# Isospin Pomeranchuk effect and the entropy of collective excitations in
twisted bilayer graphene

Yu Saito  Department of Physics, University of California at Santa Barbara,
Santa Barbara, CA 93106, USA  California Nanosystems Institute, University of
California at Santa Barbara, CA 93106 USA  Jingyuan Ge  Department of Physics,
University of California at Santa Barbara, Santa Barbara, CA 93106, USA
Takashi Taniguchi  International Center for Materials Nanoarchitectonics,
National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan
Kenji Watanabe  Research Center for Functional Materials, National Institute
for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Erez Berg
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot
76100, Israel  Andrea F. Young<EMAIL_ADDRESS>Department of Physics,
University of California at Santa Barbara, Santa Barbara CA 93106, USA

###### Abstract

In condensed matter systems, higher temperature typically disfavors ordered
phases leading to an upper critical temperature for magnetism,
superconductivity, and other phenomena. A notable exception is the Pomeranchuk
effect in 3He, in which the liquid ground state freezes upon increasing the
temperaturepomeranchuk_theory_1950 . The anomalous transition arises due to
the large spin entropy of the paramagnetic solid phase, which can overcome the
ground state energy difference at finite temperature. Here we show that a
similar mechanism describes the finite temperature dynamics of spin- and
valley- isospins in magic-angle twisted bilayer grapheneandrei_graphene_2020 ;
balents_superconductivity_2020 . In the vicinity of one hole per superlattice
unit cell, quantum oscillations show the ground state to be an isospin
unpolarized Fermi liquid. Tilted field magnetotransport measurements show that
increasing the Zeeman energy, $E_{Z}$, drives a transition from the Fermi
liquid to a spin- and valley- polarized isospin ferromagnetic state. The same
transition can be actuated by raising the temperature. Remarkably, the
temperature required to drive the transition at $E_{Z}=0$ is comparable to
$E_{Z}$ required to drive the transition at low temperature, implying an
entropy change per moiré unit cell of order $1k_{B}$. In this picture, the
finite temperature phase, while not characterized by long-range ferromagnetic
order, is distinguished from the low-temperature Fermi liquid by the presence
of large local isospin moments that fluctuate on the scale of the moiré
lattice spacing. Our findings suggest that the isospin stiffness in the
ferromagnetic phases of twisted bilayer graphene is exceptionally
smallwu_collective_2020 ; khalaf_charged_2020 , with implications for the
nature of finite temperature transportpolshyn_large_2019 ; cao_strange_2020 ;
he_tunable_2020  as well as the mechanisms underlying isospin ordering and
superconductivity in twisted bilayer graphene and related systems.

The best studied example of moiré flat band
systemsbalents_superconductivity_2020 , magic-angle twisted bilayer
graphenebistritzer_moire_2011-1  is known to host a wide array of low-
temperature phasescao_correlated_2018 ; cao_unconventional_2018 ;
yankowitz_tuning_2019 ; sharpe_emergent_2019 ; lu_superconductors_2019 ;
saito_independent_2020 ; stepanov_interplay_2019 ;
arora_superconductivity_2020 . Some among these, notably correlated
insulatorscao_correlated_2018 ; yankowitz_tuning_2019 ;
lu_superconductors_2019  and orbital Chern ferromagnetssharpe_emergent_2019 ;
lu_superconductors_2019 ; serlin_intrinsic_2020 ; tschirhart_imaging_2020 ,
are known unambiguously to arise from the effects of the Coulomb interaction.
However, the origin of other phenomena is less clear, particularly
superconductivitycao_unconventional_2018 ; yankowitz_tuning_2019 ;
lu_superconductors_2019 ; saito_independent_2020 ; stepanov_interplay_2019 ;
arora_superconductivity_2020  and the large scattering observed in the
metallic finite temperature statepolshyn_large_2019 ; cao_strange_2020 . The
prevalence of Coulomb-driven phases is suggestive of a unified origin for all
of the phenomenology. In this picture both superconductivity and the finite
temperature resistivity would arise from the interaction of charge carriers
with collective modes of the spin and valley isospin. However, theoretical
efforts based on more conventional phononic mechanisms have been successful at
capturing the basic parameters both existing experimental phenomenology of
both the superconductingdodaro_phases_2018 ; wu_theory_2018 ;
lian_twisted_2019 ; martin_moire_2020  and high-temperature metallic
stateswu_phonon-induced_2019 . In this Letter, we report on the discovery of
an entropically driven phase transition between an isospin unpolarized Fermi
liquid at low temperature and a state characterized by large, strongly
fluctuating local magnetic moments at high temperature. Whereas no such
phenomenology is expected from electron-phonon interaction alone, our results
imply the existence of low-energy collective modes of electronic origin that
couple strongly to the charge carriers. Such modes likely play a crucial role
in determining the low-temperature phase diagram as well as the finite-
temperature transport properties.

Figure 1: Zeeman stabilized isospin ferromagnetism. a, Longitudinal
resistivity $\rho_{\mathrm{xx}}$ as a function of nominal filling factor
$\nu_{0}$ and out-of-plane magnetic field $B_{\perp}$ (top, acquired at $T$ =
400 mK) and in-plane magnetic field $B_{\parallel}$ (bottom, acquired at
nominal temperature of $T$ = 20 mK). b, $\rho_{\mathrm{xx}}$ as a function of
$\nu_{0}$ at $B_{\parallel}$ = 0, 3, 6, 9, 12 T. c, $\rho_{\mathrm{xx}}$ as a
function of $\nu_{0}$ and total magnetic field $B_{\mathrm{tot}}$ oriented at
an angle with respect to the plane $\theta_{\mathrm{B}}$ of 4.1∘, d 9.6∘ and e
20.5∘. The right axis denotes corresponding $B_{\perp}$ for each
configuration. f, Hall density $\nu_{\mathrm{H}}$ and g subtracted Hall
density $\nu_{\mathrm{H}}-\nu_{0}$ expressed in electrons per superlattice
unit cell, and measured with $B_{\mathrm{tot}}$ = 0.5, 3, 6, 9, 12 T and fixed
$B_{\mathrm{\perp}}$ = 0.5 T. h, $\nu_{\mathrm{H}}-\nu_{0}$ as a function of
$B_{\mathrm{tot}}$ and $\nu_{0}$ with $\theta_{\mathrm{B}}$ = 20.5∘, measured
at nominal $T$ = 20 mK. Blue and pink circles correspond to the positions of
peaks of $\rho_{\mathrm{xx}}$ and the points of maximum descent in
$\nu_{\mathrm{H}}-\nu_{0}$ (see Fig. S3), respectively, and denote phase
boundaries between symmetry breaking isospin ferromagnets (IF1, IF2, and IF3)
and an isospin unpolarized state (IU).

We focus our study between 0 and 2 holes per superlattice cell in a high
quality twisted bilayer graphene device fabricated with inter-layer twist
angle $\theta\approx 1.12^{\circ}$. Figure 1a shows the magnetoresistance
measured at sub-kelvin temperatures, plotted as a function of the nominal
electron filling $\nu_{0}$ of the superlattice unit cell. We determine
$\nu_{0}$ from the geometric capacitance, measured from the Hall density near
charge neutrality, and the positions of the most prominent resistivity
features at $B=0$ T which we associate with actual filling factors
$\nu=-2,+2$, and $+3$. When a small magnetic field is applied perpendicular to
the sample plane, distinct sets of quantum oscillations are observed
intercepting the $B_{\perp}=0$ T axis at $\nu_{0}=-3$,$-2$, and $0$ and
showing one-fold, two-fold, and four-fold degeneracy, respectively, at
$B_{\perp}$ of 1-2 T. This is consistent with ferromagnetic interactions
favoring ground states polarized into one isospin flavor for $-4<\nu_{0}<-3$
or two flavors for $-3<\nu_{0}<-2$zondiner_cascade_2020 . Notably, no
comparable ‘fan’ is observed at low $B_{\perp}$ intercepting at $\nu_{0}=-1$,
and the quantum oscillations originating from the charge neutrality point
maintain apparent fourfold degeneracy for $-2<\nu_{0}<0$ suggesting unbroken
flavor symmetry in that regime.

Application of a sufficiently large magnetic field introduces new features, as
shown in Fig.1a-b. In particular, for in-plane magnetic field
$B_{\parallel}\gtrsim 3$ T, an additional resistance peak develops for
$-2<\nu_{0}\lesssim-1$. Simultaneously, the resistance peak at $\nu_{0}=-2$
de-pins from this filling and decreases in magnitude as it moves to larger
absolute $\nu_{0}$. The behavior of the resistivity peak is roughly
independent of the direction of the magnetic field orientation, showing nearly
identical trajectories for in-plane or partially out-of-plane magnetic fields
(Figs.1c-e and S1). Tilted field data also reveal that the resistivity peak
separates discrete domains of quantum oscillations. For
$\nu_{0}>\nu_{\mathrm{pk}}$ (where $\nu_{\mathrm{pk}}$ denotes the filling at
the peak) the oscillation minima remain qualitatively unchanged, extrapolating
to $\nu_{0}=0$, while for $\nu_{0}<\nu_{\mathrm{pk}}$ new quantum oscillations
emerge that extrapolate to $\nu_{0}=-1$ at $B=0$ T.

Figure 2: Thermally driven transition. a, $\rho_{\mathrm{xx}}$ as a function
of $T$ and $\nu_{0}$ at $B_{\mathrm{tot}}=0$ T. Blue circles correspond to
$\rho_{\mathrm{xx}}$ peak positions. b, $\rho_{\mathrm{xx}}$ traces at $T$ =
1, 5, 10, 20 and 40 K and $B_{\mathrm{tot}}=0$ T. c,
$\nu_{\mathrm{H}}-\nu_{0}$ as a function of $T$ and $\nu_{0}$ at
$B_{\mathrm{\perp}}$ = 0.5 T. Pink circles correspond to the point of maximum
descent in $\nu_{\mathrm{H}}-\nu_{0}$ for each temperature (see Fig. S3). d,
$\nu_{\mathrm{H}}-\nu_{0}$ traces at $T$ = 0.5, 2.5, 4.5, 8.0 and 20 K and
$B_{\mathrm{\perp}}$ = 0.5 T. e, $\rho_{\mathrm{xx}}$ as a function of
$\nu_{0}$ and $B_{\mathrm{tot}}$ oriented at an angle of $9.1^{\circ}$
relative to the plane, measured at 5, 10 and 15 K. Blue circles correspond to
$\rho_{\mathrm{xx}}$ peak positions. f, $\rho_{\mathrm{xx}}$ peak positions
(empty triangles) and points of maximum descent in $\nu_{\mathrm{H}}-\nu_{0}$
(filled triangles) plotted in the $\nu_{0}-B_{\mathrm{tot}}$ plane for 5, 10,
15 and 20 K.

