# A new viable mass region of Dark matter and Dirac neutrino mass generation
in a scotogenic extension of SM

Nilavjyoti Hazarika1 111E-mail<EMAIL_ADDRESS>Kalpana
<EMAIL_ADDRESS>Department Of Physics, Gauhati University,
Assam, India1

###### Abstract

We propose a scotogenic extension of the Standard Model which can provide a
scalar Dark Matter candidate in the new, theoretically previously unaddressed,
intermediate region ($200\leq M_{DM}\leq 550$ GeV) and also generate light
Dirac neutrino masses. In this framework, the standard model is extended by
three gauge singlet fermions, two singlet scalar fields, and one additional
scalar doublet, all of which are odd under $Z_{2}\times Z_{4}$ discrete
symmetry. These additional symmetries prevent the singlet fermions from
obtaining Majorana mass terms along with providing the stability to the dark
matter candidate. It is known that in the case of the scalar singlet DM model,
the only region which is not yet excluded is a narrow region close to the
Higgs resonance $m_{S}\simeq\frac{m_{h}}{2}$ \- others ruled out from
different experimental and theoretical bounds. In the case of the Inert
doublet model, the mass region ($\sim 60$-$80$ GeV) and the high mass region
(heavier than $550$ GeV) are allowed. This motivates us to explore a parameter
range in the intermediate-mass region $M_{W}\leq M_{DM}\leq 550$ GeV, which we
do in a scotogenic extension of SM with a scalar doublet and scalar singlets.
The dark matter in our model is a mixture of singlet and doublet scalars, in
freeze-out scenario. We constrain the allowed parameter space of the model
using Planck bound on present dark matter relic abundance, neutrino mass, and
the latest bound on spin-independent DM-nucleon scattering cross-section from
XENON1T experiment. Further, we constrain the DM parameters from the indirect
detection bounds arising from the global analysis of the Fermi-LAT
observations of dwarf spheroidal satellite galaxies (dSphs), Higgs invisible
decay and EWPT (Electroweak Precision Test) as well. We find that our model
findings may provide a viable DM candidate satisfying all the constraints on
DM parameters in the new, previously unexplored mass range ($200\leq
M_{DM}\leq 550$ GeV). This new window for the DM candidate could be searched
in furure experiments along with explanation of Dirac mass of neutrinos, since
so far there is no strong evidence in support of Majorana nature of neutrino
mass.

Keywords: Beyond standard model, Dark matter, Neutrino mass, XENON1T

##  I Introduction

It is a well-established fact that the majority of the matter in the universe
is made up of dark matter (DM), a non-luminous, non-baryonic form of matter
whose presence is very well evident from both astrophysical and cosmological
observations ParticleDataGroup:2018ovx . The galaxy cluster observations made
by Zwicky in 1930’s Zwicky:1933gu , observations from galaxy rotation curves
by Rubin in 1970’s Rubin:1970zza , observation of the bullet cluster
Clowe:2006eq  and the latest data from cosmological experiment PLANCK
Planck:2018vyg  suggest that approximately 27$\%$ of the present universe is
composed of DM. This is about five times more than the ordinary visible or
baryonic matter, while the rest of it is composed of dark energy. The present
abundance of DM is expressed as $\Omega_{DM}h^{2}=0.120\pm 0.001$
Planck:2018vyg , where $\Omega_{DM}$ is the DM density parameter and
$h=\boldsymbol{H}/100$ $kms^{-1}Mpc^{-1}$, where $\boldsymbol{H}$ is the
Hubble parameter. The Standard Model (SM) of particle physics which
incorporates all the particle content of the universe is by far the most
successful theory of particle physics. However, it couldn’t explain the
presence of dark matter and its particle nature. Deciphering the exact
particle nature of DM is one of the unresolved issues in the frontiers of
particle physics today. Although astrophysical and cosmological observations
suggest the presence of DM, all the experiments aimed at detecting particle
dark matter have so far reported null results. In the last few decades, this
has motivated several beyond standard model (BSM) proposals. One of the most
popular frameworks beyond SM is the so-called weakly interacting massive
particle (WIMP) paradigm Kolb:1990vq . In this framework, a DM candidate
considered is typically in the electroweak scale mass range and has
interaction rate similar to electroweak interactions which can give rise to
the correct DM relic abundance. Further, in the Standard Model neutrino is
considered to be massless. However, the evidences from Double ChooZ
DoubleChooz:2011ymz  , Daya-Bay DayaBay:2012fng , RENO RENO:2012mkc , and T2K
T2K:2011ypd  experiments hint at the neutrinos to have very small but finite
masses. For a recent update on neutrino masses, please refer to Devi:2021aaz .

The singlet scalar and inert doublet models have been very widely studied in
the literature. The singlet scalar model (SSM) is a simple extension of SM
with a singlet scalar field $S$ added to the SM Lagrangian which is odd under
a global $Z_{2}$ symmetry and all SM fields are even. This model was first
considered from a cosmological point of view by Silveria and Zee
Silveira:1985rk , where the relic abundance from thermal freeze-out and direct
detection cross-section for a stable real scalar gauge singlet was first
calculated. Later complex scalar singlets were considered in McDonald:1993ex
and collider implication on DM-self interaction effects in Burgess:2000yq .
For recent works on scalar singlet models, one could see Yaguna:2008hd ;
Mambrini:2011ik  and GAMBIT:2017gge  for the current status of the scalar
singlet dark matter models. In these works on the scalar singlet DM model, the
only region which is not yet excluded from the observed relic abundance
constraint is a narrow region close to the Higgs resonance
$m_{S}\simeq\frac{m_{h}}{2}$ \- others ruled out from different experimental
and theoretical bounds GAMBIT:2017gge . A similar model includes a scalar
doublet field being added to SM - popularly known as the Inert Doublet Model
(IDM). The lightest electromagnetically neutral component of this inert
doublet serves as a good DM candidate. It was first discussed as a particular
symmetry breaking pattern in two Higgs Doublets Models Deshpande:1977rw .
Since then, IDM has been widely explored in the literature, and for a recent
discussions on IDM, one can refer to Gustafsson:2012aj ; Arhrib:2012ia ;
Goudelis:2013uca ; Belanger:2015kga . The allowed mass region where the
observed relic abundance can be generated for IDM lies in two regions - one at
a low mass region below $W$ boson mass ($M_{DM}<M_{W}$) and the other at high
mass region (around $550$ GeV or above). The low mass region suffers strong
constraints from the direct detection experiments like LUX, PandaX-II and
XENON1T LUX:2016ggv ; PandaX-II:2016vec ; XENON:2017vdw . These constraints
reduce the allowed DM masses in the low mass region to a very narrow mass
region near $m_{DM}\simeq\frac{m_{h}}{2}$. Moreover, since so far there is no
strong evidence in support of Majorana nature of neutrino mass, one can still
consider models that can explain Dirac nature of light neutrino mass.

From above discussion, it is observed that the existing models cannot explain
the intermediate mass range of DM candidates, and this motivated us to
undertake current work. We consider scotogenic type Dirac neutrino mass model
Farzan:2012sa  which can also accommodate dark matter naturally. In this BSM
framework, the SM is extended by three singlet fermions which are odd under SM
gauge symmetries, two singlet scalars, and an inert doublet scalar, but odd
under the unbroken $Z_{2}\times Z_{4}$ symmetry. The unbroken $Z_{2}\times
Z_{4}$ symmetry leads to a stable DM candidate while the $Z_{2}\times Z_{4}$
odd particles generate light neutrino masses at the one-loop level. This model
belongs to the scotogenic extension of SM proposed by Ma in 2006 Ma:2006km .
The mixture of these singlet and doublet scalar fields plays the role of DM in
the freeze-out scenario. This mixing among new fields opens up new
annihilation and co-annihilation channels for dark matter, which contributes
towards making its relic abundance close to its observed value, for the new
range ($200\leq M_{DM}\leq 550$ GeV) of DM candidate. Thus, the novel feature
of this work is that we have been able to find a new viable mass region of DM
in a previously unexplored mass range (200-550 GeV) using the model discussed
in detail below, and the way it can also explain the origin of light neutrino
masses (Dirac mass). We constrain our model with reference to experimental
results from XENON1T (Direct detection), Fermi-LAT (indirect detection), Higgs
invisible decays from LHC, and EWPTs. If the DM is detected in this new mass
window in future experiments, then our model may provide a possible viable
theory for the same.

The paper is organised as follows. In section II, we describe the model, its
particle spectrum, particle composition of DM, and origin of light neutrino
masses. The theory of computing Dark matter abundance in the freeze-out
scenario, DM scattering cross-section, and light neutrino masses in the model
are presented in section III. Section IV contains the results of the above
analysis followed by a discussion on them. We finally summarise and present a
conclusion in section V.

##  II The Model

As stated earlier, in this work we consider a minimal extension of the
Standard Model to include Dark matter and generate neutrino masses at a one-
loop level through the scotogenic framework. The SM is extended with one
scalar doublet $\phi_{2}$ and one singlet $S$ with the usual Higgs doublet
$\phi_{1}$. In addition, another scalar singlet field $\eta$ and three copies
of vector-like fermions $N_{i}$ (i=1,2,3), apart from the SM particle content
have been included. Discrete symmetries $Z_{2}\times Z_{4}$ have also been
considered to forbid unwanted terms in the Lagrangian and to ensure the
stability of DM. Under $Z_{2}$ symmetry, all SM-fields including $\eta$ are
even (have $Z_{2}$ charge $+1$), while fields $\phi_{2}$ and $S$ and $N$ are
odd (have $Z_{2}$ charge $-1$). This $Z_{2}$ symmetry could be realised
naturally as a subgroup of a continuous gauge symmetry like $U(1)_{B-L}$,
where $B$ is the baryon quantum number and $L$ is the lepton quantum number,
with non-minimal field content Dasgupta:2014hha . The $Z_{2}$ symmetry
prevents the tree level Dirac neutrino masses term
$(\overline{L}\tilde{\phi_{1}}\nu_{R})$ involving the SM Higgs. The unbroken
$Z_{2}$ symmetry leads to a stable DM candidate and the additional discrete
symmetry $Z_{4}$ prevents Majorana mass terms of singlet fermions
$(M_{R}\nu_{R}\nu_{R})$. The particle content of our model under $U(1)_{B-L}$,
$Z_{2}$ and $Z_{4}$ symmetries is shown in Table I. The relevant Yukawa
Lagrangian involving the lepton sector is

$L_{f}=y_{l}\overline{L}\phi_{1}e_{R}+M\overline{N_{L}}N_{R}+y^{{}^{\prime}}\overline{N_{L}}S^{\dagger}\nu_{R}+y\overline{L}\tilde{\phi_{2}}N_{R}$
(1)

where $\phi_{1}$ is the SM Higgs doublet and $y$, $y^{{}^{\prime}}$s are
Yukawa couplings. The neutrinos acquire a Dirac mass at one loop level as
shown in the Feynman diagram in figure 1.

