# Revisiting the Scalar Leptoquark ($S_{1}$) Model with the Updated Leptonic
Constraints

Bibhabasu De<EMAIL_ADDRESS>Department of Physics, The ICFAI University
Tripura, Kamalghat-799210, Tripura, India

###### Abstract

The Standard Model, if extended to the energy scale of $\mathcal{O}(1)$ TeV,
the known particle spectrum could be augmented with a scalar leptoquark.
Within this minimally extended framework, explaining the anomalous magnetic
moment and electric dipole moment simultaneously for the three lepton
generations over a parameter space consistent with all the lepton flavor
violating bounds is possible. Such a model can be tested through the collider
search experiments and/or by probing the low-energy lepton phenomena. This
work studies the current prospects of the model in the presence of recent
experimental updates for the leptonic observables.

##  I Introduction

The Standard Model (SM) has already explained the color and electroweak
sectors up to a high degree of testable precision. Further, the discovery of
the 125 GeV Higgs boson at the Large Hadron Collider (LHC) has completed the
proposed particle spectrum of the SM ATLAS:2012yve ; CMS:2012qbp . However,
certain experimental observations and theoretical issues can’t be explained
within the framework of the SM and thus indicate the presence of some New
Physics (NP) yet to be explored. For example, the idea of gauge coupling
unification hints at a more fundamental theory corresponding to a single gauge
group. The SM gauge group, i.e., $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$
can be considered as its effective low-energy version obtained via a
particular symmetry-breaking chain. The list of such Grand Unified Theories
(GUT) includes $SU(4)$ Pati:1974yy , $SU(5)$ Georgi:1974sy , $SO(10)$
Georgi:1974my ; Fritzsch:1974nn , $E_{6}$ Kang:2007ib ; Hati:2015awg , etc. It
is interesting to note that within a GUT structure, quarks and leptons can
directly couple at the tree-level through a hypothetical mediator — Leptoquark
(LQ) (for recent reviews, see Refs. Dorsner:2016wpm ; Davidson:1993qk ;
Hewett:1997ce ; Nath:2006ut ). Though, in principle, within a local quantum
field theory LQs can either be scalar or vector, the scalar LQs are more
useful to study the loop-induced Beyond Standard Model (BSM) contributions
Blumlein:1996qp ; Fajfer:2015ycq ; Barbieri:2015yvd . LQs are crucial from
various phenomenological aspects. For example, an extension of the SM with a
LQ can explain several B-meson anomalies Dorsner:2013tla ; Gripaios:2014tna ;
Becirevic:2015asa ; Becirevic:2016yqi ; Crivellin:2017zlb ; Cline:2017aed ;
DiLuzio:2017chi ; Mandal:2018kau ; Aydemir:2019ynb ; Crivellin:2019dwb or can
contribute to the flavor violating processes like $\tau\rightarrow\mu\gamma$
and $h\rightarrow\tau\mu$ PhysRevD.93.015010 . LQs may also be significant for
the dark matter phenomenology Mandal:2018czf ; Choi:2018stw and the production
of scalar particles at the LHC Bhaskar:2020kdr ; Bhaskar:2022ygp ;
DaRold:2021pgn ; Agrawal:1999bk ; Enkhbat:2013oba . Note that the simplest GUT
extensions assume a heavy LQ Super-Kamiokande:2014otb ; Dorsner:2012nq to
evade the proton lifetime constraints, but they can’t be produced at the LHC.
However, there are GUT formulations that can explain the stability of proton
with a TeV-scale scalar LQ BUCHMULLER1986377 ; Murayama:1991ah ;
Dorsner:2005fq ; GEORGI1979297 ; FileviezPerez:2007bcw ; Senjanovic:1982ex .
Thus, in this paper, the later GUT motivation will be considered as the gauge
theoretical background for the new interactions, i.e., the SM will be extended
to an energy scale of $\mathcal{O}(1)$ TeV to augment the observed particle
spectrum with a scalar LQ.

Recent experiments have resulted in some remarkable observations in the lepton
sector, which can’t be addressed with the SM. For example, in 2021 a combined
result from the Fermilab-based Muon $g-2$ collaboration and Brookhaven
National Laboratory (BNL) showed a $4.2\sigma$ discrepancy between the
predicted and measured values of the anomalous magnetic moment of muon
Abi:2021gix ; Albahri:2021ixb . The result has been updated very recently on
August 2023, enhancing the significance to $5\sigma$ Muong-2:2023cdq .
Moreover, a precision measurement of the fine-structure constant using either
Cesium (Cs) Parker_2018 or Rubidium (Rb) Morel:2020dww indicates a similar
anomaly in $(g-2)_{e}$. However, note that a relative sign between the two
results leads to an experimental dispute that can’t be settled with the
present technologies. LQs can play a vital role in explaining the discrepancy
in $(g-2)_{\mu}$ Djouadi1990 ; PhysRevD.53.555 ; Cheung:2001ip ;
Dorsner:2019itg ; Greljo:2021xmg ; Kowalska:2018ulj . Moreover, in the
presence of a scalar LQ, various NP signatures, e.g., the neutrino
oscillation, $W$ mass anomaly, lepton flavor violating decays and dark matter
can be connected to the $(g-2)_{e,\,\mu}$ anomalies within a single BSM
formulation Chen:2022hle ; Choi:2018stw ; Saad:2020ihm ; Gherardi:2020qhc ;
Chen:2022hle ; Bhaskar:2022vgk ; Parashar:2022wrd ; Zhang:2021dgl ;
ColuccioLeskow:2016dox ; Mandal:2019gff . LQs can also have important
implications to explain the electric dipole moment (EDM) of leptons
Altmannshofer:2020ywf ; Dekens:2018bci ; Fuyuto:2018scm .

In this paper, a minimal extension of the SM has been considered with a scalar
LQ $S_{1}(\mathbf{\bar{3}},\,\mathbf{1},\,1/3)$ at an energy scale of
$\mathcal{O}(1)$ TeV. In Ref. Mandal:2019gff , it has already been studied in
detail that such a simple BSM framework can easily explain all the possible NP
signatures and experimental constraints in the lepton sector. However, we
shall see that the scenario could be simplified further if formulated with a
particular flavor ansatz. The present work will try to constrain the parameter
space for all the three lepton generations simultaneously considering the
current experimental updates on $(g-2)_{\ell}$ and EDM. However, due to
experimental inadequacy, the $\tau$-sector is not at all interesting compared
to $e$ and $\mu$. For $e$-sector, both experimental possibilities (i.e., the
results from the Cs and Rb experiments) will be addressed through a common
generic formulation. A direct consequence of augmenting the SM with a LQ is
opening up the 2-body and 3-body charged lepton flavor violating (CLFV) decay
channels and initiating a possibility for the lepton flavor violating Higgs
decays PhysRevD.93.015010 ; Chang:2016zll ; Husek:2021isa . However, the
experimental upper limits associated with the non-observation of these
processes can easily be explained within the considered model by adjusting the
lepton-quark couplings in a $3\times 3$ flavor basis, making the parameter
space consistent with the CLFV bounds. The paper has been organized as
follows. Sec. II introduces the new interactions arising at the TeV scale. In
Sec. III, $(g-2)_{\ell}$ and EDM have been defined along with their recent
experimental bounds. Sec. IV elaborates on the one-loop BSM contributions to
the $\ell\ell\gamma$ vertex appearing in the presence of $S_{1}$, whereas in
Sec. V, the allowed parameter space has been analyzed using numerical
techniques. Finally, the outcomes have been summarized in Sec. VI.

##  II The Model: A Minimal Extension of the SM

The considered model assumes a simple extension of the SM at a NP scale
$\Lambda\sim\mathcal{O}(1)$ TeV, where the known particle spectrum gets
augmented with a scalar Leptoquark (LQ) of electromagnetic (EM) charge $1/3$ —
usually labeled as $S_{1}\equiv S_{1}(\mathbf{\bar{3}},\,\mathbf{1},\,1/3)$.
Following the notations of Ref. Dorsner:2016wpm , the NP Lagrangian can be
cast as,

$\displaystyle\mathcal{L}_{\Lambda}=\Big{[}\lambda_{L}^{ij}\left(\bar{Q}_{L}^{Cia\beta}\epsilon^{ab}L_{L}^{jb}\right)S^{\beta}_{1}+\lambda_{R}^{ij}\left(\bar{u}_{R}^{Ci\beta}S^{\beta}_{1}\ell_{R}^{j}\right)+{\rm
h.c.}\Big{]}+\bar{M}_{S_{1}}^{2}(S_{1}^{\dagger}S_{1})+\kappa(H^{\dagger}H)(S_{1}^{\dagger}S_{1}).$
(1)

Here, the EM cahrge has been defined as $Q_{\rm EM}=T_{3}+Y$.
$Q_{L}\equiv(u_{L}\quad d_{L})^{T}$ and
$L_{L}\equiv(\nu_{L}\quad\ell_{L})^{T}$ denote the left-handed quark and
lepton doublets, whereas $u_{R}$ and $\ell_{R}$ stand for the right-handed up-
type quarks and charged leptons, respectively. The superscript $C$ defines the
charge conjugation. The indices $\\{i,\,j\\}$ and $\\{a,\,b\\}$ define the
flavor and $SU(2)$ indices, respectively. $\beta$ refers to the color index.
After electroweak symmetry breaking (EWSB) only the SM Higgs acquires a vacuum
expectation value (VEV) as,

$\displaystyle H=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}0\\\
v+h\end{array}\right),$ (4)

where $v=246$ GeV. Thus, the physical mass of $S_{1}$ can be cast as,

$\displaystyle M_{S_{1}}=\sqrt{\bar{M}_{S_{1}}^{2}+\frac{\kappa
v^{2}}{2}}~{},$ (5)

where, $\bar{M}_{S_{1}}$ is the bare mass term and $\kappa$ is a dimensionless
coupling. In principle, one should consider the kinetic term for $S_{1}$ in
Eq. (1). However, the NP contributions arising through the interaction of
$S_{1}$ with the gauge bosons (gluon and $\gamma$) 111For a detailed study,
see Ref. Bhaskar:2020kdr . are irrelevant in the lepton sector. Hence the
kinetic term can be dropped for simplicity.

