# Study on the electromagnetic properties of the $[sc][\bar{q}\bar{b}]$ and
$[sc][\bar{s}\bar{b}]$ states with $J^{P}=1^{+}$

Ulaş Özdem  0000-0002-1907-2894<EMAIL_ADDRESS>Health Services
Vocational School of Higher Education, Istanbul Aydin University, Sefakoy-
Kucukcekmece, 34295 Istanbul, Türkiye

###### Abstract

A systematic study of the electromagnetic properties of exotic states is
conducted to elucidate their nature, which continues to be the subject of
controversy and incomplete understanding in the field. In this study, the
magnetic dipole and quadrupole moments of the tetraquarks
$[sc][\bar{q}\bar{b}]$ and $[sc][\bar{s}\bar{b}]$ with spin-parity quantum
numbers $J^{P}=1^{+}$ are extracted in the context of a compact diquark-
antidiquark configuration with the help of the QCD sum rules method. The
magnetic dipole moments are given as
$\mu_{[sc][\bar{u}\bar{b}]}=-2.12^{+0.74}_{-0.59}\mu_{N}$,
$\mu_{[sc][\bar{d}\bar{b}]}=1.66^{+0.60}_{-0.46}\mu_{N}$, and
$\mu_{[sc][\bar{s}\bar{b}]}=2.01^{+0.61}_{-0.50}\mu_{N}$. The order of
magnitude of the magnetic dipole moments would suggest that these outcomes may
be achievable in forthcoming experiments. The magnetic and quadrupole moments
of hadrons represent another important observable, along with their mass and
decay width, which contribute to our understanding of the underlying quark
structure and dynamics. Therefore, we hope that the results of this study will
prove useful in theoretical and experimental investigations, which we
anticipate will be an interesting research topic.

##  I motivation

While the idea of hadrons with more intricate structures than mesons with
$q\bar{q}$ and baryons with $qqq$ has been known for a long time, the
confirmation of the existence of the exotic state designated X(3872) employing
experimental evidence was achieved by the Belle Collaboration in 2003 [1].
Since that time, numerous exotic hadron candidates have been observed by
different collaborations. With each new observation, the family of such exotic
states is expanding and enriching. The experimental discovery of these exotic
states represents a dynamic and vibrant field within hadron physics,
encompassing both experimental and theoretical methodologies. The most recent
developments in the field of exotic states, as well as experimental and
theoretical works, are listed in Refs. [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17]. Nevertheless, it is crucial to acknowledge that despite
the extensive theoretical and experimental investigations conducted, the
internal structure, decay characteristics, and quantum numbers of these exotic
states remain among the pivotal questions that remain unanswered.

A substantial majority of the experimental explorations of exotic states to
date have yielded either hidden-charm/bottom or doubly-charm outcomes. Another
class of exotic states, tetraquarks involving heavy $c\bar{b}$ diquarks, is
likely to exist. It is important to note that these states have not been
discovered experimentally. However, there are theoretical studies in the
literature that investigate their existence and study their properties [18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. One of the primary reasons for this
phenomenon is that tetraquarks constructed from $c\bar{b}$ may exhibit
stability against strong and electromagnetic decays. Due to these properties,
they are highly valuable for the study of heavy-quark dynamics and the
understanding of the dynamics of the QCD at a deeper level. Further research
is required from a range of theoretical viewpoints to gain comprehensive
insight into the properties of tetraquarks with $c\bar{b}$, which will help
future experimental studies. In addition to the mass and decay properties of
these states, it is crucial to elucidate the substructure and geometric shape
of tetraquarks with $c\bar{b}$. It would be advantageous to examine their
electromagnetic and radiative transition properties to gain a more profound
comprehension of their intrinsic nature. The literature contains several
studies on the electromagnetic properties of tetraquarks, which are designed
to elucidate their internal structure [29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50].

The purpose of this study is to systematically extract magnetic dipole and
quadrupole moments of the $[sc][\bar{q}\bar{b}]$ and $[sc][\bar{s}\bar{b}]$
tetraquarks. To this end, a compact diquark-antiquark interpolating current
with $J^{P}=1^{+}$ spin-parity quantum numbers is constructed. Magnetic dipole
and quadrupole moments are parameters of the nonperturbative domain of QCD.
Therefore, to extract these parameters a reliable method, such as the QCD
light cone sum rules (LCSR) method, is required. The LCSR method yields highly
efficacious and predictable outcomes and represents a robust non-perturbative
approach for investigating the dynamic and static attributes of conventional
and exotic hadrons. The LCSR method entails calculating the correlation
function in two distinct representations: one about hadrons (referred to as
the "hadronic part") and the other to quark-gluon degrees of freedom (referred
to as the "QCD part"). These two representations are then matched and sum
rules are derived for the respective physical quantities.

This paper is organized as follows. After Introduction, the formalism of QCD
light-cone sum rules for the multipole moments of $[sc][\bar{q}\bar{b}]$ and
$[sc][\bar{s}\bar{b}]$ tetraquarks with $J^{P}=1^{+}$ are briefly introduced
in Sec. II. We present the numerical results and discussions for these states
in Sec. III. The final section is dedicated to our concluding remarks.

##  II Deriving the QCD light-cone sum rules for multipole moments

The multipole moments of the $[sc][\bar{q}\bar{b}]$ and $[sc][\bar{s}\bar{b}]$
($Z_{c\bar{b}}$ for short) tetraquarks are obtained by the LCSR technique
through the construction of a correlation function expressed as follows:

$\Pi_{\alpha\beta}(p,q)=i\int d^{4}x\,e^{ip\cdot x}\langle
0|\mathcal{T}\\{J_{\alpha}(x)J_{\beta}^{\dagger}(0)\\}|0\rangle_{\gamma},$ (1)

where $J_{\alpha(\beta)}(x)$ is the interpolating current and the subindex
$\gamma$ represents the external electromagnetic field. The interpolating
current for the considered states is given by $J_{\alpha(\beta)}(x)$ with
quantum numbers $J^{P}=1^{+}$, which is expressed as follows:

$\displaystyle J_{\alpha}^{1}(x)$
$\displaystyle=\big{[}s^{a^{T}}(x)C\gamma_{5}c^{b}(x)\big{]}\big{[}\bar{q}^{a}(x)\gamma_{\alpha}C\bar{b}^{b^{T}}(x)+\bar{q}^{b}(x)\gamma_{\alpha}C\bar{b}^{a^{T}}(x)\big{]},$
(2)

where $a$ and $b$ are color indices, $C$ is the charge conjugation operator
and; $q(x)$ denotes the $u(x)$, $d(x)$ or $s(x)$ quarks.

At the hadronic level, we plug into the correlation function a complete set of
intermediate hadronic states with the same quantum numbers as the
interpolating current. Inserting the intermediate states, and isolating the
ground state contributions yield the following result:

$\displaystyle\Pi_{\alpha\beta}^{Had}(p,q)$ $\displaystyle={\frac{\langle
0\mid J_{\alpha}(x)\mid
Z_{c\bar{b}}(p)\rangle}{p^{2}-m_{Z_{c\bar{b}}}^{2}}}\langle
Z_{c\bar{b}}(p)\mid Z_{c\bar{b}}(p+q)\rangle_{\gamma}\frac{\langle
Z_{c\bar{b}}(p+q)\mid J_{\beta}^{\dagger}(0)\mid
0\rangle}{(p+q)^{2}-m_{Z_{c\bar{b}}}^{2}}+\mbox{higher states},$ (3)

The $\langle 0\mid J_{\alpha}(x)\mid Z_{c\bar{b}}(p)\rangle$ in Eq. (3) is
expressed regarding hadron characters, as indicated below

$\displaystyle\langle 0\mid J_{\alpha}(x)\mid
Z_{c\bar{b}}(p)\rangle=\lambda_{Z_{c\bar{b}}}\varepsilon_{\alpha}^{\theta}\,,$
(4)

with $\lambda_{Z_{c\bar{b}}}$ and $\varepsilon_{\alpha}^{\theta}\ $ are the
current coupling and the polarization vector of particles under question,
respectively. In Eq. (3), there is another matrix element, known as the
radiative transition matrix element, which is expressed as [51]

$\displaystyle\langle Z_{c\bar{b}}(p,\varepsilon^{\theta})\mid
Z_{c\bar{b}}(p+q,\varepsilon^{\delta})\rangle_{\gamma}$
$\displaystyle=-\varepsilon^{\tau}(\varepsilon^{\theta})^{\mu}(\varepsilon^{\delta})^{\nu}\Bigg{[}G_{1}(Q^{2})~{}(2p+q)_{\tau}~{}g_{\mu\nu}+G_{2}(Q^{2})~{}(g_{\tau\nu}~{}q_{\mu}-g_{\tau\mu}~{}q_{\nu})$
$\displaystyle-\frac{1}{2m_{Z_{c\bar{b}}}^{2}}G_{3}(Q^{2})~{}(2p+q)_{\tau}q_{\mu}q_{\nu}\Bigg{]},$
(5)

where $G_{i}(Q^{2})$’s being form factors of the corresponding transition,
with $Q^{2}=-q^{2}$. The correlation function is derived from Eqs. (3)-(II) in
the following manner:

$\displaystyle\Pi_{\alpha\beta}^{Had}(p,q)$
$\displaystyle=\frac{\varepsilon_{\rho}\,\lambda_{Z_{c\bar{b}}}^{2}}{[m_{Z_{c\bar{b}}}^{2}-(p+q)^{2}][m_{Z_{c\bar{b}}}^{2}-p^{2}]}\bigg{\\{}G_{1}(Q^{2})(2p+q)_{\rho}\bigg{[}g_{\alpha\beta}-\frac{p_{\alpha}p_{\beta}}{m_{Z_{c\bar{b}}}^{2}}-\frac{(p+q)_{\alpha}(p+q)_{\beta}}{m_{Z_{c\bar{b}}}^{2}}+\frac{(p+q)_{\alpha}p_{\beta}}{2m_{Z_{c\bar{b}}}^{4}}$
$\displaystyle\times(Q^{2}+2m_{Z_{c\bar{b}}}^{2})\bigg{]}+G_{2}(Q^{2})\bigg{[}q_{\alpha}g_{\rho\beta}-q_{\beta}g_{\rho\alpha}-\frac{p_{\beta}}{m_{Z_{c\bar{b}}}^{2}}\big{(}q_{\alpha}p_{\rho}-\frac{1}{2}Q^{2}g_{\alpha\rho}\big{)}+\frac{(p+q)_{\alpha}}{m_{Z_{c\bar{b}}}^{2}}\big{(}q_{\beta}(p+q)_{\rho}+\frac{1}{2}Q^{2}g_{\beta\rho}\big{)}$
$\displaystyle-\frac{(p+q)_{\alpha}p_{\beta}p_{\rho}}{m_{Z_{c\bar{b}}}^{4}}\,Q^{2}\bigg{]}-\frac{G_{3}(Q^{2})}{m_{Z_{c\bar{b}}}^{2}}(2p+q)_{\rho}\bigg{[}q_{\alpha}q_{\beta}-\frac{p_{\alpha}q_{\beta}}{2m_{Z_{c\bar{b}}}^{2}}Q^{2}+\frac{(p+q)_{\alpha}q_{\beta}}{2m_{Z_{c\bar{b}}}^{2}}Q^{2}-\frac{(p+q)_{\alpha}q_{\beta}}{4m_{Z_{c\bar{b}}}^{4}}Q^{4}\bigg{]}\bigg{\\}}\,.$
(6)

The magnetic ($F_{M}(Q^{2})$) and quadrupole ($F_{\mathcal{D}}(Q^{2})$) form
factors can be derived in the form of the previously defined form factors,
$G_{i}(Q^{2})$, as follows:

$\displaystyle F_{M}(Q^{2})=G_{2}(Q^{2})\,,$ $\displaystyle
F_{\mathcal{D}}(Q^{2})=G_{1}(Q^{2})-G_{2}(Q^{2})+\Big{(}1+\frac{Q^{2}}{4m_{\mathrm{Z_{c\bar{b}}}}^{2}}\Big{)}G_{3}(Q^{2})\,.$
(7)

In the static limit ($Q^{2}=0$), where we are dealing with the real photon,
the $F_{M}(0)$ and $F_{\mathcal{D}}(0)$ can be described as the magnetic
moment ($\mu$) and quadrupole $(\mathcal{D})$ moments by the following
relationship:

$\displaystyle\mu=\frac{e}{2m_{Z_{c\bar{b}}}}\,F_{M}(0),~{}~{}~{}~{}\mathcal{D}=\frac{e}{m^{2}_{Z_{c\bar{b}}}}\,F_{\mathcal{D}}(0).$
(8)

The results of the analysis at the hadron level are extracted in the form of a
representation of the physical parameters under study. To obtain a
representation of the analysis at the quark-gluon level, it is necessary to
proceed with further calculations.