Table 1: Particle content of the model under $U(1)_{B-L}$, $Z_{2}$ and $Z_{4}$ Particles | $U(1)_{B-L}$ | $Z_{2}$ | $Z_{4}$ |
---|---|---|---|---
$\phi_{2}$ | 0 | -1 | 1 |
$N_{L}$ | 1 | -1 | 1 |
$N_{R}$ | 1 | -1 | 1 |
$S$ | 0 | -1 | i |
$\eta$ | 0 | 1 | i |
$\nu_{R}$ | 1 | 1 | i |
$\phi_{1}$ | 1 | 1 | 1 |
$L$ | 1 | 1 | 1 |
$e_{R}$ | 1 | 1 | 1 |
$\mu_{R}$ | 1 | 1 | 1 |
$\tau_{R}$ | 1 | 1 | 1 |
$\nu_{L}$$\nu_{R}$$<\eta>$$<\phi_{1}>$$N_{R}$$N_{L}$$\phi_{2}$$S$ Figure 1:
One-loop contribution to Dirac neutrino mass

The scalar potential of our model can be written as follows,

$V=V_{\phi_{1}}+V_{S}+V_{\eta}+V_{\phi_{2}}+V_{int}$ (2)

where,

$\displaystyle\begin{array}[]{l}V_{\phi_{1}}=-\mu_{1}^{2}(\phi_{1}^{\dagger}\phi_{1})+\lambda_{H}(\phi_{1}^{\dagger}\phi_{1})^{2}\\\
V_{S}=\mu_{S}^{2}(S^{\dagger}S)+\lambda_{S}(S^{\dagger}S)^{2}\\\
V_{\eta}=-\mu_{\eta}^{2}(\eta^{\dagger}\eta)+\lambda_{\eta}(\eta^{\dagger}\eta)^{2}\\\
V_{\phi_{2}}=\mu_{2}^{2}(\phi^{\dagger}\phi)+\lambda_{\phi}(\phi^{\dagger}\phi)^{2}\end{array}$
(7)

and,

$\displaystyle\begin{array}[]{rcl}V_{int}&=&\lambda_{3}(\phi_{1}^{\dagger}\phi_{1})(\phi_{2}^{\dagger}\phi_{2})+\lambda_{4}(\phi_{1}^{\dagger}\phi_{2})(\phi_{2}^{\dagger}\phi_{1})+\frac{\lambda_{5}}{2}[(\phi_{1}^{\dagger}\phi_{2})^{2}+h.c]+\lambda_{6}(S^{\dagger}S)(\phi_{1}^{\dagger}\phi_{1})\\\
&&+\lambda_{7}(S^{\dagger}S)(\phi_{2}^{\dagger}\phi_{2})+\lambda_{8}(\eta^{\dagger}\eta)(\phi_{1}^{\dagger}\phi_{1})+\lambda_{9}(\eta^{\dagger}\eta)(\phi_{2}^{\dagger}\phi_{2})+\lambda_{10}(S^{2}\eta^{2}+h.c)\\\
&&+\lambda_{11}(S^{\dagger}S\eta^{\dagger}\eta)+\lambda_{12}(\eta^{\dagger}S\phi_{2}^{\dagger}\phi_{1}+h.c)\end{array}$
(11)

Here, $\lambda$’s and $\mu$’s are various coupling constants and mass
parameters respectively.

After spontaneous symmetry breaking, the SM Higgs $\phi_{1}$ and the inert
doublet $\phi_{2}$ can be expressed as,

$\phi_{1}=\begin{pmatrix}0\\\
\frac{1}{\sqrt{2}}(v+h)\end{pmatrix}\quad,\phi_{2}=\begin{pmatrix}H^{\pm}\\\
\frac{1}{\sqrt{2}}(H_{0}+iA_{0})\end{pmatrix}$ (12)

The vacuum expectation value ($vev$) of the neutral component of the doublet
$\phi_{1}$ is denoted by $v$. The field $h$ corresponds to the physical SM
Higgs-boson. The inert doublet consists of a neutral CP$-$even scalar $H_{0}$
with mass $m_{H_{0}}$, a pseudo-scalar $A_{0}$ with mass $m_{A_{0}}$, and a
pair of charged scalars $H^{\pm}$ with mass $m_{H^{\pm}}$. The singlets $S$
and $\eta$ of the model are assumed to be complex with real part
$S_{1},\eta_{1}$ and complex part $S_{2},\eta_{2}$. The masses of the real
parts are $m_{S_{1}},m_{\eta_{1}}$ and that of the complex parts are
$m_{S_{2}},m_{\eta_{2}}$. Further, it is assumed that the real component of
$\eta$ i.e. $\eta_{1}$ acquires a $vev$ $v_{\eta}$. In mathematical form these
singlets can be written as,

$\displaystyle\begin{array}[]{l}S=\dfrac{1}{\sqrt{2}}(S_{1}+iS_{2}),\\\
\eta=\dfrac{1}{\sqrt{2}}[(\eta_{1}+v_{\eta})+i\eta_{2}]\\\ \end{array}$ (15)

By minimising the potential $V$ we get the masses of different physical
scalars including SM Higgs and inert particles and the singlets as,

$\displaystyle\begin{array}[]{l}m_{h}^{2}=2\lambda_{H}v^{2},\\\
m_{s_{1}}^{2}=\mu_{s}^{2}+\frac{\lambda_{6}}{2}v^{2}+\frac{\lambda_{11}}{2}v_{\eta}^{2}+\lambda_{10}v_{\eta}^{2},\\\
m_{s_{2}}^{2}=\mu_{s}^{2}+\frac{\lambda_{6}}{2}v^{2}+\frac{\lambda_{11}}{2}v_{\eta}^{2}-\lambda_{10}v_{\eta}^{2},\\\
m_{\eta_{1}}^{2}=2\lambda_{\eta}v_{\eta}^{2},\\\
m_{H^{\pm}}^{2}=\mu_{2}^{2}+\frac{\lambda_{3}}{2}v^{2}+\frac{\lambda_{9}}{2}v_{\eta}^{2},\\\
m_{H_{0}}^{2}=\mu_{2}^{2}+\lambda_{L}v^{2}+\frac{\lambda_{9}}{2}v_{\eta}^{2},\\\
m_{A_{0}}^{2}=\mu_{2}^{2}+\lambda_{L}^{{}^{\prime}}v^{2}+\frac{\lambda_{9}}{2}v_{\eta}^{2}\end{array}$
(23)

where $\lambda_{L}=\frac{\lambda_{3}+\lambda_{4}+\lambda_{5}}{2}$ and
$\lambda_{L}^{{}^{\prime}}=\frac{\lambda_{3}+\lambda_{4}-\lambda_{5}}{2}$. We
consider a mixed state formed by the SM Higgs and real part of singlet field
$\eta$, i.e. between $h$ and $\eta_{1}$. The physical mass eigenstates
$\zeta_{1}$ and $\zeta_{2}$ are linear combinations of $h$ and $\eta_{1}$ and
can be written as,

$\displaystyle\begin{array}[]{l}\zeta_{1}=h\cos\theta+\eta_{1}\sin\theta\\\
\zeta_{2}=-h\sin\theta+\eta_{1}\cos\theta\end{array}$ (26)

The masses and mixing angle $\theta$ can be found from the diagonalisation of
the mass matrix

${M_{1}}^{2}=\begin{pmatrix}m_{h}^{2}&m_{\eta_{1}h}^{2}\\\
m_{\eta_{1}h}^{2}&m_{\eta_{1}}^{2}\end{pmatrix}=\begin{pmatrix}2\lambda_{H}v^{2}&\lambda_{8}vv_{\eta}\\\
\lambda_{8}vv_{\eta}&2\lambda_{\eta}v_{\eta}^{2}\end{pmatrix}\\\ $ (27)

Masses of the neutral physical scalars $\zeta_{1}$ and $\zeta_{2}$ are
computed to be,

$\displaystyle\begin{array}[]{l}m_{\zeta_{1}}^{2}=\frac{m_{h}^{2}+m_{\eta_{1}}^{2}}{2}-\frac{m_{h}^{2}-m_{\eta_{1}}^{2}}{2}\sqrt{1+\tan^{2}2\theta},\\\
\\\
m_{\zeta_{2}}^{2}=\frac{m_{h}^{2}+m_{\eta_{1}}^{2}}{2}+\frac{m_{h}^{2}-m_{\eta_{1}}^{2}}{2}\sqrt{1+\tan^{2}2\theta}\end{array}$
(31)

and the mixing angle is,

$\tan 2\theta=\frac{2m_{\eta_{1}h}^{2}}{(m_{h}^{2}-m_{\eta_{1}}^{2})}$ (32)

Similarly, the physical mass eigenstates $H_{1}$ and $H_{2}$ can be obtained
as the mixture of CP-even scalars $S_{1}$ and $H_{0}$. We consider one of the
mixed states $H_{2}$ as the lightest particle, so that it could serve as the
DM candidate. Also, $A_{1}$ and $A_{2}$ are the physical mass eigenstates
obtained from the mixing of CP-odd scalars $S_{2}$ and $A_{0}$. The mass
matrices are:

${M_{2}}^{2}=\begin{pmatrix}m_{H_{0}}^{2}&m_{S_{1}H_{0}}^{2}\\\
m_{S_{1}H_{0}}^{2}&m_{S_{1}}^{2}\end{pmatrix}=\begin{pmatrix}\mu_{\phi}^{2}+\frac{\lambda_{L}}{2}v^{2}+\frac{\lambda_{9}}{2}v_{\eta}^{2}&\frac{\lambda_{12}}{2}vv_{\eta}\\\
\frac{\lambda_{12}}{2}vv_{\eta}&\mu_{s}^{2}+\frac{\lambda_{6}}{2}v^{2}+\frac{\lambda_{11}}{2}v_{\eta}^{2}+\lambda_{10}v_{\eta}^{2}\end{pmatrix}\\\
$ (33)

${M_{3}}^{2}=\begin{pmatrix}m_{A_{0}}^{2}&m_{S_{2}A_{0}}^{2}\\\
m_{S_{2}A_{0}}^{2}&m_{S_{2}}^{2}\end{pmatrix}=\begin{pmatrix}\mu_{\phi}^{2}+\frac{\lambda_{L}^{{}^{\prime}}}{2}v^{2}+\frac{\lambda_{9}}{2}v_{\eta}^{2}&\frac{\lambda_{12}}{2}vv_{\eta}\\\
\frac{\lambda_{12}}{2}vv_{\eta}&\mu_{s}^{2}+\frac{\lambda_{6}}{2}v^{2}+\frac{\lambda_{11}}{2}v_{\eta}^{2}-\lambda_{10}v_{\eta}^{2}\end{pmatrix}$
(34)

The physical masses are obtained via the diagonalisation process of the above
mass matrices respectively as,

$\displaystyle\begin{array}[]{l}m_{H_{1}}^{2}=\frac{m_{H_{0}}^{2}+m_{S_{1}}^{2}}{2}-\frac{m_{H_{0}}^{2}-m_{S_{1}}^{2}}{2}\sqrt{1+\tan^{2}2\alpha},\\\
\\\
m_{H_{2}}^{2}=\frac{m_{H_{0}}^{2}+m_{S_{1}}^{2}}{2}+\frac{m_{H_{0}}^{2}-m_{S_{1}}^{2}}{2}\sqrt{1+\tan^{2}2\alpha}\end{array}$
(38)

and

$\displaystyle\begin{array}[]{l}m_{A_{1}}^{2}=\frac{m_{A_{0}}^{2}+m_{S_{2}}^{2}}{2}-\frac{m_{A_{0}}^{2}-m_{S_{2}}^{2}}{2}\sqrt{1+\tan^{2}2\beta},\\\
\\\
m_{A_{2}}^{2}=\frac{m_{A_{0}}^{2}+m_{S_{2}}^{2}}{2}+\frac{m_{A_{0}}^{2}-m_{S_{2}}^{2}}{2}\sqrt{1+\tan^{2}2\beta}\end{array}$
(42)

The mixing angles are obtained as,

$\displaystyle\tan
2\alpha=\frac{2m_{S_{1}H_{0}}^{2}}{(m_{H_{0}}^{2}-m_{S_{1}}^{2})}$ (43)
$\displaystyle\tan
2\beta=\frac{2m_{S_{2}A_{0}}^{2}}{(m_{A_{0}}^{2}-m_{S_{2}}^{2})}$ (44)

Various couplings of the model can be expressed in terms of the masses as,

$\displaystyle\begin{array}[]{l}\lambda_{H}=\frac{m_{h}^{2}}{2v^{2}},\\\
\lambda_{\eta}=\frac{m_{\zeta_{1}}^{2}\sin^{2}\theta+m_{\zeta_{2}}^{2}\cos^{2}\theta}{2v_{\eta}^{2}},\\\
\lambda_{8}=\frac{(m_{\zeta_{2}}^{2}-m_{\zeta_{1}}^{2})\sin\theta\cos\theta}{vv_{\eta}},\\\
\lambda_{3}=\frac{2(\lambda_{L}v^{2}+m_{H^{\pm}}^{2}-m_{H_{1}}^{2}\cos^{2}\alpha-
m_{H_{2}}^{2}\sin^{2}\alpha)}{v^{2}},\\\
\lambda_{4}=\frac{m_{H_{1}}^{2}\cos^{2}\alpha+m_{H_{2}}^{2}\sin^{2}\alpha+m_{A_{1}}^{2}\cos^{2}\beta+m_{A_{2}}^{2}\sin^{2}\beta-2m_{H^{\pm}}^{2}}{v^{2}},\\\
\lambda_{5}=\frac{m_{H_{1}}^{2}\cos^{2}\alpha+m_{H_{2}}^{2}\sin^{2}\alpha-
m_{A_{1}}^{2}\cos^{2}\beta-m_{A_{2}}^{2}\sin^{2}\beta}{v^{2}},\\\
\mu_{2}^{2}=m_{\zeta_{1}}^{2}\cos^{2}\theta+m_{\zeta_{2}}^{2}\sin^{2}\theta-\frac{\lambda_{L}}{2}v^{2}-\frac{\lambda_{9}}{2}v_{\eta}^{2},\\\
\mu_{S}^{2}=\frac{2(m_{H_{1}}^{2}\sin^{2}\alpha+m_{H_{2}}^{2}\cos^{2}\alpha)-\lambda_{6}v^{2}-\lambda_{11}v_{\eta}^{2}-2\lambda_{10}v_{\eta}^{2}}{2},\\\
\lambda_{12}=\frac{(m_{H_{2}}^{2}-m_{H_{1}}^{2})\sin 2\alpha}{vv_{\eta}}\\\
\end{array}$ (54)

###  II.1 Vacuum Stability

In order that the potential in our model is bounded from below the vacuum
stability requires,

$\displaystyle\lambda_{H},\lambda_{\phi},\lambda_{S},\lambda_{\eta}>0;\quad\lambda_{3}+2\sqrt{\lambda_{H}\lambda_{\phi}}>0$
(55)
$\displaystyle\lambda_{3}+\lambda_{4}-|\lambda_{5}|+\lambda_{H}\lambda_{\phi}>0$
$\displaystyle\lambda_{6}+2\sqrt{\lambda_{H}\lambda_{S}}>0;\quad\lambda_{7}+2\sqrt{\lambda_{\phi}\lambda_{S}}>0$

###  II.2 Light neutrino mass

Light neutrino masses arise at the one-loop level as shown in the Feynman
diagram of figure 1. To obtain Dirac neutrino mass, we consider mixing between
real and imaginary components of the fields $H_{0}$ and $S_{1}$ and also
between $A_{0}$ and $S_{2}$. Let, $H_{1}$ and $H_{2}$ be the mass eigenstates
of $H_{0}$ and $S_{1}$ sector with a mixing angle $\alpha$ . Similarly,
$A_{1}$ and $A_{2}$ be the mass eigenstates of $A_{0}$ and $S_{2}$ sector with
a mixing angle $\beta$. The contribution of the real sector $(H_{0},S_{1})$ to
one loop Dirac neutrino mass can then be written as Borah:2017dfn

$(m_{\nu})_{R_{ij}}=\frac{\sin\alpha\cos\alpha}{32\pi^{2}}\sum_{k}y_{ik}y_{kj}^{\prime}M_{N_{K}}\left[\frac{m_{H_{1}}^{2}}{m_{H_{1}}^{2}-M^{2}_{N_{K}}}\ln\frac{m_{H_{1}}^{2}}{M^{2}_{N_{K}}}-\frac{m_{H_{2}}^{2}}{m_{H_{2}}^{2}-M^{2}_{N_{K}}}\ln\frac{m_{H_{2}}^{2}}{M^{2}_{N_{K}}}\right]$
(56)

Similarly, the contribution of the imaginary sector $(A_{0},S_{2})$ to one
loop Dirac neutrino mass can be written as Borah:2017dfn

$(m_{\nu})_{I_{ij}}=\frac{\sin\beta\cos\beta}{32\pi^{2}}\sum_{k}y_{ik}y_{kj}^{\prime}M_{N_{K}}\left[\frac{m_{A_{1}}^{2}}{m_{A_{1}}^{2}-M^{2}_{N_{K}}}\ln\frac{m_{A_{1}}^{2}}{M^{2}_{N_{K}}}-\frac{m_{A_{2}}^{2}}{m_{A_{2}}^{2}-M^{2}_{N_{K}}}\ln\frac{m_{A_{2}}^{2}}{M^{2}_{N_{K}}}\right]$
(57)

where $M_{N_{k}}$ is the mass eigenvalue of the mass eigenstate $N_{k}$ in the
internal line and the indices $i,j=1,2,3$ run over the three neutrino
generations as well as three copies of $N_{i}$. $m_{H_{1}}$ and $m_{H_{2}}$
are the physical masses of the mass eigenstates $H_{1}$ and $H_{2}$, and
$m_{A_{1}}$ and $m_{A_{1}}$ are the physical masses of the mass eigenstates
$A_{1}$ and $A_{2}$. Thus, the total neutrino mass becomes

$(m_{\nu})_{ij}=(m_{\nu})_{R_{ij}}+(m_{\nu})_{I_{ij}}$ (58)

where, $(m_{\nu})_{R_{ij}}$ and $(m_{\nu})_{I_{ij}}$ are defined in Eq. 56 and
Eq. 57 respectively. For this light neutrino mass to match with experimentally
observed limits $(\sim 0.1)$ $eV$, we expect each of the terms of the above
equation to be of sub-eV scale. Considering $M_{H_{1}}=100$ $GeV$ and
$M_{N}=10$ $TeV$, the first term in the above expression becomes

$(m_{\nu})^{1}_{R_{ij}}=1.46\times 10^{-2}\sin 2\alpha\sum_{k}y_{ik}y_{kj}$

This could become possible naturally if

$\sin 2\alpha y_{ik}y_{kj}^{\prime}<10^{-8}$ (59)

This condition could be satisfied by suitable tuning of the Yukawa couplings
$y.y^{\prime}$. The mixing angles $\alpha,\beta$ depend on various masses,
vev’s and relevant couplings (Eq. 43, 44,54). Thus, one can choose masses of
various particles, and couplings of the scalar Lagrangian appropriately
leading to the required mixing angles $\alpha$ and $\beta$ so that light
neutrino masses agree with their observed values.