The NP couplings $\lambda_{L,R}^{ij}$ play a crucial role in the low-energy
lepton phenomenology. However, to explain the observed discrepancies in the
anomalous magnetic moments and the discovery prospects of the electric dipole
moments for all the three lepton generations ($e,\,\mu,\,\tau$), along with
the non-observation of the charged lepton flavor violating (CLFV) processes,
e.g., $\ell_{j}\to\ell_{i}\gamma$ Baldini:2013ke ; Aushev:2010bq ; MEG:2016leq
; BaBar:2009hkt , $\ell_{j}\to 3\ell_{i}$ Blondel:2013ia ; BELLGARDT19881 ;
Aushev:2010bq ; Hayasaka:2010np and $h\to\ell_{i}\ell_{j}$ ATLAS:2019old ;
ATLAS:2019pmk ; CMS:2017con , it is helpful to consider the following flavor
ansatz:

$\displaystyle\lambda_{L,R}=\left(\begin{array}[]{c c
c}0&0&\lambda_{L,R}^{u,\tau}\\\ 0&\lambda_{L,R}^{c,\mu}&0\\\
\lambda_{L,R}^{t,e}&0&0\end{array}\right).$ (9)

Moreover, in Sec. V, we shall see that this particular choice of quark-lepton
mixing results in an enhancement of the NP contributions to
$a_{\ell}=(g-2)_{\ell}/2$ and EDM through the chirality-flipping terms.

##  III New Physics Observables and Experimental Bounds

The most generic gauge invariant representation for the effective
$\ell\ell\gamma$ vertex corresponding to Fig. 1 is given by,

$\displaystyle\Gamma^{\mu}_{\ell\ell\gamma}=\gamma^{\mu}\mathcal{F}_{1}(q^{2})+\left\\{i\mathcal{F}_{2}(q^{2})+\mathcal{F}_{4}(q^{2})\,\gamma^{5}\right\\}\left(\frac{\sigma^{\mu\nu}q_{\nu}}{2m_{\ell}}\right)+\mathcal{F}_{3}(q^{2})(q^{\mu}\not{q}-q^{2}\gamma^{\mu})\gamma^{5},$
(10)

where $\mathcal{F}_{(1,2,3,4)}$ are the form factors and $q$ represents the
photon momentum.

Figure 1: Effective $\ell\ell\gamma$ vertex. $p_{1,\,2}$ represent the
external momenta, with $q=p_{1}-p_{2}$ being the photon momentum.

However, in the case of an off-shell photon, there should be additional
contributions in Eq. (10). Note that the form factors $\mathcal{F}_{3}$ and
$\mathcal{F}_{4}$ must vanish in any parity-conserving theory (e.g., QED) and
can only arise through the diagrams where electroweak (EW) gauge bosons appear
as the virtual particles. Thus, the renormalized vertex correction in QED
results in PhysRev.73.416 ,

$\displaystyle\mathcal{F}_{1}(0)=0,\qquad\mathcal{F}_{2}(0)=\frac{\alpha_{\rm
EM}}{2\pi},$ (11)

where $\mathcal{F}_{1}(0)$ corresponds to the correction in EM charge while
$\mathcal{F}_{2}(0)$ represents the QED contribution to the anomalous magnetic
moment of leptons at $\mathcal{O}(\alpha_{\rm EM})$. However, in the presence
of the weak gauge bosons, the $\ell\ell\gamma$ vertex gets modified as
Hollik:1991qb 222The axial vector coupling associated with the form factor
$\mathcal{F}_{3}$ vanishes for on-shell photons as a consequence of the Ward
identity Ward:1950xp .,

$\displaystyle\Gamma^{\mu}_{\rm EW}$
$\displaystyle=e\Bigg{[}\left(1+\frac{\delta
e}{e}\right)\gamma^{\mu}+\frac{i\sigma^{\mu\nu}q_{\nu}}{2m_{\ell}}\left\\{\frac{\alpha_{\rm
EM}}{2\pi}+\mathcal{F}^{EW}_{2}(0)\right\\}+\frac{\sigma^{\mu\nu}q_{\nu}}{2m_{\ell}}\gamma^{5}\mathcal{F}_{4}(0)\Bigg{]},$
(12)

where $\delta e$ denotes the sum of the charge correction at one-loop order
and the corresponding counterterm. The additional contribution to the
anomalous magnetic moment can be parametrized as Jackiw:1972jz ,

$\displaystyle\mathcal{F}_{2}^{EW}(0)=\frac{\mathcal{G}_{F}\,m_{\ell}^{2}}{8\sqrt{2}\,\pi^{2}}\left[\frac{5}{3}+\frac{1}{3}(1-4\sin^{2}\theta_{W})^{2}+\mathcal{O}\left(\frac{m_{\ell}^{2}}{M_{W}^{2}}\right)\right],$
(13)

where, $\mathcal{G}_{F}$, $\theta_{W}$, and $M_{W}$ signify the Fermi
constant, weak mixing angle, and mass of the $W$-boson, respectively.
Moreover, considering the leading order (LO) hadronic contribution, one
obtains Gourdin:1969dm ,

$\displaystyle\mathcal{F}_{2}(0)^{\rm Had}{\rm[LO]}=\left(\frac{\alpha_{\rm
EM}}{\pi\sqrt{3}}\right)^{2}\int_{m^{2}_{\pi}}^{\infty}\frac{K(s)}{s}R^{(0)}(s)\,ds,$
(14)

where, $K(s)$ stands for the QED kernel function Brodsky:1967sr and
$R^{(0)}(s)$ represents the ratio of electron-positron bare annihilation cross
into the hadrons to the cross section of muon-pair production with center of
mass energy $\sqrt{s}$. Similarly, the last term in Eq. (12), i.e.,
$\mathcal{F}_{4}(0)$ represents the leading order SM contribution to the
electric dipole moment ($d_{\ell}$) of leptons. As already mentioned in Sec.
I, despite considering all the SM contributions these leptonic observables
exhibit a sharp discrepancy with the experimental results.

###  III.1 Anomalous Magnetic Moment

The best available SM prediction for the anomalous magnetic moment of muon is
given by $a_{\mu}^{\rm SM}=116591810(43)\times 10^{-11}$ Aoyama:2020ynm ,
whereas the recent experimental data from Muon $g-2$ collaboration results in
a world average of $a_{\mu}^{\rm Exp}=116592059(22)\times 10^{-11}$
Muong-2:2023cdq , leading to a discrepancy,

$\displaystyle\Delta a_{\mu}=a^{\rm Exp}_{\mu}-a^{\rm SM}_{\mu}=(2.49\pm
0.48)\times 10^{-9}~{}(5.0\,\sigma).$ (15)

It is one of the most remarkable signatures of a possible BSM sector. Further,
in the context of electrons, experiments indicate a similar anomaly in
$(g-2)_{e}$. A precision measurement of the fine structure constant through
the recoil of ${\rm Cs}^{133}$ atoms has yielded a notable contradiction
between the measured and predicted values of $a_{e}$ as Parker_2018 ,

$\displaystyle\Delta a^{\rm(Cs)}_{e}=a^{\rm Exp\,(Cs)}_{e}-a^{\rm
SM}_{e}=(-8.8\pm 3.6)\times 10^{-13}~{}(2.4\,\sigma).$ (16)

However, for the same $a_{e}$ a Rubidium based experiment results in
Morel:2020dww ,

$\displaystyle\Delta a^{\rm(Rb)}_{e}=a^{\rm Exp\,(Rb)}_{e}-a^{\rm
SM}_{e}=(4.8\pm 3.0)\times 10^{-13}~{}(1.6\,\sigma).$ (17)

Note that, despite having a significant expectation value, the experimental
measurements for $\Delta a_{e}$ have large error bars. However, this paper is
able to address both of the results for $\Delta a_{e}$, along with the non-
zero value of $\Delta a_{\mu}$ within a common BSM framework.

Unlike the first two generations, measuring the anomalous magnetic moment of
$\tau$ is extremely challenging due to its short lifetime. Thus,
$a_{\tau}^{\rm Exp}$ can only be traced back from the secondary particles
produced through the decay of $\tau$. The latest experimental bound (95% CL)
can be quoted as DELPHI:2003nah ; ParticleDataGroup:2022pth ,

$\displaystyle-0.052<a_{\tau}<0.013,$ (18)

whereas, the corresponding SM prediction is given by, $a_{\tau}^{\rm
SM}=117721(5)\times 10^{-8}$ Eidelman:2007sb .

###  III.2 Electric Dipole Moment

The precision measurement of the electric dipole moment of leptons can be
crucial to search for the NP. EDM can be related to the form factor
$\mathcal{F}_{4}$ as, $d_{\ell}=e\mathcal{F}_{4}(0)/m_{\ell}$, for which the
SM predicts
$|\mathcal{F}^{e}_{4}(0)|<|\mathcal{F}^{\mu}_{4}(0)|<|\mathcal{F}^{\tau}_{4}(0)|\approx
10^{-23}$ Booth:1993af ; PhysRevD.54.3377 ; PhysRevLett.125.241802 ;
PhysRevD.103.013001 , i.e., $|d_{\tau}^{\rm SM}|\simeq 10^{-37}~{}e\,{\rm
cm}$. It is much smaller than the experimental sensitivity. Thus, any
observation of lepton EDM can be treated as a direct evidence of some New
Physics interaction. The experimental upper limits for the three lepton
generations can be read as ACME:2018yjb ; Muong-2:2008ebm ; Belle:2002nla ,

$\displaystyle|d_{e}|<0.11\times 10^{-28}~{}e\,{\rm cm}~{}(90\%~{}{\rm CL}),$
$\displaystyle|d_{\mu}|<1.8\times 10^{-19}~{}e\,{\rm cm}~{}(95\%~{}{\rm CL}),$
$\displaystyle{\rm Re}\,[d_{\tau}]\supset\left[-0.220,\,0.45\right]\times
10^{-16}~{}e\,{\rm cm}~{}(95\%~{}{\rm CL}),$ $\displaystyle{\rm
Im}\,[d_{\tau}]\supset\left[-0.250,\,0.08\right]\times 10^{-16}~{}e\,{\rm
cm}~{}(95\%~{}{\rm CL}).$ (19)

These experimental bounds on $\Delta a_{\ell}$ and $d_{\ell}$ will be
simultaneously considered to constrain the chosen parameter space for each
lepton generation. Moreover, one should note that the parameter space is
inherently consistent with all the CLFV bounds.

