# Mass Spectra and Semileptonic Decays of Doubly Heavy $\Xi$ and $\Omega$
Baryons

Zahra<EMAIL_ADDRESS>1 , Cheng-Ping Shen2 and
Masoumeh Moazzen Sorkhi1
1Department of Physics, Kosar University of Bojnord, Bojnourd 94156-15458,
Iran
2Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, China

###### Abstract

In the framework of a non-relativistic quark model, the mass spectra of the
ground and excited states of doubly heavy $\Xi$ and $\Omega$ baryons are
calculated. We estimate the mass difference between the $\Omega$ and
corresponding $\Xi$ baryons as $M_{\Omega}-M_{\Xi}\simeq 178$ MeV for all the
states containing $cc,~{}bc$, or $bb$ quarks. A simple form of the universal
Isgur-Wise function, as the transition form factor between the doubly heavy
baryons, is introduced. Working in the close-to-zero recoil limit, we
investigate the $b\rightarrow c$ semileptonic decay widths and branching
fractions of the doubly heavy baryons. The obtained results are compared with
other theoretical predictions.

##  1 Introduction

The investigation of doubly heavy baryons is of great interest in
understanding quantum chromodynamics (QCD) inspired potential model, the non-
relativistic QCD factorization theory, etc., at the hadronic scale. During the
last few years many theoretical progresses have been achieved in doubly heavy
baryon spectroscopy [1, 2, 3, 4, 5]. In 2002, the lightest doubly charmed
baryon $\Xi^{+}_{cc}$ was observed with a statistical significance of
6.3$\sigma$ with a measured mass of $(3519\pm 1)$ MeV in the decay mode
$\Xi^{+}_{cc}\to\Lambda_{c}^{+}K^{-}\pi^{+}$ by the SELEX collaboration [6].
This state was subsequently confirmed by the same collaboration in another
decay mode $\Xi^{+}_{cc}\to pD^{+}K^{-}$ [7]. However, negative results in
searching for the $\Xi^{+}_{cc}$ were reported by FOCUS [8], BaBar [9], or
Belle [10] collaborations. In 2017, the doubly charmed baryon $\Xi^{++}_{cc}$
was first observed by the LHCb collaboration via the decay mode
$\Xi^{++}_{cc}\to\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+}$ with a measured mass
$(3621.40\pm 0.72\pm 0.14\pm 0.27)$ MeV [11], where the uncertainties are
statistical, systematic, and from the limited knowledge of the
$\Lambda_{c}^{+}$ mass, respectively, and confirmed in another decay mode
$\Xi^{++}_{cc}\to\Xi^{+}_{c}\pi^{+}$ [12]. The measured mass is about 100 MeV
higher than that of $\Xi^{+}_{cc}$ determined by the SELEX collaboration [6].
The lifetime of the $\Xi_{c}^{++}$ was measured to be
$\tau(\Xi_{cc}^{++})=(0.256^{+0.024}_{-0.022}\pm 0.014)$ ps [13]. The
$\Xi^{++}_{cc}$ observation demonstrates how really the LHC is a powerful
discovery machine, stimulating the theoretical studies of mass spectra of
doubly heavy baryons. Therefore, the predictions for doubly heavy baryon
masses have become a subject of renewed interest.

Other than the double-charm $\Xi_{cc}$ baryons, the beauty-charm and also
double-beauty baryons are the different kinds of doubly heavy baryons, which
have not been found yet. Recently, several experimental efforts have been made
on the exclusive channels $\Xi^{0}_{bc}\rightarrow D^{0}pK^{-}$ [14] and
$\Xi^{0}_{bc}\rightarrow\Xi_{c}^{+}\pi^{-}$ [15] to search for the beauty-
charm baryons, but no signals were observed. The $\Xi_{bc}$ baryons are
expected to have smaller sizes than the $\Xi_{cc}$ states. In comparison to
the $\Xi_{cc}$, searching for the $\Xi_{bc}$ are more complicated and fewer
states could be produced at the LHC. To overcome this difficulty, an inclusive
decay channel $\Xi_{bc}\rightarrow\Xi_{cc}^{++}+X$ has been proposed [16],
where $X$ stands for all the possible particles.

For the singly charmed baryons, some semileptonic decays have been calculated
in theory [17, 18, 19, 20, 21, 22] and measured in experiments [23, 24, 25,
26, 27]. However, for the doubly heavy baryons, no experimental data on
semileptonic decays are reported and only a limited number of theoretical
calculations are available. As there are more doubly heavy baryons that may be
discovered in the future, proposing a theoretical model for their structures
is essential. The investigations of the inclusive and exclusive semileptonic
decays of doubly heavy baryons are of special interest for two basic reasons:
they play an important role in the calculation of fundamental parameters of
the electroweak standard model and provide a useful tool to extract the
Cabibbo-Kobayashi-Maskawa (CKM) matrix element $V_{cb}$. Some characteristics
of semileptonic decays, such as the transition form factors and exclusive
decay rates, also provide information about the internal structures of heavy
baryons and the strong and weak interactions inside them.

Some theoretical calculations to the semileptonic weak decays of doubly heavy
baryons have been provided based on the QCD sum rules (QCDSR) [28, 29],
covariant confined quark model [30], Bethe-Salpeter equation [31], heavy quark
spin symmetry (HQSS) [32], relativistic quark model (RQM) [33, 34], non-
relativistic quark model (NRQM) [35], or heavy diquark effective theory (EFT)
[16, 36]. However, the calculations of semileptonic decays presented by
different approaches lead to essentially different values for the decay
widths.

In this paper, we present a description of the properties of heavy baryons
containing two heavy quarks (charm or beauty quark) within the framework of
non-relativistic quark model proposed in Ref. [37]. Firstly, we calculate the
mass spectra of the ground and orbitally excited states of doubly heavy $\Xi$
and $\Omega$ baryons, including
$\Xi_{cc},~{}\Xi_{bc},~{}\Xi_{bb},~{}\Omega_{cc},~{}\Omega_{bc}$, and
$\Omega_{bb}$ states, in the hypercentral constituent quark model. Then, we
focus on the studies of $b\rightarrow c$ semileptonic decays of the ground
states of doubly heavy $\Xi$ and $\Omega$ baryons. We proceed our model close
to the zero recoil point in which the form factors of these transitions can be
expressed by a few universal functions [38]. Considering the doubly baryon
states can be produced sizably at LHC, especially LHC has started Run3 data
taking, our results can be tested in the near future.

This paper is structured as follows. In Section 2 we present our predictions
for the ground and also orbitally excited states of the doubly heavy baryons
using the results obtained in our previous work. In Section 3 we introduce a
universal function as the transition form factor to study the semileptonic
decays of doubly charm and bottom heavy baryons close to the zero recoil point
and present our numerical results for semileptonic decay widths and branching
fractions of specific modes. Section 4 includes a short summary.

##  2 Doubly heavy baryon spectra

We start with a brief review of our previous work [37] in which the mass
spectra and radiative transitions of $\Sigma_{b}$ and $\Lambda_{b}$ baryons
are studied within the hypercentral constituent quark model [39, 40, 41, 42,
43, 44]. The hypercentral quark model contains a few free parameters, whose
values are obtained by fitting the baryon spectrum. Once the parameters are
obtained, the model is completely determined and we can make our predictions
for the baryon properties. In Ref. [37] we introduced a phenomenological
potential model, and solved the baryonic three-body equation in a non-
relativistic limit by choosing the Killingbeck potential
$V(x)=ax^{2}+bx-\frac{c}{x}$. The potential parameters $a,~{}b$, and $c$ are
constant.
We apply the Ansatz method [45, 46, 47] to solve the Schrödinger equation. In
the hypercentral approach, the hyperradial part of the wave function is
determined by

$[\frac{d^{2}}{dx^{2}}+\frac{5}{x}\frac{d}{dx}-\frac{\gamma(\gamma+4)}{x^{2}}]\psi_{\gamma}(x)=-2m[E_{\gamma}-V(x)]\psi_{\gamma}(x).$
(1)

Assuming the transformation $\psi(x)=x^{-5/2}\phi(x)$ we get

$\phi_{\gamma}^{{}^{\prime\prime}}(x)+[\varepsilon-
a_{1}x^{2}-b_{1}x+\frac{c_{1}}{x}-\frac{(2\gamma+3)(2\gamma+5)}{4x^{2}}]\phi_{\gamma}(x)=0.$
(2)

We make use of the following Ansatz

${\phi_{\gamma}(x)=exp[-\frac{1}{2}\alpha x^{2}-\beta x+\delta lnx],}$ (3)

where $\alpha,\beta$ and $\delta$ parameters are determined in terms of the
potential parameters. By substitution of equation 3 into equation 2 and
comparing the coefficient of $x$ on both sides of the new equation we can
evaluate the energy eigenvalues and normalized eigenfunctions for the baryon
states as follows (for details see Refs. [37, 47]):

$E_{\gamma}=(2\gamma+6)\frac{w}{2}-\frac{2mc^{2}}{(2\gamma+5)^{2}}$ (4)

and

$\psi_{\gamma}(x)=N_{\gamma}x^{-\frac{5}{2}}\phi_{\gamma}(x)=N_{\gamma}x^{\gamma}exp(-\frac{mw}{2}x^{2}-\frac{2mc}{(2\gamma+5)}x),$
(5)

where $x$ is the hyperradius and $w=\sqrt{\frac{2a}{m}}$ is the oscillating
frequency. $m$, $\gamma$ and $N_{\gamma}$ are the reduced mass, angular
quantum number and normalization constant, respectively. Baryon mass is
obtained by summing the quark masses, energy eigenvalues, and hyperfine
interaction potential treated as a perturbation:

$M_{\rm baryon}=\sum_{i=1}^{3}m_{i}+E_{\gamma}+<H_{S}>.$ (6)

Here, $<H_{S}>$ is the expectation value of the hyperfine spin-spin
interactions given as [37, 46]

$H_{S}=\Sigma_{i<j}A_{S}(\frac{1}{\sqrt{\pi}\sigma_{S}})^{3}exp(\frac{-x^{2}}{\sigma_{S}})(\overrightarrow{s_{i}}.\overrightarrow{s_{j}}),$
(7)

where $\overrightarrow{s_{i}}$ is the spin operator of the $i^{\rm th}$ quark,
and $\sigma_{S}$ and $A_{S}$ are constant. The isospin values of the strange
and heavy quarks are zero and therefore, in the case of doubly heavy baryons
the isospin dependent terms are not included in the hyperfine interactions.
The spin-orbit interaction has values smaller than $0.01$ GeV in the hyperfine
contributions and therefore, we neglect it in our calculations. All of the
model parameters listed in Table 1 are taken from our previous work [37]. We
take the experimentally measured mass of the $\Xi_{cc}^{++}$ [12] to determine
the mass of the charm quark. Our calculated results of the masses of the
ground and excited baryon states including $P$-wave, $D$-wave, and $F$-wave
are listed in Tables 2 and 3, and compared with the ones obtained from
different models of RQM, NRQM, QCDSR, HQSS, and EFT [48, 49, 50, 51, 52, 53,
54, 55]. Table 2 shows that our results for the masses of the
$\Xi_{cc},~{}\Xi_{bc},~{}\Omega_{cc}$, and $\Omega_{bc}$ are in good agreement
with those reported in Ref. [48]. For the masses of the $\Xi_{bb}$ and
$\Omega_{bb}$ states, our predictions are close to the results from Refs. [50,
51, 52].

