# Invisible and Semi-invisible Decays of Bottom Baryons

Yong<EMAIL_ADDRESS>Jian-Nan<EMAIL_ADDRESS>corresponding author, Dong-Hao<EMAIL_ADDRESS>corresponding
author, Lei-Yi<EMAIL_ADDRESS>corresponding author, Cai-Dian
L$\ddot{\rm<EMAIL_ADDRESS>Fu-Sheng<EMAIL_ADDRESS>1MOE
Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou
730000, China
2School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000,
China
3Center for High Energy Physics, Peking University, Beijing 100871, China
4School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 101408, China
5Institute of High Energy Physics, CAS, P.O. Box 918(4) Beijing 100049, China

###### Abstract

The similar densities of dark matter and baryons in the universe imply that
they might arise from the same ultraviolet model. The B-Mesogenesis, which
assumes dark matter is charged under the baryon number, attempts to
simultaneously explain the origin of baryon asymmetry and dark matter in the
universe. In particular, the B-Mesogenesis might induce bottom-baryon decays
into invisible or semi-invisible final states, which provide a distinctive
signal for probing this scenario. In this work, we systematically study the
invisible decays of bottom baryons into dark matters, and semi-invisible
decays of bottom baryons into a meson or a photon together with a dark matter
particle. In particular, the fully invisible decay can explore the stable
particles in B-Mesogenesis. Some QCD-based frameworks are used to calculate
the hadronic matrix elements under the B-Mesogenesis model. We estimate the
constraints on the Wilson coefficients or the product of some new physics
couplings with the Wilson coefficients by the semi-invisible and invisible
decays of bottom baryons at future colliders.

##  I Introduction

Cosmology observations, such as the velocity dispersion of galaxies
Zwicky:1933gu , the Big Bang nucleosynthesis (BBN) Walker:1991ap ;
Cyburt:2015mya  and the cosmic microwave background (CMB) WMAP:2010qai ,
provide strong evidence for the existence of dark matter (DM) and baryon
asymmetry in the universe. The cosmological measurements Planck:2018vyg
indicate the relic densities of dark matter ($\Omega_{\text{DM}}$) and baryons
($\Omega_{\text{B}}$) are at the same order, i.e. $\Omega_{\text{DM}}=(5.36\pm
0.06)\ \Omega_{\text{B}}$, which implies the dark matter and baryon asymmetry
might stem from the same ultraviolet (UV) model Davoudiasl:2010am .

In recent years, a model of B-Mesogenesis was proposed to simultaneously
explain the origins of DM and baryon asymmetry by assuming the dark matter
charged under baryon number Elor:2018twp . The Sakharov conditions are
satisfied in the following ways. A long-lived particle produces bottom mesons
and their anti-mesons out of the thermal equilibrium in the early universe.
Then the neutral bottom mesons undergo $B^{0}-\overline{B}^{0}$ and
$B_{s}^{0}-\overline{B}_{s}^{0}$ oscillations which provide CP violation
naturally. Finally, the bottom mesons decay into an ordinary baryon with the
baryon number $B=+1$ and a dark fermion $\psi$ which is charged under baryon
number with $B=-1$, so that the visible baryon number is violated but the
total baryon number is conserved. Last but not least, to preserve the DM and
baryon asymmetry, the dark fermion $\psi$ should decay into a dark Majorana
fermion $\xi$ and a dark scalar baryon $\phi$. In this way, both the DM relic
abundance and the baryon asymmetry in the universe can be simultaneously
explained in this model. What’s more, most of the new particles in this
scenario locate at GeV scale, which can be precisely examined at the current
B-factories and hadron colliders or some experiments in the near future.

This model has attracted a lot of attentions in both theories and experiments
during the past a few years Nelson:2019fln ; Alonso-Alvarez:2019fym ; Alonso-
Alvarez:2021qfd ; Elahi:2021jia ; Alonso-Alvarez:2021oaj ; Elor:2022jxy ;
Berger:2023ccd ; Khodjamirian:2022vta ; Boushmelev:2023huu ; Shi:2023riy ;
Shi:2024uqs ; Belle:2021gmc ; BaBar:2023dtq ; BaBar:2023rer . From the
proposal of B-Mesogenesis scenario, most of the theoretical studies
concentrate on the semi-invisible decays of B mesons to explore this scenario.
The BABAR and Belle experiments have measured the decay of bottom mesons into
a baryon and a dark sector anti-baryon $\psi$, such as
$B^{0}\to\Lambda^{0}\psi$ and $B^{+}\to p\psi$ Belle:2021gmc ; BaBar:2023dtq ;
BaBar:2023rer . On the other hand, it is also allowed that the bottom baryons
decay into a meson and a dark baryon, such as $\Lambda_{b}(\Xi_{b})\to
P\bar{\psi}$, with $P$ denoting a light pseudo-scalar $\pi,K$ meson, which has
been mentioned in the proposal of B-Mesogenesis Elor:2018twp  and well
estimated in a recent study Shi:2024uqs . However, for bottom baryons, there
are other two processes which have not ever been studied under the
B-Mesogenesis, which are the fully invisible decay
($\Lambda_{b}^{0}\to\xi\bar{\phi}$) and the semi-invisible radiative decay
($\Lambda_{b}^{0}\to\gamma\bar{\psi}$) of $\Lambda_{b}^{0}$ baryon. In
particular, the fully invisible decay of $\Lambda_{b}^{0}$ yields a
distinctive signal to explore the stable dark particles in B-Mesogenesis. All
these processes can be explored at the future colliders with a high accuracy.
For instance, the upgrade of Belle-II Krizan:2022uir  is expected to produce
the bottom baryons pair at the threshold, while the LHCb can measure the
signals with its vertices detector Rodriguez:2021urv . What’s more, as the
Circular Electron Positron Collider (CEPC) CEPCStudyGroup:2018ghi  and Future
Circular Collider (FCC-ee) FCC:2018evy  will produce substantial bottom
baryons, these rare decays of bottom baryons are expected to be explored at
the future lepton colliders. Therefore, to precisely probe the B-Mesogenesis,
the systematical analysis on the invisible and semi-invisible decay of bottom
baryons are crucial and might shed light on the study of dark matter and
baryon asymmetry in the universe.

Except for $\Lambda_{b}^{0}\to\xi\bar{\phi}$, the fully invisible decays are
also very interesting for neutron and hyperon decays, which share the same
type of interaction as $u_{a}d_{b}d_{c}\psi$ with $a,b,c$ representing the
flavors of up- and down-type quarks. It was proposed to explain the puzzle of
neutron lifetime between the bottle method and the beam method, by allowing
neutron decaying into dark matter Fornal:2018eol . It is also allowed for the
invisible decays of $\Lambda^{0}$ hyperon in the new physics models similar to
the B-Mesogenesis and the ones for the neutron lifetime anomaly Alonso-
Alvarez:2021oaj . The BESIII collaboration has measured it as
$BR(\Lambda^{0}\to\text{invisible})<7.4\times 10^{-5}$ BESIII:2021slv . The
invisible decays of bottom baryons rely on the interaction with a bottom
quark, which is beneficial to explore the B-Mesogenesis.

When searching for the new physics in the decays of hadrons, hadronic matrix
elements are inevitable and crucial for determining the observables, except
for the interaction of UV models. Therefore, it is necessary to calculate the
corresponding matrix elements to examine the new physics or constrain the
parameters of a specific model in hadron decays. Since the matrix elements
relate to the strong interaction, we have to utilize a perturbative or non-
perturbative method to determine the hadronic matrix elements. In practice,
the decays of $\Lambda_{b}(\Xi_{b})\to P\bar{\psi}$ involve bottom baryon to
light-meson transition form factors, which have been investigated in the
light-cone sum rules (LCSR) method very recently Shi:2024uqs . Since the
baryon-to-meson form factors are a new kind of physical quantities, it
deserves more efforts of detailed discussions in the theoretical studies.
Under the heavy quark limit, these matrix elements can also be estimated
within the QCD factorization approach Beneke:1999br ; Beneke:2000ry ;
Beneke:2000wa ; Bell:2020qus . For example, this kind of form factors also
emerge in the baryon and lepton number violation decays of $\Lambda_{b}\to
P\ell$ from a Leptoquark model, and have been studied very recently within the
QCD factorization Li:2024htn . Although the higher dimensional operators of
B-Mesogenesis are different to that of Leptoquark model, the bottom baryon to
light meson transition form factors can be similarly determined within the QCD
factorization. Our results suggest that the transition form factors can be
factorized into a convolution of the perturbatively calculable hard-scattering
kernel and the non-perturbative distribution amplitudes without endpoint
divergence Li:2024htn ; Wang:2011uv . The matrix elements involved in
$\Lambda_{b}^{0}\to\gamma\bar{\psi}$ can also be calculated within the same
QCD factorization method, while that of $\Lambda_{b}^{0}\to\xi\bar{\phi}$ can
be estimated directly rely on the matrix elements in the SM. With these matrix
elements, we can calculate the branching ratios of invisible and semi-
invisible decays of bottom baryons, and predict the potential of exploring
B-Mesogenesis at the future measurements.

