# Heavy baryon decays into light meson and dark baryon within LCSR

Yu-Ji Shi1,2<EMAIL_ADDRESS>(corresponding author), Ye Xing 3
<EMAIL_ADDRESS>and Zhi-Peng Xing 4
<EMAIL_ADDRESS>1 School of Physics, East China University of
Science and Technology, Shanghai 200237, China
2 Shanghai Key Laboratory of Particle Physics and Cosmology,
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China
3 School of Materials Science and Physics, China University of Mining and
Technology, Xuzhou 221116, China
4 Department of Physics and Institute of Theoretical Physics, Nanjing Normal
University, Nanjing, Jiangsu 210023, China

###### Abstract

We studied the decays of Heavy baryon into a pseudoscalar meson and a dark
baryon in the recently developed $B$-Mesogenesis scenario, where the two types
of effective Lagrangians proposed by the scenario are both considered. The
decay amplitudes of $\Lambda_{b}^{0}$ are calculated by light-cone sum rules
using its light-cone distribution amplitudes. The decay amplitudes of
$\Xi_{b}^{0,\pm}$ are related with those of $\Lambda_{b}^{0}$ through a flavor
SU(3) analysis. The uncertainties of threshold parameter and the Borel
parameter are both considered in the numerical calculation. The values of
effective coupling constants in the $B$-Mesogenesis are taken as their upper
limits that obtained from our previous study on the inclusive decay. The upper
limits of the decay branching fractions are presented as functions of the dark
baryon mass.

##  I Introduction

The Standard Model of particle physics and cosmology have been proven to be
successful in describing the physics of the most microscopic and macroscopic
worlds. However, there is inconsistency between these two models, where two of
the most confusing problems are the existence of dark matter (DM) and the
asymmetry of matter and anti-matter. Nowadays, a number of mechanisms aiming
to solve this puzzle have been proposed according to the Sakharov conditions
for the baryogenesis Sakharov . One of the disadvantages among these
mechanisms is the existence of high energy scales and extremely massive
particles, which make them difficult to be tested in experiments. Recently, a
new mechanism called $B$-Mesogenesis is proposed by Refs. Elor:2018twp ;
Alonso-Alvarez:2021qfd ; Elahi:2021jia , which aims to explain both the relic
dark matter abundance and the baryon asymmetry without introducing high energy
scales. The $B$-Mesogenesis is testable at hadron colliders and $B$-factories
Alonso-Alvarez:2021qfd ; Borsato:2021aum , and also indirectly testable at the
Kaon and Hyperon factories Alonso-Alvarez:2021oaj ; Goudzovski:2022vbt .
Recently, the Belle-II collaboration and the LHCb collaboration have started
to search for the decays of $B$ mesons with energy missing according to the
$B$-Mesogenesis Belle:2021gmc ; Rodriguez:2021urv .

The $B$-Mesogenesis proposes a new mechanism for Baryogenesis and DM
production. In this model, during a late era in the history of the early
universe, a certain heavy scalar particle $\Phi$ decays into $b,\bar{b}$
quarks, which then form charged and neutral $B$-mesons after the universe cool
down. After that, the neutral mesons $B^{0},{\bar{B}}^{0}$ quickly undergo CP
violating oscillations, and the remained mesons continue to decay into a dark
sector baryon $\psi$ with baryon number $-1$ and visible hadron states with
baryon number $+1$. As a result, the asymmetry of the baryon and anti-baryon
number from the CP violation during $B^{0}-{\bar{B}}^{0}$ oscillations is
induced but without violating the total baryon number. Recently, there are a
number of theoretical studies on the $B$ meson decays in the $B$-Mesogenesis.
The exclusive decay $B\to p\psi$ was firstly studied by
Ref.Khodjamirian:2022vta  using leading twist light-cone sum rules (LCSR)
calculation, and a higher twist contribution are calculated in
Ref.Boushmelev:2023huu . A more complete study on $B$ meson decays into an
octet baryon or a charmed anti-triplet baryon plus $\psi$ was given in
Ref.Elor:2022jxy . In addition, a similar exclusive decay of $B$ meson into a
baryon plus missing energy are studied by Ref.Dib:2022ppx  for probing the
lightest neutralino. Besides the exclusive decays, previously we studied the
semi-inclusive decay of $B\to X_{u/c,d/s}\psi$ using heavy quark expansion
(HQE), where $X_{u/c,d/s}$ denotes any possible hadron states containing $u/c$
and $d/s$ quarks with unit baryon number Shi:2023riy . Using the experimental
upper limits on the branching fractions of $B\to X_{u/c,d/s}\psi$ from the
ALEPH experiment ALEPH:2000vvi ; ALEPH:1992zwu ; ALEPH:1994bih , we predicted
the upper limits on the coupling constants in the $B$-Mesogenesis.

Up to now, the studies on the $B$-Mesogenesis mainly focus on the $B$ decays.
However, apart from the generous production of $B$ meson in the LHCb and Belle
experiments, $\Lambda_{b}^{0}$ is the most baryons that can be produced
experimentally. In the $B$-Mesogenesis model, $\Lambda_{b}^{0}$ can also
undergo baryon number violated decays such as $\Lambda_{b}\to M\psi$, where
$M$ is a neutral pseudoscalar meson including $\pi^{0},K^{0},{\bar{D}}^{0}$.
Compared with $B\to p/n\psi$, the phase space of $\Lambda_{b}^{0}\to M\psi$ is
larger, which can amplify the decay width to a certain extent so that offer
more possibility for observing such decays in the experiments. In this work,
we will perform a theoretical study on the $\Lambda_{b}^{0}\to M\psi$ decays.
The decay amplitude will be calculated by LCSR with the use of the light-cone
distribution amplitudes (LCDAs) of $\Lambda_{b}^{0}$. One disadvantage of
$\Lambda_{b}^{0}\to M\psi$ is that since the neutral final pseudoscalar meson
makes it difficult to be detected. In contrast, the charged decay
$\Xi_{b}^{\pm}\to M\psi$ should be a more ideal channel for experimental
searching. In the flavor SU(3) symmetry limit, the decay amplitudes of these
decay channels are related with each other. The SU(3) symmetry analysis is a
powerful tools frequently used in heavy hadron decays Chau:1986jb ;
Chau:1987tk ; Chau:1990ay ; Gronau:1994rj ; Zhou:2016jkv ; Muller:2015lua ;
Shi:2017dto , where it is generally be performed in two different frameworks:
the topological diagram amplitude (TDA) and the irreducible representation
amplitude (IRA) methods. These two methods have been proven to be equivalent
in Refs.Zeppenfeld:1980ex ; Gronau:1994rj ; Muller:2015lua ; He:2018php ;
He:2018joe ; Wang:2020gmn . In this work, we will choose IRA method to obtain
the relations between the decay amplitudes of $\Xi_{b}^{\pm}\to M\psi$ and
those of $\Lambda_{b}^{0}\to M\psi$, and use them to predict the decay
branching fractions of $\Xi_{b}^{\pm}\to M\psi$.

This article is organized as follows: Section II introduces the
$B$-Mesogenesis scenario proposed by Refs. Elor:2018twp ; Alonso-
Alvarez:2021qfd ; Elahi:2021jia . Section III present a detailed LCSR
calculation for the $\Lambda_{b}^{0}\to M\psi$ decays. Section IV present a
SU(3) analysis on the decay amplitudes, which relates $\Xi_{b}^{\pm}\to M\psi$
to $\Lambda_{b}^{0}\to M\psi$. Section V present numerical calculations on the
decay amplitudes and branching fractions. Section VI is a summary of this
work.

##  II The $B$-Mesogenesis scenario

This section gives a brief introduction to the $B$-Mesogenesis scenario
Elor:2018twp ; Alonso-Alvarez:2021qfd ; Elahi:2021jia , which aims to
simultaneously explain the baryon asymmetry and the existence of dark matter
in our Universe. In the $B$-Mesogenesis scenario, the $b$ quark is possible to
decay into two light quarks and a dark baryon $\psi$. The total baryon number
in such decay process is conserved, however, due to the invisible $\psi$ the
visible decay products exhibit baryon number non-conserving phenomenon. As
proposed by Refs. Elor:2018twp ; Alonso-Alvarez:2021qfd , this kind of baryon
number violating decays can emerge from the following two types of effective
Lagrangians:

$\displaystyle\mathcal{L}_{\rm eff}^{I}=$ $\displaystyle-
y_{ub}\epsilon_{ijk}Y^{*i}{\bar{u}}_{R}^{j}b_{R}^{c,k}-y_{cb}\epsilon_{ijk}Y^{*i}{\bar{c}}_{R}^{j}b_{R}^{c,k}-y_{\psi
d}Y_{i}{\bar{\psi}}d_{R}^{c,i}-y_{\psi s}Y_{i}{\bar{\psi}}s_{R}^{c,i}+\rm
h.c,$ $\displaystyle\mathcal{L}_{\rm eff}^{II}=$ $\displaystyle-
y_{ud}\epsilon_{ijk}Y^{*i}{\bar{u}}_{R}^{j}d_{R}^{c,k}-y_{us}\epsilon_{ijk}Y^{*i}{\bar{u}}_{R}^{j}s_{R}^{c,k}-y_{cd}\epsilon_{ijk}Y^{*i}{\bar{c}}_{R}^{j}d_{R}^{c,k}-y_{cs}\epsilon_{ijk}Y^{*i}{\bar{c}}_{R}^{j}s_{R}^{c,k}$
$\displaystyle-y_{\psi b}Y_{i}{\bar{\psi}}b_{R}^{c,i}+\rm h.c,$ (1)

which correspond to the type-I and II models in the $B$-Mesogenesis,
respectively. All the quark fields are taken as right handed, the superscript
$c$ indicates charge conjugate and the $y$ s are unknown coupling constants.
$Y$ is a charged color triplet scalar with $Q_{Y}=-1/3$, which is assumed to
have large mass $M_{Y}$. In the Type-I model the $b$ quark couples with $u,c$
quarks, the dark anti-baryon $\psi$ couples with $d,s$ quarks. In the Type-II
model, the situations of the $b$ and $d,s$ quarks are interchanged. In fact,
as proposed by Ref. Alonso-Alvarez:2021qfd , there should be a third type of
effective Lagrangian with $Q_{Y}=2/3$ in $B$-Mesogenesis, which reads as

$\displaystyle\mathcal{L}_{\rm eff}^{III}=$ $\displaystyle-
y_{bd}\epsilon_{ijk}Y^{*i}{\bar{b}}_{R}^{j}d_{R}^{c,k}-y_{bs}\epsilon_{ijk}Y^{*i}{\bar{b}}_{R}^{j}s_{R}^{c,k}-y_{\psi
u}Y_{i}{\bar{\psi}}u_{R}^{c,i}-y_{\psi c}Y_{i}{\bar{\psi}}c_{R}^{c,i}+\rm
h.c.$ (2)

In this work, we will only consider the case of $Q_{Y}=-1/3$ to be consistent
with the studies of exclusive $B$ meson decay in $B$-Mesogenesis
Khodjamirian:2022vta ; Elor:2022jxy .

Integrating out the heavy boson $Y$ in Eq. (1), one can obtain the effective
Hamiltonian for the two types of models as:

$\displaystyle\mathcal{H}_{\rm eff}^{I,uq}=-\frac{y_{ub}y_{\psi
q}}{M_{Y}^{2}}i\epsilon_{ijk}({\bar{\psi}}q_{R}^{c,i})({\bar{u}}_{R}^{j}b_{R}^{c,k})+{\rm
h.c.}=-G_{(uq)}^{I}{\bar{\cal O}}_{(uq)}^{I}\psi^{c}+{\rm h.c.},$
$\displaystyle\mathcal{H}_{\rm eff}^{II,uq}=-\frac{y_{\psi
b}y_{uq}}{M_{Y}^{2}}i\epsilon_{ijk}({\bar{\psi}}b_{R}^{c,i})({\bar{u}}_{R}^{j}q_{R}^{c,k})+{\rm
h.c.}=-G_{(uq)}^{II}{\bar{\cal O}}_{(uq)}^{II}\psi^{c}+{\rm h.c.},$ (3)

where for simplicity, $q=s,d$ and we use $u$ to denote the $u$ or $c$ quarks
simultanously. The effective coupling constant and the effective operators in
the two models are defined as

$\displaystyle G_{(uq)}^{I}=\frac{y_{ub}y_{\psi q}}{M_{Y}^{2}},~{}~{}~{}{\cal
O}_{(uq)}^{I}=-i\epsilon_{ijk}(u^{iT}CP_{R}b^{j})P_{R}q^{k},$ $\displaystyle
G_{(uq)}^{II}=\frac{y_{\psi b}y_{uq}}{M_{Y}^{2}},~{}~{}~{}{\cal
O}_{(uq)}^{II}=-i\epsilon_{ijk}(u^{iT}CP_{R}q^{j})P_{R}b^{k},$ (4)

where $P_{R}=(1+\gamma_{5})/2$ and $C$ is the charge conjugate matrix. The
baryon number violated decays $\Lambda_{b}\to\pi\psi$, $\Lambda_{b}\to K\psi$
and $\Lambda_{b}\to D\psi$ are induced by $\mathcal{H}_{\rm eff}^{I,ud}$,
$\mathcal{H}_{\rm eff}^{I,us}$ and $\mathcal{H}_{\rm eff}^{I,cd}$
respectively.