Table 1: Quark-model parameters, where $q$ refers to the light quarks. Parameter | Value | Parameter | Value
---|---|---|---
$m_{q}$ | 320 MeV | $w$ | 1.557 ${\rm fm}^{-1}$
$m_{s}$ | 440 MeV | $b$ | 5.79 ${\rm fm}^{-2}$
$m_{c}$ | 1360 MeV | $A_{s}$ | 67.4 ${\rm fm}^{2}$
$m_{b}$ | 4670 MeV | $\sigma_{S}$ | 4.76 fm
Table 2: Masses of the ground states of doubly heavy baryons (in GeV). The sign $``^{*}"$ refers to the $s=\frac{3}{2}$ baryons. Baryon | Content | Our results | RQM [48] | HQSS [49] | QCDSR [50] | NRQM [51] | NRQM [52]
---|---|---|---|---|---|---|---
$\Xi_{cc}$ | $qcc$ | 3.620 | 3.620 | 3.613 |  | 3.676 | 3.685
$\Xi_{cc}^{*}$ | $qcc$ | 3.653 | 3.727 | 3.707 | 3.690 | 3.753 | 3.754
$\Xi_{bc}$ | $qbc$ | 6.958 | 6.933 | 6.928 |  | 7.020 |
$\Xi_{bc}^{*}$ | $qbc$ | 6.991 | 6.980 | 6.996 | 7.250 | 7.078 |
$\Xi_{bb}$ | $qbb$ | 10.322 | 10.202 | 10.198 |  | 10.340 | 10.314
$\Xi_{bb}^{*}$ | $qbb$ | 10.355 | 10.237 | 10.237 | 10.400 | 10.367 | 10.339
$\Omega_{cc}$ | $scc$ | 3.798 | 3.778 | 3.712 |  | 3.815 | 3.832
$\Omega_{cc}^{*}$ | $scc$ | 3.831 | 3.872 | 3.795 | 3.780 | 3.876 | 3.883
$\Omega_{bc}$ | $sbc$ | 7.137 | 7.088 | 7.013 |  | 7.147 |
$\Omega_{bc}^{*}$ | $sbc$ | 7.170 | 7.130 | 7.075 | 7.300 | 7.191 |
$\Omega_{bb}$ | $sbb$ | 10.500 | 10.359 | 10.269 |  | 10.456 | 10.447
$\Omega_{bb}^{*}$ | $sbb$ | 10.533 | 10.389 | 10.307 | 10.500 | 10.486 | 10.467
Table 3: Masses of the orbitally excited states of doubly heavy baryons (in GeV). State | Baryon | Our results | NRQM[53] | EFT[54] | Baryon | Our results | NRQM[55] | NRQM[52]
---|---|---|---|---|---|---|---|---
| $\Xi_{cc}$ | 3.928 | 3.853 | 4.028 | $\Omega_{cc}$ | 4.106 | 3.964 | 4.086
| $\Xi_{cc}^{*}$ | 3.961 | 3.862 | 4.079 | $\Omega_{cc}^{*}$ | 4.139 | 3.972 | 4.199
$P$-wave | $\Xi_{bc}$ | 7.266 | 7.140 |  | $\Omega_{bc}$ | 7.4457 | 7.375 |
| $\Xi_{bc}^{*}$ | 7.299 | 7.157 |  | $\Omega_{bc}^{*}$ | 7.478 | 7.381 |
| $\Xi_{bb}$ | 10.631 | 10.502 | 10.386 | $\Omega_{bb}$ | 10.808 | 10.634 | 10.607
| $\Xi_{bb}^{*}$ | 10.664 | 10.510 | 10.404 | $\Omega_{bb}^{*}$ | 10.841 | 10.636 | 10.796
| $\Xi_{cc}$ | 4.237 | 4.026 | 4.321 | $\Omega_{cc}$ | 4.414 | 4.133 | 4.263
| $\Xi_{cc}^{*}$ | 4.269 | 4.035 | 4.376 | $\Omega_{cc}^{*}$ | 4.474 | 4.141 | 4.265
$D$-wave | $\Xi_{bc}$ | 7.575 | 7.307 |  | $\Omega_{bc}$ | 7.754 | 7.807 |
| $\Xi_{bc}^{*}$ | 7.607 | 7.312 |  | $\Omega_{bc}^{*}$ | 7.786 | 7.812 |
| $\Xi_{bb}$ | 10.939 | 10.658 | 10.585 | $\Omega_{bb}$ | 11.1177 | 10.783 | 10.723
| $\Xi_{bb}^{*}$ | 10.972 | 10.660 | 10.610 | $\Omega_{bb}^{*}$ | 11.150 | 10.785 | 10.730
| $\Xi_{cc}$ | 4.545 | 4.185 | 4.569 | $\Omega_{cc}$ | 4.723 | 4.287 | 4.555
| $\Xi_{cc}^{*}$ | 4.577 | 4.216 | 4.644 | $\Omega_{cc}^{*}$ | 4.755 | 4.313 | 4.600
$F$-wave | $\Xi_{bc}$ | 7.883 | 7.451 |  | $\Omega_{bc}$ | 8.062 | 7.702 |
| $\Xi_{bc}^{*}$ | 7.916 | 7.468 |  | $\Omega_{bc}^{*}$ | 8.095 | 7.721 |
| $\Xi_{bb}$ | 11.274 | 10.797 | 10.757 | $\Omega_{bb}$ | 11.425 | 11.136 | 11.028
| $\Xi_{bb}^{*}$ | 11.280 | 10.805 | 10.792 | $\Omega_{bb}^{*}$ | 11.458 | 11.142 | 11.059

Using the masses from our calculations shown in Table 2, we can get the mass
differences $\Delta_{M}$ between $\Omega$ and corresponding $\Xi$ doubly heavy
baryons:

$\displaystyle M_{\Omega_{bb}}-M_{\Xi_{bb}}=178\quad{\rm MeV},$ (8)
$\displaystyle M_{\Omega_{cc}}-M_{\Xi_{cc}}=178\quad{\rm MeV},$ (9)
$\displaystyle M_{\Omega_{bc}}-M_{\Xi_{bc}}=179\quad{\rm MeV}.$ (10)

We obtain $\Delta_{M}\sim 178$ MeV for all of the $M_{\Omega}-M_{\Xi}$
splittings, compared with $\Delta_{M}=100\pm 10$ MeV reported in Ref. [28] and
$\Delta_{M}=155\sim 158$ MeV predicted in Ref. [48]. In other works referenced
in Table 2, $\Delta_{M}$ has different values for the different
$M_{\Omega}-M_{\Xi}$ splittings.

##  3 Semileptonic decays of doubly heavy baryons

To study the semileptonic transitions of baryons we need the form factors,
which can be parameterized in different approaches. Some earlier works [38,
56, 57, 58] simplified the transition form factors using different methods.
The authors of Ref. [38] studied the form factors and semileptonic decays of
doubly heavy baryons using a relativistic covariant quark model (RCQM).
According to Refs. [38, 59, 60], in the heavy quark limit, the expressions for
the decay rates can be simplified and the weak transition form factors between
doubly heavy baryons can be expressed through the single Isgur-Wise (IW)
function $\eta(\omega)$

$\displaystyle
F_{1}(\omega)=G_{1}(\omega)=\eta(\omega),\quad\quad\quad\quad\quad\quad$ (11)
$\displaystyle F_{2}(\omega)=F_{3}(\omega)=G_{2}(\omega)=G_{3}(\omega)=0.$

In Ref. [38] a closed-form expression has been derived for the IW function
$\eta(\omega)$ using a Gaussian Ansatz for the three-quark correlation
function in the heavy quark limit and close to zero recoil point. In the
present work we take the universal function $\eta(\omega)$ calculated by Ref.
[38] which depends on the kinematical parameter $\omega$ and is given by

$\eta(\omega)=exp(-3(\omega-1)\frac{m_{cc}^{2}}{\Lambda_{B}^{2}})$ (12)

with $m_{cc}=2m_{c}$ for the $bc\rightarrow cc$ weak transitions [38]. The
parameter $\Lambda_{B}$ characterizes the size of the given baryon and
represents the extension of the distribution of the constituent quarks in the
baryon. For the doubly heavy baryons the size of parameter $\Lambda_{B}$ is
allowed to vary in the range $2.5\leq\Lambda_{B}\leq 3.5$ GeV [38]. The values
of parameter $\Lambda_{B}$ are fixed using data on fundamental properties of
mesons and baryons such as leptonic decay constants, magnetic moments, and
radii. The dependence of the universal $\eta(\omega)$ function on $\omega$
with $\Lambda_{B}=3$ for the $bb\rightarrow bc$ transitions is shown in Fig.
1. For the slope $\rho^{2}$ of the $\eta(\omega)=1-\rho^{2}(\omega-1)+...$,
one obtains

$\rho^{2}=-\frac{d\eta(\omega)}{d\omega}|_{\omega=1}=3\frac{m_{cc}^{2}}{\Lambda_{B}^{2}}.$
(13)

By replacing $m_{cc}$ with $m_{bb}$ in the IW function one can obtain the
results for the $bb\rightarrow bc$ transitions. Accordingly, the slope of the
IW function for the $bb\rightarrow bc$ transitions is obtained from Eq. (12)
by replacing $m_{cc}$ with $m_{bb}$ if one uses the same size of parameter
$\Lambda_{B}$ in both cases [38]. Close to zero recoil, the IW functions for
$bb\rightarrow bc$ and $bc\rightarrow cc$ transitions explicitly contain the
flavor factors $m_{cc}$ and $m_{bb}$, and there exists only spin symmetry.
There is no dependence on the light quark masses. At zero recoil ($\omega=1$)
there exists a spin-flavor symmetry and $\eta(1)=1$ means that $bb\rightarrow
bc$ and $bc\rightarrow cc$ transitions are identical.

Figure 1:  Variation of the universal $\eta(\omega)$ function versus $\omega$
for $bb\rightarrow bc$ semileptonic transitions with $\Lambda_{B}=3$ GeV.

According to Ref. [38], in zero recoil limit, the expressions for the
semileptonic decay widths can be simplified considerably and we can get the
decay rates using the IW function $\eta(\omega)$ and the following relations:

$\Gamma_{\frac{1}{2}\rightarrow\frac{1}{2}}=\frac{G_{F}^{2}|V_{bc}|^{2}M_{i}^{5}r^{4}}{12\pi^{3}}\int_{1}^{\omega_{\rm
max}}d\omega\sqrt{\omega^{2}-1}(l^{+}_{1}(\omega)\eta^{2}(\omega)+l^{-}_{1}(\omega)\eta^{2}(\omega)),$
(14)
$\Gamma_{\frac{1}{2}\rightarrow\frac{3}{2}}=\frac{G_{F}^{2}|V_{bc}|^{2}M_{i}^{5}r^{4}}{12\pi^{3}}\int_{1}^{\omega_{\rm
max}}d\omega\sqrt{\omega^{2}-1}l_{2}(\omega)\eta^{2}(\omega),$ (15)
$\Gamma_{\frac{3}{2}\rightarrow\frac{1}{2}}=\frac{G_{F}^{2}|V_{bc}|^{2}M_{i}^{5}r^{4}}{24\pi^{3}}\int_{1}^{\omega_{\rm
max}}d\omega\sqrt{\omega^{2}-1}l_{3}(\omega)\eta^{2}(\omega),$ (16)
$\Gamma_{\frac{3}{2}\rightarrow\frac{3}{2}}=\frac{G_{F}^{2}|V_{bc}|^{2}M_{i}^{5}r^{4}}{24\pi^{3}}\int_{1}^{\omega_{\rm
max}}d\omega\sqrt{\omega^{2}-1}(l^{+}_{4}(\omega)\eta^{2}(\omega)+l^{-}_{4}(\omega)\eta^{2}(\omega)),$
(17)

where $r=M_{2}/M_{1}$, $\omega_{\rm max}=(1+r^{2})/2r$,

$l^{\pm}_{1}(\omega)=(\omega\mp 1)(3\omega_{\rm max}\pm 1-2\omega),$ (18)
$l_{2}(\omega)=2(\omega+1)(\omega_{\rm max}-\omega+\frac{\omega^{2}-1}{6r}),$
(19) $l_{3}(\omega)=2(\omega+1)(\omega_{\rm
max}-\omega+\frac{(\omega^{2}-1)r}{6}),$ (20)

and

$l^{\pm}_{4}(\omega)=\frac{4}{9}(l^{\pm}_{1}(\omega)\frac{3+2\omega^{2}}{2}\pm(\omega_{max}\pm
1)(\omega^{2}-1)).$ (21)

The parameter $G_{F}$ is the Fermi Coupling constant, $V_{bc}$ is the CKM
matrix element and its value is $V_{bc}=0.04$, $M_{1}$ and $M_{2}$ are the
initial and final state baryon masses, respectively. We take the following
values of $\omega_{\rm max}$ for different transitions:

$\displaystyle\omega_{\rm
max}[\Xi_{bb}\rightarrow\Xi_{bc}]=1.07,\quad\omega_{\rm
max}[\Omega_{bb}\rightarrow\Omega_{bc}]=1.07,$ (22) $\displaystyle\omega_{\rm
max}[\Xi_{bc}\rightarrow\Xi_{cc}]=1.22,\quad\omega_{\rm
max}[\Omega_{bc}\rightarrow\Omega_{cc}]=1.20.$ (23)

Using the obtained masses listed in Table 2 and Eqs. (14)-(21), the
semileptonic decay rates of doubly heavy $\Xi$ and $\Omega$ baryons are
calculated. The $\omega$ dependence of the semileptonic decay widths for
$\Omega_{bb}\rightarrow\Omega_{bc}\ell\bar{\nu}_{\ell}$,
$\Xi_{bb}\rightarrow\Xi_{bc}\ell\bar{\nu}_{\ell}$,
$\Omega_{bc}\rightarrow\Omega_{cc}\ell\bar{\nu}_{\ell}$, and
$\Xi_{bc}\rightarrow\Xi_{cc}\ell\bar{\nu}_{\ell}$ transitions is shown in Fig.
2 and 3, respectively, where we take $\Lambda_{B}=3$ GeV and neglect the mass
difference between the $u$ and $d$ quarks. Regarding $\omega=\omega_{\rm
max}$, the $\Lambda_{B}$ dependence of $\eta$ functions for
$\Omega_{bb}\rightarrow\Omega_{bc}$ and $\Xi_{bb}\rightarrow\Xi_{bc}$
transitions is shown in Fig. 4. Note that a smaller value of $\Lambda_{B}$
gives smaller decay rates and vice versa.