##  IV BSM Contributions to $(g-2)_{\ell}$ and EDM

As already stated in Sec. I, in the presence of a scalar LQ, there can be new
contributions to the $\ell\ell\gamma$ vertex at one-loop order. Fig. 2(a)
shows the case where the photon couples to the up-type quarks, while Fig. 2(b)
represents the situation when photon touches the $S_{1}$ propagator (magenta
line). The former will be referred to as Type-1 diagram, while the latter will
be called Type-2 for convenience.

((a)) (a) ((b)) (b)

Figure 2: BSM contributions to the $\ell\ell\gamma$ vertex, where (a) the up-
type quarks couple to the photon (Type-1 diagram), and (b) the LQ $S_{1}$
couples to the photon (Type-2 diagram). $p_{1},\,p_{2}$ represent the external
momenta.

###  IV.1 Type-1 Diagram

The correction term to $\ell_{j}\ell_{j}\gamma$ vertex due to the Type-1
diagram can be computed as,

$\displaystyle\Delta\Gamma^{\sigma}_{1}$
$\displaystyle=iN_{C}\int\frac{d^{4}k}{(2\pi)^{4}}\Bigg{[}(-\lambda_{L}^{ij}P_{L}+\lambda_{R}^{ij}P_{R})\frac{(\not{p}_{2}-\not{k}+m_{i})}{(k-p_{2})^{2}-m_{i}^{2}}(Q_{\rm
EM}^{i}\gamma^{\sigma})\frac{(\not{p}_{1}-\not{k}+m_{i})}{(k-p_{1})^{2}-m_{i}^{2}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\frac{1}{k^{2}-M_{S_{1}}^{2}}\\{-(\lambda_{L}^{ij})^{*}P_{R}+(\lambda_{R}^{ij})^{*}P_{L}\\}\Bigg{]}$
$\displaystyle\equiv iQ_{\rm
EM}^{i}\,N_{C}\int\frac{d^{4}k}{(2\pi)^{4}}\Bigg{[}\frac{\mathcal{N}_{1}^{\sigma}}{\mathcal{D}_{1}}\Bigg{]}.$
(20)

Here $N_{C}=3$ defines the color degeneracy factor, and $Q_{\rm EM}^{i}=2/3$
represents the EM charge of up-type quarks in the unit of electronic charge
$e$. $m_{i}$ denotes the up-type quark masses for $i=u,\,c,\,t$. The numerator
can be rearranged as,

$\displaystyle\mathcal{N}_{1}^{\sigma}=\frac{1}{2}\Bigg{[}\mathcal{A}_{1}\Big{\\{}$
$\displaystyle(\not{p}_{2}-\not{k})\gamma^{\sigma}(\not{p}_{1}-\not{k})+m_{i}^{2}\gamma^{\sigma}\Big{\\}}+\mathcal{A}_{2}m_{i}\Big{\\{}(\not{p}_{2}-\not{k})\gamma^{\sigma}+\gamma^{\sigma}(\not{p}_{1}-\not{k})\Big{\\}}$
$\displaystyle+\mathcal{A}_{3}\gamma^{5}\Big{\\{}(\not{p}_{2}-\not{k})\gamma^{\sigma}(\not{p}_{1}-\not{k})+m_{i}^{2}\gamma^{\sigma}\Big{\\}}+\mathcal{A}_{4}\,m_{i}\gamma^{5}\Big{\\{}(\not{p}_{2}-\not{k})\gamma^{\sigma}+\gamma^{\sigma}(\not{p}_{1}-\not{k})\Big{\\}}\Bigg{]},$
(21)

where,

$\displaystyle\mathcal{A}_{1}$
$\displaystyle=|\lambda_{R}^{ij}|^{2}+|\lambda_{L}^{ij}|^{2}\,,\qquad\qquad\mathcal{A}_{2}=-2\,{\rm
Re}[(\lambda_{L}^{ij})^{*}\lambda_{R}^{ij}],$ $\displaystyle\mathcal{A}_{3}$
$\displaystyle=|\lambda_{R}^{ij}|^{2}-|\lambda_{L}^{ij}|^{2}\,,\qquad\qquad\mathcal{A}_{4}=-2\,{\rm
Im}[(\lambda_{L}^{ij})^{*}\lambda_{R}^{ij}].$ (22)

After Feynman parametrization, the denominator can be cast as,

$\displaystyle\mathcal{D}_{1}=n^{2}-\Delta_{1}(x),$ (23)

where $n=k-yp_{1}-zp_{2}$ and
$\Delta_{1}(x)=M_{S_{1}}^{2}\Big{[}x+\rho_{i}(1-x)\Big{]}$. $x,\,y,\,z$ are
the Feynman parameters and $\rho_{i}=(m_{i}/M_{S_{1}})^{2}$. This calculation
assumes an on-shell photon and the physically viable approximation of
$(m_{\ell}/M_{S_{1}})^{2}\to 0$. $m_{\ell}$ denotes the mass of the SM
leptons. Integrating over the loop momentum $n$, the BSM contributions to the
anomalous magnetic moment ($\Delta a^{\ell}_{1}$) and electric dipole moment
($d_{1}^{\ell}$) of the SM leptons can be defined as,

$\displaystyle\Delta a^{\ell}_{1}$
$\displaystyle=\frac{1}{8\pi^{2}}\Bigg{[}\mathcal{A}_{1}\left(\frac{m_{\ell}}{M_{S_{1}}}\right)^{2}G_{1}(\rho_{i})+\mathcal{A}_{2}\left(\frac{m_{\ell}\,m_{i}}{M^{2}_{S_{1}}}\right)G_{2}(\rho_{i})\Bigg{]},$
(24) $\displaystyle d_{1}^{\ell}$
$\displaystyle=\frac{e}{8\pi^{2}}\Bigg{[}\mathcal{A}_{3}\left(\frac{m_{\ell}}{M^{2}_{S_{1}}}\right)G_{1}(\rho_{i})+\mathcal{A}_{4}\left(\frac{m_{i}}{M^{2}_{S_{1}}}\right)G_{2}(\rho_{i})\Bigg{]},$
(25)

where, the functions $G_{1}$ and $G_{2}$ are given by,

$\displaystyle G_{1}(w)$
$\displaystyle=\int_{0}^{1}\left[\frac{x(1-x)^{2}}{x+(1-x)w}\right]\,dx=\frac{2+3w-6w^{2}+w^{3}+6w\,{\rm
ln}\,w}{6(1-w)^{4}},$ $\displaystyle G_{2}(w)$
$\displaystyle=\int_{0}^{1}\left[\frac{(1-x)^{2}}{x+(1-x)w}\right]\,dx=\frac{-3+4w-w^{2}-2\,{\rm
ln}\,w}{2(1-w)^{3}}\,.$ (26)

###  IV.2 Type-2 Diagram

Fig. 2 (b) contributes to the $\ell_{j}\ell_{j}\gamma$ vertex as follows.

$\displaystyle\Delta\Gamma^{\sigma}_{2}$
$\displaystyle=iN_{C}\int\frac{d^{4}k}{(2\pi)^{4}}\Bigg{[}(-\lambda_{L}^{ij}P_{L}+\lambda_{R}^{ij}P_{R})\frac{(\not{k}+m_{i})}{k^{2}-m_{i}^{2}}.\frac{1}{(k-p_{1})^{2}-M_{S_{1}}^{2}}\,.\,Q_{\rm
EM}^{S_{1}}(p_{1}+p_{2}-2k)^{\sigma}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\frac{1}{(k-p_{2})^{2}-M_{S_{1}}^{2}}\\{-(\lambda_{L}^{ij})^{*}P_{R}+(\lambda_{R}^{ij})^{*}P_{L}\\}\Bigg{]},$
$\displaystyle\equiv iQ_{\rm
EM}^{S_{1}}\,N_{C}\int\frac{d^{4}k}{(2\pi)^{4}}\Bigg{[}\frac{\mathcal{N}_{2}^{\sigma}}{\mathcal{D}_{2}}\Bigg{]}.$
(27)

Here $Q_{\rm EM}^{S_{1}}=1/3$ is the EM charge of $S_{1}$. Recasting the
numerator of Eq. (27), one gets,

$\displaystyle\mathcal{N}_{2}^{\sigma}=\frac{1}{2}\Big{[}\\{\mathcal{A}_{1}\not{k}+\mathcal{A}_{2}m_{i}\\}+\gamma^{5}\\{\mathcal{A}_{3}\not{k}+\mathcal{A}_{4}m_{i}\\}\Big{]}(p_{1}+p_{2}-2k)^{\sigma}.$
(28)

Feynman parametrization recasts the denominator as,

$\displaystyle\mathcal{D}_{2}$
$\displaystyle=(k-yp_{1}-zp_{2})^{2}-M_{S_{1}}^{2}\Big{[}x\rho_{i}+(1-x)\Big{]}$
$\displaystyle=n^{2}-\Delta_{2}(x).$ (29)

Thus, the NP contributions to the anomalous magnetic moment and EDM, arising
from the Type-2 diagram, can be formulated as,

$\displaystyle\Delta a^{\ell}_{2}$
$\displaystyle=-\frac{1}{16\pi^{2}}\Bigg{[}\mathcal{A}_{1}\left(\frac{m_{\ell}}{M_{S_{1}}}\right)^{2}G_{3}(\rho_{i})+\mathcal{A}_{2}\left(\frac{m_{\ell}\,m_{i}}{M^{2}_{S_{1}}}\right)G_{4}(\rho_{i})\Bigg{]},$
(30) $\displaystyle d_{2}^{\ell}$
$\displaystyle=-\frac{e}{16\pi^{2}}\Bigg{[}\mathcal{A}_{3}\left(\frac{m_{\ell}}{M^{2}_{S_{1}}}\right)G_{3}(\rho_{i})+\mathcal{A}_{4}\left(\frac{m_{i}}{M^{2}_{S_{1}}}\right)G_{4}(\rho_{i})\Bigg{]},$
(31)

where,

$\displaystyle G_{3}(w)$
$\displaystyle=\int_{0}^{1}\left[\frac{x(1-x)^{2}}{xw+(1-x)}\right]\,dx=\frac{1-6w+3w^{2}+2w^{3}-6w^{2}\,{\rm
ln}\,w}{6(1-w)^{4}},$ $\displaystyle G_{4}(w)$
$\displaystyle=\int_{0}^{1}\left[\frac{x(1-x)}{xw+(1-x)}\right]\,dx=\frac{1-w^{2}+2w\,{\rm
ln}\,w}{2(1-w)^{3}}\,.$ (32)