We interpret the resistivity peak as the signature of a Zeeman-driven
transition from an isospin unpolarized (IU) paramagnetic state at low $B$ to a
spin- and valley-polarized isospin ferromagnetic (IF3) state at high $B$ in
which electrons are polarized into three of four isospin flavors. This
hypothesis is consistent with the observation of a strong Chern insulator
state (with Chern number $C=-3$) in this regime, visible at the top of Fig 1e.
This Chern insulator has been interpreted as full polarization into three
valley-projected $C=-1$ Hofstadter subbands in recent
literaturesaito_hofstadter_2020 ; wu_chern_2020 ; nuckolls_strongly_2020 ,
suggesting that the surrounding areas of the $\nu-B$ parameter space are
similarly in the IF3 phase. $B$-induced nucleation of a distinct symmetry
broken state is further supported by the behavior of
$\nu_{\mathrm{H}}=A_{\mathrm{u.c.}}n_{\mathrm{H}}$, shown in Fig. 1f (see also
Fig. S2 for data at other tilt angles), defined as the Hall density
($n_{\mathrm{H}}$) expressed in units of the superlattice unit cell area
$A_{\mathrm{u.c.}}$. As shown in Fig. 1f, in the absence of a large applied
$B$, the $\nu_{\mathrm{H}}$ is approximately linear from $\nu_{0}=0$ to $-2$
before undergoing a resetting to $\nu_{\mathrm{H}}\approx 0$ at $\nu_{0}=-2$
and again at $\nu_{0}=-3$. Under large applied $B$, $\nu_{\mathrm{H}}$
develops an additional reset near $\nu_{0}=-1$. To render this transition more
clearly, Fig. 1g shows the Hall density after subtraction of a linear
background. Resets in $\nu_{\mathrm{H}}$ appear as discrete steps between
quantized values of $\nu_{\mathrm{H}}-\nu_{0}$ corresponding to the number of
flavors lost to symmetry breaking relative to the state around charge
neutrality.

In purely out of plane $B$, the isospin symmetry breaking transition we
describe here is driven by a combination of the Zeeman effect and a Stoner-
type mechanism due to the internal structure of Hofstadter
bandssaito_hofstadter_2020 ; experimentally, it appears to be first order,
showing discontinuous jumps in transport coefficients and hysteretic
behaviorlu_superconductors_2019 . This behavior is observed near the $C=-3$
Chern insulator, while along the rest of the phase transition line the
transport coefficients evolve rapidly but smoothly. Nevertheless, we interpret
the data in terms of a first-order transition between the IU and IF3 phases,
which may be broadened by mesoscale phase separation driven by disorder or
long-range Coulomb interactionsspivak_transport_2006 . Experimentally, the
absence of clear first order signatures complicates precise identification of
the phase boundary. To address this, we empirically define the phase
boundaries as the position of the peak in $\rho_{\mathrm{xx}}$, and separately
as the point of maximum descent in $\nu_{\mathrm{H}}-\nu_{0}$ (see Fig. S3).
These are plotted in Fig. 1h, overlaid on a color map of
$\nu_{\mathrm{H}}-\nu_{0}$. Using either definition produces a similar phase
diagram, with four distinct phases appearing between $-4<\nu_{0}<0$, including
an IF2 phase polarized into two flavors for $-3\lesssim\nu_{0}\lesssim 2$ and
a fully isospin polarized IF1 phase for $-4\lesssim\nu_{0}<3$ in addition to
the IF3 and IU phases already described.

Figure 3: Isospin Pomeranchuk effect and entropy change. a, Sketch of the free
energy of the IU and IF3 phases as a function of $T$. $T^{*}$ corresponds to
the phase transition temperature where the free energy of IU equals to that of
IF3, while $\Delta E$ denotes the ground state energy difference. b, Sketch of
the entropy of the IU and IF3 phases as a function of $T$. $\Delta S$ denotes
the difference in entropy between the two phases at the phase transition
point. c, Phase boundaries for Zeeman-tuned transition ($E_{Z}^{*}/k_{B}$,
circles) and temperature tuned transition ($T^{*}$, squares). Boundaries are
shown as extracted both from $\rho_{\mathrm{xx}}$ peaks (blue symbols) and
from points of maximum descent in $\nu_{\mathrm{H}}-\nu_{0}$ (pink symbols).
d, Entropy change per superlattice unit cell across the transition, $\Delta
s/k_{B}$ (see Eq. (2)) as a function of $\nu_{0}$. Pink and blue curves are
based on the data in panel c.

Remarkably, a transition very similar to that observed as a function of
$B_{tot}$ can be actuated purely by changing the temperature. Figure 2a-d
shows data for $\rho_{\mathrm{xx}}$ and $\nu_{\mathrm{H}}-\nu_{0}$ measured in
the low-$B$ limit up to temperatures of several tens of Kelvin. At $T=10$ K,
for example, $\rho_{\mathrm{xx}}$ develops a new resistivity peak between
$\nu_{0}=-1$ and $\nu_{0}=-2$ accompanied by a new kink in
$\nu_{\mathrm{H}}-\nu_{0}$. Applying the same analysis as for the
$B$-dependent data, we find phase boundaries that behave similarly to those
observed as a function of $B$. Specifically, the high temperature extensions
of the IU and IF2 phases are disfavored by increased temperature as compared
to a new phase which we tentatively identify with the high temperature
extension of the IF3 phase. To confirm this correspondence, we perform
variable temperature measurements of $\rho_{\mathrm{xx}}$, shown in Fig. 2e
for $T=$ 5, 10, and 15 K. At the higher temperatures, the resistivity peak
separating the high temperature extensions of the IU phase from the new phase
is visible at $B_{\mathrm{tot}}=0$ T, and is observed to move towards
neutrality as a function of $B_{\mathrm{tot}}$ (see also Fig. S4)—precisely
the expected behavior if the new phase features either a new spin polarization
or large magnetic susceptibility, as would be expected from the high
temperature extension of the IF3 phase.

Our experiments are carried in a statistical ensemble in which the temperature
$T$ and the gate voltage $v_{g}$ are fixed. We parametrize $v_{g}$ by the
equivalent filling factor $\nu_{0}=\frac{1}{e}c_{g}v_{g}$, where $c_{g}$ is
the geometric capacitance of the device per moiré unit cell.
Thermodynamically, a first-order phase transition occurs when the Landau grand
potentials of the two neighboring phases per unit cell are the same (see
Methods):

$\Delta\Omega(\nu_{0},T)=\Delta\tilde{e}(\nu_{0},T)-T\Delta s(T,\nu_{0})=0.$
(1)

Here, $\Delta\tilde{e}=\Delta{e}-ev_{g}\Delta\nu$ where $\Delta e$ and
$\Delta\nu$ are the differences in the internal energy per unit cell and the
filling factor between the two phases, and $\Delta s$ is the difference in
entropy per unit cell. As the temperature increases, high entropy states are
favored. This tends to stabilize disordered states in which inter-particle
correlations play only a minor role. The most well-known exception is the
Pomeranchuk effect in 3He pomeranchuk_theory_1950 . Here, a Fermi liquid
ground state solidifies at finite temperature due to the anomalously high
entropy of the nuclear spins in the solid phase. These are very nearly
decoupled and so contribute $s=k_{B}\ln 2$ per particle even at low
temperatures where the Fermi liquid entropy, $s\propto T$, is negligible.
Electronic analogs of this effect have been proposedspivak_phases_2004  for
low density two dimensional electron systems, at the transition from a Fermi
liquid to a Wigner crystal phase.

We propose that a similar mechanism underlies the observed behavior in twisted
bilayer graphene for $-1.5\lesssim\nu_{0}\lesssim-1$. In this scenario, the IU
Fermi liquid state has lower ground state energy than the IF3 phase, but the
fluctuating moments of the IF3 state endow it with a large entropy at elevated
temperature. The transition is thus between a Fermi liquid for $T<T^{*}$ and
the fluctuating remnants of the IF3 phase for $T>T^{*}$ (see Fig. 3a). The
characteristic temperature scale at which these moments begin to fluctuate
strongly, giving rise to a large entropy, is given by the isospin stiffness,
which we denote $\rho_{\mathrm{s}}$. This scale describes the bandwidth of the
collective excitations of the spin, valley, and carbon sublattice degrees of
freedom bultinck_ground_2019  (see Fig. 3b), and has been calculated
numerically to be $\rho_{\mathrm{s}}\approx 1$ meVkhalaf_charged_2020 ;
wu_collective_2020 . Combined with the expectation that ground state energies
differ by similar energy scales, an entropically driven transition in the few
Kelvin regime is highly plausible.

In this picture, at high temperatures, the IF3 phases is distinguished from
the IU phase not by long range order but rather by the formation of local
isospin moments, correlated only on length scales comparable to the moiré
wavelength. In the low temperature IF3 phase, these moments order, but at high
temperatures they disorder with correlation lengths approaching a single unit
cell spacing. General considerations suggest that for high temperatures the
entropy will reflect the number of degrees of freedom $N$ of these moments, so
that $s/k_{B}\rightarrow\ln N\sim 1$.