Figure 2:  Behavior of semileptonic decay widths versus $\omega$ for
$\Omega_{bb}\rightarrow\Omega_{bc}\ell\bar{\nu}_{\ell}$ and
$\Xi_{bb}\rightarrow\Xi_{bc}\ell\bar{\nu}_{\ell}$ ($\ell=e$ or $\mu$)
transitions ($\Lambda_{B}=3$ GeV).  Figure 3:  Behavior of semileptonic decay
width versus $\omega$ for
$\Omega_{bc}\rightarrow\Omega_{cc}\ell\bar{\nu}_{\ell}$ and
$\Xi_{bc}\rightarrow\Xi_{cc}\ell\bar{\nu}_{\ell}$ ($\ell=e$ or $\mu$)
transitions ($\Lambda_{B}=3$ $GeV$).  Figure 4:  Behavior of $\eta$ function
versus $\Lambda_{B}$ for
$\Omega_{bb}\rightarrow\Omega_{bc}\ell\bar{\nu}_{\ell}$ and
$\Xi_{bb}\rightarrow\Xi_{bc}\ell\bar{\nu}_{\ell}$ ($\ell=e$ or $\mu$)
transitions ($\omega=\omega_{\rm max}$).

The calculated semileptonic decay widths of doubly heavy baryons and their
variations are summarized in Table 4. The uncertainties in decay widths are
due to the parameter $\Lambda_{B}$, which can take a value in the range
$2.5\leq\Lambda_{B}\leq 3.5$ GeV. The uncertainties from the quark masses and
potential parameters determined by fitting to the experimental data [37] can
be neglected. A comparison between our results and those derived by Refs. [32,
38, 48, 61, 62] is also presented. For $s=\frac{1}{2}\rightarrow
s=\frac{1}{2}$ transitions, including $\Xi_{bb}\rightarrow\Xi_{bc}$,
$\Xi_{bc}\rightarrow\Xi_{bb}$, $\Omega_{bb}\rightarrow\Omega_{bc}$, and
$\Omega_{bc}\rightarrow\Omega_{cc}$, our results are in good agreement with
those of Ref. [38]. In the case of $\Xi_{bb}\rightarrow\Xi_{bc}$ and
$\Omega_{bb}\rightarrow\Omega_{bc}$ transitions our results are close to those
of Ref. [48].

Table 4:  Semileptonic decay widths of doubly heavy baryons in units of $10^{-14}$ GeV and their variations. Decay |  Our results |  RCQM [38] |  NRQM [61] |  RQM [48] |  HQSS [32] |  LFQM111LFQM denotes the light-front quark model. [62]
---|---|---|---|---|---|---
$\Xi_{bb}\rightarrow\Xi_{bc}\ell\bar{\nu}_{\ell}$ |  $1.75\pm 0.73$ |  $1.33\pm 0.61$ | 1.37 | 1.63 | 1.92${}^{+0.25}_{-0.05}$ | 3.30
$\Xi_{bc}\rightarrow\Xi_{cc}\ell\bar{\nu}_{\ell}$ |  $4.39\pm 0.83$ |  $4.01\pm 1.21$ | 5.07 | 2.30 | 2.57${}^{+0.26}_{-0.03}$ | 4.50
$\Xi_{bb}^{*}\rightarrow\Xi_{bc}\ell\bar{\nu}_{\ell}$ |  $0.49\pm 0.18$ |  $0.25\pm 0.10$ | 0.66 | 0.28 | 0.35+0.03 |
$\Xi_{bc}^{*}\rightarrow\Xi_{cc}\ell\bar{\nu}_{\ell}$ |  $1.00\pm 0.16$ |  $0.58\pm 0.14$ | 1.16 | 0.38 | 0.43+0.06 |
$\Xi_{bb}\rightarrow\Xi_{bc}^{*}\ell\bar{\nu}_{\ell}$ |  $1.07\pm 0.43$ |  $0.61\pm 0.15$ | 1.45 | 0.53 | 0.61+0.04 |
$\Xi_{bc}\rightarrow\Xi_{cc}^{*}\ell\bar{\nu}_{\ell}$ |  $2.82\pm 0.52$ |  $1.39\pm 0.34$ | 3.32 | 0.72 | 0.75+0.06 |
$\Xi_{bb}^{*}\rightarrow\Xi_{bc}^{*}\ell\bar{\nu}_{\ell}$ |  $1.10\pm 0.48$ |  $1.62\pm 0.73$ |  | 1.92 | 2.09+0.16 |
$\Xi_{bc}^{*}\rightarrow\Xi_{cc}^{*}\ell\bar{\nu}_{\ell}$ |  $3.12\pm 0.62$ |  $4.63\pm 1.23$ |  | 2.69 | 2.63+0.40 |
$\Omega_{bb}\rightarrow\Omega_{bc}\ell\bar{\nu}_{\ell}$ |  $1.87\pm 0.76$ |  $1.92\pm 1.15$ | 2.48 | 1.70 | 2.14${}^{+0.20}_{-0.02}$ | 3.69
$\Omega_{bc}\rightarrow\Omega_{cc}\ell\bar{\nu}_{\ell}$ |  $4.70\pm 0.83$ |  $4.12\pm 1.10$ | 5.39 | 2.48 | 2.59+0.20 | 3.94
$\Omega_{bb}^{*}\rightarrow\Omega_{bc}\ell\bar{\nu}_{\ell}$ |  $0.53\pm 0.19$ |  $0.26\pm 0.10$ | 0.69 | 0.29 | 0.38${}^{+0.04}_{-0.02}$ |
$\Omega_{bc}^{*}\rightarrow\Omega_{cc}\ell\bar{\nu}_{\ell}$ |  $1.09\pm 0.16$ |  $0.59\pm 0.13$ | 1.23 | 0.40 | 0.44+0.06 |
$\Omega_{bb}\rightarrow\Omega_{bc}^{*}\ell\bar{\nu}_{\ell}$ |  $1.14\pm 0.45$ |  $0.57\pm 0.23$ | 1.53 | 0.55 | 0.67+0.08 |
$\Omega_{bc}\rightarrow\Omega_{cc}^{*}\ell\bar{\nu}_{\ell}$ |  $2.99\pm 0.52$ |  $1.78\pm 0.64$ | 3.52 | 0.74 | 0.76+0.13 |
$\Omega_{bb}^{*}\rightarrow\Omega_{bc}^{*}\ell\bar{\nu}_{\ell}$ |  $1.17\pm 0.50$ |  $1.72\pm 0.77$ |  | 2.00 | 2.29${}^{+0.31}_{-0.04}$ |
$\Omega_{bc}^{*}\rightarrow\Omega_{cc}^{*}\ell\bar{\nu}_{\ell}$ |  $3.32\pm 0.62$ |  $4.95\pm 1.26$ |  | 2.88 | 2.79+0.60 |

The absolute branching fractions of semileptonic decays of doubly heavy
baryons can be easily derived by using the following relation

${\cal B}=\Gamma\times\tau,$ (24)

where $\tau$ is the lifetime of the initial baryon. Taking
$\tau_{\Xi_{bb}}=370\times 10^{-15}s$, $\tau_{\Xi_{bc}}=244\times 10^{-15}s$
[63], $\tau_{\Omega_{bc}}=220\times 10^{-15}s$, and
$\tau_{\Omega_{bb}}=800\times 10^{-15}s$ [64, 65] as input values, the
calculated branching fractions of semileptonic decays of doubly heavy baryons
are summarized in Table 5.

Table 5: The calculated branching fractions of semileptonic decays of doubly heavy baryons. Process | Our results | NRQM [61] | LFQM [62]
---|---|---|---
$\Xi_{bc}\rightarrow\Xi_{cc}\ell\bar{\nu}_{\ell}$ | $1.63\times 10^{-2}$ | $1.11\times 10^{-2}$ | $1.67\times 10^{-2}$
$\Xi_{bb}\rightarrow\Xi_{bc}\ell\bar{\nu}_{\ell}$ | $0.98\times 10^{-2}$ | $0.28\times 10^{-2}$ | $1.86\times 10^{-2}$
$\Omega_{bc}\rightarrow\Omega_{cc}\ell\bar{\nu}_{\ell}$ | $1.57\times 10^{-2}$ | $1.10\times 10^{-2}$ | $1.32\times 10^{-2}$
$\Omega_{bb}\rightarrow\Omega_{bc}\ell\bar{\nu}_{\ell}$ | $2.27\times 10^{-2}$ |  | $4.49\times 10^{-2}$

##  4 Summary

In this paper we present a phenomenological study of the mass spectra and
semileptonic decays of doubly heavy $\Xi$ and $\Omega$ baryons. The three-body
problem of these baryons is considered in the hypercentral approach. Applying
an Ansatz method introduced in our previous work, the mass spectra of the
ground and excited states of doubly heavy $\Xi$ and $\Omega$ baryons are
obtained. By introducing a simple form for the universal IW function, we also
investigate the semileptonic decay rates and branching fractions of the
$bb\rightarrow bc$ and $bc\rightarrow cc$ baryonic transitions near to zero
recoil point. We present a comparison between our results and other available
theoretical calculations, and find that the results are acceptable. Note that
the triplet doubly heavy baryons with $s=\frac{3}{2}$ are dominated by the
strong or electromagnetic decays. If these excited states are the initial
ones, due to the smallness of the weak coupling, the weak decays can not be
observed in the experiments. Therefore, one can shelve the calculations for
the semileptonic decays of $s=\frac{3}{2}$ doubly heavy baryons, since it is
hard to perform relevant measurements in the experiments. In the current work,
we have performed the related calculations for all the ground states of doubly
heavy baryons as done in Refs. [38, 48, 32]. We hope our results are useful to
extract the value of the CKM matrix element $V_{cb}$ from future experiments
via the semileptonic decays of doubly heavy baryons.

##  5 Acknowledgments

The authors would like to thank Prof. Fu-Sheng Yu for useful discussions. This
work is supported in part by National Natural Science Foundation of China
(NSFC) under contract No. 11975076, No. 12135005, and No. 12161141008.

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# Lightfield Coordinates Adapted to Asgeirsson’s Theorem

Haotian Li  He Qin  Todor Georgiev

(Adobe Systems, San Jose, CA 95110, USA)

## Abstract

John’s differential equation and its canonical form, the ultrahyperbolic
equation, plays important role in lightfield imaging. The equation describes a
local constraint on the lightfield, that was first observed as a
“dimensionality gap” [6] in the frequency representation. Related to the
ultrahyperbolic equation, Asgeirsson’s theorems describe global properties.
These indicate new, global, constraints on the lightfield. In order to help
validate those theorems on real captured images, we introduce a coordinate
system for the lightfield, which suits better the Asgeirsson theorems, and
analyze behaviour in terms of the new coordinates.

Keywords: Lightfield, John’s equation, ultrahyperbolic PDE, Asgeirsson’s
theorems, $4D$ radiance.