This manuscript is organized as follows. In Sec. II, we give a brief
introduction to the B-Mesogenesis scenario. In Sec. III, the hadronic matrix
elements in the invisible and semi-invisible decay of $\Lambda_{b}$ and
$\Xi_{b}$ baryons are calcualted within the QCD factorization approach. In
Sec. IV, the sensitivities on the parameters of B-Mesogenesis are discussed
for the invisible and semi-invisible decay of bottom baryons at future
collider. Finally, we conclude in Sec. V.

##  II Brief Introduction to B-Mesogenesis

As the relic densities of DM and net baryon are similar, they might arise from
a same UV completion. In practice, the B-Mesogenesis Elor:2018twp ; Alonso-
Alvarez:2021qfd ; Alonso-Alvarez:2021oaj ; Elahi:2021jia , which assume dark
matter is charged under the baryon number, attempts to simultaneously explain
the existence of DM and baryon asymmetry in the Universe. In particular, this
scenario will induce the decays of bottom baryons into dark baryons, which
significantly violate the baryon number conservation in the visible sector.
The corresponding Lagrangian is given by:

$\displaystyle\mathcal{L}_{int}\supset$
$\displaystyle-y^{R}_{u_{a}d_{b}}\epsilon_{ijk}Y^{*}_{i}\bar{u}_{Ra}^{j}d_{Rb}^{k,\mathcal{C}}-y_{u_{a}d_{b}}^{L}\epsilon_{ijk}Y^{*}_{i}\bar{u}_{La}^{j}d_{Lb}^{k,\mathcal{C}}-y_{\psi
d_{c}}Y_{i}\bar{\psi}d_{Rc}^{i,\mathcal{C}}-y_{d}~{}\\!\bar{\psi}\phi\xi+h.c..$
(1)

The $u$ ($d$) represents the up-type (down-type) quarks. The indices $a,b,c$
($i,j,k$) denote the quark flavors (colors), $L$ $(R)$ represents the left-
handed (right-handed) component of quarks and $\mathcal{C}$ represents a
charge conjugation. $Y$ is a color-triplet scalar with electric charge
$Q_{Y}=-1/3$, which couples to ordinary quarks in the first two terms, and to
a right-handed quark and a dark fermion in the third term. It is a TeV-scale
particle which will be integrated out in the bottom hadron decays. $\psi$ is a
dark fermion with the baryon number $B_{\psi}=-1$, which is singlet under the
standard model gauge group so that can only couple to the right-handed down-
type quarks under that color conservation. It can be found that the total
baryon number is conserved in the interaction of
$Y_{i}\bar{\psi}d_{R_{c}}^{i,\mathcal{C}}$, while the visible baryon number is
violated. This can be illustrated in Fig.1. $\psi$ will completely decay into
stable dark matters, a dark scalar $\phi$ with $B_{\phi}=-1$ and a dark
Majorana fermion $\xi$. Their quantum numbers are summarized in TABLE 1.

Table 1: The quantum numbers of the new particles in B-Mesogenesis, including particle spin $S$, electric charge $Q$, and baryon number $B$. Field |  $S$ |  $Q$ |  $B$
---|---|---|---
$Y$ | $\quad 0$ | $-\dfrac{1}{3}$ | $-\dfrac{2}{3}$
$\psi$ | $\quad\dfrac{1}{2}$ | $0$ | $-1$
$\xi$ | $\quad\dfrac{1}{2}$ | $0$ | $0$
$\phi$ | $\quad 0$ | $0$ | $-1$

If the mass of color-triplet scalar $M_{Y}$ are much heavier than that of
bottom baryons, we can integrate out such particles and obtain an effective
Lagrangian related to the (semi-)invisible decay, which is given by

${}\mathcal{L}_{\text{EFT}}=C^{L,R}_{u_{a}d_{b},d_{c}}~{}\\!{\mathcal{O}}^{L,R}_{u_{a}d_{b},d_{c}}+C^{L,R^{*}}_{u_{a}d_{b},d_{c}}~{}\\!\bar{\mathcal{O}}^{L,R}_{u_{a}d_{b},d_{c}},$
(2)

where $C^{L,R}_{u_{a}d_{b},d_{c}}=y^{L,R^{*}}_{u_{a}d_{b}}y^{*}_{\psi
d_{c}}/M_{Y}^{2}$ is the Wilson coefficients and the effective operators
$\mathcal{O}^{L,R}_{u_{a}d_{b},d_{c}}$,
$\bar{\mathcal{O}}^{L,R}_{u_{a}d_{b},d_{c}}$ are given by

$\displaystyle{\mathcal{O}}^{L}_{u_{a}d_{b},d_{c}}=\epsilon_{ijk}\left(\bar{u}_{La}^{i,\mathcal{C}}~{}\\!d_{Lb}^{j}\right)\left(\bar{\psi}^{\mathcal{C}}~{}\\!d_{Rc}^{k}\right),$
$\displaystyle\quad\bar{\mathcal{O}}^{L}_{u_{a}d_{b},d_{c}}=\epsilon_{ijk}\left(\bar{u}_{La}^{i}~{}\\!d_{Lb}^{j,\mathcal{C}}\right)\left(\bar{\psi}~{}\\!d_{Rc}^{k,\mathcal{C}}\right),$
(3)
$\displaystyle{\mathcal{O}}^{R}_{u_{a}d_{b},d_{c}}=\epsilon_{ijk}\left(\bar{u}_{Ra}^{i,\mathcal{C}}~{}\\!d_{Rb}^{j}\right)\left(\bar{\psi}^{\mathcal{C}}~{}\\!d_{Rc}^{k}\right),$
$\displaystyle\quad\bar{\mathcal{O}}^{R}_{u_{a}d_{b},d_{c}}=\epsilon_{ijk}\left(\bar{u}_{Ra}^{i}~{}\\!d_{Rb}^{j,\mathcal{C}}\right)\left(\bar{\psi}~{}\\!d_{Rc}^{k,\mathcal{C}}\right).$

For convenience, we factorize out the external field $\psi$ from the effective
operators and rewrite the Lagrangian as:

$\displaystyle\mathcal{L}_{\text{EFT}}$
$\displaystyle=C^{L,R}_{u_{a}d_{b},d_{c}}~{}\\!\bar{\psi}^{\mathcal{C}}~{}\\!{\mathcal{O}}^{{}^{\prime}L,R}_{u_{a}d_{b},d_{c}}+h.c.,$
(4)

where the operators $\mathcal{O}^{{}^{\prime}L,R}_{u_{a}d_{b},d_{c}}$ are
shown as

$\mathcal{O}^{{}^{\prime}L}_{u_{a}d_{b},d_{c}}=\epsilon_{ijk}\
d_{Rc}^{k}\left(\bar{u}_{La}^{i,\mathcal{C}}~{}\\!d_{Lb}^{j}\right),\quad\mathcal{O}^{{}^{\prime}R}_{u_{a}d_{b},d_{c}}=\epsilon_{ijk}\
d_{Rc}^{k}\left(\bar{u}_{Ra}^{i,\mathcal{C}}~{}\\!d_{Rb}^{j}\right).$ (5)

##  III Invisible and Semi-Invisible decays of bottom baryons

Figure 1: The diagrammatic representations for the invisible and semi-
invisible decays of bottom baryons in the B-Mesogenesis. (a) and (b) are the
diagrams for the semi-invisible hadronic decay of
$\Lambda_{b}^{0}\to\pi^{0}\bar{\psi}$, (c) for $\Lambda_{b}^{0}\to
K^{0}\bar{\psi}$, (d) for $\Xi_{b}^{-}\to\pi^{-}\bar{\psi}$, (e) for
$\Xi_{b}^{-}\to K^{-}\bar{\psi}$, (f,g,h) for the semi-invisible radiative
decay of $\Lambda_{b}^{0}\to\gamma\bar{\psi}$, and (i) for the fully invisible
decay of $\Lambda_{b}^{0}\to\xi\bar{\phi}$. The dark sector particles of
$\bar{\psi}$ and $\bar{\phi}$ are charged under the baryon number of
$B_{\bar{\psi},\bar{\phi}}=+1$, so that the total baryon number are conserved
in these processes.

With the effective Lagrangian of B-Mesogenesis in Eq. (4), we can
systematically study the invisible and semi-invisible decays of bottom
baryons. To explore the B-Mesogenesis, we mainly focus on the exclusive decay
modes of

$\Lambda_{b}^{0}\to P^{0}\bar{\psi},~{}~{}\Xi_{b}^{-}\to
P^{-}\bar{\psi},~{}~{}\Lambda_{b}^{0}\to\gamma\bar{\psi},~{}~{}\Lambda_{b}^{0}\to\xi\bar{\phi}.$
(6)

The diagrammatic representation of these processes are shown in FIG. 1. In the
heavy quark limit, we calculate the relevant diagrams and apply the QCD
factorization method for these heavy baryon to meson transition form factors
at the leading order of $\alpha_{s}$. The non-perturbative inputs are the
heavy baryon and light meson distribution amplitudes, and we restrict our
calculation to the leading twist (twist-2) distribution amplitudes. Finally,
the transition matrix elements defined by effective operators
$\mathcal{O}^{{}^{\prime}L,R}_{u_{a}d_{b},d_{c}}$ can be represented as a
convolution of the perturbative part with the leading twist heavy baryon and
light meson distribution amplitudes.