Our experimental data mapping of the phase boundaries in the $\nu_{0}-B$ and
$\nu_{0}-T$ planes allows us to estimate the entropy difference at the phase
transition directly. From Eq. (1), it follows (see Methods) that the change in
entropy for a given $\nu_{0}$ across the phase transition is given by

$\Delta s=\frac{E^{*}_{Z}(\nu_{0})}{T^{*}(\nu_{0})}\frac{4+\nu_{0}}{3}$ (2)

where $T^{*}$ is the temperature of the phase boundary for vanishing magnetic
field, and $E_{Z}^{*}=g\mu_{B}B^{*}$ is the Zeeman energy at the phase
boundary for vanishing temperature (we assume that $g=2$). Fig. 3c shows these
two quantities extracted from the experimental measurements of
$\rho_{\mathrm{xx}}$ and $\nu_{\mathrm{H}}-\nu_{0}$ over a range of $\nu_{0}$,
and their ratio is plotted in Fig. 3d. Though the estimates differ
quantitatively, both produce an entropy difference of order $k_{B}$ per
superlattice unit cell over a wide range of densities.

We have focused our discussion on the density regime of $-2<\nu<-1$ where a
Pomeranchuk effect driven transition is observed. However, this range is
particular only in that the ground state is unpolarized, leading to a finite
temperature transition. The pseudospin stiffness itself is not expected to
vary strongly as a function of $\nu$. It follows from our observations that
soft neutral excitations of the electron system are likely play a key role in
the physics of flat band moiré systems. First, thermal disordering of the
internal degrees of freedom is expected to scatter electrons strongly once the
temperature exceeds the stiffness $\rho_{s}$. We speculate that the
‘superconducting-like’ transition observed in many moiré systems, in which the
resistivity rises rapidly at temperatures of a few Kelvin, may instead result
from strong scattering due to thermal fluctuations of the isospin degrees of
freedom. Some portion, if not the majority, of the large high-temperature
resistivity in flat band moiré systems likely arises from such scattering.
This appears consistent with the experimentally observed ubiquity of both
ferromagnetism across the flat bandzondiner_cascade_2020 ; wong_cascade_2020
and large resistivity at intermediate temperaturespolshyn_large_2019 ;
cao_strange_2020 . Notably, in this picture the detailed behavior of the high
temperature resistivity is expected to depend on the details of the collective
excitations and their coupling to the itinerant electrons. The precise
temperature dependence of the resistivity is therefore not expected to be
universal, consistent with experimental observation of strong
$\nu_{0}$-dependence of $\rho(T)$ (see Fig. S5). Finally, the small stiffness
implied by our measurements has important consequences for superconductivity.
In particular, $\rho_{s}$ may serve as an upper bound on $T_{c}$ in regions of
the phase diagram where isospin order is observed, either since the isospin
fluctuations act as pair breakers, or because the pairing mechanism itself
requires isospin order. For example, isospin ordering is a prerequisite for
the existence of skyrmion textures recently proposed to play a role in the
superconductivity observed in tBLGkhalaf_charged_2020 .

## Methods

Device fabrication and transport measurements
The tBLG device used in this study were fabricated using a “cut-and-stack”
technique described in Ref. saito_independent_2020, , and indeed the device is
the same as device #5 in that manuscript and the device studied in Ref.
saito_hofstadter_2020, . Prior to stacking, we first cut graphene into two
pieces using AFM to prevent the unintentional strain in tearing graphene. We
used a poly(bisphenol A carbonate) (PC)/polydimethylsiloxane (PDMS) stamp
mounted on a glass slide to pick up a 40-nm-thick hBN flake at 90$-$110∘C, and
carefully pick up the 1st half of a pre-cut graphene piece, rotate the sample
stage and pick up the 2nd half of the graphene at 25 ∘C using the hBN flake.
We rotated the graphene pieces manually by a twist angle of about
1.2${}^{\circ}-$1.3∘ before the second pickup. Finally, the 3-layer stack
(hBN-tBLG) is transferred onto another stack (hBN-graphite) containing the
bottom gate and gate dielectric, which is prepared in advance by the same dry
transfer process and cleaned by the typical solvent wash using chloroform,
acetone, methanol and IPA followed by vacuum annealing (400∘C for 8 hours) to
remove the residue of PC film on the hBN surface. Electrical connections to
the tBLG were made by CHF3/O3 etching and deposition of the Cr/Pd/Au (2/15/180
nm) metal edge-contactswang_one-dimensional_2013 .

Transport data in Figures 1a (top) and 2a-b were acquired in a top-loading
cryogen-free dilution refrigerator with a nominal base temperature of 10 mK,
using a probe with heavy RF filtering at an excitation current of 2 nA at a
frequency of 17.777 Hz. Data in Figures 1a (bottom), 1b-h, S1, S2, and S3a-c
were acquired in a different probe without filtering. Data in Figures 2e-f,
S4, and S5 were acquired using a wet, sample-in-vapor variable temperature
system without filtering at an excitation current of 10 nA at a frequency of
278 Hz. Data in Figures 2c-d and S3d-f below 4.5K was acquired in the
unfiltered dilution refrigerator probe while data above 4.5K was acquired in
the wet variable temperature system.

We measure the geometric capacitance from the Hall density near the charge
neutrality point to be be $c_{g}=58.2\pm 0.3$ nF/cm2. Separate measurements
using quantum oscillations give comparable results, $c_{g}=58.5\pm 0.1$
nF/cm2. The twist angle $\theta$ is determined from the values of charge
carrier density at which the insulating states at $n_{\nu_{0}=\pm 2}$ are
observed, following $n_{\nu_{0}=\pm 2}=\pm 4\theta^{2}/\sqrt{3}a^{2}$ , where
$a$= 0.246 nm is the lattice constant of graphene. From these measurements,
$\nu_{0}=A_{\mathrm{u.c}}c_{g}/e(v_{g}-v_{g}^{0})$ where $v_{g}^{0}$ is the
gate voltage corresponding to charge neutrality and
$A_{\mathrm{u.c.}}c_{g}/e=0.5026$ V-1.

Thermodynamic model

To describe the phase transition between the IU to the IF3 phases, we write
their free energy per moiré unit cell as:

$f_{i}(\nu,T)=e_{i}+\frac{1}{2}\left(\frac{1}{\kappa_{i}}+\frac{e^{2}}{c_{g}}\right)\nu^{2}+\mu_{i}\nu-\frac{1}{2}\gamma_{i}T^{2}-s_{i}T,$
(3)

where $i=1,2$ corresponds to the IU and IF3 phases, respectively, $e_{i}$ is
an offset energy, $c_{g}$ is the geometric capacitance to the gate per moiré
unit cell, $\mu_{i}$ is an offset chemical potential, $\gamma_{i}$ is the
specific heat coefficient (both phases are assumed to be metallic, despite the
fact that the IF3 phase has large, fluctuating magnetic moments), and
$\kappa_{i}$ is the compressibility (or quantum capacitance). $s_{i}$ is a
temperature-independent contribution of the entropy. The IU phase is a Fermi
liquid whose entropy is proportional to temperature, hence $s_{1}=0$. In the
IF3 phase, the fluctuating moments give a contribution $s_{2}>0$ to the
entropy at temperatures exceeding spin stiffness.

Since the experiment is carried out at a constant gate voltage, the phase
transition (assumed to be of first order in the absence of disorder) occurs
when the Landau grand potential $\Omega_{i}(v_{g},T)=f_{i}(\nu,T)-ev_{g}\nu$
of the two phases are equal. We minimize the grand potentials of each phase
with respect to $\nu$, and express the grand potentials in terms of the
reference filling factor $\nu_{0}\equiv\frac{1}{e}c_{g}v_{g}$. The transition
line in the $(\nu_{0},T)$ plane is then given by the condition:

$\Omega_{2}-\Omega_{1}=\Delta\tilde{e}(\nu_{0})-\frac{1}{2}\Delta\gamma
T^{2}-T\Delta s=0,$ (4)

where $\Delta\tilde{e}(\nu_{0})\approx
e_{2}-e_{1}-\frac{1}{2}\left(\frac{1}{\kappa_{2}}-\frac{1}{\kappa_{1}}\right)\nu_{0}^{2}+(\mu_{2}-\mu_{1})\nu_{0}$,
$\Delta\gamma=\gamma_{2}-\gamma_{1}$, and $\Delta s=s_{2}-s_{1}$. Here, we
have used the fact that in our setup
$\frac{e^{2}}{c_{g}}\gg\frac{1}{\kappa_{1,2}}$, and neglected terms that are
suppressed by factors of $\frac{c_{g}}{e^{2}\kappa_{1,2}}$. At sufficiently
low temperature compared to the bandwidth (estimated to be of the order of
$200-300$K), the quadratic term in $T$ is much smaller than the linear term,
giving a transition at

$T^{*}=\frac{\Delta\tilde{e}(\nu_{0})}{\Delta s}.$ (5)

To determine the entropy $\Delta s$ from the experiment, we need an estimate
of $\Delta e(\nu_{0})$. This can be obtained by examining the magnetic field
needed to trigger the transition from the IU to the IF3 phase at low
temperature (below the spin stiffness in the IF3 phase). We consider an in-
plane field, assuming that it acts primarily through the Zeeman effect. The
magnetic field induces an additional term in the grand potentials equal to
$-\int_{0}^{B}m_{i}(B^{\prime})dB^{\prime}$, where $m_{i}(B)$ is the magnetic
moment per moiré unit cell. At sufficiently low temperature, where the excess
magnetic entropy $s_{2}$ of the IF3 phase is quenched, this phase is spin
polarized, and its magnetization is nearly field-independent. To estimate the
magnetic moment in this phase, we assume that one isospin flavor whose spin is
antiparallel to the Zeeman field is completely empty (i.e., this flavor has a
filling of one hole away from charge neutrality), whereas the other three
flavors are equally populated zondiner_cascade_2020 . These considerations
give a magnetic moment of $m_{2}=2\mu_{B}\frac{4+\nu_{0}}{3}$ in the IF3
phase. In contrast, the IU phase has no magnetic moment at $B=0$. Since the IU
phase is a Fermi liquid, its magnetization is proportional to the ratio
between the Zeeman energy and the bandwidth, and is much smaller than that of
the IF3 phase. We therefore neglect the magnetization of the IU phase,
$m_{1}\approx 0$. The field-driven transition at low temperature occurs when

$\Delta\tilde{e}(\nu_{0})-2\mu_{B}\frac{4+\nu_{0}}{3}B^{*}=0.$ (6)

Combining Eqs. (5,6) gives Eq. (2) in the main text.

