# Production of the triply heavy $\Omega_{ccc}$ and $\Omega_{bbb}$ baryons at
$e^{+}e^{-}$ colliders

Su-Zhi Wua<EMAIL_ADDRESS>Pei Wub<EMAIL_ADDRESS>You-Wei Lia
<EMAIL_ADDRESS>aCollege of Science, Northwest A$\&$F University,
Yangling, Shaanxi 712100, China
bCollege of Science, Hunan University of Science and Engineering, Yongzhou,
Hunan 425199, China

###### Abstract

In this paper, we calculate the total and differential cross sections of the
processes,
$e^{+}e^{-}\rightarrow\gamma^{*}/Z^{*}\rightarrow\Omega_{ccc}\bar{c}\bar{c}\bar{c}$
and
$e^{+}e^{-}\rightarrow\gamma^{*}/Z^{*}\rightarrow\Omega_{bbb}\bar{b}\bar{b}\bar{b}$,
in the leading order at the $e^{+}e^{-}$ collider with different energies.

##  I Introduction

With the discovery of the doubly charmed baryon $\Xi_{cc}$ SELEX:2002wqn ;
LHCb:2017iph ; LHCb:2018pcs ; LHCb:2018zpl ; LHCb:2021eaf ; LHCb:2022rpd , the
triply heavy baryon has become the only one absent in the baryon family. The
masses of the triply heavy baryon have been calculated in kinds of models
theoretically Faustov:2021qqf ; Tazimi:2021ywr ; Wang:2020avt ;
Bhavsar:2018tad ; Vijande:2015faa ; Llanes-Estrada:2013rwa ; Wang:2011ae ;
Llanes-Estrada:2011gwu ; Alomayrah:2020qyw ; Yang:2019lsg ; Shah:2019jxp ;
Rai:2017hue ; Shah:2017jkr ; Martynenko:2007je ; Azizi:2014jxa ; Aliev:2012tt
; Zheng:2010zzc ; Zhang:2009re ; Patel:2008mv ; Jia:2006gw  and different
results have been gotten. Exploring the triply baryon in experiment is
important for filling the particle spectra and testing the theories. To
produce the triply heavy baryon, three pairs of heavy quarks need to be
produced. This makes it hard to calculate the cross sections of the triply
heavy baryon production. The total cross sections and the differential cross
sections of the production of these baryons at LHC have been calculated in
Chen:2011mb ; Wu:2012wj . There, the authors simplified the calculation
utilizing the characteristic of the spin, the color configurations of the
triply heavy baryons and the small relative momenta among the constituent
quarks. The production of the triply charmed baryon in $e^{+}e^{-}$
annihilation at $Z$-boson pole have been estimated in Baranov:2004er , where
the mass of the charm quark has been ignored at some places in the matrix. The
production of $\Omega_{ccc}$ from quark gluon plasma in heavy ion collisions
has been calculated in Becattini:2005hb . And the production of the triply
baryons from the fragmentation processes have been calculated in
GomshiNobary:2005ur ; MoosaviNejad:2017rvi ; GomshiNobary:2006tzy ;
GomshiNobary:2004mq ; Delpasand:2019xpk .

In this paper, we report the study on the production of the triply heavy
baryons $\Omega_{ccc}$ and $\Omega_{bbb}$ at $e^{+}e^{-}$ colliders. Here, we
calculate the direct production of these triply heavy baryons exactly. Because
the constituent quarks of the triply heavy baryon are all heavy flavored, the
NRQCD Bodwin:1994jh  can be used to describe the triply heavy baryon. And, the
production of the $\Omega_{QQQ}$ can be factorized into two parts, the short-
distance coefficient which describes the process of the production of the
triple heavy quark pairs, and the long-distance matrix element which describes
the formation of the triply heavy baryon from point-like three heavy
quarks.111$\Omega_{QQQ}$ means the triply heavy baryon $\Omega_{ccc}$ or
$\Omega_{bbb}$. And $Q$ denotes the $c$ or $b$ quark. To achieve the
calculation, we will reduce the number of the Feynman diagrams which we have
to calculate, as we have done in Chen:2011mb ; Wu:2012wj .

In the next section we will show the details on the calculation of these two
parts, the short-distance coefficient and the long-distance matrix element,
explicitly. In Sec.III, we will list the numerical results and conclusions.
And in Sec.IV, we will give a brief summary.

##  II Production of THE $\Omega_{ccc}$ and $\Omega_{bbb}$ at $e^{+}e^{-}$
colliders

As a baryon, the triply heavy baryon must be in color singlet. The color wave
function of the $\Omega_{QQQ}$ must be
$\frac{1}{\sqrt{6}}\varepsilon^{\xi_{1}\xi_{2}\xi_{3}}Q_{1\xi_{1}}Q_{2\xi_{2}}Q_{3\xi_{3}}$
with $\xi_{i}$ ($i=$1,2,3) being the color index of the valence quark $Q_{i}$.
As a ground state, the orbital angular momentum wave function is symmetrical.
The exchange antisymmetry of the identical fermions implies that the
$\Omega_{QQQ}$ must be the spin-symmetrical state and the spin of it must be
$\frac{3}{2}$.