##  1 Introduction

###  1.1 John’s Transform and John’s Equation

Given a function $f$ describing density of isotropic light sources in $3D$,
the John transform $r$ of $f$ is defined as its integral along any straight
line $\xi$:

$\displaystyle J(f)=r,\qquad r(\xi):=\int_{\xi}f(x,y,z)\mathrm{d}m(x,y,z),$
(1)

where $\mathrm{d}m$ is the Euclidean measure on the straight lines $\xi$
(notations taken from [5]). If we use two-plane parametrization for $\xi$
(Fig. 1), where $(x,y)$ gives the intersection of a light ray $\xi$ with the
first plane and $(u,v)$ indicates the angles by tracking the displacements of
$\xi$ on the second plane, then

$\displaystyle J(f)=r,\qquad
r(x,y,u,v)=\int_{-\infty}^{\infty}f(x+uz,y+vz,z)dz$ (2)

Figure 1: Two parallel planes parameterization of straight lines in $3D$.

John’s equation is derived from Eq. (2). It constrains the radiance $r$ in the
following way:

$\displaystyle(\frac{\partial}{\partial y}\frac{\partial}{\partial
u}-\frac{\partial}{\partial x}\frac{\partial}{\partial v})r=0$ (3)

###  1.2 The Ultrahyperbolic Equation

Furthermore, if we do the following reparametrization of $r(x,y,u,v)$ into
$\tilde{r}(\xi_{1},\xi_{2},\xi_{3},\xi_{4})$ (i.e. transform from $4D$
$(x,y,u,v)$-space to $4D$ $(\xi_{1},\xi_{2},\xi_{3},\xi_{4})$-space),

$\displaystyle\begin{cases}\xi_{1}=\frac{1}{2}(u+y)\\\
\xi_{2}=\frac{1}{2}(u-y)\\\ \xi_{3}=\frac{1}{2}(v+x)\\\
\xi_{4}=\frac{1}{2}(v-x)\end{cases}\qquad\qquad\begin{cases}x=\xi_{3}-\xi_{4}\\\
y=\xi_{1}-\xi_{2}\\\ u=\xi_{1}+\xi_{2}\\\ v=\xi_{3}+\xi_{4}\end{cases}$ (4)

we get the ultrahyperbolic partial differential equation

$\displaystyle(\partial_{\xi_{1}\xi_{1}}-\partial_{\xi_{2}\xi_{2}}-\partial_{\xi_{3}\xi_{3}}+\partial_{\xi_{4}\xi_{4}})\tilde{r}=0\qquad\text{
or equivalently, }(\Delta_{14}-\Delta_{23})\tilde{r}=0,$ (5)

where $\Delta_{14}$ and $\Delta_{23}$ are the Laplacians in the
$(\xi_{1},\xi_{4})$ and $(\xi_{2},\xi_{3})$ planes, respectively. (See [3]).

###  1.3 Asgeirsson’s Theorems

Next we use the ultrahyperbolic equation with four variables as shown in Eq.
(5). We base our analysis on the following theorems by Asgeirsson (see [1]).

######  Theorem 1.1.

Integral over a circle $C_{1}$ with radius $R$ in the $(1,4)$-plane is equal
to the integral over the same radius circle $C_{2}$ in the $(2,3)$-plane,
i.e.,

$\displaystyle\int_{0}^{2\pi}r(\xi_{1}+R\cos\theta,\xi_{2},\xi_{3},\xi_{4}+R\sin\theta)\mathrm{d}\theta=\int_{0}^{2\pi}r(\xi_{1},\xi_{2}+R\cos\theta,\xi_{3}+R\sin\theta,\xi_{4})\mathrm{d}\theta$
(6)

######  Theorem 1.2.

More generally, if we consider a double integral over two circles, one of
which has radius $R_{1}$ in $(1,4)$-plane and the other has radius $R_{2}$ in
$(2,3)$-plane, it is equal to the double integral over two circles with two
radii switched in the two planes, i.e.,

$\displaystyle\int_{0}^{2\pi}\int_{0}^{2\pi}r(\xi_{1}+R_{1}\cos\theta_{1},\xi_{2}+R_{2}\cos\theta_{2},\xi_{3}+R_{2}\sin\theta_{2},\xi_{4}+R_{1}\sin\theta_{1})\mathrm{d}\theta_{1}\mathrm{d}\theta_{2}$
$\displaystyle=\int_{0}^{2\pi}\int_{0}^{2\pi}r(\xi_{1}+R_{2}\cos\theta_{1},\xi_{2}+R_{1}\cos\theta_{2},\xi_{3}+R_{1}\sin\theta_{2},\xi_{4}+R_{2}\sin\theta_{1})\mathrm{d}\theta_{1}\mathrm{d}\theta_{2}$

##  2 New Coordinates

We introduce two sets of polar coordinates in the planes $(\xi_{1},\xi_{4})$
and $(\xi_{2},\xi_{3})$, respectively:

$\displaystyle\begin{cases}\xi_{1}=R_{1}\cos\theta_{1}\\\
\xi_{4}=R_{1}\sin\theta_{1}\end{cases}\qquad\begin{cases}\xi_{2}=R_{2}\cos\theta_{2}\\\
\xi_{3}=R_{2}\sin\theta_{2}\end{cases}\qquad\text{where }R_{1},R_{2}\geq
0\text{ and }\theta_{1},\theta_{2}\in[0,2\pi).$

Then, we consider the new coordinate system
$(\theta_{1},\theta_{2},R_{1},R_{2})$ for the lightfield, in which
$(\theta_{1},\theta_{2})$ are the coordinates within each new microimage, and
$(R_{1},R_{2})$ are the coordinates of the new microimages. In this coordinate
system Theorem 1.2 states that the double integral over
${(\theta_{1},\theta_{2})}$ of a microimage with coordinates $(R_{1},R_{2})$
is equal to the double integral over ${(\theta_{1},\theta_{2})}$ of a
microimage with coordinates $(R_{2},R_{1})$.

##  3 Change of Coordinates

Coordinate Transformation:
$(\theta_{1},\theta_{2},R_{1},R_{2})\longrightarrow(x,y,u,v)$

$\displaystyle\begin{cases}&x=R_{2}\sin\theta_{2}-R_{1}\sin\theta_{1}\\\
&y=R_{1}\cos\theta_{1}-R_{2}\cos\theta_{2}\\\
&u=R_{1}\cos\theta_{1}+R_{2}\cos\theta_{2}\\\
&v=R_{2}\sin\theta_{2}+R_{1}\sin\theta_{1}\\\ \end{cases}$ (7)

Inverse Coordinate Transformation:
$(x,y,u,v)\longrightarrow(\theta_{1},\theta_{2},R_{1},R_{2})$

$\displaystyle\begin{cases}&\theta_{1}=\mathrm{arctan2}(v-x,u+y)\\\
&\theta_{2}=\mathrm{arctan2}(v+x,u-y)\\\
&R_{1}=\frac{1}{2}\sqrt{(u+y)^{2}+(v-x)^{2}}\\\
&R_{2}=\frac{1}{2}\sqrt{(u-y)^{2}+(v+x)^{2}}\\\ \end{cases}$ (8)

where $\mathrm{arctan2}$ is a variation form of $\arctan$ (see Fig. 2):

$\displaystyle\mathrm{arctan2}(y,x)=\begin{cases}\arctan(\frac{y}{x})&\text{if
}x>0,\\\ \arctan(\frac{y}{x})+\pi&\text{if }x<0\text{ and }y\geq 0,\\\
\arctan(\frac{y}{x})-\pi&\text{if }x<0\text{ and }y<0,\\\
\frac{\pi}{2}&\text{if }x=0\text{ and }y>0,\\\ -\frac{\pi}{2}&\text{if
}x=0\text{ and }y<0,\\\ \text{undefined}&\text{if }x=0\text{ and }y=0.\\\
\end{cases}$ (9)

Figure 2: Function $z=\mathrm{arctan2}(y,x)$, $x$-axis (green), $y$-axis
(blue), $z$-axis (red)

##  4 Discretization

To deal with real images with pixels, discretization of the previous formulas
is necessary. Here we discretize the radii and the angles. First, we simply
discretize both radii $R_{i},$ $i=1,2$, into non-negative integers. For
example $R_{1}=0,1,2,3,\cdots$, and similar for $R_{2}$. Second, we discretize
the angles $\theta_{i}$, $i=1,2$, in a linear (in $R_{i}$) way: For
$\theta_{i}$, we evenly split the interval $[0,2\pi)$ into $7R_{i}+1$ sub-
intervals (as shown in Fig. 3). We discretize in a linear fashion in order to
produce uniform representation (see below). The Jacobian in polar coordinate
is linear w.r.t. the radius value.

Figure 3: $R_{1},\theta_{1}$ discretization in plane $(\xi_{1},\xi_{4})$. The
concentric rings represent $R_{1}=0,1,2,3.$

After discretization, we arrange the coordinates
$(\theta_{1},\theta_{2},R_{1},R_{2})$ as shown in Fig. 4. Since pixels are
equally spaced, the sizes of different microimages are different. The size of
a microimage depends on the number of pixels in it.

Now Asgeirsson’s theorems simply say that the sum of all pixels in a
microimage with coordinates $(R_{1},R_{2})$ is equal to the sum of all pixels
in the corresponding microimage with coordinates $(R_{2},R_{1})$. Note that
those two microimages are symmetric with respect to the main diagonal in Fig.
4.

Figure 4: New coordinates for the lightfield. $(\theta_{1},\theta_{2})$ are
the coordinates of pixels inside each of the small squares / rectangles.

##  5 Numerical Experiments

We use the Seagull lightfield image (Fig. 5) to generate an example of the new
lightfield representation (Fig. 6). In the experiments, we choose the pixel at
the center of the seagull’s eye as the reference point, i.e., the origin of
the $4D$ coordinates, in the Seagull lightfield.

Since the coordinate transformation (Eq. (7)) is continuous and analytic, we
generally get floating point number coordinates instead of integers when we
try to fill in the new coordinate system’s pixel values. So in Fig. 6, we use
nearest neighbor sampling to read the pixel value from the original lightfield
image.

Figure 7 is used as a map representing the coordinates in Figure 6.

Figure 8 is a zoom in into Figure 6d, where $R_{1}$ and $R_{2}$ take values
from $0$ to $4$.

The first Asgeirsson theorem says that the sum of all pixels with given
$R_{2}=k$ in the single row at the top, is equal to the sum of all pixels in
the single column on the left having $R_{1}$ equal to the same $k$.

More generally, the second Asgeirsson theorem says that in symmetric boxes
pixels sum up to the same value. Boxes are defined symmetric relative to the
main diagonal (see Section 4).

##  6 Note on Using Shift for Lightfield Imaging

One important practical point about lightfield imaging is sparsity of
sampling, and the resulting aliasing. Due to the spatio-angular resolution
tradeoff [4] and the need to produce higher spatial resolution with limited
sensor size, we often perform sparse optical sampling in the angular
dimensions. This results in aliasing artifacts when we render the final image,
or when we apply any 4D filter.

This influences our algorithms for rendering of the final image and
refocusing. It also influences the computation of derivatives in the angular
directions $u$ and $v$. The same pixel location $(x,y)$ in two neighboring
microimages $(u,v)$ and $(u+1,v)$ sample two very different rays in the
lightfield and their difference cannot represent correctly the derivative of
the radiance in $u$-direction. This is due to sparse optical sampling. We need
a microimage between $(u,v)$ and $(u+1,v)$, which however has not been
captured. Considering epipolar geometry, we substitute the pixel with spatial
coordinates $(x,y)$ in this missing microimage with a pixel from the $(u+1,v)$
microimage, with shifted $x$-coordinate. This is the same shift or “patch
size” that is used in lightfield/plenoptic rendering (see for example [2]). It
depends on depth. For the seagull in our lightfield (Fig. 5) the shift value
is $8.78$ pixels, which we approximate with $9$ in Figures 6, 7, 8.

Figure 5: Original lightfield Seagull captured with our Plenoptic 2.0 camera.
Coordinates are chosen in the traditional way: Microimage locatedion is
parametrized by $(u,v)$, pixels microimage having coordinates $(x,y)$.

