aainstitutetext: Division of Theoretical Physics, Ruđer Bošković Institute,
Bijenička cesta 54, 10000, Zagreb, Croatiabbinstitutetext: Physik Department,
Universität Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany

# Quark-hadron duality at work:
lifetimes of bottom baryons

James Gratrex b Alexander Lenz a Blaženka Melić a Ivan Nišandžić b Maria Laura
Piscopo b Aleksey V. Rusov<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In the 1990s, very low experimental values for the lifetime ratio
$\tau(\Lambda_{b})/\tau(B_{d})$ triggered a considerable amount of doubt in
the applicability of the heavy quark expansion (HQE), which is based on the
assumption of quark-hadron duality (QHD) for inclusive total decay rates.
However, these low values turned out to be the result of purely experimental
problems, and the current HFLAV average reads
$\tau(\Lambda_{b})/\tau(B_{d})=0.969(6)$. In this work, we present the
Standard Model predictions for the $b$-baryon lifetimes within the framework
of the HQE. In particular, we include for the first time the contribution of
the Darwin term and we update the estimates for the matrix elements of the
dimension-six four-quark operators. Within experimental and theoretical
uncertainties, we find excellent agreement between the data and the HQE
predictions, and thus no indication for any visible violation of QHD. Our
numerical results can be summarised by the ratios
$\tau(\Lambda_{b})/\tau(B_{d})=0.955(14)$,
$\tau(\Omega_{b}^{-})/\tau(B_{d})=1.081(42)$, and
$\tau(\Xi_{b}^{0})/\tau(\Xi_{b}^{-})=0.929(28)$.

††preprint: RBI-ThPhys-2022-48
SI-HEP-2022-38
SFB-257-P3H-22-126

##  1 Introduction

Lifetimes are among the most fundamental properties of particles. For weakly
decaying hadrons containing a heavy $b$-quark, the lifetimes can be determined
theoretically within the framework of the heavy quark expansion (HQE), whose
origin goes back to the 1980s Shifman:1986mx ; see Lenz:2014jha  for a review.
According to the HQE, the total decay rate of a bottom hadron can be described
as an expansion in inverse powers of the heavy quark mass, i.e. in
$\Lambda_{\rm QCD}/m_{b}$, with $\Lambda_{\rm QCD}$ being a typical non-
perturbative hadronic scale much smaller than the mass of the $b$-quark. The
leading term in this expansion is given by the decay of a free $b$-quark, and
is completely independent of the decaying hadron. Taking only this
contribution into account would therefore lead to the expectation of equal
lifetimes for different $b$-hadrons. Corrections to this picture, and thus
deviations of the lifetime ratios from one, are suppressed by at least two
powers of the $b$-quark mass. Without knowing the size of higher-order QCD
corrections, and with only rough estimates for the matrix elements arising in
the HQE, the naive expectation in 1986 Shifman:1986mx  was

$\displaystyle\left.\frac{\tau(B^{+})}{\tau(B_{d})}\right|^{\rm
HQE\,1986}\approx 1.1\,,\hskip
28.45274pt\left.\frac{\tau(B_{s})}{\tau(B_{d})}\right|^{\rm HQE\,1986}\approx
1\,,\hskip
28.45274pt\left.\frac{\tau(\Lambda_{b}^{0})}{\tau(B_{d})}\right|^{\rm
HQE\,1986}\approx 0.96\,.$ (1.1)

Surprisingly, early measurements of the $\Lambda_{b}^{0}$ lifetime resulted in
values which were considerably lower than the first theory expectations, as
shown in figure 1.111The $\Lambda_{b}^{0}$ baryon was discovered in 1991 in
proton-antiproton collisions by the UA1 collaboration, based on data taken in
1988/89 UA1:1991vse . The first measurement of the $\Lambda_{b}$ lifetime was
performed by the ALEPH collaboration in 1992 ALEPH:1992yid , based on LEP
$e^{+}e^{-}$ data taken in 1990/91, and resulting in
$\tau(\Lambda_{b}^{0})=\left(1.12_{-0.29}^{+0.32}(\rm stat.)\pm 0.16(\rm
syst.)\right)$ ps.

Figure 1: History of the lifetime ratio $\tau(\Lambda_{b})/\tau(B_{d})$:
experiment (lilac) vs. selected theory predictions: Shifman, Voloshin (1986)
Shifman:1986mx , Colangelo, De Fazio (1996) Colangelo:1996ta , Di Pierro,
Sachrajda, Michael (1999) DiPierro:1999tb , Huang, Liu, Zhu (1999)
Huang:1999xj , Guberina, Melic, Stefancic (1999, 2000) Guberina:1999bw ;
Guberina:1999yj , Franco et al (2002) Franco:2002fc , Gabbiani, Onishchenko,
Petrov (2004) Gabbiani:2004tp , Tarantino (2007) Tarantino:2007nf , Lenz
(2015) Lenz:2014jha , Cheng (2018) Cheng:2018rkz , and this work.

In e.g. 1996, the world average for the $\Lambda_{b}$ lifetime read
Colangelo:1996ta

$\displaystyle\tau(\Lambda_{b})=(1.18\pm 0.07)\,\mbox{ps}\,,$ (1.2)

which corresponded to a lifetime ratio of

$\displaystyle\frac{\tau(\Lambda_{b})}{\tau(B_{d})}=(0.75\pm 0.05)\,,$ (1.3)

when using the 1996 world average for the $B_{d}$ lifetime Colangelo:1996ta .
As these experimental results were more than four standard deviations below
the naive expectation in eq. (1.1), a considerable amount of interest was
triggered in the theory community, with various efforts made to accommodate
the result (1.3) within the HQE. In Huang:1999xj , the possibility of
anomalously large matrix elements of dimension-six four-quark operators in the
HQE was suggested, which was, however, in conflict with the results of
Colangelo:1996ta ; Rosner:1996fy ; DiPierro:1999tb ; while large contributions
from dimension-seven four-quark operators were considered in Gabbiani:2004tp .

Separately, the validity of the HQE itself was questioned e.g. in
Altarelli:1996gt ; Cheng:1997xba ; Ito:1997qq , with Altarelli:1996gt ;
Cheng:1997xba  suggesting a violation of local quark-hadron duality (QHD), see
e.g. Jubb:2016mvq  for a brief introduction to the concept of QHD. However,
the proposal in Altarelli:1996gt ; Cheng:1997xba  was heavily criticised since
it would have required huge $1/m_{b}$ corrections, which cannot be reconciled
with the operator product expansion approach, see e.g. Bigi:1999bq . The
notion of QHD was introduced in 1975 by Poggio, Quinn, and Weinberg
Poggio:1975af  to equate the hadronic process $e^{+}+e^{-}\to\rm hadrons$ with
the quark-level process $e^{+}+e^{-}\to\rm quarks$. In the case of the total
decay rate of a $B$-hadron, we can write

$\Gamma^{\rm tot}(B)=\\!\\!\\!\\!\\!\sum\limits_{\tiny\mbox{all possible
hadrons}}\\!\\!\\!\\!\\!\Gamma(B\to{\rm
hadrons+leptons})=\\!\\!\\!\\!\\!\sum\limits_{\tiny\mbox{all possible
quarks}}\\!\\!\\!\\!\\!\Gamma(B\to{\rm quarks+leptons})\,,$ (1.4)

and QHD-violating contributions in the HQE could correspond to non-
perturbative terms such as $\exp[-m_{b}/\Lambda_{\rm QCD}]$, e.g.
Shifman:2000jv ; Bigi:2001ys . Since an exact proof of QHD would require one
to explicitly solve QCD, which is clearly not possible currently, we can
consider two strategies in order to investigate the possible size of duality-
violating effects. Firstly, one could study simplified models of QCD, like the
’t Hooft model, a 1+1 dimension model for QCD, e.g. tHooft:1974pnl ;
Bigi:1998kc ; Lebed:2000gm ; Grinstein:2001zq ; Shifman:2000jv ; Bigi:2001ys ,
or instanton-based and resonance-based models, e.g. Chibisov:1996wf ;
Shifman:2000jv ; Bigi:2001ys . Studies of the ’t Hooft model indicate the
presence of duality-violating terms, albeit at very high orders in the HQE and
thus numerically irrelevant. Nevertheless, it is not clear what stringent
conclusions can be drawn from the study of a 1+1 dimensional model for the
real 3+1 dimensional world. The second approach is purely phenomenological,
and is based on comparing experimental data with precise HQE predictions. In
the present work, we follow this latter strategy.

Ultimately, it turned out that the low values for the $\Lambda_{b}$ lifetime
were purely an experimental problem, and the current world average for
$\tau(\Lambda_{b})/\tau(B_{d})$ Zyla:2020zbs  now agrees perfectly with the
estimates from 1986. This is clearly shown in figure 1, where we plot the time
evolution of the experimental measurements for this observable, from 1992
onwards, in comparison to selected theory predictions Shifman:1986mx ;
Colangelo:1996ta ; DiPierro:1999tb ; Huang:1999xj ; Guberina:1999bw ;
Guberina:1999yj ; Franco:2002fc ; Gabbiani:2004tp ; Tarantino:2007nf ;
Lenz:2014jha ; Cheng:2018rkz , as well as our result. Based on the
measurements in DELPHI:1995jet ; OPAL:1995nmi ; DELPHI:1996eqs ; ALEPH:1996kuy
; CDF:1996fsp ; ALEPH:1997ake ; OPAL:1997ufs ; DELPHI:1999con ; DELPHI:2005zmk
; D0:2007pfp ; CDF:2009sse ; D0:2012hfl ; ATLAS:2012cvl ; CMS:2013bcs ;
LHCb:2014qsd ; LHCb:2014wvs ; CDF:2014mon ; LHCb:2014wqn ; LHCb:2014chk ;
LHCb:2014jst ; LHCb:2016coe ; CMS:2017ygm , HFLAV Amhis:2022mac  quotes for
the lifetimes of different weakly decaying $b$-baryons the precise values
listed in table 1 and the lifetime ratios listed in table 2.

| $\Lambda_{b}^{0}$ | $\Xi_{b}^{0}$ | $\Xi_{b}^{-}$ | $\Omega_{b}^{-}$ | $B_{d}^{0}$
---|---|---|---|---|---
$\tau\,[{\rm ps}]$ | $1.471\pm 0.009$ | $1.480\pm 0.030$ | $1.572\pm 0.040$ | $1.64^{+0.18}_{-0.17}$ | $1.519\pm 0.004$
$\Gamma\,[{\rm ps}^{-1}]$ | $0.680\pm 0.004$ | $0.676\pm 0.014$ | $0.636\pm 0.016$ | $0.610^{+0.070}_{-0.066}$ | $0.636\pm 0.016$
Table 1:  HFLAV averages of the experimental determinations of $b$-baryon lifetimes Amhis:2022mac . We also include the most recent value of the $B_{d}^{0}$ meson lifetime, which we use in our predictions for the lifetime ratios with the baryons.  $\tau(\Lambda_{b}^{0})/\tau(B_{d}^{0})$ | $0.969\pm 0.006$
---|---
$\tau(\Xi_{b}^{0})/\tau(\Xi_{b}^{-})$ | $0.929\pm 0.028$
Table 2: HFLAV averages of the experimental determinations of $b$-baryon
lifetime ratios Amhis:2022mac .

In this paper, we present theory predictions for the lifetimes of baryons
containing a heavy $b$-quark, as a continuation of our work on the study of
lifetimes of $D$ mesons King:2021xqp ; Gratrex:2022xpm , $B$ mesons
Lenz:2022rbq , and charmed baryons Gratrex:2022xpm . Besides implementing for
the first time the recently determined Wilson coefficient of the Darwin
operator Mannel:2020fts ; Lenz:2020oce ; Moreno:2020rmk ; Piscopo:2021ogu , we
include radiative QCD corrections to the Wilson coefficients, where available,
and update all the relevant numerical inputs, including new estimates for the
non-perturbative matrix elements. We present predictions for the decay rates
of the $\Lambda_{b}^{0},\,\Xi_{b}^{0},\,\Xi_{b}^{-},$ and $\Omega_{b}^{-}$
baryons, and their lifetime ratios, as well as lifetime ratios of these
baryons with the $B_{d}^{0}$ meson. Within uncertainties, our results are in
excellent agreement with the experimental data. Moreover, we give predictions
for the inclusive $b$-baryon semileptonic branching fractions, although in
this case there are no current experimental determinations.

The remainder of the paper is structured as follows. In section 2 we present
the theoretical framework. Specifically, in section 2.1, we briefly describe
the structure of the HQE, followed by the discussion of short-distance
contributions in section 2.2, and the analysis of non-perturbative matrix
elements in section 2.3. Section 3 contains the description of the numerical
analysis and our predictions for the $b$-baryon lifetimes, lifetime ratios,
and semileptonic branching fractions. We conclude in section 4. Appendix A
contains numerical values of the input parameters used in the analysis, while
in appendix B, we provide the analytical expressions at LO-QCD for the
dimension-six four-quark operator contributions.

##  2 Theoretical framework

###  2.1 Effective Hamiltonian and HQE

Weak $b$-quark decays can be described by the effective Hamiltonian
Buchalla:1995vs

${\cal H}_{\rm eff}={\cal H}_{\rm eff}^{\rm NL}+{\cal H}_{\rm eff}^{\rm
SL}+{\cal H}_{\rm eff}^{\rm rare}\,.$ (2.1)

In the above equation, ${\cal H}_{\rm eff}^{\rm NL}$ parametrises the
contribution of non-leptonic $b$-quark transitions

$\displaystyle{\cal H}_{\rm eff}^{\rm
NL}=\frac{G_{F}}{\sqrt{2}}\sum_{q_{3}=d,s}\left[\,\sum_{\begin{subarray}{c}q_{1,2}=u,c\end{subarray}}\\!\\!\lambda_{q_{1}q_{2}q_{3}}\Bigl{(}C_{1}(\mu_{1})\,Q_{1}^{q_{1}q_{2}q_{3}}+C_{2}(\mu_{1})\,Q_{2}^{q_{1}q_{2}q_{3}}\Bigr{)}-\lambda_{q_{3}}\\!\\!\\!\\!\sum\limits_{j=3,\ldots,6,8}\\!\\!\\!C_{j}(\mu_{1})\,Q_{j}^{q_{3}}\right]+{\rm
h.c.}\,,$ (2.2)

where $\lambda_{q_{1}q_{2}q_{3}}=V_{q_{1}b}^{*}V_{q_{2}q_{3}}$ and
$\lambda_{q_{3}}=V_{tb}^{*}V_{tq_{3}}$ stand for the corresponding CKM
factors, $C_{i}(\mu_{1})$ denote the Wilson coefficients of the $\Delta B=1$
effective operators evaluated at the renormalisation scale $\mu_{1}\sim
m_{b}$. $Q_{1,2}^{q_{1}q_{2}q_{3}}$ and $Q_{j}^{q_{3}}$, with $j=3,\ldots,6,$
and $Q_{8}^{q}$, respectively denote the current-current,222Note that
$Q_{1}^{q_{1}q_{2}q_{3}}$ in our notation is the colour-singlet operator,
following King:2021xqp ; Lenz:2022rbq  and contrary to e.g. Buchalla:1995vs ;
Gratrex:2022xpm . penguin, and chromomagnetic operators, and are explicitly