At the quark-gluon level, the correlation function can be evaluated by the
operator product expansion (OPE) technique and expressed in terms of photon
distribution amplitudes. Upon completion of elementary mathematical
operations, the following outcome for the $Z_{c\bar{b}}$ states is derived:

$\displaystyle\Pi_{\alpha\beta}^{\mathrm{QCD}}(p,q)$ $\displaystyle=i\int
d^{4}xe^{ip\cdot x}\langle
0\mid\bigg{\\{}\mathrm{Tr}\Big{[}\gamma_{\alpha}\widetilde{S}_{b}^{b^{\prime}b}(-x)\gamma_{\beta}S_{q}^{a^{\prime}a}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{q}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)\Big{]}$
$\displaystyle+\mathrm{Tr}\Big{[}\gamma_{\alpha}\widetilde{S}_{b}^{a^{\prime}b}(-x)\gamma_{\beta}S_{q}^{b^{\prime}a}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{q}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)\Big{]}$
$\displaystyle+\mathrm{Tr}\Big{[}\gamma_{\alpha}\widetilde{S}_{b}^{b^{\prime}a}(-x)\gamma_{\beta}S_{q}^{a^{\prime}b}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{q}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)\Big{]}$
$\displaystyle+\mathrm{Tr}\Big{[}\gamma_{\alpha}\widetilde{S}_{b}^{a^{\prime}a}(-x)\gamma_{\beta}S_{q}^{b^{\prime}b}(-x)\Big{]}\mathrm{Tr}\Big{[}\gamma_{5}\widetilde{S}_{q}^{aa^{\prime}}(x)\gamma_{5}S_{c}^{bb^{\prime}}(x)\Big{]}\bigg{\\}}\mid
0\rangle_{\gamma},$ (9)

where the $S_{Q}(x)$ and $S_{q}(x)$ are the propagators of heavy and light
quarks with $\widetilde{S}_{Q(q)}^{ij}(x)=CS_{Q(q)}^{ij\rm{T}}(x)C$. The
explicit forms of these propagators are defined as follows: [52, 53],

$\displaystyle S_{q}(x)$
$\displaystyle=S_{q}^{free}(x)-\frac{\langle\bar{q}q\rangle}{12}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{4}\Big{)}-\frac{\langle\bar{q}q\rangle}{192}m_{0}^{2}x^{2}\Big{(}1-i\frac{m_{q}x\\!\\!\\!/}{6}\Big{)}+\frac{ig_{s}~{}G^{\alpha\beta}(x)}{32\pi^{2}x^{2}}\bigg{[}\hbox
to0.0pt{/\hss}{x}\sigma_{\alpha\beta}+\sigma_{\alpha\beta}\hbox
to0.0pt{/\hss}{x}\bigg{]},$ (10) $\displaystyle S_{Q}(x)$
$\displaystyle=S_{Q}^{free}(x)-\frac{m_{Q}\,g_{s}\,G^{\alpha\beta}(x)}{32\pi^{2}}\bigg{[}(\sigma_{\alpha\beta}{x\\!\\!\\!/}+{x\\!\\!\\!/}\sigma_{\alpha\beta})\frac{K_{1}\big{(}m_{Q}\sqrt{-x^{2}}\big{)}}{\sqrt{-x^{2}}}+2\sigma_{\alpha\beta}K_{0}\big{(}m_{Q}\sqrt{-x^{2}}\big{)}\bigg{]},$
(11)

with

$\displaystyle S_{q}^{free}(x)$ $\displaystyle=\frac{1}{2\pi
x^{2}}\Big{(}i\frac{x\\!\\!\\!/}{x^{2}}-\frac{m_{q}}{2}\Big{)},$ (12)
$\displaystyle S_{Q}^{free}(x)$
$\displaystyle=\frac{m_{Q}^{2}}{4\pi^{2}}\bigg{[}\frac{K_{1}\big{(}m_{Q}\sqrt{-x^{2}}\big{)}}{\sqrt{-x^{2}}}+i\frac{{x\\!\\!\\!/}~{}K_{2}\big{(}m_{Q}\sqrt{-x^{2}}\big{)}}{(\sqrt{-x^{2}})^{2}}\bigg{]},$
(13)

where $m_{0}$ being the quark-gluon mixed condensate with $m_{0}^{2}=\langle
0\mid\bar{q}\,g_{s}\,\sigma_{\alpha\beta}\,G^{\alpha\beta}\,q\mid
0\rangle/\langle\bar{q}q\rangle$, $G^{\alpha\beta}$ is the gluon field-
strength tensor, and $K_{i}$’s are the modified Bessel functions of the second
type.

To perform further calculations at the quark-gluon level, it is necessary to
consider two distinct contributions from the short- and long-distance
interactions of the photon with quarks. To ascertain the nature of the short-
distance interactions, it is necessary to proceed with the subsequent
replacement as described below:

$\displaystyle S^{free}(x)\longrightarrow\int d^{4}z\,S^{free}(x-z)\,\hbox
to0.0pt{/\hss}{\\!A}(z)\,S^{free}(z)\,.$ (14)

To include long-distance interactions in the analysis, it is convenient to
apply the formula below:

$\displaystyle
S_{\alpha\beta}^{ab}(x)\longrightarrow-\frac{1}{4}\big{[}\bar{q}^{a}(x)\Gamma_{i}q^{b}(0)\big{]}\big{(}\Gamma_{i}\big{)}_{\alpha\beta},$
(15)

where $\Gamma_{i}=\\{\textbf{1}$, $\gamma_{5}$, $\gamma_{\alpha}$,
$i\gamma_{5}\gamma_{\alpha}$, $\sigma_{\alpha\beta}/2\\}$. In calculating
short- and long-distance contributions, we employ a single propagator in the
equation, whereas the remaining propagators are treated as full propagators.
Upon the inclusion of long-distance contributions in the analysis, matrix
elements such as
$\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}G_{\alpha\beta}q(0)\right|0\rangle$
and $\langle\gamma(q)\left|\bar{q}(x)\Gamma_{i}q(0)\right|0\rangle$ emerge.
The aforementioned matrix elements, expressed in the form of photon wave
functions, are crucial parameters in the calculation of long-distance
interactions (see Ref. [54] for details on photon distribution amplitudes
(DAs)). It is important to note that the photon DAs utilized in this study
include contributions solely from light quarks. However, in principle, a
photon can be emitted long-distance from heavy quarks. However, such long-
distance photon emission from heavy quarks is highly suppressed owing to the
large mass of the heavy quarks. This kind of contribution is neglected in our
analysis. As explained in Eq. (14), only the short-distance photon emission
from heavy quarks is considered. For this reason, DAs including heavy quarks
are not considered in our analysis. The aforementioned procedures have yielded
the quark-gluon correlation function, which has been obtained through the use
of quark-gluon parameters and photon DAs.

The above-mentioned procedures give the sum rules for the magnetic and
quadrupole moments of the $[sc][\bar{q}\bar{b}]$ and $[sc][\bar{s}\bar{b}]$
tetraquarks as follows,

$\displaystyle\mu_{Z_{c\bar{b}}}\,$
$\displaystyle=\frac{e^{\frac{m_{Z_{c\bar{b}}}^{2}}{\rm{M^{2}}}}}{\lambda^{2}_{Z_{c\bar{b}}}}\,\,\rho_{1}(\rm{M^{2}},\rm{s_{0}}),~{}~{}~{}~{}~{}~{}\mathcal{D}_{\mathrm{Z_{c\bar{b}}}}=m_{\mathrm{Z_{c\bar{b}}}}^{2}\frac{e^{\frac{m_{Z_{c\bar{b}}}^{2}}{\rm{M^{2}}}}}{\lambda^{2}_{Z_{c\bar{b}}}}\,\,\rho_{2}(\rm{M^{2}},\rm{s_{0}}),$
(16)

As the explicit forms of $\rho_{i}(\rm{M^{2}},\rm{s_{0}})$ are similar, for
illustrative purposes, the result of $\rho_{1}(\rm{M^{2}},\rm{s_{0}})$ is
provided below

$\displaystyle\rho_{1}(\rm{M^{2}},\rm{s_{0}})$
$\displaystyle=-\frac{1}{2^{19}\times 3^{2}\times 5^{2}\times
7\pi^{5}}\Bigg{[}9e_{q}\Big{(}-4I[0,6]+42m_{c}m_{s}\big{(}I[0,5]+15I[1,4]\big{)}-129I[1,5]\Big{)}+14e_{b}\Big{(}38m_{c}m_{s}I[0,5]$
$\displaystyle-6I[0,6]+130m_{c}m_{s}I[1,4]-27I[1,5]\Big{)}\Bigg{]}$
$\displaystyle+\frac{\langle
g_{s}^{2}G^{2}\rangle\langle\bar{q}q\rangle}{2^{21}\times
3^{4}\pi^{3}}\Bigg{[}160(e_{c}-e_{s})m_{b}\big{(}I[0,2]-2I[1,1]\big{)}-43e_{q}m_{c}(I_{1}[\mathcal{S}]+I_{1}[\mathcal{\tilde{S}}])I[0,2]\Bigg{]}$
$\displaystyle-\frac{\langle g_{s}^{2}G^{2}\rangle}{2^{22}\times 3^{5}\times
5\pi^{5}}\Bigg{[}5(e_{c}-e_{s})\big{(}21I[0,4]-172I[1,3]\big{)}-162e_{q}\big{(}I[0,4]-4I[1,3]\big{)}-12e_{b}\big{(}3I[0,4]+32I[1,3]\big{)}\Bigg{]}$
$\displaystyle+\frac{\langle\bar{q}q\rangle}{2^{16}\times 3^{3}\times
5\pi^{3}}\Bigg{[}48e_{b}m_{b}\big{(}20m_{c}m_{s}I[0,3]-3I[0,4]\big{)}+e_{c}m_{c}\Big{(}(60m_{c}m_{s}I[0,3]-9I[0,4])(I_{4}[\mathcal{S}]-I_{4}[\mathcal{\tilde{S}}])$
$\displaystyle-4(44m_{c}m_{s}I[0,3]-9I[0,4])I_{5}[h_{\gamma}]\Big{)}\Bigg{]}$
$\displaystyle-\frac{\langle\bar{s}s\rangle}{2^{18}\times 3^{3}\times
5\pi^{3}}\Bigg{[}16\Big{(}e_{b}m_{c}(-6I[0,4]+66m_{0}^{2}I[1,2]-64I[1,3])+27e_{q}m_{c}(I[0,4]-4I[1,3])$
$\displaystyle+9e_{b}m_{s}(I[0,4]-4I[1,3])\Big{)}+e_{s}m_{c}(I_{2}[\mathcal{S}]+44I_{3}[\mathcal{S}])I[0,4]\Bigg{]}$
$\displaystyle+\frac{f_{3\gamma}}{2^{22}\times 3^{3}\times 5^{2}\times
7\pi^{3}}\Bigg{[}112e_{q}(55m_{c}m_{s}I[0,4]-9I[0,5])I_{1}[\mathcal{V}]+2511e_{s}I_{2}[\mathcal{V}]I[0,5]\Bigg{]},$
(17)

where the $I[n,m]$, and $I_{i}[\mathcal{F}]$ functions are expressed as:

$\displaystyle I[n,m]$
$\displaystyle=\int_{\mathcal{M}}^{\rm{s_{0}}}ds~{}e^{-s/\rm{M^{2}}}~{}s^{n}\,(s-\mathcal{M})^{m},$
$\displaystyle I_{1}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{2}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta^{\prime}(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{3}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{q}+\bar{v}\alpha_{g}-u_{0}),$
$\displaystyle I_{4}[\mathcal{F}]$ $\displaystyle=\int
D_{\alpha_{i}}\int_{0}^{1}dv~{}\mathcal{F}(\alpha_{\bar{q}},\alpha_{q},\alpha_{g})\delta(\alpha_{\bar{q}}+v\alpha_{g}-u_{0}),$
$\displaystyle I_{5}[\mathcal{F}]$
$\displaystyle=\int_{0}^{1}du~{}\mathcal{F}(u),$

where $\mathcal{M}=(m_{c}+m_{b}+m_{s})^{2}$ for the $[sc][\bar{q}\bar{b}]$
states and $\mathcal{M}=(m_{c}+m_{b}+2m_{s})^{2}$ for the
$[sc][\bar{s}\bar{b}]$ state; and $\mathcal{F}$ denotes the relevant DAs of
the photon. Analytical expressions for the multipole moments of the
$[sc][\bar{q}\bar{b}]$ and $[sc][\bar{s}\bar{b}]$ tetraquarks with
$J^{P}=1^{+}$ have been derived. The subsequent section will present a
numerical analysis of these quantities.

##  III Numerical analysis of the multipole moments

To conduct numerical calculations of QCD sum rules for multipole moments,
several input quantities are required, as given in Table 1. In numerical
analysis, we set $m_{u}$ =$m_{d}$ = 0 and $m^{2}_{s}=0$, but consider terms
proportional to $m_{s}$. Another crucial input parameter in the numerical
analysis is the photon DAs and the wave functions employed therein. These
expressions and the input parameters used in their explicit forms are taken
from Ref. [54].

