# Experimental review of Lepton Flavor Violation searches

M. Hernández Villanueva
on behalf of the Belle II collaboration  Deutsches Elektronen–Synchrotron,
22607 Hamburg, Germany

###### Abstract

A review of the experimental status in searches for charged Lepton Flavor
Violation (LFV) is presented. Searches for LFV in decays of leptons, heavy
mesons, and bosons are explored, with an emphasis on the experiments providing
the best constraints in each sector. In addition, prospects for upper limits
by current and upcoming experiments in the next decade are discussed.

##  I Introduction

In the Standard Model (SM), Lepton Flavor is a conserved quantity given the
left-handed chirality of massless neutrinos. The experimental results in
neutrino oscillation show that neutrinos change flavor, enabling Lepton Flavor
Violation (LFV) as an established fact. In a minimum extension of the SM where
neutrinos have a non-zero mass, charged LFV is enabled via neutrino
oscillations but heavily suppressed by GIM mechanisms with branching ratios of
$\sim 10^{-50}$, making them unobservable in current experiments. Most
extensions of the Standard Model enhance the branching ratio of LFV channels
up to 10-7, making any significant observation a clear signature of physics
Beyond Standard Model (BSM). On the other hand, setting upper limits on the
branching ratio of LFV translates into limits on the parameters associated
with BSM models.

In the literature, there are several reviews where LFV is discussed
exhaustively from the theoretical and experimental point of view Bernstein and
Cooper (2013); Cei and Nicolo (2014); Calibbi and Signorelli (2018); Ardu and
Pezzullo (2022). In this talk, an experimental overview is presented focusing
on the strongest limits set to the date. In addition, prospects on limits to
be set in future facilities are shown.

##  II LFV in muons

###  II.1 $\mu^{+}\to e^{+}\gamma$

The decay $\mu^{+}\to e^{+}\gamma$ was the first search of a LFV mode, even
before the neutrino was discovered, testing an alternative hypothesis without
neutrinos to the decay of a muon Hincks and Pontecorvo (1948). Since then,
many experiments have performed searches for the signature of $\mu^{+}\to
e^{+}\gamma$: a final state in the center-of-mass system of a back-to-back,
monochromatic positron, and photon with an energy of 52.8 MeV each. The two
main background sources are (i) the irreducible background $\mu^{+}\to
e^{+}\gamma\nu_{e}\bar{\nu}_{\mu}$ when the neutrinos are low energetic, and
(ii) the ”accidental” background $\mu^{+}\to e^{+}\nu_{e}\bar{\nu}_{\mu}$
combined with a photon from elsewhere.

The current best limit for $\mu^{+}\to e^{+}\gamma$ comes from the MEG
experiment at the PSI laboratory. It consists of a target stopping muons from
a $\mu^{+}$ beam, a drift chamber, scintillating timing counters, and a
calorimeter immersed in the magnetic field generated by a superconducting
magnet Adam et al. (2013). Figure 1 shows the schematic view of the MEG
experiment. With the full data set of $7.5\times 10^{14}$ muons stopped on
target from 2009 to 2013, MEG sets an upper limit for the branching ratio of
${\cal{B}}(\mu^{+}\to e^{+}\gamma)<4.2\times 10^{-13}$ at 90% C.L. Baldini et
al. (2016).

Figure 1: Schematic view of the MEG experiment with the decay of a muon in the
stopping target. Figure from Baldini et al. (2016).

The upgraded experiment MEG-II at PSI is designed to record a muon decay rate
twice that of the previous experiment, expecting to reach a sensitivity of
$\sim 10^{-14}$ with 3 years of data taking Baldini et al. (2018).

###  II.2 $\mu^{+}\to e^{+}e^{-}e^{+}$

The other possibility for searches of LFV in muon decays is via the channel
$\mu^{+}\to e^{+}e^{-}e^{+}$, where two positrons and an electron coming from
the same vertex are combined to search for an invariant mass matching the mass
of the muon. Main backgrounds contributions come from the decays $\mu^{+}\to
e^{+}e^{-}e^{+}\nu_{e}\bar{\nu}_{\mu}$ when the neutrinos carry low energy,
and accidental backgrounds from $\mu^{+}\to e^{+}\nu_{e}\bar{\nu}_{\mu}$ plus
an electron-positron pair coming from photon conversion.