##  III $\Lambda_{b}\to M\psi$ decays in LCSR

###  III.1 Hadron level calculation

Figure 1: The Feynman diagram of the $\Lambda_{b}\to M\psi$ decays. The green
and gray bubbles denote the $\Lambda_{b}$ and pseudoscalar $M$, respectively,
and the white crossed dot denotes the ${\cal O}_{(uq)}$ vertex.

In this section, we perform a LCSR calculation on the $\Lambda_{b}\to M\psi$
decays as shown in Fig.1. The green and gray bubbles denote the $\Lambda_{b}$
and pseudoscalar $M$, respectively, and the white crossed dot denotes the
${\cal O}_{(uq)}$ vertex. Using the effective Hamiltonians given in Eq.(3),
one can express the decay amplitude as

$\displaystyle i{\cal M}=-G_{(uq)}{\bar{u}}_{\psi}^{c}(q,s_{\psi})\langle
M(p^{\prime})|{\cal O}_{(uq)}(0)|\Lambda_{b}(p,s_{\Lambda})\rangle,$ (5)

where $u_{\psi}$ is the spinor of dark baryon, with momentum $q=p-p^{\prime}$.
The transition matrix element on the right hand side above can be
parameterized by two form factors:

$\displaystyle\langle M(p^{\prime})|{\cal
O}_{(uq)}(0)|\Lambda_{b}(p,s_{\Lambda})\rangle=P_{R}\left[F_{1}(q^{2})+\frac{\not{q}}{m_{\Lambda_{b}}}F_{2}(q^{2})\right]u_{\Lambda_{b}}(p,s_{\Lambda}).$
(6)

In the framework of LCSR, the calculation of the transition matrix element
given above starts from the following two-point correlation function

$\displaystyle\Pi(p,q)=i\int d^{4}x\ e^{ip\cdot x}q^{\mu}\langle
0|T\\{j_{\mu}^{M}(x){\cal O}_{(uq)}(0)\\}|\Lambda_{b}(p+q)\rangle,$ (7)

where $j_{\mu}^{M}$ denotes the interpolation current for the final meson,
which reads as

$\displaystyle
j_{\mu}^{\pi}=\frac{1}{\sqrt{2}}(\bar{u}\gamma_{\mu}\gamma_{5}u-\bar{d}\gamma_{\mu}\gamma_{5}d),~{}~{}~{}~{}j_{\mu}^{K}=\bar{s}\gamma_{\mu}\gamma_{5}d,~{}~{}{\text{and}}~{}~{}j_{\mu}^{D}=\bar{c}\gamma_{\mu}\gamma_{5}d$
(8)

for the final states: $M=\pi^{0},\ K^{0}$ and $D^{0}$ respectively. In LCSR,
the correlation function defined in Eq.(7) will be calculated both at the
hadron and quark-gluon level. The one at hadron level can be expressed by the
form factors defined in Eq.(6). On the other hand, the one at quark-gluon
level will be calculated explicitly in QCD, with the light-cone distribution
amplitudes (LCDAs) of $\Lambda_{b}$ taken as non-perturbative inputs. Matching
of the correlation function at these two levels enables us to extract the
transition form factors.

At the hadron level, one inserts a complete set of states with the same
quantum number of $M$ between the $j_{\mu}^{M}(q^{2})$ current and ${\cal
O}_{(uq)}(0)$. The correlation function becomes

$\displaystyle\Pi^{H}(p,q)=\frac{if_{M}}{m_{M}^{2}-p^{2}}(p\cdot
q)P_{R}\left[F_{1}(q^{2})+\frac{\not{q}}{m_{\Lambda_{b}}}F_{2}(q^{2})\right]u_{\Lambda_{b}}(p+q)+\int_{s_{\rm
th}}^{\infty}ds\frac{\rho^{H}(s,q)}{s-p^{2}},$ (9)

where the meson decay constant is defined as: $\langle
0|j_{\mu}^{M}(0)|M(p)\rangle=if_{M}p_{\mu}$. Here only the pole contribution
from $M$ is expressed explicitly, while the contribution from excited state
and continuous spectrum above threshold $s_{\rm th}$ are capsuled in the
integration of $\rho^{H}(s,q)$, which will be suppressed by Borel
transformation. Note that in the expression: $p\cdot
q=(1/2)(m_{\Lambda_{b}}^{2}-q^{2}-m_{M}^{2})+(1/2)(m_{M}^{2}-p^{2})$, the
second term will cancel the denominator of the pole contribution in Eq.(9), so
that the terms proportional to it will vanish under the Borel transformation
on $p^{2}$. As a result, the Borel transformed correlation function becomes

$\displaystyle{\cal B}\\{\Pi^{H}\\}(p,q)=$
$\displaystyle\frac{i}{2}f_{M}(m_{\Lambda_{b}}^{2}-q^{2}-m_{M}^{2})e^{-m_{M}^{2}/T^{2}}P_{R}\left[F_{1}(q^{2})+\frac{\not{q}}{m_{\Lambda_{b}}}F_{2}(q^{2})\right]u_{\Lambda_{b}}(p+q)$
$\displaystyle+\int_{s_{\rm th}}^{\infty}ds\ e^{-s/T^{2}}\rho^{H}(s,q),$ (10)

where $T$ is the Borel parameter. Now the contributions from excited state and
continuous spectrum are suppressed by the exponential term $e^{-s/T^{2}}$.
Next, the same correlation function will be calculated at the quark-gluon
level, where the result can be generally written as a dispersion integration
form

$\displaystyle\Pi^{QCD}(p,q)=\frac{1}{\pi}\int_{s_{m}}^{\infty}ds\frac{{\rm
Im}\Pi^{QCD}(s,q)}{s-p^{2}},$ (11)

with $s_{m}$ being the quark level threshold. From the assumption of quark-
hadron duality, the last term of Eq.(9), denoting the excited state and
continuous spectrum at hadron level, is equivalent with the spectrum
integration above $s_{\rm th}$ at quark-gluon level. After the Borel
transformation and subtracting the continuous spectrum contribution, we arrive
at the sum rules equation:

$\displaystyle{\cal B}\\{\Pi^{H}\\}(p,q)_{\rm pole}={\cal
B}\\{\Pi^{QCD}\\}(p,q)=\frac{1}{\pi}\int_{s_{m}}^{s_{\rm th}}ds\
e^{-s/T^{2}}{\rm Im}\Pi^{QCD}(s,q).$ (12)

Thus the form factors $F_{1,2}$ can be extracted through this equation as long
as the imaginary part of $\Pi^{QCD}$ is obtained.

###  III.2 Quark-gluon level calculation

Now we perform an quark-gluon level calculation for the correlation function
defined in Eq.(7). Here we take the type-I model and the case of
$M={\bar{D}}^{0}$ as an example to illustrate the calculation. Using the quark
level expression of $j_{\mu}^{D}$ given in Eq.(8), one can write the
correlation function as

$\displaystyle\Pi_{\Lambda_{b}\to D}^{QCD}(p,q)=$
$\displaystyle-\epsilon_{ijk}\int d^{4}xe^{ip\cdot
x}\left[P_{R}CS_{c}(0,x)\not{q}\gamma_{5}\right]_{\gamma\alpha}(P_{R})_{\tau\beta}$
$\displaystyle\times\langle
0|u_{\alpha}^{i}(q^{2})d_{\beta}^{j}(0)b_{\gamma}^{k}(0)|\Lambda_{b}(p+q)\rangle,$
(13)

where $S_{c}$ is the charm quark propagator. The non-local three quark matrix
element above is expressed by the $\Lambda_{b}$ LCDAs, which are defined as
Ball:2008fw ; Duan:2022uzm

$\displaystyle\langle
0|u_{\alpha}^{i}(x_{1})d_{\beta}^{j}(x_{2})b_{\gamma}^{k}(0)|\Lambda_{b}(v)\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{8}f_{\Lambda_{b}}^{(2)}\Psi_{2}(t_{1},t_{2})(\not{\overline{n}}\gamma_{5}C)_{\alpha\beta}u_{\Lambda_{b}\gamma}(v)+\frac{1}{4}f_{\Lambda_{b}}^{(1)}\Psi_{3}^{s}(t_{1},t_{2})(\gamma_{5}C)_{\alpha\beta}u_{\Lambda_{b}\gamma}(v)$
$\displaystyle-\frac{1}{8}f_{\Lambda_{b}}^{(1)}\Psi_{3}^{\sigma}(t_{1},t_{2})(i\sigma_{\overline{n}n}\gamma_{5}C)_{\alpha\beta}u_{\Lambda_{b}\gamma}(v)+\frac{1}{8}f_{\Lambda_{b}}^{(2)}\Psi_{4}(t_{1},t_{2})(\not{n}\gamma_{5}C)_{\alpha\beta}u_{\Lambda_{b}\gamma}(v),$
(14)

where $\sigma_{\overline{n}n}=\sigma_{\mu\nu}\overline{n}^{\mu}n^{\nu}$ and
$x_{1}=t_{1}n,x_{2}=t_{2}n$ are on the light cone. Inversely, the light cone
vectors $n,\bar{n}$ can be expressed by the coordinates as

$\displaystyle n_{\mu}=\frac{x_{\mu}}{v\cdot
x},\quad\overline{n}_{\mu}=2v_{\mu}-\frac{x_{\mu}}{v\cdot x}.$

At the quark-gluon level, the corresponding diagram for the correlation
function defined in Eq.(7) is shown in Fig.2. The green bubble denotes the
$\Lambda_{b}$ LCDAs, the white crossed and the black dots denote the ${\cal
O}_{(uq)}(x)$ at and the $j_{\mu}^{M}(0)$ vertexes, respectively.

Figure 2: The Feynman diagram of the the correlation function defined in
Eq.(7). The green bubble denotes the $\Lambda_{b}$ LCDAs, the white crossed
and the black dots denote the ${\cal O}_{(uq)}(x)$ and the $j_{\mu}^{M}(0)$
vertexes, respectively.

The $\Psi_{2},\Psi_{3}^{s},\Psi_{3}^{\sigma},\Psi_{4}$ in Eq.(14) are the
LCDAs with different twists. In the momentum space, the LCDAs can be
characterized by the total momentum $\omega$ of the two light quarks in
$\Lambda_{b}$, and the momentum fraction $u$ of one light quark:

$\Psi_{i}(t_{1},t_{2})=\int_{0}^{\infty}\omega
d\omega\int_{0}^{1}due^{-i\omega(t_{1}u+t_{2}\overline{u})}{\psi}_{i}(\omega,u),$
(15)

with $\overline{u}=1-u$, and Ball:2008fw

$\displaystyle{\psi}_{2}(\omega,u)=$
$\displaystyle\omega^{2}u(1-u)[\frac{1}{\epsilon_{0}^{4}}e^{-\omega/\epsilon_{0}}+a_{2}C_{2}^{3/2}(2u-1)\frac{1}{\epsilon_{1}^{4}}e^{-\omega/\epsilon_{1}}],$
$\displaystyle{\psi}_{3}^{s}(\omega,u)=$
$\displaystyle\frac{\omega}{2\epsilon_{3}^{3}}e^{-\omega/\epsilon_{3}},$
$\displaystyle{\psi}_{3}^{\sigma}(\omega,u)=$
$\displaystyle\frac{\omega}{2\epsilon_{3}^{3}}(2u-1)e^{-\omega/\epsilon_{3}},$
$\displaystyle{\psi}_{4}(\omega,u)=$ $\displaystyle
5\mathcal{N}^{-1}\int_{\omega/2}^{s_{0}^{\Lambda_{b}}}dse^{-s/\tau}(s-\omega/2)^{3},$
(16)

where $C_{2}^{3/2}(2u-1)$ is the Gegenbauer polynomial, and the normalization
factor $\mathcal{N}$ in $\tilde{\psi}_{4}(\omega,u)$ reads as
$\mathcal{N}=\int_{0}^{s_{0}^{\Lambda_{b}}}dss^{5}e^{-s/\tau}$. The parameters
in these four LCDAs are $\epsilon_{0}=200^{+130}_{-60}$
$\rm{MeV},\epsilon_{1}=650^{+650}_{-300}$ $\rm{MeV},\epsilon_{3}=230$
$\rm{MeV}$, $a_{2}=0.333^{+0.250}_{-0.333}$, $s_{0}^{\Lambda_{b}}=1.2$ GeV and
$\tau=0.4\sim 0.8$ GeV. Ball:2008fw .