Table 1: Parameters employed as inputs in the calculations [55, 56, 57, 24, 25]. Parameter |  Value |  Unit
---|---|---
$m_{s}$ |  $93.4^{+8.6}_{-3.4}$ |  MeV
$m_{c}$ |  $1.67\pm 0.07$ |  GeV
$m_{b}$ |  $4.78\pm 0.06$ |  GeV
$m_{[sc][\bar{s}\bar{b}]}$ |  $7.30\pm 0.77$ |  GeV
$m_{[sc][\bar{q}\bar{b}]}$ |  $7.18\pm 0.76$ |  GeV
$m_{0}^{2}$ |  $0.8\pm 0.1$ |  GeV2
$f_{3\gamma}$ |  $-0.0039$ |  GeV2
$\langle\bar{q}q\rangle$ |  $(-0.24\pm 0.01)^{3}$ |  GeV3
$\langle\bar{s}s\rangle$ |  $0.8$ $\langle\bar{q}q\rangle$ |  GeV3
$\langle g_{s}^{2}G^{2}\rangle$ |  $0.48\pm 0.14$ |  GeV4
$\lambda_{[sc][\bar{q}\bar{b}]}$ |  $0.035\pm 0.011$ |  GeV5
$\lambda_{[sc][\bar{s}\bar{b}]}$ |  $0.046\pm 0.014$ |  GeV5

Besides the aforementioned input parameters, two further parameters are
required for the calculations: the continuum threshold parameter $\rm{s_{0}}$
and the Borel mass $\rm{M^{2}}$. To acquire reliable results from QCD sum
rules, the dependence of the multipole moments on $\rm{s_{0}}$ and
$\rm{M^{2}}$ should be relatively weak, which is called working windows. The
working windows of these parameters are determined from the standard
procedures of the method used. These procedures are known as pole dominance
(PC) and convergence of the OPE (CVG), which are described by the following
formulas:

PC
$\displaystyle=\frac{\rho_{i}(\rm{M^{2}},\rm{s_{0}})}{\rho_{i}(\rm{M^{2}},\infty)},~{}~{}~{}~{}~{}~{}~{}~{}\mbox{CVG}(\rm{M^{2}},\rm{s_{0}})=\frac{\rho_{i}^{\rm{Dim7}}(\rm{M^{2}},\rm{s_{0}})}{\rho_{i}(\rm{M^{2}},\rm{s_{0}})},$
(18)

where $\rho_{i}^{\rm{Dim7}}(\rm{M^{2}},\rm{s_{0}})$ stands for the highest
dimensional term in the OPE of $\rho_{i}(\rm{M^{2}},\rm{s_{0}})$. According to
the sum rules analysis, the CVG must be sufficiently small to guarantee OPE
convergence, while the PC must be sufficiently large to maximize the
efficiency of the single-pole scheme. Under the aforementioned prerequisites,
one can obtain the working windows of the $\rm{M^{2}}$ and $\rm{s_{0}}$, which
are presented in Table 2. As a result, in the same table, the values achieved
for the PC and CVG are also listed. For completeness, Figs. 1 and 2 illustrate
the variations of the extracted multipole moments of these states concerning
$\rm{M^{2}}$ and $\rm{s_{0}}$. As illustrated by the presented figures, the
multipole moments of these states show a relatively mild dependence on the
parameter $\rm{M^{2}}$. Although the multipole moments of these states exhibit
some dependence on $\rm{s_{0}}$, they remain within the limits permitted by
this approach and constitute the primary source of uncertainty.

Predicted multipole moments of the $[sc][\bar{q}\bar{b}]$ and
$[sc][\bar{s}\bar{b}]$ states with $J^{P}=1^{+}$, which consider the
uncertainties inherent in the input parameters and the variation in the
$\rm{M^{2}}$ and $\rm{s_{0}}$ working windows, are presented in Table 2.

Table 2: Predicted multipole moments of the $[sc][\bar{q}\bar{b}]$ and $[sc][\bar{s}\bar{b}]$ states with $J^{P}=1^{+}$. Tetraquarks |  $\mu$ [$\rm{\mu_{N}}$] |  $\mathcal{D}(\times 10^{-2})$ [$\rm{fm^{2}}$] |  $\rm{M^{2}}\,\,[\rm{GeV}^{2}]$ |  $\rm{s_{0}}\,\,[\rm{GeV}^{2}]$ |  PC [$\%$] |  CVG [$\%$]
---|---|---|---|---|---|---
$[sc][\bar{u}\bar{b}]$ |  $-2.12^{+0.74}_{-0.59}$ |  $-0.66^{+0.15}_{-0.13}$ |  [4.0, 4.6] |  [59, 62] |  29.4 |  $<2.0$
$[sc][\bar{d}\bar{b}]$ |  $1.66^{+0.60}_{-0.46}$ |  $0.33^{+0.07}_{-0.07}$ |  [4.0, 4.6] |  [59, 62] |  29.1 |  $<2.0$
$[sc][\bar{s}\bar{b}]$ |  $2.01^{+0.61}_{-0.50}$ |  $0.37^{+0.07}_{-0.07}$ |  [4.4, 5.0] |  [61, 64] |  28.3 |  $<2.0$

In light of the findings presented in this study, the following key points
have been identified:

  * •

Our analysis shows that the multipole moments of these states are governed by
axial-vector diquark ([$\bar{q}\bar{b}$]) component of the interpolating
current.

  * •

When we analyze the individual quark contributions (which can be achieved
through the charge factors $e_{q}$, $e_{c}$, and $e_{b}$), it becomes evident
that the magnetic dipole moments are primarily influenced by light quarks,
which account for approximately $74\%$ of the total, while heavy quarks
represent the remaining $26\%$. In the case of the quadrupole moments, the
total contributions come from the light quarks. Upon closer examination of the
quadrupole moments, it becomes evident that the missing heavy quark
contributions are due to the terms containing heavy quarks exactly canceling
each other out.

  * •

$U$-symmetry violation between the $[sc][\bar{d}\bar{b}]$ and
$[sc][\bar{s}\bar{b}]$ states in case of the magnetic dipole moment is about
$18\%$, while in the case of the quadrupole moment, it is roughly $12\%$. A
reasonable $U$-symmetry violation is observed.

  * •

The quadrupole moment results obtained for these states are non-zero,
indicating the presence of a non-spherical charge distribution. By the
predicted signs, the geometric shape of the $[sc][\bar{u}\bar{b}]$ state is
oblate, whereas the $[sc][\bar{d}\bar{b}]$ and $[sc][\bar{s}\bar{b}]$ states
are prolate.

  * •

The order of the magnetic dipole moments could be used to give an idea of the
experimental accessibility of the corresponding physical quantities. Their
order of magnitude would suggest that these outcomes may be achievable in
forthcoming experiments.

##  IV Concluding remarks

A systematic study of the electromagnetic properties of exotic states is
conducted to elucidate their nature, which continues to be the subject of
controversy and incomplete understanding in the field. This work estimates the
multipole moments of possible $[sc][\bar{q}\bar{b}]$ and
$[sc][\bar{s}\bar{b}]$ states within the context of the QCD light-cone sum
rules. In calculating the multipole moments of the $[sc][\bar{q}\bar{b}]$ and
$[sc][\bar{s}\bar{b}]$ states, the diquark-antidiquark configuration is taken
into account with $J^{P}=1^{+}$ spin-parity quantum numbers. The order of
magnitude of the magnetic dipole moment demonstrates that it may be measured
in an experiment. The predictions derived in this study can be checked through
the use of other theoretical models, such as lattice QCD and chiral
perturbation theory. The multipole moments of the $[sc][\bar{q}\bar{b}]$ and
$[sc][\bar{s}\bar{b}]$ states reveal valuable knowledge concerning the
internal structure, size, and shape of the hadrons. The determination of these
parameters could be a significant step in our interpretation of hadron
properties concerning quark-gluon degrees of freedom and in elucidating their
nature. It is similarly crucial to ascertain the branching ratios of the
various decay modes/channels of these tetraquark states. It would be highly
intriguing to anticipate forthcoming experimental efforts that will
investigate the potential existence of $[sc][\bar{q}\bar{b}]$ and
$[sc][\bar{s}\bar{b}]$ states. These findings would then be subjected to
rigorous testing to validate and expand upon the present results. To gain a
more comprehensive understanding of these findings, further research is
recommended.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 1:  Variation of multipole moments $[sc][\bar{q}\bar{b}]$ and
$[sc][\bar{s}\bar{b}]$ states as a function of the $\rm{M^{2}}$ at different
values of $\rm{s_{0}}$.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 2:  Variation of multipole moments $[sc][\bar{q}\bar{b}]$ and
$[sc][\bar{s}\bar{b}]$ states as a function of the $\rm{s_{0}}$ at different
values of $\rm{M^{2}}$.