## Acknowledgments

The authors acknowledge discussions with S. Kivelson, A. Macdonald, B. Spivak,
and M. Zaletel, as well as experimental assistance from H. Polshyn and C.
Tshirhart. Y.S. acknowledges the support of the Elings Prize Fellowship from
the California NanoSystems Institute at University of California, Santa
Barbara. K.W. and T.T. acknowledge support from the Elemental Strategy
Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), JST.
Experimental work at UCSB was supported by the ARO under MURI
W911NF-16-1-0361. A.F.Y. acknowledges the support of the David and Lucille
Packard Foundation under award 2016-65145. EB was supported by the European
Research Council (ERC) under grant HQMAT (grant no. 817799), and by the US-
Israel Binational Science Foundation (BSF) under the NSF-BSF DMR program
(grant no. # DMR-1608055).

## Author Contributions

Y.S. and J.G. fabricated tBLG devices. Y.S. performed the measurements. Y.S.
and A.F.Y analyzed the data. A.F.Y. and E.B. constructed the thermodynamic
model. Y.S., E.B. and A.F.Y wrote the paper. T.T. and K.W. grew the hBN
crystals.

## Competing interests

The authors declare no competing financial interests.

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Supplementary Information for
Isospin Pomeranchuk effect and the entropy of collective excitations in
twisted bilayer graphene
Yu Saito, Jingyuan Ge, Kenji Watanabe, Takashi Taniguchi, Erez Berg and Andrea
F. Young

Figure S1: Landau fan diagrams for different magnetic field angles at a
nominal $T$ = 20 mK. a-g, $\rho_{\mathrm{xx}}$ as a function of nominal
filling factor $\nu_{0}$ and total magnetic field $B_{\mathrm{tot}}$ oriented
at an angle with respect to the plane $\theta_{\mathrm{B}}$ of 0.0∘ (a), 4.1∘
(b), 9.6∘ (c), 14.5∘ (d), 20.5∘ (e), 31.7∘ (f) and 90.0∘ (g). Figure S2: 2D
map of $\nu_{\mathrm{H}}-\nu_{0}$ for different magnetic field angles at a
nominal $T$ = 20 mK. a-g, $\nu_{\mathrm{H}}-\nu_{0}$ as a function of
$B_{\mathrm{tot}}$ and $\nu_{0}$ with $\theta_{\mathrm{B}}$ of 4.1∘ (a), 9.6∘
(b), 14.5∘ (c), 20.5∘ (d), 31.7∘ (e) and 90.0∘ (f). Figure S3: Comparison of
Hall density for Zeeman- and thermally driven transitions. a,
$\nu_{\mathrm{H}}$, b, $\nu_{\mathrm{H}}-\nu_{0}$, and c,
$d(\nu_{\mathrm{H}}-\nu_{0})/d\nu_{0}$ as a function of $\nu_{0}$ at
$B_{\mathrm{tot}}$ = 2, 4.5, 6, 12 T with $\theta_{\mathrm{B}}$ = 20.5∘ at $T$
= 20 mK. d, $\nu_{\mathrm{H}}$, e, $\nu_{\mathrm{H}}-\nu_{0}$, and f
$d(\nu_{\mathrm{H}}-\nu_{0})/d\nu_{0}$ as a function of $\nu_{0}$ at
$B_{\mathrm{tot}}=0.5$ T with $\theta_{\mathrm{B}}$ = 90.0∘ at $T$ = 0.5, 2.5,
4.5, 8 and 20 K. $d(\nu_{\mathrm{H}}-\nu_{0})/d\nu_{0}$ is calculated from
$\nu_{\mathrm{H}}-\nu_{0}$ using a 20-point moving average in $\nu_{0}$.
Figure S4: High temperature transport in tilted magnetic field.
$\rho_{\mathrm{xx}}$ (top) and $d(\nu_{\mathrm{H}}-\nu_{0})/d\nu_{0}$ (bottom)
as a function of $\nu_{0}$ at $T$ = 5 (a), 10 (b), 15 (c) and 20 K (d) at
$B_{\mathrm{tot}}$ = 3, 6, 9, 12, 15 T oriented at an angle of $9.1^{\circ}$
relative to the plane. $d(\nu_{\mathrm{H}}-\nu_{0})/d\nu_{0}$ is calculated
from $\nu_{\mathrm{H}}-\nu_{0}$ using a 20-point moving average in $\nu_{0}$.
Figure S5: Temperature dependence of $\rho_{\mathrm{xx}}$ and
$\rho_{\mathrm{xy}}$. a, $\rho_{\mathrm{xx}}$ as a function of temperature at
various $\nu_{0}$ for $\nu_{0}<0$ and b, $\nu_{0}>0$. c, $\rho_{\mathrm{xy}}$
as a function of temperature at various $\nu_{0}<0$ and d $\nu_{0}>0$. All
curves are offset by 2 k$\Omega$ for clarity.


# Linking resonant leptogenesis with dynamics of the inverse seesaw theory
with $A_{4}$ flavor symmetry

Maibam Ricky Devi<EMAIL_ADDRESS>Kalpana Bora<EMAIL_ADDRESS>Department of Physics, Gauhati University, Guwahati-781014, Assam, India

###### Abstract

In this paper, we analyse resonant leptogenesis in a low scale inverse seesaw
model with $A_{4}$ flavor symmetry, in a model we explored earlier to explain
light neutrino masses and mixings, and also charged lepton flavor violating
decay $\mu\rightarrow e\gamma$. Six $A_{4}$ scalar singlets and one $A_{4}$
fermion triplet are included, which are charged under the group $A_{4}\times
U(1)_{X}\times Z_{5}\times Z_{4}$, with at least two degenerate RH (Right
Handed) neutrinos. The light neutrino masses and leptogenesis both share a
same origin with the heavy right handed neutrinos. Thus, we expound the
possibility of generating resonant leptogenesis in this model at energies as
low as 1 TeV. We then analyse our findings to envision if our model inclines
more towards weak or strong washout.

##  I Introduction

Explaining origin of baryon asymmetry of the universe (BAU) is one of the most
important unsolved problems in physics. As per one school of thought, it is
believed that the BAU has its origin from the out-of-equilibrium decay of
heavy Majorana neutrinos during the early Universe. This mechanism, known as
leptogenesis, is usually implemented through the so-called hierarchical
scenario, which requires a hierarchical structure of the light neutrino masses
and CP-violating phases. However, it is also possible to generate the BAU in a
different framework, known as resonant leptogenesis (RL), which relaxes the
requirement of the heavy Majorana neutrinos being hierarchical and allows a
degeneracy between them. In the RL scenario, the CP-violating asymmetries
produced by the heavy Majorana neutrino decays are significantly larger than
in the hierarchical scenario. This is usually due to the fact that, instead of
the usual one-body decay, the heavy Majorana neutrinos can decay via a two-
body channel, which is strongly enhanced due to a resonant effect. Thus, the
BAU can be generated even for very small values of the CP-violating phases.

The baryon asymmetry of the universe can be generated by decay of the lightest
right-handed neutrino which easily follows the three Sakharov’s conditions
Sakharov:1967dj . To generate the lepton asymmetry, the Lagrangian needs to
violate C and CP symmetries as well as the lepton number. These right-handed
neutrinos can later undergo fast reactions called “Sphaleron Process”
Kuzmin:1985mm , that converts lepton-antilepton asymmetry to the observed
matter-antimatter asymmetry. If T is the temperature of the early universe and
$M_{1}$ is the mass of the lightest right-handed neutrino then any pre-
existing lepton asymmetry at $T\geq M_{1}$, can be erased by the high energy
Majorana neutrinos. However when the temperature of the universe falls below
$M_{1}$, the right-handed neutrinos can no longer follow the inverse reaction
to create thermal equilibrium, leading to a surplus of heavy Majorana
neutrinos. The event of leptogenesis Fukugita:1986hr  occurs at the point in
the timeline of the Universe when $T\leq M_{1}$. These right-handed Majorana
neutrinos are considered to be of out-of-equilibrium when the decay width of
the lightest right handed neutrino, $\Gamma(T)$ is found to be less than the
Hubble parameter $H(T)$ that determines the expansion of the universe. When
the two parameters reach an equilibrium condition, i.e., $\Gamma(T)\cong
H(T)|_{T=M_{1}}$, then this borderline condition is almost equivalent to the
condition that the effective neutrino mass $\widetilde{m_{i}}$ is equal to the
equilibrium neutrino mass $m_{*}$. We shall discuss this in details in the
next section.