###  III.1 Hadronic semi-invisible decays of $\Lambda_{b}/\Xi_{b}\to
P\bar{\psi}$

Figure 2:  The Feynman diagrams of the $\Lambda_{b}^{0}/\Xi_{b}^{-}\to
P\bar{\psi}$ decays, where the bold line represents the $b$ quark and the
circle cross denotes the currents in the $\mathcal{O}^{L,R}$ operators.

In this subsection, we construct the factorization formula and calculate the
hadronic matrix elements of $\Lambda_{b}^{0}/\Xi_{b}^{-}\to P^{0,-}\bar{\psi}$
decays explicitly in the heavy quark limit. The relevant diagrams are shown in
Fig. 2. Using the effective Lagrangian given in Eq. (4), one can express the
decay amplitude for the operator
$\mathcal{O}^{{}^{\prime}L,R}_{u_{a}d_{b},d_{c}}$ as,

${}i\mathcal{M}=C^{L,R}_{u_{a}d_{b},d_{c}}\bigg{(}\bar{v}_{\psi}^{\mathcal{C}}(q)\langle
P(p)|\mathcal{O}^{{}^{\prime}L,R}_{u_{a}d_{b},d_{c}}|\mathcal{B}(p^{\prime})\rangle\bigg{)},$
(7)

where $v_{\psi}$ is the spinor of dark baryon $\bar{\psi}$ with momentum
$q=p^{\prime}-p$, and $\mathcal{B}$ represents $\Lambda_{b}^{0}$ or
$\Xi_{b}^{-}$ with momentum $p^{\prime}=m_{\mathcal{B}}v$. The hadronic matrix
elements could be parameterized in terms of two form factors respectively,

$\displaystyle\langle
P(p)|\mathcal{O}^{{}^{\prime}L,R}_{u_{a}d_{b},d_{c}}|\mathcal{B}(p^{\prime})\rangle=P_{R}\bigg{(}\frac{\not{p}^{\prime}}{m_{\mathcal{B}}}\zeta^{\mathcal{B}\to
P(L,R)}_{1}(q^{2})+\frac{\not{q}}{m_{\mathcal{B}}}\zeta^{\mathcal{B}\to
P(L,R)}_{2}(q^{2})\bigg{)}u_{\mathcal{B}}(p^{\prime}).$ (8)

We will calculate the form factors working in the rest frame of the heavy
baryon and choose two light-cone vectors along the fast-moving final state
particles with the conventions of

$n_{\mu}=(1,0,0,1),\quad\bar{n}_{\mu}=(1,0,0,-1),\quad v^{\mu}=(1,0,0,0).$ (9)

Then any 4-vector $p^{\mu}$ can be decomposed as light-cone components with a
perpendicular component

$p^{\mu}=(n\cdot p)\frac{\bar{n}^{\mu}}{2}+(\bar{n}\cdot
p)\frac{n^{\mu}}{2}+p_{\perp}^{\mu}\equiv(n\cdot p,\bar{n}\cdot p,p_{\perp}).$
(10)

When computing the Feynman diagrams in the heavy quark limit, four different
momentum configurations show up in the heavy-to-light process under
consideration,

$\displaystyle
p^{\prime}=(m_{\mathcal{B}},m_{\mathcal{B}},0)\qquad\qquad\qquad\sim
m_{\mathcal{B}}(1,1,0),$ (11) $\displaystyle
q=(m_{\mathcal{B}},\frac{m_{\psi}^{2}}{m_{\mathcal{B}}},0)\qquad\qquad\qquad\sim
m_{\mathcal{B}}(1,\lambda^{2},0),$ $\displaystyle
p=(\frac{m^{2}_{P}}{\bar{n}\cdot
p},m_{\mathcal{B}}-\frac{m_{\psi}^{2}}{m_{\mathcal{B}}},0)\qquad\ \ \sim
m_{\mathcal{B}}(\lambda^{2},1,0),$ $\displaystyle
k=(\Lambda_{\text{QCD}},\Lambda_{\text{QCD}},\Lambda_{\text{QCD}})\qquad\sim
m_{\mathcal{B}}(\lambda,\lambda,\lambda),$

where $k$ corresponds to the momentum of the light spectator quark in the
baryon. The parameter
$\lambda\sim\frac{m_{\psi}}{m_{\mathcal{B}}}\sim\frac{m_{P}}{m_{\mathcal{B}}}$,
which is vanishing in the heavy quark limit, hence $\lambda=0$ is used in this
work. Then the contribution of the second form factor
$\zeta_{2}^{\mathcal{B}\to P(L,R)}$ in Eq. (8) to the decay amplitudes in Eq.
(7) vanishes, due to the equation of motion
$\frac{\not{q}}{m_{\mathcal{B}}}v_{\psi}=\lambda v_{\psi}=0.$

When considering only the leading twist distribution amplitudes and the tree-
level QCD calculations, the contribution of operator
$\mathcal{O}^{{}^{\prime}R}_{u_{a}d_{b},d_{b}}$ in Eq. (4) to the transition
matrix element always vanish, since there is an odd number of Dirac matrices
between the two chiral projection matrices $P_{R}=\frac{1+\gamma_{5}}{2}$
within the operator $\mathcal{O}^{{}^{\prime}R}_{u_{a}d_{b},d_{b}}$. In the
heavy quark limit, the transition amplitude in Eq. (7) also vanishes for
operators $\mathcal{O}^{L}_{ud,b}$ and $\mathcal{O}^{L}_{us,b}$ due to the
parity conservation. Take $\Lambda_{b}^{0}\to\pi^{0}\bar{\psi}$ as an example,
the partonic amplitude777The partonic amplitude defined as replacing
$|\Lambda_{b}^{0}(p^{\prime})\rangle$ and $\langle P(p)|$ by
$|u(k_{1})d(k_{2})b(m_{b}v)\rangle$ and $\langle\bar{q}(xp)q(\bar{x}p)|$
Wang:2015vgv . can be factorized as a product of two transition matrix
elements in the spin space,
$i\mathcal{M}\propto\langle\bar{\psi}(q)|\,\bar{\psi}^{\mathcal{C}}b_{R}\,|b(m_{b}v)\rangle\,\langle\bar{q}(xp)q(\bar{x}p)|\,\bar{u}^{\mathcal{C}}_{L}d_{L}\,|u(k_{1})d(k_{2})\rangle.$
Then we will find that there is a trace in the second matrix element in our
calculation, which is always zero because there is an odd number of $\gamma$
matrices. And we find that the $\gamma_{5}$ in the operator
$\bar{u}^{\mathcal{C}}_{L}d_{L}$ cannot contribute to the trace, because there
are only two independent Lorentz vectors in this process to contract with the
4-dimensional antisymmetric tensor $\epsilon_{\mu\nu\rho\sigma}$. Actually,
the $J^{P}$ quantum number of quark pair $(ud)$ in $\Lambda_{b}^{0}$ baryon is
$0^{+}$ in the leading twist approximation, and $\langle
q(xp)\bar{q}(\bar{x}p)|\,\bar{u}^{\mathcal{C}}_{L}d_{L}\,|u(k_{1})d(k_{2})\rangle$
describes a scalar decay into a pseudoscalar induced by a scalar operator,
which violate parity conservation. The hadronic effects will not affect this
conclusion cause the perturbative results in the final factorization formula
are independent of the external state Descotes-Genon:2002crx .

In the following calculation, we only consider the contribution of the
operator $\mathcal{O}^{{}^{\prime}L}_{ub,d}$ or
$\mathcal{O}^{{}^{\prime}L}_{ub,s}$, and the superscript ($L$) will be omitted
from now on. Strange number conversed or violated processes are induced by
corresponding operators, which is summarized as,

$\displaystyle\Lambda_{b}^{0}\to\pi^{0}\bar{\psi}\quad\text{and}\quad\Xi_{b}^{-}\to
K^{-}\bar{\psi},\qquad\text{induced by $\mathcal{O}^{{}^{\prime}L}_{ub,d}$},$
(12) $\displaystyle\Lambda_{b}^{0}\to
K^{0}\bar{\psi}\quad\text{and}\quad\Xi_{b}^{-}\to\pi^{-}\bar{\psi},\qquad\text{induced
by $\mathcal{O}^{{}^{\prime}L}_{ub,s}$}.$

In the framework of QCD factorization approach, the longitudinal momentum of
the quarks in the energetic final state meson is at the scale of $m_{b}$ and
the exchanged gluon have virtuality of order $m_{b}\Lambda_{\text{QCD}}$,
which can be treated perturbatively. Then we calculate the hard-scattering
contributions, such as the ones shown in Fig. 2. The decay amplitudes are
written as

${}i\mathcal{M}=\eta_{\Delta s}C^{L,R}_{u_{a}d_{b},d_{c}}\zeta^{\mathcal{B}\to
P}\bigg{(}\bar{v}_{\psi}^{\mathcal{C}}(q)P_{R}\frac{\not{n}}{2}\,u_{\mathcal{B}}(p{{}^{\prime}})\bigg{)},$
(13)

where the sign factor $\eta_{\Delta s}=-1$ for strange flavor conserved
processes $\Lambda_{b}^{0}\to\pi^{0}$ and $\Xi_{b}^{-}\to K^{-}$, while
$\eta_{\Delta s}=+1$ for strange flavor violated processes $\Lambda_{b}^{0}\to
K^{0}$ and $\Xi_{b}^{-}\to\pi^{-}$. The convention for the sign factor arises
from the inconsistency between the quark flavor order in the operator
$\mathcal{O}$ and the definition of the light-cone distribution amplitudes.