## References

  * Choi  _et al._ [2003] S. K. Choi _et al._ (Belle), Phys. Rev. Lett. 91, 262001 (2003), arXiv:hep-ex/0309032 .
  * Esposito  _et al._ [2015] A. Esposito, A. L. Guerrieri, F. Piccinini, A. Pilloni, and A. D. Polosa, Int. J. Mod. Phys. A 30, 1530002 (2015), arXiv:1411.5997 [hep-ph] .
  * Esposito  _et al._ [2017] A. Esposito, A. Pilloni, and A. D. Polosa, Phys. Rept. 668, 1 (2017), arXiv:1611.07920 [hep-ph] .
  * Olsen  _et al._ [2018] S. L. Olsen, T. Skwarnicki, and D. Zieminska, Rev. Mod. Phys. 90, 015003 (2018), arXiv:1708.04012 [hep-ph] .
  * Lebed  _et al._ [2017] R. F. Lebed, R. E. Mitchell, and E. S. Swanson, Prog. Part. Nucl. Phys. 93, 143 (2017), arXiv:1610.04528 [hep-ph] .
  * Nielsen  _et al._ [2010] M. Nielsen, F. S. Navarra, and S. H. Lee, Phys. Rept. 497, 41 (2010), arXiv:0911.1958 [hep-ph] .
  * Brambilla  _et al._ [2020] N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.-P. Shen, C. E. Thomas, A. Vairo, and C.-Z. Yuan, Phys. Rept. 873, 1 (2020), arXiv:1907.07583 [hep-ex] .
  * Agaev  _et al._ [2020] S. Agaev, K. Azizi, and H. Sundu, Turk. J. Phys. 44, 95 (2020), arXiv:2004.12079 [hep-ph] .
  * Chen  _et al._ [2016] H.-X. Chen, W. Chen, X. Liu, and S.-L. Zhu, Phys. Rept. 639, 1 (2016), arXiv:1601.02092 [hep-ph] .
  * Ali  _et al._ [2017] A. Ali, J. S. Lange, and S. Stone, Prog. Part. Nucl. Phys. 97, 123 (2017), arXiv:1706.00610 [hep-ph] .
  * Guo  _et al._ [2018] F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, and B.-S. Zou, Rev. Mod. Phys. 90, 015004 (2018), [Erratum: Rev.Mod.Phys. 94, 029901 (2022)], arXiv:1705.00141 [hep-ph] .
  * Liu  _et al._ [2019] Y.-R. Liu, H.-X. Chen, W. Chen, X. Liu, and S.-L. Zhu, Prog. Part. Nucl. Phys. 107, 237 (2019), arXiv:1903.11976 [hep-ph] .
  * Yang  _et al._ [2020] G. Yang, J. Ping, and J. Segovia, Symmetry 12, 1869 (2020), arXiv:2009.00238 [hep-ph] .
  * Dong  _et al._ [2021a] X.-K. Dong, F.-K. Guo, and B.-S. Zou, Progr. Phys. 41, 65 (2021a), arXiv:2101.01021 [hep-ph] .
  * Dong  _et al._ [2021b] X.-K. Dong, F.-K. Guo, and B.-S. Zou, Commun. Theor. Phys. 73, 125201 (2021b), arXiv:2108.02673 [hep-ph] .
  * Meng  _et al._ [2023] L. Meng, B. Wang, G.-J. Wang, and S.-L. Zhu, Phys. Rept. 1019, 1 (2023), arXiv:2204.08716 [hep-ph] .
  * Chen  _et al._ [2023] H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu, and S.-L. Zhu, Rept. Prog. Phys. 86, 026201 (2023), arXiv:2204.02649 [hep-ph] .
  * Zhang and Huang [2009] J.-R. Zhang and M.-Q. Huang, Phys. Rev. D 80, 056004 (2009), arXiv:0906.0090 [hep-ph] .
  * Zhang and Huang [2010] J.-R. Zhang and M.-Q. Huang, Commun. Theor. Phys. 54, 1075 (2010), arXiv:0905.4672 [hep-ph] .
  * Sun  _et al._ [2012] Z.-F. Sun, X. Liu, M. Nielsen, and S.-L. Zhu, Phys. Rev. D 85, 094008 (2012), arXiv:1203.1090 [hep-ph] .
  * Albuquerque  _et al._ [2012] R. M. Albuquerque, X. Liu, and M. Nielsen, Phys. Lett. B 718, 492 (2012), arXiv:1203.6569 [hep-ph] .
  * Chen  _et al._ [2014] W. Chen, T. G. Steele, and S.-L. Zhu, Phys. Rev. D 89, 054037 (2014), arXiv:1310.8337 [hep-ph] .
  * Agaev  _et al._ [2017a] S. S. Agaev, K. Azizi, and H. Sundu, Phys. Rev. D 95, 034008 (2017a), arXiv:1611.00293 [hep-ph] .
  * Agaev  _et al._ [2017b] S. S. Agaev, K. Azizi, and H. Sundu, Eur. Phys. J. C 77, 321 (2017b), arXiv:1702.08230 [hep-ph] .
  * Wang and Chen [2020] Q.-N. Wang and W. Chen, Eur. Phys. J. C 80, 389 (2020), arXiv:2002.04243 [hep-ph] .
  * Wang [2019] Z.-G. Wang, EPL 128, 11001 (2019), arXiv:1907.10921 [hep-ph] .
  * Wu  _et al._ [2019] J. Wu, X. Liu, Y.-R. Liu, and S.-L. Zhu, Phys. Rev. D 99, 014037 (2019), arXiv:1810.06886 [hep-ph] .
  * Ortega  _et al._ [2020] P. G. Ortega, J. Segovia, D. R. Entem, and F. Fernandez, Eur. Phys. J. C 80, 223 (2020), arXiv:2001.08093 [hep-ph] .
  * Özdem [2024a] U. Özdem, (2024a), arXiv:2403.16191 [hep-ph] .
  * Özdem and Azizi [2024] U. Özdem and K. Azizi, (2024), arXiv:2401.04798 [hep-ph] .
  * Mutuk [2024] H. Mutuk, Eur. Phys. J. C 84, 395 (2024), arXiv:2312.13383 [hep-ph] .
  * Wang  _et al._ [2023a] F.-L. Wang, S.-Q. Luo, and X. Liu, Phys. Rev. D 107, 114017 (2023a), arXiv:2303.04542 [hep-ph] .
  * Özdem [2024b] U. Özdem, Eur. Phys. J. C 84, 45 (2024b), arXiv:2311.11327 [hep-ph] .
  * Özdem [2024c] U. Özdem, Chin. Phys. C 48, 013101 (2024c), arXiv:2307.05028 [hep-ph] .
  * Lei and Li [2024] Y.-D. Lei and H.-S. Li, Phys. Rev. D 109, 076014 (2024), arXiv:2312.01332 [hep-ph] .
  * Zhang  _et al._ [2021] W.-X. Zhang, H. Xu, and D. Jia, Phys. Rev. D 104, 114011 (2021), arXiv:2109.07040 [hep-ph] .
  * Azizi and Özdem [2023] K. Azizi and U. Özdem, JHEP 03, 166, arXiv:2301.07713 [hep-ph] .
  * Özdem [2023] U. Özdem, Phys. Lett. B 838, 137750 (2023), arXiv:2211.10169 [hep-ph] .
  * Özdem [2022a] U. Özdem, Phys. Rev. D 105, 114030 (2022a), arXiv:2206.05196 [hep-ph] .
  * Xu  _et al._ [2020] Y.-J. Xu, Y.-L. Liu, and M.-Q. Huang, Eur. Phys. J. C 80, 953 (2020), arXiv:2007.15214 [hep-ph] .
  * Wang [2018] Z.-G. Wang, Eur. Phys. J. C 78, 297 (2018), arXiv:1712.05664 [hep-ph] .
  * Özdem [2022b] U. Özdem, Chin. Phys. C 46, 113106 (2022b), arXiv:2203.07759 [hep-ph] .
  * Wang  _et al._ [2023b] Y.-H. Wang, J. Wei, C.-S. An, and C.-R. Deng, Chin. Phys. Lett. 40, 021201 (2023b).
  * Özdem [2022c] U. Özdem, Phys. Rev. D 105, 054019 (2022c), arXiv:2112.10402 [hep-ph] .
  * Azizi and Özdem [2021] K. Azizi and U. Özdem, Phys. Rev. D 104, 114002 (2021), arXiv:2109.02390 [hep-ph] .
  * Özdem and Yıldırım [2021] U. Özdem and A. K. Yıldırım, Phys. Rev. D 104, 054017 (2021), arXiv:2104.13074 [hep-ph] .
  * Xu  _et al._ [2021] Y.-J. Xu, Y.-L. Liu, C.-Y. Cui, and M.-Q. Huang, Phys. Rev. D 104, 094028 (2021), arXiv:2011.14313 [hep-ph] .
  * Özdem and Azizi [2021] U. Özdem and K. Azizi, Eur. Phys. J. Plus 136, 968 (2021), arXiv:2102.09231 [hep-ph] .
  * Ozdem and Azizi [2018] U. Ozdem and K. Azizi, Phys. Rev. D 97, 014010 (2018), arXiv:1709.09714 [hep-ph] .
  * Ozdem and Azizi [2017] U. Ozdem and K. Azizi, Phys. Rev. D 96, 074030 (2017), arXiv:1707.09612 [hep-ph] .
  * Brodsky and Hiller [1992] S. J. Brodsky and J. R. Hiller, Phys. Rev. D 46, 2141 (1992).
  * Yang  _et al._ [1993] K.-C. Yang, W. Y. P. Hwang, E. M. Henley, and L. S. Kisslinger, Phys. Rev. D 47, 3001 (1993).
  * Belyaev and Blok [1986] V. M. Belyaev and B. Y. Blok, Z. Phys. C 30, 151 (1986).
  * Ball  _et al._ [2003] P. Ball, V. M. Braun, and N. Kivel, Nucl. Phys. B 649, 263 (2003), arXiv:hep-ph/0207307 .
  * Workman  _et al._ [2022] R. L. Workman _et al._ (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Ioffe [2006] B. L. Ioffe, Prog. Part. Nucl. Phys. 56, 232 (2006), arXiv:hep-ph/0502148 .
  * Narison [2018] S. Narison, Nucl. Part. Phys. Proc. 300-302, 153 (2018).


# “Is the Pope Catholic?” Applying Chain-of-Thought Reasoning to Understanding
Conversational Implicatures

Zae Myung Kim1 David E. Taylor2 Dongyeop Kang1
Department of Computer Science and Engineering1, Department of Philosophy2
University of Minnesota Twin Cities
<EMAIL_ADDRESS>
###### Abstract

Conversational implicatures are pragmatic inferences that require listeners to
deduce the intended meaning conveyed by a speaker from their explicit
utterances. Although such inferential reasoning is fundamental to human
communication, recent research indicates that large language models struggle
to comprehend these implicatures as effectively as the average human. This
paper demonstrates that by incorporating Grice’s Four Maxims into the model
through chain-of-thought prompting, we can significantly enhance its
performance, surpassing even the average human performance on this task.

##  1 Introduction

Recent large language models (LLMs) have shown impressive capabilities in
tasks such as language generation, question answering, and machine translation
(Bubeck et al., 2023). One question that remains unanswered is whether these
models understand the implications of conversation.

Conversational implicatures are a type of indirect meaning that is conveyed
through language in a conversation, which is not explicitly stated. For
example, suppose John asks Mary, “Are you coming to Paul’s birthday party
tonight?” and Mary responds, “I have a lot of work to catch up on.” The point
Mary is trying to convey to Paul is clear: Mary cannot come to Paul’s party.
What Mary literally says is that they [Mary] have a lot of work. But this is
not (all) that Mary means. In particular, and given the specific context of
this exchange (including common knowledge among its participants), what Mary
conversationally implies is that they will not be going to the party.

Conversational implications are present in every aspect of human
communication. As LLMs are intended to faithfully model such communication,
they must be able to understand and reason about conversational implications.

In this paper, we investigate whether LLMs like ChatGPT can comprehend
conversational implicatures by conducting quantitative experiments (on BIG-
bench and LUDWIG datasets) that “prompt” the model to give a binary (“yes” or
“no”) answer to a given conversational scenario. We find that while the model
exhibits some basic understanding of these implicatures, its performance can
be improved significantly when the reasoning process in computing the
implicatures following Grice’s Four Maxims (Grice, 1975) is demonstrated to
the model in a step-by-step manner (i.e., “chain-of-thought prompting”).

The paper is organized as follows: In Section 2, we first discuss what it
means for an AI model to understand something. Section 3 briefly introduces
what LLMs are and how they function to produce generations. Section 4
illustrates the quantitative experiments and the results. Lastly, in Section
5, we reflect on the findings and discuss their implications on the nature of
conversational implicatures.

##  2 Philosophical Perspectives on Understanding

Our question “do large language models understand conversational
implicatures?” presupposes an answer to a more fundamental question, namely,
“What is it for an AI to understand something in the first place?” The purpose
of this section is to provide an (operational) answer to the latter question
in order to investigate the former. We begin with a very brief overview of the
philosophical context that informs our conception of “understanding.”

A central question in the philosophy of mind and cognitive science is whether
genuine understanding–that is, understanding of the kind characteristic of
human cognition–requires a form of subjective experience or consciousness that
is distinct from the purely computational processes that underlie AI models.

To help frame the issue, imagine yourself in a room with a book of
instructions for manipulating a set of symbols. Symbols come in; you match
those symbols to the corresponding instructions; and you output a new string
of symbols accordingly. Now suppose that the foreign symbols are literally
symbols of a distant natural language L, one that you are not otherwise
familiar with. If the instructions are good enough, and if you are good enough
at following them, then the pattern of inputs and outputs that you manifest
from inside the room will largely match that of a native L-speaker. Given this
setup, the question is: do you–in the room by yourself with the
instructions–understand language L?

Philosophers like John Searle think that the answer is “No.” According to
Searle, understanding is something over and above dispositions to
linguistically behave in a certain way. What that extra something is a matter
of empirical discovery, but it is arguably a “biological phenomenon, and it is
as likely to be as causally dependent on the specific biochemistry of its
origins as lactation, photosynthesis, or any other biological phenomena.”
(Searle, 1980)

But not everyone is convinced by Searle’s line of reasoning. Detractors point
to facts that, first, AI models are designed to process and represent
information in ways that are very similar to the way humans do and, second,
such models can generate responses to linguistic stimuli that are contextually
appropriate and coherent.

The classic version of this broad behaviorist approach to human cognition is
the Turing test (Turing, 1950), according to which a system exhibits genuine
understanding if and only if, in theory, its responses are indistinguishable
from those one would expect from a (statistically normal) human conversant.

It is the behaviorist approach, and the Turing test in particular, that we
will adopt for the purposes of this paper. The primary rationale for this is
methodological: in order to empirically measure the ability of an AI system to
understand a natural linguistic phenomenon like implicature, we need a
relatively precise empirical operationalization of understanding, which is
precisely what the Turing Test gives us.

Therefore, in this paper, we will assume that a language model understands
conversational implicatures if it generates a response that is deemed by human
judges to match the appropriate response in a given conversational scenario.
This approach aligns with the behaviorist perspective, emphasizing observable
behavior and measurable benchmarks.

##  3 Large Language Models

LLMs are a type of artificial intelligence system that is designed to process
natural language text, understand the context of the text, and generate
appropriate responses. These models are trained on vast amounts of text data,
typically following next-token prediction task where the objective is to
approximate the joint probability of some sequences of words occurring in a
large corpus. Formally, let $y_{1},y_{2},...,y_{n}$ denote tokens in a
sentence, and $P(y_{1},y_{2},...,y_{n})$ the probability of seeing all these
tokens in this order. Using the product rule of probability (i.e., the chain
rule), we can decompose the probability of a text into conditional
probabilities of each token given the previous ones:

$\displaystyle P(y_{1},y_{2},...,y_{n})$ $\displaystyle=P(y_{1})\cdot
P(y_{2}|y_{1})\cdot\cdot\cdot P(y_{n}|y_{1},...,y_{n-1})$
$\displaystyle=\prod_{t=1}^{n}P(y_{t}|y_{<t}).$

So fundamentally, LLMs are nothing more than a probabilistic model that
predicts the next most probable token that follows the given context.
Therefore, when we let the model generate after the sentence, “What is the
capital of South Korea?” it is quite possible that the model will continue
generating questions like, “What is South Korea’s largest city? What is the
currency of South Korea?” and so on. And this is because articles on the
Internet (which the model happened to be trained on) could plausibly list out
trivia questions about South Korea.

While a well-trained language model has the ability to recognize patterns and
relationships between words and phrases that appear in diverse contexts, its
text generation solely based on the most probable tokens may not align well
with what humans consider appropriate. To reduce this disparity, recent
techniques, such as instruction tuning, have been proposed to fine-tune a pre-
trained language model to perform specific tasks (Ouyang et al., 2022). This
technique involves providing the language model with a prompt (or an
instruction), which is a specific text input that guides the model toward
generating a desired output. So when asked about the capital of South Korea,
the instruction-tuned LLM will be likely to generate a response, “The capital
of South Korea is Seoul.”

Furthermore, instruction tuning with human preferences (Ziegler et al., 2020;
Ouyang et al., 2022) has been highly effective in improving the accuracy and
relevance of the outputs generated by language models, making them more useful
for a wide range of applications. In this paper, we aim to investigate the
capacity of language models to comprehend conversational implicatures, with
the goal of shedding light on the computational nature of these implicatures.