##  III Dark Matter

As discussed earlier, the DM candidate considered in our model is the lightest
component $H_{2}$, which is a mixed singlet-doublet scalar dark matter. The
thermal relic abundance of a WIMP type DM particle $H_{2}$ is obtained by
solving the Boltzmann equation Kolb:1990vq

$\frac{dn_{DM}}{dt}+3\boldsymbol{H}n_{DM}=-<\sigma
v>[n_{DM}^{2}-(n_{DM}^{eq})^{2}]$ (60)

where $n_{DM}$ is the number density of the dark matter particle and
$n_{DM}^{eq}$ is the number density of dark matter particle when it was in
thermal equilibrium. $\boldsymbol{H}$ is the Hubble rate of expansion of the
Universe and $<\sigma v>$ is the thermally averaged annihilation cross section
of the dark matter particle. One can obtain the numerical solution of the
Boltzmann equation in terms of partial expansion $<\sigma v>=a+bv^{2}$ as
Scherrer:1985zt

$\Omega_{DM}h^{2}\approx\frac{1.04\times
10^{9}x_{F}}{M_{Pl}\sqrt{g_{\ast}}(a+\frac{3b}{x_{F}})}$ (61)

where $x_{F}=m_{DM}/T_{F}$, $T_{F}$ is the freeze-out temperature, $g_{\ast}$
is the number of relativistic degrees of freedom at the time of freeze-out.
After further simplifications, the above solution takes the form as
Jungman:1995df

$\Omega_{DM}h^{2}\approx\frac{3\times 10^{-27}cm^{3}s^{-1}}{<\sigma v>}$ (62)

###  III.1 Direct detection cross section

The thermal averaged annihilation cross section $<\sigma v>$ is given by
Gondolo:1990dk

$<\sigma
v>=\frac{1}{m_{DM}^{4}TK_{2}^{2}(\frac{m_{DM}}{T})}\int\sigma(s-4m_{DM}^{2})\surd
sK_{1}(\frac{\surd s}{T})ds$ (63)

where $K_{i}$’s are modified Bessel functions of order $i$, $m_{DM}$ is the
mass of Dark Matter particle and $T$ is the temperature of the Universe.
Further, the relevant spin-independent scattering cross-section for the scalar
dark matter $H_{2}$ mediated by SM Higgs can be expressed as Barbieri:2006dq

$\sigma_{SI}=\frac{\lambda_{L}^{2}f^{2}}{4\pi}\frac{\mu^{2}m_{n}^{2}}{m_{h}^{4}m_{DM}^{2}}$
(64)

where $m_{n}$ is the nucleon mass and $\mu=\frac{m_{n}m_{DM}}{m_{n}+m_{DM}}$
is the reduced mass of dark matter and nucleon,
$\lambda_{L}=\frac{\lambda_{3}+\lambda_{4}+\lambda_{5}}{2}$ is the quartic DM-
Higgs coupling and $f$ is the Higgs-nucleon coupling. A recent estimate of $f$
gives $f=0.32$ Giedt:2009mr  although the full range of allowed values is
$f=0.26-0.63$ Mambrini:2011ik .

###  III.2 Invisible Higgs decay

The DM-Higgs coupling $\lambda_{L}$ can be constrained from the latest LHC
constraints on the invisible decay width of the SM Higgs boson. The invisible
Higgs decay width is related to DM-Higgs coupling $\lambda_{L}$ by
ATLAS:2015ciy .

$\Gamma(h\rightarrow Invisible)=\frac{\lambda_{L}v^{2}}{64\pi
m_{h}}\sqrt{1-4m_{DM}^{2}\diagup m_{h}^{2}}$ (65)

###  III.3 Indirect detection cross section

In addition to direct detection experiments, DM parameter space can also be
constrained using results from different indirect detection experiments like
the Fermi-LAT Fermi-LAT:2015att . These experiments search for SM particles,
produced either through DM annihilations or via DM decay in the local
Universe. The final photon and neutrinos, being neutral and stable can reach
the indirect detection experiments without getting affected much by
intermediate regions. For WIMP type DM, these photons lie in the gamma-ray
regime that can be measured at space-based telescopes like the Fermi-LAT
observations of dSphs (dwarf spheroidal satellite galaxies). The observed
differential gamma ray flux produced due to DM annihilations is given by

$\dfrac{d\Phi}{dE}(\vartriangle\Omega)=\frac{1}{4\pi}<\sigma
v>\frac{J(\vartriangle\Omega)}{2M_{DM}^{2}}\dfrac{dN}{dE}$ (66)

here the solid angle corresponding to the observed region of the sky is
$\vartriangle\Omega$, $<\sigma v>$ is the thermally averaged DM annihilation
cross section, $dN/dE$ is the average gamma ray spectrum per annihilation
process. The astrophysical factor $J$ is given by

$J(\vartriangle\Omega)=\int_{\vartriangle\Omega}d\Omega^{\prime}\int_{LOS}dl\rho^{2}(l,\Omega^{\prime})$
(67)

where, $\rho$ is the DM density and LOS corresponds to line of sight. Thus,
one can constrain the DM annihilation into different final states like
$\mu^{+}\mu^{-},\tau^{+}\tau^{-},W^{+}W^{-},b\bar{b}$. Using the bounds on DM
annihilation to these final states, we constrain the DM parameters from the
global analysis of the Fermi-LAT Fermi-LAT:2015att  observations of dwarf
spheroidal satellite galaxies.

##  IV Results and discussion

In this section the results of our work and its implication have been
presented, taking into account the latest constraints on dark matter relic
density (Planck), spin-independent direct detection cross-section from XENON1T
experiment, indirect detection bounds from Fermi-LAT, Higgs invisible decays
limits from LHC, and EWPTs as well. In addition, we impose perturbativity
constraint ($\lambda_{i}<4\pi$ at all scales) on all the parameter points
uniformly. We begin our analysis by imposing the Electroweak Precision Test
(EWPT)Baek:2012uj  constraint in our study which provides an upper bound for
the value of mixing angles as a function of the DM mass. The DM mass is
allowed to vary between $5-1000$ $GeV$ and the mixing angles
$\theta,\alpha,\beta$ to vary between $0$ and $1$ as shown in figure 2. The
couplings $\lambda_{L}$, $\lambda_{\phi}$, $\lambda_{S}$, $\lambda_{7}$,
$\lambda_{9}$, $\lambda_{10}$, $\lambda_{11}$ with DM mass $M_{DM}$ is allowed
to vary between $0$ and $1$. The black line corresponds to the EWPT
(Electroweak precision test) upper bound and the uniform shaded (light orange)
portion is the area ruled out by the EWPT constraints.

Figure 2: Comparison of mixing angles $\theta,\alpha,\beta$ as a function of
DM mass. The black line corresponds to the EWPT upper bound and the uniform
shaded portion (light orange color) is the area ruled out by the EWPT
constraints Baek:2012uj .

We choose $H_{2}$ as the DM candidate, which is a linear combination of
singlet and doublet scalars and scan the parameter space which satisfies
correct relic density constraints. For computation, we have used the software
package micrOMEGA 4.3.2 Belanger:2013oya  to calculate the relic abundance and
spin-independent cross-section of DM. The results of our analysis are shown in
figures (2-7). In figure 3 we have shown the variation of DM relic abundance
with DM mass for our model. It is seen that there is a significant area of
parameter space which can produce correct relic density of DM in our model.
Moreover, there exists a funnel-shaped region around higgs resonance
$M_{DM}\approx m_{h}/2$ corresponding to the s-channel annihilation of DM into
the SM fermions mediated by the Higgs boson. The observed relic abundance is
satisfied for different sets of parameters.

Figure 3: Variation of relic density $\Omega_{DM}h^{2}$ with DM mass for
parameters $\lambda_{i}=0.01$,
$\sin{\theta}=0.1,\sin{\alpha}=0.2,\sin{\beta}=0.3$ and vev $v_{\eta}=100$ GeV

. The black line corresponds to current value of DM relic density.

We then scan the possible values of DM-Higgs coupling $\lambda_{L}$ for the
mass squared difference $(\bigtriangleup m=m_{H_{1}}^{2}-m_{H_{2}}^{2}=10$
$GeV$) and show the allowed region of parameter space in $\lambda_{L}$ vs
$M_{DM}$ plane from the requirement of satisfying the correct relic abundance
in figure 4. The black and red exclusion lines corresponds to XENON1T and LHC
limit on Higgs invisible decay respectively. The shaded region (light orange
colored) is disallowed by LHC limits ATLAS:2015ciy  on invisible decay and
direct detection of XENON1T experiment XENON:2018voc . The latest LHC
constraint on the invisible decay width of the SM Higgs boson is applicable
only for dark matter mass $m_{DM}<\frac{m_{h}}{2}$ (i.e., area to the right
side of red line is not allowed).