The strongest limit for $\mu^{+}\to e^{+}e^{-}e^{+}$ comes from the SINDRUM
experiment at PSI. A muon beam of 28 MeV/$c$ is stopped in the center of the
detector at a rate of $\sim 5\times 10^{6}\mu^{+}/s$, and the electrons and
positrons coming from muons decays are detected in five concentric multiwire
proportional chambers and a cylindrical array of 64 scintillation counters
immersed in a magnetic field of 0.33 T. With no significant excess in the
signal region, an upper limit of ${\cal{B}}(\mu^{+}\to
e^{+}e^{-}e^{+})<1.0\times 10^{-12}$ at 90% C.L. is set Bellgardt et al.
(1988).

The Mu3e experiment at PSI aims for a sensitivity of $10^{-16}$ in the upper
limit for the decay $\mu^{+}\to e^{+}e^{-}e^{+}$ by the end of the decade
Wauters (2021). Such an improvement of four orders of magnitude with respect
to the current limit comes with experimental challenges to handle the
projected rate of $10^{9}~{}\mu^{+}/s$, using low-density pixel detectors as a
tracking system and a modern data acquisition system able to handle the high
rates. Figure 2 shows the schematic view of the Mu3e experiment.

Figure 2: Schematic view of the Mu3e experiment, simulating the decay of a
muon in the stopping target. Figure from Arndt et al. (2021).

###  II.3 $\mu^{-}N\to e^{-}N$

The conversion of a muon into an electron in the field of a nucleus represents
a LFV process when no neutrinos are produced in the final state. In such a
scenario, the kinematics of the process is modeled by a two-body decay as
illustrated in Figure 3, and the final state contains a mono-energetic
electron with energy

$E_{\mu e}=m_{\mu}-E_{\rm binding}-E_{\rm recoil},$ (1)

where $m_{\mu}$ is the mass of the muon, $E_{\rm binding}$ is the binding
energy of the 1s state, and $E_{\rm recoil}$ is the nuclear recoil energy for
a muonic atom, dependent on the atomic nucleus Lee et al. (2022). The main
source of background is decay-in-orbit events where the captured muon decays
to $\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$, faking signal events when the
energy of the electron is close to $E_{\mu e}$.

Figure 3: $\mu-e$ conversion in the field of a nucleus.

The current best limit comes from SINDRUM II at PSI, setting a limit in the
rate of neutrinoless $\mu-e$ conversion relative to ordinary muon capture of

$\frac{\mu^{-}N\to e^{-}N}{\text{captured}\;\mu-N}<3.3\times 10^{-13}$ (2)

using a gold target that produces a mono-energetic electron with $E_{\mu
e}=95.56$ MeV Bertl et al. (2006).

Several collaborations are preparing experiments for improving the limits on
$\mu^{-}N\to e^{-}N$ conversions. The Mu2e experiment under construction at
Fermilab projects a sensitivity down to $O(10^{-17})$ using aluminum disks as
stoping target that produce as signal electrons with $E_{\mu e}=104.97$ MeV
Bartoszek et al. (2014). The COMET experiment at J-PARC aims to reduce the
upper limit down to $O(10^{-15})$ for Phase-I and $O(10^{-17})$ for Phase-II,
using also aluminum as a target Abramishvili et al. (2020). The DeeMe
experiment also at J-PARC is being prepared for a sensitivity of $O(10^{-13})$
using a SiC target Natori (2014).

##  III LFV in tau leptons

The $\tau$’s rate production at current colliders (109/yr) is much lower with
respect to the production of muons in coming facilities (1011/s). In addition,
the lifetime of the $\tau$ lepton is significantly shorter and its decays must
be studied in the production spot, without possibility of being transported
like in the case of muons. However, thanks to the larger mass of the $\tau$
lepton, the BSM branching ratios can be orders of magnitude larger than the
corresponding muon decays. Moreover, the $\tau$ lepton is heavy enough to
enable searches of neutrinoless semileptonic decays.