Because of the large masses of the heavy quarks, the NRQCD can be used to
describe the triply heavy baryons. In NRQCD, we describe the normalized wave
function of the $\Omega_{QQQ}$ as,

$\displaystyle|\Omega_{QQQ},\frac{3}{2},S_{Z}\rangle$
$\displaystyle=\sqrt{2M}\int\frac{d^{3}\vec{q}_{1}}{(2\pi)^{3}}\frac{d^{3}\vec{q}_{2}}{(2\pi)^{3}}\frac{1}{\sqrt{3!}}$
(1)
$\displaystyle\sum_{\xi_{1},\xi_{2},\xi_{3}}\sum_{\eta_{1},\eta_{2},\eta_{3}}\frac{\varepsilon^{\xi_{1}\xi_{2}\xi_{3}}}{\sqrt{6}}\langle\frac{3}{2},S_{Z}|\eta_{1},\eta_{2},\eta_{3}\rangle$
$\displaystyle\frac{1}{\sqrt{2E_{1}2E_{2}2E_{3}}}\psi(\vec{q}_{1},\vec{q}_{2})|Q_{1},\xi_{1},\eta_{1},\vec{q}_{1}\rangle$
$\displaystyle|Q_{2},\xi_{2},\eta_{2},\vec{q}_{2}\rangle|Q_{3},\xi_{3},\eta_{3},\vec{q}_{3}\rangle,\
\ \ \ $

where, $\vec{q}_{3}=-\vec{q}_{1}-\vec{q}_{2}$, and

$\displaystyle\langle Q_{i},\xi_{i},\eta_{i},$
$\displaystyle\vec{q}_{i}|Q_{j},\xi_{j},\eta_{j},\vec{q}_{j}\rangle$
$\displaystyle=\delta_{Q_{i}Q_{j}}\delta_{\eta_{i}\eta_{j}}\delta_{\xi_{i}\xi_{j}}(2\pi)^{3}2E_{f}\delta^{(3)}(\vec{q}_{i}-\vec{q}_{j}),$

with $\eta_{i}$ and ($E_{i}$, $\vec{q}_{i}$) ($i=1,2,3$) being the spin and
the four-momentum of the $Q_{i}$ heavy quark; $M$ being the mass of the baryon
$\Omega_{QQQ}$; $\langle\frac{3}{2},S_{Z}|\eta_{1},\eta_{2},\eta_{3}\rangle$
being the Clebsch-Gordan (C-G) coefficient; $S_{Z}$ being the third component
of the spin of the baryon, and $\psi(\vec{q}_{1},\vec{q}_{2})$ being the wave
function of the baryon in the momentum space which is normalized as follows

$\displaystyle\int\frac{d^{3}\vec{q}_{1}}{(2\pi)^{3}}\frac{d^{3}\vec{q}_{2}}{(2\pi)^{3}}\psi^{*}(\vec{q}_{1},\vec{q}_{2})\psi(\vec{q}_{1},\vec{q}_{2})=1\;.$
(2)

The production of $\Omega_{QQQ}$ at $e^{+}e^{-}$ colliders can be factorized
into two parts, the short-distance coefficient corresponding to the production
of the three $Q\bar{Q}$ pairs and the long-distance matrix element describing
the three heavy quark $Q$ coupling to the triply heavy baryon $\Omega_{QQQ}$.
In the heavy quark limit, the dependence of the short distance on the momenta,
$q_{1}$ and $q_{2}$, can be neglected in the leading order. Namely, the
momenta of the produced three identical heavy quarks are the same. As a
result, the long-distance matrix is proportional to the wave function of the
baryon at the origin,

$\displaystyle\Psi(0,0)=\int\frac{d^{3}q_{1}}{(2\pi)^{3}}\frac{d^{3}q_{2}}{(2\pi)^{3}}\psi(q_{1},q_{2})\;.$
(3)

Now, the amplitude of the process
$e^{+}e^{-}\rightarrow\Omega_{QQQ}\bar{Q}\bar{Q}\bar{Q}$ can be written as,

$\displaystyle A(e^{+}e^{-}\to\Omega_{QQQ}\bar{Q}\bar{Q}\bar{Q})=$
$\displaystyle\frac{\sqrt{2M}}{\sqrt{(2m)^{3}}}\frac{\Psi(0,0)}{\sqrt{3!}}\mathcal{M}(e^{+}e^{-}\to(QQQ)_{1}^{(\frac{3}{2},S_{Z})}\bar{Q}_{1}\bar{Q}_{2}\bar{Q}_{3})\;,\
\ \ \ $ (4)

in which, $m$ is the mass of the heavy quark $Q$, and
$\mathcal{M}(e^{+}e^{-}\to(QQQ)_{1}^{(\frac{3}{2},S_{Z})}\bar{Q}\bar{Q}\bar{Q})$
is the short-distance coefficient, namely, the matrix element of the process
$e^{+}e^{-}\to(QQQ)_{1}^{(\frac{3}{2},S_{Z})}\bar{Q}\bar{Q}\bar{Q}$.222
$(QQQ)_{1}^{(S,S_{Z})}$ means the total spin and the third component of the
total spin of the three identical $Q$-quarks are $\frac{3}{2}$ and $S_{Z}$,
respectively; the momenta of the three quarks are the same; and the three
heavy quarks couple to a color singlet.