Thus, the region of parameter space to the right of red line and below black
line satisfy both LHC and XENON1T limits. Hence, one can justify to choose
lower vales of coupling $\lambda_{L}\leq 0.1$ in our analysis. The scalar DM
in the model can give rise to DM spin-independent scattering cross section
with nucleons that are tightly constrained by the recent bounds from direct
detection experiments like LUX, PandaX-II, and XENON1T LUX:2016ggv ; PandaX-
II:2016vec ; XENON:2017vdw . It would be interesting to investigate the effect
of variation of various coupling constants. Hence, in figure 5, dependence of
spin-independent DM-nucleon cross-section $\sigma_{SI}$ ($cm^{2}$) on coupling
$\lambda_{L}$ points, allowed from correct relic density constraint is shown.
The results are compared with spin-independent DM-nucleon cross-section bounds
from latest XENON1T experiment. From the spin independent direct detection
cross section plot of figure 5, we observe that many points in the parameter
space, satisfying correct relic density constraint Planck:2018vyg  also lie in
the allowed region of XENONIT XENON:2018voc  experimental curve (below black
curve). Thus, it can be stated that the DM candidate with mass $M_{DM}\geq
200$ GeV and with small coupling $\lambda_{L}$ satisfies the experimental
constraints of both the correct relic density and the DM-nucleon scattering
cross section.

Figure 4: Relic density allowed points in $\lambda_{L}$ vs $M_{DM}$ plane. The
blues points are allowed from correct relic density constraints. The black and
red exclusion lines correspond to XENON1T and LHC limit on Higgs invisible
decay respectively (area to the right of red line and below black curve are
allowed by current experimental limits). Figure 5: Spin-independent DM-nucleon
cross-section $\sigma_{SI}$ ($cm^{2}$) for points allowed from correct relic
density constraint. The black curve is the exclusion curve from XENON1T
experiment. The blue colour points are allowed from correct relic density
constraint. The coupling $\lambda_{L}$ is varied in this plot.

Next, we study how the variation of other coupling constants affect the
analysis, and in figure 6, the variation of spin independent DM- nucleon cross
section $\sigma_{SI}$ with DM mass $M_{DM}$ is presented with the constraint
of satisfying the correct relic abundance for different couplings
$\lambda_{7}$, $\lambda_{9}$, $\lambda_{10}$, $\lambda_{\phi}$, $\lambda_{S}$.
Thus, by varying the couplings, the correct relic density could be obtained in
the allowed region from spin independent DM- nucleon cross section
$\sigma_{SI}$ constraint. Other parameters considered here are set to values
as,

$\sin{\theta}=0.1,\quad\sin{\alpha}=0.2,\quad\sin{\beta}=0.3,\quad
v_{\eta}=100\quad GeV.$ (68)

From our results in figure 6, it is clear that our model can predict DM
candidate satisfying the relic and cross section constraints, for a large
range of DM mass, and this validates our model.

Figure 6:  Spin independent DM- nucleon cross section $\sigma_{SI}$ ($cm^{2}$)
for points satisfying correct relic density. The black curve is the exclusion
curve from XENON1T experiment. The region below black curve is allowed from
experimental constraints.

As the DM in our model is a mixture of singlet and doublet scalars, hence in
addition to constraints from direct detection experiments, the DM parameter
space can also be probed in different indirect detection experiments. We
constrain the DM parameters from the indirect detection bounds arising from
the global analysis of the Fermi-LAT Fermi-LAT:2015att  observations of dwarf
spheroidal satellite galaxies (dSphs). In figure 7 we have shown DM
annihilation cross section into $\tau^{+}\tau^{-}$ , $W^{+}W^{-}$ final states
and compared the results with the latest indirect detection bounds of Fermi-
LAT Fermi-LAT:2015att . In figure 7, the regions below red curve is allowed,
and it is seen that the previously ruled out DM mass range ($200\leq
M_{DM}\leq 550$ GeV) could generate correct relic abundance in our model, and
is also allowed from the DM annihilations bounds. In low mass region the
s-channel dark matter annihilation into the SM fermions through Higgs
mediation dominates over other channels. The annihilation cross section of DM
also has additional contributions from co-annihilations between dark matter
$H_{2}$ and heavier components $H_{1}$ Edsjo:1997bg ; Bell:2013wua . We have
noted that the main processes which contribute to the relic abundance of DM
are: $H_{2}H_{2}\rightarrow f\bar{f},hh,W^{+}W^{-},ZZ,h\zeta_{2}$;
$H_{1}H_{2}\rightarrow f\bar{f},h\zeta_{2}$ (for $\zeta_{2}$, please see Eq.
(8)), and $f$ refers to SM fermions. These new annihilation and co-
annihilation channels contribute in generating the correct relic abundance of
dark matter in its new intermediate mass range. Hence, the mixed singlet-
doublet scalar DM candidate $H_{2}$ in this model can very well satisfy both
the relic abundance and direct detection cross section constraints for the new
mass range $200-550$ GeV, and this new viable mass region is the novelty of
this work.

(a) DM annihilations into $\tau^{+}\tau^{-}$

(b) DM annihilations into $W^{+}W^{-}$

Figure 7: DM annihilations into $\tau^{+}\tau^{-}$(left), $W^{+}W^{-}$ (right)
compared against the latest indirect detection bounds of Fermi-LAT. The
regions below red line are allowed.

Thus, in this model, it has been possible to obtain the feasible DM candidate
in previously unexplained mass range $200\leq M_{DM}\leq 550$, which makes the
model very interesting, and is the novelty of this work. Existence of this
medium mass range of the DM has been made possible due to opening of new
annihilation channels of DM into particles in new mass ranges (as new scalars
with varying couplings and mixings are present in the model), which brings the
DM relic density in the allowed range. Also, as is clear from Eqs. (19-22), it
can also explain the sub-eV Dirac neutrinos.

##  V Conclusions

In this work we proposed a dark matter model with the scotogenic extension of
SM, where the DM $H_{2}$ is a mixture of singlet-doublet scalars. We studied
the possibility of generating scalar dark matter in the previously disallowed
DM mass window of $M_{W}\leq M_{DM}\leq 550$ GeV along with generating small
Dirac neutrino mass (since Majorana nature of neutrinos is not yet confirmed).
This region was disallowed in the Inert Doublet model from various
experimental bounds as mentioned earlier. We impose constraints on the
couplings from perturbativity ($\lambda_{i}<4\pi$ at all scales)
considerations on all the points. Also the mixing angles $\theta,\alpha,\beta$
taken are allowed by the EWPT constraints. Further, the DM-Higgs coupling is
chosen such that it satisfies the limits from XENON1T (DD) and LHC limit on
Higgs invisible decay. We scanned the parameter space of our model which
satisfies the latest relic density and spin-independent DM-nucleon scattering
cross-section constraints. It is observed from our results in the figures
(2-7) that DM mass in the new range, i.e. $200\leq M_{DM}\leq 550$ GeV which
was previously not viable, satisfies the relic density bound (from Planck
experiment) as well as the spin-independent cross section bound from XENON1T
experiment. Further, the DM parameters very well satisfy the constraints from
the indirect detection bounds arising from the global analysis of the Fermi-
LAT observations of dSphs. In our opinion, feasibility of this new mass range
of the DM has been made possible due to opening of new annihilation and co-
annihilation channels of DM in the new mass ranges. Moreover, it is observed
that by choosing different mass terms and couplings of the scalar Lagrangian
appropriately one can obtain the Dirac neutrino mass in the sub-eV scale as
well. It may be noted that Majorana nature of neutrinos has not been
established experimentally so far. Hence, this model can explain the origin of
DM in this new intermediate-mass window if detected in future experiments,
which was previously unaddressed in currently available scalar DM models.

##  VI Acknowledgment

We acknowledge the RUSA and FIST grants of Govt. of India for support in
upgrading computer laboratory of the Physics Department of Gauhati University,
where this work was completed. We also thank Dr. Debasish Borah of IIT
Guwahati for his valuable discussions and suggestions, during initial stages
of the work.

##  VII Declarations

Conflicts of interests: The authors declare no potential conflict of
interests.

##  VIII References

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# Quantum advantage beyond entanglement in Bayesian game theory

A. Lowe<EMAIL_ADDRESS>Aston University, School of Informatics & Digital
Engineering, Birmingham B4 7ET, UK

(August 30, 2024)

###### Abstract

Quantum discord has been utilised in order to find quantum advantage in an
extension of the Clauser, Horne, Shimony, and Holt (CHSH) game. By writing the
game explicitly as a Bayesian game, the resulting game is modified such the
payoff’s are different, and crucially restrictions are imposed on the
measurements that Alice and Bob can perform. By imposing these restrictions,
it is found that there exists quantum advantage beyond entanglement for a
given quantum state. This is shown by decomposing the expected payoff into a
classical and quantum term. Optimising over the expected payoff, results in
the classical limit being surpassed. This gives an operational framework in
order to witness and determine quantum discord.

##  I Introduction

The search for quantum advantage has spanned a range of different fields, with
particular focus on finding systems where non-locality offers significant
quantum advantage Arute  _et al._ (2019); Zhong  _et al._ (2020); Brunner and
Linden (2013). However, there has been little effort in attempting to use
other forms of quantum correlations in order to witness quantum advantage. In
particular, quantum discorded states remain under utilised in the literature,
both in quantum game theory and quantum networks. Quantum discord was proposed
in tandem by Ollivier and Zurek Ollivier and Zurek (2001) and Henderson and
Vedral Henderson and Vedral (2001). Quantum discord proposes that quantum
correlations can exist even when there is no entanglement, which is often the
case for mixed separable states. An additional characteristic of quantum
discord is that it is relatively robust against perturbations Werlang  _et
al._ (2009) compared to entanglement and non-locality, which means states
which are discorded are more experimentally feasible.

A significant drawback is that quantum discord is difficult to compute
analytically, so various methods have been discussed in an attempt to
approximate it Hunt  _et al._ (2019). One method is to use geometric discord
Dakić  _et al._ (2010), which has found to be bounded by Fisher Information
for two qubits Lowe and Yurkevich (2022). However, the benefit of geometric
discord is limited as it can increase under local reversible operations Chang
and Luo (2013). There has been an increased focus in recent years on designing
methods for witnessing quantum discord both theoretically, Wang  _et al._
(2023) and even experimentally at the LHC Afik and de Nova (2023). Therefore,
it is of paramount importance that further techniques are explored in order to
witness quantum correlations beyond entanglement, where there is particular
focus on devising experimentally feasible protocols.