$Q_{1}^{q_{1}q_{2}q_{3}}=\left(\bar{b}^{i}\,\Gamma_{\mu}\,q_{1}^{i}\right)\left(\bar{q}_{2}^{j}\,\Gamma^{\mu}\,q_{3}^{j}\right)\,,\qquad
Q_{2}^{q_{1}q_{2}q_{3}}=\left(\bar{b}^{i}\,\Gamma_{\mu}\,q_{1}^{j}\right)\left(\bar{q}_{2}^{j}\,\Gamma^{\mu}\,q_{3}^{i}\right)\,,$
(2.3) $\displaystyle Q_{3}^{q_{3}}$
$\displaystyle=(\bar{b}^{i}\,\Gamma_{\mu}\,q_{3}^{i})\sum_{q}(\bar{q}^{j}\,\Gamma^{\mu}\,q^{j})\,,\qquad
Q_{4}^{q_{3}}=(\bar{b}^{i}\,\Gamma_{\mu}\,q_{3}^{j})\sum_{q}(\bar{q}^{j}\,\Gamma^{\mu}\,q^{i})\,,$
$\displaystyle Q_{5}^{q_{3}}$
$\displaystyle=(\bar{b}^{i}\,\Gamma_{\mu}\,q_{3}^{i})\sum_{q}(\bar{q}^{j}\,\Gamma_{+}^{\mu}\,q^{j})\,,\qquad
Q_{6}^{q_{3}}=(\bar{b}^{i}\,\Gamma_{\mu}\,q_{3}^{j})\sum_{q}(\bar{q}^{j}\,\Gamma_{+}^{\mu}\,q^{i})\,,$
(2.4)
$Q_{8}^{q_{3}}=\frac{g_{s}}{8\pi^{2}}m_{b}\left(\bar{b}^{i}\,\sigma^{\mu\nu}(1-\gamma_{5})t^{a}_{ij}\,q_{3}^{j}\right)G^{a}_{\mu\nu}\,,$
(2.5)

with $\Gamma_{\mu}=\gamma_{\mu}(1-\gamma_{5})$,
$\Gamma_{+}^{\mu}=\gamma^{\mu}(1+\gamma_{5})$, and
$\sigma_{\mu\nu}=(i/2)[\gamma_{\mu},\gamma_{\nu}]$, while $i,j=1,2,3,$ are
SU(3)c indices for the quark fields. Moreover, in eq. (2.5), $g_{s}$ denotes
the strong coupling, and $G_{\mu\nu}=G^{a}_{\mu\nu}t^{a}$ for $a=1,\ldots,8$
is the gluon field strength tensor. A comparison of the values of the Wilson
coefficients for different choices of the scale $\mu_{1}$ at LO- and NLO-QCD
Buchalla:1995vs  is shown in table 8 in appendix A.

The second term in eq. (2.1) describes the contribution to the effective
Hamiltonian due to semileptonic $b$-quark decays, i.e.

${\cal H}_{\rm eff}^{\rm
SL}=\frac{G_{F}}{\sqrt{2}}\sum_{q_{1}=u,c\,}\sum_{\,\ell=e,\mu,\tau}V_{q_{1}b}^{*}\,Q^{q_{1}\ell}+{\rm
h.c.}\,,$ (2.6)

with the semileptonic operator

$Q^{q_{1}\ell}=\left(\bar{b}^{i}\,\Gamma_{\mu}\,q_{1}^{i}\right)\left(\bar{\nu}_{\ell}\,\Gamma^{\mu}\,\ell\right)\,.$
(2.7)

Finally, ${\cal H}_{\rm eff}^{\rm rare}$ in eq. (2.2) encodes the contribution
due to suppressed $b$-quark transitions, which are only relevant for the study
of rare decays such as $\Lambda_{b}\to\Lambda\gamma$ or
$\Lambda_{b}\to\Lambda\ell^{+}\ell^{-}$. These modes have very small branching
fractions, below the current theoretical sensitivity for lifetimes, and so the
effect of ${\cal H}_{\rm eff}^{\rm rare}$ is neglected in this work.

The total decay width of a $b$-baryon ${\cal B}$, with mass $M_{\cal B}$ and
four-momentum $p_{\cal B}$, reads

$\Gamma({\cal B})=\frac{1}{2M_{\cal B}}\sum_{X}\int\limits_{\rm
PS}(2\pi)^{4}\delta^{(4)}(p_{\cal B}-p_{X})\,\,|\langle X(p_{X})|{\cal H}_{\rm
eff}|H(p_{\cal B})\rangle|^{2},$ (2.8)

where a summation over all possible final states $X$ into which the $b$-baryon
can decay has been performed, with PS denoting the corresponding phase space
integration. Using the optical theorem, $\Gamma({\cal B})$ can be related to
the imaginary part of the forward scattering matrix element of the time-
ordered product of the double insertion of the effective Hamiltonian, i.e.

$\Gamma({\cal B})=\frac{1}{2M_{\cal B}}{\rm Im}\langle{\cal B}|{\cal T}|{\cal
B}\rangle\,,$ (2.9)

with the transition operator defined as

${\cal T}=i\int d^{4}x\,T\left\\{{\cal H}_{\rm eff}(x)\,,{\cal H}_{\rm
eff}(0)\right\\}\,.$ (2.10)

The non-local operator in eq. (2.10) can then be evaluated by exploiting the
fact that the $b$-quark is heavy, i.e. $m_{b}\gg\Lambda_{\rm QCD}$, where
$\Lambda_{\rm QCD}$ defines a typical non-perturbative hadronic scale. In the
framework of the HQE Khoze:1983yp ; Shifman:1984wx ; Shifman:1986mx ;
Bigi:1991ir ; Bigi:1992su ; Blok:1992hw ; Blok:1992he ; Bigi:1993ex ;
Beneke:1998sy ; Lenz:2014jha , the $b$-quark momentum is decomposed as

$p_{b}^{\mu}=m_{b}v^{\mu}+k^{\mu}\,,$ (2.11)

where $v=p_{\cal B}/M_{\cal B}$ is the four-velocity of the $b$-baryon. The
residual momentum $k$ in (2.11) accounts for non-perturbative interactions of
the $b$-quark with the light degrees of freedom, i.e. soft gluons and quarks,
inside the hadron, so $k\sim\Lambda_{\rm QCD}$. Moreover, the heavy $b$-quark
field is parametrised as

$b(x)=e^{-im_{b}v\cdot x}b_{v}(x)\,,$ (2.12)

by factoring out the large component of its momentum and introducing a
rescaled field $b_{v}(x)$, which contains only low oscillation frequencies of
order $k$. This field satisfies

$iD_{\mu}b(x)=e^{-im_{b}v\cdot x}(m_{b}v_{\mu}+iD_{\mu})b_{v}(x)\,,$ (2.13)

so that the action of the covariant derivative
$D_{\mu}=\partial_{\mu}-ig_{s}A_{\mu}^{a}\,t^{a}$ also contains a large
contribution proportional to the heavy quark mass alongside a residual term of
order $\Lambda_{\rm QCD}$. The rescaled field $b_{v}(x)$ is related to the
heavy quark effective theory (HQET) field $h_{v}(x)$, see e.g. Neubert:1993mb
, by

$b_{v}(x)=h_{v}(x)+\frac{i\not{D}_{\perp}}{2m_{b}}h_{v}(x)+{\cal
O}\left(\frac{1}{m_{b}^{2}}\right)\,,$ (2.14)

with $D_{\perp}^{\mu}=D^{\mu}-(v\cdot D)\,v^{\mu}$. Finally, taking into
account eqs. (2.11)-(2.13), the total decay width in eq. (2.9) can be
systematically expanded in inverse powers of the heavy $b$-quark mass, leading
to the HQE series, which schematically reads

$\Gamma({\cal B})=\Gamma_{3}+\Gamma_{5}\frac{\langle{\cal
O}_{5}\rangle}{m_{b}^{2}}+\Gamma_{6}\frac{\langle{\cal
O}_{6}\rangle}{m_{b}^{3}}+...+16\pi^{2}\left(\tilde{\Gamma}_{6}\frac{\langle\tilde{\mathcal{O}}_{6}\rangle}{m_{b}^{3}}+\tilde{\Gamma}_{7}\frac{\langle\tilde{\mathcal{O}}_{7}\rangle}{m_{b}^{4}}+...\right).$
(2.15)

Here, the $\Gamma_{d}$ are short-distance functions, which can be computed
perturbatively in QCD, i.e.

$\Gamma_{d}=\Gamma_{d}^{(0)}+\frac{\alpha_{s}}{4\pi}\Gamma_{d}^{(1)}+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\Gamma_{d}^{(2)}+\ldots\,,$
(2.16)

while $\langle{\cal O}_{d}\rangle\equiv\langle{\cal B}|{\cal O}_{d}|{\cal
B}\rangle/(2M_{\cal B})$ denote the matrix elements of the corresponding
$\Delta B=0$ operators ${\cal O}_{d}$ of dimension $d$ in the effective
theory. Note that, starting from order $1/m_{b}^{3}$, both two- and four-quark
operator contributions appear. The latter originate from loop-enhanced
diagrams, as reflected by the explicit factor of $16\pi^{2}$ in eq. (2.15),
and, to avoid confusion in the notation, we use a tilde to label them.

###  2.2 Short-distance contributions

In this section, we give a brief summary of the short-distance contributions,
cf. eqs. (2.15, 2.16), included in our analysis. For more details we refer to
the recent studies King:2021xqp ; Gratrex:2022xpm ; Lenz:2022rbq .333There are
some differences in the structure of the HQE for charmed hadrons King:2021xqp
; Gratrex:2022xpm  as opposed to the $b$ sector; see also Fael:2019umf ;
Mannel:2021uoz .

The coefficients $\Gamma_{d},\tilde{\Gamma}_{d}$ are analytic functions of the
masses of the internal fermions running in the loops. In our analysis, we only
include the contribution of the charm-quark and tau-lepton masses, expressed
in terms of the two dimensionless parameters

$x_{c}=\frac{m_{c}^{2}}{m_{b}^{2}}\,,\qquad
x_{\tau}=\frac{m_{\tau}^{2}}{m_{b}^{2}}\,.$ (2.17)

As $m_{s}^{2}/m_{b}^{2}\approx m_{\mu}^{2}/m_{b}^{2}\sim 0.05\%$, the effect
of non-vanishing strange-quark and muon masses is far below the current
theoretical accuracy, and hence can be safely neglected.444 However, we do
include strange quark mass corrections in the non-perturbative input, where
these effects are much more pronounced, in order to account for
SU(3)F-breaking. The leading contribution to the $b$-baryon total width,
$\Gamma_{3}$ in eq. (2.15), is obtained by computing the free $b$-quark decay,
and can be compactly expressed as

$\Gamma_{3}=\Gamma_{0}\,c_{3}=\Gamma_{0}\left(c_{3}^{(0)}+\frac{\alpha_{s}}{4\pi}c_{3}^{(1)}+\ldots\right)\,,$
(2.18)

where

$\Gamma_{0}=\frac{G_{F}^{2}\,m_{b}^{5}}{192\pi^{3}}|V_{cb}|^{2}\,,$ (2.19)

and

$c_{3}={\cal C}_{3,\rm SL}+3\,C_{1}^{2}\,{\cal C}_{3,11}+2\,C_{1}C_{2}\,{\cal
C}_{3,12}+3\,C_{2}^{2}\,{\cal C}_{3,22}+C_{i}\,C_{j}\,{\cal C}^{P}_{3,ij}\,.$
(2.20)

Above, a summation over all possible non-leptonic and semileptonic modes of
the $b$-quark is implicitly assumed, and we have denoted by ${\cal
C}_{3,ij}^{P}$, with $i=1,2,$ and $j=3,\ldots,6,8,$ the contribution due to
the mixed insertion of the current-current and penguin or chromomagnetic
operators. For semileptonic modes, $\alpha_{s}^{3}$-corrections have been
computed Fael:2020tow ; Czakon:2021ybq ; however, as the accuracy for non-
leptonic modes reaches only NLO-QCD, we perform our analysis consistently at
this order and do not include the new results for ${\cal C}_{3,{\rm SL}}$.
Moreover, following a common counting adopted in the literature Lenz:1997aa ;
Lenz:1998qp , the contribution of the penguin and chromomagnetic operators is
treated as a next-to-leading order effect, i.e. ${\cal C}^{P}_{3,ij}=0$ at LO-
QCD, owing to the small size of the corresponding Wilson coefficients. The
result for $c_{3}$ at LO can be found e.g. in Lenz:2020oce ; Gratrex:2022xpm .
As for the NLO corrections, the analytical expressions for ${\cal C}_{3,11}$,
${\cal C}_{3,22}$, and ${\cal C}_{3,{\rm SL}}$ can be extracted from
Hokim:1983yt , where the computation has been performed for three different
final state masses, while those for ${\cal C}_{3,12}$ are derived from the
results presented in Krinner:2013cja  in the case of the $b\to c\bar{c}s$
transition, and in Bagan:1994zd  for the remaining modes. Finally the results
for ${\cal C}_{3,ij}^{P}$ are taken from Krinner:2013cja .

Power corrections due to two-quark operators are obtained by including the
effect of soft gluons as well as the $1/m_{b}$-expansion of lower-dimensional
matrix elements. At order $1/m_{b}^{2}$, the corresponding contribution can be
schematically written as

$\Gamma_{5}\frac{\langle{\cal
O}_{5}\rangle}{m_{b}^{2}}=\Gamma_{0}\left[c_{\pi}\frac{\langle{\cal
O}_{\pi}\rangle}{m_{b}^{2}}+c_{G}\,\frac{\langle{\cal
O}_{G}\rangle}{m_{b}^{2}}\right]\,,$ (2.21)

where the matrix elements of the kinetic and chromomagnetic operators555Note
that, with a little abuse of notation, we refer to both $Q^{q_{3}}_{8}$ and
${\cal O}_{G}$ as chromomagnetic operators. However, as they arise
respectively in the $\Delta B=1$ and $\Delta B=0$ effective theory, it should
be clear from the context to which one we refer., given explicitly in eqs.
(2.56, 2.57), are discussed in section 2.3. In our analysis, again for
consistency, we include the coefficients $c_{\pi}$ and $c_{G}$ only at LO-QCD,
since $\alpha_{s}$-corrections have so far been determined only for the
semileptonic channels Mannel:2015jka . The coefficient of the kinetic operator
is related to that of dimension-three by a purely numerical factor,
$c_{\pi}=-c_{3}^{(0)}/2$, while the coefficient $c_{G}$ can be decomposed as

$c_{G}={\cal C}_{G,\rm SL}+3\,C_{1}^{2}\,{\cal C}_{G,11}+2\,C_{1}C_{2}\,{\cal
C}_{G,12}+3\,C_{2}^{2}\,{\cal C}_{G,22}\,,$ (2.22)

where again a summation over all possible $b$-quark modes is implied. The
expressions for the non-leptonic channels ${\cal C}_{G,ij}$, originally
computed in Blok:1992he ; Blok:1992hw ; Bigi:1992ne , can be found e.g. in
Lenz:2020oce ; Gratrex:2022xpm , while the semileptonic coefficient ${\cal
C}_{G,SL}$ is taken from the general result for two different final state
masses presented e.g. in the appendix of Mannel:2017jfk , and first determined
in Balk:1993sz ; Falk:1994gw .