Using the LCDAs defined in Eq.(14), and the free charm quark propagator, one
can express the correlation function as a convolution of the perturbative part
and the LCDAs. Firstly, the twist-3s contribution can be straightforwardly
written as

$\displaystyle\Pi_{\Lambda_{b}\to D}^{QCD(3s)}(p,q)=$
$\displaystyle\frac{i}{4}f_{\Lambda_{b}}^{(1)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\
\omega\int_{0}^{1}du\ \psi_{3s}(\omega,u)\frac{1}{(u\omega
v-p)^{2}-m_{c}^{2}}P_{R}\not{q}(u\omega\not{v}-\not{p})u_{\Lambda_{b}}(v).$
(17)

To extract the imaginary part of correlation function, one has to calculate
its discontinuity across the complex plane of $p^{2}$. The denominator can be
rewritten as

$\displaystyle(u\omega
v-p)^{2}-m_{c}^{2}=\left(1-\frac{u\omega}{m_{\Lambda_{b}}}\right)(p^{2}-\Sigma_{c}),$
$\displaystyle\Sigma_{c}=\frac{1}{m_{\Lambda_{b}}-u\omega}\left[u\omega(m_{\Lambda_{b}}^{2}-q^{2})+m_{\Lambda_{b}}(m_{c}^{2}-u^{2}\omega^{2})\right].$
(18)

Now the imaginary part comes from the term

$\displaystyle{\rm Im}\left[\frac{1}{(u\omega
v-p)^{2}-m_{c}^{2}}\right]=\frac{1}{2i}(-2\pi
i)\frac{m_{\Lambda_{b}}}{m_{\Lambda_{b}}-u\omega}\delta(s-\Sigma_{c}),$ (19)

where $s=p^{2}$. Expressing $\Pi_{\Lambda_{b}\to D}^{QCD(3s)}$ as a dispersion
integration in $s_{m}<s<s_{\rm th}$, performing the Borel transformation, and
integrating out $s$ by the delta function in Eq.(19), one arrives at

$\displaystyle{\cal B}\\{\Pi_{\Lambda_{b}\to D}^{QCD(3s)}\\}(T,q)=$
$\displaystyle\frac{i}{4}f_{\Lambda_{b}}^{(1)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
\theta(\Sigma_{c}-s_{m})\theta(s_{\rm th}-\Sigma_{c})e^{-\Sigma_{c}/T^{2}}$
$\displaystyle\times\psi_{3s}(\omega,u)P_{R}\left[m_{\Lambda_{b}}\omega\not{q}-\frac{m_{\Lambda_{b}}\omega}{m_{\Lambda_{b}}-u\omega}q^{2}\right].$
(20)

The calculation for the twist-3$\sigma$ contribution is more involved. The
corresponding correlation function can be firstly written as

$\displaystyle\Pi_{\Lambda_{b}\to D}^{QCD(3\sigma)}(p,q)=$
$\displaystyle-\frac{i}{4}f_{\Lambda_{b}}^{(1)}\int d^{4}x\ e^{ip\cdot
x}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\ \omega\int_{0}^{1}du\ e^{-iu\omega
v\cdot x}\int\frac{d^{4}k}{(2\pi)^{4}}e^{ik\cdot x}$
$\displaystyle\times\psi_{3s}(\omega,u)P_{R}(1-\not{n}\not{v})\not{q}\frac{1}{\not{k}}u_{\Lambda_{b}}(v),$
(21)

where the 4-velocity of $\Lambda_{b}$ is $m_{\Lambda_{b}}v=p+q$. It can be
found that the term containing no $\not{n}$ above has almost the same form as
that of $\Pi_{\Lambda_{b}\to D}^{QCD(3s)}$. The corresponding contribution to
the Borel transformed $\Pi_{\Lambda_{b}\to D}^{QCD(3\sigma)}$ is

$\displaystyle{\cal B}\\{\Pi_{\Lambda_{b}\to D}^{QCD(3\sigma(1))}\\}(T,q)=$
$\displaystyle-\frac{i}{4}f_{\Lambda_{b}}^{(1)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
e^{-\Sigma_{c}/T^{2}}\theta(\Sigma_{c}-s_{m})\theta(s_{\rm th}-\Sigma_{c})$
$\displaystyle\times\psi_{3\sigma}(\omega,u)P_{R}\left[m_{\Lambda_{b}}\omega\not{q}-\frac{m_{\Lambda_{b}}\omega}{m_{\Lambda_{b}}-u\omega}q^{2}\right].$
(22)

Next we consider the term proportional to $\not{n}$ in Eq.(21). Note that
$n_{\mu}=x_{\mu}/v\cdot x$, to eliminate the $1/v\cdot x$ one can define
modified LCDAs as

$\displaystyle{\bar{\psi}}_{i}(\omega,u)=\int_{0}^{\omega}d\tau\tau\psi_{i}(\tau,u),$
(23)

with $i=2,3s,3\sigma,4$. Thus the $1/v\cdot x$ can be eliminated through
integration by part:

$\displaystyle\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\
\omega\psi_{i}(\omega,u)e^{-iu\omega v\cdot x}\frac{x_{\mu}}{v\cdot
x}=iu\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega{\bar{\psi}}_{i}(\omega,u)e^{-iu\omega
v\cdot x}x_{\mu}.$ (24)

Here the boundary term has been omitted since large $\omega$ in the
exponential induces high frequency oscillation under the integration of $x$,
which suppresses its contribution. The $x_{\mu}$ can be written as
$-i\partial/\partial p^{\mu}$, and thus the $\not{n}$ term in Eq.(21)
contributes:

$\displaystyle\Pi_{\Lambda_{b}\to D}^{QCD(3\sigma(n))}(p,q)=$
$\displaystyle\frac{i}{4}f_{\Lambda_{b}}^{(1)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\
\omega\int_{0}^{1}du\ {\bar{\psi}}_{3\sigma}(\omega,u)$ $\displaystyle\times
P_{R}\frac{\partial}{\partial
p^{\mu}}\left[\frac{1}{(p-u\omega)^{2}-m_{c}^{2}}\gamma^{\mu}\not{v}\not{q}(u\omega\not{v}-\not{p})\right]u_{\Lambda_{b}}(v).$
(25)

After dispersion integration, performing the Borel transformation, and
introducing auxiliary mass to lower the higher power of denominators:
$1/(p^{2}-\Sigma_{c})^{2}=(\partial/\partial
M^{2})[1/(p^{2}-\Sigma_{c}-M^{2})]|_{M^{2}=0}$, one obtains the $\not{n}$ term
contribution as

$\displaystyle{\cal B}\\{\Pi_{\Lambda_{b}\to D}^{QCD(3\sigma(n))}\\}(T,q)$
$\displaystyle=$
$\displaystyle-\frac{i}{2}f_{\Lambda_{b}}^{(1)}\frac{\partial}{\partial
M^{2}}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\ u\
{\bar{\psi}}_{3\sigma}(\omega,u)\theta[(\Sigma_{c}+M^{2})-s_{m}]\theta[s_{\rm
th}-(M^{2}+\Sigma_{c})]e^{-(\Sigma_{c}+M^{2})/T^{2}}$ $\displaystyle\times
P_{R}\left(\frac{m_{\Lambda_{b}}}{m_{\Lambda_{b}}-u\omega}\right)^{2}\left[A(p^{2},q^{2})+(2v\cdot
q)m_{c}^{2}+B(p^{2},q^{2})\not{q}\right]u_{\Lambda_{b}}(v){\Big{|}}_{M^{2}=0,\
p^{2}=\Sigma_{c}+M^{2}}\ ,$ (26)

where

$\displaystyle A(p^{2},q^{2})$ $\displaystyle=(2v\cdot
q)(m_{\Lambda_{b}}-u\omega)(m_{\Lambda_{b}}+u\omega-2v\cdot
p)-2(m_{\Lambda_{b}}-v\cdot p)q^{2},$ $\displaystyle B(p^{2},q^{2})$
$\displaystyle=q^{2}-(m_{\Lambda_{b}}-u\omega)(m_{\Lambda_{b}}+u\omega-2v\cdot
p),$ (27)

and note that all the Lorentz invariants involved above should be expressed by
$p^{2},q^{2}$ as

$\displaystyle v\cdot q$
$\displaystyle=\frac{1}{2m_{\Lambda_{b}}}(m_{\Lambda_{b}}^{2}+q^{2}-p^{2}),~{}~{}~{}v\cdot
p=\frac{1}{2m_{\Lambda_{b}}}(m_{\Lambda_{b}}^{2}+p^{2}-q^{2}),$ $\displaystyle
p\cdot q$ $\displaystyle=\frac{1}{2}(m_{\Lambda_{b}}^{2}-q^{2}-p^{2})$ (28)

The twist-2 and 4 contributions to the correlation function can be obtained
similarly as that of twist-3. The corresponding Borel transformed correlation
function is

$\displaystyle{\cal B}\\{\Pi_{\Lambda_{b}\to D}^{QCD(2)+(4)}\\}(p,q)$
$\displaystyle=$
$\displaystyle\frac{i}{4}m_{c}f_{\Lambda_{b}}^{(2)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
u\ e^{-\Sigma_{c}/T^{2}}\theta(\Sigma_{c}-s_{m})\theta(s_{\rm th}-\Sigma_{c})$
$\displaystyle\times{\psi}_{2}(\omega,u)\left(\frac{\omega
m_{\Lambda_{b}}}{m_{\Lambda_{b}}-u\omega}\right)P_{R}\left[2v\cdot
q-\not{q}\right]u_{\Lambda_{b}}(v)|_{M^{2}=0}$ $\displaystyle+$
$\displaystyle\frac{i}{4}m_{c}f_{\Lambda_{b}}^{(2)}\frac{\partial}{\partial
M^{2}}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\ u\
e^{-(\Sigma_{c}+M^{2})/T^{2}}\theta(\Sigma_{c}+M^{2}-s_{m})\theta(s_{\rm
th}-M^{2}-\Sigma_{c})({\bar{\psi}}_{2}-{\bar{\psi}}_{4})(\omega,u)$
$\displaystyle\times\left(\frac{m_{\Lambda_{b}}}{m_{\Lambda_{b}}-u\omega}\right)^{2}P_{R}\left[2p\cdot
q+q^{2}-2u\omega v\cdot
q-(m_{\Lambda_{b}}-u\omega)\not{q}\right]u_{\Lambda_{b}}(v){\Big{|}}_{M^{2}=0,\
p^{2}=\Sigma_{c}+M^{2}}\ .$ (29)

It can be found that here the twist-2 and 4 contributions are proportional to
$m_{c}$. In the case of $\Lambda_{b}\to\pi,K+\psi$ decays, the twist-2 and 4
contributions are proportional to $m_{u}=m_{d}=0$ and thus vanish. For the
$\Lambda_{b}\to\pi,K+\psi$ decays in the type-I model, the calculation of the
correlation function in Eq.(7) is almost the same, which can be found in the
Appendix A.

Generally, as Eq.(10) shows, the correlation function has two independent
spinor structures: $1$ and $\not{q}$. In the type-I model both of them exists
in the correlation function. However, it can be found that the $\not{q}$ term
is absent in the type-II model. As a result, only $F_{1}$ contributes to the
$\Lambda_{b}^{0}\to\pi^{0},K^{0},{\bar{D}}^{0}+\psi$ decays, while $F_{2}$
vanishes up to the twist-3 LCDA contributions. On the other hand, in the type-
II model the $\Lambda_{b}^{0}\to\pi^{0}+\psi$ decay is forbidden due to the
flavor SU(3) limit, which will be discussed in the next section. The
corresponding analytical results in the type-II models are given in the
Appendix B.

##  IV SU(3) analysis

In the above study we have only focused on the decay processes
$\Lambda_{b}^{0}\to\pi^{0},K^{0},{\bar{D}}+\psi$, while the various decay
channels $\Xi_{b}^{0,-}\to\pi^{0,-},K^{0,-}+\psi$ and
$\Lambda_{b}^{0},\Xi_{b}^{0}\to\eta+\psi$ have not been considered. In
principle, the amplitudes of these missed processes can also be calculated as
soon as the LCDAs of $\Xi_{b}$ or $\eta$ are known, which have not been
studied as well as the LCDAs of $\Lambda_{b}$. However, with the use of flavor
SU(3) symmetry, one can still able to predict the decay widths of these missed
channels as long as the decay widths of $\Lambda_{b}^{0}\to\pi^{0},K^{0}+\psi$
are known.

As shown in Eq.(3), the effective Hamiltonian: $-G_{uq}{\cal O}_{(uq)}$
contains two light quark fields $u,\ d$ or $u,\ s$. In the flavor SU(3)
representation, this Hamiltonian can be represented by a rank-two tensor
$H_{ij}$, where $i,j$ are the flavor indexes. Its non-vanishing components are
$H_{12}=G_{ud}$ and $H_{13}=G_{us}$ for $b\to u,d$ and $b\to u,s$ transitions,
respectively. Note that $H_{ij}$ is reducible so that can be further be
reduced into three SU(3) irreducible representations: a symmetric and
traceless tensor $\bar{S}_{\\{ij\\}}$, an anti-symmetry tensor $T_{[ij]}$ and
a trace term $I\delta_{ij}$:

$\displaystyle H_{ij}$
$\displaystyle=\bar{S}_{\\{ij\\}}+T_{[ij]}+I_{\\{ij\\}},$ (30)

where

$\displaystyle\bar{S}_{\\{ij\\}}=\frac{1}{2}H_{ij}+\frac{1}{2}H_{ji}-\frac{1}{3}\delta_{ij}H_{kk},~{}~{}~{}T_{[ij]}=\frac{1}{2}H_{ij}-\frac{1}{2}H_{ji}.$
(31)

The non-vanishing components are $\bar{S}_{12}=\bar{S}_{21}=\frac{1}{2}$,
$T_{12}=-T_{21}=\frac{1}{2}$ for $b\to u,d$, and
$\bar{S}_{13}=\bar{S}_{31}=\frac{1}{2}$, $T_{13}=-T_{31}=\frac{1}{2}$ for
$b\to u,s$. It should be mentioned that in the type-II model, as shown by Eq.4
the two light flavors are anti-symmetrized, so that $\bar{S}_{ij}$ vanishes in
this case.