Now, let us consider the short-distance coefficient. The contribution to the
production of the three heavy quark pairs comes from the $e^{+}e^{-}$
annihilation process mainly,

$\displaystyle e^{-}(k_{1})+e^{+}(k_{2})\rightarrow
Z^{*}/\gamma^{*}\rightarrow Q(p_{1},\xi_{1})+Q(p_{2},\xi_{2})$
$\displaystyle+Q(p_{3},\xi_{3})+\bar{Q}(p_{4},\chi_{1})+\bar{Q}(p_{5},\chi_{2})+\bar{Q}(p_{6},\chi_{3}),\
\ \ \ \ \ $ (5)

where, $k_{1}$ and $k_{2}$ are the 4-momenta of the electron and position;
$p_{i}$ ($i$=1,6) are the 4-momenta of the produced $Q$ and $\bar{Q}$ quarks,
with $p_{1}=p_{2}=p_{3}$; $\xi_{j}$ and $\chi_{j}$($j=$1,2,3) are the color
indices of the $Q_{j}$-quarks and $\bar{Q}_{j}$-quarks, respectively. The
produced three pairs of heavy quarks in process (II) have six permutations
denoted as, $(Q_{1}\bar{Q}_{i}Q_{2}\bar{Q}_{j}Q_{3}\bar{Q}_{k})$ with
$i,j,k$=1,2,3 and $i\neq j\neq
k$.333$(Q_{1}\bar{Q}_{i}Q_{2}\bar{Q}_{j}Q_{3}\bar{Q}_{k})$ ($i,j,k$=1,2,3 and
$i\neq j\neq k$) means the quark $Q_{1}$ in the same fermion line with the
antiquark $\bar{Q}_{i}$, the quark $Q_{2}$ in the same fermion line with the
antiquark $\bar{Q}_{j}$ and the quark $Q_{3}$ in the same fermion line with
the antiquark $\bar{Q}_{k}$. In the calculation, we will disregard the
contribution of the electroweak interaction between the heavy quarks for the
process (II). As a result, there are seven inequivalent topology structures
for each of the six permutations as the same as the one in Chen:2011mb . So,
there are 42 topology structures totally for the produced heavy quarks.
Inserting the $e^{+}e^{-}\gamma^{*}$ vertex and $e^{+}e^{-}Z^{*}$ vertex into
the 42 topology structures in all the allowed positions in the tree level, we
get 576 Feynman diagrams for the process (II).

As pointed out above, the color configuration of the produced three heavy
quarks is
$\frac{1}{\sqrt{6}}\varepsilon^{\xi_{1}\xi_{2}\xi_{3}}Q_{1\xi_{1}}Q_{2\xi_{2}}Q_{3\xi_{3}}$.
Setting $T^{a}=\frac{\lambda^{a}}{2}$ with $\lambda^{a}$ $(a=1,\ldots,8)$
being the Gell-Mann matrices and using the color-flow method in Chen:2011mb ,
we can get the color factors of the 576 Feynman diagrams. The color factors of
the last two diagrams in Fig.1 are both,

$\displaystyle\sum_{a,b,c}\sum_{\xi_{1},\xi_{2},\xi_{3}}\frac{1}{\sqrt{6}}\varepsilon^{\xi_{1}\xi_{2}\xi_{3}}f^{abc}(T^{a})_{\xi_{l}\chi_{i}}(T^{b})_{\xi_{m}\chi_{j}}(T^{b})_{\xi_{n}\chi_{k}}=0\;.$

in which, $f^{abc}$ ($(a,b,c=1,\ldots,8)$) is the structure constant of the
$SU(3)$ group, and the index $\xi_{i}$ ($i=1,2,3$) appears twice, in
$\varepsilon^{\xi_{1}\xi_{2}\xi_{3}}$ directly and as one of the indices
$\xi_{l}$, $\xi_{m}$ and $\xi_{n}$ indirectly. And we see the result is
independent onto the indices $i,j,k,l,m,$ and $n$. As a result, we get the
conclusion that the total contribution to the amplitude of the process (II)
from the 74 Feynman diagrams involving a three-gluon vertex vanishes because
of the color factors of these Feynman diagrams are all zero. The number of the
Feynman diagrams which we need to consider reduces to 504. The color factors
of the first seven Feynman diagrams in Fig.1 are all the same,