Therefore, within this minimally extended BSM framework, the complete NP
contribution to the leptonic observables can be defined as,

$\displaystyle\Delta a_{\ell}=\Delta a^{\ell}_{1}+\Delta
a^{\ell}_{2}\,,\qquad|d_{\ell}|=\left|d_{1}^{\ell}+d_{2}^{\ell}\right|.$ (33)

##  V Numerical Analysis and Results

In this section, we shall try to identify the allowed region of the parameter
space through flavor-specific constraints. $\Delta a_{\ell}$ and the
experimental upper bound on EDM will be considered simultaneously as the
constraining factors for each generation. For completeness, one can enlist the
free parameters of this model as follows:

$\displaystyle\Big{\\{}M_{S_{1}},\,\lambda_{L,R}^{u,\tau},\,\lambda_{L,R}^{c,\mu},\,\lambda_{L,R}^{t,e}\Big{\\}}.$

Note that, respecting the LHC constraints at $\sqrt{s}=13$ TeV one has to
choose $M_{S_{1}}\geq 1.5$ TeV CMS:2018ncu ; ATLAS:2020dsk ; ATLAS:2019ebv ;
CMS:2018lab ; CMS:2018svy . However, the NP couplings can be varied freely
within the bounds of perturbative unitarity Allwicher:2021rtd .

((a)) (a) ((b)) (b)

((c)) (c)

Figure 3: Allowed parameter space for the first two lepton generations
constrained through (a) $\Delta a_{e}^{\rm(Cs)}$ & $|d_{e}|$, (b) $\Delta
a_{e}^{\rm(Rb)}$ & $|d_{e}|$, and (c) $\Delta a_{\mu}$ & $|d_{\mu}|$. The four
different colors correspond to $M_{S_{1}}=1.5$ TeV (violet), 2.0 TeV (golden),
2.5 TeV (sky blue), and 3.0 TeV (red)

Fig. 3(a) shows the allowed parameter space in the
$\lambda_{L}^{t,e}-\lambda_{R}^{t,e}$ plain for a set of four $M_{S_{1}}$
values: $M_{S_{1}}=1.5$ TeV (violet), 2.0 TeV (golden), 2.5 TeV (sky blue),
and 3.0 TeV (red). The depicted region simultaneously satisfies the observed
$\Delta a^{\rm(Cs)}_{e}$ value and the experimental bound on $|d_{e}|$.
However, it is a notable feature of this considered framework that even if one
assumes the $\Delta a^{\rm(Rb)}_{e}$ results instead of Cs, a valid parameter
space can be obtained [see Fig. 3(b)]. Similarly, for the muon sector
$\lambda_{L}^{c,\mu}-\lambda_{R}^{c,\mu}$ plain has been constrained through
the $\mu$-specific observables, i.e., $\Delta a_{\mu}$ and $|d_{\mu}|$ [see
Fig. 3(c)]. Note that, numerically, the same exercise can be repeated for the
$\tau$-sector to constrain the $\lambda_{L}^{u,\tau}-\lambda_{R}^{u,\tau}$
region. However, the present experimental sensitivity is inadequate to probe
the NP effects to $a_{\tau}$ and $d_{\tau}$ that one can obtain from Fig. 2.
Thus, no significant conclusion can be drawn in this case and the entire
parameter space is effectively available.

Fig. 3 leads to two interesting observations:

  * •

With increasing $M_{S_{1}}$ value, the magnitude of the couplings shifts to
the higher side. This particular behavior can be understood by analyzing the
$M_{S_{1}}$-dependence of $\Delta a_{\ell}$ and $d_{\ell}$ for a fixed set of
fermion masses. From Eqs. (24)-(25) and (30)-(31) it is clear that there is an
overall $M_{S_{1}}^{2}$ suppression. However, the complete
$M_{S_{1}}$-dependence can only be noted by studying the individual variation
of the functions $G_{\\{1,2,3,4\\}}$. Fig. 4 shows the variation of the $G$
functions with respect to $M_{S_{1}}$. For illustration,
$m_{\ell}=m_{\mu}=0.105$ GeV and $m_{i}=m_{c}=1.275$ GeV have been assumed
ParticleDataGroup:2022pth .

Figure 4: Variation of $G_{1},\,G_{2},\,G_{3}$ and $G_{4}$ as a function of
$M_{S_{1}}$ for $m_{\ell}=m_{\mu}=0.105$ GeV and $m_{i}=m_{c}=1.275$ GeV.
Here, $\rho_{c}=(m_{c}/M_{S_{1}})^{2}$.

Fig. 4 clearly indicates that $G_{1},\,G_{3},\,G_{4}$ do not exhibit any
notable variation with the increasing $M_{S_{1}}$ value, while $G_{2}$ shows
only a slight increment. Thus, to a good approximation, one can conclude that
for a given set of quark and lepton masses, $\Delta a_{\ell}$ and $|d_{\ell}|$
decreases quadratically with $M_{S_{1}}$. Therefore, for compensating this
suppression, the couplings must rise to match the experimental observations.

  * •

In Fig. 3(a) the product $\lambda_{L}^{t,e}\times\lambda_{R}^{t,e}$ is
positive, whereas it flips to a negative value in Fig. 3(b). This is a direct
consequence of the oppositely aligned values of $\Delta a^{\rm(Cs)}_{e}$ and
$\Delta a^{\rm(Rb)}_{e}$. From Fig. 4, one can see that the function $G_{2}$
produces the leading contribution over the entire parameter space. The effect
is further enhanced due to the chosen flavor ansatz [see Eq. (9)] as it
connects the lightest lepton with the heaviest quark and vice versa. Thus, the
sign of the term
$\mathcal{A}_{2}\left(\frac{m_{\ell}\,m_{i}}{M^{2}_{S_{1}}}\right)G_{2}(\rho_{i})$
[see Eq. (24)], or to be more specific, the sign of $\mathcal{A}_{2}$
effectively decides the sign of $\Delta a_{e}$ in the theory. The same
argument goes for the negative values of
$\lambda_{L}^{c,\mu}\times\lambda_{R}^{c,\mu}$ in Fig. 3(c).

##  VI Conclusions

This paper has considered a minimal extension of the Standard Model with a
TeV-scale scalar Leptoquark $S_{1}$ transforming as
$(\mathbf{\bar{3}},\,\mathbf{1},\,1/3)$ under the SM gauge group. In the
presence of $S_{1}$, there can be corrections to the $\ell\ell\gamma$ vertex
at the one-loop level, which may lead to New Physics contributions to the
lepton $(g-2)$ and EDM. A particular flavor structure has been chosen to
suppress the CLFV processes while enhancing the BSM contributions to other
low-energy lepton phenomena. The new one-loop contributions have been computed
analytically, followed by a numerical scan to determine the parameter space
allowed under the recent $(g-2)_{\ell}$ and EDM constraints for each of the
lepton generations. Four different LQ masses have been considered to
understand the phenomenological implication of the NP scale on flavor-specific
low-energy observables. For the electron sector, viable parameter spaces have
been found corresponding to both of the experimental results, i.e., $\Delta
a^{\rm(Cs)}_{e}$ [see Eq. (16)] and $\Delta a^{\rm(Rb)}_{e}$ [see Eq. (17)].
Note that it is a significant feature of this work that it can explain both
positive ($\Delta a^{\rm(Rb)}_{e}$ & $\Delta a_{\mu}$) and negative ($\Delta
a^{\rm(Cs)}_{e}$) discrepancies in the anomalous magnetic moment of leptons by
simply rotating the parameter space while keeping the entire scenario
consistent with the respective EDM discovery limits. Though the $\tau$-sector
has also been analyzed but due to lower experimental sensistivity the complete
parameter space is allowed within the perturbative bounds. However, the
assumed model structure can explain any future update on $a_{\tau}$ and/or
$d_{\tau}$ which can probe the BSM contributions to the $\tau$ phenomenology.

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# Quantum coherence in neutrino oscillation in matter

Z. Askaripour Ravari  Department of Physics, Islamic Azad University, North
Tehran Branch, Tehran 165115-3311, I.R. Iran  M. M. Ettefaghi
<EMAIL_ADDRESS>Department of Physics, University of Qom, Ghadir Blvd.,
Qom 371614-6611, I.R. Iran  S. Miraboutalebi  Department of Physics, Islamic
Azad University, North Tehran Branch, Tehran 165115-3311, I.R. Iran

###### Abstract

A closer and more detailed study of neutrino oscillation, in addition to
assisting us in founding physics beyond the standard model, can potentially be
used to understand the fundamental aspects of quantum mechanics. In
particular, we know that the neutrino oscillation occurs because the quantum
states of the produced and detected neutrinos are a coherent superposition of
the mass eigenstates, and this coherency is maintained during the propagation
due to the small mass difference of neutrinos. In this paper, we consider the
decoherence due to the neutrino interaction in the material medium with
constant density in addition to the decoherence coming from the localization
properties. For this purpose, we use $l_{1}\text{-norm}$ in order to quantify
the coherence and investigate its dependence on the matter density. According
to our results, in general, the coherence in material medium is less than
vacuum. However, there exist exceptions; for some matter densities, the
localization coherence lengths become infinite. So, for these cases,
$l_{1}\text{-norm}$ in matter is more than the vacuum.

Neutrino oscillation, matter potential, coherence, $l_{1}\text{-norm}$

##  I Introduction

Neutrino oscillation is one of the most interesting phenomena in quantum
mechanics which has been experimentally established rev.est . It could provide
potentially a suitable room for quantum foundation explorations. The quantum
approaches to neutrino oscillation are based on the existence of non-zero and
non-degenerate neutrino masses. Although the differences between their masses
are non-zero, these differences are so smaller than the energy uncertainty in
the creation and detection processes that the neutrino mass eigenstate cannot
be distinguished. This point is used in the quantum mechanics approach; the
states of the created and detected neutrinos (well-known as flavor
eigenstates) are written as a coherent superposition of mass eigenstates par .
This coherency is preserved for large macroscopic distances. Therefore, using
the neutrino oscillation phenomenon, one could study quantum coherence, which
is a microscopic quantum effect, in large distances even up to several hundred
kilometers away.