The factorization formula of the form factor $\zeta^{\mathcal{B}\to P}$ is
given by

$\displaystyle\zeta^{\mathcal{B}\to P}$
$\displaystyle=f_{P}f_{\mathcal{B}}^{(2)}\int_{0}^{\infty}d\omega\,\omega\int_{0}^{1}du\int_{0}^{1}dx\mathcal{J}^{\mathcal{B}\to
P}(x,u,\omega,\mu)\phi_{P}(x)\psi_{2}(u,\omega),$ (14)

where $f_{P(\mathcal{B})}$ is the decay constant of particle $P(\mathcal{B})$,
$\mathcal{J}^{\mathcal{B}\to P}(x,u,\omega,\mu)$ is the corresponding hard-
scattering kernel and $\phi_{P}(x)$ and $\psi_{2}(u,\omega)$ are the leading
twist light-cone distribution amplitudes of the $P$ meson and $\mathcal{B}$
baryon Ball:2008fw ; Bell:2013tfa ; Beneke:2000ry ; Grossman:2015cak ,

$\left\langle
P(p)|\,[\bar{q}(t\bar{n})]_{A}\,[t\bar{n},0]\,[q(0)]_{B}\,|0\right\rangle=\dfrac{if_{P}}{4}\,\bar{n}\cdot
p\,\bigg{[}\,\dfrac{\not{n}}{2}\,\gamma_{5}\,\bigg{]}_{BA}\int_{0}^{1}dx\,e^{ixt\bar{n}\cdot
p}\,\phi_{P}(x,\mu),$ (15)

with the momentum fraction $x$ of anti-quark $\bar{q}$ in the meson $P$, and

$\displaystyle\bra{0}[u_{i}(t_{1}\bar{n})]_{A}$
$\displaystyle\,[0,t_{1}\bar{n}]\,[d_{j}(t_{2}\bar{n})]_{B}\,[0,t_{2}\bar{n}]\,[b_{k}(0)]_{C}\ket{\Lambda_{b}^{0}(p{{}^{\prime}})}$
(16) $\displaystyle=$
$\displaystyle\dfrac{1}{6}\dfrac{\epsilon^{ijk}}{4}\,f_{\Lambda_{b}}^{(2)}(\mu)\,[u_{\Lambda_{b}}(p{{}^{\prime}})]_{C}\,\bigg{[}\,\dfrac{\not{\bar{n}}}{2}\,\gamma_{5}\,C^{T}\,\bigg{]}_{BA}\int_{0}^{\infty}d\omega\,\omega\int_{0}^{1}du\,e^{-i\omega(t_{1}u+it_{2}\bar{u})}\,\psi_{2}(u,\omega),$
$\displaystyle\bra{0}[d_{i}(t_{1}\bar{n})]_{A}$
$\displaystyle\,[0,t_{1}\bar{n}]\,[s_{j}(t_{2}\bar{n})]_{B}\,[0,t_{2}\bar{n}]\,[b_{k}(0)]_{C}\ket{\Xi_{b}^{-}(p{{}^{\prime}})}$
$\displaystyle=$
$\displaystyle\dfrac{1}{6}\dfrac{\epsilon^{ijk}}{4}\,f_{\Xi_{b}}^{(2)}(\mu)\,[u_{\Xi_{b}^{-}}(p{{}^{\prime}})]_{C}\,\bigg{[}\,\dfrac{\not{\bar{n}}}{2}\,\gamma_{5}\,C^{T}\,\bigg{]}_{BA}\int_{0}^{\infty}d\omega\,\omega\int_{0}^{1}du\,e^{-i\omega(t_{1}u+it_{2}\bar{u})}\,\psi_{2}(u,\omega),$

where $\omega_{i}=(n\cdot k_{i})$ represent the momentum components of the
spectator quark respectively. We use the convention $\omega_{1}=u\omega$,
$\omega_{2}=\bar{u}\omega$ with $\omega=\omega_{1}+\omega_{2}$, where $u$ and
$\bar{u}$ represent the momentum fraction of the spectator quark in the light
quark pair. The coupling constants $f_{\mathcal{B}}$ for $\Lambda_{b}^{0}$ and
$\Xi_{b}^{-}$, in the heavy-quark limit, are defined by Bell:2013tfa ,
respectively,

$\displaystyle\epsilon_{ijk}\langle
0|u^{iT}_{\alpha}(0)d^{j}_{\beta}(0)b^{k}_{\gamma}(0)|\Lambda_{b}^{0}(p{{}^{\prime}})\rangle$
$\displaystyle\equiv\frac{1}{4}\left\\{f_{\Lambda_{b}}^{(1)}[\gamma_{5}C^{T}]_{\beta\alpha}+f_{\Lambda_{b}}^{(2)}[\not{v}\gamma_{5}C^{T}]_{\beta\alpha}\right\\}\left[u_{\Lambda_{b}}(p{{}^{\prime}})\right]_{\gamma}$
(17) $\displaystyle\epsilon_{ijk}\langle
0|d^{iT}_{\alpha}(0)s^{j}_{\beta}(0)b^{k}_{\gamma}(0)|\Xi_{b}^{-}(p{{}^{\prime}})\rangle$
$\displaystyle\equiv\frac{1}{4}\left\\{f_{\Xi_{b}}^{(1)}[\gamma_{5}C^{T}]_{\beta\alpha}+f_{\Xi_{b}}^{(2)}[\not{v}\gamma_{5}C^{T}]_{\beta\alpha}\right\\}\left[u_{\Xi_{b}}(p{{}^{\prime}})\right]_{\gamma}.$

The gauge links in Eq. (16) is used to preserve the gauge invariance,

$[t\bar{n},0]=\mathrm{P}\,\mathrm{exp}\bigg{[}\,ig\int_{0}^{t}dx\,\bar{n}\cdot
A(x\bar{n})\,\bigg{]},$ (18)

which do not affect our calculations at the leading order of QCD. The leading-
twist LCDAs of pseudoscalar mesons can be expanded using Gegenbauer
polynomials Braun:2003rp ,

$\phi_{P}(x,\mu)=6x(1-x)\bigg{[}\,1+\sum_{n=1}^{\infty}a_{n}^{P}(\mu)\,C_{n}^{(3/2)}(2x-1)\,\bigg{]},$
(19)

and the leading-twist wave function for the $\Lambda_{b}^{0}$ baryon is given
by,

$\psi_{2}(u,\omega)=u\,(1-u)\,\omega^{2}\,\dfrac{1}{\omega_{0}^{4}}\,e^{-\omega/\omega_{0}}.$
(20)

In the limit of the SU(3) flavor symmetry, the wave function of $\Xi_{b}^{-}$
is the same as that of $\Lambda_{b}^{0}$. TABLE 2 summarizes the numerical
input parameters.

Table 2: The input parameters with renormalization scale $\mu_{0}=1\,\mathrm{GeV}$. $\quad m_{\Lambda_{b}}=5.6196\,\mathrm{GeV}$ | ParticleDataGroup:2022pth  | $\quad\tau_{\Lambda_{b}}=1.471\,\mathrm{ps}$ | ParticleDataGroup:2022pth
---|---|---|---
$\quad m_{\Xi_{b}}=5.7970\,\mathrm{GeV}$ | ParticleDataGroup:2022pth  | $\quad\tau_{\Xi_{b}}=1.572\,\mathrm{ps}$ | ParticleDataGroup:2022pth
$\quad\omega_{0}=0.280\,_{-0.038}^{+0.047}\,\mathrm{GeV}$ |  Wang:2015ndk  | $\quad f_{\Lambda_{b}}^{(2)}(\mu_{0})=0.030\pm 0.005\,\mathrm{GeV}^{3}$ |  Wang:2015ndk
$\quad f_{\pi}=0.1304\pm 0.0002\,\mathrm{GeV}$ |  Grossman:2015cak  | $\quad f_{K}=0.1562\pm 0.0007\,\mathrm{GeV}$ | Grossman:2015cak
$\quad a_{1}^{\pi}(\mu_{0})=0$ | Grossman:2015cak  | $\quad a_{2}^{\pi}(\mu_{0})=0.29\pm 0.08$ | Grossman:2015cak
$\quad a_{1}^{K}(\mu_{0})=-0.07\pm 0.04$ |  Grossman:2015cak  | $\quad a_{2}^{K}(\mu_{0})=0.24\pm 0.08$ | Grossman:2015cak

The corresponding hard-scattering functions are calculated as follows,

$\displaystyle\mathcal{J}^{\Lambda_{b}^{0}\to\pi^{0}}(x,u,\omega,\mu)$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}\mathcal{J}_{u}(x,u,\omega,\mu)+\mathcal{J}_{\bar{u}}(x,u,\omega,\mu)\big{)},$
(21) $\displaystyle\mathcal{J}^{\Lambda_{b}^{0}\to K^{0}}(x,u,\omega,\mu)$
$\displaystyle=\mathcal{J}^{\Xi_{b}^{-}\to
K^{-}}(x,u,\omega,\mu)=\mathcal{J}_{\bar{u}}(x,u,\omega,\mu),$
$\displaystyle\mathcal{J}^{\Xi_{b}^{-}\to\pi^{-}}(x,u,\omega,\mu)$
$\displaystyle=\mathcal{J}_{u}(x,u,\omega,\mu),$

with

$\mathcal{J}_{u}(x,u,\omega,\mu)=\frac{1}{9}\frac{\pi\alpha_{s}(\mu)}{u\omega^{2}x},\qquad\mathcal{J}_{\bar{u}}(x,u,\omega,\mu)=\frac{1}{9}\frac{\pi\alpha_{s}(\mu)}{\bar{u}\omega^{2}x},$
(22)

where the $\frac{1}{\sqrt{2}}$ is isospin factor comes from
$|\pi^{0}\rangle=\frac{1}{\sqrt{2}}(|u\bar{u}\,\rangle-|d\bar{d}\,\rangle)$.
Noted that the asymptotic behaviors of the leading twist light-cone
distribution amplitudes in the endpoint region are