$\displaystyle\sum_{a,b}\sum_{\xi_{1},\xi_{2},\xi_{3}}\frac{1}{\sqrt{6}}\varepsilon^{\xi_{1}\xi_{2}\xi_{3}}(T^{a})_{\xi_{l}\chi_{i}}$
$\displaystyle(T^{a}T^{b})_{\xi_{m}\chi_{j}}(T^{b})_{\xi_{n}\chi_{k}}$
$\displaystyle=(-1)^{N}\frac{4}{9}\frac{1}{\sqrt{6}}\varepsilon^{\chi_{i}\chi_{j}\chi_{k}}\;,$

in which $N$ is the number of permutations of transforming the set
$(\xi_{l},\xi_{m},\xi_{n})$ to the set $(\xi_{1},\xi_{2},\xi_{3})$. For each
of the Feynman diagrams, there is a Feynman factor $(-1)^{N^{*}}$ where
$N^{*}$ equals the total number of permutations of transforming the set
$(l,m,n)$ to the set $(1,2,3)$ and transforming the set $(i,j,k)$ to the set
$(1,2,3)$. Subsuming the Feynman factor into the corresponding color factor,
we find that all the color factors of the remaining 504 Feynman diagrams are
the same,

$\displaystyle
C_{col}=(-1)^{N^{*}}(-1)^{N}\frac{4}{9}\frac{1}{\sqrt{6}}\varepsilon^{\chi_{i}\chi_{j}\chi_{k}}=\frac{4}{9}\frac{1}{\sqrt{6}}\varepsilon^{\chi_{1}\chi_{2}\chi_{3}}\;.$

Figure 1: Nine typical Feynman diagrams for the process (II). The indices
$i,j,k,l,m,n=1,2,3$ (with $i\neq j\neq k$ and $l\neq m\neq n$).  Figure 2:
Distributions of the differential production cross section versus the
transverse momentum of the $\Omega_{ccc}$ produced at CEPC with
$\sqrt{s}=91.2\;{\rm GeV}$.  Figure 3: Distributions of the differential
production cross section versus the transverse momentum of the $\Omega_{bbb}$
produced at CEPC with $\sqrt{s}=91.2\;{\rm GeV}$.  Figure 4: The production
cross section of the $\Omega_{ccc}$ produced by $e^{+}e^{-}$ annihilation at
different energies.  Figure 5: The production cross section of the
$\Omega_{bbb}$ produced by $e^{+}e^{-}$ annihilation at different energies.

Now, let’s consider the remained 504 Feynman diagrams. For each one of the six
permutations given above, $(Q_{1}\bar{Q}_{i}Q_{2}\bar{Q}_{j}Q_{3}\bar{Q}_{k})$
with $i,j,k$=1,2,3, there are 84 Feynman diagrams, in which 42 ones correspond
to the process $e^{-}+e^{+}\to Z^{*}\to
Q_{1}Q_{2}Q_{3}\bar{Q}_{1}\bar{Q}_{2}\bar{Q}_{3}$ and the other 42 ones
correspond to the process $e^{-}+e^{+}\to\gamma^{*}\to
Q_{1}Q_{2}Q_{3}\bar{Q}_{1}\bar{Q}_{2}\bar{Q}_{3}$. In our calculation, the
momenta of the produced identical heavy quarks are considered to be equal. As
a result, we find that the contribution of every 84 Feynam diagrams
corresponding to each one of the six permutations are the same. So, the total
amplitude for the process
$e^{+}e^{-}\to(QQQ)_{1}^{(\frac{3}{2},S_{Z})}\bar{Q}\bar{Q}\bar{Q}$

$\displaystyle\mathcal{M}(e^{+}e^{-}\to(QQQ)_{1}^{(\frac{3}{2},S_{Z})}\bar{Q}\bar{Q}\bar{Q})$
$\displaystyle=$ $\displaystyle\sum_{k=1}^{504}C_{col}\,\Gamma_{k}$ (6)
$\displaystyle=$ $\displaystyle
3!\sum_{k=1}^{84}C_{col}\,\Gamma^{{}^{\prime}}_{k}\;,$

where $\Gamma_{k}$ ($k=1,2....,504$) are the matrices without color factors of
the Feynman diagrams, and $\Gamma^{{}^{\prime}}_{k}$ ($k=1,2....,84$) are the
matrices without color factors of the 84 Feynman diagrams corresponding to the
permutation $(Q_{1}\bar{Q}_{1}Q_{2}\bar{Q}_{2}Q_{3}\bar{Q}_{3})$.