Furthermore, neutrinos interact with the matter inside the medium in which
they propagate. It is noticeable that due to the presence of electrons in
matter, the matter effect leads to an additional potential energy $V$
($V=\sqrt{2}G_{F}N_{e}(x)$ where $G_{F}$ and $N_{e}(x)$ are the Fermi constant
and the electron density in the medium, respectively) only for electron
neutrinos. In fact, neutrino oscillation is caused by the interference effects
of coherent neutrino flavor waves. If the potential $V$ is attributed to the
elastic forward scattering of neutrinos off electrons, protons and neutrons
within the medium via the electroweak interaction, the coherency of neutrino
wave may be preserved and the interference effect does not cease generally. In
particular, when the matter density in the medium varies adiabatically with
the coordinate $x$, and consequently time $t$ ($x\approx t$), a resonance
phenomenon occurs. Thus if the primary neutrino state is the electron
neutrino, it is possible an almost complete transition to the other flavors to
take place at the resonance point even for small mixing angles. Historically,
Wolfenstein first showed the difference in neutrino oscillation in the
material medium in comparison with the vacuum wolf . Later on, Smirnov and
Mikheev showed that if the electron density in the matter is such a way that
we are at the point of resonance, we have the highest flavor conversion mikh .
This effect is well-known as the MSW effect, which explains the oscillation of
solar neutrinos. However, the interaction of neutrinos with the matter can
cause the survival and transition probabilities of the electron neutrinos to
cease when the density of the matter is much larger than the resonances
density. This behavior is nontrivial since corresponding to the two
distinguished mass squared differences, there are two values for potential
which lead to the resonance for the flavor transition. In this paper, we
analyze the oscillation probability of neutrinos propagating in a medium with
constant matter density for three flavors and study the decoherence effects on
the neutrino oscillation due to the interaction with matter in the medium.

Quantum coherence as a basic ingredient for quantum technologies can be
quantified using an appropriate quantum resource theory plenio ; plenio2 . In
these frameworks, some meaningful and applicable methods are introduced to
quantify the desired quantum resource and provide us with measures in order to
determine the quantumness. Among the several measures for quantum coherence, a
very intuitive one is related to the off-diagonal elements of the considered
quantum state. Therefore, the $l_{1}\text{-norm}$ which is defined by the sum
of the absolute values of all the off-diagonal elements of the density matrix

$\textit{c}(\rho)=\sum_{i\neq j}|\rho_{ij}|,$ (1)

is desirable. It has been shown that this quantity can satisfy all conditions
of measures in the quantum resource theories plenio ; plenio2 . On the other
hand, we know that the neutrino oscillation can take place provided that we
have non-zero off-diagonal elements in the corresponding density matrix in the
flavor representation basis glashow . Hence, the non-zero $l_{1}\text{-norm}$
or equivalently non-zero coherence is a necessary condition for the neutrino
oscillation phenomenon. In this direction, in Ref. song , the authors have
quantified the coherence for neutrino oscillation by the $l_{1}\text{-norm}$
and have compared it with the experimentally observed data.

Furthermore, in order to have a complete and thorough description of neutrino
oscillation, we should use the wave packet (WP) approach, since the
production, propagation and detection of neutrino should be considered as
localized processes and this localization is very well fulfilled using the WP
approach in neutrino oscillation par ; kayser ; giunti ; ettplb ; ettscripta ;
qftrev ; matterwp . In this approach, for a baseline distance more than the
coherence length, the oscillation probabilities diminish. Therefore, we obtain
the coherence length which in this case, depends not only on the neutrino
states properties (mass squared difference, energy and its dispersion) but
also on the matter potential Holanda ; Kersten ; smir2021 . Consequently, we
obtain the $l_{1}\text{-norm}$ and analyze its behavior in terms of the
baseline distance. In fact, according to its definition by Eq. (1), this
parameter depends on all the oscillation probabilities. Therefore, we expect
the $l_{1}\text{-norm}$ is affected nontrivially by the value of $V$.

In the next section, we will extend the theory developed in Ref. 1  for
neutrino oscillation in a uniform matter to include the localization
properties and analyze the probabilities of oscillation in terms of the
baseline distance for some values of the matter potential. The
$l_{1}\text{-norm}$ will be investigated in section III. Finally, we will
summarize and discuss on the results in the last section. In Appendix, we will
give a brief review of the neutrino oscillation in the uniform matter
according to the theory developed in Ref. 1 .

##  II Analyzing oscillation probabilities in matter

As a proper explanation of the neutrino oscillation phenomenon in the quantum
mechanics framework, we describe produced and detected neutrinos by the
localized weak interaction processes and suppose that they propagate as a
localized WP in the position space par . Hence, we consider that neutrinos of
flavor $\alpha$ are produced at the origin of the space-time coordinates and
are detected after a time $T$ by a detector which is located at a distance $L$
and is sensitive to neutrinos of flavor $\beta$. Furthermore, we suppose that
neutrinos propagate in a medium with constant matter density. In general,
their interactions with matter cause the dispersion relation between the
momentum $|\mathbf{p}|$ and the energy $E$ of neutrinos to be changed.
Therefore, the investigation of time evolution of the realistic three
generations scheme becomes complicated, generally speaking. Meanwhile, more
simplified and accurate arguments have been performed for the oscillations in
matter with constant density versus baseline divided by neutrino energy plane
by using a perturbative framework in Ref. 1 . We give a brief review of this
theory in Appendix A. Accordingly, this theory is based on the effective
Hamiltonian matrix which is achievable by using the unjustified either “same
energy” or “same momentum” assumption at the plane wave (PW) description
level. Nevertheless, one can obtain the standard formula for the probability
of neutrino oscillations by this assumption if the decoherence effects due to
the WP separation are negligible and the emitted and absorbed neutrino are
sufficiently well localized par . However, the main purpose of this paper is
related to considering the decoherence issues. Alternatively, in most
situations the decoherence effects can be reliably estimated based on the
standard oscillation formula and simple physical considerations.

By assumption a Gaussian wave function for the produced and detected
neutrinos, one can obtain the probability of transition in vacuum by using WP
approach as follows: giunti :

$P_{\alpha\beta}(\mathbf{L})\simeq\sum_{i,j}U^{*}_{\alpha i}U_{\beta
i}U_{\alpha j}U^{*}_{\beta j}\exp\left[-2\pi
i\dfrac{L}{L_{ij}^{\text{osc}}}-(\dfrac{L}{L_{ij}^{\text{coh}}})^{2}-2\pi^{2}\rho^{2}(\dfrac{\sigma_{x}}{L_{ij}^{\text{osc}}})^{2}\right],$
(2)

in which $U_{\alpha i}$ denotes the mixing matrix elements. This matrix is
unitary and can be parameterized by the well-known PMNS matrix

$U_{\text{PMNS}}=\begin{bmatrix}c_{13}c_{12}&c_{13}s_{12}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-e^{i\delta}c_{12}s_{23}s_{13}&c_{12}c_{23}-e^{i\delta}s_{12}s_{23}s_{13}&c_{13}s_{23}\\\
s_{12}s_{23}-e^{i\delta}c_{12}c_{23}s_{13}&-c_{12}s_{23}-e^{i\delta}s_{12}c_{23}s_{13}&c_{13}c_{23}\\\
\end{bmatrix}.$ (3)

Here, we use the abbreviation $c_{ij}$ and $s_{ij}$ for $\cos\theta_{ij}$ and
$\sin\theta_{ij}$ in which $\theta_{ij}$ stands for the mixing angles. In Eq.
(2), $\sigma_{x}$ is the position uncertainty given by

$\sigma_{x}^{2}\equiv\sigma_{xD}^{2}+\sigma_{xP}^{2},$ (4)

where $\sigma_{xP}$ and $\sigma_{xD}$ denote the produced and detected
neutrino WP widths, respectively, and $\rho$ is a parameter of order unity,
related to the energy-momentum conservation of the production process.
Moreover, in this equation, the following definition for $L_{ij}^{\text{osc}}$
and $L_{ij}^{\text{coh}}$ are obtained by direct calculation:

$L_{ij}^{\text{osc}}\equiv\dfrac{4\pi E}{\Delta
m_{ij}^{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}L_{ij}^{\text{coh}}\equiv\dfrac{4\sqrt{2}\sigma_{x}E^{2}}{\mid\Delta
m_{ij}^{2}\mid},$ (5)

where the mass square differences are defined by $\Delta m_{ij}^{2}\equiv
m_{i}^{2}-m_{j}^{2}$. In this manner, we have two more factors in comparison
to the PW approach for the probability of oscillation. The first factor is the
Gaussian factor which describes the coherence condition. According to this
condition, the oscillation can only take place if the overlapping of the mass
eigenstates is not destroyed; which means $L<L^{\text{coh}}_{ij}$. The second
factor is due to the last term in the exponent which emphasizes that for
seeing the oscillation we must have $\sigma_{x}\ll L_{ij}^{\text{osc}}$.