At order $1/m_{b}^{3}$, both two- and four-quark operators contribute, cf. eq.
(2.16). For the former, we can compactly write666Formally, at dimension-six
the basis also includes the spin-orbit operator ${\cal O}_{\rm LS}$. However,
by adopting definitions in terms of full covariant derivatives rather than
transversal ones, the contribution of ${\cal O}_{\rm LS}$ to the total decay
width vanishes. For more detail, see e.g. Dassinger:2006md .

$\displaystyle\Gamma_{6}\frac{\langle{\cal
O}_{6}\rangle}{m_{b}^{3}}=\Gamma_{0}\,c_{\rho_{D}}\frac{\langle{\cal
O}_{D}\rangle}{m_{b}^{3}}\,,$ (2.23)

where the matrix element of the Darwin operator is defined in eq. (2.58),
while the corresponding short-distance coefficient can be decomposed as

$c_{\rho_{D}}={\cal C}_{\rho_{D},\rm SL}+3\,C_{1}^{2}\,{\cal
C}_{\rho_{D},11}+2\,C_{1}C_{2}\,{\cal C}_{\rho_{D},12}+3\,C_{2}^{2}\,{\cal
C}_{\rho_{D},22}\,,$ (2.24)

summing again over all $b$-quark decay modes. As NLO-QCD corrections are only
available for semileptonic decays Mannel:2019qel ; Mannel:2021zzr ;
Moreno:2022goo , the accuracy in our analysis again extends to only LO-QCD,
identically to the dimension-five contributions. The complete expressions of
${\cal C}_{\rho_{D},ij}$ for all non-leptonic channels have been obtained
recently in Lenz:2020oce ; Mannel:2020fts ; Moreno:2020rmk ; Piscopo:2021ogu ,
while the coefficient ${\cal C}_{\rho_{D},{SL}}$, first computed in
Gremm:1996df , can be read off the general results for the case of two
different final state masses presented e.g. in Rahimi:2022vlv ; Moreno:2022goo
. It is worth emphasising that the coefficient of the Darwin operator is one
order of magnitude larger than the corresponding ones at dimension-five.
However, as shown in detail in Lenz:2020oce , this in fact follows from an
accidental suppression of the dimension-five coefficients, rather than an
enhancement of the Darwin term. Therefore, the contribution of the Darwin
operator, neglected in previous phenomenological studies, turns out to be an
important ingredient in the theoretical prediction of the $b$-baryon
lifetimes, see section 3, and of $B$ meson lifetimes Lenz:2022rbq .

Figure 2: Diagrams corresponding, from left to right, to the int-, exc, and
int+ topologies at LO-QCD. The numbering scheme for quarks follows that in eq.
(2.3), so that $q_{1,2}$ are up-type quarks and $q_{3}$ is down-type. Note
that semileptonic contributions only arise in the ${\rm int}^{+}$ topology.
Disregarding the second, non-interacting spectator quark, these topologies are
related by crossing to those in meson decays, cf. footnote 7.

The short-distance coefficients due to four-quark operators are obtained by
computing, at LO-QCD, the discontinuity of the one-loop diagrams shown in
figure 2, commonly denoted in the literature as destructive Pauli interference
(int-), weak-exchange (exc), and constructive Pauli interference (int+),
respectively.777For $B$ mesons, the corresponding topologies are respectively
denoted by weak-exchange (WE), Pauli interference (PI) and weak annihilation
(WA). Hence, when translating results from baryons to mesons and vice-versa,
the following replacements should be adopted: int${}^{-}\leftrightarrow$ WE,
exc $\leftrightarrow$ PI, and int${}^{+}\leftrightarrow$ WA Gratrex:2022xpm .
Taking into account the different topologies, the dimension-six contribution
from four-quark operators can be compactly written as

$\displaystyle 16\pi^{2}\,\tilde{\Gamma}_{6}\frac{\langle\tilde{\cal
O}_{6}\rangle}{m_{b}^{3}}=\\!\\!\\!\sum_{q_{1},q_{2},q_{3}}\Biggl{[}\tilde{\Gamma}_{6,{\rm
int}^{-}}^{q_{3}}(x_{q_{1}},x_{q_{2}})+\tilde{\Gamma}_{6,{\rm
exc}}^{q_{2}}(x_{q_{1}},x_{q_{3}})+\tilde{\Gamma}_{6,{\rm
int}^{+}}^{q_{1}}(x_{q_{2}},x_{q_{3}})\Biggr{]}\\!+\\!\sum_{q_{1},\ell}\tilde{\Gamma}_{6,{\rm
int}^{+}}^{q_{1}}(x_{\ell},x_{\nu_{\ell}})\,,$ (2.25)

where $\tilde{\Gamma}_{6,T}^{q}(x_{f_{1}},x_{f_{2}})$ denotes the imaginary
part of the diagram with topology $T$, with external light quark $q$ and
internal fermions $f_{1},f_{2}$, while $x_{f_{i}}=m_{f_{i}}^{2}/m_{b}^{2}$,
where $m_{f_{i}}$ is the corresponding fermion mass. Note that eq. (2.25)
implies that, contrary to the corrections described so far, contributions to
specific $b$-baryons differ not only due to different states appearing in the
respective matrix elements, but also due to different short-distance
coefficients. In light of this, and of the formal loop enhancement with
respect to two-quark operators, the effect of four-quark operators was
expected to give the dominant correction to the total widths, and in
particular to the lifetime ratios, see e.g. Uraltsev:1996ta ; Neubert:1996we .

The functions $\tilde{\Gamma}_{6,T}^{q}(x_{f_{1}},x_{f_{2}})$ in eq. (2.25)
can be further decomposed as follows:

$\displaystyle\tilde{\Gamma}_{6,T}^{q}(x_{f_{1}},x_{f_{2}})=\Gamma_{0}\sum_{i=1}^{4}c^{i}_{6,T}(x_{f_{1}},x_{f_{2}})\,\frac{\langle{{\cal
O}}_{i}^{q}\rangle}{m_{b}^{3}}\,,$ (2.26)

with ${\cal O}^{q}_{1,\ldots,4}$ denoting an appropriate set of four-quark
operators, cf. eqs. (2.32)-(2.33), and recall the short-hand notation
$\langle{\cal O}_{i}^{q}\rangle\equiv\langle{\cal B}|{\cal O}_{i}^{q}|{\cal
B}\rangle/(2M_{\cal B})$. For a comprehensive discussion of these matrix
elements, we refer to section 2.3.888Note that when comparing with the basis
presented in section 2.3, one should make the identifications ${\cal
O}^{q}_{3}=\tilde{\cal O}^{q}_{1}$ and ${\cal O}^{q}_{4}=\tilde{\cal
O}^{q}_{2}$. The complete expressions for the coefficients $c^{\,i}_{6,{\rm
int}^{-}}(x_{q_{1}},x_{q_{2}})$ and $c^{\,i}_{6,\rm exc}(x_{q_{1}},x_{q_{3}})$
up to NLO-QCD corrections, including also the contribution of the penguin and
chromomagnetic operators, have been computed in Franco:2002fc  for four-quark
operators defined in HQET.999Partial NLO results in the case of operators
defined in QCD can be found in Franco:2002fc ; Beneke:2002rj . The results for
$c^{\,i}_{6,{\rm int}^{+}}(x_{q_{2}},x_{q_{3}})$ can be obtained, by means of
a Fierz transformation, from the corresponding ones for $c^{\,i}_{6,{\rm
int}^{-}}(x_{q_{1}},x_{q_{2}})$ by replacing $C_{1}\leftrightarrow C_{2}$,
while for semileptonic modes, the NLO-corrections to the coefficients
$c^{\,i}_{6,{\rm int}^{+}}(x_{\ell},x_{\nu_{\ell}})$ have been determined in
Lenz:2013aua . Because of the different terminology used to denote the same
loop diagrams in baryons and mesons, in appendix B we present the LO-QCD
expressions for the functions $\tilde{\Gamma}_{6,T}^{q}(x_{f_{1}},x_{f_{2}})$
given in eq. (2.25).

Considering all possible contractions in the time-ordered product in eq.
(2.10), the complete dimension-six four-quark operator contributions to
$\Gamma(\cal B)$, included in our analysis, respectively read

$\displaystyle 16\pi^{2}\,\tilde{\Gamma}_{6}\frac{\langle\tilde{\cal
O}_{6}\rangle_{\Lambda_{b}^{0}}}{m_{b}^{3}}$
$\displaystyle=\Bigl{[}\tilde{\Gamma}^{u}_{6,\textrm{exc}}(x_{c},x_{d})+\tilde{\Gamma}^{u}_{6,\textrm{exc}}(x_{c},x_{s})+\tilde{\Gamma}^{u}_{6,\textrm{exc}}(x_{u},x_{d})+\tilde{\Gamma}^{u}_{6,\textrm{exc}}(x_{u},x_{s})$
$\displaystyle+\tilde{\Gamma}^{d}_{6,\textrm{int}^{-}}(x_{c},x_{u})+\tilde{\Gamma}^{d}_{6,\textrm{int}^{-}}(x_{c},x_{c})+\tilde{\Gamma}^{d}_{6,\textrm{int}^{-}}(x_{u},x_{u})+\tilde{\Gamma}^{d}_{6,\textrm{int}^{-}}(x_{u},x_{c})$
$\displaystyle+\tilde{\Gamma}^{u}_{6,\textrm{int}^{+}}(x_{u},x_{d})+\tilde{\Gamma}^{u}_{6,\textrm{int}^{+}}(x_{c},x_{s})+\tilde{\Gamma}^{u}_{6,\textrm{int}^{+}}(x_{u},x_{s})+\tilde{\Gamma}^{u}_{6,\textrm{int}^{+}}(x_{c},x_{d})$
$\displaystyle+\tilde{\Gamma}^{u}_{6,\textrm{int}^{+}}(x_{\tau},x_{\nu_{\tau}})+\tilde{\Gamma}^{u}_{6,\textrm{int}^{+}}(x_{\mu},x_{\nu_{\mu}})+\tilde{\Gamma}^{u}_{6,\textrm{int}^{+}}(x_{e},x_{\nu_{e}})\Bigr{]}_{\Lambda_{b}^{0}}\,,$
(2.27) $\displaystyle 16\pi^{2}\,\tilde{\Gamma}_{6}\frac{\langle\tilde{\cal
O}_{6}\rangle_{\Xi_{b}^{0}}}{m_{b}^{3}}$
$\displaystyle=\Bigl{[}\tilde{\Gamma}^{u}_{6,\textrm{exc}}(x_{c},x_{d})+\tilde{\Gamma}^{u}_{6,\textrm{exc}}(x_{c},x_{s})+\tilde{\Gamma}^{u}_{6,\textrm{exc}}(x_{u},x_{d})+\tilde{\Gamma}^{u}_{6,\textrm{exc}}(x_{u},x_{s})$
$\displaystyle+\tilde{\Gamma}^{s}_{6,\textrm{int}^{-}}(x_{c},x_{c})+\tilde{\Gamma}^{s}_{6,\textrm{int}^{-}}(x_{c},x_{u})+\tilde{\Gamma}^{s}_{6,\textrm{int}^{-}}(x_{u},x_{c})+\tilde{\Gamma}^{s}_{6,\textrm{int}^{-}}(x_{u},x_{u})$
$\displaystyle+\tilde{\Gamma}^{u}_{6,\textrm{int}^{+}}(x_{u},x_{d})+\tilde{\Gamma}^{u}_{6,\textrm{int}^{+}}(x_{c},x_{s})+\tilde{\Gamma}^{u}_{6,\textrm{int}^{+}}(x_{u},x_{s})+\tilde{\Gamma}^{u}_{6,\textrm{int}^{+}}(x_{c},x_{d})$
$\displaystyle+\tilde{\Gamma}^{u}_{6,\textrm{int}^{+}}(x_{\tau},x_{\nu_{\tau}})+\tilde{\Gamma}^{u}_{6,\textrm{int}^{+}}(x_{\mu},x_{\nu_{\mu}})+\tilde{\Gamma}^{u}_{6,\textrm{int}^{+}}(x_{e},x_{\nu_{e}})\Bigr{]}_{\Xi_{b}^{0}}\,,$
(2.28) $\displaystyle 16\pi^{2}\,\tilde{\Gamma}_{6}\frac{\langle\tilde{\cal
O}_{6}\rangle_{\Xi_{b}^{-}}}{m_{b}^{3}}$
$\displaystyle=\Bigl{[}\tilde{\Gamma}^{d}_{6,\textrm{int}^{-}}(x_{c},x_{u})+\tilde{\Gamma}^{s}_{6,\textrm{int}^{-}}(x_{c},x_{c})+\tilde{\Gamma}^{s}_{6,\textrm{int}^{-}}(x_{c},x_{u})+\tilde{\Gamma}^{d}_{6,\textrm{int}^{-}}(x_{c},x_{c})$
$\displaystyle+\tilde{\Gamma}^{d}_{6,\textrm{int}^{-}}(x_{u},x_{u})+\tilde{\Gamma}^{s}_{6,\textrm{int}^{-}}(x_{u},x_{c})+\tilde{\Gamma}^{s}_{6,\textrm{int}^{-}}(x_{u},x_{u})+\tilde{\Gamma}^{d}_{6,\textrm{int}^{-}}(x_{u},x_{c})\Bigr{]}_{\Xi_{b}^{-}}\,,$
(2.29) $\displaystyle 16\pi^{2}\,\tilde{\Gamma}_{6}\frac{\langle\tilde{\cal
O}_{6}\rangle_{\Omega_{b}^{-}}}{m_{b}^{3}}$
$\displaystyle=\Bigl{[}\tilde{\Gamma}^{s}_{6,\textrm{int}^{-}}(x_{c},x_{c})+\tilde{\Gamma}^{s}_{6,\textrm{int}^{-}}(x_{c},x_{u})+\tilde{\Gamma}^{s}_{6,\textrm{int}^{-}}(x_{u},x_{c})+\tilde{\Gamma}^{s}_{6,\textrm{int}^{-}}(x_{u},x_{u})\Bigr{]}_{\Omega_{b}^{-}}\,,$
(2.30)

where we have now explicitly indicated the specific baryon appearing in the
corresponding matrix elements. We stress that the results in eqs.
(2.27)-(2.30) do not take into account contributions in which the light quark
in the four-quark operators differs from the spectator quarks in the
$b$-baryon, the so-called ‘eye contractions’. These have been recently
computed for mesons in King:2021jsq , but they are still unknown for baryons.
However, as they constitute subleading corrections to the dimension-six
contribution, we expect their effect to go beyond the current accuracy of our
study. Moreover, in our numerical analysis we only keep non-vanishing the
masses of the charm quark and of the tau-lepton, i.e. we set
$x_{u,d,s}=x_{\mu,e,\nu_{\ell}}=0$, cf. eq. (2.17).

