), especially the electroweak sector, as well as
searching for New Physics. The properties of its production and decay have
been actively studied at the Tevatron and LHC, and will also be among the core
physical programs at future high-energy lepton colliders ILC:2013jhg ;
CLICdp:2018esa ; FCC:2018evy ; CEPCStudyGroup:2018ghi .

The large $t$-quark mass $m_{t}$, currently measured to be $172.69\pm
0.30\,$GeVParticleDataGroup:2022pth  usually interpreted as the pole mass, not
only sets a sufficiently-large scale to apply perturbative Quantum
Chromodynamics (QCD), but also makes for a unique feature relevant for QCD
phenomenology: the $t$-quark decays almost exclusively to $W$ and bottom-quark
before it hadronizes. The current most precise measurement for the $t$-quark
decay width $\mathrm{\Gamma}_{t}$ comes from the CMS CMS:2014mxl  which gives
$1.36\pm 0.02\mathrm{(stat.)}^{+0.14}_{-0.11}\mathrm{(syst.)}\,$GeV. The
anticipated experimental uncertainties in the measurement of
$\mathrm{\Gamma}_{t}$ at the future hadron and lepton colliders can be reduced
to about $20\sim 26\,$MeVHoriguchi:2013wra ; CLICdp:2018esa ; Baskakov:2018huw
; Li:2022iav . To fully taking advantage of the data, theoretical error should
at least smaller than one third of the experimental error, say less than
$7\,$MeV. Although QCD corrections to $t$-quark decay have been calculated
perturbatively up to next-to-next-to leading order (NNLO) in the strong
coupling $\alpha_{s}$, e.g. in refs.Jezabek:1988iv ; Czarnecki:1990kv ;
Li:1990qf ; Czarnecki:1998qc ; Chetyrkin:1999ju ; Fischer:2001gp ;
Blokland:2004ye ; Blokland:2005vq ; Czarnecki:2010gb ; Gao:2012ja ;
Brucherseifer:2013iv ; Meng:2022htg ; Chen:2022wit , we will see later that
theoretical error at NNLO cannot meet this request. It is thus desirable to
explicitly determine the complete QCD corrections at next-to-next-to-next-to
leading order (NNNLO) in $\alpha_{s}$, which we accomplish for the first time
in this Letter.

One particularly interesting phenomenon in $t$-quark decay is that the $W$
produced is polarized even if the $t$-quark is unpolarized, due to the chiral
structure of the weak interaction. The current measurements ATLAS:2022rms  of
fractions of longitudinal ($f_{0}$), left-handed ($f_{\mathrm{L}}$) and right-
handed ($f_{\mathrm{R}}$) polarisation states of $W$ at the ATLAS detector are
found to be $f_{0}=0.684\pm 0.005\,\mathrm{(stat.)}\pm
0.014\,\mathrm{(syst.)}$, $f_{\mathrm{L}}=0.318\pm 0.003\,\mathrm{(stat.)}\pm
0.008\,\mathrm{(syst.)}$ and $f_{\mathrm{R}}=-0.002\pm
0.002\,\mathrm{(stat.)}\pm 0.014\,\mathrm{(syst.)}$. The current uncertainties
in $f_{0}$ and $f_{\mathrm{L}}$ are dominated by the experimental
uncertainties in $m_{t}$, while the uncertainty in $f_{\mathrm{R}}$ is
dominated by uncertainties in $\alpha_{s}$ and $m_{b}$. All these parameters
are expected to be significantly reduced at future lepton colliders
ILC:2013jhg ; CLICdp:2018esa ; FCC:2018evy ; CEPCStudyGroup:2018ghi ;
Vos:2016til ; Schwienhorst:2022yqu , and thus experimental errors of
$f_{\mathrm{L,R,0}}$ can be expected to be comparable or smaller than
theoretical errors based on NNLO QCD calculations Czarnecki:2010gb ;
Gao:2012ja ; Brucherseifer:2013iv .

It is thus also desirable to improve the accuracy of the theoretical
predictions for these precision observables at higher orders in QCD. However,
unlike $\mathrm{\Gamma}_{t}$, these quantities are notoriously difficult to be
predicted to high precision from a differential calculation with Monte Carlo
methods, although the expansion method of ref. Czarnecki:2010gb  does offer a
systematically improvable way. In this Letter we formulate a different
efficient method to tackle this problem, which not only allows us to obtain
high-precision results at one order higher in QCD but also provides us with
the ingredients that would be needed to arrive at a fully differential
calculation of any infrared-safe observables for $t$-quark decay process at
NNNLO later on.

The method. —  Aiming for obtaining NNNLO QCD corrections to the semi-
inclusive distributions related to the $W$ produced in $t\rightarrow
b+W^{+}+X_{\mathrm{\tiny QCD}}$, from which inclusive observables such as
$\mathrm{\Gamma}_{t}$ and $f_{\mathrm{L,R,0}}$ can be composed as well, we
write $t$-quark decay width in term of the semi-inclusive hadronic tensor
$\mathcal{W}^{\,\mu\nu}_{tb}$ integrated over the $W$ momentum $k$ as follows,

$\mathrm{\Gamma}_{t}=\frac{1}{2\,m_{t}}\,\int\frac{\mathrm{d}^{d-1}k}{(2\pi)^{d-1}2E}\,\mathcal{W}_{tb}^{\,\mu\nu}\sum_{\lambda}^{L,R,0}\,\varepsilon^{*}_{\mu}(k,\lambda)\,\varepsilon_{\nu}(k,\lambda)\,,$
(1)

with fixed $t$-quark momentum $p$. For an on-shell $W$ with mass $m_{W}$, the
polarization-sum
$\sum_{\lambda}\,\varepsilon^{*}_{\mu}(k,\lambda)\,\varepsilon_{\nu}(k,\lambda)$
can be reduced to $\big{(}g^{\mu\nu}-k^{\mu}k^{\nu}\big{)}/m_{W}^{2}$. For $W$
with a definite helicity $\lambda$, the corresponding projectors can be found
in ref. Fischer:2000kx  and see also ref. Chen:2019wyb . Anticipating
potential infrared-soft and/or collinear (IR) divergence in the phase-space
integration over $k$ in the region where $E$ reaches its maximum, we have
introduced the dimensional regularization (DR) with the spacetime dimension
denoted by $d\equiv 4-2\epsilon$.

$\mathcal{W}_{tb}^{\,\mu\nu}$ can be decomposed to 5 linearly-independent
Lorentz-tensor structures:

$\displaystyle\mathcal{W}_{tb}^{\,\mu\nu}(p,k)$ $\displaystyle=$
$\displaystyle
W_{1}\,g^{\mu\nu}\,+\,W_{2}\,p^{\mu}p^{\nu}+\,W_{3}\,k^{\mu}k^{\nu}$ (2)
$\displaystyle+$ $\displaystyle
W_{4}\,\big{(}p^{\mu}k^{\nu}\,+\,k^{\mu}p^{\nu}\big{)}\,+\,W_{5}\,i\epsilon^{\mu\nu\rho\sigma}\,p_{\rho}\,k_{\sigma}\,,$

where the Levi-Civita tensor $\epsilon^{\mu\nu\rho\sigma}$ appears due to the
chiral structure of the weak $tbW$-vertex. Since the $\gamma_{5}$ from the
$tbW$-vertex always appears on an open fermion chain of the contributing QCD
amplitudes, $\gamma_{5}$ can be treated fully anticommutively Bardeen:1972vi ;
Chanowitz:1979zu ; Gottlieb:1979ix  in a straightforward manner.