On the other hand, the initial anti-triplet baryons $T_{\bf{b\bar{3}}}$ and
the final light mesons $M$ can also be expressed by SU(3) irreducible
representations as

$\displaystyle(T_{\bf{b\bar{3}}}^{ij})=\left(\begin{array}[]{ccc}0&\Lambda_{b}^{0}&\Xi_{b}^{0}\\\
-\Lambda_{b}^{0}&0&\Xi_{b}^{-}\\\
-\Xi_{b}^{0}&-\Xi_{b}^{-}&0\end{array}\right)_{ij},\;\;\;M^{i}_{j}=\begin{pmatrix}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&\pi^{+}&K^{+}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{6}}&{K^{0}}\\\
K^{-}&\overline{K}^{0}&-\frac{2\eta}{\sqrt{6}}\end{pmatrix}_{ij}.$ (35)

Using $T_{\bf{b\bar{3}}}$, $M^{i}_{j}$, $\bar{S}_{ij}$ and $T_{ij}$, one can
construct SU(3) invariant amplitude as

$\displaystyle{\cal A}=s_{1}\
T_{\bf{b\bar{3}}}^{ij}{\bar{S}}_{jk}(M^{T})^{k}_{i}\psi+t_{1}\
T_{\bf{b\bar{3}}}^{ij}T_{jk}(M^{T})^{k}_{i}\psi+t_{2}\
T_{\bf{b\bar{3}}}^{ij}T_{ij}(M^{T})^{k}_{k}\psi.$ (36)

Here we have not considered the color singlet $\eta_{1}$ in the construction
of $M^{i}_{j}$, thus the third term above vanishes due to $M^{k}_{k}=0$. The
unknown amplitudes $s_{1},t_{1},t_{2}$ contain all the information about the
strong interaction in the decays. All the decay amplitudes of anti-triplet
bottom baryon decays into a light meson and dark baryon are listed in Table 1,
where the left and right two columns correspond to $b\to d$ and $b\to s$
transitions.

Table 1: Decay amplitudes of anti-triplet bottom baryon decays into a light meson and dark baryon from Eq.(36), where the left and right columns correspond to $b\to d$ and $b\to s$ transitions, respectively. Channel | Amplitude | Channel | Amplitude
---|---|---|---
$\Lambda_{b}^{0}\to\pi^{0}\psi$ | $\sqrt{2}s_{1}G_{ud}$ | $\Lambda_{b}^{0}\to K^{0}\psi$ | $-(s_{1}+t_{1})G_{us}$
$\Xi_{b}^{0}\to{\bar{K}}^{0}\psi$ | $-(s_{1}+t_{1})G_{ud}$ | $\Xi_{b}^{0}\to\pi^{0}\psi$ | $\frac{1}{\sqrt{2}}(s_{1}-t_{1})G_{us}$
$\Xi_{b}^{-}\to K^{-}\psi$ | $-(s_{1}-t_{1})G_{ud}$ | $\Xi_{b}^{-}\to\pi^{-}\psi$ | $(s_{1}-t_{1})G_{us}$
$\Lambda_{b}^{0}\to\eta\psi$ | $-\sqrt{\frac{2}{3}}t_{1}G_{ud}$ | $\Xi_{b}^{0}\to\eta\psi$ | $\Big{(}\frac{1}{\sqrt{6}}t_{1}-\sqrt{\frac{3}{2}}s_{1}\Big{)}G_{us}$

Note that the amplitude of $\Lambda_{b}^{0}\to\pi^{0}+\psi$ is proportional to
$s_{1}$ which is absent in the type-II model. Therefore, in the type-II model
the $\Lambda_{b}^{0}\to\pi^{0}+\psi$ decay is forbidden, and we have $s_{1}=0$
in Table 1 so that the amplitudes of all the decay channels are proportional
to $t_{1}$. On the other hand, in the type-I model, using the decay amplitudes
of $\Lambda_{b}^{0}\to\pi^{0}\psi$ and $\Lambda_{b}^{0}\to K^{0}\psi$
calculated in this work by LCSR, we can determine the fraction
$\xi_{K/\pi}=t_{1}/s_{1}$ as

$\displaystyle-\frac{1}{\sqrt{2}}(1+\xi_{K/\pi})\lambda_{s/d}=\frac{{\cal
A}(\Lambda_{b}^{0}\to K^{0}\psi)}{{\cal A}(\Lambda_{b}^{0}\to\pi^{0}\psi)},$
(37)

with $\lambda_{s/d}=G_{us}/G_{ud}$. This enables us to predict all the decay
amplitudes listed in Table 1.

##  V Numerical Results

The hadron masses are taken as: $m_{\Lambda_{b}}=5.62$ GeV, $m_{\pi}=0.135$
GeV, $m_{K}=0.498$ GeV and $m_{D}=1.86$ GeV. The quark masses are taken as
$m_{u}=m_{d}=0$ and $m_{c}=1.1$ GeV at $\mu=3$ GeV ParticleDataGroup:2022pth .
$\tau=0.6$ GeV is taken as the center value in its range $\tau=0.4\sim 0.8$
GeV. The threshold parameters should be above the lowest state while nearly
below the next lowest state. The next lowest states corresponding to $\pi,D,K$
are $\pi(1300),K(1460),D(2550)$ ParticleDataGroup:2022pth  and $s_{\rm th}$
can be parameterized as

$\displaystyle s_{\rm th}^{\pi,K,D}=(1-\lambda)m_{\pi,K,D}^{2}+\lambda\
m_{\pi(1300),K(1460),D(2550)}^{2},$ (38)

where $\tau=0$ and $\tau=1$ correspond to the lowest and the next lowest
states, respectively. In this work, to estimate the uncertainties from the
threshold parameter, we choose the range $0.6<\lambda<1.0$ for error analysis
in numerical calculations.

Figure 3: The form factors $F_{1},F_{2}$ as functions of $T^{2}$ in the type-I
model, with $q^{2}=0$. The band width denotes the uncertainty of $s_{\rm th}$:
$0.6<\lambda<1.0$. Figure 4: The form factors $F_{1},F_{2}$ as functions of
$T^{2}$ in the type-II model, with $q^{2}=0$. The band width denotes the
uncertainty of $s_{\rm th}$: $0.6<\lambda<1.0$.

The form factors $F_{1},F_{2}$ as functions of $T^{2}$ are shown in Fig.3 and
Fig.4 for the type-I and II models respectively, where $q^{2}=0$. In
principle, the form factors should be independent of the unphysical Borel
parameter. Therefore, one has to choose a window for $T^{2}$ where the
behavior of $F_{1},F_{2}$ are almost stable. From Fig.3, it can be found that
the stable regions are at large $T^{2}$, and the $T^{2}$ windows are chosen as

$\displaystyle\Lambda_{b}\to\pi\psi,\ K\psi:$ $\displaystyle~{}~{}~{}4\ {\rm
GeV}^{2}<T^{2}<12\ {\rm GeV}^{2},$ $\displaystyle\Lambda_{b}\to D\psi:$
$\displaystyle~{}~{}~{}12\ {\rm GeV}^{2}<T^{2}<28\ {\rm GeV}^{2}$ (39)

for the both type-I and II models.

On the other hand, note that the $F_{1,2}(q^{2})$ obtained by LCSR are only
reliable in the small positive $q^{2}$ or $q^{2}<0$ regions, instead of the
physical region $q^{2}>0$. Therefore, one has to fit the form factors in the
$q^{2}<0$ region by a suitable parameterization function, and extent the form
factors to physical region. In this work, we use the z-series formula
Bourrely:2008za  with single pole structure to perform the fitting, which
reads as

$\displaystyle F_{1,2}(q^{2})=\frac{F_{1,2}(0)}{1-\frac{q^{2}}{m_{\rm
pole}^{2}}}\left[1+b_{1,2}(z(q^{2})-z(0))+c_{1,2}(z(q^{2})-z(0))^{2}\right],$
(40)

where $m_{\rm pole}$ is the mass of the lowest baryon state that can be
created by ${\cal O}_{uq}^{\dagger}$ from the vaccum. For the
$\Lambda_{b}\to\pi,K,D$ transitions, $m_{\rm pole}$ is chosen as
$m_{\Lambda_{b}},m_{\Xi_{b}}=5.79\text{GeV},m_{\Xi_{bc}}=6.94\text{GeV}$
respectively, where $m_{\Xi_{bc}}$ is taken from QCD sum rules calculation
Hu:2017dzi . The $F_{1,2}(0),b_{1,2}$ and $c_{1,2}$ in Eq.(40) are three
fitting parameters, and the $z$ function is defined as

$\displaystyle z(q^{2})$
$\displaystyle=\frac{\sqrt{t^{+}-q^{2}}-\sqrt{t^{+}-t_{0}}}{\sqrt{t^{+}-q^{2}}+\sqrt{t^{+}-t_{0}}}$
(41)

with $t^{\pm}=(m_{\Lambda_{b}}\pm m_{M})^{2}$ and
$t_{0}=t^{+}\left(1-\sqrt{1-t^{-}/t^{+}}\right)$. The fitting results of
$F_{1,2}(0),b_{1,2}$ and $c_{1,2}$ are listed in Table 2, while the form
factors extended to the physical region are shown in Fig.5 and Fig.6 for the
type I and II models, respectively.

Figure 5: The form factors $F_{1},F_{2}$ as functions of $q^{2}$ in the type-I model, with $q^{2}=0$. The band width denotes the combined uncertainties of the Borel parameter and $s_{\rm th}$: $0.6<\lambda<1.0$. Figure 6: The form factors $F_{1},F_{2}$ as functions of $q^{2}$ in the type-II model, with $q^{2}=0$. The band width denotes the combined uncertainties of the Borel parameter and $s_{\rm th}$: $0.6<\lambda<1.0$. Table 2: Fitting parameters of the $F_{1,2}$ for $\Lambda_{b}\to\pi,K,D$ transitions in the Type-I and II models. Type-I | $F_{1}(0)$ | $b_{1}$ | $c_{1}$ | $F_{2}(0)$ | $b_{2}$ | $c_{2}$
---|---|---|---|---|---|---
$\Lambda_{b}^{0}\to\pi^{0}$ | $-0.02\pm 0.005$ | $-15.77$ | 16.41 | $0.07\pm 0.014$ | -3.81 | 4.43
$\Lambda_{b}^{0}\to K^{0}$ | $0.04\pm 0.006$ | -5.74 | 12.34 | $-0.04\pm 0.006$ | -5.88 | 13.32
$\Lambda_{b}^{0}\to{\bar{D}}^{0}$ | $0.01\pm 0.001$ | 26.79 | -35.06 | $0.06\pm 0.003$ | -4.0 | 11.89
Type-II | $F_{1}(0)$ | $b_{1}$ | $c_{1}$ | $F_{2}(0)$ | $b_{2}$ | $c_{2}$
$\Lambda_{b}^{0}\to K^{0}$ | $0.003\pm 0.001$ | 6.21 | -5.76 | – | – | –
$\Lambda_{b}^{0}\to{\bar{D}}^{0}$ | $0.07\pm 0.0$ | -1.01 | -17.6 | – | – | –

With the use of form factors $F_{1,2}$ obtained above, we can calculate the
decay widths of $\Lambda_{b}\to M\psi$ decays. The decay width formula reads
as

$\displaystyle\Gamma[\Lambda_{b}\to M\psi]$ $\displaystyle=$
$\displaystyle\frac{G_{(uq)}^{2}|\vec{q}|}{8m_{\Lambda_{b}}^{2}(2\pi)^{5}}\left[(m_{\Lambda_{b}}^{2}+m_{\psi}^{2}-m_{M}^{2})\left(F_{1}^{2}(m_{\psi}^{2})+\frac{m_{\psi}^{2}}{m_{\Lambda_{b}}^{2}}F_{2}^{2}(m_{\psi}^{2})\right)+4m_{\psi}^{2}F_{1}(m_{\psi}^{2})F_{2}(m_{\psi}^{2})\right],$
(42)

which is a function of the dark baryon mass $m_{\psi}$. $|\vec{q}|$ is the
3-momentum magnitude of the dark baryon in the center of mass frame:

$\displaystyle|\vec{q}|=\frac{1}{2m_{\Lambda_{b}}}\sqrt{(m_{\Lambda_{b}}^{2}-(m_{\psi}+m_{M})^{2})(m_{\Lambda_{b}}^{2}-(m_{\psi}-m_{M})^{2})}.$
(43)

The upper limits of the unknown coupling constants $G_{(uq)}$ have been
determined in Ref.Shi:2023riy  through a study of $B$ meson semi-inclusive
decays into baryons and a dark baryon, which read as

$\displaystyle{\rm Type\ I}:~{}~{}~{}~{}$ $\displaystyle G_{ud}^{2}<(1.8\pm
0.35)\times 10^{-14}{\rm GeV}^{-4},~{}~{}~{}G_{us}^{2}<(3.75\pm 0.74)\times
10^{-14}{\rm GeV}^{-4},$ $\displaystyle G_{cd}^{2}<(1.06\pm 0.21)\times
10^{-12}{\rm GeV}^{-4},~{}~{}~{}G_{cs}^{2}<(1.63\pm 0.33)\times 10^{-12}{\rm
GeV}^{-4};$ $\displaystyle{\rm Type\ II}:~{}~{}~{}~{}$ $\displaystyle
G_{us}^{2}<(1.07\pm 0.21)\times 10^{-11}{\rm
GeV}^{-4},~{}~{}~{}G_{cs}^{2}<(3.62\pm 0.72)\times 10^{-10}{\rm GeV}^{-4}.$
(44)

Here we take $G_{(uq)}$ as their upper limit values to calculate the decay
width. Accordingly, the upper limits of branching fractions for
$\Lambda_{b}\to M\psi$ as functions of $m_{\psi}$ are shown in Fig.7 and Fig.8
in the type I and II models, respectively. The blue and red bands denote the
uncertainties from the Borel parameter and the $G_{(uq)}$ from Eq.(44).