##  III Numerical results and discussions

Now, the cross section of the progress (II) can be written as

$\displaystyle\sigma=$
$\displaystyle\frac{1}{3!}\int\sum_{S_{Z},\varsigma_{i}}\frac{(2\pi)^{4}}{2\hat{s}}\delta^{4}(k_{1}+k_{2}-P-p_{4}-p_{5}-p_{6})$
(7) $\displaystyle
d\Pi_{4}\frac{1}{4}\sum_{s_{1},s_{2},\chi_{i}}|\mathcal{A}(e^{+}e^{-}\to\Omega_{QQQ}\bar{Q}\bar{Q}\bar{Q})|^{2},$

with

$\displaystyle
d\Pi_{4}=\frac{d^{3}P}{(2\pi)^{3}2E}\frac{d^{3}p_{4}}{(2\pi)^{3}2E_{p_{4}}}\frac{d^{3}p_{5}}{(2\pi)^{3}2E_{p_{5}}}\frac{d^{3}p_{6}}{(2\pi)^{3}2E_{p_{6}}},\
\ \ \ $

where $\varsigma_{i}$ ($i=1,2,3$), $s_{1}$ and $s_{2}$ are the spins of the
antiquarks $\bar{Q}_{i}$, $e^{+}$ and $e^{-}$, respectively.

To do the numerical calculation, the parameters are taken as follows:

$\displaystyle m_{c}=1.5\,{\rm GeV},\;m_{b}=4.9\,{\rm GeV},\;m_{Z}=91.18\,{\rm
GeV},$ $\displaystyle sin^{2}\theta_{w}=0.224,\;\Gamma_{{}_{Z}}=2.49\,{\rm
GeV},\;\alpha(m_{Z})=1/127.95,\;$
$\displaystyle|\Psi_{\Omega_{ccc}}(0,0)|^{2}=0.36\cdot 10^{-3}\,{\rm
GeV^{6}}\;$ $\displaystyle\text{and}\ \
|\Psi_{\Omega_{bbb}}(0,0)|^{2}=0.189\,{\rm GeV^{6}}.$ (8)

For the electromagnetic coupling constant, we adopt

$\displaystyle\alpha(q)$ $\displaystyle=$
$\displaystyle\frac{\alpha(m_{Z})}{1-\frac{\alpha(m_{Z})}{3\pi}{\rm
log}(\frac{q^{2}}{m_{Z}^{2}})},$ (9)

where, $q$ denotes the energy scale of the electromagnetic coupling constant,
and we take $q=\frac{\sqrt{s}}{2}$ in this paper with $\sqrt{s}$ being the
colliding energy in the center-of-mass frame. And the strong coupling
constant, we adopt

$\displaystyle\alpha_{s}(\mu)$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}(m_{Z})}{1+\frac{b_{0}}{2\pi}\alpha_{s}(m_{Z}){\rm
log}(\frac{\mu}{m_{Z}})},$ (10) $\displaystyle\text{with}\ \ b_{0}$
$\displaystyle=$ $\displaystyle 11-\frac{2}{3}n_{f},$

where $\mu$ is the energy scale and $\alpha_{s}(m_{Z})=0.118$. For the
production of the $\Omega_{ccc}$, $n_{f}$ is taken to be 4 when $\mu\leq
2m_{b}$ and 5 when $\mu$ is larger than 2$m_{b}$. And for the production of
the $\Omega_{bbb}$, $n_{f}$ is taken to be 5. For comparison, we take two
different values for $\mu$, i.e., $\mu$=$\sqrt{s}/2$ and $\mu$=$\mu_{R}/2$,
where $\mu_{R}$ is the transverse mass of the produced baryon, namely
$\mu_{R}^{2}$=$p_{T}^{2}$+$M^{2}$.444Here, $p_{T}$ means the transverse
momentum of the produced triply heavy baryon.

SuperKEKB, the asymmetric beam energy $e^{-}e^{+}$ collider, is an upgrade of
the KEKB accelerator facility. The target integrated luminosity of 50
$ab^{-1}$ to be collected by the Belle II experiment. And a high luminosity
$e^{+}e^{-}$ collider named the Circular Electron-Positron Collider (CEPC) has
been proposed by the Chinese particle physics community. The CEPC is designed
to operate at three different modes, as a Higgs factory at $\sqrt{s}=240\,{\rm
GeV}$, as a $Z$ factory at $\sqrt{s}=91.2\,{\rm GeV}$ and performing $WW$
threshold scans around $\sqrt{s}=160\,{\rm GeV}$ An:2018dwb . We calculate the
production cross sections of the baryons $\Omega_{ccc}$ and $\Omega_{bbb}$ at
CEPC with the energy $\sqrt{s}=91.2\,{\rm GeV}$ and $\sqrt{s}=160\,{\rm GeV}$,
and at Belle II with the center-of-mass energy $\sqrt{s}=10.58\,{\rm GeV}$.
The results are shown in TABLE. 1.