Each form factor $W_{i}$ is a function of $m_{t},\,m_{W}$ and the $W$-energy
$E$, as well as $m_{b}$ in case of a massive bottom-quark, and receives both
virtual and real-radiation type QCD corrections. Loop integrals are reduced to
relatively simpler master integrals using integration-by-parts (IBP)
identities Chetyrkin:1981qh , employing the recently released package Blade
www:Blade  armed with the strategy of block-triangular formLiu:2018dmc ;
Guan:2019bcx  (For other reduction packages on the market, see refs.
Anastasiou:2004vj ; Smirnov:2008iw ; Smirnov:2013dia ; Smirnov:2014hma ;
Smirnov:2019qkx ; Maierhofer:2017gsa ; Maierhofer:2018gpa ; Klappert:2020nbg ;
Lee:2012cn ; Lee:2013mka ; Studerus:2009ye ; vonManteuffel:2012np ; Wu:2023upw
). The resulting master integrals are computed using the differential equation
method Kotikov:1990kg  based on series expansion (See e.g. refs.Remiddi:1997ny
; Caffo:2008aw ; Czakon:2008zk ; Lee:2014ioa ; Moriello:2019yhu ;
Hidding:2020ytt ; Armadillo:2022ugh ). To fix the boundary conditions for
these differential equations, we utilize the auxiliary mass flow method
Liu:2017jxz ; Liu:2020kpc ; Liu:2021wks ; Liu:2022mfb  implemented in AMFlow
Liu:2022chg . The phase-space integrals over the momenta of final-state
particles, except for $k$ being fixed, are treated in the same manner as loop
integrals taking advantage of the reverse unitarity Anastasiou:2002yz ;
Anastasiou:2002qz ; Anastasiou:2003yy . To give an idea of the complexity
encountered in our calculation, the number of integrals in the
$\mathcal{O}(\alpha_{s}^{3})$ corrections to $\mathcal{W}_{tb}^{\,\mu\nu}$ is
about $7\times 10^{4}$, which are reduced to 2988 master integrals to be
solved using the aforementioned method. Armed with all these highly efficient
techniques, we are able to construct a piecewise series expansion
representation (PSE) for each master integral, deeply-expanded up to about 200
orders in $E$, using the differential equation solver in AMFlow. Consequently,
a high-precision result for $\mathcal{W}_{tb}^{\,\mu\nu}$ is obtained in the
form of deeply-expanded PSE.

Here comes a tricky point when completing the phase-space integration of
$\mathcal{W}_{tb}^{\,\mu\nu}$ over $k$ for calculating physical observables:
$W_{i}$ in (2) contain IR divergence in the region with $E$ reaching its
maximum, and the integration over $k$ shall be done with proper regularization
which we take to be DR. To make the calculation easier, the phase-space
integration of $\mathcal{W}_{tb}^{\,\mu\nu}$ over the IR-dangerous regions in
$k$ is done using its PSE with $\epsilon$ assigned with non-zero numbers to
regularize the potential IR divergences Liu:2022mfb ; Liu:2022chg . The fit
regarding the $\epsilon$-dependence is done only at the very end for the final
finite physical objects of interest, which can be $\mathrm{\Gamma}_{t}$ and
$f_{\mathrm{L,R,0}}$ as well as distributions.

We validated the above method using the simplest inclusive observable of the
process, the $\mathrm{\Gamma}_{t}$, by comparing the numerical results
composed from $\mathcal{W}_{tb}^{\,\mu\nu}$, to be presented below, against
those calculated by applying the optical theorem (where the integration over
$k$ can be done using the reversed unitarity Anastasiou:2002yz ;
Anastasiou:2002qz ; Anastasiou:2003yy ). We find a perfect agreement between
the two method within our high numerical precision.

The inclusive decay width. —  We are now ready to present our numerical
results and begin with $\mathrm{\Gamma}_{t}$. The QCD effects on
$\mathrm{\Gamma}_{t}$ in SM can be parameterized as

$\mathrm{\Gamma}_{t}=\Gamma_{0}\,\Big{[}\mathbf{c}_{0}+\frac{\alpha_{s}}{\pi}\mathbf{c}_{1}+\Big{(}\frac{\alpha_{s}}{\pi}\Big{)}^{2}\mathbf{c}_{2}+\Big{(}\frac{\alpha_{s}}{\pi}\Big{)}^{3}\mathbf{c}_{3}+\mathcal{O}(\alpha_{s}^{4})\Big{]}\,,$
(3)

where we introduced a prefactor
$\Gamma_{0}\equiv\frac{G_{F}\,\,m_{W}^{2}\,m_{t}\,|V_{tb}|^{2}}{12\sqrt{2}}$
with $G_{F}$ the Fermi constant, the CKM matrix element $V_{tb}$ taken to be 1
in the following numerical results. The perturbative coefficients
$\mathbf{c}_{i}$ are functions of the kinematic ratio $m_{W}^{2}/m_{t}^{2}$ in
the limit $m_{b}=0$, and also the renormalization scale $\mu$ in general.
Below we present the numerical results explicitly in a scheme where $m_{t}$ is
renormalized in the on-shell scheme and $\alpha_{s}$ is
$\overline{\mathrm{MS}}$-renormalized with 5 massless quark flavors.

At the scale $\mu=m_{t}/2$, close to the kinetic energy $m_{t}-m_{W}-m_{b}$ of
the final state, we obtain

$\displaystyle\mathbf{c}_{0}=1.93851\,,\quad\mathbf{c}_{1}=-4.85519\,,$
$\displaystyle\mathbf{c}_{2}=-21.2260\,,~{}\mathbf{c}_{3}=-174.265\,,$ (4)

using the SM input parameters $m_{t}=172.69$ GeV and $m_{W}=80.377$ GeV. In
(Top-Quark Decay at Next-to-Next-to-Next-to-Leading Order in QCD) we have
truncated our internal high-precision results for the perturbative
coefficients $\mathbf{c}_{i}$ to the first 6 digits for the sake of display
(Results with higher precision can be obtained from the supplementary file
associated with this Letter.) Up to NNLO our result (Top-Quark Decay at Next-
to-Next-to-Next-to-Leading Order in QCD) agree with those in refs.
Blokland:2004ye ; Blokland:2005vq ; Brucherseifer:2013iv ; Chen:2022wit . With
$G_{F}=1.166379\times 10^{-5}$ GeV-2 and $\alpha_{s}(\mu=m_{t}/2)\approx
0.1189$, obtained by solving the renormalization-group equation for the
running $\alpha_{s}$ at four-loop order vanRitbergen:1997va ; Chetyrkin:1999pq
; Czakon:2004bu ; Chetyrkin:2004mf  with input $\alpha_{s}(\mu=m_{Z})=0.1179$
at the Z-pole mass $m_{Z}=91.1876$ GeV, we further obtain

$\displaystyle\mathrm{\Gamma}_{t}$ $\displaystyle=$ $\displaystyle
1.48642-0.140877-0.023305-0.007240\,\,\text{GeV}$ (5) $\displaystyle=$
$\displaystyle 1.31500\,\,\text{GeV}\,,$

where the first line provides the decomposition of the total NNNLO result
according to the $\alpha_{s}$ order. Therefore the QCD corrections continue to
decreases the Born-level result for $\mathrm{\Gamma}_{t}$ up to
$\mathcal{O}(\alpha_{s}^{3})$.