##  4 Gricean Experiments

###  4.1 Conversational Implicatures

Conversational implicatures are a type of implied meaning that arises from a
speaker’s use of language in a particular conversational context. The standard
theory of such implicatures comes from philosopher H. P. Grice. The theory is
built on the idea that conversation is, at bottom, a goal-driven, rational
activity. The goal may vary from conversation to conversation. However, to the
extent that one is involved in a given conversation, one will try to pursue
that conversation’s goal along with other participants (Grice, 1975).

Specifically, Grice identified four maxims that guide this cooperative
behavior of conversation:

  * •

Quality: only say what you reasonably believe to be true.

  * •

Quantity: only provide information that is needed to achieve the purpose of
the conversation.

  * •

Relevance: make your contribution relevant to the conversation and its
purpose.

  * •

Manner: use relatively clear and concise language, within reason.

Conversational implicatures arise on certain occasions when a person appears
to break one of these four maxims. Suppose for example that speakers A and B
are both fully aware that the weather in B’s location is particular unpleasant
this time of year.

> A: How’s the weather over there?
>
> B: Delightful, just as predicted!

What B literally says is that the weather is delightful; but what they
mean–what they conversationally imply–is that the weather is unpleasant, as
expected. The implicature arises because (i) speaker A has flouted the maxim
of quality by saying something the falisity of which is common knowledge
between A and B; (ii) the only way for B to reconcile what A says with A’s
presumed cooperation in the conversation is to suppose that what A really
means is not what they say but rather something like “the weather is
terrible.”

Gricean conversational implicatures are important for understanding the
nuanced and context-dependent nature of language use in communication. They
also present a challenge for language models, which must be able to infer the
intended meaning of a speaker based on the context and any implicatures that
arise.

###  4.2 Previous Studies and Datasets

Zheng et al. (2021) were the first to look at the LLMs’ capability to
understand conversational implicatures by constructing a synthetic dataset
called GRICE, highlighting the difficulty of the task for LLMs. Also, similar
efforts were made by researchers developing the “Beyond the Imitation Game
Benchmark (BIG-bench),” (Srivastava et al., 2022) which is a collaborative
benchmark consisting of 200 tasks intended to probe large language models.
Comprehending conversational implicatures is one of those tasks. More
recently, Ruis et al. (2022) approached the problem with a new dataset
(LUDWIG) consisting of naturally occurring implicatures and experimented with
recent instruction-tuned LLMs such as ChatGPT. In this paper, we conducted
experiments with the test sets from BIG-bench and LUDWIG datasets, which
consist of 492 and 600 test cases (i.e., conversations), respectively.

###  4.3 Experimental Setting

Following Ruis et al. (2022), we conducted experiments in zero-shot and few-
shot prompting. Additionally, in this paper, chain-of-thought (CoT) prompting
was also utilized to verify the calculability argument for conversational
implicatures.

#### Zero-Shot Prompting

refers to generating natural language text using LLMs without providing any
explicit training examples related to the specific task of concern. In zero-
shot prompting, the LLM is tasked with generating responses based on its pre-
existing knowledge and context learned during training on a diverse range of
tasks. An example of a zero-shot prompt looks as follows:

> Esther and Juan are engaging in a conversation. Esther asked “Does it bother
> you that your wife goes away on long business trips?” and Juan responded,
> “Absence makes the heart grow fonder.” What does Juan mean by his response?
> Answer by either “yes” or “no”

#### Few-Shot Prompting

provides the LLM with a few related examples (called shots) in the prompt to
steer the model to produce generations in the way we want it to do. An example
prompt is:

> The following examples are coherent sentences:
>
> Esther asked “Have you found him yet?” and Juan responded “They’re still
> looking”, which means “no.”
>
> Esther asked “Are you having fun?” and Juan responded “Is the pope
> Catholic?” which means “yes.”
>
> Similarly, please, finish the following sentence by either “yes” or “no”:
> Esther asked “Does it bother you that your wife goes away on long business
> trips?” and Juan responded “Absence makes the heart grow fonder.” which
> means

#### Chain-of-Thought Prompting

takes one step further and explicitly demonstrates the inferential reasoning
steps involved in completing the task. The main idea here is that by
explaining to LLMs the detailed reasoning process in the examples, they will
also follow and show the reasoning process when answering the prompt, and this
often leads to more accurate results. In our case, following the calculability
argument put forward by Grice, we demonstrate a detailed step-by-step
inferential reasoning process in examples included in the prompt:

> Esther and Juan are engaging in a conversation. Esther asked “Juan, are you
> going to Paul’s party?” and Juan responded “I have to work late.” What does
> Juan mean by his response? Answer by either “yes” or “no”
>
> Esther wants to know an answer to her question: “Juan, are you going to
> Paul’s party?” Juan responds: “I have to work late.” Juan’s response, in the
> literal sense, does not count as a direct answer to the question. On its
> face, Juan appears to be talking about something irrelevant to the question.
> Yet Esther has no reason to believe that Juan is opting out of the operation
> of the cooperative principle which assumes that participants in a
> conversation cooperate with each other and attempt to be truthful,
> informative, relevant, and clear in order to facilitate successful
> communication. Juan must therefore intend for Esther to infer an answer from
> “what was said” and background knowledge. What would be the relevant
> background knowledge in this situation? It is the fact that work-related
> duties typically take precedence over temporally co-located social
> gatherings. Juan must therefore intend for Esther to infer that he will not
> attend the party due to him having to work late. Thus, Juan means “no” from
> his response.
>
> Answer: No.
>
> [Another example for positive case is shown here in a similar fashion]
>
> Esther and Juan are engaging in a conversation. Esther asked “Does it bother
> you that your husband goes away on long business trips?” and Juan responded
> “Absence makes the heart grow fonder.” What does Juan mean by his response?
> Answer by either “yes” or “no”

In this manner, we generated three sets of test prompts for each test
conversation in the BIG-bench and LUDWIG datasets accordingly.

Table 1: ChatGPT’s performance on the BIG-bench test set. | prec. | rec. | F1 | acc.
---|---|---|---|---
zero-shot | 0.78 | 0.68 | 0.64 | 0.68
few-shot | 0.80 | 0.77 | 0.76 | 0.77
chain-of-thought | 0.91 | 0.91 | 0.91 | 0.91

Table 2: ChatGPT’s performance on the LUDWIG test set. | prec. | rec. | F1 | acc.
---|---|---|---|---
zero-shot | 0.74 | 0.65 | 0.60 | 0.63
few-shot | 0.75 | 0.72 | 0.70 | 0.71
chain-of-thought | 0.87 | 0.87 | 0.87 | 0.87

###  4.4 Experimental Results

Table 2 and 2 show the precision, recall, F1, and accuracy scores of
predicting binary answers for the three prompting scenarios using the ChatGPT
(text-davinci-003) model released by OpenAI. We can see that the performance
increases from zero-shot to few-shot and to CoT settings. It is impressive to
note that the CoT reasoning drastically improves the performance,
outperforming the average human scores of 86.2% accuracy reported in Ruis et
al. (2022).

Indeed, via the CoT reasoning, we verify that the model was able to identify
the missing background knowledge correctly in many cases:111More examples (and
code for experiments) can be found at
https://github.com/minnesotanlp/conversational-implicatures.

> Esther wants to know an answer to her question: “Can he walk now?” Juan
> responds: “He is as strong as an ox.” … What would be the relevant
> background knowledge in this situation? It is the fact that “being as strong
> as an ox” connotes exceptional physical strength, which is indicative of
> sufficient mobility to walk. Juan must therefore intend for Esther to infer
> that he is indeed able to walk now. … Answer: Yes.

##  5 Reflections

As mentioned in Section 3, research in LLMs has seen a huge breakthrough with
the development of instruction tuning with human feedback. While pre-trained
LLMs were already exposed to a vast amount of texts and thus capable of
understanding diverse contexts, the notion of correct or preferred generations
was missing. And the instruction tuning filled that gap by nudging the LLMs
towards the preferred direction.

Ruis et al. (2022) found that even the instruction-tuned LLMs still struggle
with understanding conversational implicatures, falling behind the performance
of average humans. This is due to the pragmatic nature of these implicatures
where additional missing background knowledge or assumptions must be inferred
and this may not be apparent to the LLMs. However, in this paper, we found
that by detailing the step-by-step process for inferential reasoning, the LLMs
became much better at the task, confirming the calculability argument for
these implicatures.

Does the fact that these reasoning steps had to be demonstrated undermine the
capability of LLMs? Not necessarily so. While it is true that the models may
require some further fine-tuning to accurately respond in the zero-shot
setting, the generated results obtained from the CoT setting demonstrated the
model’s ability to deduce the missing logical connection which led to much-
improved performance, beating the average human scores. Therefore, from a
behaviorist perspective, we can say that the recent LLMs coupled with CoT
prompting are capable of understanding conversational implicatures.