(a) shift = 0

(b) shift = 2

(c) shift = 5

(d) shift = 9

Figure 6: New coordinate representations of lightfield Seagull (Fig. 5) with
different shifts. The important role of shifts is explained in section 6. See
Fig. 7 for colormap and description of our coordinates.

(a) shift = 0

(b) shift = 2

(c) shift = 5

(d) shift = 9

Figure 7: Color mapping of the new coordinate representation of lightfield
Seagull (Fig. 6) with different shifts. Different tones of green represent
values of $R_{1}$ from $0$ (darkest) to $7$ (lightest), and similar for red
(representing $R_{2}$). Pixels inside each rectangle have coordinates
$\theta_{1}$ and $\theta_{2}$ ranging from $0$ to $2\pi$. Gray represents
pixels that are outside the captured lightfield (in our case outside Fig. 5).
Figure 8: A zoom in into Fig. 6d, where $R_{1}$ and $R_{2}$ take values from
$0$ to $4$. Figure 9: Color mapping in original coordinates with shift 9.

## References

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  * [3] T. Georgiev, H. Qin, and H. Li, John transform and ultrahyperbolic equation for lightfields, arXiv preprint arXiv:1907.01186, (2019).
  * [4] T. Georgiev, K. C. Zheng, B. Curless, D. Salesin, S. Nayar, and C. Intwala, Spatio-angular resolution tradeoff in integral photography, Proceedings of Eurographics Symposium on Rendering, (2006).
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# An improved bound on the chromatic number of the Pancake graphs

Leen Droogendijk<EMAIL_ADDRESS>Elena V. Konstantinova
<EMAIL_ADDRESS>Sobolev Institute of Mathematics, Ak. Koptyug av. 4,
Novosibirsk, 630090, Russia  Novosibisk State University, Pirogova str. 2,
Novosibirsk, 630090, Russia

###### Abstract

In this paper an improved bound on the chromatic number of the Pancake graph
$P_{n},n\geqslant 2$, is presented. The bound is obtained using a
subadditivity property of the chromatic number of the Pancake graph. We also
investigate an equitable coloring of $P_{n}$. An equitable $(n-1)$-coloring
based on efficient dominating sets is given and optimal equitable
$4$-colorings are considered for small $n$. It is conjectured that the
chromatic number of $P_{n}$ coincides with its equitable chromatic number for
any $n\geqslant 2$.

###### keywords:

Pancake graph; chromatic number; equitable coloring

###### MSC:

[2010] 05C15, 05C25, 05C69

††journal: arXiv

##  1 Introduction

The Pancake graph $P_{n},\ n\geqslant 2$, is defined as the Cayley graph over
the symmetric group $\mathrm{Sym}_{n}$ with the generating set of all
prefix–reversals $r_{i},\ 2\leqslant i\leqslant n$, inverting the order of any
substring $[1,i]$ of a permutation when multiplied on the right. It is a
connected vertex–transitive $(n-1)$-regular graph without loops and multiple
edges of order $n!$. It contains all cycles $C_{l}$ of length $l$, where
$6\leqslant l\leqslant n!$ [14, 23].

A mapping $c:V(\Gamma)\rightarrow\\{1,2,\ldots,k\\}$ is called a proper
$k$–coloring of a graph $\Gamma=(V,E)$ if $c(u)\neq c(v)$ whenever the
vertices $u$ and $v$ are adjacent. The chromatic number $\chi(\Gamma)$ of a
graph $\Gamma$ is the least number of colors needed to properly color vertices
of $\Gamma$. A subset of vertices assigned to the same color forms an
independent set, i.e. a proper $k$–coloring is the same as a partition of the
vertex set into $k$ independent sets. The trivial lower and upper bounds on
the chromatic number of the Pancake graphs are given as follows:

$3\leqslant\chi(P_{n})\leqslant n-1{\mbox{ for any }}n\geqslant 4.$ (1)

Indeed, the graph $P_{n}$ is $(n-1)$–regular, hence by Brooks’ theorem [4] we
have the upper bound. Moreover, $\chi(P_{3})=2$ since $P_{3}\cong C_{6}$, and
$\chi(P_{4})=3$ since there are $7$–cycles in $P_{n}$ for any $n\geqslant 4$
[19] which gives us the lower bound. The Brooks’ bound is improved by $1$ for
graphs with $\omega\leqslant(\Delta-1)/2$, where $\omega$ and $\Delta$ are the
size of the maximum clique and the maximum degree of the graph (see [5, 6]).
Since $\omega(P_{n})=2$, then $\chi(P_{n})\leqslant n-2$ for any $n\geqslant
6$. Moreover, there is a proper $3$–coloring of $P_{5}$ [18]. Thus, we have:

$\chi(P_{n})\leqslant n-2{\mbox{ for any }}n\geqslant 5.$ (2)

Catlin’s bound for $C_{4}$–free graphs [7], that is
$\chi\leqslant\frac{2}{3}\left(\Delta+3\right)$, gives one more bound for any
$n\geqslant 8$:

$\chi(P_{n})\leqslant\frac{2}{3}\left(n+2\right).$ (3)

Using structural properties of $P_{n}$, the following bounds were obtained in
[18]:

${\mbox{\it}for}\ 5\leqslant n\leqslant 8,\
\chi(P_{n})\leqslant\left\\{\begin{array}[]{ll}n-k,&\hbox{if $n\equiv
k\,(${\rm mod} $4)$ for $k=1,3$;}\\\ n-2,&\hbox{if $n$ is
even;}\end{array}\right.$ (4)

${\mbox{\it}for}\ 9\leqslant n\leqslant 16,\
\chi(P_{n})\leqslant\left\\{\begin{array}[]{ll}n-(k+2),&\hbox{if $n\equiv
k\,(${\rm mod} $4)$ for $k=1,3$;}\\\ n-4,&\hbox{if $n$ is
even;}\end{array}\right.$ (5)

${\mbox{\it}for}\ n\geqslant 17,\
\chi(P_{n})\leqslant\left\\{\begin{array}[]{ll}n-(k+4),&\hbox{if $n\equiv
k\,(${\rm mod} $4)$ for $k=1,2,3$;}\\\ n-8,&\hbox{if $n\equiv 0\,(${\rm mod}
$4)$.}\end{array}\right.$ (6)

These bounds improve (2) for $n\geqslant 7$, however Catlin’s bound (3) is
still better for all $n>28$ and some smaller $n$ (for example,
$n=21,25,26,27$). Thus, they are far from good. Meanwhile, the asymptotic
bound $\chi(P_{n})\leqslant O\left(\frac{n-1}{log(n-1)}\right)$ holds for the
Pancake graphs which follows from the results for $C_{3},C_{4}$–free graphs
[10, 15].

In this paper in Section 2 we present a new upper bound which improves
Catlin’s bound (3). The new bound is obtained using a subadditivity property
of the chromatic number of $P_{n}$ and known chromatic numbers for $n\leqslant
9$. We have $\chi(P_{3})=2$ since $P_{3}\cong C_{6}$, and $\chi(P_{4})=3$
since there are $7$–cycles in $P_{n},\,n\geqslant 4$. An example of a proper
$3$–coloring for $P_{5}$ was given in [18]. An optimal $4$-coloring for
$P_{6}$ was computed by Tomaž Pisanski, University of Primorska, Koper,
Slovenia, and Jernej Azarija, University of Ljubljana, Slovenia, so
$\chi(P_{6})=4$. Since $P_{n-1}$ is an induced subgraph of $P_{n}$, then
$\chi(P_{7})$ is at least $4$, and from (4) we have $\chi(P_{7})=4$. Optimal
$4$-colorings for $P_{8}$ and $P_{9}$ were computed by A. H. Ghodrati, Sharif
University, Tehran, Iran. By (5), $4\leqslant\chi(P_{n})\leqslant 12$, where
$10\leqslant n\leqslant 16$, however, proper $4$-colorings in these cases are
unknown. The known chromatic numbers are presented in the Table 1.

In Section 3 an equitable coloring is considered. A graph $\Gamma$ is said to
be equitably $k$-colorable if $\Gamma$ has a proper $k$-coloring such that the
sizes of any two color classes differ by at most one. The equitable chromatic
number $\chi_{=}(\Gamma)$ is the smallest integer $k$ such that $\Gamma$ is
equitably $k$-colorable. Equitable coloring was introduced by W. Meyer in
$1973$ due to scheduling problems [21]. Moreover, it was conjectured that
every connected graph with maximum degree $\Delta$ has an equitable coloring
with $\Delta$ or fewer colors, with the exceptions of complete graphs and odd
cycles. A strengthened version [8] of the conjecture states that each such
graph has an equitable coloring with exactly $\Delta$ colors, with one
additional exception, a complete bipartite graph in which both sides of the
bipartition have the same odd number of vertices. A survey on equitable
colorings can be found in [20].

In Section 3.1 an equitable $(n-1)$-coloring based on efficient dominating
sets in the Pancake graphs $P_{n},n>2$, is presented. Moreover, in Section 3.2
simple optimal equitable $4$-colorings for $P_{5},P_{6}$ and $P_{7}$ are
described.

Let us note that any equitable coloring of $P_{n}$ with at most $n$ colors has
the property that the sizes of all color classes are equal since every integer
at most $n$ divides $n!$. Thus, we have a strongly equitable coloring [12].

Since equitable coloring is a proper coloring with an additional condition,
the inequality $\chi(P_{n})\leqslant\chi_{=}(P_{n})$ holds for any $n\geqslant
2$. However, since all above optimal colorings are strongly equitable we have
conjectured.

###### Conjecture 1

For any $n\geqslant 2$,

$\chi(P_{n})=\chi_{=}(P_{n}).$

$n$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---
$\chi(P_{n})$ | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4
Table. 1: Chromatic numbers of the Pancake graphs $P_{n},\ 2\leqslant
n\leqslant 9$.

##  2 Improved upper bound

Our main result is given by the following theorem.

###### Theorem 1

For any $n\geqslant 9,$ the following holds for the Pancake graph $P_{n}$:

$\chi(P_{n})\leqslant
4\left\lfloor\frac{n}{9}\right\rfloor+\chi\left(P_{n\pmod{9}}\right).$ (7)

To prove this result we need more notation. Let $[n]=\\{1,2,\ldots,n\\}$. We
consider a permutation $\pi=[\pi_{1}\pi_{2}\ldots\pi_{n}]$ written as a string
in one-line notation, where $\pi_{i}=\pi(i)$ for any $i\in[n]$. For
$K\subset[n]$, let $P_{n,K}$ be the induced subgraph of $P_{n}$ whose vertex
set consists of all permutations $\pi$ with $\pi_{1}\in K$. By the symmetry of
$P_{n}$, for any $k$-element subset $K$ of $[n]$, the induced subgraph
$P_{n,K}$ is isomorphic to $P_{n,[k]}$, which is abbreviated to $P_{n,k}$.

We define a map $f_{n,k}:P_{n,k}\to P_{k}$ by removing the elements that are
not in $[k]$. For example, for $n=5$ and $k=3$, the vertex $[14352]$ of
$P_{5,3}$ is mapped to the vertex $[132]$ of $P_{3}$. It is clear that this
mapping is a graph homomorphism. Note that $f_{n,k}$ is surjective, but not
necessarily an isomorphism. In fact, $P_{n,k}$ is not even connected unless
$k=n$ or $n=2$.

Since an $r$-coloring of a graph $\Gamma$ is equivalent to a graph
homomorphism from $\Gamma$ to the complete graph $K_{r}$, this property
implies that

$\chi(P_{n,k})\leqslant\chi(P_{k}).$ (8)

Since $P_{n,k}$ always contains a subgraph isomorphic to $P_{k}$ (e. g. the
subgraph of all vertices that end with $k+1,k+2,\ldots,n$) it even follows
that $\chi(P_{n,k})=\chi(P_{k})$.

One more useful property says that all fibers $f_{n+1,n}:P_{n+1,n}\to P_{n}$
are of size $n$, which means that $\left|f_{n+1,n}^{-1}(v)\right|=n$ for every
$v\in V(P_{n})$. Indeed, for any permutation of length $n$ one can insert
$n+1$ at $n$ different positions: the first position is forbidden since the
vertices of $P_{n+1,n}$ start with an element from $[n]$.

The following theorem immediately gives the general upper bound of (7) with
taking into account $\chi(P_{9})=4$.