Upper limits for LFV modes in $\tau$ lepton decays are mostly dominated by
B-Factories Belle Abashian et al. (2002) and BaBar Aubert et al. (2002),
designed to collide with high-intensity electrons and positrons at the
$\Upsilon(4S)$ energy in the center-of-mass system. The B-Factories share as
common characteristics a well-defined initial state up to initial-state
radiation (ISR), high vertex resolution, and dedicated subsystems for
calorimetry and particle identification. The cross-section for production of
$\tau$ lepton pairs is of the same order than for the production of B mesons,
making the B-Factories optimal experiments for searches of rare $\tau$ lepton
decays.

###  III.1 $\tau^{-}\to\ell^{-}\gamma$

Even in presence of bounds for $\mu\to e\gamma$, rates for the counterparts
$\tau^{-}\to\ell^{-}\gamma$ with $\ell^{-}=e^{-},\mu^{-}$ can be higher by
orders of magnitude. Furthermore, $\tau^{-}\to\mu^{-}\gamma$ is considered one
of the golden modes for the search of LFV in $\tau$ decays, having the largest
branching ratio in models where the LFV is induced by heavy BSM particles in
one-loop diagrams Aushev et al. (2010). However, searches for
$\tau^{-}\to\ell^{-}\gamma$ in $e^{+}e^{-}$ colliders are affected by strong
backgrounds ilustrated in Figure 4. An irreducible background coming from
$\tau^{-}\to\ell^{-}\bar{\nu}_{\ell}\nu_{\tau}$ decays plus a photon coming
from elsewhere, and Bhabha or di-muon events from electron-positron collision
misidentified as $\tau$ lepton pairs.

(a) Irreducible background

(b) Mis-id tagging

Figure 4: Main background contributions in searches of the decay
$\tau^{+}\to\ell^{+}\gamma$ in $e^{+}e^{-}$ collisions.

Both B-Factories BaBar and Belle have searched for the LFV decays
$\tau^{-}\to\ell^{-}\gamma$ with a recorded integrated luminosity of 516 fb-1
and 988 fb-1 respectively. Search for signal events is performed in a 2D
region, where the kinematics of the 2-body decay establishes a well-defined
signal region. BaBar identifies signal candidates with the beam-energy
constrained $\tau$ mass ($m_{\rm EC}$), defined from a kinematic fit after
requiring the center-of-mass $\tau$ energy to be $\sqrt{s}/2$ and assigning
the origin of $\gamma$ to the point of closest approach of the signal lepton
to the $e^{+}e^{-}$ axis, simultaneously with $\Delta
E=(E^{CM}_{\ell\gamma}-\sqrt{s}/2)$; Belle uses the beam-constrained invariant
mass ($M_{bc}$) vs $\Delta E$ phase space, being $M_{bc}=\sqrt{(E^{CM}_{\rm
beam})^{2}-|\vec{p}^{CM}_{\ell\gamma}|^{2}}$. Figure 5 shows remaining events
after the selection criteria is applied for $\tau^{-}\to e^{-}\gamma$ and
$\tau^{-}\to\mu^{-}\gamma$ candidates in BaBar and Belle respectively. As no
significant excess has been found in the signal region Aubert et al. (2010);
Abdesselam et al. (2021), the best 90% C.L. upper limits have been set as

$\displaystyle{\cal{B}}(\tau^{-}\to e^{-}\gamma)$ $\displaystyle<3.3\times
10^{-8}{\rm\hskip 14.22636pt(BaBar)},$ (3)
$\displaystyle{\cal{B}}(\tau^{-}\to\mu^{-}\gamma)$ $\displaystyle<4.2\times
10^{-8}{\rm\hskip 14.22636pt(Belle)}.$

Figure 5: Distribution of events in the 2D planes described in the text for
$\tau\to e\gamma$ candidates at BaBar (left) and $\tau\to\mu\gamma$ candidates
at Belle (right). The $2\sigma$ elliptical signal region is shown in both
cases. Figures from Aubert et al. (2010) and Abdesselam et al. (2021).