## References

  * Bubeck et al. (2023) Sébastien Bubeck, Varun Chandrasekaran, Ronen Eldan, Johannes Gehrke, Eric Horvitz, Ece Kamar, Peter Lee, Yin Tat Lee, Yuanzhi Li, Scott Lundberg, Harsha Nori, Hamid Palangi, Marco Tulio Ribeiro, and Yi Zhang. 2023.  Sparks of artificial general intelligence: Early experiments with gpt-4.
  * Grice (1975) Herbert P Grice. 1975.  Logic and conversation.  In _Speech acts_ , pages 41–58. Brill.
  * Ouyang et al. (2022) Long Ouyang, Jeff Wu, Xu Jiang, Diogo Almeida, Carroll L. Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, John Schulman, Jacob Hilton, Fraser Kelton, Luke Miller, Maddie Simens, Amanda Askell, Peter Welinder, Paul Christiano, Jan Leike, and Ryan Lowe. 2022.  Training language models to follow instructions with human feedback.
  * Ruis et al. (2022) Laura Ruis, Akbir Khan, Stella Biderman, Sara Hooker, Tim Rocktäschel, and Edward Grefenstette. 2022.  Large language models are not zero-shot communicators.
  * Searle (1980) John R Searle. 1980.  Minds, brains, and programs.  _Behavioral and brain sciences_ , 3(3):417–424.
  * Srivastava et al. (2022) Aarohi Srivastava, Abhinav Rastogi, Abhishek Rao, Abu Awal Md Shoeb, Abubakar Abid, Adam Fisch, Adam R. Brown, Adam Santoro, Aditya Gupta, Adrià Garriga-Alonso, Agnieszka Kluska, Aitor Lewkowycz, Akshat Agarwal, Alethea Power, Alex Ray, Alex Warstadt, Alexander W. Kocurek, Ali Safaya, Ali Tazarv, Alice Xiang, Alicia Parrish, Allen Nie, Aman Hussain, Amanda Askell, Amanda Dsouza, Ambrose Slone, Ameet Rahane, Anantharaman S. Iyer, Anders Andreassen, Andrea Madotto, Andrea Santilli, Andreas Stuhlmüller, Andrew Dai, Andrew La, Andrew Lampinen, Andy Zou, Angela Jiang, Angelica Chen, Anh Vuong, Animesh Gupta, Anna Gottardi, Antonio Norelli, Anu Venkatesh, Arash Gholamidavoodi, Arfa Tabassum, Arul Menezes, Arun Kirubarajan, Asher Mullokandov, Ashish Sabharwal, Austin Herrick, Avia Efrat, Aykut Erdem, Ayla Karakaş, B. Ryan Roberts, Bao Sheng Loe, Barret Zoph, Bartłomiej Bojanowski, Batuhan Özyurt, Behnam Hedayatnia, Behnam Neyshabur, Benjamin Inden, Benno Stein, Berk Ekmekci, Bill Yuchen Lin, Blake Howald, Cameron Diao, Cameron Dour, Catherine Stinson, Cedrick Argueta, César Ferri Ramírez, Chandan Singh, Charles Rathkopf, Chenlin Meng, Chitta Baral, Chiyu Wu, Chris Callison-Burch, Chris Waites, Christian Voigt, Christopher D. Manning, Christopher Potts, Cindy Ramirez, Clara E. Rivera, Clemencia Siro, Colin Raffel, Courtney Ashcraft, Cristina Garbacea, Damien Sileo, Dan Garrette, Dan Hendrycks, Dan Kilman, Dan Roth, Daniel Freeman, Daniel Khashabi, Daniel Levy, Daniel Moseguí González, Danielle Perszyk, Danny Hernandez, Danqi Chen, Daphne Ippolito, Dar Gilboa, David Dohan, David Drakard, David Jurgens, Debajyoti Datta, Deep Ganguli, Denis Emelin, Denis Kleyko, Deniz Yuret, Derek Chen, Derek Tam, Dieuwke Hupkes, Diganta Misra, Dilyar Buzan, Dimitri Coelho Mollo, Diyi Yang, Dong-Ho Lee, Ekaterina Shutova, Ekin Dogus Cubuk, Elad Segal, Eleanor Hagerman, Elizabeth Barnes, Elizabeth Donoway, Ellie Pavlick, Emanuele Rodola, Emma Lam, Eric Chu, Eric Tang, Erkut Erdem, Ernie Chang, Ethan A. Chi, Ethan Dyer, Ethan Jerzak, Ethan Kim, Eunice Engefu Manyasi, Evgenii Zheltonozhskii, Fanyue Xia, Fatemeh Siar, Fernando Martínez-Plumed, Francesca Happé, Francois Chollet, Frieda Rong, Gaurav Mishra, Genta Indra Winata, Gerard de Melo, Germán Kruszewski, Giambattista Parascandolo, Giorgio Mariani, Gloria Wang, Gonzalo Jaimovitch-López, Gregor Betz, Guy Gur-Ari, Hana Galijasevic, Hannah Kim, Hannah Rashkin, Hannaneh Hajishirzi, Harsh Mehta, Hayden Bogar, Henry Shevlin, Hinrich Schütze, Hiromu Yakura, Hongming Zhang, Hugh Mee Wong, Ian Ng, Isaac Noble, Jaap Jumelet, Jack Geissinger, Jackson Kernion, Jacob Hilton, Jaehoon Lee, Jaime Fernández Fisac, James B. Simon, James Koppel, James Zheng, James Zou, Jan Kocoń, Jana Thompson, Jared Kaplan, Jarema Radom, Jascha Sohl-Dickstein, Jason Phang, Jason Wei, Jason Yosinski, Jekaterina Novikova, Jelle Bosscher, Jennifer Marsh, Jeremy Kim, Jeroen Taal, Jesse Engel, Jesujoba Alabi, Jiacheng Xu, Jiaming Song, Jillian Tang, Joan Waweru, John Burden, John Miller, John U. Balis, Jonathan Berant, Jörg Frohberg, Jos Rozen, Jose Hernandez-Orallo, Joseph Boudeman, Joseph Jones, Joshua B. Tenenbaum, Joshua S. Rule, Joyce Chua, Kamil Kanclerz, Karen Livescu, Karl Krauth, Karthik Gopalakrishnan, Katerina Ignatyeva, Katja Markert, Kaustubh D. Dhole, Kevin Gimpel, Kevin Omondi, Kory Mathewson, Kristen Chiafullo, Ksenia Shkaruta, Kumar Shridhar, Kyle McDonell, Kyle Richardson, Laria Reynolds, Leo Gao, Li Zhang, Liam Dugan, Lianhui Qin, Lidia Contreras-Ochando, Louis-Philippe Morency, Luca Moschella, Lucas Lam, Lucy Noble, Ludwig Schmidt, Luheng He, Luis Oliveros Colón, Luke Metz, Lütfi Kerem Şenel, Maarten Bosma, Maarten Sap, Maartje ter Hoeve, Maheen Farooqi, Manaal Faruqui, Mantas Mazeika, Marco Baturan, Marco Marelli, Marco Maru, Maria Jose Ramírez Quintana, Marie Tolkiehn, Mario Giulianelli, Martha Lewis, Martin Potthast, Matthew L. Leavitt, Matthias Hagen, Mátyás Schubert, Medina Orduna Baitemirova, Melody Arnaud, Melvin McElrath, Michael A. Yee, Michael Cohen, Michael Gu, Michael Ivanitskiy, Michael Starritt, Michael Strube, Michał Swędrowski, Michele Bevilacqua, Michihiro Yasunaga, Mihir Kale, Mike Cain, Mimee Xu, Mirac Suzgun, Mo Tiwari, Mohit Bansal, Moin Aminnaseri, Mor Geva, Mozhdeh Gheini, Mukund Varma T, Nanyun Peng, Nathan Chi, Nayeon Lee, Neta Gur-Ari Krakover, Nicholas Cameron, Nicholas Roberts, Nick Doiron, Nikita Nangia, Niklas Deckers, Niklas Muennighoff, Nitish Shirish Keskar, Niveditha S. Iyer, Noah Constant, Noah Fiedel, Nuan Wen, Oliver Zhang, Omar Agha, Omar Elbaghdadi, Omer Levy, Owain Evans, Pablo Antonio Moreno Casares, Parth Doshi, Pascale Fung, Paul Pu Liang, Paul Vicol, Pegah Alipoormolabashi, Peiyuan Liao, Percy Liang, Peter Chang, Peter Eckersley, Phu Mon Htut, Pinyu Hwang, Piotr Miłkowski, Piyush Patil, Pouya Pezeshkpour, Priti Oli, Qiaozhu Mei, Qing Lyu, Qinlang Chen, Rabin Banjade, Rachel Etta Rudolph, Raefer Gabriel, Rahel Habacker, Ramón Risco Delgado, Raphaël Millière, Rhythm Garg, Richard Barnes, Rif A. Saurous, Riku Arakawa, Robbe Raymaekers, Robert Frank, Rohan Sikand, Roman Novak, Roman Sitelew, Ronan LeBras, Rosanne Liu, Rowan Jacobs, Rui Zhang, Ruslan Salakhutdinov, Ryan Chi, Ryan Lee, Ryan Stovall, Ryan Teehan, Rylan Yang, Sahib Singh, Saif M. Mohammad, Sajant Anand, Sam Dillavou, Sam Shleifer, Sam Wiseman, Samuel Gruetter, Samuel R. Bowman, Samuel S. Schoenholz, Sanghyun Han, Sanjeev Kwatra, Sarah A. Rous, Sarik Ghazarian, Sayan Ghosh, Sean Casey, Sebastian Bischoff, Sebastian Gehrmann, Sebastian Schuster, Sepideh Sadeghi, Shadi Hamdan, Sharon Zhou, Shashank Srivastava, Sherry Shi, Shikhar Singh, Shima Asaadi, Shixiang Shane Gu, Shubh Pachchigar, Shubham Toshniwal, Shyam Upadhyay, Shyamolima, Debnath, Siamak Shakeri, Simon Thormeyer, Simone Melzi, Siva Reddy, Sneha Priscilla Makini, Soo-Hwan Lee, Spencer Torene, Sriharsha Hatwar, Stanislas Dehaene, Stefan Divic, Stefano Ermon, Stella Biderman, Stephanie Lin, Stephen Prasad, Steven T. Piantadosi, Stuart M. Shieber, Summer Misherghi, Svetlana Kiritchenko, Swaroop Mishra, Tal Linzen, Tal Schuster, Tao Li, Tao Yu, Tariq Ali, Tatsu Hashimoto, Te-Lin Wu, Théo Desbordes, Theodore Rothschild, Thomas Phan, Tianle Wang, Tiberius Nkinyili, Timo Schick, Timofei Kornev, Timothy Telleen-Lawton, Titus Tunduny, Tobias Gerstenberg, Trenton Chang, Trishala Neeraj, Tushar Khot, Tyler Shultz, Uri Shaham, Vedant Misra, Vera Demberg, Victoria Nyamai, Vikas Raunak, Vinay Ramasesh, Vinay Uday Prabhu, Vishakh Padmakumar, Vivek Srikumar, William Fedus, William Saunders, William Zhang, Wout Vossen, Xiang Ren, Xiaoyu Tong, Xinran Zhao, Xinyi Wu, Xudong Shen, Yadollah Yaghoobzadeh, Yair Lakretz, Yangqiu Song, Yasaman Bahri, Yejin Choi, Yichi Yang, Yiding Hao, Yifu Chen, Yonatan Belinkov, Yu Hou, Yufang Hou, Yuntao Bai, Zachary Seid, Zhuoye Zhao, Zijian Wang, Zijie J. Wang, Zirui Wang, and Ziyi Wu. 2022.  Beyond the imitation game: Quantifying and extrapolating the capabilities of language models.
  * Turing (1950) A. M. Turing. 1950.  Computing machinery and intelligence.  _Mind_ , 59(236):433–460.
  * Zheng et al. (2021) Zilong Zheng, Shuwen Qiu, Lifeng Fan, Yixin Zhu, and Song-Chun Zhu. 2021.  GRICE: A grammar-based dataset for recovering implicature and conversational rEasoning.  In _Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021_ , pages 2074–2085, Online. Association for Computational Linguistics.
  * Ziegler et al. (2020) Daniel M. Ziegler, Nisan Stiennon, Jeffrey Wu, Tom B. Brown, Alec Radford, Dario Amodei, Paul Christiano, and Geoffrey Irving. 2020.  Fine-tuning language models from human preferences.


componentwise action. Once again, $\alpha$ induces a permutation $\pi\in
S_{k}$, where

$j^{\pi}=m\text{ if }\mathbf{c}_{j}^{\alpha}=\mathbf{c}_{m}.$

Then by arguing as above, we deduce that $(\alpha,\dots,\alpha)\pi\in
G_{(\Delta)}$.

Finally, assume $(x,y)^{\alpha}=(x^{-1},y^{-1})$ and note that

$\\{\mathbf{c}_{j}:j\in[k]\\}^{\alpha}=\\{(x^{-1},y^{-1})\mathbf{c}_{j}:j\in[k]\\}.$

Here $\alpha$ also induces a permutation $\pi\in S_{k}$, where

$j^{\pi}=m\text{ if }\mathbf{c}_{j}^{\alpha}=(x^{-1},y^{-1})\mathbf{c}_{m},$

and thus $t_{0,j}^{\alpha}=x^{-1}t_{0,j^{\pi}}$ and
$t_{1,j}^{\alpha}=y^{-1}t_{1,j^{\pi}}$ for all $j\in[k]$, noting that $\pi\neq
1$ if $\alpha=1$. Now we have

$D\mathbf{a}_{0}^{(\alpha,\dots,\alpha)\pi}=D(\varphi_{t_{i,1^{\pi^{-1}}}^{\alpha}},\dots,\varphi_{t_{i,k^{\pi^{-1}}}^{\alpha}})=D(\varphi_{x^{-1}}\varphi_{t_{i,1}},\dots,\varphi_{x^{-1}}\varphi_{t_{i,k}})=D\mathbf{a}_{0}$

and similarly, $D\mathbf{a}_{1}^{(\alpha,\dots,\alpha)\pi}=D\mathbf{a}_{1}$.
This completes the proof. ∎

######  Proposition 5.13.

If $P\in\\{A_{k},S_{k}\\}$ and $k=|T|^{2}-2$, then

$b(G)=\begin{cases}4&\mbox{if $T\in\\{A_{5},A_{6}\\}$ and
$G=T^{k}.(\mathrm{Out}(T)\times S_{k})$;}\\\ 3&\mbox{otherwise.}\end{cases}$

###### Proof.

By Lemmas 5.10 and 5.12, we may assume that $S_{k}\leqslant G$, and $G$ is not
$T^{k}.(\mathrm{Out}(T)\times S_{k})$ if $T\in\\{A_{5},A_{6}\\}$. That is,
$G=T^{k}.(O\times S_{k})$ for some $O\leqslant\mathrm{Out}(T)$, with
$O\neq\mathrm{Out}(T)$ if $T\in\\{A_{5},A_{6}\\}$. We will prove that $b(G)=3$
by constructing a base of size $3$.

Write $K=\mathrm{Inn}(T).O\leqslant\mathrm{Aut}(T)$. Note that there exist
$x,y\in T$ such that $C_{K}(x)\cap C_{K}(y)=1$ and there is no $\alpha\in K$
with $(x,y)^{\alpha}=(x^{-1},y^{-1})$. This can be obtained by Corollary 5.4
when $T\notin\\{A_{5},A_{6}\\}$, and the cases where $T\in\\{A_{5},A_{6}\\}$
can be checked using Magma (note that $K<\mathrm{Aut}(T)$ if
$T\in\\{A_{5},A_{6}\\}$). Now let $\mathcal{P}=\\{\mathcal{P}_{t}:t\in T\\}$
be a partition of $[k]$ with $|\mathcal{P}_{1}|=|\mathcal{P}_{x}|=|T|-1$, and
$|\mathcal{P}_{t}|=|T|$ if $t\notin\\{1,x\\}$. And we label the elements in
$T$ by $T=\\{g_{1},\dots,g_{|T|}\\}$, where $g_{1}=1$ and $g_{|T|}=y$. Define
$\mathbf{a}_{0}=(\varphi_{t_{0,1}},\dots,\varphi_{t_{0,k}})\in\mathrm{Inn}(T)^{k}$,
where $t_{0,j}=t$ if $j\in\mathcal{P}_{t}$, and define
$\mathbf{a}_{1}=(\varphi_{t_{1,1}},\dots,\varphi_{t_{1,k}})\in\mathrm{Inn}(T)^{k}$
by setting

$t_{1,j}=\begin{cases}g_{h}&\mbox{if $t\neq 1$ and $j$ is the $h$-th smallest
number in $\mathcal{P}_{t}$;}\\\ g_{h+1}&\mbox{if $j$ is the $h$-th smallest
number in $\mathcal{P}_{1}$.}\end{cases}$

Now we claim that $\Delta=\\{D,D\mathbf{a}_{0},D\mathbf{a}_{1}\\}$ is a base
for $G$.

Suppose $(\alpha,\dots,\alpha)\pi\in G_{(\Delta)}$, noting that $\alpha\in K$.
By Lemma 2.2(i), we have $\pi\in P_{\\{\mathcal{P}\\}}$, so either $\pi\in
P_{\\{\mathcal{P}_{1}\\}}\cap P_{\\{\mathcal{P}_{x}\\}}$, or
$\mathcal{P}_{1}^{\pi}=\mathcal{P}_{x}$, hence there are two cases to
consider.