Figure 7: The upper limits of branching fractions for $\Lambda_{b}\to M\psi$
as functions of $m_{\psi}$ in the type-I model. The blue and red bands denote
the uncertainties from the Borel parameter and the $G_{(uq)}$ from Eq.(44).

Figure 8: The upper limits of branching fractions for $\Lambda_{b}\to M\psi$
as functions of $m_{\psi}$ in the type-II model. The blue and red bands denote
the uncertainties from the Borel parameter and the $G_{(uq)}$ from Eq.(44).

In Sec.IV, a SU(3) analysis is performed to predict the amplitudes of all the
anti-triplet bottom baryon decays into a meson and dark baryon. In the type-I
model, using Eq.(37) one can further determine $\xi_{K/\pi}$ by the ratio of
branching fractions:

$\displaystyle\frac{1}{2}(1+\xi_{K/\pi})^{2}\lambda_{s/d}^{2}=\frac{{\cal
B}(\Lambda_{b}^{0}\to K^{0}\psi)}{{\cal B}(\Lambda_{b}^{0}\to\pi^{0}\psi)},$
(45)

if the phase space volume of $\Lambda_{b}^{0}\to K^{0}\psi$ is assumed to be
the same as that of $\Lambda_{b}^{0}\to\pi^{0}\psi$. The $\xi_{K/\pi}$ as a
function of $m_{\psi}$ is shown in the left diagram of Fig.9, where the range
of $m_{\psi}$ is chosen as $0<m_{\psi}<(m_{\Lambda_{b}}-m_{K})$ and
$G_{ud},G_{us}$ are taken as the center values in Eq.(44).

Figure 9: The $\xi_{K/\pi}$ as a function of $m_{\psi}$ from Eq.(45), where
the range of $m_{\psi}$ is chosen as $0<m_{\psi}<(m_{\Lambda_{b}}-m_{K})$
(left). The branching fractions of $\Xi_{b}^{-}\to K^{-}\psi$ as functions of
$m_{\psi}$ in the type-I model (right). The blue and red bands denote the
uncertainties from the Borel parameter and the $G_{(ud)}$ from Eq.(44).

Now the branching fractions of all the decay channels listed in Table 1 can be
expressed in the units of ${\cal B}(\Lambda_{b}^{0}\to\pi^{0}\psi)$ and ${\cal
B}(\Lambda_{b}^{0}\to K^{0}\psi)$ for the type-I and II models respectively,
which are shown in Table 3. Particularly, the branching fraction of the
charged decay $\Xi_{b}^{-}\to K^{-}\psi$ in the type-I model is shown in the
right diagram of Fig.9. The branching fraction of $\Xi_{b}^{-}\to\pi^{-}\psi$
has the same shape as $\Xi_{b}^{-}\to K^{-}\psi$ but a factor
$\lambda_{s/d}^{2}$ should be timed on it.

Table 3: The amplitudes of all the anti-triplet bottom baryon decays into a light meson and dark baryon are expressed in the units of ${\cal B}(\Lambda_{b}^{0}\to\pi^{0}\psi)$ and ${\cal B}(\Lambda_{b}^{0}\to K^{0}\psi)$ for the type-I and II models, respectively. Channel | Type-I Br. | Channel | Type-I Br.
---|---|---|---
$\Lambda_{b}^{0}\to\pi^{0}\psi$ | 1 | $\Lambda_{b}^{0}\to K^{0}\psi$ | $\frac{1}{2}(1+\xi_{K/\pi})^{2}\lambda_{s/d}^{2}$
$\Xi_{b}^{0}\to{\bar{K}}^{0}\psi$ | $\frac{1}{2}(1+\xi_{K/\pi})^{2}$ | $\Xi_{b}^{0}\to\pi^{0}\psi$ | $\frac{1}{4}(1-\xi_{K/\pi})^{2}\lambda_{s/d}^{2}$
$\Xi_{b}^{-}\to K^{-}\psi$ | $\frac{1}{2}(1-\xi_{K/\pi})^{2}$ | $\Xi_{b}^{-}\to\pi^{-}\psi$ | $\frac{1}{2}(1-\xi_{K/\pi})^{2}\lambda_{s/d}^{2}$
$\Lambda_{b}^{0}\to\eta\psi$ | $\frac{1}{3}\xi_{K/\pi}^{2}$ | $\Xi_{b}^{0}\to\eta\psi$ | $\frac{1}{12}(1-3\xi_{K/\pi})^{2}\lambda_{s/d}^{2}$
Channel | Type-II Br. | Channel | Type-II Br.
$\Lambda_{b}^{0}\to\pi^{0}\psi$ | 0 | $\Lambda_{b}^{0}\to K^{0}\psi$ | 1
$\Xi_{b}^{0}\to{\bar{K}}^{0}\psi$ | $\lambda_{s/d}^{-2}$ | $\Xi_{b}^{0}\to\pi^{0}\psi$ | $1/2$
$\Xi_{b}^{-}\to K^{-}\psi$ | $\lambda_{s/d}^{-2}$ | $\Xi_{b}^{-}\to\pi^{-}\psi$ | $1$
$\Lambda_{b}^{0}\to\eta\psi$ | $\frac{2}{3}\lambda_{s/d}^{-2}$ | $\Xi_{b}^{0}\to\eta\psi$ | $1/6$

The branching fractions of the charged decays in the type-II model are simply
proportional to ${\cal B}(\Lambda_{b}^{0}\to K^{0}\psi)$ so we do not plot
them here.

##  VI Conclusion

In this work, we have studied the decays of Heavy baryon into a pseudoscalar
meson and a dark baryon using the recently developed $B$-Mesogenesis scenario,
where the two types of effective Lagrangians proposed by the scenario are both
considered. The decay amplitudes of $\Lambda_{b}^{0}$ have been calculated by
LCSR using its light-cone distribution amplitudes. The decay amplitudes of
$\Xi_{b}^{0,\pm}$ has been related with those of $\Lambda_{b}^{0}$ through a
flavor SU(3) analysis. In the numerical calculation, the uncertainties of
threshold parameter and the Borel parameter are both considered. The values of
effective coupling constants in the $B$-Mesogenesis are taken as their upper
limits that obtained from our previous study on the inclusive decay. The upper
limits of the decay branching fractions of $\Lambda_{b}^{0},\Xi_{b}^{0,\pm}\to
M\psi$ are presented as functions of the dark baryon mass, which will be
tested by future experimental detections.

## Acknowledgements

The work of Y.J. Shi is supported by Natural Science Foundation of China under
Grant No.12305103, and Opening Foundation of Shanghai Key Laboratory of
Particle Physics and Cosmology under Grant No.22DZ2229013-2. The work of Y.
Xing is supported by National Science Foundation of China under Grant
No.12005294. The work of Z.P. Xing is supported by China Postdoctoral Science
Foundation under Grant No.2022M72210.

##  Appendix A Analytical Results in the Type-I model

In this appendix, we present the analytical results of the quark-gluon level
calculation for the correlation function defined in Eq.(7). In the type-I
model, the twist-3s contribution to the correlation function for the
$\Lambda_{b}\to\pi$ decay is

$\displaystyle{\cal B}\\{\Pi_{\Lambda_{b}\to\pi}^{QCD(3s)}\\}(T,q)=$
$\displaystyle\frac{i}{4\sqrt{2}}f_{\Lambda_{b}}^{(1)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
\theta(\Sigma-s_{m})\theta(s_{\rm th}-\Sigma)e^{-\Sigma/T^{2}}$
$\displaystyle\times\psi_{3s}(\omega,u)P_{R}\left[m_{\Lambda_{b}}\omega\not{q}-\frac{m_{\Lambda_{b}}\omega}{m_{\Lambda_{b}}-u\omega}q^{2}\right]$
$\displaystyle+\frac{i}{4\sqrt{2}}f_{\Lambda_{b}}^{(1)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
\theta({\bar{\Sigma}}-s_{m})\theta(s_{\rm
th}-{\bar{\Sigma}})e^{-{\bar{\Sigma}}/T^{2}}$
$\displaystyle\times\psi_{3s}(\omega,u)P_{R}\left[(\bar{u}\omega(2q\cdot
v)-2p\cdot q-q^{2})\frac{\omega
m_{\Lambda_{b}}}{m_{\Lambda_{b}}-\bar{u}\omega}+m_{\Lambda_{b}}\omega\not{q}\right]u_{\Lambda_{b}}(v),$
(46)

where $\Sigma=\Sigma_{c}|_{m_{c}=0}$ and
${\bar{\Sigma}}=\Sigma|_{u\to\bar{u}}$. The twist-$3\sigma$ contributions are

$\displaystyle{\cal B}\\{\Pi_{\Lambda_{b}\to\pi}^{QCD(3\sigma)}\\}(p,q)=$
$\displaystyle-\frac{i}{4\sqrt{2}}f_{\Lambda_{b}}^{(1)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
\theta(\Sigma-s_{m})\theta(s_{\rm th}-\Sigma)e^{-\Sigma/T^{2}}$
$\displaystyle\times\psi_{3s}(\omega,u)P_{R}\left[m_{\Lambda_{b}}\omega\not{q}-\frac{m_{\Lambda_{b}}\omega}{m_{\Lambda_{b}}-u\omega}q^{2}\right]u_{\Lambda_{b}}(v)$
$\displaystyle-\frac{i}{4\sqrt{2}}f_{\Lambda_{b}}^{(1)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
\theta({\bar{\Sigma}}-s_{m})\theta(s_{\rm
th}-{\bar{\Sigma}})e^{-{\bar{\Sigma}}/T^{2}}$
$\displaystyle\times\psi_{3\sigma}(\omega,u)P_{R}\left[(\bar{u}\omega(2q\cdot
v)-2p\cdot q-q^{2})\frac{\omega
m_{\Lambda_{b}}}{m_{\Lambda_{b}}-\bar{u}\omega}+m_{\Lambda_{b}}\omega\not{q}\right]u_{\Lambda_{b}}(v)$
$\displaystyle-\frac{i}{2\sqrt{2}}f_{\Lambda_{b}}^{(1)}\frac{\partial}{\partial
M^{2}}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
{\bar{\psi}}_{3\sigma}(\omega,u)e^{-(\Sigma+M^{2})/T^{2}}$
$\displaystyle\times\theta((\Sigma+M^{2})-s_{m})\theta(s_{\rm
th}-(\Sigma+M^{2}))\left(\frac{m_{\Lambda_{b}}}{m_{\Lambda_{b}}-u\omega}\right)^{2}$
$\displaystyle\times
P_{R}[A(p^{2},q^{2})+B(p^{2},q^{2})\not{q}]u_{\Lambda_{b}}(v){\Big{|}}_{M^{2}=0,\
p^{2}=\Sigma+M^{2}}$
$\displaystyle+\frac{i}{\sqrt{2}}f_{\Lambda_{b}}^{(1)}\frac{\partial}{\partial
M^{2}}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
{\bar{\psi}}_{3\sigma}(\omega,u)e^{-({\bar{\Sigma}}+M^{2})/T^{2}}$
$\displaystyle\times\theta(({\bar{\Sigma}}+M^{2})-s_{m})\theta(s_{\rm
th}-({\bar{\Sigma}}+M^{2}))\left(\frac{m_{\Lambda_{b}}}{m_{\Lambda_{b}}-\bar{u}\omega}\right)^{2}$
$\displaystyle\times(p\cdot q-\bar{u}\omega v\cdot
q)P_{R}(m_{\Lambda_{b}}-\bar{u}\omega-\not{q})u_{\Lambda_{b}}(v){\Big{|}}_{M^{2}=0,\
p^{2}=\bar{\Sigma}+M^{2}}\ ,$ (47)

The twist-3s and twist-$3\sigma$ contributions to the correlation function for
the $\Lambda_{b}\to K$ decay are

$\displaystyle{\cal B}\\{\Pi_{\Lambda_{b}\to K}^{QCD(3s)}\\}(T,q)=$
$\displaystyle-\frac{i}{4}f_{\Lambda_{b}}^{(1)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
\theta({\bar{\Sigma}}-s_{m})\theta(s_{\rm
th}-{\bar{\Sigma}})e^{-{\bar{\Sigma}}/T^{2}}$
$\displaystyle\times\psi_{3s}(\omega,u)P_{R}\left[(\bar{u}\omega(2q\cdot
v)-2p\cdot q-q^{2})\frac{\omega
m_{\Lambda_{b}}}{m_{\Lambda_{b}}-\bar{u}\omega}+m_{\Lambda_{b}}\omega\not{q}\right]u_{\Lambda_{b}}(v),$
$\displaystyle{\cal B}\\{\Pi_{\Lambda_{b}\to K}^{QCD(3\sigma)}\\}(p,q)=$
$\displaystyle\frac{i}{4}f_{\Lambda_{b}}^{(1)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
\theta({\bar{\Sigma}}-s_{m})\theta(s_{\rm
th}-{\bar{\Sigma}})e^{-{\bar{\Sigma}}/T^{2}}$
$\displaystyle\times\psi_{3\sigma}(\omega,u)P_{R}\left[(\bar{u}\omega(2q\cdot
v)-2p\cdot q-q^{2})\frac{\omega
m_{\Lambda_{b}}}{m_{\Lambda_{b}}-\bar{u}\omega}+m_{\Lambda_{b}}\omega\not{q}\right]u_{\Lambda_{b}}(v)$
$\displaystyle-if_{\Lambda_{b}}^{(1)}\frac{\partial}{\partial
M^{2}}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
{\bar{\psi}}_{3\sigma}(\omega,u)e^{-({\bar{\Sigma}}+M^{2})/T^{2}}$
$\displaystyle\times\theta(({\bar{\Sigma}}+M^{2})-s_{m})\theta(s_{\rm
th}-({\bar{\Sigma}}+M^{2}))\left(\frac{m_{\Lambda_{b}}}{m_{\Lambda_{b}}-\bar{u}\omega}\right)^{2}$
$\displaystyle\times(p\cdot q-\bar{u}\omega v\cdot
q)P_{R}(m_{\Lambda_{b}}-\bar{u}\omega-\not{q})u_{\Lambda_{b}}(v){\Big{|}}_{M^{2}=0,\
p^{2}=\bar{\Sigma}+M^{2}}.$ (48)

The twist-2, 4 contributions to the correlation function for the
$\Lambda_{b}\to\pi,K$ decays vanishes in the chiral limit $m_{u}=m_{d}=0$.