In the case of propagation in the material medium, the energy eigenvalues and
mixing matrix are modified in comparison with the corresponding one in the
vacuum. In general, these parameters alter with the position provided that the
density of matter depends on it. The WP treatment of the neutrino oscillation
in a material medium with variable density is given in Ref. matterwp .
However, it is complicated to obtain the mixing matrices and energy
eigenvalues when we consider the general case of three generations schema. In
the case of uniform matter density, these parameters are constant. Therefore,
using a treatment similar to Ref. matterwp , we can generalize Eq. (2) for the
transition probability as follows:

$\displaystyle P^{m}_{\alpha\beta}(\mathbf{L})$
$\displaystyle\\!\\!\simeq\\!\\!$ $\displaystyle\sum_{i,j}{U^{m}}^{*}_{\alpha
i}{U^{m}}_{\beta i}{U^{m}}_{\alpha j}{U^{m}}^{*}_{\beta j}$ (6)
$\displaystyle\\!\\!\times\\!\\!$ $\displaystyle\exp\left[-2\pi
i\dfrac{L}{{L^{m}}_{ij}^{\text{osc}}}-(\dfrac{L}{{L^{m}}_{ij}^{\text{coh}}})^{2}-2\pi^{2}\rho^{2}(\dfrac{\sigma_{x}}{{L^{m}}_{ij}^{\text{osc}}})^{2}\right],$

in which the superscript $m$ refers to the related parameters in the material
medium. ${L^{m}}_{ij}^{\text{osc}}$ and ${L^{m}}_{ij}^{\text{coh}}$ are
defined in this manner by

${L^{m}}_{ij}^{\text{osc}}\equiv\dfrac{2\pi}{\Delta
E^{m}_{ij}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{L^{m}}_{ij}^{\text{coh}}\equiv\dfrac{2\sqrt{2}\sigma_{x}}{\mid\Delta
v^{m}_{ij}\mid},$ (7)

where $\Delta E^{m}_{ij}\equiv E^{m}_{i}-E^{m}_{j}$ and $\Delta
v^{m}_{ij}\equiv v^{m}_{i}-v^{m}_{j}$ in which $E^{m}_{i}$’s and $v^{m}_{i}$’s
are the Hamiltonian eigenvalues and the group velocities in matter,
respectively. Here, we suppose that the wavelengths of neutrinos are very
small compared to the position uncertainties of the production and detection
processes or equivalently, the momenta of neutrinos are very large compared to
the corresponding uncertainties. This means that the wave functions of the
produced and detected neutrinos in momentum space are very localized around
the corresponding averaged momenta. Therefore, we can use approximately the
results of Ref. 1  for the mixing matrix and Hamiltonian eigenvalues in Eq.
(6). In this manner, the mixing matrix $U^{m}_{\alpha i}$ has the
$U_{\text{PMNS}}$ form with differences that $\theta_{12}$ and $\theta_{13}$
are replaced by $\theta^{m}_{13}$ and $\theta^{m}_{12}$ which are given,
respectively, by

$\theta_{13}^{m}=\dfrac{1}{2}\arctan[\dfrac{\epsilon\sin{2\theta_{13}}}{\epsilon\cos{2\theta_{13}}-2EV}],$
(8)

and

$\theta_{12}^{m}=\dfrac{1}{2}\arctan[\Delta
m_{21}^{2}\dfrac{\sin{2\theta_{12}}}{l_{2}-l_{1}}\cos(\theta_{13}-\theta_{13}^{m})],$
(9)

where $\epsilon=\Delta m_{31}^{2}-\Delta m_{21}^{2}\sin^{2}{\theta_{12}}$.
Moreover, the Hamiltonian eigenvalues $E^{m}_{i}$ are as follows:

$E^{m}_{1,2}=\frac{1}{4E}[(l_{1}+l_{2})\mp\sqrt{(l_{1}-l_{2})^{2}+\Delta
m_{21}^{4}\sin^{2}{2\theta_{12}}\cos^{2}({\theta_{13}}-\theta_{13}^{m}})],$
(10)

and

$E^{m}_{3}=\frac{l_{3}}{2E}.$ (11)

Here $l_{i}$’s are given by

$l_{1,3}=\dfrac{1}{2}[(\Delta m_{31}^{2}+2EV+\Delta
m^{2}_{21}\sin^{2}\theta_{12})\mp\sqrt{(2EV)^{2}+\epsilon^{2}-4EV\epsilon\cos{2\theta_{13}}}],$
(12)

and

$l_{2}=\Delta m_{21}^{2}\cos^{2}{\theta_{12}}.$ (13)

According to Eq. (7), in order to obtain the coherence lengths, we need to
know the differences of the group velocities of matter eigenstates which are
given by

$\Delta v^{m}_{ij}=\frac{d\Delta E^{m}_{ij}}{dE}.$ (14)

For three generations schema, there exist two independent values for $\Delta
v^{m}_{ij}$ which are obtained as

$\Delta v^{m}_{21}=\dfrac{1}{8E}(-8\Delta E_{21}^{m}+\xi),$ (15)

and

$\Delta v^{m}_{32}=\dfrac{1}{8E}\left(6\frac{dl_{3}}{dE}-4V-8\Delta
E_{32}^{m}-\dfrac{1}{2}\xi\right),$ (16)

where

$\xi=\dfrac{1}{2}\left(\dfrac{1}{4EE_{2}^{m}-(l_{1}+l_{2})}\right)\left(8(l_{1}-l_{2})\\\
\frac{dl_{1}}{dE}+\dfrac{\zeta}{(\epsilon\cos{2\theta_{13}}-2EV)^{2}}\right),$
(17)

with

$\zeta=4\Delta m_{21}^{4}\epsilon
V\cos^{2}{2\theta_{13}^{m}}\sin^{2}{2\theta_{12}}\sin{2\theta_{13}}\sin{2(\theta_{13}-\theta_{13}^{m})}.$
(18)

Using Eq. (12), one can obtain

$\frac{dl_{1,3}}{dE}=V\left(1\mp\dfrac{(2EV-\epsilon\cos{2\theta_{13}})}{\sqrt{\epsilon^{2}+(2EV)^{2}-4EV\epsilon\cos{2\theta_{13}}}}\right).$
(19)

For the third the differences of the group velocities, we have $\Delta
v^{m}_{31}=\Delta v^{m}_{32}+\Delta v^{m}_{21}$. There are three different
values for the density of matter or equivalently for the corresponding
potential through which the $\Delta v^{m}_{ij}$ diminishes and consequently,
the related coherence length becomes infinite.

In fact, it is noticeable that in addition to the dependence on the baseline
distance $L$, neutrino energies and the WP widths, the probabilities in medium
are function of the matter density through the dependence of $\theta^{m}_{ij}$
and $\Delta E^{m}_{ij}$ on it. There exist two resonance points in matter
($V^{\text{res}}_{1}=3.196\times 10^{-16}eV$ and
$V^{\text{res}}_{2}=2.577\times 10^{-14}eV$) at which the maximal flavor
conversion occurs. The resonances take place provided that the density of
matter in the medium is in such a way that the matter mixing angles
$\theta_{13}^{m}$ and $\theta_{12}^{m}$ become $\frac{\pi}{4}$. We take
$E=4.5\times 10^{10}eV$, $\Delta m_{21}^{2}=7.39\times 10^{-5}eV^{2}$, $\Delta
m_{31}^{2}=2.451\times 10^{-3}eV^{2}$, $\sigma_{x}=0.5\times 10^{-9}m$,
$\theta_{12}=33.82^{\circ}$, $\theta_{13}=8.61^{\circ}$,
$\theta_{23}=49.7^{\circ}$ and $\delta=217^{\circ}$ ie . In the following, we
analyze the oscillation probabilities for the two following situations:

  * •

Let us consider the initially produced neutrinos to be $\nu_{e}$. We
illustrate the survival (diagram a) and transition to $\nu_{\mu}$ (diagram b)
probabilities versus $L$ for various matter densities in Fig. (1). Explicitly,
we plot the probability in terms of $L$ for five cases; vacuum ($V=0$), for
three values of potential which correspond to three infinity coherence lengths
and for a matter dominated potential. The purple curve corresponds to
$V=2.824\times 10^{-14}eV$ at which ${L^{m}}^{\text{coh}}_{32}$ becomes
infinite. Since this coherence length does not play significant role in the
transition of $\nu_{e}$, we see that the behavior of survival and transition
probabilities is similar to the corresponding situation in vacuum (the green
curve) with the difference that damping appears at smaller distance. The red
and blue curves depict the survival and transition probabilities given that
$V=2.242\times 10^{-15}eV$ and $V=1.099\times 10^{-14}eV$, respectively.
Although with these values of potential, ${L^{m}}^{\text{coh}}_{21}$, which is
relevant for transition $\nu_{e}$ to $\nu_{\mu}$, becomes infinite, the
amplitudes of oscillation behavior are limited. For the brown curve, we take a
matter dominated density ($V=1.00\times 10^{-12}eV$). In this case, the
transition probability vanishes.

Figure 1: Survival (a) and transition to $\nu_{\mu}$ (b) probability of
$\nu_{e}$. In addition to vacuum (green), we take $V=2.242\times 10^{-15}eV$
(red), $V=1.099\times 10^{-14}eV$(blue) and $V=2.824\times 10^{-14}eV$(purple)
which correspond to infinity coherence lengths. We also take $V=1.00\times
10^{-12}eV$(brown) corresponding to matter dominated.

  * •

Now, we consider the initially produced neutrinos to be $\nu_{\mu}$. We know
that the eigenstates $\nu_{\mu}$ and $\nu_{\tau}$ interact with matter only
through neutral current interactions in contrast to the electron neutrinos.
Therefore, it may be imagined that their oscillatory behavior is not affected
by the interaction with matter. Hence, we also illustrate the survival
(diagram a) and transition to $\nu_{e}$ (diagram b) probabilities for
$\nu_{\mu}$ versus $L$ for matter densities similar to what were considered in
Fig. (1). We see that if $V=2.824\times 10^{-14}eV$ in which
${L^{m}}^{\text{coh}}_{32}$ becomes infinite (the purple curve), the survival
probability does not diminish for the assumed interval distance. Meanwhile,
the damping behaviors of survival probabilities for the other considered
potentials are roughly similar to the vacuum one. In the case of transition to
$\nu_{e}$, we see that the statement is different since
${L^{m}}^{\text{coh}}_{32}$ is not the relevant coherence length.

Figure 2: Survival (a) and transition to $\nu_{e}$ (b) probability of
$\nu_{\mu}$. In addition to vacuum (green), we take $V=2.242\times 10^{-15}eV$
(red), $V=1.099\times 10^{-14}eV$(blue) and $V=2.824\times 10^{-14}eV$(purple)
which correspond to infinity coherence lengths. We also take $V=1.00\times
10^{-12}eV$(brown) corresponding to matter dominated.