High scale seesaw models to explain light neutrino mass are not testable, and
hence low scale seesaw models are considered to be very interesting. Moreover,
there is no theory so far that can authenticate the origin of Baryon Asymmetry
of the Universe. With this motivation, in this work we study resonant
leptogenesis Pilaftsis:1997dr ; Pilaftsis:1997jf ; Pilaftsis:2003gt  in the
inverse seesaw model developed and presented in Ref. Devi:2021aaz ;
Bora:2022jdp ; Devi:2022scm , as a possible explanation of the baryon
asymmetry of the universe. Low-scale seesaw models can be tested in the
current and future experiments such as BASE experiment BASE:2015ity . Some of
the previous works on resonant leptogenesis at low scale can be found in
Dolan:2018yqy ; Dev:2015cxa ; Eijima:2017anv ; Drewes:2016lqo ; Drewes:2016jae
; Lopez-Pavon:2015cga ; Boubekeur:2004ez ; Chakraborty:2021azg . In this low-
scale inverse seesaw model the CP asymmetry is resonantly enhanced due to
presence of two RH Majorana neutrinos of nearly degenerate masses. Testable
resonant leptogenesis at GeV-TeV scale in the context of flavor symmetry and
GUT has been discussed in Fong:2021tqj , whereas some studies of gravitino
constraints, which are important in the case of non-SUSY models to specify
assumptions on the physical conditions in the very early Universe can be found
Khlopov:1984pf , balestra1984p , Khlopov:1993ye ,and studies of physical and
cosmological effects of broken symmetry of families, unifying PQ symmetry and
mechanism of neutrino Majorana mass generation Sakharov:1994pr ,
Berezhiani:1989fp , Berezhiani:1990jj . We consider a hierarchical structure
of the light neutrino masses, with the lightest one being much smaller than
the other two, and we assume that the heavy Majorana neutrinos have a quasi-
degenerate spectrum. We calculate the CP-violating asymmetry produced by the
heavy Majorana neutrino decays as a function of the oscillation parameters and
determine the values of the oscillation parameters for which the resonance
condition is satisfied. We then calculate the BAU for those values of the
oscillation parameters and study the dependence of the BAU on the oscillation
parameters. We compare this our results with the recent data released by the
PLANCK Experiment Planck:2018vyg . One of the interesting features of this
scenario is that the resonant leptogenesis connects the neutrino physics with
cosmology. The Baryon Asymmetry of the Universe can be either enhanced or
suppressed for a particular set of neutrino oscillation parameters whose
relevant BAU values seems to be in agreement with the observed values. Thus
the neutrino oscillation parameters can significantly effect the observed
Baryon Asymmetry of the Universe.
The paper has been organised as follows. In Section 2, we give a brief review
about resonant leptogenesis. Section 3 contains details about our inverse
seesaw model Devi:2021aaz ; Bora:2022jdp ; Devi:2022scm  , and also about
degenerate heavy RH Majorana neutrinos. Numerical analysis about the work has
been presented in Section 4, and Section 5 contains results and discussion. We
summarise and conclude the work in Section 6.

##  II Resonant Leptogenesis

In the scenario of resonant leptogenesis which is typically like the low-
energy thermal leptogenesis, the Davidson-Ibarra bound can be circumvented
when at least two of the RH neutrinos (RHN) have degenerate masses. Lepton
number asymmetry can be sufficiently generated even for RH neutrino masses of
$\sim$ the TeV scale provided that the self-energy effects from the RHNs get
dominated in the lepton CP asymmetry. In this special case, the mass splitting
between the nearly degenerate RHNs is equivalent to the decay width of the
Majorana neutrinos. The Sphaleron process produces the baryon asymmetry
through CP asymmetry as they preserve the B-L quantum number.In the inverse
seesaw scenario, the resonant enhancement of the lepton asymmetry are achieved
by the degenerate pseudo-Dirac neutrinos in S-N sectors, as the matrix
$\mu_{s}$ controls the small mass splitting between the two nearly degenerate
RHNs. The CP asymmetry induced by the self-energy contribution in the decay of
the lightest RHNs into lepton of flavor $\alpha$ is given by Pilaftsis:1997jf
; Pilaftsis:2003gt ; Pilaftsis:1998pd ; Aoki:2015owa

$\displaystyle\epsilon_{i,j}$ $\displaystyle=$
$\displaystyle\frac{\sum_{\alpha}\left[\Gamma(N_{i,j}\rightarrow
L_{\alpha}+H^{\ast})-\Gamma(N_{i,j}\rightarrow
L_{\alpha}^{c}+H)\right]}{\sum_{\alpha}\left[\Gamma(N_{i,j}\rightarrow
L_{\alpha}+H^{\ast})+\Gamma(N_{i,j}\rightarrow L_{\alpha}^{c}+H)\right]}$ (1)
$\displaystyle\simeq$ $\displaystyle\frac{{\rm
Im}(Y_{\nu}^{N{\dagger}}Y_{\nu}^{S})^{2}_{ij}}{(Y_{\nu}^{S{\dagger}}Y_{\nu}^{S})_{ii}(Y_{\nu}^{N{\dagger}}Y_{\nu}^{N})_{jj}}\frac{(m^{2}_{N_{i}}-m^{2}_{N_{j}})m_{N_{i}}\Gamma_{j}}{(m^{2}_{N_{i}}-m^{2}_{N_{j}})^{2}+m^{2}_{N_{i}}\Gamma^{2}_{j}}$

The decay width of the neutrinos $N_{i,j}$ is given by
$\Gamma_{i,j}=A_{i,j}m_{N_{i,j}}/(8\pi)$ where

$\displaystyle
A_{i}=(Y_{\nu}^{S{\dagger}}Y_{\nu}^{S})_{ii},~{}~{}~{}A_{j}=(Y_{\nu}^{N{\dagger}}Y_{\nu}^{N})_{jj}.$
(2)

.
The baryon asymmetry $\eta_{B}$ can thus be determined from the lepton
asymmetry $\epsilon_{i,j}$ as Aoki:2015owa

$\eta_{B}=-\frac{28}{79}\frac{0.3\epsilon_{i,j}}{g_{*}\kappa_{i,j}(ln\kappa_{i,j})^{0.6}}$
(3)

where $g_{*}=106.75$ is the number of relativistic degrees of freedom and the
Hubble constant $H(T)=1.66\sqrt{g_{\ast}}T^{2}/m_{\rm Pl}$, and $m_{\rm Pl}$
is the Planck mass. The decay parameter $\kappa$ can be used so as to estimate
the expected baryon asymmetry as follows Pilaftsis:2003gt

$\kappa_{i,j}=\dfrac{\Gamma_{i,j}}{H(T)}|_{T=m_{N_{i,j}}}=\dfrac{\widetilde{m}}{m_{*}}$
(4)

where
$\widetilde{m}=\Sigma\widetilde{m_{l}}=\dfrac{(Y_{l1}Y^{\dagger}_{l1})_{ii}\upsilon^{2}}{m_{N_{1}}}\sim
m_{\nu}$ and
$m_{*}=\dfrac{8\pi\upsilon^{2}}{m^{2}_{N_{1}}}H|_{T=m_{N_{1}}}=1.08\times
10^{-3}$ eV. Here, $N_{1}$ is the mass of lightest RHN, whose decay produces
the lepton asymmetry. The baryon asymmetry is enhanced for
$(m_{N_{i}}-m_{N_{j}})\sim\Gamma_{i,j}/2$ Pilaftsis:2003gt .

##  III Inverse seesaw model

For the sake of thoroughness, we review here the inverse seesaw model
Mohapatra:1986aw ; Mohapatra:1986bd , which was developed and studied in our
earlier work Devi:2021aaz . We then discuss how it limits the characteristics
of heavy neutrinos. We use cyclic groups $Z_{4}$ and $Z_{5}$ along with
$A_{4}$ group and $U(1)_{X}$ global symmetry Devi:2021aaz ,. In addition to
the SU(2) singlet neutrinos, $N$, our model also includes a family of three
singlet sterile neutrinos, designated as S. The neutrino mass matrix generated
from the seesaw model’s Lagrangian in the basis ($\nu^{c}_{L},N,S$) can be
expressed as follows Wyler:1982dd :

$\left(M_{\nu}\right)_{iss}=\begin{bmatrix}0&m_{D}&0\\\ m^{T}_{D}&0&M\\\
0&M^{T}&\mu_{s}\end{bmatrix}\textrm{.}$ (5)

Here, $m_{D}$ is the Dirac neutrino mass, M is the mass matrix of RH Majorana
neutrinos, and $\mu_{s}$ is the mass matrix of the sterile neutrinos. The
prerequisite $m_{D},M\gg\mu_{s}$ must be met in order to implement the inverse
seesaw mechanism. The light neutrino mass matrix is doubly suppressed by M and
depends on the previous condition to produce it at $\mathcal{O}(eV)$ scale as

$m_{\nu}=m_{D}(M^{T})^{-1}\mu_{s}M^{-1}m^{T}_{D}\textrm{ .}$ (6)

The lepton number conservation is broken down in this instance by the
$\mu_{s}$ term of the low-scale seesaw mechanism, as noted in tHooft:1980xss .
In Table (1), where $\Phi_{s}$, $\Phi_{t}$ are triplet scalar fields and
$\eta$, $\xi$, $\tau$, $\rho$, $\rho^{\prime}$, $\rho^{\prime\prime}$ are
singlet scalar fields under $A_{4}$ group transformation, we show the particle
content of the various fields included in the model. $H$ is the SM Higgs
doublet, and $L$ is the SM LH leptonic doublet.

| L | H | $e_{R}$ | $\mu_{R}$ | $\tau_{R}$ | N | S | $\Phi_{T}$ | $\Phi_{s}$ | $\eta$ | $\xi$ | $\tau$ | $\rho$ | $\rho^{\prime}$ | $\rho^{\prime\prime}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$A_{4}$ | 3 | 1 | 1 | $1^{\prime\prime}$ | $1^{\prime}$ | 3 | 3 | 3 | 3 | 1 | $1^{\prime}$ | $1^{\prime\prime}$ | 1 | 1 | 1
$Z_{4}$ | 1 | 1 | i | i | i | i | 1 | i | -i | -i | -i | -i | i | i | 1
$Z_{5}$ | 1 | 1 | $\omega$ | $\omega$ | $\omega$ | $\omega^{2}$ | 1 | $\omega$ | 1 | 1 | 1 | 1 | $\omega^{2}$ | 1 | 1
$U(1)_{X}$ | -1 | 0 | -1 | -1 | -1 | -1 | 1 | 0 | -1 | -1 | -1 | -1 | 0 | -4 | -3
Table 1: Particle content under $A_{4}\times Z_{4}\times Z_{5}\times U(1)_{X}$
symmetry for inverse seesaw model, taken from our work Devi:2021aaz

Applying the $A_{4}$ product rules now to the aforementioned fields yields the
following charged lepton’s Lagrangian:

$\mathcal{L}_{c.l.}\supset\dfrac{y_{e}}{\Lambda}(\bar{L}\Phi^{\dagger}_{T})He_{R}+\dfrac{y_{\mu}}{\Lambda}(\bar{L}\Phi^{\dagger}_{T})^{\prime}H\mu_{R}+\dfrac{y_{\tau}}{\Lambda}(\bar{L}\Phi^{\dagger}_{T})^{\prime\prime}H\tau_{R}\textrm{
.}$ (7)