First assume that $\mathcal{P}_{1}^{\pi}=\mathcal{P}_{x}$. There exists a
unique $g\in T$ such that $t_{0,j}^{\alpha}=gt_{0,j^{\pi}}$ for all $j\in[k]$,
and by taking $j\in\mathcal{P}_{1}$ we have $g=x^{-1}$. This implies that
$x^{\alpha}=x^{-1}$ by taking $j\in\mathcal{P}_{x}$. Let
$\mathcal{Q}=\\{\mathcal{Q}_{t}:t\in T\\}$ be the partition of $[k]$ defined
by setting $j\in\mathcal{Q}_{t}$ if $t_{1,j}=t$. Then
$|\mathcal{Q}_{1}|=|\mathcal{Q}_{y}|=|T|-1$, and $|\mathcal{Q}_{t}|=|T|$ if
$t\notin\\{1,y\\}$. By arguing as above, either $\pi\in
P_{\\{\mathcal{Q}_{1}\\}}\cap P_{\\{\mathcal{Q}_{y}\\}}$ or
$\mathcal{Q}_{1}^{\pi}=\mathcal{Q}_{y}$. If the former holds, then

$(\mathcal{P}_{1}\cap\mathcal{Q}_{1})^{\pi}=\mathcal{P}_{x}\cap\mathcal{Q}_{1}.$

However, as can be seen from the definitions of $\mathbf{a}_{0}$ and
$\mathbf{a}_{1}$, we have $|\mathcal{P}_{1}\cap\mathcal{Q}_{1}|=0$, while
$|\mathcal{P}_{x}\cap\mathcal{Q}_{1}|=1$. This implies that
$\mathcal{Q}_{1}^{\pi}=\mathcal{Q}_{y}$, so $y^{\alpha}=y^{-1}$ as above. By
our assumptions on $x$ and $y$, there is no $\alpha\in K$ with
$(x,y)^{\alpha}=(x^{-1},y^{-1})$, which gives a contradiction.

Finally, suppose that $\pi\in P_{\\{\mathcal{P}_{1}\\}}\cap
P_{\\{\mathcal{P}_{x}\\}}$. First note that $t_{0,j}^{\alpha}=t_{0,j^{\pi}}$
for all $j\in[k]$, so $x^{\alpha}=x$. Similarly, we have $\pi\in
P_{\\{\mathcal{Q}_{1}\\}}\cap P_{\\{\mathcal{Q}_{y}\\}}$ and $y^{\alpha}=y$.
This implies that $\alpha\in C_{K}(x)\cap C_{K}(y)=1$, and thus
$t_{i,j}=t_{i,j^{\pi}}$ for all $i\in\\{0,1\\}$ and $j\in[k]$, which yields
$\pi=1$ and completes the proof. ∎

######  Proposition 5.14.

If $\ell\geqslant 3$, $k=|T|^{\ell}-2$ and $P\in\\{A_{k},S_{k}\\}$, then
$b(G)=\ell+1$.

###### Proof.

In view of Theorem 2.3(iii), it suffices to construct a base for $G$ of size
$\ell+1$. First note that there exist $x,y,z\in T$ such that

$C_{\mathrm{Aut}(T)}(x)\cap C_{\mathrm{Aut}(T)}(y)\cap
C_{\mathrm{Aut}(T)}(z)=1$

and there is no $\alpha\in\mathrm{Aut}(T)$ with

$(x,y,z)^{\alpha}=(x^{-1},y^{-1},z^{-1}).$

To see this, if $T\notin\\{A_{5},A_{6}\\}$ then we apply Corollary 5.4, and if
$T\in\\{A_{5},A_{6}\\}$ then it can be checked using Magma. Let
$\mathcal{P}=\\{\mathcal{P}_{t}:t\in T\\}$ be a partition of $[k]$ with
$|\mathcal{P}_{1}|=|\mathcal{P}_{x}|=|T|^{\ell-1}-1$ and
$|\mathcal{P}_{t}|=|T|^{\ell-1}$ if $t\notin\\{1,x\\}$. Write
$T^{\ell-1}=\\{\mathbf{b}_{1},\dots,\mathbf{b}_{|T|^{\ell-1}}\\}$, where
$\mathbf{b}_{h}=(a_{1,h},\dots,a_{\ell-1,h})$, and we may assume
$\mathbf{b}_{1}=(1,\dots,1)$ and $\mathbf{b}_{|T|^{\ell-1}}=(y,z,\dots,z)$.
Now define $\mathbf{a}_{i}=(\varphi_{t_{i,1}},\dots,\varphi_{t_{i,k}})$ for
$i\in\\{0,\dots,\ell-1\\}$, where $t_{0,j}=t$ if $j\in\mathcal{P}_{t}$, and if
$i\geqslant 1$,

$t_{i,j}=\begin{cases}a_{i,h}&\mbox{if $t\neq 1$ and $j$ is the $h$-th
smallest number in $\mathcal{P}_{t}$;}\\\ a_{i,h+1}&\mbox{if $j$ is the $h$-th
smallest number in $\mathcal{P}_{1}$.}\end{cases}$

We claim that
$\Delta=\\{D,D\mathbf{a}_{0},D\mathbf{a}_{1},\dots,D\mathbf{a}_{\ell-1}\\}$ is
a base for $G$.

We argue as in the proof of Proposition 5.13. Suppose
$(\alpha,\dots,\alpha)\pi\in G_{(\Delta)}$, noting that $\pi\in
P_{\\{\mathcal{P}\\}}$ by Lemma 2.2(i). It follows that either $\pi\in
P_{\\{\mathcal{P}_{1}\\}}\cap P_{\\{\mathcal{P}_{x}\\}}$ or
$\mathcal{P}_{1}^{\pi}=\mathcal{P}_{x}$.

First assume that $\mathcal{P}_{1}^{\pi}=\mathcal{P}_{x}$. Note that there
exists a unique $g\in T$ such that $t_{0,j}^{\alpha}=gt_{0,j^{\pi}}$ for all
$j\in[k]$. Now $g=x^{-1}$ by taking $j\in\mathcal{P}_{1}$, and thus
$x^{\alpha}=x^{-1}$ by taking $j\in\mathcal{P}_{x}$. Let
$\mathcal{Q}=\\{\mathcal{Q}_{t}:t\in T\\}$ be the partition of $[k]$ defined
by setting $j\in\mathcal{Q}_{t}$ if $t_{1,j}=t$. Then
$|\mathcal{Q}_{1}|=|\mathcal{Q}_{y}|=|T|^{\ell-1}-1$, and
$|\mathcal{Q}_{t}|=|T|^{\ell-1}$ if $t\notin\\{1,y\\}$. By applying Lemma
2.2(i) again, we have either $\pi\in P_{\\{\mathcal{Q}_{1}\\}}\cap
P_{\\{\mathcal{Q}_{y}\\}}$ or $\mathcal{Q}_{1}^{\pi}=\mathcal{Q}_{y}$. If
$\pi\in P_{\\{\mathcal{Q}_{1}\\}}\cap P_{\\{\mathcal{Q}_{y}\\}}$, then

$(\mathcal{P}_{1}\cap\mathcal{Q}_{1})^{\pi}=\mathcal{P}_{x}\cap\mathcal{Q}_{1},$

which is impossible since
$|\mathcal{P}_{1}\cap\mathcal{Q}_{1}|=|T|^{\ell-2}-1$, while
$|\mathcal{P}_{x}\cap\mathcal{Q}_{1}|=|T|^{\ell-2}$. Hence, we have
$\mathcal{Q}_{1}^{\pi}=\mathcal{Q}_{y}$, and thus $y^{\alpha}=y^{-1}$ with the
same argument as above. Now suppose $i\geqslant 2$ and let
$\mathcal{R}=\\{\mathcal{R}_{t}:t\in T\\}$ be the partition of $[k]$ defined
by setting $j\in\mathcal{R}_{t}$ if $t_{i,j}=t$. Then
$|\mathcal{R}_{1}|=|\mathcal{R}_{z}|=|T|^{\ell-1}-1$, and
$|\mathcal{R}_{t}|=|T|^{\ell-1}$ if $t\notin\\{1,z\\}$. By arguing as above,
we have $z^{\alpha}=z^{-1}$. However, by our assumptions on $x$, $y$ and $z$,
there is no automorphism of $T$ simultaneously inverting all three elements,
which gives a contradiction.

It follows that $\pi\in P_{\\{\mathcal{P}_{1}\\}}\cap
P_{\\{\mathcal{P}_{x}\\}}$, and with the same reason, we have $\pi\in
P_{\\{\mathcal{Q}_{1}\\}}$ and $\pi\in P_{\\{\mathcal{R}_{1}\\}}$. Hence,
$t_{i,j}^{\alpha}=t_{i,j^{\pi}}$ for all $i\in\\{0,\dots,\ell-1\\}$ and
$j\in[k]$. This implies that

$\alpha\in C_{\mathrm{Aut}(T)}(x)\cap C_{\mathrm{Aut}(T)}(y)\cap
C_{\mathrm{Aut}(T)}(z),$

so $\alpha=1$. Moreover, note that if $j,j^{\prime}\in\mathcal{P}_{t}$ for
some $t\in T$ and $j\neq j^{\prime}$, then there exists
$i\in\\{1,\dots,\ell-1\\}$ such that $t_{i,j}\neq t_{i,j^{\prime}}$. Hence,
$\pi=1$ and so $\Delta$ is a base for $G$. ∎

We conclude that the proof of Theorem 3 is complete by combining Theorem 1
with Propositions 5.3, 5.8, 5.9, 5.11, 5.13 and 5.14.

##  6\. Proofs of Theorems 6 and 7

In this final section, we will prove Theorems 6 and 7. As introduced in
Section 1, let $\mathbb{Q}_{k}(T)$ be the probability that a random
$k$-element subset of $T^{\\#}$ has a non-trivial setwise stabiliser in
$\mathrm{Aut}(T)$. That is,

$\mathbb{Q}_{k}(T):=\frac{|\\{R\in\mathscr{S}_{k}(T):\mathrm{Aut}(T,R)\neq
1\\}|}{|\mathscr{S}_{k}(T)|},$

where $\mathscr{S}_{k}(T)$ is the set of $k$-subsets of $T^{\\#}$ (we will
simply write $\mathscr{S}_{k}$ if $T$ is clear from the context). Consider the
diagonal type group $G=T^{k}.(\mathrm{Out}(T)\times
S_{k})\leqslant\mathrm{Sym}(\Omega)$ and recall that

$\mathbb{P}_{k}(T):=\frac{|\\{(t_{1},\dots,t_{k-1})\in
T^{k-1}:\\{D,D(\varphi_{t_{1}},\dots,\varphi_{t_{k-1}},1)\\}\mbox{ is a base
for $G$}\\}|}{|T|^{k-1}},$

which is the probability that a random element in $\Omega$ is in a regular
orbit of $G_{D}=D$.

The following is [25, Theorem 1.5].

######  Theorem 6.1.

Let $k\geqslant 5$ and let $(T_{n})$ be a sequence of non-abelian finite
simple groups such that $|T_{n}|\rightarrow\infty$ as $n\rightarrow\infty$.
Then $\mathbb{P}_{k}(T_{n})\rightarrow 1$ as $n\rightarrow\infty$.

######  Lemma 6.2.

For any $k\geqslant 4$, we have $\mathbb{Q}_{k}(T)\leqslant
1-\mathbb{P}_{k+1}(T)$.

###### Proof.

First, by Lemma 2.15, we have
$\\{D,D(\varphi_{t_{1}},\dots,\varphi_{t_{k}},1)\\}$ is a base for $G$ if and
only if $t_{1},\dots,t_{k}\in T^{\\#}$ are distinct and
$\mathrm{Hol}(T,\\{t_{1},\dots,t_{k},1\\})=1$. The latter condition implies
that $\mathrm{Aut}(T,\\{t_{1},\dots,t_{k}\\})=1$, so

$\mathbb{P}_{k+1}(T)\leqslant\frac{|\\{(t_{1},\dots,t_{k})\in(T^{\\#})^{k}:t_{1},\dots,t_{k}\mbox{
are distinct and }\mathrm{Aut}(T,\\{t_{1},\dots,t_{k}\\})=1\\}|}{|T|^{k}}.$

Note that the numerator of the expression on the right-hand side is

$k!\cdot|\\{R\in\mathscr{S}_{k}:\mathrm{Aut}(T,R)=1\\}|.$

Thus, we have

$\mathbb{P}_{k+1}(T)\leqslant\frac{k!\cdot|\\{R\in\mathscr{S}_{k}:\mathrm{Aut}(T,R)=1\\}|}{|T|^{k}}$

and it suffices to show that

$|T|^{k}\geqslant k!\cdot|\mathscr{S}_{k}|.$

This is clear, as $|\mathscr{S}_{k}|={|T|-1\choose k}$. ∎

Theorem 6 now follows by combining Theorem 6.1 and Lemma 6.2. Finally, we
establish Theorem 7. Recall that $\mathscr{P}_{k}$ is the set of $k$-subsets
of $T$, and

$\mathrm{fix}(\sigma,\mathscr{P}_{k})=\\{S\in\mathscr{P}_{k}:\sigma\in\mathrm{Hol}(T,S)\\}$

is the set of fixed points of $\sigma\in\mathrm{Hol}(T)$ on $\mathscr{P}_{k}$.