###### Theorem 2

The chromatic number of the Pancake graph is subadditive, i. e.

$\chi(P_{n+m})\leqslant\chi(P_{n})+\chi(P_{m})$ (9)

for all positive integers $n$ and $m$.

Proof. Let the vertices of $P_{n+m}$ be partitioned into sets $U$ and $W$ such
that $U$ contains permutations whose first element is in $[n]$ and $W$
contains permutations whose first element is in $\\{n+1,\ldots,n+m\\}$. The
subgraphs $P_{n+m,U}$ and $P_{n+m,W}$ are isomorphic to $P_{n+m,n}$ and
$P_{n+m,m}$, respectively. Hence, by (8) the graph $P_{n+m,U}$ is
$\chi(P_{n})$-colorable, and $P_{n+m,W}$ is $\chi(P_{m})$-colorable. Using
disjoint color sets on both subgraphs proves the desired inequality. $\square$

##  3 Equitable coloring and optimal colorings

It was shown in Introduction that there are optimal colorings of the Pancake
graphs found by computer experiments. Such computations do not provide us any
structural insight. In this section we consider colorings of the Pancake
graphs $P_{n},n\geqslant 3$, using their structural properties. More
precisely, we present equitable colorings of $P_{n}$ in $(n-1)$ colors.

An equitable coloring is not the same as a perfect coloring for which the
multiset of colors of all neighbors of a vertex depends only on its own color
[11]. This type of coloring gives a partition known as an equitable partition
[13] which are used in algebraic combinatorics, graph theory and coding
theory. In coding theory such kind of partitions are known as perfect codes
[1, 16]. Some general information about equitable partitions can be found in
[2].

The notion of perfect codes was generalized to the Pancake graphs in a natural
way in [9]. An independent set $D$ of vertices in a graph $\Gamma$ is an
efficient dominating set (or $1$-perfect code) if each vertex not in $D$ is
adjacent to exactly one vertex in $D$. There are $n$ efficient dominating sets
in $P_{n}$ [9, 22] given by:

$D_{i}=\\{[i\,\pi_{2}\,\ldots\,\pi_{n}]\\},$ (10)

where $\pi_{k}\in[n]\backslash\\{i\\},\,k\in[n]\backslash\\{1\\},\,i\in[n]$.
It is obvious that $|D_{i_{1}}\bigcap D_{i_{2}}|=\emptyset$,
$i_{1},i_{2}\in[n],\ i_{1}\neq i_{2}$, which immediately gives a proper
$n$-coloring. Moreover, this coloring is perfect and equitable.

To present a proper $(n-1)$-coloring of the Pancake graphs based on efficient
dominating sets we need to define such sets for induced subgraphs $P_{n-1}$ of
$P_{n}$.

Due to the hierarchical structure, for any $n\geqslant 3$ the graph $P_{n}$
has $n$ copies of $P_{n-1}(i)$ with the vertex set
$V_{i}=\\{[\pi_{1}\ldots\pi_{n-1}i]\\}$, where
$\pi_{k}\in[n]\backslash\\{i\\},k\in[n-1]$, $|V_{i}|=(n-1)!$. Any two copies
$P_{n-1}(i),P_{n-1}(j),i\neq j$, are connected by $(n-2)!$ edges
$\\{[i\pi_{2}\ldots\pi_{n-1}j],\,[j\pi_{n-1}\ldots\pi_{2}i]\\}$, where
$[i\pi_{2}\ldots\pi_{n-1}j]r_{n}=[j\pi_{n-1}\ldots\pi_{2}i]$. Prefix–reversals
$r_{j},2\leqslant j\leqslant n-1$, define internal edges in all $n$ copies
$P_{n-1}(i)$, and the prefix–reversal $r_{n}$ defines external edges between
copies.

Efficient dominating sets of $P_{n-1}(j),j\in[n]$, contain all permutations
with the last element fixed to $j$ and the first element fixed [17], namely:

$D_{i}^{j}=\\{[i\,\pi_{2}\,\ldots\,\pi_{n-1}\,j]\\},$ (11)

where $i,j\in[n],i\neq j$,
$\pi_{k}\in[n]\backslash\\{i,\,j\\},\,k\in[n]\backslash\\{1,n\\}$. For any
$i\in[n]$, the sets (10) and (11) are given by the following obvious
relationship:

$D_{i}=\bigcup_{i=1,j\neq i}^{n}D_{i}^{j}.$ (12)

###  3.1 Equitable $(n-1)$-coloring

We now present an equitable $(n-1)$-coloring based on efficient dominating
sets. Let

$D=\\{D_{i}^{j}:i,j\in[n],i\neq j\\},\ \ |D|=n\,(n-1),$ (13)

and

$D^{j}=\\{D_{i}^{j}:i\in[n],i\neq j\\},\ \ j\in[n],\ \ |D^{j}|=n-1,$ (14)

where

$|D_{i}^{j}|=(n-2)!.$ (15)

Note that $D$ partitions the vertices of $P_{n}$, and $D^{j}$ partitions the
vertices of $P_{n}$ that end with $j$. We now define a graph $Q_{n}$ whose
vertices are the elements of $D$, and $X,Y\in D$ are adjacent in $Q_{n}$ if
and only if a vertex of $X$ is adjacent to a vertex of $Y$ in $P_{n}$. From
the properties of the Pancake graphs we immediately see that vertices
$D_{i}^{j}$ and $D_{i^{\prime}}^{j^{\prime}}$ are adjacent in $Q_{n}$ if and
only if one of the following statements is true:

  1. (A1)

$j=j^{\prime}$ and $i\neq i^{\prime}$.

  2. (A2)

$i=j^{\prime}$ and $j=i^{\prime}$.

It is obvious that a proper coloring $c$ of $Q_{n}$ trivially extends to a
proper coloring of $P_{n}$ in such a way that any vertex of $P_{n}$ belongs to
exactly one efficient dominating set $X\in D$ and we give it the color $c(X)$.

We now have reduced the problem to finding a proper ($n-1$)-coloring for
$Q_{n}$. First let us show an idea of such colorings for the graphs $Q_{4}$
and $Q_{6}$. The graphs are presented on Figures 1 and 2 such that the
vertices corresponding to the set $D_{i}^{j},\ i,j\in[n],i\neq j$, are denoted
by labels $ij$. The vertices are arranged in a hamiltonian cycle such that all
vertices with the same last element are grouped together and form $3$\- and
$5$-cliques, respectively. Within each clique the first element of labeling is
cyclically incremented. Obviously, the elements of each clique must all have
different colors, but the pictures suggest that we can ‘almost’ cyclically
repeat a color pattern chosen on the first clique. The only collisions occur
with the ‘long’ chords that connect antipodal vertices. We see that if we
exchange the color of one end of each long chord with the color of the vertex
counterclockwise next to it on the cycle, we obtain a proper coloring. It is
clear that the proper colorings of $Q_{4}$ and $Q_{6}$ are equitable. Indeed,
by (13)-(15) and from the construction $Q_{4}$ has $3$ color classes of
cardinality $4$ each, and $Q_{6}$ has $5$ color classes of cardinality $6$
each. However, they are not perfect since the multisets of colors of all
neighbors are different for different vertices having the same color. For
example, in $Q_{4}$ the red vertex $(14)$ has two green and one blue
neighbors, while the red vertex $(32)$ has two blue and one green neighbors.
Similar, in $Q_{6}$ despite the red and the purple vertices have the same
multiset of colors of their neighbors, the green, the blue and the dark blue
vertices do not meet this condition to be perfect.

Note that this coloring is exactly the greedy coloring for the vertex sequence
that starts with $(1n)$ and then counterclockwise follows the cycle.

Figure 1: The equitable $3$-coloring of $Q_{4}$ Figure 2: The equitable
$5$-coloring of $Q_{6}$

Now we formalize and prove this observation. First we define a map:

$f:D\to[n-1]$ (16)

such that

$\left\\{\begin{array}[]{ll}f(D_{i}^{j})=i-j,&\hbox{if $i>j$};\\\
f(D_{i}^{j})=n+i-j,&\hbox{otherwise}.\end{array}\right.$ (17)

Note that $f$ indeed has all its values in $[n-1]$, and that the restriction
of $f$ to $D_{j}$ is injective, i.e.

$f(D_{i}^{j})=f(D_{i^{\prime}}^{j})\mbox{ if and only if }i=i^{\prime}.$

Next we let $k=\frac{n}{2}$ and define a coloring

$c:D\to[n-1]$ (18)

by

$c(D_{i}^{j})=f(D_{i}^{j})+\varepsilon,$ (19)

where

$\varepsilon=\left\\{\begin{array}[]{ll}+1,&\hbox{if $j>k$ and
$f(D_{i}^{j})=k$;}\\\ -1,&\hbox{if $j>k$ and $f(D_{i}^{j})=k+1$;}\\\
0,&\hbox{otherwise.}\end{array}\right.$ (20)

If $n$ is odd, the first two cases cannot occur, since $f(D_{i}^{j})$ is an
integer, but $k$ is not. The restriction of $c$ to $D^{j}$ is still injective,
since it is either equal to $f$ for $j\leqslant k$ or it is equal to $f$ with
at most two adjacent function values exchanged.

In terms of the intuitive approach above, $f$ is the coloring that cyclically
repeats along the cycle, and $c$ is the coloring that exchanges the colors
near the ends of long chords.

###### Theorem 3

The coloring $c$ is proper and equitable for $P_{n},n>2$.

Proof. Indeed, suppose that $X=D_{i}^{j}$ and $Y=D_{i^{\prime}}^{j^{\prime}}$
are adjacent. If $j=j^{\prime}$, then $i\neq i^{\prime}$ by (A1). By the
injectivity of $c$ for fixed $j$, $X$ and $Y$ must have different colors. By
(A2) the only other possibility is that $Y=D_{j}^{i}$. Without loss of
generality we assume that $j<i$.

First we handle the case that $n$ is odd. Then
$c(X)=c(D_{i}^{j})=f(D_{i}^{j})=i-j$, and
$c(Y)=c(D_{j}^{i})=f(D_{j}^{i})=n+j-i$. The equality $c(X)=c(Y)$ implies
$n=2i-2j$, so $n$ is even. There is a contradiction.

In the last part of the proof we assume that $n$ is even, so $k$ is an
integer. We do a case analysis based on the position of $k$.

If $j<i\leqslant k$ then $c(X)=i-j$, and $c(Y)=n+j-i$. The equality
$c(X)=c(Y)$ implies $2i=n+2j=2k+2j$, so $i=k+j>k$, and this gives a
contradiction.

If $j\leqslant k<i$ then $c(X)=i-j$, and $c(Y)=n+j-i+\varepsilon$. The
equality $c(X)=c(Y)$ implies $i-j=n+j-i+\varepsilon$, so
$2k+2j=2i-\varepsilon$. Then $\varepsilon$ must be even, hence
$\varepsilon=0$, and $i=k+j$ or $f(Y)=f(D_{j}^{i})=n+j-i=k$. Since $i>k$ we
have $\varepsilon=1$ by (20) which gives a contradiction.

If $k<j<i$ then $c(X)=i-j+\varepsilon_{1}$ and $c(Y)=n+j-i+\varepsilon_{2}$.
The equality $c(X)=c(Y)$ implies $i-j+\varepsilon_{1}=n+j-i+\varepsilon_{2}$,
so $2i+\varepsilon_{1}=2k+2j+\varepsilon_{2}$, and hence $\varepsilon_{1}$ and
$\varepsilon_{2}$ have the same parity. There are the following possibilities.

(i) If $\varepsilon_{1}=\varepsilon_{2}$ then $i-j=k$. Since $k<j<i\leqslant
n=2k$ we have $k=i-j<2k-k=k$ which gives a contradiction.

(ii) If $\varepsilon_{1}=-1$, $\varepsilon_{2}=1$. As in case (i), we have
$k+1=i-j<2k-k=k$ which leads to a contradiction.

(iii) If $\varepsilon_{1}=1$, $\varepsilon_{2}=-1$ then $i-j=k-1$. Again,
since $k<j<i\leqslant 2k$, this is only possible if $j=k+1$ and $i=n=2k$. Then
$f(D_{i}^{j})=k-1$, so $\varepsilon_{1}=0$ by (20), and we have a
contradiction.