Following a different approach in order to witness quantum advantage, there
has been increasing use of game theory. This is a branch of mathematics which
focuses on optimising players decisions in order to achieve their respective
best payoffs. Game theory was developed in the middle of the $20^{\text{th}}$
century by Von Neumann and Morgenstern von Neumann and Morgenstern (1947) and
then extended upon by Nash Nash (1950). Typically, game theory allows only
classical correlations between the players. However, it is possible to design
a game where the players have access to quantum correlations, and thus can use
the properties of quantum mechanics in order to achieve a better payoff
relative to the respective classical counterpart. This is the origin of
quantum game theory. It was pioneered by Meyer Meyer (1999), and then Eisert,
Wilkens, and Lewenstein Eisert  _et al._ (1999) where the first demonstration
of quantum advantage using game-theoretic techniques was proposed. Since then,
quantum game theory has been implemented in evolutionary models Iqbal and Toor
(2000, 2002), and developed onto quantum protocols Ikeda and Lowe (2022);
Ikeda and Aoki (2021) with applications in quantum networks.

Therefore, it is of significant interest to attempt to unite these two fields
and subsequently witness quantum advantage using game-theoretic techniques. In
particular, it is important to attempt to devise a game which can utilise
quantum discord in order to create quantum advantage, as this will be of
practical benefit due its robustness against disorder, and therefore it will
have real-world application. The type of game that will be studied is a
Bayesian game which has only witnessed quantum advantage using a non-local
quantum correlation. This was first proposed for the CHSH game Clauser  _et
al._ (1969) which allows a witness for violation of Bell’s inequality Bell
(1964).

In this paper, an extended quantum game is proposed which utilises the
properties of quantum discord, in order to witness quantum advantage under a
specific set of conditions. The fundamental difference between classical and
quantum correlations is found in the differing probability distributions, thus
it is beneficial to introduce probability in quantum mechanics.

In quantum mechanics, the probability is computed using Born’s rule, which is
defined as

$P_{i}=\rm{tr}\Pi_{i}\rho,$ (1)

where $P_{i}$ is the probability, $\Pi_{i}$ is the projective measurement, and
$\rho$ is the density matrix. For the scenario considered in this paper, the
systems considered are bi-partite two particle spin systems, where the density
matrix is mixed separable. Given this, the probability to measure the spin of
one qubit up or down ($\sigma=\pm 1$) along $\bf a$ and another qubit
($\sigma^{\prime}=\pm 1$) along ${\bf b}$ (where $\bf a$ and $\bf b$ are Alice
and Bob’s respective detector settings) is given by;

$P(\sigma,\sigma^{\prime}|{\bf a},{\bf b})=\rm{tr}\,\Pi_{\sigma|{\bf
a}}\otimes\Pi_{\sigma^{\prime}|{\bf b}}\,\rho\,,$ (2)

where

$\Pi_{\sigma|{\bf a}}=\frac{1}{2}\left[1+\sigma\,{\bf
a}\cdot{\bm{\sigma}}\right]\,,$ (3)

where $\bm{\sigma}$ is the vector of Pauli matrices such that
$\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$. It is important to stress
that projective measurements must take place in each subsystem, such that they
are separable. If the projector was not separable, then it would entangle the
state, and thus any quantum advantage could be due to entanglement.
Subsequently, any claims about the quantum correlations being inherently
related to quantum discord, would be redundant. Throughout this paper, the
measurements take place on the Bloch sphere such that
${\bf{a}}=(\cos\phi_{a}\sin\theta_{a},\sin\phi_{a}\sin\theta_{a},\cos\theta_{a})$
with $|a|=1$, and the assumption that $\phi_{a}=0$ . Similarly these
assumptions are used for $\bf a^{\prime},b,b^{\prime}$.

##  II Quantum discord in a modified CHSH game

The CHSH game is traditionally a Bayesian game, where the players are
cooperative. The Bayesian nature arises due to each player having a private
prior probability, where the other players are unaware of this. This prior
probability is used to compute their expected payoff. For this game, the Nash
equilibrium for the game is the maximum payoff that the players can achieve,
as any unilateral deviation to their strategy would result in a worse payoff
due to the cooperative nature of the game. However, the game has only been
studied using either classical states, or non-local states. After outlining
the relevant aspects of quantum discord, the modified form of the CHSH game is
introduced. From this, the modified game is shown to yield quantum advantage,
when considering a mixed separable state which has a non-zero quantum discord
in a given measurement space.

###  II.1 Quantum Discord

Quantum discord for a given state quantifies quantum correlations by taking
the difference between the mutual information between two parties $A$ and $B$
before and after a local projective measurement. This definition reveals how
quantum correlations can persist even in the case of zero entanglement. This
implies mixed separable states can possess non-zero discord, and therefore
non-zero quantum correlation. By defining the mutual information as

$I(\rho^{AB})=S(\rho^{A})+S(\rho^{B})-S(\rho^{AB}),$ (4)

where
$S(\rho^{AB})=-\tr\rho^{AB}\log\rho^{AB}=-\sum_{i}\lambda_{i}\ln\lambda_{i}$,
$\rho^{AB}$ is the density matrix of the system, and $\lambda_{i}$ are the
eigenvalues of the density matrix. In order to define the quantum mutual
information from player A’s perspective given a measurement on player B’s
subsystem, the sum over all local projective measurements is taken. Therefore,
the density matrix after measurement has the form

$\rho_{A|\Pi_{\mu}^{B}}=\frac{1}{p_{\mu}}\tr_{B}(\mathbb{1}\otimes\Pi_{\mu}^{B})\rho^{AB}(\mathbb{1}\otimes\Pi_{\mu}^{B}),$
(5)

where $\Pi_{\mu}^{B}=(1/2)(\mathbb{1}+\mu\bf{n}.\bm{\sigma})$, $\mu=\pm 1$,
where $\bf{n}$ is the Bloch vector. Additionally

$p_{\mu}=\tr(\mathbb{1}\otimes\Pi_{\mu}^{B})\rho^{AB}.$ (6)

Note the projector used here only acts in one of the subsystems, not both as
defined previously. Therefore quantum discord is defined using local
projective measurements in one subsystem. If the quantum discord was computed
using $B$’s subsystem given measurements on $A$’s subsystem, then the
projective measurements would take place in $A$’s subspace. From these
definitions the conditional entropy can be computed as

$S(A|\Pi_{\mu}^{B})=\sum_{\mu}p_{\mu}S(\rho_{A|\Pi_{\mu}^{B}}).$ (7)

Subsequently, the quantum mutual information is

$J_{A}(\rho^{AB})=S(\rho^{A})-S(A|\Pi_{\mu}^{B}).$ (8)

Therefore the quantum discord can be defined as

$\begin{split}D_{A}(\rho^{AB})&=\min\limits_{\Pi_{\mu}^{B}}[I(\rho^{AB})-J_{A}(\rho^{AB})]\\\
&=\min\limits_{\Pi_{\mu}^{B}}S(A|\Pi_{\mu}^{B})+S(\rho^{B})-S(\rho^{AB}).\end{split}$
(9)

Note that due to the minimisation over all projective measurements of player
B, any quantum correlations which exist must be due to the initial state, and
not due to a particular choice of projector. The only states which yield zero
quantum discord take the form

$\rho^{AB}=\rho^{A}\otimes\rho^{B}.$ (10)

Therefore, generally, states of the form

$\rho^{AB}=\sum_{\mu=0}^{N}q_{\mu}\rho_{\mu}^{A}\otimes\rho_{\mu}^{B},$ (11)

result in non-zero discord, where $\sum_{\mu=0}^{N}q_{\mu}=1$. Note how eq.
(11) is clearly a mixed separable state as each subsystem acts in its own
space. This reveals that quantum correlations can exist beyond entanglement,
as will be demonstrated in the solution of the game.

There are a few extra comments about this definition. It is clear that discord
is NP-hard to compute since it is generally a discrete optimisation problem
which scales exponentially with the dimensionality of the system. The
definition given is the quantum discord for player A. Therefore, the quantum
discord for one subsystem is not necessarily the same as for the other
subsystem, which implies that quantum discord is in general not symmetric,
i.e. $D_{A}(\rho^{AB})\neq D_{B}(\rho^{AB})$. However, to derive the
definition for player B follows a similar procedure, but the projective
measurements will take place in player A’s subspace. Given this definition of
quantum discord, the original CHSH game can be introduced to explicitly show
how quantum correlations can be revealed and understood using game-theoretic
techniques.

###  II.2 Description of the game

The general operational procedure for the CHSH game can be described as
follows: a referee, Charlie sends bits either 0 or 1 to Alice, and similarly
Charlie sends bits either 0 or 1 to Bob. When Alice receives bit 0, she sets
her detector setting to ${\bf{a}}$ and then performs a measurement on her
shared state with Bob. When Alice receives bit 1, she sets her detector
setting to ${\bf{a^{\prime}}}$ and performs a measurement on her shared state.
When Bob receives bit 0, he sets his detector setting to ${\bf{b}}$, and
performs a measurement on their shared state, and when Bob receives bit 1, he
sets his detector setting to ${\bf{b^{\prime}}}$ and performs a measurement.
Alice and Bob do not communicate what bits they have received, so they are
unaware which game they are playing. Given Alice and Bob’s combined
measurements, we can assign a payoff. For example, if they both receive bit 0,
and therefore are in the type of game given by $({\bf{a}},{\bf{b}})$, and they
both measure spin up, then they are assigned a payoff of 1, as shown by the
payoff table in Table I. These payoff’s are specifically for the CHSH game,
therefore any deviation from Table I will result in a modified version of the
original game.