In matrix form, Eq. (7) can be written as:

$M_{c.l.}=\dfrac{\upsilon^{\dagger}_{t}\upsilon_{h}}{\Lambda}\begin{bmatrix}y_{1}&0&0\\\
0&y_{2}&0\\\ 0&0&y_{3}\end{bmatrix}\textrm{,}$ (8)

Here, $y_{e}$, $y_{\mu}$ and $y_{\tau}$ are the coupling constants, the normal
cut-off scale of the theory is denoted as $\Lambda$, and the VEVs of the
standard model Higgs are taken as $\langle h\rangle=\upsilon_{h}$, and
$\langle\Phi_{t}\rangle=\upsilon_{t}$. After using the $A_{4}$ product rules,
which are $1^{\prime}\times 1^{\prime}=1^{\prime\prime}$, $1^{\prime}\times
1^{\prime\prime}=1,1^{\prime\prime}\times 1^{\prime\prime}=1^{\prime}$ and
$3\times 3=1+1^{\prime}+1^{\prime\prime}+3_{A}+3_{S}$ Altarelli:2010gt , we
may move on to writing the neutrino sector Lagrangian which can be expressed
as

$\mathcal{L}_{\rm Y}\supset
Y_{D}\frac{\bar{L}\tilde{H}N\rho^{\dagger}}{\Lambda}+Y_{M}NS\rho^{\dagger}+Y_{\mu}SS[\frac{\rho^{\prime}\rho^{\prime\prime^{\dagger}}(\Phi_{s}+\eta+\xi+\tau)}{\Lambda^{2}}]+h.c.\textrm{.}$
(9)

From Eqn. (9), we get various mass matrices as Devi:2021aaz :

$M_{D}=\dfrac{Y_{D}\upsilon_{h}\upsilon^{\dagger}_{\rho}}{\Lambda}\left(\begin{matrix}1&0&0\\\
0&0&1\\\ 0&1&0\end{matrix}\right)\Rightarrow
Y_{D}=M_{D}\dfrac{\Lambda}{\upsilon_{h}\upsilon^{\dagger}_{\rho}}\left(\begin{matrix}1&0&0\\\
0&0&1\\\ 0&1&0\end{matrix}\right)\textrm{, }$ (10)
$M=Y_{M}\upsilon^{\dagger}_{\rho}\left(\begin{matrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{matrix}\right)\textrm{, }$ (11) $\textrm{and,
}\mu_{s}=\dfrac{Y_{\mu_{s}}\upsilon_{\rho^{\prime}}\upsilon^{\dagger}_{\rho_{\prime\prime}}}{\Lambda^{2}}\left(\begin{array}[]{ccc}v_{\Omega}+2v_{s}\phi_{a}&v_{\xi}-v_{s}\phi_{c}&v_{\tau}-v_{s}\phi_{b}\\\
v_{\xi}-v_{s}\phi_{c}&v_{\tau}+2v_{s}\phi_{b}&v_{\Omega}-v_{s}\phi_{a}\\\
v_{\tau}-v_{s}\phi_{b}&v_{\Omega}-v_{s}\phi_{a}&v_{\xi}+2v_{s}\phi_{c}\\\
\end{array}\right)\textrm{.}$ (12) $\Rightarrow
m_{\nu}=F_{1}\left(\begin{array}[]{ccc}v_{\Omega}+2v_{s}\phi_{a}&v_{\xi}-v_{s}\phi_{c}&v_{\tau}-v_{s}\phi_{b}\\\
v_{\xi}-v_{s}\phi_{c}&v_{\tau}+2v_{s}\phi_{b}&v_{\Omega}-v_{s}\phi_{a}\\\
v_{\tau}-v_{s}\phi_{b}&v_{\Omega}-v_{s}\phi_{a}&v_{\xi}+2v_{s}\phi_{c}\\\
\end{array}\right)\textrm{,}$ (13)

where,
$F_{1}=\dfrac{Y^{2}_{D}Y_{\mu_{s}}}{Y^{2}_{M}}[\dfrac{v^{2}_{h}v_{\rho^{\prime}}v^{\dagger}_{\rho^{\prime\prime}}}{\Lambda^{4}}]$.
Here, $Y_{D}$, $Y_{M}$, $Y_{\mu_{s}}$ are the dimensionless coupling constants
which are usually complex. The non-zero VEVs of scalars can be represented as:
$\langle H\rangle=v_{h}$, $\langle\Omega\rangle=v_{\Omega}$,
$\langle\rho\rangle=v_{\rho}$,
$\langle\rho^{\prime}\rangle=v_{\rho^{\prime}}$,
$\langle\rho^{\prime\prime}\rangle=v_{\rho^{\prime\prime}}$,
$\langle\xi\rangle=v_{\xi}$, $\langle\tau\rangle=v_{\tau}$,
$\langle\Phi_{S}\rangle=v_{s}(\Phi_{a},\Phi_{b},\Phi_{c})$.

Thus from Eqns. (11), (12) we get the mass of degenerate RH Majorana neutrinos
$M_{R}=M\pm\mu_{s}/2$, i.e.

$M_{R_{\pm}}=Y_{M}\upsilon^{\dagger}_{\rho}\left(\begin{matrix}1&0&0\\\
0&0&1\\\
0&1&0\end{matrix}\right)\pm\dfrac{K}{2}\left(\begin{array}[]{ccc}A+2\phi_{a}&C-\phi_{c}&B-\phi_{b}\\\
C-\phi_{c}&B+2\phi_{b}&A-\phi_{a}\\\ B-\phi_{b}&A-\phi_{a}&C+2\phi_{c}\\\
\end{array}\right)$ (14)

where each non-zero element is M $\sim\mathcal{O}(10^{12})$ eV,
$K=\dfrac{Y_{\mu_{s}}v_{s}\upsilon_{\rho^{\prime}}\upsilon^{\dagger}_{\rho_{\prime\prime}}}{\Lambda^{2}}\sim\mathcal{O}(1)$
eV. We scan the parameters $A,B,C\sim(1-9)\times 10^{3}$ where
$A=v_{\Omega}/v_{s}$, $C=v_{\xi}/v_{s}$, $B=v_{\tau}/v_{s}$ and are
dimensionless parameters.

##  IV Numerical analysis

We compute the BAU in the low-scale resonant leptogenesis scenario as
described in section II. On comparing Eqn. (6) with the light neutrino mass
matrix obtained using the global best fit values with a parametrisation of
$U_{PMNS}$ matrix, we get a system of equations for the triplet flavon
components. Then we scan the unknown parameters space (as input) in a viable
range.The information obtained by numerically solving this system of equations
can be used to compute BAU in our model, using equations (1-3). We have done
our computation up to the tolerance $<10^{-5}$.

As the scale of flavor symmetry breaking is still not known, therefore, in
Eqn. (10), we take different values of
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}$ for different cases, and compute
$\epsilon$ and $\eta_{B}$ accordingly. Mass matrix $M_{D}$ is parametrised by
fitting the light-neutrino data in the Extended Casas Ibarra parametrisation
Dolan:2018qpy ; Dolan:2018yqy

$\Rightarrow M_{D}=Um^{1/2}R\mu_{s}^{-1/2}M^{T}$ (15)

here, R is taken as complex orthogonal matrix Dolan:2018yqy
$RR^{T}=\mathbf{1}_{3\times 3}$ with
$\theta_{n}=Re(\theta_{n})+iIm(\theta_{n})$

$R(\theta_{1},\theta_{2},\theta_{3})=R(\theta_{1})R(\theta_{2})R(\theta_{3})$
(16) $R(\theta_{1})=\left(\begin{matrix}c_{1}&-s_{1}&0\\\ s_{1}&c_{1}&0\\\
0&0&1\end{matrix}\right),R(\theta_{2})=\left(\begin{matrix}c_{2}&0&-s_{2}\\\
0&1&0\\\
s_{2}&0&c_{2}\end{matrix}\right),R(\theta_{3})=\left(\begin{matrix}1&0&0\\\
0&c_{3}&-s_{3}\\\ 0&s_{3}&c_{3}\end{matrix}\right)$ (17)

where $c_{1}=cos\theta_{1}$, $c_{2}=cos\theta_{2}$, $c_{3}=cos\theta_{3}$ and
$s_{1}=sin\theta_{1}$, $s_{2}=sin\theta_{2}$, $s_{3}=sin\theta_{3}$. The
values of the real and imaginary components of $\theta_{1}$, $\theta_{2}$ and
$\theta_{3}$ (randomly chosen) are fixed as:

$Re[\theta_{1}]=\dfrac{\pi}{5},Re[\theta_{2}]=\dfrac{5\pi}{6},Re[\theta_{3}]=\dfrac{4\pi}{7},Im[\theta_{1}]=Im[\theta_{3}]=0,Im[\theta_{2}]=10^{-2}.$
(18)

where there is no specific significance to the chosen value of the above
angles. The resonant leptogenesis parameters are not sensitive to these values
and hence any value of above angles can be used.

##  V Results and Discussions

By examining whether the deviation from thermal equilibrium during
baryogenesis took place through a freeze-out or free-in scenario, one may
categorise the type of leptogenesis one has to deal with. Typically, heavy
neutrinos with degenerate Majorana masses in the freeze-out framework are
associated with resonant leptogenesis. The masses of heavy Majorana neutrinos,
coupling constants and mass splitting between degenerate Majorana masses
decide the magnitude of the BAU. In what follows, we discuss some parameters
and issues that affect the value of BAU.