The Belle II experiment is the major upgrade and successor of Belle, expected
to collect 50 ab-1 by the end of its operation. The large statistics will
allow a better understanding of the physics backgrounds for LFV modes. In
addition, an increase in signal efficiency will be achieved at Belle II thanks
to higher trigger efficiencies and improvements in the reconstruction
algorithms. On the other hand, beam backgrounds are a potentially more serious
concern than in the predecessor Belle. The sensitivity expected at Belle II
for the $\tau\to\ell\gamma$ modes is at the $O(10^{-9})$ level with the full
data set Altmannshofer et al. (2019).

###  III.2 $\tau^{-}\to\ell^{-}\ell^{+}\ell^{-}$

Belle and BaBar have also searched for the LFV decay
$\tau^{-}\to\ell^{-}\ell^{+}\ell^{-}$ in six possible combinations when
$\ell=e\;{\rm or}\;\mu$, with Belle imposing the strongest upper limits
Hayasaka et al. (2010) listed at Table 1. Signal events are defined by four
reconstructed tracks, combining 3 charged lepton candidates with invariant
mass $m_{\ell\ell\ell}$ around the mass of the $\tau$ lepton and with $\Delta
E$ around 0. An advantage of the 3 lepton LFV decay channels with respect to
$\tau\to\ell\gamma$ modes is they are ”background-free”, in the sense that no
background events are expected in the signal region. To illustrate, Figure 6
shows remaining events in the $m_{\ell\ell\ell}$ vs $\Delta E$ at Belle for
$\tau^{-}\to e^{-}e^{+}e^{-}$ and $\tau^{-}\to\mu^{-}\mu^{+}\mu^{-}$ after
selection criteria is applied. It is clear the low background when compared to
the plots shown for $\tau\to\ell\gamma$ and the absence of events close to the
signal region.

Table 1: Current best upper limits in the branching ratios of LFV $\tau$ decays to three leptons, imposed by the Belle experiment Hayasaka et al. (2010). Mode |  U.L. (90% C.L.)
---|---
$\tau^{-}\to e^{-}e^{+}e^{-}$ | $2.7\times 10^{-8}$
$\tau^{-}\to\mu^{-}\mu^{+}\mu^{-}$ | $2.1\times 10^{-8}$
$\tau^{-}\to\mu^{-}e^{+}e^{-}$ | $1.8\times 10^{-8}$
$\tau^{-}\to\mu^{-}\mu^{+}e^{-}$ | $2.7\times 10^{-8}$
$\tau^{-}\to e^{+}\mu^{-}\mu^{-}$ | $1.7\times 10^{-8}$
$\tau^{-}\to\mu^{+}e^{-}e^{-}$ | $1.5\times 10^{-8}$
Figure 6: Distribution of events in the 2D $m_{\ell\ell\ell}$ vs $\Delta E$
planes for $\tau^{-}\to e^{-}e^{+}e^{-}$ (left) and
$\tau^{-}\to\mu^{-}\mu^{+}\mu^{-}$ (right) candidates at Belle. The elliptical
signal region is shown in both cases. Figure from Hayasaka et al. (2010).

Of particular interest is the decay $\tau^{-}\to\mu^{-}\mu^{+}\mu^{-}$, which
can be searched in proton-proton collisions at the LHC experiments thanks to a
clean signature of muons in the trigger systems. Searches are performed via
$W^{-}$ or heavy hadron decays containing a $\tau$ lepton with the subsequent
LFV decay. The strongest limit from proton collisions is set by the LHCb
experiment, searching $\tau^{-}\to\mu^{-}\mu^{+}\mu^{-}$ events in $\tau$
leptons coming from semileptonic decays of b and c hadrons. Signal events are
reconstructed from 3 muon candidates, and two multi-variable analysis
classifiers are used to perform signal discrimination. Without a significant
excess in the defined signal region, the 90% C.L. upper limit set by LHCb is
${\cal{B}}(\tau^{-}\to\mu^{-}\mu^{+}\mu^{-})~{}<~{}4.6~{}\times~{}10^{-8}$
Aaij et al. (2015).