Results for $\mathrm{\Gamma}_{t}$ at other scales can be readily derived from
(5) using the renormalization-group equation method, which are provided as a
function in the supplementary file associated with this Letter where the value
of $m_{t}$ can be changed as well. In FIG. 1 we plot the scale dependence of
our results for $\mathrm{\Gamma}_{t}$ for $\mu/m_{t}\in[0.1,1]$ up to NNNLO.

Figure 1: The scale dependence of the fixed-order results for
$\mathrm{\Gamma}_{t}$ in $\mu/m_{t}\in[0.1,1]$

It can be found that the scale dependence has been further improved at NNNLO,
as expected. However, due to the turning point of the NNLO green curve (at
about $\mu/m_{t}=0.14$) in FIG. 1, its scale variation can never cover the
NNNLO result at any scales less than $\mu/m_{t}=0.6$, including the central
value $\mu/m_{t}=0.5$ chosen in this Letter. This clearly demonstrates that
the NNLO results can underestimate the theoretical error by simply studying
the $\mu$ dependence, and thus determining the $\mathcal{O}(\alpha_{s}^{3})$
corrections explicitly is very important.

We close this discussion by a quick remark that at the upper scale
$\mu=m_{t}$, typically chosen in literature, the pure
$\mathcal{O}(\alpha_{s}^{3})$ correction decreases $\mathrm{\Gamma}_{t}$ by
about $10\,$MeV of the previous NNLO result at $\mu=m_{t}$, significantly
exceeding the usual NNLO scale uncertainty.

The $W$-helicity fractions and angular asymmetries. —  As the first example
beyond the inclusive $\mathrm{\Gamma}_{t}$ at NNNLO, we consider the
$W$-helicity fractions $f_{\mathrm{L,R,0}}$ resulting from a $t$-quark decay,
unequal between each other due to the chiral structure of the weak
$tbW$-vertex in SM.

To present our results for QCD corrections to $f_{\mathrm{L,R,0}}$, following
from their very definitions, we introduce further $f^{[n]}_{\lambda}$ accurate
to the $n$-th order in $\alpha_{s}$ as in ref. Czarnecki:2010gb ,
$f^{[n]}_{\lambda}=\frac{\sum_{i=0}^{n}\Gamma^{[n]}_{\lambda}}{\sum_{i=0}^{n}\Gamma^{[n]}_{t}}$
where $\Gamma^{[n]}_{t}=\sum_{\lambda=0,L,R}\Gamma^{[n]}_{\lambda}$ is the
total top-decay width at $\mathcal{O}(\alpha_{s}^{n})$, expressed as a sum
over partial width $\Gamma^{[n]}_{\lambda}$ with a polarized $W$ defined by
$\Gamma^{[n]}_{\lambda}=\frac{1}{2\,m_{t}}\int\frac{\mathrm{d}^{d-1}k}{(2\pi)^{d-1}2E}\,\mathcal{W}_{tb}^{\,\mu\nu}\varepsilon^{*}_{\mu}(k,\lambda)\,\varepsilon_{\nu}(k,\lambda)$
at $\mathcal{O}(\alpha_{s}^{n})$ with the projector for
$\varepsilon^{*}_{\mu}(k,\lambda)\,\varepsilon_{\nu}(k,\lambda)$ taken from
ref. Fischer:2000kx .

Choosing default parameters in the last section, we obtain the following
numerical results for the NNNLO QCD corrections:

$\displaystyle f_{0}^{[3]}$ $\displaystyle=$ $\displaystyle
0.697706-0.008401-0.001954-0.000613\,,$ $\displaystyle=$ $\displaystyle
0.686737\,,$ $\displaystyle f_{\mathrm{L}}^{[3]}$ $\displaystyle=$
$\displaystyle 0.302294+0.007254+0.001799+0.000586\,,$ $\displaystyle=$
$\displaystyle 0.311933\,,$ $\displaystyle f_{\mathrm{R}}^{[3]}$
$\displaystyle=$ $\displaystyle 0.+0.001147+0.000155+0.000027\,,$ (6)
$\displaystyle=$ $\displaystyle 0.001330\,,$

at $\mu=m_{t}/2$. When evaluated at $\mu=m_{t}$, our results up to NNLO are in
good agreement with the numbers given in ref. Czarnecki:2010gb . Defined as
ratios, the scale uncertainties of $f_{\mathrm{L,R,0}}$ are naturally quite
small.

The three $f_{\mathrm{L,R,0}}$ with numerical results in (Top-Quark Decay at
Next-to-Next-to-Next-to-Leading Order in QCD) govern the semi-inclusive
$\cos{\theta^{*}}$ angular distribution through Aguilar-Saavedra:2015yza

$\frac{1}{\mathrm{\Gamma}_{t}}\frac{\mathrm{d}\mathrm{\Gamma}_{t}}{\mathrm{d}\cos\theta^{*}}=\frac{3}{4}(\sin^{2}\theta^{*})\,f_{0}+\frac{3}{8}(1-\cos\theta^{*})^{2}\,f_{\mathrm{L}}+\frac{3}{8}(1+\cos\theta^{*})^{2}\,f_{\mathrm{R}}\,,$

where $\theta^{*}$ is the angle between the momentum direction of the charged
lepton from the $W$-decay in the $W$-rest frame and the $W$-momentum direction
in the $t$-quark rest frame. In FIG. 2 we plot our results for the
$\cos{\theta^{*}}$ angular distribution up to NNNLO in QCD, confined to the
window $\pi/4<\theta^{*}<3\pi/4$ for a better display.

Figure 2: Angular distribution of the charged lepton from $t$-quark decay in
the $W$ rest frame.

For reducing measurement uncertainties when extracting results on
$f_{\mathrm{L,R,0}}$, ref. Aguilar-Saavedra:2006qvv  suggested a generalized
angular asymmetry based on this $\cos{\theta^{*}}$ distribution,

$\mathcal{A}_{z}=\frac{\mathrm{N}(\cos{\theta^{*}}>z)-\mathrm{N}(\cos{\theta^{*}}<z)}{\mathrm{N}(\cos{\theta^{*}}>z)+\mathrm{N}(\cos{\theta^{*}}<z)},$

where
$\mathrm{N}(\cos{\theta^{*}}>z)=\int_{z}^{1}\frac{\mathrm{d}\mathrm{\Gamma}_{t}}{\mathrm{d}\cos\theta^{*}}\,\mathrm{d}\cos\theta^{*}$,
propartional to the number of charged leptons in $t$-quark decay observed with
$\cos{\theta^{*}}$ larger than a given value $z$ (and similarly for
$\mathrm{N}(\cos{\theta^{*}}<z)$). The so-called forward-backward asymmetry of
the charged lepton in $\theta^{*}$ with respect to $\theta^{*}=\pi/2$
corresponds to the special case of $\mathcal{A}_{z}$ at $z=0$, which equals to
$\frac{3}{4}\big{(}f_{\mathrm{L}}-f_{\mathrm{R}}\big{)}$. Our results (Top-
Quark Decay at Next-to-Next-to-Next-to-Leading Order in QCD) can be readily
used to determine the QCD corrections to these quantities up to NNNLO.