($\alpha,\beta$) ${\bf{b}}$ ${\bf{b^{\prime}}}$ ${\bf{a}}$ $\uparrow$
$\downarrow$ $\uparrow$ 1 0 $\downarrow$ 0 1 $\uparrow$ $\downarrow$
$\uparrow$ 1 0 $\downarrow$ 0 1 ${\bf{a^{\prime}}}$ $\uparrow$ $\downarrow$
$\uparrow$ 1 0 $\downarrow$ 0 1 $\uparrow$ $\downarrow$ $\uparrow$ 0 1
$\downarrow$ 1 0

Table 1: This table shows the payoff matrix for the traditional CHSH game.
Based on what bits they receive, a,b,a’,b denote how Alice and Bob’s detectors
are set up. From this, they then perform measurements on their shared states
and based on their results, they are assigned a payoff. Also,
$\alpha=\\{{\bf{a}},{\bf{a^{\prime}}}\\}$,
$\beta=\\{{\bf{b}},{\bf{b^{\prime}}}\\}$ and therefore denotes what bits the
players receive, either 0 or 1.

The general expected payoff to player A for an arbitrary state for this type
of game is given by

$u_{A}=\sum_{\alpha,\beta,\sigma,\sigma^{\prime}}u_{A,\sigma,\sigma^{\prime}}^{\alpha,\beta}P_{A}(\alpha,\beta)P(\sigma,\sigma^{\prime}|\alpha,\beta),$
(12)

where $u_{A,\sigma,\sigma^{\prime}}^{\alpha,\beta}$ is a tensor of payoffs for
player $A$, $P_{A}(\alpha,\beta)$ is player A’s prior belief of which type of
game they will be in, and $P(\sigma,\sigma^{\prime}|\alpha,\beta)$ represents
the conditional probability for measuring a given spin depending on what type
of game the players are in. A similar equation written for player $B$’s payoff
is given by

$u_{B}=\sum_{\alpha,\beta,\sigma,\sigma^{\prime}}u_{B,\sigma,\sigma^{\prime}}^{\alpha,\beta}P_{B}(\alpha,\beta)P(\sigma,\sigma^{\prime}|\alpha,\beta),$
(13)

where $u_{B,\sigma,\sigma^{\prime}}^{\alpha,\beta}$ are the payoff’s for
player $B$ and $P_{B}(\alpha,\beta)$ is player B’s prior belief of which type
of game they will be in. For the traditional CHSH game, the payoffs assigned
to Alice and Bob are the same, as well as each player’s prior beliefs,
therefore $u_{A}=u_{B}$. Using classical statistics, the best payoff Alice and
Bob can achieve is $0.75$. This can be seen if they choose their detector
settings such that they will always measure spin up, regardless of the type of
game they are in. However, if the players know that they are allowed to share
a Bell state, given by

$\rho=\ket{\psi}\bra{\psi},\hskip
28.45274pt\ket{\psi}=\frac{1}{\sqrt{2}}\big{(}\ket{\uparrow\uparrow}+\ket{\downarrow\downarrow}\big{)},$
(14)

then the optimum payoff they can achieve is $(1/4)(2+\sqrt{2})\approx 0.85$.
In both scenarios, the prior probability is given by
$P(\alpha,\beta)=P(\alpha)P(\beta)$ as Alice and Bob are independent, and they
each believe $P(\alpha)=P(\beta)=1/2$, therefore $P(\alpha,\beta)=1/4$. This
is justifiable as they have no prior information on what type of game they
will be in, so they can only assume each game will be played uniformly. It is
important to note that strict quantum advantage (i.e. surpassing 0.75) can
only be found in the CHSH game for non-local states. However this does not
necessarily mean there is no quantum advantage that can be gained from a mixed
separable quantum correlated state.

Consider an extension of the game, where the payoff’s are different such that
the payoff table is given in Table II.

($\alpha,\beta$) ${\bf{b}}$ ${\bf{b^{\prime}}}$ ${\bf{a}}$ $\uparrow$
$\downarrow$ $\uparrow$ -1 1 $\downarrow$ 1 -1 $\uparrow$ $\downarrow$
$\uparrow$ -1 1 $\downarrow$ 1 -1 ${\bf{a^{\prime}}}$ $\uparrow$ $\downarrow$
$\uparrow$ -1 1 $\downarrow$ 1 -1 $\uparrow$ $\downarrow$ $\uparrow$ 1 -1
$\downarrow$ -1 1

Table 2: The numbers denote Alice’s and Bob’s payoff. As before, based on what
bits they receive, a,b,a’,b’ denote how Alice and Bob’s detectors are set up.
From this, they then perform measurements on their shared states and based on
their results, they are assigned a payoff.

This scenario is similar to the traditional CHSH game, where it is a
cooperative game and thus both players maximising their payoff’s is the Nash
equilibrium, as their payoff’s are the same. Unlike the previous CHSH game, a
non-entangled state will be used in order to witness quantum advantage
utilising quantum correlations. The shared state between Alice and Bob is
discorded, where it is defined as

$\rho^{AB}(x)\equiv\rho_{x}^{AB}=\frac{1}{2}\Big{[}\ket{\uparrow\uparrow}\bra{\uparrow\uparrow}+\ket{xx}\bra{xx}\Big{]},$
(15)

where $\ket{x}=\cos(x/2)\ket{\uparrow}+\sin(x/2)\ket{\downarrow}$. This is a
mixed separable state as it can be written in the same form as eq. (11), and
therefore does not possess any non-locality or entanglement. When numerically
computing the discord, the result is given by Fig. (1).

Figure 1: This shows the numerical discord for the state in eq. (15). It is
clear there is zero quantum correlation at $x=0,\pi$. Elsewhere, there is non-
zero quantum correlation, with the maximum occurring around $x\approx
n\pi+\pi/2$, where $n\in\mathbb{Z}$. The minimisation was over $\theta$, and
$\phi$ as defined in the projectors. This can only occur for a 2-qubit state,
as the parameterisation for $\theta$ and $\phi$ is only valid for a two
particle system. Note $\phi$ is included here for the optimisation to compute
the discord, but for the optimisation in the game, $\phi=0$ to allow
analytical tractability.

Since both Alice and Bob know the shared state beforehand, they can tune their
respective detector settings in order to maximise their respective payoffs.
However, this optimisation includes optimising over each others detector
settings. Therefore each others maximum payoff depends on what the other
player will do. Fortunately, the cooperative nature of this game yields a
solution in which both players would choose the optimum detector settings
which would benefit each other.

###  II.3 Solution to the game

At this stage, it is important to emphasise that both Alice and Bob have
complete knowledge of each other’s payoff tables, and the state which they
share. Their only sources of lack of knowledge arise since they do not know
what the value of the parameter $x$ will be, and what type of game they will
end up playing. Also, it is assumed that the prior $P(\alpha,\beta)=1/4$ as
before. This is because the players have no knowledge of what type of game
they will play. Additionally, the state used to compute the conditional
probability is given in eq. (15). Given this knowledge that both Alice and Bob
share, they can compute their expected payoff’s as functions of each other’s
choices and the discord parameter $x$. These are given by

$u_{A}=u_{B}=f(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}},x),$
(16)

where

$\begin{split}&f(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}},x)=-\frac{1}{16}\Big{[}2\cos\theta_{a,b}^{-}\\\
&+2\cos\theta_{a^{\prime},b}^{-}+2\cos\theta_{a,b^{\prime}}^{-}-2\cos\theta_{a^{\prime},b^{\prime}}^{-}+\cos\theta_{a,b}^{+}\\\
&\cos\theta_{a^{\prime},b}^{+}+\cos\theta_{a,b^{\prime}}^{+}-\cos\theta_{a^{\prime},b^{\prime}}^{+}\\\
&+\cos(\theta_{a,b}^{+}-2x)+\cos(\theta_{a^{\prime},b}^{+}-2x)\\\
&+\cos(\theta_{a,b^{\prime}}^{+}-2x)-\cos(\theta_{a^{\prime},b^{\prime}}^{+}-2x)\Big{]},\end{split}$
(17)

where $\theta_{\alpha,\beta}^{-}=\theta_{\alpha}-\theta_{\beta}$ and
$\theta_{\alpha,\beta}^{+}=\theta_{\alpha}+\theta_{\beta}$. This expected
payoff can be re-written by adding and subtracting
$\cos\theta_{a,b}^{+}+\cos\theta_{a,b^{\prime}}^{+}+\cos\theta_{a^{\prime},b}^{+}-\cos\theta_{a^{\prime},b^{\prime}}^{+}$
such that eq. (17) becomes

$\begin{split}&f(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}},x)=-\frac{1}{16}\Big{\\{}2\Big{[}\cos\theta_{a,b}^{-}\\\
&+\cos\theta_{a^{\prime},b}^{-}+\cos\theta_{a,b^{\prime}}^{-}-\cos\theta_{a^{\prime},b^{\prime}}^{-}+\cos\theta_{a,b}^{+}\\\
&\cos\theta_{a^{\prime},b}^{+}+\cos\theta_{a,b^{\prime}}^{+}-\cos\theta_{a^{\prime},b^{\prime}}^{+}\Big{]}\\\
&+\cos(\theta_{a,b}^{+}-2x)+\cos(\theta_{a^{\prime},b}^{+}-2x)\\\
&+\cos(\theta_{a,b^{\prime}}^{+}-2x)-\cos(\theta_{a^{\prime},b^{\prime}}^{+}-2x)\\\
&-\cos\theta_{a,b}^{+}-\cos\theta_{a,b^{\prime}}^{+}-\cos\theta_{a^{\prime},b}^{+}+\cos\theta_{a^{\prime},b^{\prime}}^{+}\Big{\\}},\end{split}$
(18)

It is important to emphasise that this has not changed the result, but now
allows eq. (18) to be written into a purely classical term, and a purely
quantum term. Finally, the expected payoff can be written as

$\begin{split}f(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}},x)&=f_{Cl}(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}})\\\
&+f_{Q}(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}},x),\end{split}$
(19)

where the classical term is given by

$\begin{split}&f_{Cl}(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}})=-\frac{1}{4}\Big{[}\cos\theta_{a}(\cos\theta_{b}\\\
&+\cos\theta_{b^{\prime}})+\cos\theta_{a^{\prime}}(\cos\theta_{b}-\cos\theta_{b^{\prime}})\Big{]},\end{split}$
(20)

and the quantum term has the form

$\begin{split}&f_{Q}(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}},x)=-\frac{\sin
x}{8}\Big{[}\sin(\theta_{a,b}^{+}-x)\\\
&+\sin(\theta_{a^{\prime},b}^{+}-x)+\sin(\theta_{a,b^{\prime}}^{+}-x)\\\
&-\sin(\theta_{a^{\prime},b^{\prime}}^{+}-x)\Big{]}.\end{split}$ (21)

Due to the $\sin x$ term, eq. (21) becomes zero at $x=0,\pi$ and non-zero
elsewhere. This is a truly quantum contribution, which can be seen by noting
that the quantum correlations for the shared state vanish when $x=0,\pi$, seen
explicitly in Fig. 1.