- | $\sqrt{s}$ | $\mu$ | $\Omega_{ccc}$ | $\Omega_{bbb}$
---|---|---|---|---
CEPC | 91.2 GeV | $\sqrt{s}/2$ | 0.00204(5) | $0.332(4)\times 10^{-3}$
- | - | $\mu_{R}/2$ | 0.0124(3) | $0.952(4)\times 10^{-3}$
- | 160 GeV | $\sqrt{s}/2$ | $0.214(9)\times 10^{-5}$ | $0.61(2)\times 10^{-6}$
- | - | $\mu_{R}/2$ | $0.108(9)\times 10^{-4}$ | $0.197(5)\times 10^{-5}$
Belle II | 10.58 GeV | $\sqrt{s}/2$ | $0.249(1)\times 10^{-5}$ | -
- | - | $\mu_{R}/2$ | $0.707(3)\times 10^{-5}$ | -
Table 1: Production cross sections (in unit $fb$) of the $\Omega_{ccc}$ and
$\Omega_{bbb}$ at the CEPC and Belle II.

Here, we also calculate the differential cross sections $d\sigma/dp_{T}$ of
the production of the $\Omega_{ccc}$ and $\Omega_{bbb}$ at $\sqrt{s}=91.2\ \
{\rm GeV}$, which are shown in Figs. 2 and 3. The line for the $p_{T}$
distribution are not smooth due to the Vegas simulation error which can be
improved by increasing the event numbers. We calculate the production of the
$\Omega_{ccc}$ and $\Omega_{bbb}$ at the $e^{+}e^{-}$ collider with different
colliding energies, which are shown in Fig.4 and Fig.5.

As we see that both the integrated cross sections and differential cross
sections are proportional to $|\Psi(0,0)|^{2}$, $\alpha^{2}(q)$and
$\alpha_{s}^{4}(\mu)$. So the numerical results can be changed by one even two
orders using different the wave function at the origin of the triply heavy
baryons, different running coupling constants and different energy scale
choices. We can get the conclusion from the numerical results shown in Figs.4
and 5, that the production cross sections of the $\Omega_{ccc}$ and
$\Omega_{bbb}$ take their maximum values at $\sqrt{s}=m_{Z}$. From the
numerical results in Table 1, we know that it is impossible to observe
$\Omega_{ccc}$ at SuperKEKB, if there is no other models to produce them at
the $e^{+}e^{-}$ colliders. Also, we see there are around 32-198 events for
the production of $\Omega_{ccc}$ and 5-15 events for the production of
$\Omega_{bbb}$, at CEPC with $16\,ab^{-1}$ integrated luminosity running at
$91.2\,{\rm GeV}$.

##  IV Summary

As a summary, we have studied the production of the $\Omega_{ccc}$ and
$\Omega_{bbb}$ at the SuperKEKB and the CEPC. From the numerical results, we
conclude that it is impossible to find the triply heavy baryons at SuperKEKB,
and it is hard to find $\Omega_{ccc}$ and $\Omega_{bbb}$ at the CEPC because
of the fewer events.

Acknowledgments: The authors thank Doctor Jian-Wei Xu for helpful discussions
and important suggestions on the manuscript. The work of S. Z. Wu was
supported by the Natural Science Foundation of China under Grants No.
11347200.

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#  _TemporAI_ : Facilitating Machine Learning Innovation in Time Domain Tasks
for Medicine

Evgeny S. Saveliev
DAMTP, University of Cambridge
<EMAIL_ADDRESS>
Mihaela van der Schaar
DAMTP, University of Cambridge
The Alan Turing Institute
<EMAIL_ADDRESS>

###### Abstract

_TemporAI_ is an open source Python software library for machine learning (ML)
tasks involving data with a time component, focused on medicine and healthcare
use cases. It supports data in time series, static, and eventmodalities and
provides an interface for prediction, causal inference, and time-to-event
analysis, as well as common preprocessing utilities and model interpretability
methods. The library aims to facilitate innovation in the medical ML space by
offering a standardized temporal setting toolkit for model development,
prototyping and benchmarking, bridging the gaps in the ML research, healthcare
professional, medical/pharmacological industry, and data science communities.
_TemporAI_ is available on
GitHub111https://github.com/vanderschaarlab/temporai and we welcome community
engagement through use, feedback, and code contributions.

_K_ eywords Machine Learning $\cdot$ Time Series $\cdot$ Medicine

##  1 Time domain is crucial for ML in medicine

Data with a time component222Depending on the context, referred to
alternatively as: temporal, longitudinal, or time series data. are ubiquitous
in modern healthcare and medicine: from patient electronic health records
(EHRs) [1], to data streams from Internet-of-Things (IoT) devices and consumer
wearables [2], to large public health datasets [3], naming just a few key
growing areas. In fact, since patient information is typically associated with
a particular time point, the vast majority of healthcare data is temporal, and
may be viewed as a time series. Furthermore, availability of open access data
in this field is also improving [4, 5, 6], attracting significant attention
from the artificial intelligence (AI), machine learning (ML) and deep learning
(DL) research, as well as the medical data science communities [7, 8, 9]. A
such, it is evident that the temporal setting is becoming the cornerstone for
ML in healthcare and medicine, with a significant potential for impact.