The $W$-energy distribution. —  Having the results for
$\mathcal{W}_{tb}^{\,\mu\nu}$, we are able to calculate the NNNLO QCD
corrections to the $W$-energy distribution in $t$-quark decay observed in the
$t$-quark rest frame, shown in FIG. 3.

Figure 3: The $W$-energy distribution in $t$-quark decay observed in the
$t$-quark rest frame.

The distribution of $W$-energy $E$ increases as $E$ becomes larger, and
becomes singular in the limit of $E$ reaching its maximum $E_{\mathrm{max}}$
where the QCD radiations have to be either soft and/or collinear to the
bottom-quark. The fixed-order prediction breaks down in this region, and we
take an average over a $1\,$GeV-bin in the rightmost end of the distribution,
where the QCD corrections up to $\mathcal{O}(\alpha_{s}^{3})$ decrease the
Born-level result in a way similar to $\mathrm{\Gamma}_{t}$.

From the lower panel of FIG. 3, one sees that the QCD corrections are positive
and quite sizable, in particular, the pure $\mathcal{O}(\alpha_{s}^{3})$
correction modifies the lowest order result by about $7\sim 14\%$ for
$E\in[94,104]\,$GeV. The scale variation of the next-to-leading (NLO) curve is
solely determined by the change of $\alpha_{s}$, and thus the band is
independent of $E$. Starting from NNLO, the appearance of large logarithm
$\ln\big{(}(E_{\mathrm{max}}-E)/E\big{)}$ leads to the increase of corrections
as $E$ approaches $E_{\mathrm{max}}$. At NNNLO, the change in the scale-
variation band in this region becomes more visible due to a poor convergence
of the perturbative series contaminated by the aforementioned large
logarithmic structure.

Miscellaneous effects. —  With the expression for $\mathrm{\Gamma}_{t}$ as a
deeply-expanded PSE w.r.t. $m_{W}^{2}/m_{t}^{2}$ at our hand, valid in the
whole physical region, we can investigate the so-called off-shell $W$ effect
up to $\mathcal{O}(\alpha_{s}^{3})$. The standard definition of
$\tilde{\Gamma}_{t}$ with off-shell $W$ effect (See e.g. ref.Jezabek:1988ja )
is obtained by replacing the $W$ propagator by a Breit-Wigner distribution,
where the effective mass of $W$ is essentially the invariant-mass of the
lepton pair produced from $W$-decay. $\tilde{\Gamma}_{t}$ assumes a similar
perturbative parametrization as in (3) but with the $i$-th order coefficient
denoted as $\tilde{\mathbf{c}}_{i}$ for distinction. With $W$ total decay
width $\Gamma_{W}=2.085$ GeV,
$\delta_{i}\equiv\big{(}\tilde{\mathbf{c}}_{i}-\mathbf{c}_{i}\big{)}/\mathbf{c}_{i}$
are found to be quite small and decrease, albeit very slowly, as the
$\alpha_{s}$-order increases: $\delta_{i}$ takes
$-1.54\%,-1.53\%,-1.38\%,-1.23\%$ respectively for $i=0,1,2,3$.

For the finite bottom-quark mass effect, we denote the $i$-th order
coefficient for $\mathrm{\Gamma}_{t}$ with a non-zero $m_{b}$ as
$\mathbf{c}^{m_{b}}_{i}$. With $m_{b}=4.78\,$GeV we find that
$\big{(}\mathbf{c}^{m_{b}}_{1}-\mathbf{c}_{1}\big{)}/\mathbf{c}_{1}\approx\big{(}\mathbf{c}^{m_{b}}_{2}-\mathbf{c}_{2}\big{)}/\mathbf{c}_{2}\approx-1.47\%$.
This strongly indicates that the small non-zero $m_{b}$ effect at
$\mathcal{O}(\alpha_{s}^{3})$ may observe a similar small ratio, well-below
sub-per-mille level for the total $\mathrm{\Gamma}_{t}$.

Taking into account the aforementioned finite $m_{b}$ and off-shell $W$
effects, as well as the NLO electroweak corrections Bohm:1986rj ;
Denner:1990cpz ; Do:2002ky , we obtain our final result for
$\mathrm{\Gamma}_{t}$:

$\displaystyle\mathrm{\Gamma}_{t}=1.3148^{+0.003}_{-0.005}+0.027\,(m_{t}-172.69)\,\text{GeV}\,,$
(7)

where the second term parameterizes the main source of error on
$\mathrm{\Gamma}_{t}$ originated from the experimental uncertainty of the
input $t$-quark mass value. The numbers in the super- and sub-script of (7)
correspond to a conservative estimate of the QCD scale uncertainty obtained by
varying $\mu/m_{t}$ in $[0.1,1]$.

Being ratios, the $m_{b}$ effects on $f_{\mathrm{L,R,0}}$ are even smaller,
below one per-mille, except for the small $f_{\mathrm{R}}$ whose Born-level
expression vanishes at $m_{b}=0$. After incorporating the NLO electroweak
correction Do:2002ky  our final results for $f_{\mathrm{L,R,0}}$ read

$\displaystyle f_{0}^{[3]}$ $\displaystyle=0.686^{+0.002}_{-0.003}\,,~{}$
$\displaystyle f_{\mathrm{L}}^{[3]}$
$\displaystyle=0.312^{+0.001}_{-0.002}\,,~{}$ $\displaystyle
f_{\mathrm{R}}^{[3]}$ $\displaystyle=0.00157^{+0.00002}_{-0.00002}\,.$ (8)

Since the scale uncertainties of $f_{\mathrm{L,R,0}}$ are naturally quite
small due to them being ratios, the errors for the above results, given in
numbers in super- and sub-script are obtained by adding the errors of the
following two main sources: the conservative QCD-scale-uncertainty error for
$\mathrm{\Gamma}_{t}$ determined in (7) and the error induced by the input
top-quark mass $172.69\pm 0.30\,$GeV and bottom-quark mass
$4.78^{+0.02}_{-0.03}\,$GeVParticleDataGroup:2022pth .

Our results for $\mathrm{\Gamma}_{t}$ and $f_{\mathrm{L,R,0}}$ in (7) and
(Top-Quark Decay at Next-to-Next-to-Next-to-Leading Order in QCD) respectively
are thus sufficient to meet the potential precision requirement on the
theoretical predictions for these observables to be measured at the future
hadron and lepton collidersHoriguchi:2013wra ; CLICdp:2018esa ;
Baskakov:2018huw ; Li:2022iav , as quoted in the Introduction.