##  Appendix B Analytical Results in the Type-II model

In the type-II model, the $\Lambda_{b}\to\pi$ decay is forbidden in the flavor
SU(3) limit. The twist-3s and twist-$3\sigma$ contributions to the correlation
function for the $\Lambda_{b}\to K$ decay are

$\displaystyle{\cal B}\\{\Pi_{\Lambda_{b}\to K}^{QCD(3s)}\\}(T,q)=$
$\displaystyle-\frac{i}{2}f_{\Lambda_{b}}^{(1)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
\theta({\bar{\Sigma}}-s_{m})\theta(s_{\rm
th}-{\bar{\Sigma}})e^{-{\bar{\Sigma}}/T^{2}}$
$\displaystyle\times\psi_{3s}(\omega,u)\frac{\omega
m_{\Lambda_{b}}}{m_{\Lambda_{b}}-\bar{u}\omega}P_{R}[\bar{u}\omega v\cdot
q-p\cdot q]u_{\Lambda_{b}}(v),$ $\displaystyle{\cal B}\\{\Pi_{\Lambda_{b}\to
K}^{QCD(3\sigma)}\\}(p,q)=$
$\displaystyle\frac{i}{2}f_{\Lambda_{b}}^{(1)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
\theta({\bar{\Sigma}}-s_{m})\theta(s_{\rm
th}-{\bar{\Sigma}})e^{-{\bar{\Sigma}}/T^{2}}$
$\displaystyle\times\psi_{3\sigma}(\omega,u)\frac{\omega
m_{\Lambda_{b}}}{m_{\Lambda_{b}}-\bar{u}\omega}P_{R}[\bar{u}\omega v\cdot
q-p\cdot q]u_{\Lambda_{b}}(v)$
$\displaystyle-2if_{\Lambda_{b}}^{(1)}\frac{\partial}{\partial
M^{2}}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
{\bar{\psi}}_{3\sigma}(\omega,u)e^{-({\bar{\Sigma}}+M^{2})/T^{2}}$
$\displaystyle\times\theta(({\bar{\Sigma}}+M^{2})-s_{m})\theta(s_{\rm
th}-({\bar{\Sigma}}+M^{2}))\left(\frac{m_{\Lambda_{b}}}{m_{\Lambda_{b}}-\bar{u}\omega}\right)^{2}$
$\displaystyle\times(v\cdot p-\bar{u}\omega)(p\cdot q-\bar{u}\omega v\cdot
q)P_{R}u_{\Lambda_{b}}(v){\Big{|}}_{M^{2}=0,\ p^{2}=\bar{\Sigma}+M^{2}}.$ (49)

The twist-2+4 contributions to the correlation function for the
$\Lambda_{b}\to D$ decay are

$\displaystyle{\cal B}\\{\Pi_{\Lambda_{b}\to D}^{QCD(2+4)}\\}(p,q)=$
$\displaystyle\frac{i}{2}m_{c}f_{\Lambda_{b}}^{(2)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
\theta(\Sigma_{c}-s_{m})\theta(s_{\rm th}-\Sigma_{c})e^{-\Sigma_{c}/T^{2}}$
$\displaystyle\times\psi_{2}(\omega,u)\frac{\omega
m_{\Lambda_{b}}}{m_{\Lambda_{b}}-\bar{u}\omega}(v\cdot
q)P_{R}u_{\Lambda_{b}}(v)$
$\displaystyle+\frac{i}{2}m_{c}f_{\Lambda_{b}}^{(2)}\frac{\partial}{\partial
M^{2}}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
({\bar{\psi}}_{2}-{\bar{\psi}}_{4})(\omega,u)e^{-({\Sigma_{c}}+M^{2})/T^{2}}$
$\displaystyle\times\theta(({\Sigma_{c}}+M^{2})-s_{m})\theta(s_{\rm
th}-({\Sigma_{c}}+M^{2}))\left(\frac{m_{\Lambda_{b}}}{m_{\Lambda_{b}}-\bar{u}\omega}\right)^{2}$
$\displaystyle\times(p\cdot q-u\omega v\cdot
q)P_{R}u_{\Lambda_{b}}(v){\Big{|}}_{M^{2}=0,\ p^{2}=\Sigma_{c}+M^{2}}.$ (50)

The twist-3 contributions to the correlation function for the $\Lambda_{b}\to
D$ decay are

$\displaystyle{\cal B}\\{\Pi_{\Lambda_{b}\to D}^{QCD(3s)}\\}(p,q)=$
$\displaystyle-
if_{\Lambda_{b}}^{(1)}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
\theta(\Sigma_{c}-s_{m})\theta(s_{\rm th}-\Sigma_{c})e^{-\Sigma_{c}/T^{2}}$
$\displaystyle\times(\psi_{3s}+\psi_{3\sigma})(\omega,u)\frac{\omega
m_{\Lambda_{b}}}{m_{\Lambda_{b}}-\bar{u}\omega}(u\omega v\cdot q-p\cdot
q)P_{R}u_{\Lambda_{b}}(v)$
$\displaystyle+2if_{\Lambda_{b}}^{(1)}\frac{\partial}{\partial
M^{2}}\int_{0}^{2s_{0}^{\Lambda_{b}}}d\omega\int_{0}^{1}du\
{\bar{\psi}}_{3\sigma}(\omega,u)e^{-({\Sigma_{c}}+M^{2})/T^{2}}$
$\displaystyle\times\theta(({\Sigma_{c}}+M^{2})-s_{m})\theta(s_{\rm
th}-({\Sigma_{c}}+M^{2}))\left(\frac{m_{\Lambda_{b}}}{m_{\Lambda_{b}}-\bar{u}\omega}\right)^{2}$
$\displaystyle\times(v\cdot p-u\omega)(p\cdot q-u\omega v\cdot
q)P_{R}u_{\Lambda_{b}}(v){\Big{|}}_{M^{2}=0,\ p^{2}=\Sigma_{c}+M^{2}}.$ (51)

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Also for all $w\in[-1/2,1/2]$ we have $\limsup_y \rho(y,w) = +\infty$. Indeed for any fixed  $n\ge 2$, let $j$ such that $w \in  I_{n,j}$. Then letting

\[
y_n = L_{n,j} + n/2,
\]

we have $\rho(y_n, w) = w + n/(4\pi) \ge n/(4\pi) -1/2.$  The  sequence $y_n$ converges to $\infty$ and is  such that $\rho(y_n,w)\to \infty$ as $n\to \infty$,  which proves the claim.

We now verify that the conditions gathered in (<ref>) hold.

* First for all $y\ge 0$,

\begin{align*}
\EE[\rho(y,W)]
& =  1 + \EE[W ] +
\frac{n }{4 \pi} \sin\Big( \pi (y - L_{n,j}) / n \Big)
\left[ 1/n - (n-1)/(n(n-1))\right] =1.
%\frac{1}{n(y)} \frac{n(y) }{4 \pi} \sin\Big( \pi (y - L_{n(y),j(y)}) /  n(y) \Big) \\
% &\quad -  \frac{n(y) - 1}{n(y)}
%   \frac{n(y) }{4(n(y)-1) \pi} \sin\Big( \pi (y - L_{n(y),j(y)}) /  n(y) \Big) \\
\end{align*}

* We show that for all $y$, $\rho(y,W) \ge 1/3$ almost surely. By construction, $\rho(y,W) \ge 1/2  - \frac{n(y) }{4(n(y)-1) \pi} $. Since
$m/(m-1)\le 2$  for $m\ge 2$, we obtain

\[\rho(y,W) \ge 1/2  - \frac{2 }{4 \pi} \ge 1/2-1/6 = 1/3 . \]

* We now show that $y\mapsto S(y)\rho(y,w)$ is non increasing for all $w\in[-1/2,1/2]$.  Since both  $S$ and $\rho$ are continuous functions of $y$, with derivatives from the right which we denote respectively $S'(y)$ and $\rho'(y,w)$, we need to show that $\rho'(y,w) < - \rho(y,w)S'(y)/S(y)$. Here $S'(y)/S(y) = - 1$, and from the above point we obtain $- \rho(y,w)S'(y)/S(y)\ge 1/3$.
To conclude we show that
\[\forall y\ge 0, w\in [-1/2,1/2],  \rho'(y,w) \le 1/4. \]
Let $y>0$ and $(n,j) = (n(y), j(y))$ as above.  On the one hand if
$w \in I_{n(y), j(y)}$
we have
$0\le \rho'(y,W) \le 1/4$. On the other hand if
$w \notin I_{n(y),j(y)}$, we have $\rho'(y,w)<0$. In both cases $\rho'(y,w)\le 1/4\le 1/3 \le -\rho(y,w) S'(y)/S(y)$, which conludes the argument.

* Finally we verify that $\lim_{y\to\infty} \rho(y,W)S(y)=0$,
almost surely. To see this, notice that for all
$y>0, w\in[-1/2,1/2], |\rho(y,w)|\le 3/2 + \frac{n(y)}{4\pi}$. Now since $L_n \ge n^2$,
$\{n: L_n\le y \} \subset \{n: n^2\le y \}$, so that

$  n(y)  =  \sup\{n: L_n\le y \}  \le \sup\{n: n^2\le y \}  \le \sqrt{y}. $

Thus $|\rho(y,w)|e^{-y} \le (3/2 + \sqrt{y}/(4\pi) )e^{-y} \to 0$ as $y\to \infty$.

§.§.§ Construction of $q(y,v,w)$

Recall $n(y)$ from the beginning of the above paragraph. Define

\begin{equation}
\begin{aligned}
q(y,v,w) =& 1 + v \bigg[\un\Big\{w>\frac{1}{2} - \frac{1}{n(y)}\Big\}+
\un\Big\{w\le \frac{1}{2} - \frac{1}{n(y)} \Big\}
\exp\Big(- \frac{y +  \left\lceil\frac{ 1}{1/2 - w }\right\rceil}{4} \Big)\bigg]
\end{aligned}
% \begin{cases}
%   1 + v\; & \text{ if } w \in (1/2 - 1/n(y),  1/2) \\
%   1 + v \;  h \Big(  y -  \left\lceil\frac{ 1}{1/2 - w }\right\rceil \Big) & \text{ if } w \le 1/2 - 1/ n(y)
% \end{cases}
\label{eq:def_q}
\end{equation}

We now verify that  the function $y\mapsto q(y,v,w) S(y,w) = S(y,v,w)$ is non increasing. Notice already that all the other constraints gathered in (<ref>) are satisfied. Since for fixed $(v,w)$, both  $y\mapsto q(y,v,w)$ and  $y\mapsto S(y,w)$ are continuous,  it is enough to verify that the derivative from the right of $y\mapsto q(y,v,w) S(y,w)$ is negative or null, that is (since $q\ge 1/2$ is positive), we need to ensure that

\begin{equation}\frac{q'(y,v,w)}{q(y,v,w)} \le - \frac{S'(y,w)}{S(y,w)}. \label{eq:condmonot}
\end{equation}

With our definition of $S(y,w)$ from Subsection <ref>,

\begin{align*}
- S'(y,w)/S(y,w) &= (\rho(y,w) - \rho'(y,w))/\rho(y,w)
= 1 - \rho'(y,w)/\rho(y,w)
\ge 1 - \frac{1/4}{1/3}
= 1/4. \end{align*}

If we denote $y(w) =   y -  \left\lceil\frac{ 1}{1/2 - w }\right\rceil$, we have

\[
q'(y,v,w)/q(y,v,w) = -\frac{1}{4} \un\Big\{w\le \frac{1}{2} - \frac{1}{n(y)}\Big\} v \exp(- y(w)/4) / (1 + v \exp(-y(w)/4) ).
\]

The above display is always less than $1/4$ so that (<ref>) holds for all $y>0$ and    $v,w \in [-1/2,1/2]$ and (<ref>) is satisfied.  This fact combined with the argument in Subsection <ref> implies that the functions $(S, \rho,q)$ define a proper joint distribution for $(Y,V,W)$.