$\phi_{P}(x)\sim x(1-x)\qquad\psi_{2}(u,\omega)\sim u(1-u)\omega^{2},$ (23)

which will cancel the divergent behavior in hard-scattering function
$\mathcal{J}_{u}(x,u,\omega,\mu)$ and $\mathcal{J}_{\bar{u}}(x,u,\omega,\mu)$.
Then the convolution integral in Eq. (14) converges as noticed already in
Li:2024htn ; Wang:2011uv . Next, we use the QCD coupling constants as
$\alpha_{s}(\small{2~{}\mathrm{GeV}})\approx 0.3$ and other inputs summarized
in TABLE. 2. The numerical results for the form factors are

$\displaystyle\zeta^{\Lambda_{b}^{0}\to\pi^{0}}$
$\displaystyle=(1.43\,_{-0.54}^{+0.46})\times 10^{-2}\,[\mathrm{GeV}^{2}],$
(24) $\displaystyle\zeta^{\Lambda_{b}^{0}\to K^{0}}$
$\displaystyle=(1.33\,_{-0.47}^{+0.40})\times 10^{-2}\,[\mathrm{GeV}^{2}],$
$\displaystyle\zeta^{\Xi_{b}^{-}\to K^{-}}$
$\displaystyle=(1.33\,_{-0.47}^{+0.40})\times 10^{-2}\,[\mathrm{GeV}^{2}],$
$\displaystyle\zeta^{\Xi_{b}^{-}\to\pi^{-}}$
$\displaystyle=(1.01\,_{-0.38}^{+0.33})\times 10^{-2}\,[\mathrm{GeV}^{2}].$

Note that the mass dimension of form factor is $[\zeta]=2$, since the
dimension of the baryon coupling constant $[f_{\mathcal{B}}]=3$.

Keep in mind that we take the heavy quark limit during the calculation for
heavy baryon to light meson transition form factors, and the reliability of
our results is best when the momentum transfer $q^{2}=0$. To extrapolate form
factors $\zeta(q^{2}=0)$ toward $\zeta(q^{2}=m_{\psi}^{2})$, we introduce the
simplest single-pole model to describe the $q^{2}$ dependence of form factor
$\zeta(q^{2})$ Wirbel:1985ji ; Shi:2024uqs ,

$\zeta(q^{2})=\frac{1}{1-q^{2}/m^{2}_{\text{pole}}}\zeta(q^{2}=0).$ (25)

Here the denominator is introduced to describe the threshold Behaviour below
the onset of the continuum. Frankly, the form factor has a pole at the mass of
the lowest state that related to the operators in Eq. (12). For instance,
$m_{\rm pole}$ is the mass of $\Lambda_{b}^{0}(\frac{1}{2}^{+})$ for
$\Lambda_{b}^{0}\to\pi^{0}\,/\Xi^{-}_{b}\to K^{-}$ transition, as
$\Lambda_{b}^{0}(\frac{1}{2}^{+})$ is the lowest state of operator
$\mathcal{O}^{{}^{\prime}L}_{ub,d}$, while $m_{\rm pole}$ is the mass of
$\Xi_{b}^{0}(\frac{1}{2}^{+})$ for $\Lambda_{b}^{0}\to
K^{0}\,/\Xi^{-}_{b}\to\pi^{-}$ transition.

The branching fractions of $\mathcal{B}\to P\bar{\psi}$ are expressed as

$\mathcal{BR}(\mathcal{B}\to
P\bar{\psi})=\frac{|\vec{q}_{\psi}|}{4\pi\Gamma_{\mathcal{B}}}\left|C^{L}_{u_{a}d_{b},d_{c}}\right|^{2}\left|\zeta^{\mathcal{B}\to
P}(q^{2})\right|^{2},$ (26)

where $(u_{a}d_{b},d_{c})$ represents $(ub,d)$ or $(ub,s)$ for effective
operators $\mathcal{O}^{L}_{ub,d}$ or $\mathcal{O}^{L}_{ub,s}$,
$|\vec{q}_{\psi}|$ is the momentum of the dark baryon $\bar{\psi}$

$|\vec{q}_{\psi}|=\frac{\sqrt{(m_{\mathcal{B}}^{2}-(m_{\psi}+m_{P})^{2})(m_{\mathcal{B}}^{2}-(m_{\psi}-m_{P})^{2})}}{2m_{\mathcal{B}}},$
(27)

and $\Gamma_{\mathcal{B}}$ ($m_{\mathcal{B}}$) is the decay width (mass) of
the baryon $\mathcal{B}$.

###  III.2 Radiative semi-invisible decay of
$\Lambda_{b}^{0}\to\gamma\bar{\psi}$

Figure 3: The Feynman diagram for $\Lambda_{b}^{0}\to\gamma\bar{\psi}$ with
the bold line representing the $b$ quark.

For the radiative decay processes $\Lambda_{b}^{0}\to\gamma\bar{\psi}$ showed
in FIG. 3, the decay amplitudes can be expressed as,

${}i\mathcal{M}=C^{L,R}_{u_{a}d_{b},d_{c}}\bigg{(}\bar{v}_{\psi}^{\mathcal{C}}(q)\langle\gamma(p)|\mathcal{O}^{{}^{\prime}L,R}_{u_{a}d_{b},d_{c}}|\Lambda_{b}^{0}(p^{\prime})\rangle\bigg{)}.$
(28)

The transition amplitudes associated with the right-handed operators
$\mathcal{O}_{ub,d}^{{}^{\prime}R}$ and $\mathcal{O}_{ud,b}^{{}^{\prime}R}$
always vanish for the same reason we have mentioned previously in the hadronic
semi-invisible decays. The hadronic matrix elements in Eq. (28) could be
parameterized in terms of two form factors,

$\langle\gamma(p)|\mathcal{O}^{{}^{\prime}L}_{u_{a}d_{b},d_{c}}|\Lambda_{b}^{0}(p^{\prime})\rangle=\bigg{(}\zeta^{\Lambda_{b}\to\gamma}_{1}\frac{\not{p}}{m_{\Lambda_{b}}^{2}}i\sigma^{\mu\nu}P_{R}+\zeta^{\Lambda_{b}\to\gamma}_{2}\frac{\not{q}}{m_{\Lambda_{b}}^{2}}i\sigma^{\mu\nu}P_{L}\bigg{)}u_{\Lambda_{b}}(p^{\prime})\epsilon^{*}_{\mu}p_{\nu}.$
(29)

In the heavy quark limit, $\zeta^{\Lambda_{b}\to\gamma}_{2}$ is negligible due
to the equation of motion of $\bar{\psi}$, similarly to the case of hadronic
semi-invisible decays in the previous subsection. Additionally, only the
diagram involving the coupling of the photon to the $d$ quark contributes to
the final result. The photon radiation from $b$ quark is suppressed by $m_{b}$
in heavy quark propagator. The contribution of the radiation from $u$ quark is
also neglected, which is proportional to $\not{q}$ rather than
$\not{p}^{\prime}$, thus is irrelevant to the form factor $\zeta_{1}$.

In the heavy quark limit, the transition amplitude for operator
$\mathcal{O}^{{}^{\prime}L}_{ud,b}$ vanishes in our calculation. We can also
arrive at this conclusion by considering the amplitude at the partonic level,
which can be factorized as a product of two transition matrix elements in the
spin space,
$i\mathcal{M}\propto\langle\bar{\psi}(q)|\,\bar{\psi}^{\mathcal{C}}b_{R}\,|b(m_{b}v)\rangle\,\langle\gamma(p)|\,\bar{u}^{\mathcal{C}}_{L}d_{L}\,|u(k_{1})d(k_{2})\rangle$.
The second matrix element describes the transition of a quark pair to the
photon, with the initial momentum $k$, the transfer momentum $r$ and the final
state momentum $p$. Its amplitude is proportional to the polarization vector
$\epsilon_{\mu}$ of the photon. Considering momentum conservation $k+r=p$ and
on-shell condition $p^{2}=0$, there are two independent Lorentz-invariant
amplitudes proportional to $p\cdot\epsilon$ and $r\cdot\epsilon$ respectively.
In the heavy quark limit, we neglect the momentum $k\sim$
$\Lambda_{\text{QCD}}$ of the spectator quarks. Therefore, the second
transition matrix element must be proportional to
$p\cdot\epsilon=r\cdot\epsilon$, which is zero after implementing the Ward
Identity $p\cdot\epsilon=0$.