Conclusion. —  We have perfomed the complete NNNLO QCD corrections to the
inclusive $t$-decay width $\mathrm{\Gamma}_{t}$, $W$-helicity fractions
$f_{\mathrm{L,R,0}}$ as well as the related asymmetries and $W$-energy
distribution in this Letter, in which the off-shell-$W$ and finite $m_{b}$
effects are addressed as well. At $\mu=m_{t}/2$, our best prediction for
$\mathrm{\Gamma}_{t}$ reads $\mathrm{\Gamma}_{t}=1.3148\,$GeV with a
conservative QCD scale-uncertainty corresponding to $[+3,-5]\,$MeV. The QCD
effects in $f_{\mathrm{L,R,0}}$ are found to be quite small and thus the
corresponding theoretical error is very small. For the $W$-energy
distribution, we see that it receives quite sizable QCD corrections. These
results represent the most precise theoretical predictions and will be useful
for the purpose of testing the SM and probing New Physics. Moreover, the
$W$-energy distribution can serve as an ingredient to arrive at a fully
differential calculation of any infrared-safe observables for $t$-quark decay
process at NNNLO in QCD where only NNLO IR-subtraction terms are needed.

The calculations accomplished in this Letter have demonstrated the efficiency
of our method formulated. It can be readily applied to the decay of polarized
$t$-quarks at NNNLO, where phenomenologically more interesting observables can
be constructed, as well as to the mixed QCD-electroweak corrections needed to
achieve a reliable sub-percent level phenomenological analysis for this
process. Needless to say, our method can be equally applied in studies of
other heavy-to-light quark decays, in particular the B-meson semi-leptonic
decay.

###### Acknowledgements.

Acknowledgments. —  The work of L.C. was supported by the Natural Science
Foundation of China under contract No. 12205171, No. 12235008. The work of
X.C., X.G. and Y.Q.M. were supported in part by the National Natural Science
Foundation of China (Grants No. 11875071, No. 11975029), the National Key
Research and Development Program of China under Contracts No. 2020YFA0406400.
We acknowledge the computational support from the High-performance Computing
Platform of Shandong University and Peking University. Note added: Parallel to
the calculations done in this Letter, the leading-color part of the $t$-quark
inclusive decay width $\mathrm{\Gamma}_{t}$ was computed independently in
CLLWWW , and a perfect agreement was found regarding this piece.

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# Digraphs with exactly one Eulerian tour

Luz Grisales1, Antoine Labelle2, Rodrigo Posada1, Stoyan Dimitrov3 1
Massachusetts Institute of Technology  2 Collège de Maisonneuve  3 University
of Illinois at Chicago<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We give two combinatorial proofs of the fact that the number of loopless
directed graphs (digraphs) on the vertex set $[n]$ with no isolated vertices
and with exactly one Eulerian tour up to a cyclic shift is
$\frac{1}{2}(n-1)!C_{n}$, where $C_{n}$ denotes the $n$-th Catalan number. We
construct a bijection with a set of labeled rooted plane trees and with a set
of valid parenthesis arrangements.

##  1\. Introduction and main facts

Richard Stanley has a list containing nearly 250 problems and facts, for which
he asks of combinatorial proofs [1]. This work describes two such proofs for
one of the problems in the list, namely Problem 199, without a known
combinatorial proof. First, we recall some definitions following [2].

A (finite) directed graph (or digraph) $D$ consists of a vertex set
$V=\\{v_{1},\ldots,v_{n}\\}$ and an edge set $E=\\{e_{1},\ldots,e_{q}\\}$,
together with a function $\phi:V\to V$ determining the direction of each edge.
If $\phi(e)=(u,v)$, then we think of $e$ as an arrow from $u$ to $v$. We will
call $u$ \- initial vertex and $v$ \- final vertex. The outdegree of a vertex
$v$, denoted $outdeg(v)$, is the number of edges of $D$ with initial vertex
$v$. Similarly, the indegree of $v$, denoted $indeg(v)$, is the number of
edges of $D$ with final vertex $v$. A loop is an edge $e$ for which
$\phi(e)=(v,v)$ for some vertex $v$. A digraph is balanced if
$indeg(v)=outdeg(v)$ for each of its vertices $v$. An oriented path in a
digraph $D$ is a sequence of vertices $v_{1},\ldots,v_{m}$, where
$(v_{i},v_{i+1})$ is an edge of $D$ for each $i\in[m-1]$. If the vertices
$v_{1},\ldots,v_{m}$ are all different, then we call the path _simple_. If we
have a simple path and $v_{m}=v_{1}$, then we have an _oriented simple cycle_.

######  Definition 1.

An Eulerian tour in a directed graph $D$ is a sequence of vertices
$a_{1}a_{2}\cdots a_{k}$ such that $(a_{1},a_{2}),\
(a_{2},a_{3}),\cdots,(a_{k-1},a_{k}),\ (a_{k},a_{1})$ are all the distinct
directed edges of $D$.

Any cyclic shift $a_{i}a_{i+1}\cdots a_{k}a_{1}\cdots a_{i-1}$ of an Eulerian
tour is also an Eulerian tour and we will say that these tours are equivalent
up to a cyclic shift.

######  Definition 2.

An Eulerian digraph is a digraph which has no isolated vertices and contains
exactly one Eulerian tour (and its equivalents under cyclic shift).

Our goal is to prove the following claim.

######  Theorem 1.

If $A_{n}$ is the set of loopless Eulerian digraphs on the vertex set $[n]$,
then
$|A_{n}|=\frac{1}{2}(n-1)!C_{n}$ (sequence A102693 in OEIS [5]), where
$C_{n}=\frac{1}{n+1}\binom{2n}{n}$ denotes the $n$-th Catalan number.

For example, $|A_{3}|=5$. Indeed, there are two such digraphs that look like
triangles and three that consist of two 2-cycles with a common vertex (see
Figure 1). The related OEIS sequence, A102693, was created by Richard Stanley.
As a reference, he points out to an unpublished work of him. Thus, we can
assume that Theorem 1 was first proved there.

123132132213321 Figure 1. The five digraphs with three vertices and a unique
Eulerian tour.

It is not difficult to show that every Eulerian graph must be connected and
balanced [2, Theorem 10.1]. The BEST theorem that we recall below gives us a
formula for the total number of Eulerian tours in a digraph. In order to
understand this result, one should be familiar with the term _oriented tree_
(see Figure 2). An oriented tree with root $v$ is a finite digraph $T$ with
$v$ as one of its vertices, such that there is a unique directed path from any
other vertex of $T$ to $v$. This means that the underlying undirected graph
(after we erase all the arrows of the edges of $T$) is a tree.

124356 Figure 2. Example of an oriented tree.

######  Theorem 2 (BEST theorem, [2]).

Let $D$ be a connected balanced digraph with vertex set $V$. Fix an edge $e$
in $V$ and let $v$ be the initial vertex of that edge. Let $\tau(D,v)$ denote
the number of oriented (spanning) subtrees of $D$ with root $v$, and let
$\epsilon(D,e)$ denote the number of Eulerian tours of $D$ starting with the
edge $e$. Then

$\epsilon(D,e)=\tau(D,v)\prod_{u\in V}(outdeg(u)-1)!$

######  Corollary 1 (from Theorem 2).

A digraph $D\in A_{n}$ if and only if

  1. (1)

For every vertex $v$, $D$ has exactly one oriented (spanning) subtree with
root $v$.

  2. (2)

The outdegree of an arbitrary vertex of $D$ is 1 or 2.