###  V.1 Prerequisites and assessment of parameter analysis

We solve a set of flavon equations of triplet scalar $\Phi_{s}$ that is
obtained after comparing Eqn. (6) derived from our model with the light
neutrino mass matrix obtained after standard parametrisation
$U_{PMNS}m_{\nu_{diag}}U^{T}_{PMNS}$. In these flavon equations, whose
solutions provide us with the values of the unknown neutrino oscillation
parameters, we do a parameter scan for the known sets of neutrino oscillation
parameters involved. We examined Devi:2021aaz  the potential parameter space
for the normal and inverted hierarchies within the 3$\sigma$ range of the
known parameters, while taking into account various VEV alignments of the
triplet scalar flavon $\Phi_{s}$. It was shown in Devi:2021aaz  that only six
VEVs are valid in particular mass hierarchies, namely (0,1,1)/(0,-1,1) in NH,
(-1,1,1)/(1,-1,-1) in NH, and (0,1,-1)/(0,-1,1) in IH. To compute the baryon
asymmetry of the universe while taking into account the 23 elements of mass
matrix in equation (14) as $\sim 1$ TeV, we use the data of the oscillation
parameters for each of the permissible VEV alignments for the purpose. From
equations (1-4), the connection between leptogenesis and heavy Majorana
neutrinos of about 1 TeV mass is evident, and we can use this to trace the
evolution of the baryon asymmetry.

Since so far, there is no evidence in support of a particular value of scale
of flavour symmetry breaking, in the models one can take a viable range of
this scale and investigate its consequences on various physical observables.
In our earlier works Devi:2021aaz ; Bora:2022jdp ; Devi:2022scm , we discussed
about this. In this work therefore, we compute lepton asymmetry $\epsilon$ and
and BAU $\eta_{B}$ for four different values of
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}$, for all the three allowed VEVs
(as shown in our earlier work Devi:2021aaz ) of the triplet scalar flavon,
$\Phi_{s}$, in Table (2).

$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}$ | VEV (-1,1,1) in NH
---|---
Range of $\epsilon$ | Range of $\eta_{B}$
0.1 | ${-4.40209\times 10^{-6}\rightarrow 0.000151268}$ | ${2.06889\times 10^{-13}\rightarrow 1.84782\times 10^{-9}}$
0.3 | ${-0.0000396188\rightarrow 0.00136141}$ | ${1.862\times 10^{-12}\rightarrow 1.66304\times 10^{-8}}$
0.5 | ${-0.000110052\rightarrow 0.0037817}$ | ${5.17222\times 10^{-12}\rightarrow 4.61955\times 10^{-8}}$
0.7 | ${-0.000215703\rightarrow 0.00741213}$ | ${1.01375\times 10^{-11}\rightarrow 9.05433\times 10^{-8}}$
(a) VEV (-1,1,1) in NH

$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}$ | VEV (0,1,1) in NH
---|---
Range of $\epsilon$ | Range of $\eta_{B}$
0.1 | ${-1.65326\times 10^{-6}\rightarrow 7.87137\times 10^{-6}}$ | ${8.80658\times 10^{-19}\rightarrow 6.61007\times 10^{-11}}$
0.3 | ${-0.0000148793\rightarrow 0.0000708423}$ | ${7.92592\times 10^{-18}\rightarrow 5.94906\times 10^{-10}}$
0.5 | ${-0.0000413314\rightarrow 0.000196784}$ | ${2.20165\times 10^{-17}\rightarrow 1.65252\times 10^{-9}}$
0.7 | ${-0.0000810096\rightarrow 0.000385697}$ | ${4.31523\times 10^{-17}\rightarrow 3.23894\times 10^{-9}}$
(b) VEV (0,1,1) in NH

$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}$ | VEV (0,1,-1) in IH
---|---
Range of $\epsilon$ | Range of $\eta_{B}$
0.1 | ${-3.13823\times 10^{-7}\rightarrow 0.0000242336}$ | ${3.22169\times 10^{-15}\rightarrow 2.08037\times 10^{-10}}$
0.3 | ${-2.82441\times 10^{-6}\rightarrow 0.000218103}$ | ${2.89952\times 10^{-14}\rightarrow 1.87234\times 10^{-9}}$
0.5 | ${-7.84558\times 10^{-6}\rightarrow 0.000605841}$ | ${8.05422\times 10^{-14}\rightarrow 5.20093\times 10^{-9}}$
0.7 | ${-0.0000153773\rightarrow 0.00118745}$ | ${1.57863\times 10^{-13}\rightarrow 1.01938\times 10^{-8}}$
(c) VEV (0,1,-1) in IH

Table 2: Ranges of parameters $\epsilon$ and $\eta_{B}$ for VEV (-1,1,1) in
NH, VEV (0,1,1) in NH and VEV (0,1,-1) in IH are shown in Tables (2a), (2b)
and (2c) respectively

We have also obtained the correlation plots ($\eta_{B}\times 10^{10}$ vs
$Log[\kappa]$), ($\eta_{B}\times 10^{10}$ vs $Log[\epsilon]$),
($\eta_{B}\times 10^{10}$ vs $m_{3}$) and ($\eta_{B}\times 10^{10}$ vs
$(Sin\delta)^{2}$), for the three allowed VEV alignments of the triplet scalar
flavon, and are shown in Figs. (1\- 9), where (\- -) dashed and (—) bold lines
represents the limits given by $5.8\leq\eta_{B}\times 10^{-10}\leq 6.5$Riemer-
Sorensen:2017vxj ; Cooke:2017cwo  and $\eta^{obs}_{B}=(6.5\pm 0.08)\times
10^{-10}$, Planck:2018vyg  respectively.

###  V.2 Impact of Coupling constant and flavon VEV alignment

One of the the primary factors affecting the production and decay rates of RH
neutrinos is the magnitude of the relevant Yukawa couplings. Small Yukawa
couplings result in lesser production of RH neutrinos and smaller decay rates,
which might result in a BAU value that is lower than the observed value as is
evident from Eqn. (1). Fortunately, this limit can be circumvented for special
choices of parameters such as flavon VEVs $\upsilon^{\dagger}_{\rho}$ and the
cut-off scale of the theory related to seesaw mechanism $\Lambda$, for which
the size of the Yukawa couplings can be sufficient enough, to produce
baryogenesis via Sphaleron processes within its global fit parameter space.
From Eqn. (10), we can infer that there is a decrease in the production of
right-handed Majorana neutrinos as the ratio of
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}$ increases. This can be justified
because the Yukawa couplings gradually decrease with the increasing value of
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}$. The computed information on
dynamics of the theory, i.e. VEV of triplet flavon field and values of the
Yukawa couplings, for which the light neutrino mass matrix of equation (13)
lies in its experimentally allowed $3\sigma$ range, are shown in Table (3).

$\upsilon^{\dagger}_{\rho}/{\Lambda}$ | Range of Yukawa Coupling
---|---
VEV (-1,1,1)in NH | VEV (0,-1,1)in IH | VEV (0,1,1)in NH
0.1 | ${0.0485142\rightarrow 0.431652}$ | ${0.117593\rightarrow 0.549414}$ | ${0.000250288\rightarrow 0.518041}$
0.3 | ${0.0161714\rightarrow 0.143884}$ | ${0.0391976\rightarrow 0.0183138}$ | ${0.0000834292\rightarrow 0.17268}$
0.5 | ${0.00970285\rightarrow 0.0863305}$ | ${0.023185\rightarrow 0.109883}$ | ${0.000000575\rightarrow 0.103608}$
0.7 | ${0.00693061\rightarrow 0.0616646}$ | ${0.016799\rightarrow 0.0784878}$ | ${0.0000357554\rightarrow 0.0740058}$
Table 3: Ranges of Yukawa couplings for different VEV alignments of the
triplet flavon field

###  V.3 Effect of Washout on Leptogenesis

In addition to the mass splittings and decay width of the RH neutrinos, the
amount of BAU generated by low-scale leptogenesis also depends on how quickly
any asymmetry is eluded through washout processes.

(a)

(b)

(c)

(d)

Figure 1: Different correlation plots between neutrino oscillation parameters
and resonant leptogenesis parameters, as computed in our work. The plots (1a),
(1b), (1c) and (1d) show the correlation between (i) $\eta_{B}.10^{10}$ vs
Log[$\kappa$], (ii) $\eta_{B}.10^{10}$ vs Log[$\epsilon$], (iii)
$\eta_{B}.10^{10}$ vs $m_{1}$ and (iv) $\eta_{B}.10^{10}$ vs $(Sin\delta)^{2}$
for (-1,1,1) NH with $\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}\sim 0.1$
respectively. The (\- -) dashed and (—) bold lines represents the limits given
by $5.8\leq\eta_{B}\times 10^{-10}\leq 6.5$Riemer-Sorensen:2017vxj ;
Cooke:2017cwo  and $\eta^{obs}_{B}=(6.5\pm 0.08)\times 10^{-10}$,
Planck:2018vyg  respectively.

(a)

(b)

(c)

(d)

Figure 2: The plots (2a), (2b), (2c) and (2d) show the correlation between (i)
$\eta_{B}.10^{10}$ vs Log[$\kappa$], (ii) $\eta_{B}.10^{10}$ vs
Log[$\epsilon$], (iii) $\eta_{B}.10^{10}$ vs $m_{1}$ and (iv)
$\eta_{B}.10^{10}$ vs $(Sin\delta)^{2}$ for (-1,1,1) NH with
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}\sim 0.3$ respectively.

(a)

(b)

(c)

(d)

Figure 3: The plots (3a), (3b), (3c) and (3d) show the correlation between (i)
$\eta_{B}.10^{10}$ vs Log[$\kappa$], (ii) $\eta_{B}.10^{10}$ vs
Log[$\epsilon$], (iii) $\eta_{B}.10^{10}$ vs $m_{1}$ and (iv)
$\eta_{B}.10^{10}$ vs $(Sin\delta)^{2}$ for (-1,1,1) NH with
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}\sim 0.5$ respectively.

(a)

(b)

(c)

(d)

Figure 4: The plots (4a), (4b), (4c) and (4d) show the correlation between (i)
$\eta_{B}.10^{10}$ vs Log[$\kappa$], (ii) $\eta_{B}.10^{10}$ vs
Log[$\epsilon$], (iii) $\eta_{B}.10^{10}$ vs $m_{1}$ and (iv)
$\eta_{B}.10^{10}$ vs $(Sin\delta)^{2}$ for (-1,1,1) NH with
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}\sim 0.7$ respectively.