In all cases the assumption of color equality leads to a contradiction, which
finishes the proof that $c$ is a proper coloring. Moreover, by (18) we have
$(n-1)$ color classes each of which has a cardinality
$|D^{j}|\cdot|D^{j}_{i}|=n(n-2)!$ (see (14), (15)) which finishes the proof of
the statement. $\square$

###  3.2 Optimal colorings

It is obvious that above equitable $(n-1)$-coloring produces an optimal
coloring for $P_{3}$ and $P_{4}$ (see Table 1). However, for $n>4$ a proper
coloring of $Q_{n}$ can never produce an optimal coloring for $P_{n}$. Indeed,
by (2) for $n>4$ we have $\chi(P_{n})<n-1$ and $\chi(Q_{n})\geqslant n-1$
since $Q_{n}$ contains an ($n-1$)-clique.

Now we give a simple optimal $4$-coloring of $P_{5}$, $P_{6}$ or $P_{7}$. We
define an even (odd) prefix–reversal $r_{i},\ 2\leqslant i\leqslant n$, if it
corresponds to an even (odd) permutation. By [18, Lemma 4], if $i\equiv
0,1\pmod{4}$ then $r_{i}$ is an even prefix–reversal. Similar, $r_{i}$ is an
odd prefix–reversal if $i\equiv 2,3\pmod{4}$. By [18, Lemma 6], the Pancake
graph $P_{n},n\geqslant 3$, has $n!/\ell$ independent even $\ell$–cycles where
$6\leqslant\ell\leqslant 2n$.

Let $\Gamma$ be one of $P_{5}$, $P_{6}$, $P_{7}$. Then the subgraph $H$
generated by the even prefix–reversals $r_{4}$ and $r_{5}$ is a spanning
subgraph of $\Gamma$ consisting of disjoint $10$-cycles $C_{10}=(r_{5}\
r_{4})^{5}$. Since even prefix–reversals preserve parity, all vertices of $H$
that are on the same cycle have the same parity. Since all other edges of
$\Gamma$ correspond to odd permutations, they have one endpoint on an ‘even’
$10$-cycle and the other endpoint on an ‘odd’ $10$-cycle. Thus, we can
$2$-color the even cycles and $2$-color the odd cycles using two other colors.
This results in an equitable $4$-coloring of $\Gamma$ where each of the color
classes has $n!/4$ vertices. Since $\chi(P_{6})=\chi(P_{7})=4$ this coloring
is optimal for $P_{6}$ and $P_{7}$.

##  4 Discussion

There are trivial examples of graphs for which Conjecture $1$ holds such as
even cycles, bipartite graphs with equal parts. Any Cayley graph over the
symmetric group generated by a set of transpositions gives a bipartite graph
with two equal color classes. The statement holds for the Hamming graphs
$H(d,q)$ whose (equitable) chromatic number is $q$. A regular graph with a
Hoffman coloring always gives a strongly equitable coloring [3]. Hoffman’s
lower bound is known as $\chi(\Gamma)\geqslant 1-\lambda_{1}/\lambda_{v}$,
where $\lambda_{1}$ and $\lambda_{v}$ are the largest and the smallest
eigenvalues of $\Gamma$. If equality holds, an optimal coloring of $\Gamma$ is
called a Hoffman coloring.

## Acknowledgements

The research work of the second author is supported by Mathematical Center in
Akademgorodok, the agreement with Ministry of Science and High Education of
the Russian Federation number 075-15-2019-1613. The authors thank Alexander
Kostochka and Sergey Avgustinovich for useful discussions on equitable
colorings.

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4*         4*1599      4*11     4*0.5       1                  0.973 $\pm$ 0.080      0.983 $\pm$ 0.090      0.929 $\pm$ 0.067      0.962 $\pm$ 0.079               0.936 $\pm$ 0.069               1.008 $\pm$ 0.162

2                  0.983 $\pm$ 0.082      0.990 $\pm$ 0.087      0.915 $\pm$ 0.063      0.974 $\pm$ 0.082               0.922 $\pm$ 0.067               1.081 $\pm$ 0.193

4                  0.999 $\pm$ 0.085      1.004 $\pm$ 0.090      0.915 $\pm$ 0.064               0.973 $\pm$ 0.081               0.908 $\pm$ 0.064      1.774 $\pm$ 0.468

8                  1.016 $\pm$ 0.093      1.023 $\pm$ 0.095      0.953 $\pm$ 0.075               0.988 $\pm$ 0.084               0.938 $\pm$ 0.072               0.927 $\pm$ 0.100

Average test NLL in nats on regression datasets (errors are $\pm 1$ standard error). Bold results indicate the best performance. Here, $N$ is the size of dataset, $D$ is the number of input dimensions, $\sigma^2_{\epsilon}$ is the noise variance, and Depth is the number of hidden layers of the MLP.

Data set     Classes     $N_{\text{train}}$     $N_{test}$     $D$     prior                 prior           prior             prior                    Deep Ensemble

2                    10980                  4000           14      0.404 $\pm$ 0.120     0.406 $\pm$ 0.129     0.179 $\pm$ 0.046     0.179 $\pm$ 0.075     0.150 $\pm$ 0.053     0.240 $\pm$ 0.097

2                    12898                  5000           8       0.071 $\pm$ 0.007     0.072 $\pm$ 0.007     0.066 $\pm$ 0.008     0.068 $\pm$ 0.007     0.066 $\pm$ 0.008     0.067 $\pm$ 0.008

2                    14020                  5000           10      0.316 $\pm$ 0.006     0.312 $\pm$ 0.005     0.286 $\pm$ 0.005     0.298 $\pm$ 0.004     0.284 $\pm$ 0.005     0.294 $\pm$ 0.005

2                    120064                 10000          50      0.218 $\pm$ 0.004     0.215 $\pm$ 0.004     0.179 $\pm$ 0.007     0.168 $\pm$ 0.004     0.165 $\pm$ 0.004     0.207 $\pm$ 0.004

26                   15000                  5000           16      0.445 $\pm$ 0.008     0.409 $\pm$ 0.008     0.166 $\pm$ 0.006     0.128 $\pm$ 0.005     0.115 $\pm$ 0.005     0.147 $\pm$ 0.006

11                   48509                  10000          48      0.098 $\pm$ 0.002     0.088 $\pm$ 0.002     0.028 $\pm$ 0.001     0.023 $\pm$ 0.001     0.022 $\pm$ 0.001     0.049 $\pm$ 0.002

5                    68095                  10000          37      0.060 $\pm$ 0.002     0.050 $\pm$ 0.002     0.032 $\pm$ 0.002     0.027 $\pm$ 0.001     0.021 $\pm$ 0.001     0.040 $\pm$ 0.002

Average test NLL in nats on classification datasets (errors are $\pm 1$ standard error). Bold results indicate the best performance. Here, Classes is the number of classes; the  $N_{\text{train}}$, $N_{test}$ is the sizes of training set and test set, respectively; $D$ is the number of input dimensions.

§.§ Additional results with full-batch Hamiltonian Monte Carlo

<ref> shows a comparison between full-batch HMC and SGHMC using the and our priors on small regression datasets.
We use the NUTS extension <cit.> of HMC with the NumPyro's implementation <cit.>.
NUTS adaptively sets the trajectory length of HMC, which along with the adaptation of the mass matrix and the step size.
We have simulated $4$ chains with a burn-in phase of $200$ iterations and $200$ collected samples for each chain.
We see that SGHMC performs remarkably similar to a carefully tuned HMC algorithm, despite the discretization error.

1c||     1c|     2c|prior     2cprior

Data set                   $\sigma^2_{\epsilon}$     1cHMC      1c|         1cHMC     1c
0.1                       3.065 $\pm$ 1.006                  3.093 $\pm$ 1.001                   2.821 $\pm$ 0.907                 2.835 $\pm$ 0.922

0.1                       5.369 $\pm$ 0.294                  5.488 $\pm$ 0.253                   4.715 $\pm$ 0.431                 4.801 $\pm$ 0.416

0.001                     0.386 $\pm$ 0.064                  0.389 $\pm$ 0.062                   0.339 $\pm$ 0.075                 0.343 $\pm$ 0.071

0.05                      3.931 $\pm$ 0.165                  4.008 $\pm$ 0.168                   3.438 $\pm$ 0.201                 3.723 $\pm$ 0.183

0.5                       0.637 $\pm$ 0.043                  0.641 $\pm$ 0.044                   0.606 $\pm$ 0.046                 0.609 $\pm$ 0.046

Average test RMSE results of full-batch HMC and on regression datasets (errors are $\pm 1$ standard error).
We use a MLP with two hidden layers of $100$ neurons.
$\sigma^2_{\epsilon}$ is the noise variance.

§.§ Additional discussion on the optimization of Wasserstein distance

In the <ref>, we have opted to separate the two optimization procedures for the Lipschitz function $\phi_{\mbtheta}$ and the Wasserstein distance.
We acknowledge that the two could have been optimized jointly in a single loop, as <ref> defines a minimax problem.
However, our choice allows $\phi_{\mbtheta}$ to be stabilized before a single Wasserstein minimization step takes place.
In fact, this is a common trick to make convergence more stable (see e.g., the original <cit.> paper, which suggests to allow more training of the discriminator for each step of the generator).
<ref> illustrates the convergence behavior of these two algorithmic choices measured by the squared MMD between the target GP prior and the optimized BNN prior on the regression datasets (see <ref> for the experimental protocol).
Our optimization strategy demonstrates a much more stable convergence compared to the joint optimization approach.

Comparison between strategies to optimize the Lipschitz function and the Wasserstein distance: ([baseline=-.65ex] [line width=3pt, draw=color2] plot (0, 0) –+(.5, 0);) our strategy of separating these two operations; and ([baseline=-.65ex] [line width=3pt, draw=color1] plot (0, 0) –+(.5, 0);) the strategy of joint optimization.
Here, the convergence is measured by the squared MMD between the target GP prior and the optimized BNN prior.

Convergence of Wasserstein optimization for two-layer MLP on the regression datasets.

Convergence of Wasserstein optimization for two-layer MLP on the classification datasets.

Convergence of Wasserstein optimization for CNN on the dataset.

§ A PRIMER ON WASSERSTEIN DISTANCE

Given two Borel's probability measures $\pi(\mbx)$ and $\nu(\mby)$ defined on the Polish space $\cX$ and $\cY$ (i.e. any complete separable metric space such as a subset of $\bbR^d$), the $p$-Wasserstein distance is defined as follows

\begin{align}
W_p(\pi, \nu) = \left(\inf_{\gamma \in \Gamma(\pi, \nu)} \int_{\cX\times\cY}D(\mbx,\mby)^p \gamma(\mbx,\mby) \d{\mbx}\d{\mby}  \right)^{1/p}\,,
\end{align}

where $D(\mbx,\mby)$ is a proper distance metric between two points $\mbx$ and $\mby$ in the space $\cX\times\cY$ and $\Gamma(\pi, \nu)$ is the set of functionals of all possible joint densities whose marginals are indeed $\pi$ and $\nu$.

When the space of $\mbx$ and $\mby$ coincides (i.e. $\mbx,\mby\in\cX\subseteq\bbR^d$), the most used formulation is the 1-Wasserstein distance with Euclidian norm as distance,

\begin{align}
\label{eq:wasserstein-1-append}
W(\pi, \nu) = \inf_{\gamma \in \Gamma(\pi, \nu)} \int_{\cX\times\cX}\|\mbx - \mby\| \gamma(\mbx,\mby) \d{\mbx}\d{\mby}\,,
\end{align}

This is also known in the literature as the Earth-Mover distance.
Intuitively, here $\gamma$ measures how much mass must be transported from $\mbx$ to $\mby$ in order to transform the distributions $\pi$ into the distribution $\nu$.
Solving the Wasserstein distance means computing the minimum mass that needs to be moved. The question “How?” is answered by looking at the optimal transport plan (not the focus of these notes).

The remaining part of these notes will be dedicated to the proof of the dual formulation for <ref>.
It is well known in the literature of optimization that linear programming problem with convex constrains admits a dual formulation.
Kantorovich introduced the dual formulation of the Wasserstein distance in 1942.