Note that, in eqs. (2.27)-(2.30), the non-leptonic contributions have been
ordered by topology, and within each topology we have listed the terms in
order of their CKM hierarchy. In particular, the leading contributions to the
$\Lambda_{b}^{0}$ and $\Xi_{b}^{0}$ decay widths arise from the
$\textrm{int}^{-}$ and exc topologies. As for the semileptonic contributions,
they can only arise in the int+ topology, and, since we do not include the eye
contractions, they only enter the decay width of the $\Lambda_{b}^{0}$ and
$\Xi_{b}^{0}$ baryons, see the last line of eqs. (2.27), (2.28), and not that
of the $\Xi_{b}^{-}$ or $\Omega_{b}^{-}$. However, the semileptonic
contributions give a negligible numerical effect to the total widths, and in
particular do not generate any significant splitting between the semileptonic
branching fractions of $b$-baryons, as expected because of the strong CKM
suppression $|V_{ub}|^{2}\ll|V_{cb}|^{2}$. Thus, within our current
sensitivity, any difference between the semileptonic branching fractions of
$b$-baryons can arise only from SU(3)F effects in the matrix elements of the
two-quark operators.

In section 3, we present our predictions for the lifetimes ratios of the
$b$-baryons with the $B_{d}$ meson. For completeness, in order to facilitate
the comparison, the corresponding leading dimension-six four-quark
contribution for the latter is Lenz:2022rbq

$\displaystyle 16\pi^{2}\,\tilde{\Gamma}_{6}\frac{\langle\tilde{\cal
O}_{6}\rangle_{B_{d}}}{m_{b}^{3}}$
$\displaystyle=\Bigl{[}\tilde{\Gamma}^{d}_{6,{\rm
WE}}(x_{c},x_{u})+\tilde{\Gamma}^{d}_{6,{\rm
WE}}(x_{c},x_{c})+\tilde{\Gamma}^{d}_{6,{\rm
WE}}(x_{u},x_{u})+\tilde{\Gamma}^{d}_{6,{\rm
WE}}(x_{u},x_{c})\Bigr{]}_{B_{d}}.$ (2.31)

Finally, at order $1/m_{b}^{4}$, the short-distance contributions due to four-
quark operators are also known in the literature, albeit only at LO-QCD, see
e.g. King:2021xqp . They have been determined in Gabbiani:2003pq ;
Gabbiani:2004tp  for operators defined in QCD101010Some inconsistencies in the
expressions of Gabbiani:2003pq ; Gabbiani:2004tp  were identified in
Lenz:2022rbq , cf. footnote 8 therein. and also in Lenz:2013aua ; Lenz:2022rbq
for the HQET operators. However, as compared with our previous studies
King:2021xqp ; Gratrex:2022xpm ; Lenz:2022rbq , we do not include the
subleading $1/m_{b}$ corrections to the four-quark matrix elements in our
central values for the total widths, preferring instead to treat these
contributions as part of the uncertainty estimate. The reason for this is the
absence of a consistent procedure to determine the corresponding matrix
elements for baryons, particularly in HQET, due to a proliferation of the
dimension-seven operators, see e.g. Neubert:1993mb , and in contrast to the
case of mesons, where the vacuum insertion approximation (VIA) provides a
first estimate. This problem was extensively discussed in Gratrex:2022xpm .
Moreover, other $1/m_{b}^{4}$ corrections are also missing, namely those due
to two-quark operators, which so far are known only for semileptonic $b$-quark
decays Dassinger:2006md ; Mannel:2010wj ; Gambino:2016jkc ; Mannel:2018mqv ,
and those to the dimension-six matrix elements, see section 2.3. As a result,
a complete analysis of the $b$-baryon lifetimes up to this order is currently
not possible. Given that, in the $b$-system, power corrections prove to be
well under control, we consider it more justified in this work to treat the
$1/m_{b}^{4}$ contributions as an additional source of uncertainty, rather
than trying to include them in the central values for lifetimes with only
partial, and potentially misleading, estimates for the dimension-seven matrix
elements.

###  2.3 Non-perturbative Matrix Elements

In this section, we present our determinations of the hadronic parameters. It
is convenient to first consider the matrix elements of the four-quark
operators, followed by the discussion of the two-quark matrix elements
$\mu_{\pi}^{2}(\mathcal{B}),\,\mu_{G}^{2}(\mathcal{B}),\,\rho_{D}^{3}(\mathcal{B})$.

A basis of dimension-six four-quark operators in HQET suitable for the
$b$-baryons is Neubert:1996we 111111The recent study Gratrex:2022xpm  made use
of the QCD basis of operators instead.

$\displaystyle\mathcal{O}_{1}^{q}$
$\displaystyle=(\bar{h}_{v}^{i}\gamma_{\mu}(1-\gamma_{5})q^{i})(\bar{q}^{j}\gamma^{\mu}(1-\gamma_{5})h_{v}^{j})\,,\qquad$
$\displaystyle\mathcal{O}_{2}^{q}$
$\displaystyle=(\bar{h}_{v}^{i}(1-\gamma_{5})q^{i})(\bar{q}^{j}(1+\gamma_{5})h_{v}^{j})\,,$
(2.32) $\displaystyle\tilde{\mathcal{O}}_{1}^{q}$
$\displaystyle=(\bar{h}_{v}^{i}\gamma_{\mu}(1-\gamma_{5})q^{j})(\bar{q}^{j}\gamma^{\mu}(1-\gamma_{5})h_{v}^{i})\,,\qquad$
$\displaystyle\tilde{\mathcal{O}}_{2}^{q}$
$\displaystyle=(\bar{h}_{v}^{i}(1-\gamma_{5})q^{j})(\bar{q}^{j}(1+\gamma_{5})h_{v}^{i})\,,$
(2.33)

with $q$ labeling the light quark in the corresponding operator, i.e.
$q=u,d,s$. Note that the colour-rearranged operators $\tilde{\cal
O}_{1,2}^{q}$ are related to the colour-octet ones commonly adopted in studies
of heavy meson lifetimes, see e.g. King:2021xqp ; Gratrex:2022xpm ;
Lenz:2022rbq , by the completeness property of the SU(3)c generators

$t_{ij}^{a}t^{a}_{lm}=\frac{1}{2}\Bigl{(}\delta_{im}\delta_{jl}-\frac{1}{N_{c}}\delta_{ij}\delta_{lm}\Bigr{)}\,.$
(2.34)

The usefulness of the choice of basis in eqs. (2.32), (2.33) is exhibited by
the relations

$\langle\mathcal{B}|\tilde{\mathcal{O}}_{i}^{q}|\mathcal{B}\rangle=-\tilde{B}_{i}^{q}\langle\mathcal{B}|\mathcal{O}_{i}^{q}|\mathcal{B}\rangle\,,\qquad
i=1,2\,,$ (2.35)

where, assuming the valence quark approximation, the total colour antisymmetry
of the baryon wave function imposes $\tilde{B}_{i}^{q}=1$ Neubert:1996we . In
our study, we consider a universal parameter
$\tilde{B}_{i}^{q}\equiv\tilde{B}$,121212In general,
$\tilde{B}_{1}^{q}=\tilde{B}_{2}^{q}+{\cal O}(1/m_{b})$, and
$\tilde{B}^{u,d}\neq\tilde{B}^{s}$, but we neglect these subleading
corrections. with $\tilde{B}=1$ valid at a typical hadronic scale $\mu_{h}\ll
m_{b}$. When performing the numerical analysis, we vary this scale in the
range $1\,{\rm GeV}\leq\mu_{h}\leq 1.5\,{\rm GeV}$, while taking as our
reference value $\mu_{h}=1.5\,$ GeV.

In order to estimate the matrix elements on the r.h.s. of eq. (2.35), we adopt
the non-relativistic constituent quark model (NRCQM), according to which the
matrix elements of the colour-singlet four-quark operators can be expressed,
in terms of baryon wave functions evaluated at the origin, as

$\displaystyle\cfrac{\langle{\cal T}_{b}|\mathcal{O}_{1}^{q}|{\cal
T}_{b}\rangle}{2M_{{\cal T}_{b}}}$ $\displaystyle=-|\Psi^{{\cal
T}_{b}}(0)|^{2}\,,\quad\quad\quad\cfrac{\langle{\cal
T}_{b}|\mathcal{O}_{2}^{q}|{\cal T}_{b}\rangle}{2M_{{\cal
T}_{b}}}=\frac{1}{2}|\Psi^{{\cal T}_{b}}(0)|^{2}\,,$ (2.36)

for the SU(3)F triplet ${\cal
T}_{b}=(\Lambda_{b}^{0},\Xi_{b}^{-},\Xi_{b}^{0})$, and

$\displaystyle\cfrac{\langle\Omega_{b}^{-}|\mathcal{O}_{1}^{s}|\Omega_{b}^{-}\rangle}{2M_{\Omega_{b}^{-}}}$
$\displaystyle=-6|\Psi^{\Omega_{b}^{-}}(0)|^{2}\,,\quad\quad\quad\cfrac{\langle\Omega_{b}^{-}|\mathcal{O}_{2}^{s}|\Omega_{b}^{-}\rangle}{2M_{\Omega_{b}^{-}}}=-|\Psi^{\Omega_{b}^{-}}(0)|^{2}\,,$
(2.37)

for the $\Omega_{b}^{-}$. It should be emphasised that the constituent quark
picture provides access only to the valence quark contributions, for which the
field of a light quark within the operator matches at least one of the baryon
valence quarks. The missing non-valence contributions are, however, expected
to provide subleading corrections. Hence, in (2.36), it should be understood
that the relations are valid only when the light quark $q$ in the operator
$\mathcal{O}_{i}^{q}$ matches one of the valence quarks in the baryon ${\cal
T}_{b}$, and the matrix element is otherwise taken to be zero, and similarly
in (2.37) for the $\Omega_{b}^{-}$.

We stress that, apart from the exploratory study in DiPierro:1999tb , which
has never been followed up, there are no lattice determinations for the four-
quark baryonic matrix elements available. A computation for the
$\Lambda_{b}^{0}$, within HQET sum rules, was performed in Colangelo:1996ta .
In contrast to the case of $B$ mesons, where one can set up a sum rule for the
small deviation of the bag parameter from one King:2021jsq ; King:2019lal ;
Kirk:2017juj ; Grozin:2016uqy , for baryons one can only write down sum rules
for the whole matrix element. Thus, the baryon case may be sensitive to
stability issues often associated with three-point sum rules Braun:1999dp .
Moreover, the sum rule work in Colangelo:1996ta  does not yet include NLO-QCD
effects. These corrections can be large, as was shown in the HQET sum rule
calculation of the two-point correlator Yakovlev:1996bs , entering also the
computation of the four-quark matrix element, where the
$\alpha_{s}$-contributions appear to be of a similar size to the leading
contribution. Very recently, the four-quark $\Lambda_{b}^{0}$ matrix elements
were also determined with QCD sum rules Zhao:2021lzd , confirming the
relatively small values obtained by the HQET sum rules in Colangelo:1996ta
.131313A separate sum rule calculation in Huang:1999xj  was able to
accommodate the then very low experimental values of the
$\tau(\Lambda_{b}^{0})/\tau(B_{d})$ lifetime ratio, at the expense of an
anomalously large four-quark contribution; see figure 1. We are not aware of
sum rule determinations of the matrix elements of the $\Xi_{b}^{0}$,
$\Xi_{b}^{-}$, or $\Omega_{b}^{-}$ baryons. Therefore, in this work, we choose
to consistently apply the NRCQM to calculate the matrix elements of the
dimension-six four-quark operators for all the baryons considered. For
comparison, however, we briefly discuss the numerical impact of the sum rule
determination from Colangelo:1996ta  on the $\Lambda_{b}^{0}$ lifetime in
section 3.

Following the standard approach proposed by de Rujula, Georgi, and Glashow
DeRujula:1975qlm , the baryon wave functions can be extracted from the known
values of hyperfine mass splittings Barger:1979pd ; Cortes:1980jz ;
Blok:1991st ; Rosner:1996fy . In the NRCQM, the hyperfine splittings are
controlled by the short-distance gluon exchange between the constituent
quarks. For a generic hadron $H$, the mass $M_{H}$ can be expressed as

$M_{H}=M_{0}+\langle H_{\text{spin}}\rangle\,,$ (2.38)

where $M_{0}$ contains the spin-independent contributions, including the
constituent quark masses and the binding energies. The spin-dependent terms
are, for the ground state $(L=0)$ hadrons, given as

$\displaystyle H_{\rm spin,\,baryons}$
$\displaystyle=\sum_{i>j}\frac{16\pi\alpha_{s}}{9}\frac{(\vec{s}_{i}\cdot\vec{s}_{j})}{m_{i}^{\,b}\,m_{j}^{\,b}}\delta^{3}(\vec{r}_{ij})\,,$
(2.39) $\displaystyle H_{\rm spin,\,mesons}$
$\displaystyle=\frac{32\pi\alpha_{s}}{9}\frac{(\vec{s}_{i}\cdot\vec{s}_{j})}{m_{i}^{\,m}\,m_{j}^{\,m}}\delta^{3}(\vec{r}_{ij})\,,$
(2.40)

where $i,j,$ label the constituent quarks in the hadron, with masses
$m^{\,b}_{i}$ and $m^{\,m}_{i}$ respectively for baryons and mesons, while
$\vec{s}_{i}$ denotes the corresponding quark spin operator. When evaluating
the expectation value in eq. (2.38) for a given hadronic state, the delta
functions in eqs. (2.39), (2.40) result in the modulus squared of the hadron
wave function at the origin, $|\Psi^{H}(0)|^{2}$, and the light quarks in
$b$-baryons are taken to form a diquark spin state. Note that we do not assume
the constituent quark masses within mesons and baryons to be equal, i.e.
$m_{i}^{\,m}\neq m_{i}^{\,b}$, but instead take their values as used in the
fit to hadronic masses Karliner:2014gca .

Following the approach of Rosner Rosner:1996fy , the wave functions appearing
in eqs. (2.36), (2.37) are extracted using the hyperfine splittings between
the positive-parity spin-3/2 and spin-1/2 bottom baryons. For example, for the
$\Lambda_{b}$ baryon, this results in the relation141414Applying a similar
relation for mesons, and using this to estimate the decay constant, would lead
to the estimates
$f_{B}=0.188(14)\,\textrm{GeV},\,f_{B_{s}}=0.241(18)\,\textrm{GeV}$, where the
uncertainty arises from varying the scale of $\alpha_{s}(\mu_{h})$ between 1.0
and 1.5 GeV. These values are consistent with those obtained from lattice
computations, supporting the applicability of the NRCQM to baryons.