The non-vanishing amplitude $\Lambda_{b}^{0}\to\gamma\bar{\psi}$ defined by
$\mathcal{O}_{ub,d}^{{}^{\prime}L}$ is given by,

$\displaystyle i\mathcal{M}$
$\displaystyle=C^{L}_{ub,d}\zeta^{\Lambda_{b}^{0}\to\gamma}\bigg{(}\bar{v}_{\psi}^{\mathcal{C}}(q)P_{R}\dfrac{\not{n}}{2}\not{\epsilon}\dfrac{\not{\bar{n}}}{2}u_{\Lambda_{b}}(p{{}^{\prime}})\bigg{)}.$
(30)

The form factors can be factorized as

$\displaystyle\zeta^{\Lambda_{b}^{0}\to\gamma}=f_{\Lambda_{b}}^{(2)}\int_{0}^{\infty}d\omega\,\omega\int_{0}^{1}du\mathcal{J}^{\Lambda_{b}^{0}\to\gamma}(u,\omega,\mu)\psi_{2}(u,\omega),$
(31)

where $\mathcal{J}^{\Lambda_{b}^{0}\to\gamma}(u,\omega,\mu)$ is the hard-
scattering kernel for $\Lambda_{b}\to\gamma$ transition,

$\displaystyle\mathcal{J}^{\Lambda_{b}^{0}\to\gamma}(u,\omega,\mu)$
$\displaystyle=\dfrac{1}{4}\bigg{(}\dfrac{Q_{d}(\mu)}{\bar{u}\omega}\bigg{)},$
(32)

where the electric charge of the light quark $d$ is
$Q_{d}(\mu)=-\frac{1}{3}e(\mu)$. The numerical result for the form factor of
the spectator processes $\Lambda_{b}^{0}\to\gamma\bar{\psi}$ is,

$\displaystyle\zeta^{\Lambda_{b}^{0}\to\gamma}=(2.74\,_{-0.65}^{+0.59})\times
10^{-3}\,[\mathrm{GeV^{2}}],$ (33)

whose input is shown in TABLE 2 with $\alpha_{em}(2~{}\mathrm{GeV})\approx
1/133$ Erler:1998sy . The single-pole model is also used to describe the
$q^{2}$ dependence of form factor $\zeta(q^{2})$,

$\zeta(q^{2})=\frac{1}{1-q^{2}/m^{2}_{\Lambda_{b}}}\zeta(q^{2}=0),$ (34)

where $m_{\Lambda_{b}}$ is the mass of $\Lambda_{b}^{0}(\frac{1}{2}^{+})$.
Then the branching ratio for $\Lambda_{b}^{0}\to\gamma\bar{\psi}$ with the
operator $\mathcal{O}_{ub,d}^{{}^{\prime}L}$ is estimated as

$\displaystyle\mathcal{BR}({\Lambda_{b}^{0}\to\gamma\bar{\psi}})$
$\displaystyle=\frac{|\vec{q}_{\psi}|}{8\pi\Gamma_{\Lambda_{b}}}\
|C^{L}_{ub,d}|^{2}\left|\zeta^{\Lambda_{b}^{0}\to\gamma}(q^{2}=m_{\psi}^{2})\right|^{2}.$
(35)

###  III.3 Fully invisible decay for $\Lambda_{b}^{0}$

Figure 4: The Feynman diagram for $\Lambda_{b}^{0}\to\xi\bar{\phi}$ process,
with the bold line denoting the $b$ quark.

Under the B-Mesogenesis scenario, $\Lambda_{b}^{0}\to\xi\bar{\phi}$ channel is
governed by both effective operators ${\mathcal{O}}_{ud,b}^{{}^{\prime}L,R}$
and ${\mathcal{O}}_{ub,d}^{{}^{\prime}L,R}$. The Feynman diagram is shown in
FIG. 4. The decay amplitudes could be respectively expressed as

$\displaystyle i\mathcal{M}_{ud,b}^{L,R}$
$\displaystyle=y_{d}C^{L,R}_{ud,b}\bigg{(}\bar{u}_{\xi}^{\mathcal{C}}(q)\frac{\not{p}{{}^{\prime}}+m_{\psi}}{p^{\prime
2}-m_{\psi}^{2}}\langle
0|\mathcal{O}^{{}^{\prime}L,R}_{ud,b}|\Lambda_{b}^{0}(p{{}^{\prime}})\rangle\bigg{)},$
(36) $\displaystyle i\mathcal{M}_{ub,d}^{L,R}$
$\displaystyle=y_{d}C^{L,R}_{ub,d}\bigg{(}\bar{u}_{\xi}^{\mathcal{C}}(q)\frac{\not{p}{{}^{\prime}}+m_{\psi}}{p^{\prime
2}-m_{\psi}^{2}}\langle
0|\mathcal{O}^{{}^{\prime}L,R}_{ub,d}|\Lambda_{b}^{0}(p{{}^{\prime}})\rangle\bigg{)},$

where $u_{\xi}(q)$ is the spinor for $\xi$ with the four-momentum $q$. To
estimate the branching ratios of the invisible baryon decay, we have to
firstly calculate the corresponding hadronic matrix elements in Eq. (36),
which could be parameterized in terms of the coupling constants defined by

$\displaystyle\langle
0|\mathcal{O}^{{}^{\prime}L,R}_{ud,b}|\Lambda_{b}^{0}(p{{}^{\prime}})\rangle=\lambda_{ud,b}^{L,R}\
P_{R}u_{\Lambda_{b}}(p{{}^{\prime}}),\quad\langle
0|\mathcal{O}^{{}^{\prime}L,R}_{ub,d}|\Lambda_{b}^{0}(p{{}^{\prime}})\rangle=\lambda_{ub,d}^{L,R}\
P_{R}u_{\Lambda_{b}}(p{{}^{\prime}}).$ (37)

We found that the coupling $\lambda_{ud,b}^{L,R}$ and $\lambda_{ub,d}^{L,R}$
could be related to the coupling $f_{\Lambda_{b}}^{(1,2)}$ defined in Eq. (17)
by multiply corresponding matrix element. For instance,

$\displaystyle\langle
0|\mathcal{O}^{{}^{\prime}L,R}_{ud,b}|\Lambda_{b}^{0}(p{{}^{\prime}})\rangle$
$\displaystyle=\epsilon_{ijk}\langle
0|\left[u^{iT}\,CP_{L,R}\,d^{j}\right]b_{R}^{k}|\Lambda_{b}^{0}\left(p{{}^{\prime}}\right)\rangle$
(38)
$\displaystyle=\sum_{\rho}(CP_{L,R})^{\alpha\beta}(P_{R})^{\rho\gamma}\left\\{\epsilon_{ijk}\langle
0|u^{iT}_{\alpha}d^{j}_{\beta}b^{k}_{\gamma}|\Lambda_{b}^{0}(p{{}^{\prime}})\rangle\right\\}$
$\displaystyle=\frac{1}{4}\left\\{f^{(1)}_{\Lambda_{b}}\text{Tr}[CP_{L,R}\gamma_{5}C^{T}]+f^{(2)}_{\Lambda_{b}}\text{Tr}[CP_{L,R}\not{v}\gamma_{5}C^{T}]\right\\}P_{R}u_{\Lambda_{b}}(p{{}^{\prime}}).$

The last line is the result after using Eq. (17). After the trace, we can
obtain the relations,

$\displaystyle\lambda^{L}_{ud,b}$
$\displaystyle=-\frac{1}{2}f_{\Lambda_{b}}^{(1)},\quad\lambda^{R}_{ud,b}$
$\displaystyle=+\frac{1}{2}f_{\Lambda_{b}}^{(1)}.$ (39)

The same operation can be used for $\lambda^{L,R}_{ub,d}$, which can obtain,

$\displaystyle\lambda^{L}_{ub,d}$
$\displaystyle=-\frac{1}{4}f_{\Lambda_{b}}^{(2)},\quad\lambda^{R}_{ub,d}$
$\displaystyle=+\frac{1}{4}f_{\Lambda_{b}}^{(1)}.$ (40)

For the numerical value of the couplings we quote the result of the NLO QCD
sum rule analysis in Ref. Groote:1997yr :

$f_{\Lambda_{b}}^{(2)}\approx f_{\Lambda_{b}}^{(1)}=0.030\pm 0.005\
[\text{GeV}^{3}].$ (41)

Then, we could estimate the branching ratios of the invisible baryon decay.
Assuming the new physics individually exist in either the left-handed or
right-handed component, the branching ratios of $\Lambda_{b}\to\xi\bar{\phi}$
is given by

$\displaystyle\mathcal{BR}({\Lambda_{b}\to\xi\bar{\phi}})^{L,R}_{u_{a}d_{b},d_{c}}$
$\displaystyle=\frac{\left|\vec{q}_{\xi}\right|}{8\pi
m_{\Lambda_{b}}\Gamma_{\Lambda_{b}}}\bigg{(}\left|y_{d}\right|^{2}\left|C^{L,R}_{u_{a}d_{b},d_{c}}\right|^{2}\left|\lambda_{{}_{u_{a}d_{b},d_{c}}}^{L,R}\right|^{2}\bigg{)}\frac{E_{\xi}\left(m_{\Lambda_{b}}^{2}+m^{2}_{\psi}\right)+2m_{\Lambda_{b}}m_{\psi}m_{\xi}}{\left(m_{\Lambda_{b}}^{2}-m_{\psi}^{2}\right)^{2}},$
(42)

where $(u_{a}d_{b},d_{c})$ represents $(ud,b)$ or $(ub,d)$ for different
effective operators $\mathcal{O}^{L,R}_{ud,b}$ and $\mathcal{O}^{L,R}_{ub,d}$
convention in Eq. (3) respectively,
$m_{i}~{}\\!(i=\Lambda_{b}^{0},\bar{\psi},\bar{\phi},\xi)$ and
$\Gamma_{\Lambda_{b}}$ are the mass of particle $i$ and the total width of
heavy baryons. The $\vec{q}_{\xi}$ and
$E_{\xi}=\frac{m_{\Lambda_{b}}^{2}+m_{\xi}^{2}-m_{\phi}^{2}}{2m_{\Lambda_{b}}}$
are the momentum and kinetic energy of $\xi$ in heavy baryon $\Lambda_{b}^{0}$
rest frame.