Numerous novel methods have been developed to tackle medically-relevant tasks
in the time domain, such as: prediction [10, 11], causal inference [12, 13,
14], time-to-event analysis [15, 16, 17], clustering [18, 19]333In medical and
other contexts, these tasks may also be referred to as, respectively:
forecasting, (individualized) treatment effect estimation, survival analysis,
phenotyping. The descriptor “temporal” may be used to contrast with the static
task setting., as well as data imputation [20, 21], and model interpretability
[22, 23] methods, among others. Yet currently a significant limitation exists
in the lack of standardization of both data representation and model
benchmarking [7, 9]. _TemporAI_ addresses these limitations as the first
toolkit for development, prototyping and benchmarking of ML models on
medically-relevant tasks with time series, static, and eventdata modalities.

##  2 _TemporAI_ library

###  2.1 The ecosystem of temporal domain ML for medicine

While a vibrant software ecosystem is developing for time series data science
and applied machine learning, as exemplified by a range of libraries seen in
Table 1, few are primarily focused on the healthcare and medicine applications
and problem settings. _clairvoyance_ [24] developed by the authors previously
was an attempt at filling this gap, but the library had significant drawbacks
in  (i) insufficient modularity in the code base,  (ii) lack of robust testing
and software engineering best practices,  (iii) absence of continued support
or proactive engagement with the community.  _TemporAI_ aims to address these
issues, and to create a new ecosystem for temporal domain ML in medicine,
aimed at – and hoping to bridge the gap between – the various communities in
this field: ML researchers, research engineers, and data scientists,
healthcare and pharmacology industry experts, as well as clinical
professionals.

The design of _TemporAI_ maps the different data modalities and medically-
relevant use cases to corresponding algorithm implementations, with their
appropriate roles in the research workflow, and applicable evaluation metrics
– which we conceptually illustrate in Figure 1.

Table 1: _TemporAI_ compared to a range of other time series frameworks

| _seglearn_ | _tsfresh_ | _sktime_ | _darts_ | _Kats_ |  | _scikit-_
---
_survival_
| _clair-_
---
_voyance_
_TemporAI_
| [25] | [26] | [27] | [28] | [29] | [30] | [24] |
Primary focus: |  |  |  |  |  |  |  |
ML | ✗ | ✗ | • | • | ✓ | ✗ | ✓ | ✓
Medical domain | ✗ | ✗ | ✗ | ✗ | ✗ | • | ✓ | ✓
Software: |  |  |  |  |  |  |  |
Modular design | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓
Unit and integration tests | • | ✓ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓
Predictive tasks: |  |  |  |  |  |  |  |
prediction | ✓ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓ | ✓
causal inference | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ | ✓
time-to-event analysis | ✗ | ✗ | ✗ | ✗ | ✗ | ✓∗ | ✗ | ✓
_Additional functionality:_ |  |  |  |  |  |  |  |
Preprocessing | ✓ | ✓ | ✓ | ✓ | ✓ | • | • | ✓
data imputation | ✗ | • | ✗ | ✓ | ✗ | ✗ | ✓ | ✓
model interpretability | ✗ | ✓ | ✗ | ✗ | ✓ | ✗ | ✗ | ✓

  * •

∗ Only with static covariates.

  * •

Key: ✗ no, • partial, ✓ yes

Figure 1: _TemporAI_ connects problem setting elements (data modality and use
cases) to the appropriate algorithms, roles in the workflow, hyperparameters,
and evaluation metrics.

###  2.2 Data modalities

_TemporAI_ currently handles three data modalities: _time series_ , _static_
and _event_ :

  1. 1.

_static_ : data comprised of features not associated with a time point. Each
data point is described by $(i,f_{s},x_{s})$ where $i\in[N]$ is the sample
index, $f_{s}\in[F_{s}]$ is the static feature index, and $x_{s}$ is the data
point value. Patient blood type and year of birth are examples of static data
modality features of a patient.

  2. 2.

_time series_ : data comprised of features, each of which is a sequence of
values at certain time points. Each data point is described by
$(i,f_{t},t_{if},x_{t})$ where $i\in[N]$ is the sample index,
$f_{t}\in[F_{t}]$ is the time series feature index,
$t_{if}\in\\{t_{if}^{0},\ldots,t_{if}^{max}\\}$ is the time index of the data
point (for sample $i$ and feature $f_{t}$), and $x_{t}$ is the data point
value. Note that this representation is very flexible, and it means that:  (i)
feature sequences may differ in length,  (ii) time indexes need not be
regular,  (iii) and – in the broadest case – features of the same sample may
not be aligned on their time indexes. A patient’s systolic blood pressure
measurements and white blood cell counts are examples of time series data
modality features.

  3. 3.

_event_ : data comprised of features, each of which is represented by a time
index and a value. Each data point is described by $(i,f_{e},e_{if},x_{e})$
where $i\in[N]$ is the sample index, $f_{e}\in[F_{e}]$ is the event feature
index, $e_{if}$ is the time index of the event (for sample $i$ and feature
$f_{e}$), and $x_{e}$ is the data point value. Patient death or a myocardial
infarction are examples of the event modality.