##  III Study quantum coherence through $l_{1}\text{-norm}$ calculation

In order to have the oscillation phenomena for neutrinos, they must be
coherent during the production, propagation, and detection processes. In the
quantum resource framework and using $l_{1}\text{-norm}$ as a measure of
coherence, this means that the off-diagonal elements of the corresponding
density matrix must be non-zero. This happens when the produced neutrino state
is a coherent superposition of mass eigenstates, holds it during the
propagation and finally is detected coherently. It is noticeable that during
the propagation, one can attribute a non-zero entanglement between the various
modes of the neutrino flavors which can be evaluated by the usual entanglement
measures such as concurrence bla ; mass . Meanwhile, the coherence of
propagation and the entanglement of flavor modes are not independent quantum
correlations mm . In this section, we will investigate the effects of the
charged current elastic forward scattering of electron neutrinos off electrons
(matter potential $V$) during the propagation on the $l_{1}\text{-norm}$. When
this quantity is non-zero, we conclude that there exists quantum coherence and
one can observe a quantum interference which is the neutrino oscillation.

One can write the density matrices for the primary electron neutrinos as

$\rho_{e}^{m}(t)=\begin{bmatrix}|A_{ee}^{m}(t)|^{2}&A_{ee}^{m}(t)A_{e\mu}^{m*}(t)&A_{ee}^{m}(t)A_{e\tau}^{m*}(t)\\\
A_{e\mu}^{m}(t)A_{ee}^{m*}(t)&|A_{e\mu}^{m}(t)|^{2}&A_{e\mu}^{m}(t)A_{e\tau}^{m*}(t)\\\
A_{e\tau}^{m}(t)A_{ee}^{m*}(t)&A_{e\tau}^{m}(t)A_{e\mu}^{m*}(t)&|A_{e\tau}^{m}(t)|^{2}\\\
\end{bmatrix},$ (20)

and muon neutrinos as

$\rho_{\mu}^{m}(t)=\begin{bmatrix}|A_{\mu\mu}^{m}(t)|^{2}&A_{\mu\mu}^{m}(t)A_{\mu
e}^{m*}(t)&A_{\mu\mu}^{m}(t)A_{\mu\tau}^{m*}(t)\\\ A_{\mu
e}^{m}(t)A_{\mu\mu}^{m*}(t)&|A_{\mu e}^{m}(t)|^{2}&A_{\mu
e}^{m}(t)A_{\mu\tau}^{m*}(t)\\\
A_{\mu\tau}^{m}(t)A_{\mu\mu}^{m*}(t)&A_{\mu\tau}^{m}(t)A_{\mu
e}^{m*}(t)&|A_{\mu\tau}^{m}(t)|^{2}\\\ \end{bmatrix},$ (21)

where the superscript $m$ emphases that the propagation takes place in the
material medium. According to Eq. (1), the $l_{1}\text{-norm}$s corresponding
to the density matrices given in Eqs. (20) and (21) in terms of the
oscillation probabilities are

$\textit{c}(\rho_{e}^{m})=2(\sqrt{P_{ee}^{m}P_{e\mu}^{m}}+\sqrt{P_{ee}^{m}P_{e\tau}^{m}}+\sqrt{P_{e\mu}^{m}P_{e\tau}^{m}}),$
(22)

and

$\textit{c}(\rho_{\mu}^{m})=2(\sqrt{P_{\mu\mu}^{m}P_{\mu
e}^{m}}+\sqrt{P_{\mu\mu}^{m}P_{\mu\tau}^{m}}+\sqrt{P_{\mu
e}^{m}P_{\mu\tau}^{m}}),$ (23)

respectively. In general, the maximum possible value for $c(\rho)$ is
$c_{\text{m}ax}=d-1$, where $d$ is the dimension of the corresponding density
matrix plenio2 . Thus it is equal to two here.

We illustrate the behavior of $\textit{c}(\rho_{e}^{m})$ (a) and
$\textit{c}(\rho_{\mu}^{m})$ (b) versus the baseline distance $L$ through
Figs. (3). Similar to the oscillation probability’s illustrations in Figs. (1)
and (2), we do these for five cases; vacuum ($V=0$), for three values of
potential which correspond to three infinity coherence lengths and for a
matter dominated potential. We see that when the initial neutrinos are
$\nu_{e}$ (diagram a in Fig. (3)), the $l_{1}$-norm behavior shows that the
coherence condition is conserved if $V=2.242\times 10^{-15}eV$ (the red curve)
which cause to ${L^{m}}^{\text{coh}}_{21}$ become infinite. It is reasonable
because this coherence length is relevant in the transition probabilities. For
the other potentials considered in this diagram, the coherence lengths are
smaller than vacuum one. When the initial neutrinos are $\nu_{\mu}$, the
$l_{1}$-norm related to $V=2.824\times 10^{-14}eV$ (the purple curve), which
leads ${L^{m}}^{\text{coh}}_{32}$ to become infinite, is larger than the
others. Of course, this coherence length is relevant for the probability of
transition to $\nu_{\tau}$. $l_{1}$-norms for the other cases are smaller than
the vacuum one. This shows that except for the two cases mentioned,
decoherence in matter is larger than vacuum.

Figure 3: $l_{1}\text{-norm}$ for the initial neutrinos to be $\nu_{e}$ (a)
and $\nu_{\mu}$ (b). In addition to vacuum (green), we take $V=2.242\times
10^{-15}eV$ (red), $V=1.099\times 10^{-14}eV$(blue) and $V=2.824\times
10^{-14}eV$(purple) which correspond to infinity coherence lengths. We also
take $V=1.00\times 10^{-12}eV$(brown) corresponding to matter dominated.

Furthermore, we compare the results obtained using PW (a) and WP (b)
approaches by Figs. (4) and (5). In Figs. (4), $\textit{c}(\rho_{e}^{m})$ is
illustrated versus $V$ for two distances $L=10^{15}m$ (green curve) and
$L=10^{17}m$ (blue curve), using both PW and WP approaches by the diagrams a
and b, respectively. For the former distance (green curve) which is shorter
than the coherence length, $\textit{c}(\rho_{e}^{m})$ obtained by both
approaches have similar behavior. In the case of the latter distance,
$\textit{c}(\rho_{e}^{m})$ obtained using PW approach achieves the maximum
values for two regions of the potential values. However, in the case of WP
approach, $\textit{c}(\rho_{e}^{m})$ achieves a maximum smaller than 2 in a
region of the potential values. This is due to the decoherence effects coming
from the separation of neutrino WPs. We also give a similar illustration for
$\textit{c}(\rho_{\mu}^{m})$ in Fig. (5). The statement in this Fig. is
similar to previous one.

Overall, we see that when there exists a perfect decoherence due to the wave
packets separation, as well as due to the matter dominated potential, the
$l_{1}$-norm tends to an almost constant value smaller than 1. Therefore, the
other important point resulting from this study is that the $L$-independence
of $l_{1}$-norm can be mooted as an indication of the perfect coherency loss.

Figure 4: Plots of $\textit{c}(\rho_{e}^{m})$ against the potential $V$
obtained by PW (a) and WP (b) approach. We take $L=10^{15}m$(green) and
$L=10^{17}m$(blue).

Figure 5: Plots of $\textit{c}(\rho_{\mu}^{m})$ against the potential $V$
obtained by PW (a) and WP (b) approach. We take $L=10^{15}m$(green) and
$L=10^{17}m$(blue).

##  IV Summary and discussions

Neutrino oscillation is a complex and astonishing quantum phenomenon that
researchers have been studying for more than 60 years. So far, after many
experiments, we could get a lot of information in this regard, but many
aspects are still unknown such as neutrino mass ordering, leptonic CP
violation phase and so on. Furthermore, neutrino experiment data can also be
utilized as a diverse platform to study the fundamental aspects of quantum
mechanics, for instance, see 2016 . In particular, neutrino oscillation can be
considered as a good illustration of quantum coherence song . There exist a
fundamental reason for the loss of coherency in neutrino oscillation; taking
into account the localization properties of production and detection
processes, one can find the decoherence effects due to the separation of WPs
describing neutrinos par ; kayser ; giunti ; ettplb ; ettscripta . In this
paper, in addition to considering this issue, we have investigated the effect
of neutrino propagation in the material medium with a constant density on
quantum coherence. For this purpose, we have used the approach introduced in
Ref. 1  in order to diagonalize the related Hamiltonian. We know that in
addition to neutral current interactions for all neutrinos, the electron
neutrino has an additional charge current interaction. Therefore, it is
expected that propagation in the material medium leads to a decoherency, which
depends on the density of matter in the medium. In fact, this efficient effect
is due to the elastic forward scattering of neutrinos off electrons, which
causes the electron neutrinos to feel an interaction potential $V$, which is
proportional to the density of matter, more than others. For two values of
$V$, we have resonance in the transition processes. However, the coherence
length for some values of potentials becomes infinite Holanda ; Kersten ;
smir2021 . This effect is due to the larger mixing angle in the material
medium.

We have illustrated the dependence of the oscillation probabilities and
$l_{1}\text{-norm}$ as a measure for quantum coherence on $V$ through Figs.
(1), (2), (3), (4), and (5). We see that the perfect coherence loss due to the
wave packets separation, as well as due to the matter dominated potential,
causes the $l_{1}$-norm to tend to almost constant value smaller than 1. In
fact, the $L$-independence of $l_{1}$-norm can be interpreted as an indication
of the perfect coherency loss. In general, interaction with matter leads to
the faster damping in neutrino oscillation probabilities comparison to vacuum
except two values111We should notice that there are two values for potential
($V=2.242\times 10^{-15}eV$ and $V=1.099\times 10^{-14}eV$) in which
${L^{m}}^{\text{coh}}_{21}$ becomes infinite. However, in the case of larger
value, the decoherence due to the interaction with matter cause this effect to
be irrelevant. of potential in which the coherence lengths
${L^{m}}^{\text{coh}}_{21}$ and ${L^{m}}^{\text{coh}}_{32}$ become infinite.
When the primary neutrinos are electron neutrinos $\nu_{e}$,
${L^{m}}^{\text{coh}}_{21}$ plays a significant role in the transition
process. Therefore, provided that $V$ has the value in which this coherence
length become infinite, the maximum value of $l_{1}$-norm ,
$\textit{c}(\rho_{e}^{m})$, is conserved for the interval distance very larger
than the coherence length in vacuum (please see Fig. (3) plot a). Also, when
the primary neutrinos are muon neutrinos $\nu_{\mu}$, we have roughly a
similar situation for $\textit{c}(\rho_{\mu}^{m})$ (please see Fig. (3) plot
b). Furthermore, Figs. (4), and (5) show that $l_{1}$-norm is naturally
independent of the decoherency due to the localization properties when the
baseline distance is smaller than coherence lengths. Meanwhile, when the
baseline distance is of the order of coherence length, the behaviors of
$l_{1}$-norms obtained by PW approach are nontrivially different from the
corresponding one obtained by WP approach. In particular, we find that
$\textit{c}(\rho_{e}^{m})$ obtained by PW description has two maximum
corresponding to two resonance potentials while the second maximum occurs
weakly in the case of WP treatment.