##  IV Probe B-Mesogenesis with invisible and semi-invisible decay of bottom
baryons

With the hadronic matrix elements obtained in the above section, we can
explore the B -Mesogenesis via the invisible and semi-invisible decays of
bottom baryons at the future lepton colliders. The reconstructions of the
invisible and semi-invisible decays of bottom baryons are always difficult in
experiments. But it can be overcome by the double tag method MARK-III:1989dea
. In the $e^{+}e^{-}$ collisions, the initial energy is well known in each
event. Suppose a good resolution in the detectors, we reconstruct all the
recoiling particles except for $\Lambda_{b}^{0}/\Xi_{b}^{0,-}$ in the
colliding event, which helps to determine the invariant mass of the bottom-
baryon candidate using the energy-momentum conservation. It would be
identified as a production of bottom baryon, if the invariant mass is
consistent with the mass of $\Lambda_{b}^{0}/\Xi_{b}^{0,-}$ and the
conservation laws are satisfied for the electric charge, the baryon number,
the bottom number and the strange number. Then we are able to measure the
invisible and semi-invisible decays of bottom baryons. The double tag method
has been widely used by BESIII, BABAR and Belle. For example, BESIII measured
the absolute branching fraction of $\Lambda_{c}^{+}\to\Lambda^{0}e^{+}\nu_{e}$
at the $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ threshold, which determines
$\Lambda_{c}^{+}$ by reconstructing the charmed anti-baryon via its hadronic
decays such as $\bar{\Lambda}_{c}^{-}\to\bar{p}K^{+}\pi^{-}$ BESIII:2015ysy .
It is also possible to use the double tag method at the energy much higher
than the baryon-antibaryon threshold. The Belle collaboration measured the
absolute branching fractions of $\Lambda_{c}^{+}$ decays with reconstructing
the recoiling $D^{(*)-}\bar{p}\pi^{+}$ system in the event of $e^{+}e^{-}\to
D^{(*)-}\bar{p}\pi^{+}\Lambda_{c}^{+}$ at the colliding energy around 10 GeV
Belle:2013jfq . Therefore, the invisible and semi-invisible decays of
$\Lambda_{b}^{0}/\Xi_{b}^{-}$ can be measured using the double tag method at
the $e^{+}e^{-}$ colliders, such as BelleII at the bottom baryon-antibaryon
threshold, or CEPC and FCC-ee at a higher energy as well.

In practice to estimate the sensitivity on the parameters of B-Mesogenesis, we
define the significance $s$ of the signal at $95\%$ C.L. as

$s=\frac{n_{S}}{\sqrt{n_{S}+n_{B}}}=\sqrt{n_{S}}=\sqrt{N_{\mathcal{B}}\times
Br(\mathcal{B}\to X)}=2,$ (43)

where $n_{S}$, $n_{B}$ denote the events number of signal and background at
future lepton colliders. The background events are negligible using the double
tag method, since the three kinds of processes studied in this work are all
forbidden in the Standard Model by the visible baryon number violation. The
branching fractions $Br(\mathcal{B}\to X)$ with
$X=\pi\bar{\psi},K\bar{\psi},\gamma\bar{\psi},\xi\bar{\phi}$ are given in Eqs.
(26,35,42), respectively. $N_{\mathcal{B}}$ represents the production number
of bottom baryons detected by the double tag method. Notice its difference
from the total production of bottom baryons in the collisions. Besides, we
assume an efficiency of reconstruction as $100\%$ for pion, kaon and photon in
the semi-invisible decays. In this work, we assume the production number of
bottom baryons by the double tag method are
$N_{\mathcal{B}}=N_{\Lambda_{b},\Xi_{b}}=10^{8}$ as a benchmark for the future
lepton colliders. The corresponding constraints can directly scale up to the
results respected to the actual production number of bottom baryons at future
lepton colliders, which are still indistinct in the current theoretical study.

There are some constraints on the masses of the dark particles under the
B-Mesogenesis scenario. To prevent the decay of proton into the dark fermion,
the mass of dark fermion $m_{\psi}$ must satisfy

$m_{\psi}>m_{p}-m_{e}\simeq 937.8\;\text{MeV}.$ (44)

In addition, the stability of dark matter requires the mass difference between
the dark scalar $\phi$ and dark Majorana fermion $\xi$ must obey Alonso-
Alvarez:2021qfd

${}|m_{\xi}-m_{\phi}|<m_{p}+m_{e}\simeq 938.8\ \text{MeV}.$ (45)

###  IV.1 Probe B-Mesogenesis in semi-invisible decays of bottom baryons.

Firstly, we estimate the sensitivity on the B-Mesogenesis by searching for the
semi-invisible decay of bottom baryons at future lepton colliders. Their
branching ratios are only sensitive to the mass $m_{\psi}$ and Wilson
coefficients $|C^{L}_{ub,d}|$ or $|C^{L}_{ub,s}|$ of the scenario. The
constraints on the Wilson coefficients $|C^{L}_{ub,d}|$ and $|C^{L}_{ub,s}|$
versus $m_{\psi}$ at $95\%$ C.L. are illustrated in FIG. 5, where the
production number of bottom baryons by the double tag method are assuming to
be $N_{\Lambda_{b},\Xi_{b}}=10^{8}$. The constraints on the right-handed
operators $\mathcal{O}^{R}_{ud_{a},d_{b}}$ are omitted as they could not
project out the leading-twist wave function of $\Lambda_{b}$ and $\Xi_{b}$
baryons.

Figure 5: The constraints on the Wilson coefficients at $95\%$ C.L. versus
$m_{\psi}$ in the semi-invisible decays of bottom baryons. The left panel
illustrates the constraints on $|C^{L}_{ub,d}|$, while the right panel display
the constraints on $|C^{L}_{ub,s}|$. The enhancements at the large mass region
arise from the evolution of form factor. The declines of the constraints at
the tail result from the suppression of phase space. When dark fermion mass
$m_{\psi}$ close to the mass of $\Lambda_{b}^{0}$ baryon, the soft photon
radiation in $\Lambda_{b}^{0}\to\gamma\bar{\psi}$ channel will induce a
serious infrared divergence, which invalid the leading order estimation in
this region. Hence, we omit the constraints in
$\Lambda_{b}^{0}\to\gamma\bar{\psi}$ process when $m_{\psi}>5$ GeV (blue dot
dashed line).

For $\Lambda_{b}(\Xi_{b})\to P\bar{\psi}$ processes, we found that the semi-
invisible decay $\Lambda_{b}^{0}\to\pi^{0}\bar{\psi}$ is most sensitive to
operator $\mathcal{O}^{L}_{ub,d}$, while the $\Xi^{-}_{b}\to\pi^{-}\bar{\psi}$
process is most sensitive to operator $\mathcal{O}^{L}_{ub,s}$. When assuming
the production number of bottom baryons by the double tag method is
$N_{\Lambda_{b},\Xi_{b}}=10^{8}$, the constraints on the Wilson coefficient
can reach $\mathcal{O}(10^{-8})$ GeV-2. The enhancements of the constraints at
the large mass region arise from the evolution of form factors. And the
declines of the Wilson coefficients at the tail result from the suppression of
phase space. Note that the form factors and the hadrons mass in semi-invisible
decay are similar, the constraints on the Wilson coefficients are adjacent in
these processes.

For the $\Lambda_{b}^{0}\to\gamma\bar{\psi}$ channel, the constraints on the
Wilson coefficients is $\mathcal{O}(10^{-7})$ GeV-2. This constraint is looser
than that in the hadronic semi-invisible decay, due to the form factors of
photon radiation is smaller than those of meson final states. What’s more,
when the dark fermion mass $m_{\psi}$ close to the mass of initial baryons
$m_{\Lambda_{b}}$, the divergence in the evolution of form factors invalid the
estimation of branching ratio. Therefore, we omit the constraints in
$\Lambda_{b}^{0}\to\gamma\bar{\psi}$ channel when $m_{\psi}>5$ GeV.

Last but not least, since the final state hadrons in $\Xi_{b}^{-}$ semi-
invisible decay are all electric charged, these decay modes could be probed at
higher energy future lepton colliders (i.e. CEPC or FCC-ee) with a displaced
vertex method. Therefore, we expect a well exploration on these semi-invisible
decay of bottom baryons at future lepton colliders.