The following data point value types are currently supported: continuous
$x\in\mathbb{R}$, integer $x\in\mathbb{Z}$, and categorical
$x\in\\{c_{1},\ldots,c_{n}\\}$. A sentinel value $*$ is allowed, representing
a missing data point (and in the context of the event modality, this naturally
represents censoring).

_TemporAI_ allows for the capacity to support different data containers (e.g.
pandas.DataFrame or numpy.ndarray), and different ways of representing the
data modalities within these.

Finally, _TemporAI_ distinguishes three roles any set of data features may
fulfil: _covariates_ X, _targets_ Y, and _treatments_ (interventions) A.

###  2.3 Workflow

The _TemporAI_ workflow builds on the familiar fit/transform/predict API of
_scikit-learn_ [31, 32]. Additional methods are introduced when necessary to
cover the relevant use cases, e.g. predict_counterfactuals for causal
inference. Figure 2 demonstrates a typical workflow. It is worth highlighting
that _TemporAI_ introduces a novel role in the workflow: that of a “wrapper”
model. This allows for implementation of algorithms which encapsulate other
models, as may be the case in some interpretability or clustering use cases.

Figure 2: _TemporAI_ dataset and a typical workflow illustrated as a block
diagram.

###  2.4 Components

For an up-to-date information on the available library components (algorithms,
data formats, etc.), please refer to project
documentation444https://temporai.readthedocs.io/.

###  2.5 Design principles

The library adheres to the software engineering design principles of
_modularity_ , _encapsulation_ , and _abstraction_ , and emphasizes the ease
of _extensibility_ – as developing into a benchmarking environment in the
machine learning for medicine temporal domain is one of its goals.

##  3 Medical use cases

_TemporAI_ can be used to address a wide range of questions in the medical
domains of acute and critical care, hospital medicine, transplantation, and
preventative care, among others. Broadly, problems within the temporal setting
involving predicting or forecasting, risk scoring (potentially with competing
risks), individualized treatment effects, and patient phenotyping are in the
scope of this tool. We maintain a reference of healthcare questions
categorized by their medical domain, problem type, canonical formalism, and
the most applicable software package at van der Schaar lab _Hub for
Healthcare_ 555https://www.vanderschaar-lab.com/hub-for-healthcare/, which we
encourage the reader to brows. This resource is being updated as we gather
more feedback and guidance from medical experts. A discussion of illustrative
examples corresponding to the major areas of _TemporAI_ follows.

###  3.1 forecasting

The forecasting (prediction) use case is well-exemplified by the intensive
care setting. A typical problem of interest may be “How can machine learning
improve early warning scores using all the data clinicians have access to?”
The approach would be to utilize the available static, event, and time series
data modalities of a patient to predict either a static or a time series
outcome which denotes the risk variable – _TemporAI_ prediction module
provides the tooling needed for training, inference, and evaluation of models
in such a setting.

###  3.2 survival analysis

The survival analysis (time-to-event analysis) use case can be illustrated by
an example in transplantation. One may wish to find out “How can I identify
those patients who will need transplantation at an earlier stage of their
illness?” or “How can we use the most up to date data to adapt predictions
(usually based on data from 10-15 years prior to the time the prediction is
made) so that they reflect probable future outcomes?” Here, making use of the
available patient static, event, and time series data modalities, the goal
would be to obtain accurate risk scores or survival curves, to aid in
answering such questions. _TemporAI_ survival module provides the tooling
necessary for this setting.

###  3.3 treatment effect estimation

A question matching the (individualized) treatment effect estimation, or
causal inference, use case in the critical care domain may be “Can we predict
whether and when a patient on mechanical ventilation can be safely extubated
if a particular course of antibiotics is given?” Starting with the static,
event, and time series data modalities of a patient, the goal in this
formalism is to predict reliable counterfactual target (static or time series)
variables, given alternative treatment plan inputs – such counterfactuals then
act to aid in the decision-making process. _TemporAI_ treatment module
provides the algorithms for this use case.

###  3.4 Making the most of data and models

It is often the case that algorithms for a particular use case alone are
insufficient for real life application of ML. The data may require significant
preprocessing such as missing value imputation, and, especially in the high
stakes clinical setting, the user may wish to go beyond predictions and to
gain additional insights from, e.g. understanding the most important
contributing features, or the allocation of patients to clusters based on
their likely outcomes. With this in mind, _TemporAI_ provides tools for
preprocessing (e.g. preprocessing.imputation module), interpretability
(interpretability module), and clustering (clustering module)666Some of these
modules are still under active development. to build up a software ecosystem
suitable for ML researchers, data scientists, and healthcare professionals
alike.

## Acknowledgments

We thank Prof. Eoin McKinney (Cambridge Institute of Therapeutic Immunology
and Infectious Disease) for many invaluable discussions of medical
applications and of clinically-relevant problems. We also thank Nick Maxfield
for his tireless work on designing the van der Schaar Lab _Hub for Healthcare_
and for his diligent curation of the topics therein.

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