######  Proposition 6.3.

Let $m>0$ be a real number. Then $\mathbb{Q}_{k}(T)<1/m$ if

${|T|\choose
k}>m\sum_{\sigma\in\mathcal{R}}|\mathrm{fix}(\sigma,\mathscr{P}_{k})|,$ (42)

where $\mathcal{R}$ is the set of elements of prime order in
$\mathrm{Hol}(T)$.

###### Proof.

As noted in Section 3.1, we have

$|\\{S\in\mathscr{P}_{k}:\mathrm{Hol}(T,S)\neq
1\\}|\leqslant\sum_{\sigma\in\mathcal{R}}|\mathrm{fix}(\sigma,\mathscr{P}_{k})|,$

which implies that $\mathrm{Hol}(T)$ has

$r>\frac{m-1}{m|\mathrm{Hol}(T)|}{|T|\choose k}$

regular orbits on $\mathscr{P}_{k}$. Then

$|\\{R\in\mathscr{S}_{k}:\mathrm{Hol}(T,R)=1\\}|=r(|T|-k)|\mathrm{Aut}(T)|>\frac{(m-1)(|T|-k)}{m|T|}{|T|\choose
k}$

and thus

$\mathbb{Q}_{k}(T)=\frac{|\\{R\in\mathscr{S}_{k}:\mathrm{Aut}(T,R)\neq
1\\}|}{|\mathscr{S}_{k}|}<1-\frac{(m-1)(|T|-k)}{m|T|}\cdot\frac{{|T|\choose
k}}{{|T|-1\choose k}}=\frac{1}{m},$

as desired. ∎

###### Proof of Theorem 7..

Note that if $T=A_{5}$, then $5\log|T|<k<|T|-5\log|T|$ implies that $k=30$, in
which case we can check the theorem using Magma. Now assume $|T|\geqslant
168$, so $5\log|T|<|T|/4$. It suffices to show that (42) holds for $m=|T|$ and
$5\log|T|<k\leqslant|T|/2$, and we can do this by arguing as in the proof of
Proposition 3.7. More precisely, if $|T|/4\leqslant k\leqslant|T|/2$ then (42)
holds for $m=|T|$ if

$2t_{0}^{|T|}>\sqrt{30}e^{\frac{1}{8}}|T|^{\frac{10}{3}},$

where

$t_{0}=4\cdot 3^{-\frac{3}{4}}\cdot 2^{-\frac{1}{2}-\frac{1}{10}}=1.1577....$

This inequality is valid for all $|T|\geqslant 168$. And if $k<|T|/4$ then
(42) holds for $m=|T|$ if $(5/3)^{k}>|T|^{10/3}$, which holds true for all
$k>5\log|T|$. ∎

######  Remark 6.4.

By Proposition 6.3, we have $\mathbb{Q}_{k}(T)<1/2$ if (8) holds. We refer the
reader to the proofs in Section 4 for a wider range of $k$ satisfying (8) for
each class of simple groups. For example, the proof of Proposition 4.9 shows
that if $T=A_{n}$ and $n\geqslant 7$ then (8) holds for all $n\leqslant
k\leqslant 4\log|T|$, which implies that $\mathbb{Q}_{k}(T)<1/2$ for all
$n\leqslant k\leqslant|T|-n$.

## References

  * [1] L. Babai, Finite digraphs with given regular automorphism groups, Period. Math. Hungar. 11 (1980), 257–270.
  * [2] L. Babai and C.D. Godsil, On the automorphism groups of almost all Cayley graphs, European J. Combin. 3 (1982), 9–15.
  * [3] R.F. Bailey and P.J. Cameron, Base size, metric dimension and other invariants of groups and graphs, Bull. Lond. Math. Soc. 43 (2011), 209–242.
  * [4] K.D. Blaha, Minimum bases for permutation groups: the greedy approximation, J. Algorithms 13 (1992), 297–306.
  * [5] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symb. Comput. 24 (1997), 235–265.
  * [6] T. Breuer, _The GAP Character Table Library, Version 1.3.1_, GAP package,
http://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib, 2020.

  * [7] T. Breuer, R.M. Guralnick and W.M. Kantor, Probabilistic generation of finite simple groups, II, J. Algebra 320 (2008), 443–494.
  * [8] T.C. Burness, Base sizes for primitive groups with soluble stabilisers, Algebra Number Theory 15 (2021), 1755–1807.
  * [9] T.C. Burness, _Simple groups, fixed point ratios and applications_ , in Local representation theory and simple groups, 267–322, EMS Ser. Lect. Math., Eur. Math. Soc., Zürich, 2018.
  * [10] T.C. Burness, Fixed point ratios in actions of finite classical groups. II, J. Algebra 309 (2007), 80–138.
  * [11] T.C. Burness, On base sizes for actions of finite classical groups, J. Lond. Math. Soc. 75 (2007), 545–562.
  * [12] T.C. Burness and M. Giudici, _On the Saxl graph of a permutation group_ , Math. Proc. Cambridge Philos. Soc. 168 (2020), 219–248.
  * [13] T.C. Burness and M. Giudici, Classical groups, derangements and primes, Australian Mathematical Society Lecture Series, vol. 25, Cambridge University Press, Cambridge, 2016.
  * [14] T.C. Burness, R.M. Guralnick and J. Saxl, On base sizes for symmetric groups, Bull. Lond. Math. Soc. 43 (2011), 386–391.
  * [15] T.C. Burness and S. Harper, Computations concerning the uniform domination number of a finite simple group, available at http://seis.bristol.ac.uk/~tb13602/udncomp.pdf.
  * [16] T.C. Burness and H.Y. Huang, On base sizes for primitive groups of product type, J. Pure Appl. Algebra 227 (2023), Paper No. 107228, 43 pp.
  * [17] T.C. Burness and H.Y. Huang, On the Saxl graphs of primitive groups with soluble stabilisers, Algebr. Comb. 5 (2022), 1053–1087.
  * [18] T.C. Burness, M.W. Liebeck and A. Shalev, Base sizes for simple groups and a conjecture of Cameron, Proc. Lond. Math. Soc. 98 (2009), 116–162.
  * [19] T.C. Burness and A.R. Thomas, The classification of extremely primitive groups, Int. Math. Res. Not. IMRN 2022, 10148–10248.
  * [20] T.C. Burness, E.A. O’Brien and R.A. Wilson, Base sizes for sporadic simple groups, Israel J. Math. 177 (2010), 307–333.
  * [21] P.J. Cameron, Permutation groups, London Math. Soc. Student Texts, vol. 45, Cambridge University Press, 1999.
  * [22] H. Chen and S. Du, On the Burness-Giudici conjecture, Comm. Algebra 51 (2023), 5019–5045.
  * [23] H. Duyan, Z. Halasi and A. Maróti, A proof of Pyber’s base size conjecture, Adv. Math. 331 (2018), 720–747.
  * [24] J.B. Fawcett, Bases of twisted wreath products, J. Algebra 607 (2022), 247–271.
  * [25] J.B. Fawcett, The base size of a primitive diagonal group, J. Algebra 375 (2013), 302–321.
  * [26] J. Fulman and R.M. Guralnick, The number of regular semisimple conjugacy classes in the finite classical groups, Linear Algebra Appl. 439 (2013), 488–503.
  * [27] J. Fulman and R.M. Guralnick, Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements, Trans. Amer. Math. Soc. 364 (2012), 3023–3070.
  * [28] P.X. Gallagher, The number of conjugacy classes in a finite group, Math. Z. 118 (1970), 175–179.
  * [29] D. Garzoni and N. Gill, On the number of conjugacy classes of a primitive permutation group, Proc. Roy. Soc. Edinburgh Sect. A 153 (2023), 115–136.
  * [30] C.D. Godsil, On the full automorphism group of a graph, Combinatorica 1 (1981), 243–256.
  * [31] D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups. Number 3, Mathematical Surveys and Monographs, vol. 40. American Mathematical Society, Providence, RI., 1998.
  * [32] R. Gow, Commutators in finite simple groups of Lie type, Bull. London Math. Soc. 32 (2000), 311–315.
  * [33] R.M. Guralnick and W.M. Kantor, Probabilistic generation of finite simple groups, J. Algebra 234 (2000), 743–792.
  * [34] R.M. Guralnick and G. Malle, Simple groups admit Beauville structures, J. Lond. Math. Soc. 85 (2012), 694–721.
  * [35] Z. Halasi, M.W. Liebeck and A. Maróti, Base sizes of primitive groups: bounds with explicit constants, J. Algebra 521 (2019), 16–43.
  * [36] Z. Halasi and K. Podoski, Every coprime linear group admits a base of size two, Trans. Amer. Math. Soc. 368 (2016), 5857–5887.
  * [37] G.A. Jones, Finite simple automorphism groups of edge-transitive maps, J. Algebra 607 (2022), 454–472.
  * [38] W.M. Kantor, A. Lubotzky and A. Shalev, Invariable generation and the Chebotarev invariant of a finite group, J. Algebra 348 (2011), 302–314.
  * [39] P.B. Kleidman and M.W. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Series, vol. 129, Cambridge University Press, 1990.
  * [40] M. Lee and T. Popiel, Saxl graphs of primitive affine groups with sporadic point stabilisers, Internat. J. Algebra Comput. 33 (2023), 369–389.
  * [41] D. Leemans and M.W. Liebeck, Chiral polyhedra and finite simple groups, Bull. Lond. Math. Soc. 49 (2017), 581–592.
  * [42] M.W. Liebeck, C.E. Praeger and J. Saxl, _On the O’Nan-Scott theorem for finite primitive permutation groups_ , J. Austral. Math. Soc. 44 (1988), 389–396.
  * [43] M.W. Liebeck and G.M. Seitz, Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, Mathematical Surveys and Monographs, vol. 180, Amer. Math. Soc., 2012.
  * [44] M.W. Liebeck and A. Shalev, _Bases of primitive permutation groups_ , in Groups, combinatorics & geometry (Durham, 2001), 147–154, World Sci. Publ., River Edge, NJ, 2003.
  * [45] M.W. Liebeck and A. Shalev, Simple groups, permutation groups, and probability, J. Amer. Math. Soc. 12 (1999), 497–520.
  * [46] F. Lübeck, _Centralisers and numbers of semisimple classes in exceptional groups of Lie type_ ,
http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/CentSSClasses

  * [47] A. Lucchini, M. Morigi and M. Moscatiello, Primitive permutation IBIS groups, J. Combin. Theory Ser. A 184 (2021), Paper No. 105516, 17 pp.
  * [48] G.A. Miller, On the groups generated by two operators, Bull. Amer. Math. Soc. 7 (1901), 424–426.
  * [49] J. Morris and P. Spiga, Asymptotic enumeration of Cayley digraphs, Israel J. Math. 242 (2021), 401–459.
  * [50] M. Neunhöffer, F. Noeske, E.A. O’Brien and R.A. Wilson, Orbit invariants and an application to the Baby Monster, J. Algebra 341 (2011), 297–305.
  * [51] L. Pyber, Asymptotic results for permutation groups, in Groups and Computation (eds. L. Finkelstein and W. Kantor), DIMACS Series, vol. 11, pp.197–219, 1993.
  * [52] Á. Seress, Permutation group algorithms, Cambridge Tracts in Math. 152, Cambridge University Press, 2003\.
  * [53] Á. Seress, Primitive groups with no regular orbits on the set of subsets, Bull. London Math. Soc. 29 (1997), 697–704.
  * [54] Á. Seress, The minimal base size of primitive solvable permutation groups, J. London Math. Soc. 53 (1996), 243–255.
  * [55] N. Spaltenstein, Caractères unipotents de ${{}^{3}}D_{4}(\mathbb{F}_{q})$, Comment. Math. Helv. 57 (1982), 676–691.
  * [56] P. Stănică, Good lower and upper bounds on binomial coefficients, JIPAM. J. Inequal. Pure Appl. Math. 2 (2001), Article 30, 5 pp.
  * [57] M. Suzuki, On a class of doubly transitive groups, Annals of Math. 75 (1962), 105–145.
  * [58] G. Verret and B. Xia, Oriented regular representations of out-valency two for finite simple groups, Ars Math. Contemp. 22 (2022), Paper No. 7, 6 pp.
  * [59] R.A. Wilson et al., _A World-Wide-Web Atlas of finite group representations_ ,
http://brauer.maths.qmul.ac.uk/Atlas/v3/

  * [60] B. Xia, Graphical regular representations of $(2,p)$-generated groups, submitted (2023), arXiv:2304.00541.
  * [61] B. Xia, S. Zheng and S. Zhou, Cubic graphical regular representations of some classical simple groups, J. Algebra 612 (2022), 256–280.