$M_{\Sigma_{b}^{\ast}}-M_{\Sigma_{b}}=\frac{16\pi\alpha_{s}}{m_{b}^{\,b}\,m_{\tilde{q}}^{\,b}}\,\frac{3}{2}|\Psi^{{\Lambda_{b}}}(0)|^{2}\,,$
(2.41)

with $\tilde{q}=u,d,$ while the corresponding relations for
$|\Psi^{\Xi_{b}}(0)|^{2}$, and $|\Psi^{\Omega_{b}}(0)|^{2}$, involve the
hyperfine splittings $M_{\Xi_{b}^{\ast}}-M_{\Xi^{\prime}_{b}}$, and
$M_{\Omega_{b}^{\ast}}-M_{\Omega_{b}}$, respectively. After normalising these
relations to the analogous expressions involving the meson mass splittings, we
can express the matrix elements in eqs. (2.36), (2.37), in terms of $B$-meson
wave functions151515Note that we use interchangeably the notation
$B_{u}=B_{d}\equiv B$, in the limit of exact isospin symmetry. as

$\displaystyle\displaystyle\frac{\langle\Lambda_{b}|\mathcal{O}_{1}^{\tilde{q}}|\Lambda_{b}\rangle}{2M_{\Lambda_{b}}}$
$\displaystyle=-y_{\tilde{q}}\,\frac{4}{3}\frac{M_{\Sigma_{b}^{\ast}}-M_{\Sigma_{b}}}{M_{B^{\ast}}-M_{B}}|\Psi^{{B}}(0)|^{2}\,,$
(2.42)
$\displaystyle\frac{\langle\Xi_{b}^{0}|\mathcal{O}_{1}^{u}|\Xi_{b}^{0}\rangle}{2M_{\Xi_{b}}}$
$\displaystyle=\frac{\langle\Xi_{b}^{-}|\mathcal{O}_{1}^{d}|\Xi_{b}^{-}\rangle}{2M_{\Xi_{b}}}=-y_{\tilde{q}}\,\frac{4}{3}\frac{M_{\Xi_{b}^{\ast}}-M_{\Xi^{\prime}_{b}}}{M_{B^{\ast}}-M_{B}}|\Psi^{{B}}(0)|^{2}\,,$
(2.43)
$\displaystyle\frac{\langle\Xi_{b}^{-}|\mathcal{O}_{1}^{s}|\Xi_{b}^{-}\rangle}{2M_{\Xi_{b}}}$
$\displaystyle=\frac{\langle\Xi_{b}^{0}|\mathcal{O}_{1}^{s}|\Xi_{b}^{0}\rangle}{2M_{\Xi_{b}}}=-y_{s}\,\frac{4}{3}\frac{M_{\Xi_{b}^{\ast}}-M_{\Xi^{\prime}_{b}}}{M_{B_{s}^{\ast}}-M_{B_{s}}}|\Psi^{B_{s}}(0)|^{2}\,,$
(2.44)
$\displaystyle\frac{\langle\Omega_{b}^{-}|\mathcal{O}_{1}^{s}|\Omega_{b}^{-}\rangle}{2M_{\Omega_{b}}}$
$\displaystyle=-y_{s}\,6\,\frac{4}{3}\frac{M_{\Omega_{b}^{\ast}}-M_{\Omega_{b}}}{M_{B_{s}^{\ast}}-M_{B_{s}}}|\Psi^{B_{s}}(0)|^{2}\,,$
(2.45)

where $y_{\tilde{q}}$, $y_{s}$, denote ratios of the constituent quark masses
in baryons and mesons Karliner:2014gca

$y_{\tilde{q}}=\frac{m_{b}^{\,b}\,m_{\tilde{q}}^{\,b}}{m_{b}^{\,m}\,m_{\tilde{q}}^{\,m}}\simeq
1.18\,,\qquad\qquad
y_{s}=\frac{m_{b}^{\,b}\,m_{s}^{\,b}}{m_{b}^{\,m}\,m_{s}^{\,m}}\simeq 1.12\,.$
(2.46)

The ratios of the mass splittings,

$r_{q}(\mathcal{B})\equiv\frac{4}{3}\frac{M_{\mathcal{B}^{\ast}}-M_{\mathcal{B}}}{M_{B_{q}^{\ast}}-M_{B_{q}}}\,,$
(2.47)

are key inputs for the evaluation of the matrix elements.161616In
Jenkins:1996de ; Jenkins:1996rr , both $1/m_{Q}$ and $1/N_{c}$ expansions were
employed simultaneously, leading to the exact relation $r_{q}(\mathcal{B})\to
2/3$ in the $m_{Q},N_{c}\to\infty$ limit. This holds quite accurately in the
charm sector, and could also be expected to apply, in principle to an even
greater degree of accuracy, in the $b$ sector. However, the relationship is
potentially sensitive to higher-order corrections in the $1/N_{c}$ expansion.
In our numerical analysis we use the experimental values of meson and baryon
mass splittings, when available Workman:2022ynf , and assume exact isospin
symmetry within the hyperfine splittings, i.e.
$M_{B^{\ast}_{d}}-M_{B_{d}}=M_{B^{\ast}_{u}}-M_{B_{u}}$. As for the ratio
$r_{s}(\Omega_{b}^{-})$, since the mass of the $\Omega_{b}^{\ast}$ has not yet
been measured, we employ the result for the splitting
$M_{\Omega^{\ast}_{b}}-M_{\Omega_{b}}$ from Karliner:2014gca , consistently
with the use of the values of the constituent mass ratios in eq. (2.46). This
leads to

$r_{s}(\Omega_{b})=0.66\pm 0.22\,.$ (2.48)

A comparison between the predictions for $r_{q}$, based both on NRCQM fits and
lattice QCD evaluations, alongside the corresponding available experimental
results, is shown in table 3. We note that for the $B$-meson mass splittings,
we use the averages of the experimental values reported in Workman:2022ynf .

Quantity | Experiments Workman:2022ynf  | Lattice QCD Brown:2014ena  | NRCQM Karliner:2014gca
---|---|---|---
$r_{\tilde{q}}(\Sigma_{b})$ | $0.58\pm 0.01$ | $0.62\pm 0.26$ | $0.63\pm 0.24$
$r_{\tilde{q}}(\Xi^{\prime}_{b})$ | $0.60\pm 0.00$ | $0.79\pm 0.27$ | $0.67\pm 0.24$
$r_{s}(\Xi^{\prime}_{b})$ | $0.56\pm 0.02$ | $0.74\pm 0.25$ | $0.63\pm 0.22$
$r_{s}(\Omega_{b})$ | unknown | $0.78\pm 0.22$ | $0.66\pm 0.22$
Table 3: Comparisons of the NRCQM results for the $r_{q}(\mathcal{B})$ to
available experimental data and lattice QCD evaluations. For the $B$-meson
mass splittings, we use the measured values reported in Workman:2022ynf .

Having the ratios of hadron mass splittings under control, we proceed by
relating the meson wave functions in eqs. (2.42)-(2.45) to the static decay
constants via

$|\Psi^{B_{q}}(0)|^{2}=\frac{F_{B_{q}}^{2}(\mu_{0})}{12}\,,$ (2.49)

with

$\langle
0|\bar{q}\gamma^{\mu}\gamma_{5}h_{v}|B_{q}\rangle_{\text{HQET}}=i\,F_{B_{q}}(\mu_{0})\sqrt{M_{B_{q}}}\,v^{\mu}\,,$
(2.50)

following the conventions for the HQET states used in King:2021xqp . Assuming
the constituent-quark relations for the matrix elements of the operators
$\mathcal{O}^{q}_{i}$ in eqs. (2.36), (2.37), as well as the valence quark
approximation result $\tilde{B}=1$ in (2.35), to be satisfied at a low
hadronic scale $\mu_{h}$, in eq. (2.49) we set
$\mu_{0}=\mu_{h}=1.5\,\text{GeV}$, the same hadronic scale that was used in
the HQET sum rule derivation of the corresponding bag parameters in $B$ mesons
King:2021jsq . The value of the static decay constant at the scale $\mu_{h}$
can be extracted using its relation Neubert:1992fk  to the QCD decay constant
in the static limit $\hat{f}_{B_{q}}$,

$\hat{f}_{B_{q}}=\frac{F_{B_{q}}(\mu_{0})}{\sqrt{M^{\phantom{-}}_{B_{q}}}}\bigg{[}1+\frac{\alpha_{s}(\mu_{0})}{2\pi}\bigg{(}\ln\frac{\mu_{b}^{2}}{\mu_{0}^{2}}-\frac{4}{3}\bigg{)}\bigg{]}\,,$
(2.51)

where $\hat{f}_{B_{q}}$ differs from the full QCD decay constant $f_{B_{q}}$
used for meson lifetimes, by the terms of order $\mathcal{O}(1/m_{b})$, and
$\mu_{b}=4.5\,\text{GeV}$. The parameter $\hat{f}_{B_{q}}$ is available from
lattice QCD simulations Aoki:2014nga , from which we take the numerical values

$\hat{f}_{B}=(219\pm 17)\,\text{MeV}\,,\qquad\qquad\hat{f}_{B_{s}}=(264\pm
19)\,\text{MeV}\,,$ (2.52)

which result in

$\begin{split}F_{B}(\mu_{h}=1.5\,\text{GeV})=(0.48\pm
0.04)\,\text{GeV}^{3/2}\,,\\\ F_{B_{s}}(\mu_{h}=1.5\,\text{GeV})=(0.58\pm
0.04)\,\text{GeV}^{3/2}\,,\end{split}$ (2.53)

as compared to $F_{B}(\mu_{b})=(0.53\pm 0.04)\,\text{GeV}^{3/2}$ and
$F_{B_{s}}(\mu_{b})=(0.64\pm 0.05)\,\text{GeV}^{3/2}$. With this ingredient in
place, we list in table 4 the numerical values of the relevant matrix elements
of the operator ${\cal O}_{1}^{q}$ at the scale $\mu_{h}$.

Matrix elements at $\mu_{h}=1.5\,\text{GeV}$ | Value in units $\text{GeV}^{3}$
---|---
$\displaystyle{\langle\mathcal{O}_{1}^{u}\rangle_{\Lambda_{b}^{0}}=\langle\mathcal{O}_{1}^{d}\rangle_{\Lambda_{b}^{0}}}$ | $-0.013\pm 0.002\pm{0.004}$
$\displaystyle{\langle\mathcal{O}_{1}^{u}\rangle_{\Xi_{b}^{0}}}=\langle\mathcal{O}_{1}^{d}\rangle_{\Xi_{b}^{-}}$ | $-0.014\pm 0.002\pm{0.004}$
$\displaystyle{\langle\mathcal{O}_{1}^{s}\rangle_{\Xi_{b}^{0}}=\langle\mathcal{O}_{1}^{s}\rangle_{\Xi_{b}^{-}}}$ | $-0.018\pm 0.003\pm{0.005}$
$\displaystyle{\langle\mathcal{O}_{1}^{s}\rangle_{\Omega_{b}^{-}}}$ | $-0.126\pm 0.046\pm{0.038}$
Table 4: Numerical values for the matrix elements of the operator
$\mathcal{O}_{1}^{q}$ at the scale $\mu_{h}=1.5\,\text{GeV}$, following the
notation $\langle{\cal O}_{i}^{q}\rangle_{\mathcal{B}}\equiv\langle{\cal
B}|{\cal O}_{i}^{q}|{\cal B}\rangle/(2M_{\cal B})$. The first errors are
obtained by varying the input parameters, and the second ones by adding a
conservative $30\%$ model uncertainty. The remaining matrix elements are, at
the scale $\mu_{h}$, related to the matrix elements of $\mathcal{O}^{q}_{1}$
via eqs. (2.35), (2.36), and (2.37), with $\tilde{B}_{i}^{q}(\mu_{h})=1$.

Using the results for the renormalisation group evolution of the matrix
elements of the dimension-six four-quark operators within HQET Shifman:1984wx
; Voloshin:1986dir ; Politzer:1988wp ; Neubert:1996we , for
$\mu_{h}=1.5\,\,\textrm{GeV}$ and $\mu_{b}=4.5\,\,\textrm{GeV}$, we obtain

$\begin{pmatrix}\langle{\cal O}_{1}^{q}\rangle\\\ \langle{\cal
O}_{2}^{q}\rangle\\\ \langle\tilde{\cal O}_{1}^{q}\rangle\\\
\langle\tilde{\cal
O}_{2}^{q}\rangle\end{pmatrix}\\!(\mu_{b})=\begin{pmatrix}1.29&&0&&-0.09&&0\\\
0&&1.29&&0&&-0.09\\\ 0&&0&&1&&0\\\
0&&0&&0&&1\end{pmatrix}\begin{pmatrix}\langle{\cal O}_{1}^{q}\rangle\\\
\langle{\cal O}_{2}^{q}\rangle\\\ \langle\tilde{\cal O}_{1}^{q}\rangle\\\
\langle\tilde{\cal O}_{2}^{q}\rangle\end{pmatrix}\\!(\mu_{h})\,.$ (2.54)

Then, at the scale $\mu_{b}$, the matrix elements for the triplet ${\cal
T}_{b}$ and the $\Omega_{b}$ baryon read respectively

$\begin{split}\begin{pmatrix}\langle\mathcal{O}_{1}^{q}\rangle\\\
\langle\mathcal{O}_{2}^{q}\rangle\\\
\langle\tilde{\mathcal{O}}_{1}^{q}\rangle\\\
\langle\tilde{\mathcal{O}}_{2}^{q}\rangle\end{pmatrix}_{\mathcal{T}_{b}}\\!\\!\\!\\!(\mu_{b})=\begin{pmatrix}1.38\,\langle\mathcal{O}_{1}^{q}\rangle\\\
-0.69\,\langle\mathcal{O}_{1}^{q}\rangle\\\
-\langle\mathcal{O}_{1}^{q}\rangle\\\
\frac{1}{2}\langle\mathcal{O}_{1}^{q}\rangle\end{pmatrix}_{\mathcal{T}_{b}}\\!\\!\\!\\!(\mu_{h})\,,\qquad\qquad\begin{pmatrix}\langle\mathcal{O}_{1}^{s}\rangle\\\
\langle\mathcal{O}_{2}^{s}\rangle\\\
\langle\tilde{\mathcal{O}}_{1}^{s}\rangle\\\
\langle\tilde{\mathcal{O}}_{2}^{s}\rangle\end{pmatrix}_{\Omega_{b}}\\!\\!\\!\\!\\!(\mu_{b})=\begin{pmatrix}1.38\,\langle\mathcal{O}_{1}^{s}\rangle\\\
0.23\,\langle\mathcal{O}_{1}^{s}\rangle\\\
-\langle\mathcal{O}_{1}^{s}\rangle\\\
-\frac{1}{6}\langle\mathcal{O}_{1}^{s}\rangle\end{pmatrix}_{\Omega_{b}}\\!\\!\\!\\!\\!(\mu_{h})\,,\end{split}$
(2.55)

which amounts, for both the triplet and the $\Omega_{b}$, to a modification of
the parameter $\tilde{B}$ from the value $\tilde{B}(\mu_{h})=1$ to
$\tilde{B}(\mu_{b})=1.38$.171717Choosing the value for the initial scale
$\mu_{h}=1\,\text{GeV}$ instead of $\mu_{h}=1.5\,\text{GeV}$ results in
$\tilde{B}(\mu_{b})=1.66$. At the same time, the one-loop running preserves
the ratios between the matrix elements of the operators $\mathcal{O}_{1}^{q}$
and $\mathcal{O}_{2}^{q}$.