###  III.3 Semileptonic $\tau$ LFV modes

The $\tau$ is the only lepton heavy enough to decay into hadrons, enabling
semileptonic LFV modes. For the searches of $\tau$ LFV modes, semileptonic
decays are classified as a lepton and a pseudoscalar meson:
$\tau^{-}\to\ell^{-}P^{0}$ with $S^{0}=\pi^{0},\eta,\eta^{\prime},K_{S}^{0}$;
a lepton and a scalar meson $\tau^{-}\to\ell^{-}f_{0}(980)$; a lepton and a
neutral vector meson: $\tau^{-}\to\ell^{-}V^{0}$ with
$V^{0}=\rho,\omega,K^{*}(892),\phi$; or a lepton and two mesons
$\tau^{-}\to\ell^{-}h_{1}h_{2}$. Figure 7 show the 52 neutrinoless 2-body and
3-body decay modes searched by CLEO Andrews et al. (1983), the B-Factories and
the LHC experiments, as well as projections for the upper limits to be set at
Belle II with 5 and 50 ab-1 of integrated luminosity recorded.

Figure 7: Current observed upper limits at CLEO, BaBar, Belle, ATLAS, CMS and
LHCb experiments for the 52 LFV 2-body and 3-body $\tau$ lepton decays, as
well as projections for expected upper limits at Belle II. Figure from
Banerjee et al. (2022).

##  IV Searches of LFV with B mesons

Recent hints of lepton non-universality observed in semileptonic decays of B
mesons and $b\to s\ell\ell$ transitions Crivellin and Hoferichter (2021);
Ferreres-Solé (2022) could be explained by BSM scenarios that enable LFV in B
meson decays, such as models with leptoquarks Bečirević et al. (2016) or a new
$Z^{\prime}$ boson Crivellin et al. (2015).

###  IV.1 $B^{0}_{(s)}\to e\mu$ decays

In scenarios with BSM interactions, the branching ratio for $B^{0}_{(s)}\to
e^{\pm}\mu^{\mp}$ decays can be enhanced up to O($10^{-11})$. Best upper
limits for $B^{0}_{(s)}\to e^{\pm}\mu^{\mp}$ are imposed by the LHCb
experiment. Search is performed with 3 fb-1 of data collected from $pp$
collisions at 7 and 8 TeV. Signal candidates are identified by combining two
tracks identified as a muon and an electron, with a common vertex and
invariant mass around the mass of the $B^{0}_{(s)}$ meson. Major background
contributions come from B decays with misidentified or missing final states. A
boosted decision tree (BDT) is used to discriminate signal events, using both
candidates with and without bremsstrahlung correction. Without significant
excess of events in the signal region Aaij et al. (2018), the 90% C.L. upper
limits set by LHCb are

$\displaystyle{\cal{B}}(B^{0}\to e^{\pm}\mu^{\mp})$ $\displaystyle<1.3\times
10^{-9},$ (4) $\displaystyle{\cal{B}}(B_{s}^{0}\to e^{\pm}\mu^{\mp})$
$\displaystyle<6.3\times 10^{-9}.$

###  IV.2 $B^{0}_{(s)}\to\tau\ell$ decays

Another possibility is to explore two-lepton LFV decays of the B mesons with a
$\tau$ lepton included. Final states involving a $\tau$ make the
reconstruction harder due to the presence of missing energy in the signal
candidate.