##  Appendix A A method for diagonalizing neutrino Hamiltonian in uniform
matter

The scattering of neutrino off electrons in the matter causes the electron
neutrinos to feel an additional potential $V$ which varies with position,
consequently $t$, when the density of matter varies with the position. In
general, $V(x)$ leads the dispersion relation between the momentum $|\vec{p}|$
and the energy $E$ of neutrinos to be changed. Therefore, the investigation of
time evolution of the realistic three generations scheme becomes complicated,
generally speaking. Many authors have considered this problem paul ; joh ; wil
; smir ; niro ; ohl ; his ; shu ; akh21 ; 1 ; 2 ; 3 . In particular, neutrino
oscillation in the matter has been treated comprehensively in Refs. 2 ; niro .
More simplified and accurate arguments have been performed for the
oscillations in uniform matter density versus baseline divided by neutrino
energy plane by using a perturbative framework in Ref. 1 . As was said, we use
the method developed in Ref. 1  to diagonalize the Hamiltonian for a uniform
matter. Here we give a brief of it.

We suppose the flavor eigenstates are written in terms of mass eigenstates as
follows:

$\left|\nu_{\alpha}\right\rangle=\sum_{i=1}^{3}U^{*}_{\alpha
i}\left|\nu_{i}\right\rangle,$ (24)

where $U_{\alpha i}$ is giving in Eq. 3. One can decompose $U_{\text{PMNS}}$
as follows:

$U_{\text{PMNS}}=U_{23}(\theta_{23})I_{\delta}U_{13}(\theta_{13})I_{\delta}^{*}U_{12}(\theta_{12}),$
(25)

in which

$U_{23}(\theta_{23})=\begin{bmatrix}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\\\ \end{bmatrix},\,\,\,\,\,I_{\delta}=\begin{bmatrix}1&0&0\\\
0&1&0\\\ 0&0&e^{i\delta}\\\
\end{bmatrix},\,\,\,\,\,U_{13}(\theta_{13})=\begin{bmatrix}c_{13}&0&s_{13}\\\
0&1&0\\\ -s_{13}&0&c_{13}\\\
\end{bmatrix},\,\,\,\,\,U_{12}(\theta_{12})=\begin{bmatrix}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\\\ \end{bmatrix}.$ (26)

The time evolution of the neutrino state in the presence of matter potential
is governed by the following Schrodinger equation:

$i\partial_{t}|\nu_{\alpha},t\rangle=\frac{1}{2E}[U_{\text{PMNS}}{\text{diag}}(0,\Delta
m_{21}^{2},\Delta
m_{31}^{2})U_{\text{PMNS}}^{\dagger}+{\text{diag}(2EV(x),0,0)}]|\nu_{\alpha},t\rangle,$
(27)

where $V(x)=\sqrt{2}G_{F}N_{e}(x)$. $N_{e}$ denotes the density of electron in
the medium and hereafter we assume it to be constant. We know that the
neutrino interactions with matter in the medium do not alter CP phase $\delta$
and mixing angle $\theta_{23}$. So Eq. (27) is covariant under the inverse
transformation $U^{-1}_{23}(\theta_{23})$ and $I^{-1}_{\delta}$. Therefore, we
can write the Hamiltonian as follows

$\acute{H}=\frac{1}{2E}[U_{13}(\theta_{13})U_{12}(\theta_{12}){\text{diag}}(0,\Delta
m_{21}^{2},\Delta
m_{31}^{2})U_{12}^{\dagger}(\theta_{12})U_{13}^{\dagger}(\theta_{13})+{\text{diag}(2EV,0,0)}].$
(28)

Now, one can diagonalize $\acute{H}$ by applying two consecutive rotations in
the matter. First, one needs to do the $\theta_{13}^{m}$ rotation:

$H^{\prime}=U_{13}^{\dagger}(\theta_{13}^{m})\acute{H}U_{13}(\theta_{13}^{m}),$
(29)

where

$U_{13}^{\dagger}(\theta_{13}^{m})=\begin{bmatrix}\cos\theta_{13}^{m}&0&-\sin\theta_{13}^{m}\\\
0&1&0\\\ \sin\theta_{13}^{m}&0&\cos\theta_{13}^{m}\\\ \end{bmatrix},$ (30)

with

$\theta_{13}^{m}=\dfrac{1}{2}\arctan[\dfrac{\epsilon\sin{2\theta_{13}}}{\epsilon\cos{2\theta_{13}}-2EV}].$
(31)

Here, $\epsilon$ is given by

$\epsilon=\Delta m_{31}^{2}-\Delta m_{21}^{2}\sin^{2}{\theta_{12}}.$ (32)

Therefore, $H^{\prime}$ is obtained as follows:

$\displaystyle{}H^{\prime}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2E}\begin{bmatrix}l_{1}&0&0\\\ 0&l_{2}&0\\\
0&0&l_{3}\\\ \end{bmatrix}$ (33) $\displaystyle+$ $\displaystyle\dfrac{\Delta
m_{21}^{2}\sin{2\theta_{12}}}{4E}\begin{bmatrix}0&\cos(\theta_{13}-\theta_{13}^{m})&0\\\
\cos(\theta_{13}-\theta_{13}^{m})&0&-\sin(\theta_{13}-\theta_{13}^{m})\\\
0&-\sin(\theta_{13}-\theta_{13}^{m})&0\\\ \end{bmatrix},$

in which

$l_{2}=\Delta m_{21}^{2}\cos^{2}{\theta_{12}},$ (34)
$l_{1,3}=\dfrac{1}{2}[(\Delta m_{31}^{2}+2EV+\Delta
m^{2}_{21}\sin^{2}\theta_{12})\mp\sqrt{(2EV)^{2}+\epsilon^{2}-4EV\epsilon\cos{2\theta_{13}}}].$
(35)

Second, a rotation with $\theta_{12}^{m}$ angle must be performed such a way
that $H^{\prime}$ is transformed as follows:

$H^{\prime\prime}=U_{12}^{\dagger}(\theta_{12}^{m})H^{\prime}U_{12}(\theta_{12}^{m})=H_{0}+H_{1},$
(36)

with

$H_{0}=\begin{bmatrix}E^{m}_{1}&0&0\\\ 0&E^{m}_{2}&0\\\ 0&0&E^{m}_{3}\\\
\end{bmatrix},$ (37)

where

$E^{m}_{3}=\frac{l_{3}}{2E},$ (38)
$E^{m}_{1,2}=\frac{1}{4E}[(l_{1}+l_{2})\mp\sqrt{(l_{1}-l_{2})^{2}+\Delta
m_{21}^{4}\sin^{2}{2\theta_{12}}\cos^{2}({\theta_{13}}-\theta_{13}^{m}})],$
(39)

and

$H_{1}=\dfrac{1}{4E}\Delta
m_{21}^{2}\sin{2\theta_{12}}\sin(\theta_{13}-\theta_{13}^{m})\begin{bmatrix}0&0&\sin{\theta_{12}^{m}}\\\
0&0&-\cos{\theta_{12}^{m}}\\\
\sin{\theta_{12}^{m}}&-\cos{\theta_{12}^{m}}&0\\\ \end{bmatrix}.$ (40)

$U_{12}^{\dagger}(\theta_{12}^{m})$ is defined by

$U_{12}^{\dagger}(\theta_{12}^{m})=\begin{bmatrix}\cos\theta_{12}^{m}&-\sin\theta_{12}^{m}&0\\\
\sin\theta_{12}^{m}&\cos\theta_{12}^{m}&0\\\ 0&0&1\\\ \end{bmatrix},$ (41)

in which

$\theta_{12}^{m}=\dfrac{1}{2}\arctan[\Delta
m_{21}^{2}\dfrac{\sin{2\theta_{12}}}{l_{2}-l_{1}}\cos(\theta_{13}-\theta_{13}^{m})].$
(42)

In Eq. (36), $H_{0}$ has the main contribution in the total Hamiltonian
$H^{\prime\prime}$ and one can treat perturbatively with $H_{1}$. However, in
this paper we ignore the $H_{1}$ contribution because it will have no
remarkable effects on our results.

In the case of anti-neutrino, the following changes are necessary:

$I_{\delta}\longrightarrow I_{\delta}^{*},$ (43)

and

$V\longrightarrow-V.$ (44)

Therefore, the final parameters of the anti-neutrino oscillation in matter are
given by

$\bar{\theta}_{13}^{m}=\dfrac{1}{2}\arctan[\dfrac{\epsilon\sin{2\theta_{13}}}{\epsilon\cos{2\theta_{13}}+2EV}],$
(45) $\bar{\theta}_{12}^{m}=\dfrac{1}{2}\arctan[\Delta
m_{21}^{2}\cos(\theta_{13}-\bar{\theta}_{13}^{m})\dfrac{\sin{2\theta_{12}}}{\bar{l}_{2}-\bar{l}_{1}}],$
(46)

${\bar{E}}^{m}_{3}=\frac{\bar{l}_{3}}{2E},$ (47)
${\bar{E}}^{m}_{1,2}=\dfrac{1}{4E}[(\bar{l}_{1}+\bar{l}_{2})\mp\sqrt{(\bar{l}_{1}-\bar{l}_{2})^{2}+\Delta
m_{21}^{4}\sin^{2}{2\theta_{12}}\cos^{2}({\theta_{13}}-\bar{\theta}_{13}^{m}})],$
(48)

with

$\bar{l}_{2}=l_{2}=\Delta m_{21}^{2}\cos^{2}{\theta_{12}},$ (49)
$\bar{l}_{1,3}=\dfrac{1}{2}[(\Delta m_{31}^{2}-2EV+\Delta
m_{21}^{2}\sin^{2}{\theta_{12}})\mp\sqrt{(2EV)^{2}+\epsilon^{2}+4EV\epsilon\cos{2\theta_{13}}}].$
(50)

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