We now turn to discuss the remaining, non-spectator matrix elements
Bigi:1993ex ; Bigi:1994ga ; Mannel:1994kv ; Dassinger:2006md ,

$\displaystyle\mu_{\pi}^{2}(\mathcal{B})$
$\displaystyle=-\frac{1}{2M_{\mathcal{B}}}\langle\mathcal{B}|\bar{b}_{v}(iD_{\mu})(iD^{\mu})b_{v}|\mathcal{B}\rangle\,,$
(2.56) $\displaystyle\mu_{G}^{2}(\mathcal{B})$
$\displaystyle=\frac{1}{2M_{\mathcal{B}}}\langle\mathcal{B}|\bar{b}_{v}(iD_{\mu})(iD_{\nu})(-i\sigma^{\mu\nu})b_{v}|\mathcal{B}\rangle\,,$
(2.57) $\displaystyle\rho_{D}^{3}(\mathcal{B})$
$\displaystyle=\frac{1}{2M_{\mathcal{B}}}\langle\mathcal{B}|\bar{b}_{v}(iD_{\mu})(iv\cdot
D)(iD^{\mu})b_{v}|\mathcal{B}\rangle\,,$ (2.58)

which correspond to the kinetic, chromomagnetic, and Darwin parameters
respectively. Following Dassinger:2006md , we define the operators in terms of
the field $b_{v}(x)$, rather than the HQET field $h_{v}(x)$, with differences
due to this choice arising only at order $1/m_{b}^{4}$. These parameters can
be further related to the heavy-quark expansion of the hadron mass Falk:1992wt
; Falk:1992ws ; Bigi:1994ga ; Neubert:1996qg ,

$M_{\mathcal{B}}=m_{b}+\bar{\Lambda}+\frac{\mu_{\pi}^{2}(\mathcal{B})}{2m_{b}}-\frac{\mu_{G}^{2}(\mathcal{B})}{2m_{b}}+\mathcal{O}\left(\frac{1}{m_{b}^{2}}\right)\,,$
(2.59)

where $\bar{\Lambda}\sim 0.5\,\textrm{GeV}$. Applying the expansion (2.59) to
the mass difference between hyperfine partners, and taking into account the
proportionality of the chromomagnetic parameter to the spin factor $d_{\cal
B}$, we have

$\mu_{G}^{2}(\mathcal{B})=d_{\mathcal{B}}\frac{M_{\mathcal{B}^{*}}^{2}-M_{\mathcal{B}}^{2}}{d_{\mathcal{B}}-d_{\mathcal{B}^{*}}}\,,$
(2.60)

with

$d_{\mathcal{B}}=-2\left(S_{\mathcal{B}}(S_{\mathcal{B}}+1)-S_{b}(S_{b}+1)-S_{l}(S_{l}+1)\right)\,,$
(2.61)

and $S_{X}$ denoting the spin of the particle $X$. As only
$d_{\Omega_{b}^{-}}$ is non-zero, with $d_{\Omega_{b}^{-}}=4$ and
$d_{{\Omega_{b}^{-}}^{*}}=-2$, it follows that $\mu_{G}^{2}(\mathcal{B})=0$
for the triplet ${\cal T}_{b}$, while, using the masses and splitting from
Karliner:2014gca , we obtain

$\mu_{G}^{2}(\Omega_{b}^{-})=(0.193\pm 0.065\pm 0.019)\,\textrm{GeV}^{2}\,.$
(2.62)

Here, the first uncertainty is parametric, while the second one corresponds to
our $10\%$ uncertainty estimate from missing higher-order $1/m_{b}$
corrections.

Concerning the kinetic parameter, one can relate
$\mu_{\pi}^{2}(\Lambda_{b}^{0})$ to $\mu_{\pi}^{2}(B)$ via

$\overline{M}_{B}-M_{\Lambda_{b}^{0}}=\bar{\Lambda}_{B}-\bar{\Lambda}_{\Lambda_{b}^{0}}+\frac{\mu_{\pi}^{2}(B)-\mu_{\pi}^{2}(\Lambda_{b}^{0})}{2m_{b}}+\mathcal{O}\left(\frac{1}{m_{b}}\right)\,,$
(2.63)

where we differentiate between the parameter $\bar{\Lambda}$ for mesons and
baryons, and $\overline{M}_{B}$ denotes the spin-averaged meson mass
$\overline{M}_{B}=(M_{B}+3M_{B^{\ast}})/4$. To proceed, we assume the equality
of the difference $\bar{\Lambda}_{B_{q}}-\bar{\Lambda}_{\mathcal{B}}$ in the
bottom and charmed sectors, as well as $\mu_{\pi}^{2}(B)=\mu_{\pi}^{2}(D)$ and
$\mu_{\pi}^{2}(\Lambda_{b}^{0})=\mu_{\pi}^{2}(\Lambda_{c}^{+})$, resulting in
the expression Bigi:1992su ; Bigi:1995jr

$\left(\overline{M}_{D}-M_{\Lambda_{c}^{+}}\right)-\left(\overline{M}_{B}-M_{\Lambda_{b}^{0}}\right)=\left(\frac{1}{2m_{c}}-\frac{1}{2m_{b}}\right)\left(\mu_{\pi}^{2}(B)-\mu_{\pi}^{2}(\Lambda_{b}^{0})\right)\\\
+\mathcal{O}\left(\frac{1}{m_{b}},\frac{1}{m_{c}}\right)\,,$ (2.64)

where $\overline{M}_{D}=(M_{D}+3M_{D^{\ast}})/4$, and for the inputs on the
left-hand side, we have used the isospin-averaged hadron masses. Unlike in the
charm sector, however, there have been analyses of inclusive semileptonic
$B\to X_{c}\,\ell\nu_{\ell}$ decays Alberti:2014yda ; Gambino:2016jkc ;
Bordone:2021oof ; Bernlochner:2022ucr  in order to extract the values of the
parameter $\mu_{\pi}^{2}(B)$ from fits to experimental data. For our numerical
analysis, we use the value obtained in Bordone:2021oof :

$\mu_{\pi}^{2}(B)=\left(0.477\pm 0.056\right)\,\textrm{GeV}^{2}\,.$ (2.65)

Furthermore, we adopt the spectroscopic estimate of the size of
SU(3)F-breaking from Bordone:2022qez ; Lenz:2022rbq :

$\mu_{\pi}^{2}(B_{s})-\mu_{\pi}^{2}(B)=\left(0.04\pm
0.02\right)\,\textrm{GeV}^{2}\,.$ (2.66)

| $\Lambda_{b}^{0}$ | $\Xi_{b}^{0,-}$ | $\Omega_{b}^{-}$
---|---|---|---
$\mu_{G}^{2}(\mathcal{B})/\,\textrm{GeV}^{2}$ | 0 | 0 | $0.193\pm 0.068$
$\mu_{\pi}^{2}(\mathcal{B})/\,\textrm{GeV}^{2}$ | $0.50\pm 0.06$ | $0.54\pm 0.06$ | $0.56\pm 0.06$
$\rho_{D}^{3}(\mathcal{B})/\,\textrm{GeV}^{3}$ | $0.031\pm 0.009$ | $0.037\pm 0.009$ | $0.050\pm 0.021$
Table 5: Non-perturbative parameters for the non-spectator contributions used
in our analysis. Values for $\mu_{\pi}^{2}(\mathcal{B})$ follow from the
relations derived in eqs. (2.64) and (2.68), with $\mu_{\pi}^{2}(B)$ taken
from the fit value in Bordone:2021oof  and $\mu_{\pi}^{2}(B_{s})$ obtained
using the SU(3)F-breaking estimate from Bigi:2011gf . Values for
$\rho_{D}^{3}(\mathcal{B})$ follow from employing the equation of motion
(2.73). The errors quoted here are obtained by combining in quadrature the
parametric uncertainty and the uncertainty due to missing power corrections.

For the $\Omega_{b}$, the analogous relation, derived for the first time in
Gratrex:2022xpm , is

$\mu_{\pi}^{2}(\Omega_{b}^{-})\left(\frac{1}{2m_{b}}-\frac{1}{2m_{c}}\right)\simeq
m_{c}-m_{b}+\frac{1}{3}\left(\left(M_{\Omega_{b}^{-}}+2M_{{\Omega_{b}^{-}}^{*}}\right)-\left(M_{\Omega_{c}^{0}}+2M_{{\Omega_{c}^{0}}^{*}}\right)\right)+\mathcal{O}\left(\frac{1}{m_{b}},\frac{1}{m_{c}}\right)\,.$
(2.67)

This can be recast, using a similar relation for the $B_{s}$ meson, as

$\left(\mu_{\pi}^{2}(\Omega_{b}^{-})-\mu_{\pi}^{2}(B_{s})\right)\left(\frac{1}{2m_{b}}-\frac{1}{2m_{c}}\right)\simeq\overline{M}_{D_{s}}-\overline{M}_{B_{s}}+\frac{1}{3}\left(\left(M_{\Omega_{b}^{-}}+2M_{{\Omega_{b}^{-}}^{*}}\right)-\left(M_{\Omega_{c}^{0}}+2M_{{\Omega_{c}^{0}}^{*}}\right)\right)\,,$
(2.68)

up to corrections of order $1/m_{b,c}$, and where $\overline{M}_{B_{s}}$,
$\overline{M}_{D_{s}}$ are again spin-averaged meson masses. Concerning the
quark masses, we use their values in the kinetic scheme, i.e. $m_{b}^{\rm
kin}(\mu_{\text{cut}}=1\,\,\textrm{GeV})=4.57\,\,\textrm{GeV}$, as extracted
from the fit in Bordone:2021oof ; Bordone:2022qez , and $m^{\rm
kin}_{c}(\mu_{\text{cut}}=0.5\,\textrm{GeV})=1.40\,\,\textrm{GeV}$
Fael:2020iea ; Chetyrkin:2000yt ; Herren:2017osy . The values for the
differences of the kinetic parameters turn out to be small, and we
obtain181818This small separation between $\mu_{\pi}^{2}(B)$ and
$\mu_{\pi}^{2}(\Lambda_{b}^{0})$ is consistent with the sum rules calculation
in Colangelo:1995qp .

$\displaystyle\mu_{\pi}^{2}(\Lambda_{b}^{0})-\mu_{\pi}^{2}(B)$
$\displaystyle=(0.029\pm 0.001\pm 0.015)\,\textrm{GeV}^{2}\,,$ (2.69)
$\displaystyle\mu_{\pi}^{2}(\Xi_{b})-\mu_{\pi}^{2}(B)$
$\displaystyle=(0.061\pm 0.002\pm 0.030)\,\textrm{GeV}^{2}\,,$ (2.70)
$\displaystyle\mu_{\pi}^{2}(\Omega_{b}^{-})-\mu_{\pi}^{2}(B_{s})$
$\displaystyle=\left(0.040\pm 0.023\pm 0.020\right)\,\textrm{GeV}^{2}\,,$
(2.71)

where again the first quoted errors represent the parametric uncertainties,
while the second ones follow from our assignment of $50\%$ uncertainties to
account for possibly sizeable $1/m_{c}$ corrections. Combining the above
results with those in (2.65), (2.66) leads to our estimates for the baryonic
kinetic parameters presented in table 5.

As for the Darwin parameter $\rho_{D}^{3}(\mathcal{B})$, this can be related,
up to $\mathcal{O}(1/m_{b})$ corrections, to the four-quark matrix elements by
the equation of motion for the gluon field strength tensor,

$\left[iD_{\mu},iD_{\nu}\right]=ig_{s}G_{\mu\nu}\,,\qquad[D^{\mu},G_{\mu\nu}]=-g_{s}t^{a}\sum_{q=u,d,s}\bar{q}\gamma_{\nu}t^{a}q\,,$
(2.72)

which leads to the relation

$2M_{\mathcal{B}}\,\rho_{D}^{3}(\mathcal{B})=g_{s}^{2}\sum_{q=u,d,s}\langle\mathcal{B}|\left(-\frac{1}{8}\mathcal{O}_{1}^{q}+\frac{1}{24}\tilde{\mathcal{O}}_{1}^{q}+\frac{1}{4}\mathcal{O}_{2}^{q}-\frac{1}{12}\tilde{\mathcal{O}}_{2}^{q}\right)|\mathcal{B}\rangle+\mathcal{O}\left(\frac{1}{m_{b}}\right)\,,$
(2.73)

in terms of the operator basis defined in (2.32). We evaluate the right-hand
side of eq. (2.73) using the matrix elements of the four-quark operators
renormalised at the scale $\mu_{0}=\mu_{b}$, which, together with
$\alpha_{s}(\mu_{b})=0.22$, results in the values for the Darwin parameter
shown in table 5.191919For comparison, the values obtained using instead
$\alpha_{s}=1$, and the four-quark matrix elements at the low hadronic scale
$\mu_{h}$, read
$(\rho_{D}^{3}(\Lambda_{b}),\rho_{D}^{3}(\Xi_{b}),\rho_{D}^{3}(\Omega_{b}))\simeq(0.11,0.13,0.18)\,\,\textrm{GeV}^{3}$.
However, the limit on the size of $\rho_{D}^{3}(B)$ derived in Chow:1995mz
supports a lower value of $\alpha_{s}$, consistent with the results in table
5.

##  3 Numerical Analysis and Results

In this section, we present our predictions for the total decay widths of
$b$-baryons and their lifetime ratios, as well as for the values of their
lifetimes normalised to $\tau(B_{d})$, as summarised in table 6 and figure 3.
We also provide results for the semileptonic decay widths and inclusive
$b$-baryon semileptonic branching fractions, shown in eqs. (3.11) and (3.12),
(3.13).

The values of the non-perturbative parameters used in our numerical analysis
are displayed in tables 4 and 5 of section 2.3, while all remaining inputs are
collected in appendix A. Note that the renormalisation scales $\mu_{1}$ and
$\mu_{0}$ are varied independently, both in the same interval
$\mu_{b}/2\leq\mu_{0,1}\leq 2\mu_{b}$, with $\mu_{b}=4.5\,\textrm{GeV}$, and
using as central values $\mu_{0}=\mu_{1}=\mu_{b}$. In addition, in order to
account for possible uncertainties in our assumption for the “factorisation”
scale $\mu_{h}$, we vary this between $1\,\textrm{GeV}$ and
$1.5\,\textrm{GeV}$, fixing its central value to $1.5\,\textrm{GeV}$.

As we present results for the lifetime ratios of $b$-baryons with the $B_{d}$
meson, a couple of comments with respect to our recent study Lenz:2022rbq  are
in order. Firstly, as discussed in section 2.2, in our analysis of the
$b$-baryon total widths, we treat the dimension-seven contributions as an
additional source of uncertainty, and do not provide any estimates for their
central values. Hence, for consistency, here we have adopted the same
treatment also for the total width of the $B_{d}$ meson, differently from
Lenz:2022rbq .202020However, for the $B_{d}$ meson, the dimension-seven four-
quark contribution turns out to be negligible, see eq. (3.4) of Lenz:2022rbq .
Secondly, as the value of the Darwin parameter $\rho_{D}^{3}({\cal B})$ for
baryons is obtained using the equations of motion for the gluon field strength
tensor evaluated at the scale $\mu_{b}$, see eq. (2.73), we again follow the
same procedure for the $B_{d}$ meson and use212121This value is consistent
with the experimental fit in Bernlochner:2022ucr , rather than that in
Bordone:2021oof .

$\rho_{D}^{3}(B_{d})=(0.028\pm 0.010)\,{\rm GeV}^{3}.$ (3.1)

Our predictions for the total widths are determined from eq. (2.15), while the
lifetime ratios are obtained using the relation

$\frac{\tau(H_{1})}{\tau(H_{2})}=1+\left[\Gamma(H_{2})-\Gamma(H_{1})\right]^{\rm
HQE}\tau(H_{1})^{\rm exp}\,,$ (3.2)

where the difference $\Gamma(H_{2})-\Gamma(H_{1})$ is computed from eq.
(2.15), and we use as input the experimental value for the lifetime of the
$H_{1}$ hadron.