On the same setup as before, the Wasserstein distance defined as

\begin{align}
\label{eq:theo-wasserstein}
W(\pi, \nu) = \inf_{\gamma \in \Gamma(\pi, \nu)} \int_{\cX\times\cX}\|\mbx - \mby\| \gamma(\mbx,\mby) \d{\mbx}\d{\mby}\,,
\end{align}

admits the following dual form

\begin{align}
\label{eq:theo-wasserstein-dual-appendix}
W(\pi, \nu) = \sup_{\|f\|_L\leq1} \int_\cX f(\mbx)\pi(\mbx)d\mbx - \int_\cX f(\mby)\nu(\mby)d\mby
\end{align}

where $f$ is a 1-Lipschitz continuous function defined on $\cX\rightarrow\bbR$.

§.§ Step 1: Kantorovich duality

First of all we start with the Kantorovich duality, which defines a dual form for the generic 1-Wasserstein.

Given a nonnegative measurable function $D:\cX\times\cX\rightarrow\bbR$, the 1-Wasserstein is computed as follows,

\begin{align}
W(\pi, \nu) = \inf_{\gamma \in \Gamma(\pi, \nu)} \int D(\mbx,\mby) \gamma(\mbx,\mby) \d{\mbx}\d{\mby}  \,,
\end{align}

The Kantorovich duality proves that this is equal to the following constrained optimization problem,

\begin{align}
W(\pi, \nu) = \sup_{\substack{f,g \\f(\mbx)+g(\mby)\leq D(\mbx, \mby)}} \int f(\mbx)\pi(\mbx)\d{\mbx} + \int g(\mby)\nu(\mby)\d{\mby}\,.
\end{align}

We define $\iota_\Gamma(\gamma)$ the following quantity

\begin{align}
\iota_\Gamma(\gamma) & = \sup_{f,g} \left[ \int f(\mbx)\pi(\mbx)\d{\mbx} + \int g(\mby)\nu(\mby)\d{\mby} - \iint \left[f(\mbx)+g(\mby)\right]\gamma(\mbx, \mby)  \d{\mbx}\d{\mby} \right] \nonumber
\end{align}

and we observe that

\begin{align}
\iota_\Gamma(\gamma) = \left\{
\begin{aligned}
0 \quad       & \text{if}\;\gamma\in\Gamma(\pi, \nu)\,, \\
+\infty \quad & \text{otherwise}\,.
\end{aligned}
\right. \nonumber
\end{align}

This is true because given the definition of $\Gamma$, if $\gamma\in\Gamma(\pi, \nu)$ then $\pi(\mbx)=\int \gamma(\mbx,\mby)\d{\mby}$ and $\nu(\mby)=\int \gamma(\mbx,\mby)\d{\mbx}$.
By substituiting these quantities, it follows that

\begin{align}
\int f(\mbx)\pi(\mbx)\d{\mbx} + \int g(\mby)\nu(\mby)\d{\mby} & = \int f(\mbx)\int \gamma(\mbx,\mby)\d{\mby}\d{\mbx} + \int g(\mby)\int \gamma(\mbx,\mby)\d{\mbx}\d{\mby}  \nonumber \\
& = \iint \left[f(\mbx)+g(\mby)\right]\gamma(\mbx, \mby) \d{\mbx}\d{\mby} \,. \nonumber                                \end{align}

In other cases, $f$ and $g$ can be chosen such that the supremum becomes $+\infty$.
Given this property and the constrain on $\gamma$, we can add $\iota_\Gamma(\gamma)$ to the formulation of the Wasserstein distance in <ref>,

\begin{align}
& W(\pi, \nu)  = \inf_{\gamma \in \Gamma(\pi, \nu)} \left[\int D(\mbx, \mby) \gamma(\mbx,\mby) \d{\mbx}\d{\mby}\right] + \iota_\Gamma(\gamma) =                                                                                \nonumber \\
& = \inf_{\gamma} \left[\int D(\mbx, \mby) \gamma(\mbx,\mby) \d{\mbx}\d{\mby} + \sup_{f,g} \left[ \int f(\mbx)\pi(\mbx)\d{\mbx} + \int g(\mby)\nu(\mby)\d{\mby} - \right.\right.\\
&\qquad\qquad\left.\left. \iint \left[f(\mbx)+g(\mby)\right]\gamma(\mbx, \mby)  \d{\mbx}\d{\mby} \right]\right]\,, \nonumber
\end{align}

Now, the original integral of the Wasserstein distance does not depend on $f$ and $g$; therefore the supremum can be moved in front,

\begin{align}
W(\pi, \nu) &= \inf_{\gamma} \sup_{f,g} \Upsilon(\gamma, (f, g)) \\
\Upsilon(\gamma, (f, g)) &\defeq
\int D(\mbx, \mby) \gamma(\mbx,\mby) \d{\mbx}\d{\mby} +  \int f(\mbx)\pi(\mbx)\d{\mbx} + \int g(\mby)\nu(\mby)\d{\mby} - \nonumber\\
&\qquad \iint \left[f(\mbx)+g(\mby)\right]\gamma(\mbx, \mby)  \d{\mbx}\d{\mby} \nonumber
\end{align}

Under certain conditions stated by the minimax theorem, i.e. $\Upsilon(\gamma, (f, g))$ is convex-concave function ($\Upsilon$ is concave for fixed $(f,g)$ while convex for fixed $\gamma$ ), we can swap the infinum and the supremum and rewrite the definition as follows,

\begin{align}
W(\pi, \nu) = \sup_{f,g}{\inf_{\gamma} \int \left[ D(\mbx, \mby) - f(\mbx)-g(\mby)\right] \gamma(\mbx,\mby) \d{\mbx}\d{\mby}} +  \int f(\mbx)\pi(\mbx)\d{\mbx} + \int g(\mby)\nu(\mby)\d{\mby}\nonumber
\end{align}

Proofs that the hypothesis used for the minimax theorem hold for this case are presented in Theorem 1.9 of “Topics in Optimal Transport” <cit.>.
Focusing on the infimum part, we can write

\begin{align}
\inf_{\gamma} \int \left[ D(\mbx, \mby) - f(\mbx)-g(\mby)\right] \gamma(\mbx,\mby) \d{\mbx}\d{\mby} = \left\{
\begin{aligned}
0  \quad      & \text{if}\;{f(\mbx)+g(\mby)\leq  D(\mbx, \mby)} \,, \\
-\infty \quad & \text{otherwise}\,.
\end{aligned}
\right.\nonumber
\end{align}

If the function $\zeta(\mbx, \mby) =  D(\mbx, \mby) - (f(\mbx)+g(\mby))$ takes a negative value at some point $(\mbx_0, \mby_0)$, then by choosing $\gamma=\lambda\delta(\mbx_0, \mby_0)$ with $\lambda\rightarrow+\infty$ (i.e. a Dirac delta in $(\mbx_0, \mby_0)$), we see that the infimum is infinite.
On the other hand, is $\zeta(\mbx, \mby)$ is nonnegative, then the infimum is obtained for $\gamma=0$.
Finally, this constrains can be added to the previous conditions making thus recovering the formulation in <ref>.

§.§ Step 2: D-Transforms

The next challenge is to find $f$ and $g$ such that we can easily recover the constrain optimization above.
We approach this problem by supposing to have chosen some $f(\mbx)$. This means that the objective is to find a good $g(\mby)$ that for all $\mbx,\mby$ satisfy the condition

\begin{align}
f(\mbx) + g(\mby) \leq D(\mbx, \mby)\,.\nonumber
\end{align}

The trivial solution is $g(\mby) \leq D(\mbx, \mby) - f(\mbx)$.
This must be true for all $\mbx$, also in the worst case (when we take the infimum),

\begin{align}
g(\mby) \leq \inf_\mbx \left[D(\mbx, \mby) - f(\mbx)\right].\nonumber
\end{align}

At this point, we observe that for a given $f$, if we want the supremum in Eq. 5 we cannot get a better $g$ then taking the equality,

\begin{align}
\bar f(\mby) := \inf_\mbx \left[D(\mbx, \mby) - f(\mbx)\right].\nonumber
\end{align}

We therefore have the following formulation of the Wasserstein distance,

\begin{align}
W(\pi, \nu) = \sup_{f}\left[\int f(\mbx)\pi(\mbx)\d{\mbx} + \int \bar f(\mby)\nu(\mby)\d{\mby} \right]\nonumber
\end{align}

If now we suppose to choose $g$, by following the same reasoning the best $f$ that we can get is defined

\begin{align}
\bar{\bar f}(\mbx) = \bar g(\mbx) := \inf_\mby \left[D(\mbx, \mby) - g(\mby)\right].\nonumber
\end{align}

If we replace $g(\mby)$ with Eq. 17 we have yet another recursive definition of the Wasserstein distance,

\begin{align}
W(\pi, \nu) = \sup_{f}\left[\int \bar{\bar f}(\mbx)\pi(\mbx)\d{\mbx} + \int \bar f(\mby)\nu(\mby)\d{\mby} \right]\nonumber
\end{align}

If we constrain $f$ to be $D$-concave, then $\bar{\bar f} = f$.

§.§ Step 2.1: Euclidean distance

It's worth mentioning that this formulation is valid for any nonnegative measurable function $D$.
For the Euclidian distance this simplify even further.

When $D(\mbx, \mby) = \|\mbx - \mby\|$ and $f$ is 1-Lipschitz, $f$ is $D$-concave if and only if $\bar f = -f$

We prove the necessity condition of such result.
First of all, we observe that if $f$ is 1-Lipschitz then $\bar f$ is 1-Lipschitz too. This is true because for any given $\mbx$

\begin{align}
\bar f_\mbx(\mby) = \| \mbx - \mby \| - f(\mbx)\nonumber
\end{align}

is 1-Lipschitz and therefore the infimum of $\bar f(\mby) = \inf_{\mbx} \| \mbx - \mby \| - f(\mbx)$ is 1-Lipschitz.
Since $\bar f$ is 1-Lipschitz, for all $\mbx$ and $\mby$ we have

\begin{align}
& \left|\bar f(\mby) - \bar f(\mbx)\right| \leq \|\mby - \mbx\|                   \nonumber \\
\implies  - & \bar f(\mbx) \leq \|\mbx-\mby\| - \bar f(\mby)\nonumber
\end{align}

Since this is true for all $\mby$,

\begin{align}
& - \bar f(\mbx) \leq \inf_\mby\|\mbx-\mby\| - \bar f(\mby)\,                                                     \nonumber  \\
& - \bar f(\mbx) \leq \underbrace{\inf_\mby\|\mbx-\mby\| - \bar f(\mby)}_{\bar{\bar f}\equiv f} \leq - \bar f(\mbx)\nonumber
\end{align}

where the right inequality follows by choosing $\mby = \mbx$ in the infimum.
We know that $\bar{\bar f}\equiv f$. This means that $-\bar f(\mbx)$ must be equal to $f(\mbx)$ for the last equation to hold.

§.§ Step 3. Putting everything together

We started our discussion by proving the Kantovich duality, which states that

\begin{align}
\inf_{\gamma \in \Gamma(\pi, \nu)} \int D(\mbx,\mby) \gamma(\mbx,\mby) \d{\mbx}\d{\mby} = \sup_{\substack{f,g \\f(\mbx)+g(\mby)\leq D(\mbx, \mby)}} \int f(\mbx)\pi(\mbx)\d{\mbx} + \int g(\mby)\nu(\mby)\d{\mby}\,,\nonumber
\end{align}

We then proved that

\begin{align*}
\sup_{\substack{f,g \\f(\mbx)+g(\mby)\leq D(\mbx, \mby)}} \left[\int f(\mbx)\pi(\mbx)\d{\mbx} + \int g(\mby)\nu(\mby)\d{\mby}\right] = \nonumber\\
= \sup_{\substack{f   \\\bar f= \inf_\mbx D - f}}\left[\int f(\mbx)\pi(\mbx)\d{\mbx} + \int \bar f(\mby)\nu(\mby)\d{\mby} \right] \,,\nonumber
\end{align*}

Finally, given $D(\mbx, \mby)$ to be the Euclidean distance, we discussed the shape of $\bar f$ when we restrict $f$ to be 1-Lipschitz, showing that $\bar f = -f$.
Putting everything together, we obtain the dual 1-Wasserstein distance in <ref>,

\begin{align}
W(\pi, \nu) = \sup_{\|f\|_L\leq1} \int f(\mbx)\pi(\mbx)d\mbx - \int f(\mby)\nu(\mby)d\mby\nonumber
\end{align}