The LHCb experiment has set the strongest limits for the branching ratio of
$B^{0}_{(s)}\to\tau^{\pm}\mu^{\mp}$. With $\tau$ lepton candidates
reconstructed from $\tau^{-}\to\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$, a
simultaneous fit on 4 regions of the response of a BDT is used to search for
an excess of events around the mass of the $B$ meson. Assuming no cross-feed
contamination between both FLV channels given the overlap between the $B^{0}$
and $B^{0}_{s}$ signal regions, the upper limits

$\displaystyle{\cal{B}}(B^{0}\to\tau^{\pm}\mu^{\mp})$ $\displaystyle<1.4\times
10^{-5},$ (5) $\displaystyle{\cal{B}}(B_{s}^{0}\to\tau^{\pm}\mu^{\mp})$
$\displaystyle<4.2\times 10^{-5},$

at 95% C.L. are obtained Aaij et al. (2019a).

In the case of the LFV decay to a $\tau$ lepton and an electron, the Belle
experiment imposes the strongest constrain for the branching ratio of
$B^{0}\to\tau^{\pm}e^{\mp}$. Since the initial state for the $B$-pair is well
defined, a hadronic B tag is used to infer the momentum of the B meson on the
opposite side and the $\tau$ lepton candidate does not require to be
reconstructed. Instead, the missing invariant mass of the event is determined
and signal candidates are searched around the mass of the $\tau$ lepton.
Without an excess with respect to the background level expectation, Belle sets
an upper limit at 90% C.L. of ${\cal{B}}(B^{0}\to\tau^{\pm}e^{\mp})<1.6\times
10^{-5}$ Atmacan et al. (2021).

###  IV.3 $b\to s\ell\ell^{\prime}$ transitions

Extensions of the SM that enable LFV $b\to s\ell\ell^{\prime}$ transitions
predict branching ratios up to the range $10^{8}-10^{10}$. Table 2 lists best
limits to the date for the transitions with the three generations of leptons
involved, showing less stringent limits when $\tau$ leptons are included in
the final state due to the missing energy in the reconstruction process.
Models of LFV can produce signatures with different charge configurations, and
both limits are provided in addition to the sum.

Upgrades I and II at LHCb, as well as searches with the full data set to be
collected by Belle II, project an improvement in the upper limits of one order
of magnitude for the next decade. Early results with the full-event
interpretation algorithm at Belle II show higher tag-side reconstruction
efficiency Abudinén et al. (2021); Dattola (2021), opening the possibility for
further constraints in the limits with inclusive tagging techniques.

Table 2: Limits in branching ratios for $b\to s\ell\ell^{\prime}$ transitions. Mode | U.L. (90% C.L.) | Experiment
---|---|---
$B^{+}\to K^{+}\mu^{-}e^{+}$ | $7.0\times 10^{-9}$ | LHCb Aaij et al. (2019b)
$B^{+}\to K^{+}\mu^{+}e^{-}$ | $6.4\times 10^{-9}$ | LHCb Aaij et al. (2019b)
$B^{0}\to K^{0}\mu^{\pm}e^{\mp}$ | $3.8\times 10^{-8}$ | Belle Choudhury et al. (2021)
$B^{0}\to K^{*0}\mu^{+}e^{-}$ | $5.7\times 10^{-9}$ | LHCb LHCb (2022)
$B^{0}\to K^{*0}\mu^{-}e^{+}$ | $6.7\times 10^{-9}$ | LHCb LHCb (2022)
$B^{0}\to K^{*0}\mu^{\pm}e^{\mp}$ | $9.9\times 10^{-9}$ | LHCb LHCb (2022)
$B^{0}\to\phi\mu^{\pm}e^{\mp}$ | $1.6\times 10^{-8}$ | LHCb LHCb (2022)
$B^{+}\to K^{+}\tau^{\pm}\mu^{\mp}$ | $4.8\times 10^{-5}$ | BaBar Lees et al. (2012)
$B^{+}\to K^{+}\tau^{\pm}e^{\mp}$ | $3.0\times 10^{-5}$ | BaBar Lees et al. (2012)

##  V LFV searches with bosons

Searches of LFV modes in the decays of the $Z$ and Higgs bosons are accessible
only at high-energy colliders and complementary to LFV searches in leptons and
mesons. On the other hand, any BSM mechanism that produces LFV interactions in
bosons would enable them also in low-energy processes. In consequence, limits
from LFV decays of muons and tau leptons set indirect limits on the maximum
branching ratios for $Z/H\to\ell\ell^{\prime}$ decays Calibbi et al. (2021).