In order to understand the size of each of the contributions in the HQE
included in our analysis, below we show our results for the decomposition of
the total widths of $b$-baryons, explicitly indicating the LO- and NLO-QCD
corrections when the latter are present. For central values of the input
parameters, we obtain

$\displaystyle\Gamma(\Lambda_{b}^{0})$
$\displaystyle=\Gamma_{0}\Biggr{[}(\,\underbrace{5.97}_{\rm
LO}-\underbrace{0.44}_{\rm\Delta
NLO})-0.14\,\frac{\mu_{\pi}^{2}(\Lambda_{b}^{0})}{\,\textrm{GeV}^{2}}-1.35\,\frac{\rho_{D}^{3}(\Lambda_{b}^{0})}{\,\textrm{GeV}^{3}}-\big{(}\underbrace{10.6}_{\rm
LO}+\underbrace{5.04}_{\rm\Delta
NLO}\\!\big{)}\frac{\langle\mathcal{O}_{1}^{q}\rangle_{\Lambda_{b}^{0}}}{\,\textrm{GeV}^{3}}\Biggl{]}\,,$
(3.3) $\displaystyle\Gamma(\Xi_{b}^{0})$
$\displaystyle=\Gamma_{0}\Biggr{[}(\,\underbrace{5.97}_{\rm
LO}-\underbrace{0.44}_{\rm\Delta
NLO})-0.14\,\frac{\mu_{\pi}^{2}(\Xi_{b}^{0})}{\,\textrm{GeV}^{2}}-1.35\,\frac{\rho_{D}^{3}(\Xi_{b}^{0})}{\,\textrm{GeV}^{3}}$
$\displaystyle\qquad\qquad\qquad-\big{(}\underbrace{18.2}_{\rm
LO}+\underbrace{4.02}_{\rm\Delta
NLO}\\!\big{)}\frac{\langle\mathcal{O}_{1}^{q}\rangle_{\Xi_{b}^{0}}}{\,\textrm{GeV}^{3}}-\big{(}\underbrace{-7.31}_{\rm
LO}+\underbrace{1.48}_{\rm\Delta
NLO}\\!\big{)}\frac{\langle\mathcal{O}_{1}^{s}\rangle_{\Xi_{b}^{0}}}{\,\textrm{GeV}^{3}}\Biggl{]}\,,$
(3.4) $\displaystyle\Gamma(\Xi_{b}^{-})$
$\displaystyle=\Gamma_{0}\Biggr{[}(\,\underbrace{5.97}_{\rm
LO}-\underbrace{0.44}_{\rm\Delta
NLO})-0.14\,\frac{\mu_{\pi}^{2}(\Xi_{b}^{-})}{\,\textrm{GeV}^{2}}-1.35\,\frac{\rho_{D}^{3}(\Xi_{b}^{-})}{\,\textrm{GeV}^{3}}$
$\displaystyle\qquad\qquad\qquad-\big{(}\underbrace{-7.62}_{\rm
LO}+\underbrace{1.02}_{\rm\Delta
NLO}\\!\big{)}\frac{\langle\mathcal{O}_{1}^{q}\rangle_{\Xi_{b}^{-}}}{\,\textrm{GeV}^{3}}-\big{(}\underbrace{-7.31}_{\rm
LO}+\underbrace{1.48}_{\rm\Delta
NLO}\\!\big{)}\frac{\langle\mathcal{O}_{1}^{s}\rangle_{\Xi_{b}^{-}}}{\,\textrm{GeV}^{3}}\Biggl{]}\,,$
(3.5) $\displaystyle\Gamma(\Omega_{b}^{-})$
$\displaystyle=\Gamma_{0}\Biggr{[}(\,\underbrace{5.97}_{\rm
LO}-\underbrace{0.44}_{\rm\Delta
NLO})-0.14\,\frac{\mu_{\pi}^{2}(\Omega_{b}^{-})}{\,\textrm{GeV}^{2}}-0.24\,\frac{\mu_{G}^{2}(\Omega_{b}^{-})}{\,\textrm{GeV}^{2}}-1.35\,\frac{\rho_{D}^{3}(\Omega_{b}^{-})}{\,\textrm{GeV}^{3}}$
$\displaystyle\qquad\qquad\qquad-\big{(}\underbrace{-3.81}_{\rm
LO}+\underbrace{0.72}_{\rm\Delta
NLO}\\!\big{)}\frac{\langle\mathcal{O}_{1}^{s}\rangle_{\Omega_{b}^{-}}}{\,\textrm{GeV}^{3}}\Biggl{]}\,,$
(3.6)

with $q=u,d$. The total decay widths are clearly dominated by the dimension-
three contribution, with the radiative corrections giving a ${\sim}10$%
effect. Among the power-suppressed terms, the largest contribution comes from
dimension-six four-quark operators, and in particular from the exc topology,
which enters the $\Lambda_{b}^{0}$ and $\Xi_{b}^{0}$ widths. Radiative
corrections also play an important role, and range from ${\sim}10$% to
${\sim}50$% of the four-quark contribution depending on the specific topology.
The Darwin term gives the next dominant power correction, and in some cases
partially compensates the contribution of four-quark operators, as for example
in the $\Lambda_{b}^{0}$, eq. (3.3).

For completeness, we also show the decomposition for the total width of the
$B_{d}$ meson, cf. eq. (3.4) of Lenz:2022rbq :

$\displaystyle\Gamma(B_{d}^{0})$ $\displaystyle=$
$\displaystyle\Gamma_{0}\biggl{[}(\,\underbrace{5.97}_{\rm
LO}-\underbrace{0.44}_{\rm\Delta NLO})-\,0.14\,\frac{\mu_{\pi}^{2}(B)}{\rm
GeV^{2}}-0.24\,\frac{\mu_{G}^{2}(B)}{\rm
GeV^{2}}-1.35\,\frac{\rho_{D}^{3}(B)}{\rm GeV^{3}}$ (3.7)
$\displaystyle\quad-\,(\,\underbrace{0.012}_{\rm
LO}+\underbrace{0.022}_{\rm\Delta
NLO}\,)\,{\tilde{B}}_{1}^{q}+(\,\underbrace{0.012}_{\rm
LO}+\underbrace{0.020}_{\rm\Delta
NLO}\,)\,{\tilde{B}}_{2}^{q}-\,(\,\underbrace{0.74}_{\rm
LO}+\underbrace{0.03}_{\rm\Delta NLO})\,{\tilde{B}}_{3}^{q}$
$\displaystyle\quad+\,(\,\underbrace{0.78}_{\rm
LO}-\underbrace{0.01}_{\rm\Delta
NLO})\,{\tilde{B}}_{4}^{q}-0.14\,\tilde{\delta}^{qq^{\prime}}_{1}+0.02\,\tilde{\delta}^{qq^{\prime}}_{2}-2.29\,\tilde{\delta}^{qq^{\prime}}_{3}+0.00\,\tilde{\delta}^{qq^{\prime}}_{4}$
$\displaystyle\quad-\,0.01\,\tilde{\delta}^{sq}_{1}+0.01\,\tilde{\delta}^{sq}_{2}-0.69\,\tilde{\delta}^{sq}_{3}+0.78\,\tilde{\delta}^{sq}_{4}\,\biggr{]}\,,$

where $\tilde{B}_{i}^{q}$ and
$\tilde{\delta}_{i}^{qq^{\prime}},\tilde{\delta}_{i}^{sq}$ denote,
respectively, the $B$ meson dimension-six Bag parameters and the ‘eye
contractions’, see Lenz:2022rbq  for details. Their numerical values, as well
as of those for $\mu_{\pi}^{2}(B)$ and $\mu_{G}^{2}(B)$, are taken to be the
same as in Lenz:2022rbq .

Our HQE predictions for the $b$-baryon lifetimes and their ratios, together
with the corresponding experimental values, are presented in table 6 and
visualised in figure 3. The quoted theoretical errors are obtained by
combining uncertainties due to variation of the input parameters and of the
renormalisation scales $\mu_{0},\mu_{1}$, and $\mu_{h}$, as well as an
additional $15\%$ uncertainty added to the dimension-six contribution to
account for missing $1/m_{b}^{4}$ corrections. Overall, we find excellent
agreement between the HQE predictions and the experimental data for all the
observables considered.

It is important to point out that computing the lifetime ratios entirely
within the HQE, i.e. without using the experimental values for
$\tau(H_{1})^{\rm exp}$ in eq. (3.2), leads to very similar results as those
in table 6, albeit with slightly larger uncertainties. Furthermore, when using
the HQET sum rules result for the four-quark matrix elements Colangelo:1996ta

$\displaystyle{\langle\mathcal{O}_{1}^{u}\rangle_{\Lambda_{b}}=\langle\mathcal{O}_{1}^{d}\rangle_{\Lambda_{b}}}=-(3.2\pm
1.6)\times 10^{-3}{\rm\,\textrm{GeV}}^{3}\,,$ (3.8)

we obtain a larger value for the lifetime ratio
$\tau(\Lambda_{b}^{0})/\tau(B_{d}^{0})$, namely

$\tau(\Lambda_{b}^{0})/\tau(B_{d}^{0})=0.976\pm 0.012\,,$ (3.9)

which however is consistent, within uncertainties, with the value shown in
table 6.

Finally, we also present HQE predictions for the inclusive semileptonic decay
rates $\Gamma_{\rm SL}({\cal B})$, defined as

$\Gamma_{\rm SL}({\cal B})\equiv\Gamma({\cal B}\to
X_{c+u}\ell\bar{\nu}_{\ell})\,,$ (3.10)

with a massless lepton $\ell=e,\mu$. We obtain

$\displaystyle\Gamma_{\rm SL}({\cal T}_{b})$
$\displaystyle=0.075^{+0.004}_{-0.003}\,\,{\rm
ps}^{-1}\,,\qquad\qquad\Gamma_{\rm
SL}(\Omega_{b})=0.073^{+0.004}_{-0.003}\,\,{\rm ps}^{-1}\,,$ (3.11)

which leads to the following results for the inclusive semileptonic branching
fractions ${\rm BR}_{\rm SL}({\cal B})$:

$\displaystyle{\rm BR}_{\rm SL}(\Lambda_{b}^{0})$
$\displaystyle=(11.0^{+0.6}_{-0.5})\,\%\,,\qquad\qquad{\rm BR}_{\rm
SL}(\Xi_{b}^{-})=(11.7^{+0.7}_{-0.6})\,\%,$ (3.12) $\displaystyle{\rm BR}_{\rm
SL}(\Xi_{b}^{0})$ $\displaystyle=(11.1^{+0.6}_{-0.6})\,\%\,,\qquad\qquad{\rm
BR}_{\rm SL}(\Omega_{b}^{-})=(12.0^{+1.4}_{-1.4})\,\%\,,$ (3.13)

where

${\rm BR}_{\rm SL}({\cal B})=\Gamma_{\rm SL}({\cal B})\,\tau({\cal B})^{\rm
exp}\,.$ (3.14)

Note that the value for ${\rm BR}_{\rm SL}(\Lambda_{b}^{0})$ in eq. (3.12)
perfectly agrees with the result obtained in the recent study
Colangelo:2020vhu . Although measurements of inclusive $b$-baryon semileptonic
branching fractions are extremely difficult at present machines, the
theoretical predictions might still prove useful in Monte Carlo simulations.

Observable |  HQE prediction |  Experimental value
---|---|---
$\Gamma(\Lambda_{b}^{0})$ | $0.671^{+0.108}_{-0.071}\,{\rm ps^{-1}}$ | $(0.680\pm 0.004)\,{\rm ps^{-1}}$
$\Gamma(\Xi_{b}^{0})$ | $0.670^{+0.108}_{-0.071}\,{\rm ps^{-1}}$ | $(0.678\pm 0.014)\,{\rm ps^{-1}}$
$\Gamma(\Xi_{b}^{-})$ | $0.622^{+0.104}_{-0.067}\,{\rm ps^{-1}}$ | $(0.636\pm 0.016)\,{\rm ps^{-1}}$
$\Gamma(\Omega_{b}^{-})$ | $0.591^{+0.108}_{-0.071}\,{\rm ps^{-1}}$ | $0.610^{+0.070}_{-0.066}\,{\rm ps^{-1}}$
$\tau(\Lambda_{b}^{0})/\tau(B_{d}^{0})$ | $0.955\pm 0.014$ | $0.969\pm 0.006$
$\tau(\Xi_{b}^{0})/\tau(B_{d}^{0})$ | $0.956\pm 0.023$ | $0.974\pm 0.020^{\,*}$
$\tau(\Xi_{b}^{-})/\tau(B_{d}^{0})$ | $1.029\pm 0.015$ | $1.035\pm 0.027^{\,*}$
$\tau(\Omega_{b}^{-})/\tau(B_{d}^{0})$ | $1.081\pm 0.042$ | $1.080^{+0.118\,*}_{-0.112}$
$\tau(\Xi_{b}^{0})/\tau(\Lambda_{b}^{0})$ | $1.002\pm 0.023$ | $1.006\pm 0.021^{\,*}$
$\tau(\Xi_{b}^{-})/\tau(\Lambda_{b}^{0})$ | $1.078\pm 0.021$ | $1.069\pm 0.028^{\,*}$
$\tau(\Omega_{b}^{-})/\tau(\Lambda_{b}^{0})$ | $1.132\pm 0.047$ | $1.115^{+0.122\,*}_{-0.116}$
$\tau(\Xi_{b}^{0})/\tau(\Xi_{b}^{-})$ | $0.929\pm 0.028$ | $0.929\pm 0.028$
Table 6:  Comparison between our predictions based on the HQE and the data.
The theoretical uncertainties are obtained by combining uncertainties due to
input parameters, the renormalisation scales $\mu_{0},\mu_{1}$, and $\mu_{h}$,
and missing $1/m_{b}^{4}$ corrections. The experimental numbers marked with an
asterisk are obtained by dividing the corresponding values shown in table 1,
and do not take into account possible experimental correlations.

Figure 3: Graphical representation of the results presented in table 6.

##  4 Conclusions

We have performed a phenomenological study of the lifetimes of $b$-baryons,
including for the first time the contribution of the Darwin operator and a new
extraction of the matrix elements of the four-quark operators within the
framework of the non-relativistic constituent quark model. Overall, we observe
an excellent agreement between our predictions and the experimental data. For
the lifetime ratios $\tau(\Lambda_{b}^{0})/\tau(B_{d})$,
$\tau(\Xi_{b}^{0})/\tau(B_{d})$, and $\tau(\Xi_{b}^{0})/\tau(\Xi_{b}^{-})$,
the theoretical and experimental uncertainties are comparable, while for the
total decay rates the theoretical errors dominate, although, in the case of
the $\Omega_{b}^{-}$ baryon, the experimental uncertainties are also still
quite sizeable.

In particular, we find

$\displaystyle\frac{\tau(\Lambda_{b}^{0})}{\tau(B_{d}^{0})}^{\rm
HQE}=1-(0.045\pm 0.014)\,,$
$\displaystyle\frac{\tau(\Lambda_{b}^{0})}{\tau(B_{d}^{0})}^{\rm
Exp.}=1-(0.031\pm 0.006)\,,$ (4.1)

showing that the measured suppression of the $\Lambda_{b}^{0}$ lifetime by
$(-3.1\pm 0.6)\%$ compared to $\tau(B_{d})$ is impressively confirmed by the
corresponding theory prediction of $(-4.5\pm 1.4)\%$. Therefore, we do not see
any indications for visible violations of quark-hadron duality affecting the
HQE, as applied to the $\Lambda_{b}^{0}$ baryon.