§.§.§ Conclusion

We have constructed a joint distribution for $(Y,V,W)$ in Sections <ref>, <ref>,

such that $P_{V,W}$-almost surely, $q(y,V,W)\to 1$ as $y\to\infty$, as can be seen immediately from the definition of $q$ in (<ref>). Thus $(Y,V,W)$ satisfy  .

However, for all $n\ge 0$, let  $y_n= L_n + n/2$ (see Subsection <ref>), so that  by construction $n(y_n) = n$. Notice that  $I_{n,0} = [1/2-1/n, 1/2]$ and for $w\in I_{n,0}$, we have $\rho(y_n,w) = 1+ w + n/(4\pi)\ge n/16$ and $q(y_n,v,w) = 1 + v$. Thus

\begin{align*}
\EE[|R(y_n,V,W)|] &=  \EE[ | q(y_n,V,W) -1| \rho(y_n,W)]\\
& \ge \EE[ | q(y_n,V,W) -1| \rho(y_n,W) \un\{W >1/2-1/n\} |]  \\
& =  \PP[W>1/2-1/n]  \EE[| q(y_n,V,W) -1| \rho(y_n,W) \given W \in I_{n,0}] \\
& \ge  \frac{1}{n}  \EE[ |V|  n /16] \ge \EE[|V|] /16 = 1/64.
\end{align*}

We have shown that $
\limsup_{y\to\infty} \EE[|R(y_,V,W)|] >0, $
so that does not hold, which concludes the counter-example.

§.§ Additive Mixture Model (Remark 2 in the main paper)

We end this section devoted to examples with a  full derivation of the  additive mixture example mentioned in Remark 2 from the main paper.
We consider here an additive mixture $Y = Y_1 + Y_2$. The first (light-tailed) component is  $Y_1 = V \in [a,b]$ ($-\infty <a<b<\infty$) ; and  the second (heavy-tailed) one is $Y_2 = W \xi$ where $ W\in [c,d]$ with $0<c<d<\infty$, and  $\xi$ has a continuous survival function   $S_\xi(y):=1-F_{\xi}(y)$ satisfying $q(y)=y^\alpha S_\xi(y)\to C $ as $y\to\infty$, for some $\alpha, C>0$.
In addition, we assume that $V$ and $W$ are independent.

We show that holds, that is $\tailindep{Y}{V }{W}$.
Introducing the function

\[g(v , w ,y) = w^{-\alpha}y^{\alpha} %\big((y - x_1)/w\big)^{-\alpha}
S_\xi[(y - v) / w] ,\]

we have that

\begin{equation}
\label{eq:unifCVex2}
\begin{aligned}
&\sup_{[v,w] \in [a,b]\times [c,d] } \big|g(v,w,y)-C| \\
&= \sup_{v,w\in [a,b]\times [c,d]}  \Big| \Big(1 - \frac{v}{y}\Big)^{-\alpha}
\Big(\frac{y - v}{w} \Big)^\alpha S_\xi\Big[\frac{y - v}{ w} \Big] - c \Big| \xrightarrow[y\to\infty]{} 0. \quad %\text{ as }y\to \infty.
\end{aligned}
\end{equation}

The last limit relation follows from  $\frac{y - v}{ w}  \to \infty$ and
$1 - v/y\to 1$, uniformly for $v,w\in [a,b]\times [c,d]$ as $y\to\infty$. We have that

\begin{align*}
&\mathbb P(Y>y | V,W) = S_\xi( (y - V) / W)  = W^\alpha y^{-\alpha} g(V,W,y) \\%\big((1 - V/y) / W\big)^{-\alpha} q( (y - V) / W) ,\\
& \mathbb P(Y>y | W) %= \int_a^b S_\xi( (y - x) / W) f_1(x) \ud  x
= W^\alpha  y^{-\alpha} \int_a^b  g(v, W,y)% \big((1 - v/y) / W\big)^{-\alpha} q( (y - v) / W)
f_1(v) \ud v,\\
& \mathbb P(Y>y )
% = \int_c^d\int_a^b S_\xi( (y - v) / w) f_1(v) f_2(w) \ud  v \ud w
= y^{-\alpha} \int_c^d\int_a^b w^{\alpha} g(v,w,y)
% \big((1 - v/y) / w\big)^{-\alpha} q( (y - v) / w)
f_1(v) f_2(w)\ud v\ud w.
\end{align*}

\begin{align*}
\frac{\PP[Y>y |X] - \PP[Y>y|W]}{\PP[Y>y]} & = \frac{W^{\alpha}\left\{g(v,W,y) - \int_a^b  g(v, W,y) f_1(v) \ud v\right\}}{\int_c^d\int_a^b w^{\alpha} g(v,w,y)
f_1(v) f_2(w)\ud v\ud w}
\end{align*}

By (<ref>) and dominated convergence, $\int_c^d\int_a^b g(v,w,y)
f_1(v) f_2(w)\ud v\ud w \to c \EE(W^\alpha)$ as $y\to\infty$. Regarding the  numerator, Cauchy's inequality implies that

\begin{align*}
&\EE \Big| W^{\alpha}\left\{g(v,W,y) - \int_a^b  g(v, W,y) f_1(v) \ud  v\right\}\Big|\\
\leq & \sqrt{\EE W^{2\alpha}}\sqrt{\EE\left\{g(v,W,y) - \int_a^b  g(v, W,y) f_1(v) \ud  v\right\}^2}.
\end{align*}

The right-hand side tends to zero by noting that $\EE W^{2\alpha}<\infty$ and applying the dominated convergence theorem twice to the second term. The proof is complete.

§ PROOF OF THEOREM <REF>

We need to show that

\begin{equation}
\label{eq:toshowTheorem2}
Q_e \EE[ ZZ^\top -I  \given Y>y]  \xrightarrow[y\to\supy]{} 0.
\end{equation}

Notice first that from (LC) (<ref>) and (CCV) (<ref>) it holds that
$Q_e(\Var{Z \,|\, P_e Z} -I_p) =  - Q_e P_e = 0$. Thus also
\[Q_e\EE[ZZ^\top -I_p \given P_e Z] = Q_e(\Var{Z \,|\, P_e Z} -I_p) + Q_e E[Z\,|\, P_e Z] E[Z\,|\, P_e Z]^\top = Q_e P_e = 0.\]

As a consequence

\begin{align}
& Q_e \EE[(ZZ^\top - I_p)\un\{Y>y\}]  = Q_e\EE[ \EE\Big((ZZ^\top - I_p)\un\{Y>y\} \,|\, P_e Z, Y  \Big) ]\nonumber \\
& = Q_e\EE\Big[\Big(\EE[ZZ^\top - I_p \given P_e Z, Y]  - %\dots \\
% & \dots
\EE[ZZ^\top -I_p \given P_e Z]\Big) \un\{Y>y\}\Big] \nonumber\\
& = Q_e \EE\Big[\Big(\EE[ZZ^\top  \given P_e Z, Y] - \EE[ZZ^\top  \given P_e Z]\Big) \un\{Y>y\}\Big] \nonumber
%\label{eq:identityAboveu} % \\
\end{align}

Thus in order to show (<ref>) it is sufficient to show that  for all pair $(i,j)\in\{1,\ldots, p\}^2$, writing $p_y = \PP[Y>y]$,

\begin{equation}
\label{eq:toshowth2-a}
p_y^{-1} \EE\Big[\Big(\EE[Z_iZ_j \given P_e Z, Y] - \EE[Z_iZ_j  \given P_e Z]\Big) \un\{Y>y\}\Big] \xrightarrow[y\to\supy]{} 0
\end{equation}

Fixing $i,j\le p$ and   following the same path as in Theorem <ref> we decompose the left-hand side of (<ref>) for any $A>0$ as a sum $C_1(A,y) + C_2(A,y)$ where

\begin{equation*}
%  \label{eq:decomposeA}
\begin{aligned}
C_1(A,y) &=   p_y^{-1} \EE\Big[\Big(\EE[Z_iZ_j \un\{\|Z\| \le A\}\given P_e Z, Y] - \dots \\
& \dots
\EE[Z_iZ_j \un\{ \|Z\| \le A\} \given P_e Z]\Big) \un\{Y>y\}\Big] \;,\\
C_2(A,y)& =     p_y^{-1} \EE\Big[\Big(\EE[Z_iZ_j \un\{\|Z\|  > A\}\given P_e Z, Y] -\dots \\
& \dots    \EE[Z_iZ_j \un\{ \|Z\| >  A\} \given P_e Z]\Big) \un\{Y>y\}\Big].
\end{aligned}
\end{equation*}

Point (iii)
of Proposition <ref> with
$h = 1$ and $g(Z) = Z_iZ_j\un\{ \|Z\| \le   A\}$ ensures that $C_1(A,y)\to 0$ as $y\to\supy$ for any fixed $A$. On the other hand, using that $|Z_iZ_j|\le \frac{1}{2} (|Z_i|^2 + |Z_j|^2) \le \frac{1}{2}\|Z\|_2^2 \le c \|Z\|^2$ for some constant $c$  we may bound $|C_2(A,y)|$ as   follows,

\begin{align*}
| C_2(A,y) |& \le p_y^{-1}c  \EE\Big[\EE[ \|Z\|^2 \un\{\|Z\|  > A\}\given P_e Z, Y] \un\{Y>y\}\Big]  + \dots \\
& \dots  p_y^{-1}c  \EE\Big[\EE[ \|Z\|^2 \un\{\|Z\|  > A\}\given P_e Z] \un\{Y>y\}\Big]\\
& = p_y^{-1}c \bigg(\EE[ \|Z\|^2 \un\{\|Z\|  > A\}\un\{Y>y\}] +
\EE\Big(\EE[ \|Z\|^2 \un\{\|Z\|  > A\}\given P_e Z] \un\{Y>y\}\Big)\bigg)\\
&= c \EE[h_{1,A}(Z) \given Y>y] + \EE[h_{2,A}(Z) \given Y>y].
\end{align*}

Hence, in view of condition (<ref>) for any $\epsilon>0$ there exists some $A>0$ such that
$$\limsup_{y\to \supy} |C_2(A,y)| \le \epsilon,$$
$\limsup_{y\to \supy} |C_2(A,y)| + |C_1(A,y)|\le \epsilon$, which shows (<ref>) and completes the proof.

§ PROOFS AND AUXILIARY RESULTS FOR SECTION <REF>

§.§ Inverse of empirical 's and order statistics

The following general fact is used on several occasions in our proofs:

For $u\in (0,1], \hat H^-(u) = T_{( \lceil nu \rceil )}$ and for $z \in [0,n-1]$:

\[( \hat H(T_i) < (z+1)/n) \, \Leftrightarrow  (T_i  \leq  \hat H^-(z/n)).\]

The first statement follows from the definition of $\hat H^-$. Thus, using (<ref>),

\[
\begin{aligned}
\hat H(T_i) < (z+1)/n) & \iff T_i < \hat H^-((z+1)/n ) = T_{( \lceil z +1\rceil) } = T_{( \lceil z \rceil + 1)} \\
&\iff T_i \le T_{( \lceil z \rceil )} =   \hat H^-(z /n).
\end{aligned}
\]

§.§ Vervaat's Lemma

We quote Lemma 4.3 in [58], which is a variant of “Vervaat's lemma”, i.e., the functional delta method for the mapping sending a monotone function to its inverse.

Let $G : \mathbb{R} \to [0, 1]$ be a continuous distribution function. Let $0 < r_n \to \infty$ and let $\hat{G}_n$ be a sequence of random distribution functions such that, in $\ell^\infty(\mathbb{R})$, we have $r_n ( \hat{G}_n - G) \leadsto \beta \circ G$, as $n \to \infty$, where $\beta$ is a random element of $\ell^\infty([0, 1])$ with continuous trajectories. Then $\beta(0) = \beta(1) = 0$ almost surely and as $n \to \infty$,

\begin{equation*}
% \label{eq:Delta:op1}
\sup_{u \in [0, 1]} r_n |  (G \{ \hat{G}_n^-(u)\} - u ) +  ( \hat{G}_n \{ G^-(u)\} - u ) | = o_{{\mathbb P}}(1).
\end{equation*}

§.§ Proof of Lemma <ref>

Because we only need to show that for any $u\in \mathbb R^p$, $v^T \widetilde \Gamma  \leadsto v ^T \widetilde  W $, to prove that $\widetilde \Gamma$ is asymptotically tight  we may  consider the case where $q=1$, i.e., $h(V)\in \mathbb R$.
Denoting by $\psi$ the derivative of $x\mapsto \EE[h(V)^2 \un{\{U\leq x\}}]$, there exist by assumption positive constants $(c_0,\delta_0)$ such that for all $\delta\leq \delta_0$, $\psi(\delta) \leq c_0 $. Similarly, because we assume  the existence of

$\Xi = S(0)$ (Assumption 2. in Theorem <ref>), there exist positive constants $(c_1,\delta_1)$ such that for all $\delta\leq \delta_1$, $\mathbb E [h(V)^2 \,|\, U<\delta_1 ]\leq c_1 $.
We assume in the following argument  that $k/n \leq \delta_0 \wedge \delta_1$.