Using Corollary 1, we will characterize the digraphs in $A_{n}$ by two other
conditions that will be used later.

######  Lemma 1.

A digraph $D\in A_{n}$ if and only if

  1. i)

There exists a unique oriented simple path between any two vertices of $D$.

  2. ii)

Every vertex of $D$ is part of exactly one or two simple oriented cycles.

###### Proof.

$[$First part: $D\in A_{n}$ $\Longrightarrow$ conditions $i)$ and $ii)]$ Let
$D\in A_{n}$. Let $u$ and $v$ be two arbitrary vertices in $D$. By condition
$(1)$ of Corollary 1, there exists exactly one oriented spanning subtree
$T_{u}$ of $D$ with root $u$. We know that $v$ has to be a vertex of $T_{u}$
and that there exists a unique simple path $\mathcal{P}$ from $v$ to $u$ in
$T_{u}$, which is a simple oriented path in $D$. Assume that there exists
another path $\mathcal{P^{\prime}}\neq\mathcal{P}$ between $v$ and $u$ in $D$.
Begin from $v$ and follow $\mathcal{P}^{\prime}$. Let $f^{\prime}$ be the
first edge in $\mathcal{P}^{\prime}$ which is not part of $\mathcal{P}$ and
let $f$ be the edge in $\mathcal{P}$ with the same initial vertex as
$f^{\prime}$. If you delete $f$ from $T_{u}$ and add $f^{\prime}$ to it, you
will obtain a graph $T^{\prime}_{u}$. One can easily see that $T^{\prime}_{u}$
is an oriented spanning subtree of $D$, different from $T_{u}$ (see Figure 3).
This is a contradiction. Thus, we showed that condition $i)$ holds.

Corollary 1 implies that each vertex of $D$ can be part of at most two simple
oriented cycles since its outdegree is $1$ or $2$. It remains to show that
each vertex of $D$ is part of at least one such cycle. Let $v$ be an arbitrary
vertex of $D$ and let $(u,v)$ be an edge of $D$ (such an edge exists since the
indegree of $v$ is $1$ or $2$). We showed that there is a unique oriented
simple path between $v$ and $u$. This path together with the edge $(u,v)$
forms a simple oriented cycle. Thus condition $ii)$ holds.

12345678911121310$f^{\prime}$$f$ Figure 3. The tree $T_{u}$ in the first part
of the proof of Lemma 1; $u=1$, $v=11$, $\mathcal{P}=11,9,6,5,1$,
$\mathcal{P^{\prime}}=11,9,6,2,1$, $f^{\prime}=(6,2)$ and $f=(6,5)$. Delete
$f$ and add $f^{\prime}$ to obtain another tree $T^{\prime}_{u}$.

$[$second part: conditions $i)$ and $ii)$ $\Longrightarrow$ $D\in A_{n}]$ Take
a digraph $D$ for which conditions $i)$ and $ii)$ hold. We have to show that
conditions $(1)$ and $(2)$ from Corollary 1 also hold. Let $v$ be an arbitrary
vertex of $D$. Condition $ii)$ implies that $outdeg(v)\geq 1$. We will show
that $outdeg(v)<3$. Condition $i)$ implies that no edge of $D$ can be part of
two different simple cycles. Indeed, assume that $(u,w)$ is an edge of $D$,
which is a part of two different simple oriented cycles. Then, we must have at
least two different simple oriented paths between $w$ and $u$, which
contradicts condition $i)$. Now, assume that $outdeg(v)\geq 3$ and let
$(v,u_{1})$, $(v,u_{2})$ and $(v,u_{3})$ are three different edges of $D$. We
know that there exist simple paths between $u_{i}$ and $v$, for $i=1,2,3$.
Thus, $v$ participates in simple cycles through $u_{i}$, for $i=1,2,3$ and no
two of these simple cycles share and edge. Therefore these three cycles are
different. This is a contradiction with condition $ii)$.

It remains to show that condition $(1)$ from Corollary 1 holds. Take an
arbitrary vertex $v$ of $D$. We have a unique oriented simple path from each
of the other vertices of $D$ to $v$. Take the union of these paths. The graph
that you will obtain is an oriented spanning subtree $T_{v}$ of $D$ with root
$v$. Assume that there exists another such subtree $T^{\prime}_{v}$. Then,
since $T_{v}\neq T^{\prime}_{v}$, we must have a vertex $w$ in $D$, for which
the unique oriented path from $w$ to $v$ in $T^{\prime}_{v}$ is different from
the unique oriented path from $w$ to $v$ in $T_{v}$. These are two different
oriented paths from $w$ to $v$ in $D$, which is a contradiction. ∎

######  Corollary 2.

If $D\in A_{n}$, then no pair of cycles in $D$ have an edge in common.

###### Proof.

Assume $C_{1}$ and $C_{2}$ are two different cycles in $D$, which have the
edge $(u,v)$ in common. Then, we will have at least two oriented paths from
$v$ to $u$ in $D$ \- one following $C_{1}$ and another one following $C_{2}$.
This is a contradiction with Corollary 1. ∎

##  2\. Bijection with a set of labeled rooted plane trees

We will construct a bijection between the digraphs in $A_{n}$ and a set of
labeled rooted plane trees on $n+1$ vertices. Let $U_{n}$ be the set of the
unlabeled rooted plane trees with $n+1$ vertices. It is well-known that
$|U_{n}|=C_{n}$ (see [3, Theorem 1.5.1]). Let $L_{n}$ be the set of the
labeled rooted plane trees with $n+1$ vertices such that the root is always
labeled as $0$ and the left-most child of the root is always labeled as $1$.
We have $|L_{n}|=(n-1)!|U_{n}|=(n-1)!C_{n}$. Finally, let
$L^{\prime}_{n}\subset L_{n}$ be the set of the labeled trees in $L_{n}$, such
that the vertex $2$ is in the subtree with root $1$.

First, we define a map $f$ over $L_{n}$, such that
$f(L^{\prime}_{n})=L_{n}\setminus L^{\prime}_{n}$ and $f(f(T))=T$, i.e., an
involution. This shows that $|L^{\prime}_{n}|=|A_{n}|$. Then, we define a map
$g:L^{\prime}_{n}\to A_{n}$, which is shown to be a bijection. Below, we will
denote by $T_{(x,j)}$ the $j$-th child (from left to right) of the vertex $x$
of a tree $T\in L_{n}$.

######  Definition 3.

Let $f:L_{n}\rightarrow L_{n}$ be a map, which switches the places of the
subtree with root $1$ (excluding $1$) and the subtree with root $0$ (excluding
$0$ and the subtree with root $1$), for every tree in $L_{n}$ (see Figure 4).
Formally, if $T\in L_{n}$, then $f(T)$ has the following properties.
For every $j\geq 1$:

  * •

If the vertex $v$ is the (j+1)-th child of the vertex $0$ in $T$, then $v$ is
the j-th child of vertex $1$ in $f(T)$, i.e., $T_{(0,j+1)}=f(T)_{(1,j)}$.

  * •

If the vertex $v$ is the j-th child of the vertex $1$ in $T$, then $v$ is the
(j+1)-th child of vertex $0$ in $f(T)$, i.e., $T_{(1,j)}=f(T)_{(0,j+1)}$.