It is interesting to note that the theory estimate from 1986 Shifman:1986mx
led to almost exactly the same central value as the one obtained in our study.
The authors of Shifman:1986mx  included in their analysis: LO-QCD corrections
to the free-quark decay, $\Gamma_{3}^{(0)}$ in eq. (2.15), taking into account
charm quark mass dependence; LO-QCD corrections to the spectator effects,
$\tilde{\Gamma}_{6}^{(0)}$ in eq. (2.15), without charm quark mass dependence;
and estimates of the matrix elements of the four-quark operators based on a
simplified version of the non-relativistic constituent quark model. They
neglected corrections of order $1/m_{b}^{2}$, i.e. $\Gamma_{5}^{(0)}$ in eq.
(2.15), as well as $1/N_{c}$ corrections in the free quark decay. Furthermore,
the NLO-QCD corrections to the $\Delta B=1$ Wilson coefficients, to the free-
quark decay, $\Gamma_{3}^{(1)}$ in eq. (2.15), and to the spectator effects,
$\tilde{\Gamma}_{6}^{(1)}$ in eq. (2.15), as well as the contribution of the
Darwin operator, $\Gamma_{6}^{(0)}$ in eq. (2.15), were unknown in 1986.
Shifman and Voloshin correctly predicted 36 years ago a small negative
deviation of $\tau(\Lambda_{b}^{0})/\tau(B_{d}^{0})$ from one, however, the
perfect matching of their result with our post-diction from 2023 is a kind of
numerical coincidence, since the effect of their approximations seems to have
cancelled with the low value of the decay constant used in 1986, $f_{B}=110$
MeV, resulting in $\left(f_{B}^{(1986)}/f_{B}^{(2023)}\right)^{2}\approx
0.34$.

Moreover, we confirm the experimentally observed lifetime splitting of the
$\Xi_{b}^{0}$ and $\Xi_{b}^{-}$ baryons

$\displaystyle\frac{\tau(\Xi_{b}^{0})}{\tau(\Xi_{b}^{-})}^{\rm
HQE}=1-(0.071\pm 0.028)\,,$
$\displaystyle\frac{\tau(\Xi_{b}^{0})}{\tau(\Xi_{b}^{-})}^{\rm
Exp.}=1-(0.071\pm 0.028)\,,$ (4.2)

coincidentally obtaining the same central value and uncertainty estimate. For
the $\Omega_{b}^{-}$ baryon we predict a larger lifetime compared to the
$B^{0}_{d}$ meson, although here a clear experimental confirmation is still
missing.

Our results also agree, within uncertainties, with the most recent estimate of
$b$-baryon lifetimes presented in Cheng:2018rkz . This agreement holds in
spite of the fact that NLO-corrections in dimension-six four-quark
contributions, as well as the Darwin contribution, which was at the time
unknown, are missing from the theoretical expression in Cheng:2018rkz , while
the uncertainties in Cheng:2018rkz  are artificially small, as they arise only
from the variation of $\mu_{h}$ and from the bag parameters entering the four-
quark contribution to $\tau(B_{d}^{0})$, and do not include other parametric
and scale uncertainties.

Concerning the lifetime hierarchy, our calculations indicate

$\displaystyle\tau(\Lambda_{b}^{0})\approx\tau(\Xi_{b}^{0})<\tau(\Xi_{b}^{-})\leq\tau(\Omega_{b}^{-})\,,$
(4.3)

which is confirmed by data. Note that this hierarchy was already predicted in
e.g. Guberina:1999bw ; Cheng:2018rkz .

Finally, we have presented numerical updates for the inclusive semileptonic
branching fractions of the $b$-baryons, which currently seem to be difficult
to measure at LHCb, and are not possible at $\Upsilon(4S)$ runs with Belle II.
However, they might be feasible for the flavour physics programme at the high
luminosity upgrade of the LHC Cerri:2018ypt  or even further in the future at
FCC-ee, see e.g. Lenz:2021bkv ; Lenz:2022kal .

In order to further improve the theoretical precision in the lifetime ratios,
the following calculations can be performed in the future:

  * $*$

Non-perturbative, and in particular lattice QCD, determinations of the matrix
elements of the four-quark operators of dimension-six,
$\langle\tilde{\mathcal{O}}_{6}\rangle$, and of dimension-seven,
$\langle\tilde{\mathcal{O}}_{7}\rangle$.

  * $*$

NNLO-QCD corrections to the dimension-six spectator contributions,
$\tilde{\Gamma}_{6}^{(2)}$.

  * $*$

Complete determination of LO-QCD dimension-seven contributions,
$\Gamma_{7}^{(0)}$.

As for the total decay rates, the HQE prediction is dominated by the free-
quark decay. In this case, the theoretical uncertainties could be
significantly reduced if the complete NNLO-QCD contributions, i.e.
$\Gamma_{3}^{(2)}$, were available. Hence, the computation of the missing
$\alpha_{s}^{(2)}$-corrections to non-leptonic $b$-quark decays is highly
desirable.

In conclusion, combined with our recent studies on charmed hadrons
Gratrex:2022xpm ; King:2021xqp  and on $B$ mesons Lenz:2022rbq , the results
of this work confirm that the HQE provides a consistent framework to predict
inclusive decay rates of heavy hadrons.

### Acknowledgements

We would like to thank Olivier Schneider for providing us with old HFAG
averages for the $\Lambda_{b}$ lifetime. We would also like to thank Johannes
Albrecht and Tim Gershon for useful correspondence on the prospects for
measuring inclusive semileptonic decay rates of $b$-baryons at LHCb. We
acknowledge support from the Alexander von Humboldt Foundation in the
framework of the Research Group Linkage Programme, funded by the German
Federal Ministry of Education. JG and BM have also been supported by the
Croatian Science Foundation (HRZZ) project “Heavy hadron decays and lifetimes”
IP-2019-04-7094. The work of MLP was funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - project number
500314741. JG and IN wish to thank the theoretical particle physics group at
the University of Siegen for the kind hospitality shown during their recent
stay, where part of this work was undertaken.

##  Appendix A Numerical inputs

Here we collect the values of the parameters (table 7) and of the $\Delta B=1$
Wilson coefficients (table 8) used in our analysis.

Parameter | Value | Source | Parameter | Value | Source
---|---|---|---|---|---
$M_{B^{+}}$ | $5.27934\,\textrm{GeV}$ | Workman:2022ynf  | $|V_{us}|$ | $0.22500^{+0.00024}_{-0.00021}$ | Charles:2004jd
$M_{B_{d}}$ | $5.27965\,\textrm{GeV}$ | $\displaystyle\frac{|V_{ub}|}{|V_{cb}|}$ | $0.08848^{+0.00224}_{-0.00219}$
$M_{B_{s}}$ | $5.36688\,\textrm{GeV}$ | $V_{cb}$ | $0.04145^{+0.00035}_{-0.00061}$
$M_{\Lambda_{b}}$ | $5.61960\,\textrm{GeV}$ | $\delta$ | $\left(65.5^{+1.3}_{-1.2}\right)^{\circ}$
$M_{\Xi_{b}^{-}}$ | $5.7970\,\textrm{GeV}$ | $m_{b}^{\rm kin}$ | $(4.573\pm 0.012)\,\textrm{GeV}$ | Bordone:2021oof
$M_{\Xi_{b}^{0}}$ | $5.7919\,\textrm{GeV}$ | $\bar{m}_{c}(\bar{m}_{c})$ | $(1.27\pm 0.02)\,\textrm{GeV}$ | Workman:2022ynf
$M_{\Omega_{b}}$ | $6.9452\,\textrm{GeV}$ | $f_{B}$ | $(0.1900\pm 0.0013)\,\textrm{GeV}$ | Aoki:2019cca
$\alpha_{s}(M_{Z})$ | $0.1179\pm 0.0010$ | $f_{B_{s}}$ | $(0.2303\pm 0.0013)\,\textrm{GeV}$
Table 7: Summary of inputs used in the numerical analysis. Values of the non-perturbative parameters for $b$-baryons are presented in tables 4 and 5.  $\mu_{1}\text{[GeV]}$ |  2.5 |  4.2 |  4.5 |  4.8 |  9
---|---|---|---|---|---
$C_{1}(\mu_{1})$ |  1.13 |  1.08 |  1.08 |  1.07 |  1.04
|  (1.17) |  (1.12) |  (1.11) |  (1.11) |  (1.07)
$C_{2}(\mu_{1})$ |  $-$0.27 |  $-$0.19 |  $-$0.18 |  $-$0.17 |  $-$0.11
|  ($-$0.36) |  ($-$0.27) |  ($-$0.26) |  ($-$0.25) |  ($-$0.17)
$C_{3}(\mu_{1})$ |  0.02 |  0.01 |  0.01 |  0.01 |  0.01
|  (0.02) |  (0.01) |  (0.01) |  (0.01) |  (0.01)
$C_{4}(\mu_{1})$ |  $-$0.05 |  $-$0.04 |  $-$0.03 |  $-$0.03 |  $-$0.02
|  ($-$0.04) |  ($-$0.03) |  ($-$0.03) |  ($-$0.03) |  ($-$0.02)
$C_{5}(\mu_{1})$ |  0.01 |  0.01 |  0.01 |  0.01 |  0.01
|  (0.01) |  (0.01) |  (0.01) |  (0.01) |  (0.01)
$C_{6}(\mu_{1})$ |  $-$0.06 |  $-$0.04 |  $-$0.04 |  $-$0.04 |  $-$0.03
|  ($-$0.05) |  ($-$0.03) |  ($-$0.03) |  ($-$0.03) |  ($-$0.02)
$C_{8}^{\rm eff}(\mu_{1})$ |  ($-$0.17) |  ($-$0.15) |  ($-$0.15) |  ($-$0.15) |  ($-$0.14)
Table 8: Values of the Wilson coefficients at NLO(LO)-QCD for different
choices of $\mu_{1}$.

##  Appendix B Dimension-six four-quark operator contributions at LO-QCD

The analytical expressions for the functions
$\tilde{\Gamma}_{6,T}^{q}(x_{f_{1}},x_{f_{2}})$, introduced in eq. (2.25), are
provided explicitly below at LO-QCD. For non-leptonic transitions $b\to
q_{1}\bar{q}_{2}q_{3}$, with $q_{1,2}=u,c$ and $q_{3}=d,s$, they read
respectively

$\displaystyle\tilde{\Gamma}_{6,{\rm int}^{-}}^{q_{3}}(x_{q_{1}},x_{q_{2}})$
$\displaystyle=\frac{G_{F}^{2}}{12\pi}|V_{q_{1}b}|^{2}|V_{q_{2}q_{3}}|^{2}m_{b}^{2}\sqrt{\lambda(1,x_{q_{1}},x_{q_{2}})}\,\Bigg{\\{}k_{1}\Big{[}\,\omega_{1}(x_{q_{1}},x_{q_{2}})\,\langle\mathcal{O}_{1}^{q_{3}}\rangle-2\omega_{2}(x_{q_{1}},x_{q_{2}})\langle\mathcal{O}_{2}^{q_{3}}\rangle\Big{]}$
$\displaystyle+k_{2}\Big{[}\omega_{1}(x_{q_{1}},x_{q_{2}})\langle\tilde{\mathcal{O}}_{1}^{q_{3}}\rangle-2\omega_{2}(x_{q_{1}},x_{q_{2}})\langle\tilde{\mathcal{O}}_{2}^{q_{3}}\rangle\Big{]}\Bigg{\\}}\,,$
(B.1) $\displaystyle\tilde{\Gamma}_{6,{\rm exc}}^{q_{2}}(x_{q_{1}},x_{q_{3}})$
$\displaystyle=\frac{G_{F}^{2}}{2\pi}|V_{q_{1}b}|^{2}|V_{q_{2}q_{3}}|^{2}m_{b}^{2}\sqrt{\lambda(1,x_{q_{1}},x_{q_{3}})}\,(1-x_{q_{1}}-x_{q_{3}})\,\Big{[}k_{3}\langle\mathcal{O}_{1}^{q_{2}}\rangle+k_{4}\langle\tilde{\mathcal{O}}_{1}^{q_{2}}\rangle\Big{]}\,,$
(B.2) $\displaystyle\tilde{\Gamma}_{6,{\rm
int}^{+}}^{q_{1}}(x_{q_{2}},x_{q_{3}})$
$\displaystyle=\frac{G_{F}^{2}}{12\pi}|V_{q_{1}b}|^{2}|V_{q_{2}q_{3}}|^{2}m_{b}^{2}\sqrt{\lambda(1,x_{q_{3}},x_{q_{2}})}\,\Bigg{\\{}k_{5}\Big{[}\,\omega_{1}(x_{q_{3}},x_{q_{2}})\,\langle\mathcal{O}_{1}^{q_{1}}\rangle-2\omega_{2}(x_{q_{3}},x_{q_{2}})\langle\mathcal{O}_{2}^{q_{1}}\rangle\Big{]}$
$\displaystyle+k_{6}\Big{[}\omega_{1}(x_{q_{3}},x_{q_{2}})\langle\tilde{\mathcal{O}}_{1}^{q_{1}}\rangle-2\omega_{2}(x_{q_{3}},x_{q_{2}})\langle\tilde{\mathcal{O}}_{2}^{q_{1}}\rangle\Big{]}\Bigg{\\}}\,,$
(B.3)

while for semileptonic transitions $b\to q_{1}\bar{\nu}_{\ell}\ell$, with
$q_{1}=u,c$ and $\ell=e,\mu,\tau$, the explicit expression is

$\displaystyle\tilde{\Gamma}_{6,{\rm
int}^{+}}^{q_{1}}(x_{\ell},x_{\nu_{\ell}})$
$\displaystyle=\frac{G_{F}^{2}}{12\pi}|V_{q_{1}b}|^{2}m_{b}^{2}\sqrt{\lambda(1,x_{\ell},x_{\nu_{\ell}})}\,\Big{[}\,\omega_{1}(x_{\ell},x_{\nu_{\ell}})\,\langle\mathcal{O}_{1}^{q_{1}}\rangle-2\omega_{2}(x_{\ell},x_{\nu_{\ell}})\langle\mathcal{O}_{2}^{q_{1}}\rangle\Big{]}\,,$
(B.4)

where $x_{f}=m_{f}^{2}/m_{b}^{2}$ and $\lambda(a,b,c)=(a-b-c)^{2}-4bc$ is the
Källen function. Moreover, in eqs. (B.1)-(B.4) we have introduced the
functions $\omega_{1,2}(a,b)$, symmetric in their arguments, with

$\displaystyle\omega_{1}(a,b)=(a-b)^{2}+a+b-2\,,\qquad\omega_{2}(a,b)=2\,(a-b)^{2}-(1+a+b)\,,$
(B.5)

while $k_{1},\ldots,k_{6},$ denote the following combinations of Wilson
coefficients:

$\displaystyle k_{1}=2\,C_{1}C_{2}+N_{c}\,C_{2}^{2}\,,\qquad$ $\displaystyle
k_{2}=\,C_{1}^{2}\,,$ (B.6) $\displaystyle k_{3}=2\,C_{1}C_{2}\,,\qquad$
$\displaystyle k_{4}=\,\Big{(}C_{1}^{2}+C_{2}^{2}\Big{)}\,,$ (B.7)
$\displaystyle k_{5}=N_{c}\,C_{1}^{2}+2\,C_{1}C_{2}\,,\qquad$ $\displaystyle
k_{6}=\,C_{2}^{2}\,.$ (B.8)

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