We  apply Theorem 2.11.23 in [63] (Classes of functions changing with $n$) with

\begin{align*}
&f_{n,u}(V,U) = \sqrt{\frac{n}{k}} h(V) \un{\{U\leq uk/n\}},\\
& \mathcal F_n = \{f_{n,u} \,:\, u\in[0,1]\},\\
& F_n(V,U) = \sqrt{\frac{n}{k}} | h(V) | \un{\{U\leq k/n\}}.
\end{align*}

We start by verifying equation 2.11.21 in [63]. First, we have
$$\EE[ F_{n}(V,U)^2] \leq  c_1.$$
Second, for any $\eta>0 $ and $M>0$, it holds that (for $n,k$ large enough)

\begin{align*}
&\EE[ F_{n}(V,U)^2 \un{\{ F_n(V,U) >\eta \sqrt n \}} ] \\
& = \left(\frac n k \right)   \EE[ |h(V)|^2  \un{\{U\leq k/n\}}    \un{\{ |h(V)| \un{\{U\leq k/n\}} >\eta \sqrt k \}} ]\\
&\leq \left(\frac n k \right)   \EE[ |h(V)|^2 \un{\{U\leq k/n\}}  \un{\{ |h(V)|  >\eta \sqrt k \}} ]\\
&\leq \left(\frac n k \right)   \EE[ |h(V)|^2 \un{\{U\leq k/n\}}  \un{\{ |h(V)|  >M \}} ] .
\end{align*}

\begin{align*}
\limsup_{n\to \infty} \EE[ F_{n}(V,U)^2 \un{\{ F_n(U) >\eta \sqrt n \}} ]\leq  S(M).
\end{align*}

But $M$ is arbitrary so the latter display is arbitrarily small. Third,  by the mean value theorem, whenever $u\leq t$, $\exists \tilde t\in (u,t)$ such that

\begin{align*}
\EE[ (f_{n,u} (V,U) - f_{n,t}(V,U))^2 ] &= \left({\frac{n}{k}}\right) \EE[ h(V)^2 \un{\{  uk/n \leq U \leq tk/n \}} ] \\
&= \psi(\tilde t k/n) ( t-u)\\
& \leq c_0 (t-u).
%&  =   v  \EE[ |V| ^2 | U\leq vk/n  ] - u   \EE[ |V| ^2 | U\leq uk/n  ]   \\
%& = (v -u) \EE[ |V| ^2 | U\leq vk/n  ] + u (  \EE[ |V| ^2 | U\leq vk/n  ] -  \EE[ |V| ^2 | U\leq uk/n  ] )\\
%& = (v -u) \left( \EE[ |V| ^2 | U\leq vk/n  ] + u h(\tilde v k/n) (k/n) \right)\\
%& \leq (v -u) (c_0  +  h_1 (k/n))
\end{align*}

This implies that

\begin{align*}
\sup_{|u-t|\leq \delta_n }  \EE[ (f_{n,u} (V,U) - f_{n,t}(V,U))^2 ] \to 0 , \text{ as } \delta_n \to 0.
\end{align*}

It remains to check the entropy condition for the class $\mathcal F_n$. Let $0<\epsilon< 1$, and denote by $u_i = i \epsilon$, $i = 0, \ldots , N$ and $u_{N+1} = 1$ with $N = \lfloor 1/\epsilon\rfloor $. Denote respectively by $f_{n,u}^+$ and $f_{n,u}^-$ the positive and negative parts of $f_{n,u}$ and by $\mathcal{F}_n^+, \mathcal{F}_n^-$ the associated classes. The functions $(f_{n,u_i }^+)$ (resp. $(f_{n,u_i }^-)$)  forms an $(\epsilon,L_2) $-bracketing of $\mathcal F_n^+ $ (resp. $\mathcal F_n^-$), i.e., for any $u\in [0,1]$, there exists $i$ such that

\begin{align*}
f_{n,u_i} ^+\leq f_{n,u}^+ \leq f_{n,u_{i+1}}^+ ,
\end{align*}

\begin{align*}
\EE[ (f_{n,u_{i+1}}^+ (V, U)  - f_{n,u_i}^+(V,U) )^2 ]\leq c_0 \epsilon.
\end{align*}

Similar inequalities remain valid for $\mathcal F_n^-$.
Hence  considering the functions $f_{n,i} = f_{n,u_i}^+  - f_{n, u_{i+1} }^- $, we have that for $u \in [u_i,u_{i+1}]$, $i = 0,\ldots, N$,

\[
f_{n, u}(\point) = f_{n,u}^+(\point) - f_{n,u}^-(\point) \in [f_{n,i}(\point) , f_{n, i+1}(\point) ]\,,
\]

thus  there exists $ C>0$ such that

\begin{align*}
\mathcal N_{[\,]} (\epsilon\|F_n\|_{L_2(P)} , \mathcal F_n , L_2(P)) \leq  C/\epsilon^2.
\end{align*}

The entropy condition is satisfied as for all $\delta_n\to 0$,

\begin{align}
\int_0^{\delta_n} \sqrt{ \log \mathcal N_{[\,]} (\epsilon\|F_n\|_{L_2(P)} , \mathcal F_n , L_2(P)) } \,d \epsilon  \to  0.\label{eq:entropyCondit}
\end{align}

Consequently, the process $\widetilde \Gamma$ is tight.
Finally the covariance functions at $s\le t$ are given by

\[
\begin{aligned}
\Cov{\widetilde \Gamma_h(s), \widetilde\Gamma_h(t)} & =
\EE[n/k h(V)h(V)^\top \un\{U \le s k/n \}] - \dotsb \\
&\qquad \qquad n/k \EE[h(V)\un\{U \le s k/n \}]\EE[h(V)\un\{U \le t k/n \}]\\
& = s\EE[ h(V)h(V)^\top\given U\le s k/n ]  - \dotsb \\
& \qquad \qquad k/n \; st \EE[h(V) \given U\le sk/n ]  \EE[h(V) \given U\le tk/n ]
\end{aligned}
\]

The first term in the right-hand side converges to $s\; \Xi  = (s\wedge t )\; \Xi$ while the second term goes to zero from assumption 3. in Theorem <ref>'s statement. This concludes the proof.

§.§ Proof of Lemma <ref>

We apply  Lemma <ref> (Vervaat) to the distribution functions

\begin{align*}
\hat G_n  (u) = \left\{ \begin{array}{ll}
0 &\text{ for }  u<0  \\
{\hat F_U(u  k/n )} / {\hat F_U(k/n)}& \text{ for }  0\leq u  \leq  1  \\
1 & \text{ for }  1<u
\end{array}\right.  , \qquad G(u) = \left\{ \begin{array}{ll}
0 &\text{ for }  u<0  \\
u & \text{ for }  0\leq u  \leq  1  \\
1 & \text{ for }  1<u
\end{array}\right..
\end{align*}

The quantile functions of $\hat G_n$ and $G$ are respectively, for any $u\in [0,1]$,

\begin{align*}
\hat G_n ^-  (u) = \frac{\hat F_U^-(u \hat F_U( k/n))}{ k/n } ,\qquad G^-(u) = u,
\end{align*}

Now we prove that the conditions of  Lemma <ref> are satisfied with $r_n = \sqrt k $ and $\beta$ a Brownian bridge with covariance function $u_1\wedge u_2 - u_1u_2$.

\begin{align*}
&  a_n = \frac{ k/n }{ \hat F_U( k/n)}
\end{align*}

and write

\begin{align*}
\sqrt k (\hat G_n  (u)  - u) &= \left(\frac{ \sqrt k}{\hat F_U(k/n)}\right) \left(\hat F_U(uk/n)   - u \hat F_U(k/n) \right)\\
&=a_n  \sqrt k  \left( \frac{n}{k} \hat F_U(uk/n)   - u\frac{n}{k}  \hat F_U(k/n) \right)\\
&=a_n  \sqrt k \left( (\frac{n}{k} \hat F_U(uk/n) - u ) -  u ( \frac{n}{k}  \hat F_U(k/n) - 1) \right)\\
&=a_n \left( \hat \gamma_1(u) -  u \hat \gamma_1(1) \right)\\
& = a_n \hat \gamma_2(u),
\end{align*}

where $\hat\gamma_1$ is defined in (<ref>) and

\begin{align}
\label{eq:gamma2}
\hat \gamma_2(u) = \hat \gamma_1(u) -  u \hat \gamma_1(1).
\end{align}

Now use that $a_n\to 1$ in probability
and that $\sup_{u\in[0,1] } |\hat \gamma_2(u)|= O_{\mathbb P} (1)$ (both are consequences of Corollary <ref>) to conclude (invoking Slutsky's lemma)  that

\begin{align}
\sqrt k (\hat G_n  (u)  - u)  &= \hat \gamma_2(u) + (a_n - 1 ) \hat \gamma_2(u) \nonumber  \\
& = \hat \gamma_2(u)  + o_{{\mathbb P}}(1),\label{eq:Gnu}
\end{align}

where the stochastic convergence  $o_{{\mathbb P}}(1)$ is uniform in $ u\in [0,1]$.  In particular that $\sqrt k (\hat G_n  (u)  - u) $ weakly converges to a Brownian bridge with covariance function $u_1\wedge u_2 - u_1u_2$. The conclusion of Lemma <ref> is that

\begin{align*}
\sup_{u\in (0,1] }  \left|  \hat \gamma_3 (u)  + % (n/k)\hat F_U^-(u \hat F_U( k/n))  - u \right)+
\sqrt k  (\hat G_n  (u)  - u) \right|  = o_{{\mathbb P}}(1),
\end{align*}

\begin{equation}
\label{eq:gamma3}
\hat \gamma_3 (u)  =  \sqrt{k} ( \hat G_n^-(u) - G^-(u))  =\sqrt k \left( \frac{n}{k}\hat F_U^-(u \hat F_U( k/n))  - u \right) .
\end{equation}

Consequently, using (<ref>),

\begin{align}\label{eq:gam1_2}
\sup_{u\in (0,1] }   \left|  \hat \gamma_3 (u)   + \hat \gamma_2(u) \right|  = o_{{\mathbb P}}(1).
\end{align}

Remark that

\begin{align}\label{eq:useful}
\sqrt k \left( (n/k)\hat F_U^-(u k/n )  - u \right)   =  \hat \gamma_3 (u  a_n  ) + u \sqrt k (a_n - 1).
\end{align}

and that, as $\hat \gamma_1(1) = \sqrt k  ( (n/k) \hat F_U(k/n) - 1)$,

\begin{align}
\sqrt k (a_n - 1) & =- a_n \hat\gamma_1(1) \label{eq:an}
\end{align}

Using the triangle inequality and (<ref>), we get

\begin{align*}
&\left|\sqrt k   \left( (n/k)\hat F_U^-(u k/n )  - u \right) + \hat \gamma_1(u)   \right|  \\
&=  \left| \hat \gamma_3(u  a_n  )  +u \sqrt k (a_n - 1) +   \hat \gamma_1(u)\right|  \\
&\leq \left| \hat \gamma_3(u  a_n  )  + \hat \gamma_2(ua_n)  \right| +  \left| u \sqrt k (a_n - 1) +   \hat \gamma_1(u) -\hat \gamma_2(ua_n)   \right| \\
%& \leq \left| \hat \gamma_3(u  a_n  )  + \hat \gamma_2(ua_n)  \right|  + | \hat \gamma_1(u   ) - \hat \gamma_1(u  a_n  ) | + | ua_n \hat \gamma_1 (1) + u \sqrt k (a_n - 1)  | \\
&=   \left| \hat \gamma_3(u  a_n  )  + \hat \gamma_2(ua_n)  \right|  + | \hat \gamma_1(u   ) - \hat \gamma_1(u  a_n  ) | ,
\end{align*}

where the last line is deduced from (<ref>) and $\hat \gamma_2(u) = \hat \gamma_1 (u)- u \hat \gamma_1(1)$. Whenever $u\in [0,1/2]$, we have, with probability going to $1$, that $u  a_n \in [0,1]$.
Moreover, because $a_n\to 0$ in probability, there exists $\delta_n\to 0$ such that the event $ |u - u a_n |\leq |a_n| \leq \delta_n$ has probability going to $1$.
On these events, it holds

\begin{align*}
&\sup_{u\in (0,1/2]} \left| \hat \gamma_3(u  a_n  )  + \hat \gamma_2(ua_n)  \right|  \leq  \sup_{u\in (0,1]} \left| \hat \gamma_3(u   )  + \hat \gamma_2(u)  \right| =   o_{\mathbb P}(1)\\
&\sup_{u\in (0,1/2]}   | \hat \gamma_1(u   ) - \hat \gamma_1(u  a_n  ) |  = \sup_{u\in (0,1],v\in(0,1], |u-v|\leq \delta_n}   | \hat \gamma_1(u   ) - \hat \gamma_1(v  ) |  =  o_{\mathbb P}(1).
\end{align*}

We have used (<ref>)  and the asymptotic equicontinuity of $\hat \gamma_1$. Consequently we have shown that, whenever $n\to \infty $, $k\to \infty$, we have

\begin{align*}
\sup_{u\in (0,1/2]}&  \left|\sqrt k \left( (n/k)\hat F_U^-(u k/n )  - u \right) + \hat \gamma_1(u)   \right|  =o_{\mathbb P}(1) .
\end{align*}

To obtain the stated result, apply this  with $2k$ in place of $k$.

§ EXTENSION TO NON-STANDARDIZED COVARIATES