  * •

All the other directed edges are left the same for both trees, i.e.,
$T_{(u,j)}=f(T)_{(u,j)}$ for $u\not\in\\{0,1\\}$.

$T$236541$0$$f(T)$645321$0$ Figure 4. Example of the action of the map $f$.

Note that $f(L^{\prime}_{n})=L_{n}\setminus L^{\prime}_{n}$ and $f(f(T))=T$,
i.e., $f^{-1}=f$. Therefore,
$|L^{\prime}_{n}|=\frac{|L_{n}|}{2}=\frac{(n-1)!C_{n}}{2}=|A_{n}|$.

######  Definition 4.

Let $g:L^{\prime}_{n}\to A_{n}$ be a map, such that if $T\in L^{\prime}_{n}$,
$g(T)=D^{\prime}$ is a digraph with $V(D^{\prime})=[n]$ and $E(D^{\prime})$,
such that if $x$ is a vertex of $T$ with $r$ children, then:

  1. (1)

For every $i\in[1,r),\ (T_{(x,i)},T_{(x,i+1)})\in E(D^{\prime})$.

  2. (2)

If $x=0$, then $(T_{(x,r)},T_{(x,1)})\in E(D^{\prime})$.

  3. (3)

If $x\not=0$, then $(T_{(x,r)},x)\in E(D^{\prime})$ and $(x,T_{(x,1)})\in
E(D^{\prime})$.

$T$237986541$0$$g(T)$279836541 Figure 5. Example of the action of the map $g$.

######  Lemma 2.

For every $T\in L^{\prime}_{n}$, $g(T)\in A_{n}$, i.e., $g(T)$ has exactly one
Eulerian tour.

###### Proof.

First, note that every vertex $x$ of $g(T)$ belongs to one or two cycles:

  * •

The cycle where $x$ and its parent from $T$ both belong.

  * •

In case $x$ has children in $T$, the cycle formed by $x$ and its children in
$g(T)$.

By Lemma 1, it remains to show that there exists a unique oriented simple path
between any two vertices of $g(T)$, for arbitrary $T\in L^{\prime}_{n}$. To
see this, observe that if $(v,w)\in E(T)$ and $v,w\neq 0$ or if both $v$ and
$w$ are children of the root $0$ in $T$, then we have a unique oriented simple
path between $v$ and $w$ in $g(T)$, which is part of a single cycle. For
instance, the edge $(1,3)$ in the graph $T$ shown at Figure 5 corresponds to
the oriented simple path $1273$ in $g(T)$, whereas the path $514$ in $g(T)$
corresponds to the pair of children $4$ and $5$ of the root of $T$.

We will show that since we have a unique non-oriented path $\mathcal{P}$
between any two vertices $u$ and $v$ in $T$, where $u,v\neq 0$, we will also
have a unique oriented simple path $\mathcal{P}^{\text{or}}$ between $u$ and
$v$ in $g(T)$. If the vertex $0$ is part of $\mathcal{P}$, then we must have
vertices $h_{1}$ and $h_{2}$ in $T$, such that the edges $(h_{1},0)$ and
$(0,h_{2})$ are part of $\mathcal{P}$ (since $v,w\neq 0$). Hence, $h_{1}$ and
$h_{2}$ are two children of the root $0$. Replace the edges $(h_{1},0)$ and
$(0,h_{2})$ of $\mathcal{P}$ with the unique oriented simple path between
$h_{1}$ and $h_{2}$ in $g(T)$. Replace all the other edges of $\mathcal{P}$
with the corresponding oriented simple paths to obtain
$\mathcal{P}^{\text{or}}$. For example, the unique path $\mathcal{P}$ between
$3$ and $8$ in the graph $T$ on Figure 5 is comprised of the edges $(3,1)$,
$(1,7)$, $(7,8)$. The oriented paths corresponding to these edges are $31$,
$127$ and $798$, respectively. The union of these paths, namely $312798$,
gives the unique path $\mathcal{P}^{\text{or}}$ between $3$ and $8$ in $g(T)$.
Another example is the path $71056$ in $T$, which transforms to the path
$73156$ in $g(T)$. ∎

######  Lemma 3.

For every digraph $D\in A_{n}$, there exists a unique labeled tree $T\in
L^{\prime}_{n}$, for which $g(T)=D$. i.e., $g$ has an inverse.

###### Proof.

Let $D$ be an arbitrary digraph in $A_{n}$. Below, we describe the procedure
$build-subtree(r,C,T,D)$ that will be used to obtain the tree $T$, for which
$g(T)=D$. The first argument, $r$, is a vertex in $D$ and the second argument,
$C$, is a cycle in $D$ that contains $r$.

$build-subtree(r,C,T,D)$:
\+ For each $v\in C$:
\+ if $v\neq r$:
\+ add an edge $(r,v)$ to $E(T)$.
if $v$ is part of a cycle $C_{1}\neq C$:
\+ $build-subtree(v,C_{1},T)$. Initially, let $T$ be an empty tree with $n+1$
vertices, i.e., let $V(T)=\\{0,1,\ldots,n\\}$ and let $E(T)=\emptyset$. Take
the vertex with label $1$ in $D$. By Lemma 1, this vertex belongs to one or
two simple oriented cycles. Suppose that the vertex belongs to one such cycle
and let $C$ denotes this cycle. Then, run $build-subtree(1,C,T,D)$ and add an
edge $(0,1)$ to $E(T)$. One can easily show that the resulting graph, $T$, is
a tree in $L^{\prime}_{n}$. First, $T$ is connected since the execution of
$build-subtree(1,C,T,D)$ will reach every vertex of $D$ and connect this
vertex to an already reached vertex. In addition, if we have two different
paths between two vertices $u$ and $v$ of $T$, then we will be able to find
two simple oriented paths between $u$ and $v$ in $D$, which contradicts Lemma
1. Finally, $n\geq 2$ and the only vertex of $T$, which is not in the subtree
with root $1$, is the vertex $0$. Thus, $2$ is in that subtree and $T\in
L^{\prime}_{n}$.

Now, suppose that the vertex $1$ belongs to two different cycles $C_{1}$ and
$C_{2}$. Then, find the unique oriented simple path between $1$ and $2$ in
$D$. This path has to have an edge in common with either $C_{1}$ or $C_{2}$,
but not with both. Otherwise, we will have a contradiction with Corollary 2.
Without lose of generality, let this be $C_{2}$. Execute $build-
subtree(0,C_{1},T,D)$. The graph $T$, obtained at the end, will be a tree in
$L^{\prime}_{n}$ (see Figure 6). ∎

$g(T)$152$C_{2}$463$C_{1}$$T$152364$0$ Figure 6. Example of the action of the
inverse map $g^{-1}$.

Lemmas 2 and 3 imply that the map $g$ is a bijection.