###  IV.2 Probe B-Mesogenesis in invisible decay of bottom baryons.

As shown in Ref. Elor:2018twp ; Elor:2022jxy ; Khodjamirian:2022vta ;
Boushmelev:2023huu ; Shi:2023riy ; Shi:2024uqs , previous studies mainly focus
on searching for the dark fermion $\psi$ in the semi-invisible decay of bottom
baryons or mesons. To keep the dark matter to be stable, the B-Mesogenesis
must involve a dark Majorana fermion $\xi$ and a dark scalar baryon $\phi$ and
requires all the dark fermion $\psi$ completely decay into these two invisible
particles, with $\mathcal{BR}(\psi\to\phi\xi)=100\%$. Therefore, the coupling
of $y_{d}$ and the interaction of $\bar{\psi}\phi\xi$ are very hard to be
explored in the semi-invisible decays where $\psi$ is always on shell. The
width of $\psi$ depending on $y_{d}$ cannot be measured in experiments since
the final states $\phi$ and $\xi$ are not measurable. On the contrary, the
$\bar{\psi}$ is off-shell in the fully invisible decay of
$\Lambda_{b}^{0}\to\xi\bar{\phi}$, seen in FIG. 1(i) or FIG. 4. Then it is
able to explore the stable dark matters $\xi$ or $\bar{\phi}$, and measure the
interaction of $y_{d}\bar{\psi}\phi\xi$ as shown in the branching fraction of
$\Lambda_{b}^{0}\to\xi\bar{\phi}$ in Eq. (42). So we suggest that the fully
invisible decay of $\Lambda_{b}^{0}\to\xi\bar{\phi}$ is crucial for probing
the B-Mesogeneisis as it provides a distinctive signal to examine the stable
dark particles in this scenario.

Assuming the production number of $\Lambda_{b}^{0}$ baryon using the double-
tag method as $N_{\Lambda_{b}}=10^{8}$, the constraints on the parameters
$|y_{d}~{}\\!C^{L,R}_{ud_{a},d_{b}}|$ can be given straightforwardly from Eq.
(42), depending on the different masses of $\bar{\psi}$, $\bar{\phi}$ and
$\xi$. The constraints with $95\%$ C.L. are displayed in FIG. 6 and 7, for the
variations of $m_{\psi}$ and $m_{\xi}+m_{\phi}$, respectively.

In FIG. 6, the constraints of the parameters depending on $m_{\psi}$ are given
for $|y_{d}~{}\\!C_{ud,b}^{L,R}|$ in the left panel, and for
$|y_{d}~{}\\!C_{ub,d}^{L,R}|$ in the right panel. Three cases of masses of
$\xi$ and $\phi$ are shown in the blue, green and red curves, with (i)
$m_{\xi}=m_{\phi}=0.4m_{\psi}$, (ii) $m_{\xi}=0.7m_{\psi}$,
$m_{\phi}=0.1m_{\psi}$, and (iii) $m_{\xi}=0.1m_{\psi}$,
$m_{\phi}=0.7m_{\psi}$. Since the DM stability in Eq. (45) prohibits a large
mass difference between dark fermion $\xi$ and dark scalar baryon $\phi$, it
imposes a truncation at the large mass region with the specific mass
distribution. We found that the constraints on the parameters of B-Mesogenesis
can reach $\mathcal{O}(10^{-7})$ GeV-2. Since the values of form factors arose
from the left- and right-handed operators are the same except for a different
sign, the constraints on the operators with different chirality are equal to
each other. For the operators with different flavour structures, a factor of
$2$ in the constraint on $|y_{d}~{}\\!C^{L,R}_{ub,d}|$ arises from the half
suppression of the decay constants $\lambda^{L,R}_{ub,d}$ in the comparison
between Eqs.(39) and (40). In the mass region which satisfies the DM
stability, the constraints on the Wilson coefficients with different mass
distribution are similar, even one of the stable particles carry most of the
$\bar{\psi}$ mass. The drop of the constraints at $m_{\psi}$ close to
$m_{\Lambda_{b}}$ arises from a nearly on-shell transition of
$\Lambda_{b}^{0}\to\bar{\psi}$, which significantly enlarge the invisible
decay rate.

Similarly, the constraints of parameters on the mass of
$m_{\xi\phi}=m_{\xi}+m_{\phi}$ is shown in FIG. 7, for
$|y_{d}~{}\\!C^{L,R}_{ud,b}|$ and $|y_{d}~{}\\!C^{L,R}_{ub,d}|$ in the left
and right panels, respectively. The three cases of (i) $m_{\phi}=m_{\xi}$,
(ii) $m_{\xi}=m_{\phi}+0.9$ GeV, and (iii) $m_{\phi}=m_{\xi}+0.9$ GeV, are
displayed in blue, green and red curves, respectively, with the mass of $\psi$
fixed as 4.0 GeV. It is interesting that the curves are roughly flat at the
region of $m_{\xi\phi}$ between 1 GeV and 5 GeV. Furthermore, the differences
between the three curves in each figure are also very small. These phenomenons
indicate that the constraints on the parameters are insensitive to the mass
distribution of $m_{\xi}$ and $m_{\phi}$. This is beneficial to search for the
stable particles in B-Mesogenesis. If there is a deviation in the semi-
invisible decays of bottom baryons or mesons, the fully invisible decay of
$\Lambda_{b}^{0}$ can further explore the stable components which is inherent
in B-Mesogenesis. The declines of the constraints at the large mass region
result from the suppression of phase space.

Figure 6: The constraints on the parameters $|y_{d}~{}\\!C^{L,R}_{ud,b}|$ and
$|y_{d}~{}\\!C^{L,R}_{ub,d}|$ versus $m_{\psi}$ from the invisible decay of
$\Lambda_{b}^{0}$ with $95\%$ C.L. We estimate the constraints in three
different mass distribution of dark matters. The left and right panels
illustrate the constraints related to operators $\mathcal{O}_{ud,b}^{L,R}$ and
$\mathcal{O}_{ub,d}^{L,R}$, respectively. The dark matter (DM) stability in
Eq. (45) prohibits a large mass difference between dark fermion $\xi$ and dark
scalar $\bar{\phi}$, which imposes a truncation at large mass region in the
first two distributions (red and green lines).

Figure 7:  The constraints on the parameters $|y_{d}~{}\\!C^{L,R}_{ud,b}|$ and
$|y_{d}~{}\\!C^{L,R}_{ub,d}|$ versus total mass $m_{\xi\phi}=m_{\xi}+m_{\phi}$
with $95\%$ C.L.. As the DM stability prohibits a large mass splitting of dark
fermion and dark baryons, we estimate the constraints in three different mass
distribution of dark matters.

##  V Conclusion

The similar densities of baryons and dark matter in the universe imply that
they might arise from a same UV complete model. The B-Mesogensis, which assume
dark matter can be charged under the baryon number, attempts to simultaneously
explain the origins of dark matter and baryon asymmetry. Most of previous
studies mainly concentrate on exploring the semi-invisible decays of $B$
mesons, since this model was proposed. Actually, the invisible and semi-
invisible decays of bottom baryons are also crucial for exploring the
B-Mesogenesis at the GeV scale. In particular, as the B-Mesogenesis must
involves a dark Majorana fermion and a dark scalar baryon to keep the DM and
baryon asymmetry stable, the fully invisible decay of $\Lambda_{b}^{0}$ yield
a distinctive signal to directly probe the stable particles of this scenario.
What’s more, when exploring the B-Mesogenesis in the decays of bottom baryons,
the hadronic matrix elements are inevitable and vital for determining the
physical observables. We systematically study the hadronic semi-invisible
decays of $\Lambda_{b}^{0}\to\pi^{0}(K^{0})\bar{\psi}$ and
$\Xi_{b}^{-}\to\pi^{-}(K^{-})\bar{\psi}$, the radiative semi-invisible decay
of $\Lambda_{b}^{0}\to\gamma\bar{\psi}$, and the fully invisible decay of
$\Lambda_{b}^{0}\to\xi\bar{\phi}$. The relevant hadronic matrix elements are
calculated within the QCD factorization method in the heavy quark limit.
Finally, we analyze the sensitivities on the parameters of B-Mesogenesis by
searching for the invisible and semi-invisible decay of bottom baryons at
future lepton colliders. Assuming the production of bottom baryons by the
double-tag method $N_{\Lambda_{b},\Xi_{b}}=10^{8}$, we found that the
constraints on the Wilson coefficients $|C^{L}_{ub,d}|$ and $|C^{L}_{ub,s}|$
in semi-invisible decay of bottom baryons can reach $\mathcal{O}(10^{-8})$
GeV-2, while the constraints on $|y_{d}~{}\\!C^{L,R}_{ud,b}|$ and
$|y_{d}~{}\\!C^{L,R}_{ub,d}|$ can reach $\mathcal{O}(10^{-7})$ GeV-2 from the
fully invisible decay of $\Lambda_{b}^{0}$. Although the invisible decay of
$\Lambda_{b}^{0}$ baryon sensitive to more new physics parameters, we want to
emphasize that this decay mode provides a distinctive signal to directly
explore the stable dark particles of B-Mesogenesis.

##  VI Acknowledgments

We are grateful to Sheng-Qi Zhang and Man-Qi Ruan for useful discussions. The
work is partly supported by the National Natural Science Foundation of China
with Grant No.12335003, 12275277 and the National Key Research and Development
Program of China under Contract No.2020YFA0406400 and 2023YFA1606000.

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