(a)

(b)

(c)

(d)

Figure 5: The plots (5a), (5b), (5c) and (5d) show the correlation between (i)
$\eta_{B}.10^{10}$ vs Log[$\kappa$], (ii) $\eta_{B}.10^{10}$ vs
Log[$\epsilon$], (iii) $\eta_{B}.10^{10}$ vs $m_{1}$ and (iv)
$\eta_{B}.10^{10}$ vs $(Sin\delta)^{2}$ for (0,1,1) NH with
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}\sim 0.5$ respectively.

(a)

(b)

(c)

(d)

Figure 6: The plots (6a), (6b), (6c) and (6d) show the correlation between (i)
$\eta_{B}.10^{10}$ vs Log[$\kappa$], (ii) $\eta_{B}.10^{10}$ vs
Log[$\epsilon$], (iii) $\eta_{B}.10^{10}$ vs $m_{1}$ and (iv)
$\eta_{B}.10^{10}$ vs $(Sin\delta)^{2}$ for (0,1,1) NH with
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}\sim 0.7$ respectively.

(a)

(b)

(c)

(d)

Figure 7: The plots (7a), (7b), (7c) and (7d) show the correlation between (i)
$\eta_{B}.10^{10}$ vs Log[$\kappa$], (ii) $\eta_{B}.10^{10}$ vs
Log[$\epsilon$], (iii) $\eta_{B}.10^{10}$ vs $m_{3}$ and (iv)
$\eta_{B}.10^{10}$ vs $(Sin\delta)^{2}$ for (0,1,-1) IH with
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}\sim 0.3$ respectively.

(a)

(b)

(c)

(d)

Figure 8: The plots (8a), (8b), (8c) and (8d) show the correlation between (i)
$\eta_{B}.10^{10}$ vs Log[$\kappa$], (ii) $\eta_{B}.10^{10}$ vs
Log[$\epsilon$], (iii) $\eta_{B}.10^{10}$ vs $m_{3}$ and (iv)
$\eta_{B}.10^{10}$ vs $(Sin\delta)^{2}$ for (0,1,-1) IH with
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}\sim 0.5$ respectively.

(a)

(b)

(c)

(d)

Figure 9: The plots (9a), (9b), (9c) and (9d) show the correlation between (i)
$\eta_{B}.10^{10}$ vs Log[$\kappa$], (ii) $\eta_{B}.10^{10}$ vs
Log[$\epsilon$], (iii) $\eta_{B}.10^{10}$ vs $m_{3}$ and (iv)
$\eta_{B}.10^{10}$ vs $(Sin\delta)^{2}$ for (0,1,-1) IH with
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}\sim 0.7$ respectively.

It is seen from Table (2) that in all the cases listed, we get baryon
asymmetry of the universe within the allowed region of $\eta_{B}$
Fields:2014uja , except for (0,1,1) in NH and (0,1,-1) in IH for
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}\sim 0.1$. The decay parameter
$\kappa$ is found to be greater than 1 for all three allowed cases. This
indicates a strong washout scenario where the heavy Majorana neutrinos even at
temperature $T<M_{1}$ do not decouple from the thermal bath immediately, and
the forward and inverse decay reactions thermalise very quickly at $T\sim
M_{1}$ Biondini:2016hhn . This removes any pre-existing asymmetry before
leptogenesis occurs. Thus, our model indicate towards strong washout leading
to highly predictive resonant leptogenesis that doesn’t depend on the initial
conditions Biondini:2016hhn . Although the generation of matter-antimatter
asymmetry at $T<M_{1}$ through the decay of the lightest Majorana neutrinos is
applicable for both the weak and strong washout scenarios, however, it is
worth mentioning that in the case of strong washout, the lepton asymmetry
begins immediately after $T\sim M_{1}$. Whereas, for weak washout, the
asymmetry is gradually generated at later times Biondini:2016hhn . From
results presented in Table (2) and Figs. (1\- 9), we conclude that for
$0.1\leq\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}\leq 0.7$, the BAU lies
within the experimental bounds which imply that resonant leptogenesis can be
successfully enhanced in our inverse seesaw scenario. Leptogenesis can have a
sizeable parameter space, as seen from our results in Figs (1\- 9) with a
total of six yet unknown parameters. These variables are the lightest neutrino
mass $m_{1}$, the Dirac and Majorana phases of the PMNS matrix and the lepton
asymmetry and baryon number density.

By carefully examining the plots (1\- 9), it is evident that

1\. With increasing value of $\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}$, the
number of points in the allowed region of $\eta_{B}$ increases for (0,1,1)/NH
and (0,-1,1)/IH, and decreases for (-1,1,1)/NH .

2\. There are no points in the allowed region of $\eta_{B}$, for
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}=$0.1 and 0.3, for (0,1,1)/NH and
(-1,1,1)/IH, as well as $\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}=0.1$ for
(-1,1,1)/NH.

3\. Hence, we can say that the value of BAU depends on the decay parameter
$\kappa$, triplet flavon alignment and the CP asymmetry.

4\. From the plot of light neutrino mass versus baryon number density, in
figures (1c, 2c, 3c, 4c, 5c, 6c, 7c, 8c, 9c), it is observed that range of
this mass has a dependence on triplet flavon VEV alignment.

Hence, in future, if the mass hierarchy and the absolute value of mass of
light neutrinos is measured with precision, the results presented in this work
can throw some light on scale of flavor symmetry breaking as well as favored
VEV alignment of the triplet flavon of $A_{4}$ symmetry. Thus, BAU can be
connected with dynamics of flavor symmetry.

##  VI Summary and Conclusion

The baryon asymmetry of the universe may be explained by the mechanism of
leptogenesis, which states that the excess of matter over antimatter in the
universe is due to the decay of the lightest Majorana neutrinos. In order for
this mechanism to be able to produce the observed baryon asymmetry of the
universe, the neutrino oscillation parameters must lie within a certain range.
In this study, we investigated the effect of the neutrino oscillation
parameters on the baryon asymmetry of the universe in the context of resonant
leptogenesis, and the BAU lies within its experimental bounds. We considered
our previously developed ISS model with $A_{4}$ flavor symmetry. We found that
the BAU of the universe is sensitive to the lightest neutrino mass, which must
lie within a certain range in order to achieve the observed asymmetry. It is
also observed that the BAU is significantly affected by the mass splitting
between the two heaviest neutrinos and CP-violating phases.

We observed that the decay parameter $\kappa$ is found to be greater than 1
for all three allowed cases. It indicates that the theory of resonant
leptogenesis follows a strong washout scenario where the heavy Majorana
neutrinos even at temperature $T<M_{1}$ don’t decouple from the thermal bath
immediately and the forward and inverse decay reactions thermalise very
quickly. It removes any pre-existing asymmetry before leptogenesis occurs,
leading to highly predictive resonant leptogenesis that doesn’t depend on the
initial conditions. BAU in our model is found to lie within currently allowed
limits for a large parameter space with six unknown parameters, including the
lightest neutrino mass $m_{1}$, Dirac and Majorana phases of the PMNS matrix,
lepton asymmetry and baryon number density. The permitted mixing angles of the
heavy neutrinos are constrained by the requirement of recreating both the
baryon asymmetry of the universe and the light neutrino masses through our
inverse seesaw model.

From these results, we find that more favorable cases from BAU point of view
(VEV alignment of triplet flavon and scale of symmetry breaking of $A_{4}$
flavor symmetry, and other parameters) are as follows:

VEV of $\Phi_{s}$ | Scale $\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}$ | Favoured ($\checkmark$) / Disfavoured ($\times$)
---|---|---
| 0.1 | $\checkmark$
| 0.3 | $\checkmark$
(-1,1,1) in NH | 0.5 | $\checkmark$
| 0.7 | $\checkmark$
| 0.1 | $\times$
| 0.3 | $\times$
(0,1,1) in NH | 0.5 | $\checkmark$
| 0.7 | $\checkmark$
| 0.1 | $\times$
| 0.3 | $\checkmark$
(0,-1,1) in IH | 0.5 | $\checkmark$
| 0.7 | $\checkmark$
Table 4: Favoured and disfavoured flavor symmetry breaking scale for allowed
VEV alignment of triplet scalar flavon for which BAU computed in our model
lies within its current 3$\sigma$ range.

Upon analyzing Table (4) and Figures (1-10), it is evident that a decrease in
the value of $\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}$ results in a lower
probability of locating a correlation between the model parameters and BAU
parameters. The VEV (-1,1,1) NH has a higher probability of probing the BAU
for $\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}$ values ranging from
$(0.1\rightarrow 0.7)$, while (0,1,1) NH has the lowest probability of probing
the BAU. Also, an increase in the value of
$\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}$ leads to an increase in the value
of $\eta_{B}.10^{10}$.

We would like to highlight that our analysis can give important insights in
understanding the role of the model parameters to find BAU and its parameters.
By identifying the VEVs that have a higher probability of probing the BAU,
researchers can focus on the specific VEV alignment where the BAU can be
highly predicted. Furthermore, the observed increase in $\eta_{B}.10^{10}$
with an increase in $\dfrac{\upsilon^{\dagger}_{\rho}}{\Lambda}$ highlights
the importance of considering these parameters in future studies. In
conclusion, our model has successfully predicted the BAU through resonant
leptogenesis, and it is found that the BAU depends on VEV alignment of the
triplet flavon and scale of symmetry breaking of the $A_{4}$ flavor symmetry,
the lightest neutrino mass, the mass splitting between the two heaviest
neutrinos and the CP-violating phases. Thus, one can pinpoint their favored
values, and can obtain some information on hitherto unknown dynamics of flavor
symmetry. Hence, our model is testable when all the light neutrino oscillation
parameters will be fixed in future, and it has been successfully able to
explain several phenomenon like light neutrino oscillation, charged lepton
flavor violation and BAU, in a unified manner. The findings reported in this
work will provide a benchmark for future studies on neutrino parameters and
BAU in the framework of resonant leptogenesis.

## Acknowledgment

The authors are grateful to Professor M. P. Bora of the Department of Physics
at Gauhati University for his assistance in creating a figure.

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