##  3\. A bijection with parentheses arrangements

In this section, we give a second combinatorial proof of Theorem 1, via a
bijection between the digraphs in $A_{n}$ and a set of valid parentheses
arrangements. Suppose that you have $n$ pairs of opening and closing
parenthesis, such that the two parenthesis in each pair are labeled with the
numbers in $[n]$. A valid labeled parentheses arrangement is an ordering of
these $2n$ parenthesis, such that we cannot have two interlaced pairs, e.g.,
$(_{i}(_{j})_{i})_{j}$ for some $i,j\in[n]$. A valid unlabeled parentheses
arrangement is a sequence of unlabeled opening and closing parentheses that
can be obtained by forgetting the labels of a valid labeled arrangement. One
can easily check that a sequence of $n$ opening and $n$ closing parentheses is
valid if and only if every prefix of the sequence has at least as many opening
parentheses as closing parentheses. The number of valid arrangements of $n$
unlabeled pairs of parentheses is $C_{n}$ [3, Theorem 1.5.1]. Thus the number
of such arrangements for labeled pairs is $n!C_{n}$.

To construct a bijection with the set of digraphs $A_{n}$, let us first note
that one can assume that all vertices of an Eulerian digraph have indegree and
outdegree $2$, provided that we allow digraphs with loops.

######  Lemma 4.

If $B_{n}$ is the set of all Eulerian digraphs on the vertex set $[n]$
(possibly with loops) with all vertices of indegree and outdegree $2$, then
$|B_{n}|=|A_{n}|$

###### Proof.

Given a digraph $D$ in $A_{n}$, Corollary 1 implies that all vertices have
outdegree $1$ or $2$ and the same indegree. Moreover, the single Eulerian tour
of $D$ passes exactly once through each vertex of outdegree $1$. Hence, adding
a loop to every vertex of outdegree $1$ gives an element of $B_{n}$.

Conversely, given a digraph $D^{\prime}$ in $B_{n}$, deleting all loops gives
an element of $A_{n}$. Indeed, the loopless digraph still has a unique
Eulerian tour, which is just the tour for $D^{\prime}$ without the loops (the
uniqueness follows because adding back loops must give an Eulerian tour for
$D^{\prime}$). These two maps are inverses of each other and thus give a
bijection between $A_{n}$ and $B_{n}$. ∎

Now, let $B_{n}^{*}$ be the set of digraphs in $B_{n}$ together with an
identified edge. Since all digraphs in $B_{n}$ have $2n$ edges, we have
$|B_{n}^{*}|=2n|B_{n}|=2n|A_{n}|$. We will give a bijection between
$B_{n}^{*}$ and the set of valid arrangements of $n$ labeled pairs of
parentheses, which will show that $2n|A_{n}|=n!C_{n}$, that is,
$|A_{n}|=\frac{1}{2}(n-1)!C_{n}$.

######  Theorem 3.

There exists a bijection between $B_{n}^{*}$ and the set of valid arrangements
of $n$ labeled pairs of parentheses.

###### Proof.

[first part: Digraphs in $B_{n}^{*}$ $\to$ valid parentheses arrangements].

Let $D$ be a digraph in $B_{n}^{*}$ with identified edge $e$. We define a
parentheses arrangement $h(D)$ as follows:

Following the unique Eulerian tour of $D$, starting at $e$, open the $i$ \- th
pair of parentheses when you pass through the vertex $i$ for the first time
and close the $i$ \- th pair of parentheses when you pass through the vertex
$i$ for the second time (see Figure 7 below) .

213 Figure 7. The digraph in $B_{3}^{*}$ (with the edge $2\to 1$ being
identified) that yields the string $(_{1})_{1}(_{2}(_{3})_{3})_{2}$

To show that the resulting string of parentheses is valid, we have to show
that we cannot have two interlaced pairs of parentheses, e.g.,

$\cdots(_{i}\cdots(_{j}\cdots)_{i}\cdots)_{j}\cdots.$

In other words, the unique tour cannot have the form

$i\xrightarrow[]{a}j\xrightarrow[]{b}i\xrightarrow[]{c}j\xrightarrow[]{d}i$

for some walks $a,b,c,d$. But this is clearly impossible, because otherwise we
would have a second Eulerian tour
$i\xrightarrow[]{a}j\xrightarrow[]{d}i\xrightarrow[]{c}j\xrightarrow[]{b}i$.

[second part: Valid parentheses arrangements $\to$ Digraphs in $B_{n}^{*}$]
Given a valid parentheses arrangement $w=(_{x}\cdots)_{y}$, we obtain a
digraph $h^{-1}(w)\in B_{n}^{*}$ by putting an edge between the corresponding
vertices of any pair of consecutive parentheses (from the first parenthesis to
the second) and an edge from $y$ to $x$. The identified edge is $y\to x$.

Clearly, every vertex in $h^{-1}(w)$ has indegree and outdegree $2$ and that
there exists an Eulerian tour $T$, given by the order of the parentheses’
labels in $w$. Hence, we just have to show that $T$ is the unique Eulerian
tour of $h^{-1}(w)$. Let $i\in[n]$ and let

$w=\cdots?_{\ell}(_{i}(_{j}\cdots)_{i}?_{k}\cdots,$

where $?$ represent either a closing or an opening parenthesis (if $(_{i}$ and
$)_{i}$ are consecutive we let $j=i$, if $(_{i}$ is the first parenthesis of
$w$ we let $\ell$ be the label of the last one and if $)_{i}$ is the last
parenthesis, we let $k$ be the label of the first one). We have to show that
if an Eulerian tour enters the vertex $i$ for the first time from $\ell$, this
tour must exit the vertex $i$ towards $j$ and not $k$. Indeed, if this is true
for all $i$, then the Eulerian tour is entirely determined by its first edge.
Thus, this tour and $T$ are equal up to a cyclic shift. Suppose, for the sake
of contradiction, that there exists an Eulerian tour $T^{\prime}$ of
$h^{-1}(w)$, which exits $i$ towards $k$, after entering $i$ for the first
time, through $\ell$.

Note that, by the properties of valid parentheses arrangements, the two
parentheses corresponding to any vertex $v\neq i$ are either both between
$(_{i}$ and $)_{i}$ (then we will say that $v$ is of type $A$) or both outside
(type $B$). Clearly, all edges of the graph $h^{-1}(w)$ with initial vertex of
type $A$ (respectively $B$) have a final vertex either $i$ or of type $A$
(respectively $B$), so the only way to go from a vertex of type $A$ to a
vertex of type $B$ is through $i$ and vice-versa. Therefore, since $k$ is of
type $B$, we must eventually enter $i$ in $T^{\prime}$ through a vertex of
type $B$, in order to access vertices of type $A$. The only way to do so,
however, is through the edge $\ell\to i$, which was already used in
$T^{\prime}$. This is a contradiction, so the uniqueness of the Eulerian tour
is proved. The two described maps $h$ and $h^{-1}$ are obviously inverses of
each other, so the proof is complete.

∎

##  4\. Acknowledgements

We are thankful to Jan Kyncel for the suggestions in the mathoverflow post
related to the considered problem [4].

## References

  * [1] Stanley, Richard P. ”Bijective proof problems.” URL: http://www-math.mit.edu/rstan/bij.pdf (2009).
  * [2] Stanley, R.P., 2013. Algebraic combinatorics. Springer
  * [3] Stanley, Richard P. Stanley, R. P. (2015). Catalan numbers. Cambridge University Press.
  * [4] https://mathoverflow.net/questions/366383/digraphs-with-exactly-one-eulerian-tour
  * [5] OEIS Foundation Inc. (2020), The On-Line Encyclopedia of Integer Sequences