Given it is possible to split the expected payoff into two contributions,
coming from classical correlations and quantum correlations, opens up the
possibility that there may be some advantage to be gained from the extra
quantum term. By maximising over all the variables with no restrictions, the
maximum expected payoff is $0.5$ which coincides with the largest classical
payoff. This is expected, as the state considered is a mixed separable state,
and surpassing this bound would imply non-locality. However, by restricting
the measurement space available to the players, such that they are only
allowed to set their measurement angles from 0 to $\pi/2$, yields an
unexpected quantum advantage for the players. Numerically maximising eq. (19),
it is found that

$f^{max}(\theta_{a}^{*},\theta_{a^{\prime}}^{*},\theta_{b}^{*},\theta_{b^{\prime}}^{*},x^{*})=0.30178\equiv
f^{*},$ (22)

where $\theta_{a}^{*}=\theta_{b}^{*}=\pi/2$,
$\theta_{a^{\prime}}^{*}=\theta_{b^{\prime}}^{*}=0$, and crucially
$x^{*}=7\pi/8$. The key result is that a non-zero $x$ gave the maximum for the
expected payoff. For comparison, by considering a purely classical scenario
($x=0$), the maximum expected payoff is 0.25. This is emphasised in Fig. 2.

Figure 2: This plot reveals how there are regions where there is quantum
advantage, for given different values of $x$, such that the expected payoff is
greater than $0.25$. The maxima of this function are found at $7\pi/8$ and
$15\pi/8$, which yields 0.30178.

It is also possible to consider scenarios where the quantum correlation
dominates the classical correlation. Defining the ratio between the quantum
contribution and the classical contribution, yields

$\begin{split}&\kappa(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}},x)=\absolutevalue{\frac{f_{Q}(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}},x)}{f_{Cl}(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}})}}\\\
&\equiv\kappa,\end{split}$ (23)

where the quantum contribution must obey
$f_{Q}(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}},x)\geq 0$
and similarly the classical part is constrained such that
$f_{Cl}(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}})\geq 0$.
Similar ratios could be defined for differing signs. Therefore, when
$\kappa>1$, this implies the quantum contribution is larger than the classical
contribution, and conversely $\kappa<1$ implies the classical contribution is
dominating the quantum contribution. When the contributions from both the
classical and quantum part are equal, then $\kappa=1$. This ratio gives a
measure of how much quantum correlation is in a given quantum game relative to
the classical counterpart. It is important to emphasise, the total correlation
may not reveal quantum advantage in the particular quantum game. However, it
is possible that the classical correlation is zero, and there is a non-zero
quantum correlation resulting in an infinite $\kappa$. This is the case for
$\theta_{a}=\pi/2$, $\theta_{a^{\prime}}=\pi/4$, and
$\theta_{b}=\theta_{b^{\prime}}=0$. Whether it is possible to develop a
hierarchy between classical and quantum correlation below the CHSH bound
remains an open question.

###  II.4 Relations

There are interesting relations which can be derived from the results of the
game. These can be analysed by considering eqns. (20) and (21) individually,
which are given by

$\begin{split}&f_{Cl}^{max}(\theta_{a}^{*}=\pi/2,\theta_{a^{\prime}}^{*}=0,\theta_{b}^{*}=\pi/2,\theta_{b^{\prime}}=0)\\\
&=0.25=\frac{4}{16},\end{split}$ (24)

and

$\begin{split}&f_{Q}^{max}(\theta_{a}^{*}=\pi/2,\theta_{a^{\prime}}^{*}=0,\theta_{b}^{*}=\pi/2,\theta_{b^{\prime}}=0,\\\
&x=7\pi/8)\approx 0.05178=\frac{0.828}{16},\end{split}$ (25)

which interestingly when combined can be written as

$\begin{split}f^{max}(\theta_{a}^{*},\theta_{a^{\prime}}^{*},\theta_{b}^{*},\theta_{b^{\prime}}^{*},x^{*})&\approx\frac{1}{16}\big{(}4+0.828\big{)}\\\
&=\frac{1}{16}\big{(}2+2\sqrt{2}\big{)}.\end{split}$ (26)

This can be directly compared with the result from the CHSH game using a non-
local state, where the differences are due to the denominator, and an extra
factor of 2 in front of $\sqrt{2}$. The apparent connection to the maximum
bound for the CHSH inequality of $2\sqrt{2}$ was not expected as the state
considered for maximum violation is strictly non-local. Despite this, it is
clear that the extra quantum term in the game considered here gives rise to an
extra contribution which can be utilised for quantum advantage in this
Bayesian game.

When calculating the maximum expected payoff, it was noted that an interesting
relation occurs between this expected payoff and the trace of the Hessian
matrix. Given the trigonometric nature of the expected payoff, it is clear
that the second derivatives will return the same functions with the signs
changed. The sum of the second derivatives with respect to their corresponding
variables is proportional to the expected payoff such that

$\tr\hat{H}=-2f(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}},x),$
(27)

where $\hat{H}$ is the Hessian matrix. Since

$\tr\hat{H}=\sum_{i}\phi_{i},$ (28)

where $\phi_{i}$ are the eigenvalues of the Hessian matrix yields the relation
that

$f(\theta_{a},\theta_{a^{\prime}},\theta_{b},\theta_{b^{\prime}},x)=-\frac{1}{2}\sum_{i}\phi_{i}.$
(29)

Therefore, optimising over the eigenvalues of the Hessian matrix is equivalent
to optimising over the expected payoff function. It is important to stress,
this has only been found for this particular payoff function, but whether
there is an underlying relation between the eigenvalues of the Hessian and the
expected payoff in Bayesian game theory remains an interesting direction for
future research.

##  III Discussion

The proposed game offers tangible benefit for experimentalists when building
quantum discorded states. Given the algorithmic procedure of this game, this
operational method gives a framework in which the experimentalists could
verify that the state they have attempted to build is correct. For example, an
experimentalist would know the state they have designed gives quantum
advantage when performing the proposed analysis of this paper using the given
constraints. This work could be extended such that, when attempting to build
other states, similar calculations can be performed and further quantum
advantage could be witnessed.

There are two significant limitations of the analysis performed in this paper.
The main drawback is that quantum advantage was demonstrated for a specific
discorded state, rather than proving quantum advantage for a general discorded
state. However, this paper opens up the avenue for developing methodology
which can witness quantum advantage using a general discorded state in a
quantum game. Subsequently, this will be a key focus of future research.

Another significant limitation, is due to the restrictions imposed on Alice
and Bob’s measurements. Whilst it is beneficial for witnessing quantum
correlations, the maximum classical expected payoff will always be the upper
bound for mixed separable states. To further generalise the results,
$\phi_{a}$ would also be generally non-zero, similarly for
${\bf{a^{\prime}}},{\bf{b}}$ and ${\bf{a^{\prime}}}$. Whether this allows
further quantum advantage would be a significant question to investigate.

It would be interesting to determine if there is a generic relation between
the measurement space and the ability to witness quantum advantage. The same
logic could be applied; if given a specific quantum state, is there a
measurement space which would witness the quantum correlation.

From a game-theoretic perspective, it would be interesting to modify this game
such that Alice and Bob are competing, thus no longer cooperative. Given this
change, how would the quantum correlations affect the Nash equilibrium
compared to the classical solution. This may offer further insight into the
benefit of utilising quantum correlations in game theory. As a further
extension of this, this quantum game could be implemented onto a quantum
network, where a quantum protocol could be devised which could offer
technological benefit by utilising quantum technologies. It is also worth
studying whether quantum advantage beyond entanglement can be witnessed in
extensive form games, as quantum advantage has already been shown for non-
local states Ikeda (2023).

It was also interesting to note the similarities between the maximum expected
payoff in the CHSH game, and the result in eq. (26). Given the discorded state
is inherently local, it is intriguing to see a factor of $2\sqrt{2}$
contributing to the quantum advantage. A further unexpected result that arose
from the analysis was the relation between the spectra of the eigenvalues of
the Hessian matrix, and the expected payoff function. Further investigation is
required to determine whether this is a general relation.

##  IV Conclusion

Quantum advantage is witnessed in a modified version of the CHSH game, where
the advantage is entirely due to quantum correlations beyond entanglement.
This proof of concept result was done by taking a specific discorded state,
and imposing measurement restrictions on the players. Using these constraints,
the players were able to surpass the classical limit for the given
restrictions, entirely due to the quantum correlations. By utilising game-
theoretic techniques, this protocol could be readily implemented in
experiments, in order to witness and characterise quantum advantage.

This is the first theoretical result using quantum discord for quantum
advantage in Bayesian game theory. Future research arising from this work will
extend quantum discord into a wide range of quantum games, in order to witness
and quantify quantum advantage, with specific focus on the importance of the
measurement space.

##  V Acknowledgements

The author thanks David Lowe for insightful discussions about incomplete
information games, Igor Yurkevich for assistance with probability in quantum
mechanics and Kazuki Ikeda for discussing the applications of quantum game
theory. This work was supported by the Leverhulme Trust Grant No.
RPG-2019-317.

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