###  V.1 $Z^{0}\to\ell\ell^{\prime}$ decays

In the case of the LFV mode with leptons of first and second generation only,
reconstruction of $Z^{0}\to e^{\pm}\mu^{\mp}$ candidates is performed from
tracks associated with an electron and a muon, and searching for an excess in
the invariant mass $m_{e\mu}$ of the signal candidates around the mass of the
$Z^{0}$ boson. The ATLAS experiment has set the best limit for $Z\to
e^{\pm}\mu^{\mp}$ using 20.3 fb-1 of data collected in $pp$ collisions at 8
TeV. Main background comes from $Z^{0}\to\tau^{+}\tau^{-}$ with the subsequent
leptonic decay the $\tau$ lepton, as illustrated by Figure 8. The mass
spectrum is consistent with the background expectations, and an upper limit of
${\cal{B}}(Z^{0}\to e^{\pm}\mu^{\mp})<7.5\times 10^{-7}$ at 95% C.L. is set
Aad et al. (2014).

Figure 8: Invariant mass distribution of $Z^{0}\to e\mu$ candidates in data
collected by ATLAS, with background expectation from MC processes after the
selection cuts are applied. Figure from Aad et al. (2014).

Limits with a $\tau$ lepton in the final state are still dominated by the LEP
experiments OPAL and DELPHI. The reconstruction of
$Z^{0}\to\tau^{\pm}\ell^{\mp}$ decays with $\ell=e,\mu$ is performed with
criteria optimized for the identification of an electron or a muon and a
charged cone associated with a $\tau$ decay. A cone consists of charged tracks
and electromagnetic clusters within a cone of half-angle 35∘. The background
events are composed mostly of $Z^{0}\to\tau^{+}\tau^{-}$ events with
subsequent $\tau^{-}\to\ell^{-}\bar{\nu}_{\ell}\nu_{\tau}$ in one of the
leptons. The observed candidates are consistent with the expected background,
and limits at the 95% C.L. are set Akers et al. (1995); Abreu et al. (1997):

$\displaystyle{\cal{B}}(Z^{0}\to\tau^{\pm}e^{\mp})$ $\displaystyle<9.8\times
10^{-6}{\rm\hskip 14.22636pt(OPAL)},$ (6)
$\displaystyle{\cal{B}}(Z^{0}\to\tau^{\pm}\mu^{\mp})$ $\displaystyle<1.2\times
10^{-5}{\rm\hskip 14.22636pt(DELPHI)}.$

In all $Z^{0}$ LFV modes, an improvement in the upper limits by a factor of 10
is expected with the data collected by the high-luminosity LHC.

###  V.2 $H\to\ell\ell^{\prime}$ decays

The best limit for the LFV mode for a Higgs boson $H\to e^{\pm}\mu^{\mp}$ is
imposed by the ATLAS experiment, using 139 fb-1 of data collected in $pp$
collisions at 13 TeV. The search is performed with a similar strategy as
described in the previous section, combining an electron and a muon candidates
and performing a fit in the invariant mass $m_{e\mu}$ distribution. Events
from top quarks are suppressed by identifying b-hadrons in the final state,
using a multivariate algorithm with calorimeter and tracking information, and
the remaining background events come from $Z\to\tau\tau$ and W + jets. Without
a significant excess in the signal, an upper limit of ${\cal{B}}(H\to
e^{\pm}\mu^{\mp})<6.1\times 10^{-7}$ at 95% C.L. is established Aad et al.
(2020a).

The most stringent limits for $H\to\tau^{\pm}e^{\mp}$ and
$H\to\tau^{\pm}e^{\mp}$ are also provided by ATLAS, based on a data set of
$pp$ collisions at 13 TeV corresponding to 36 fb-1. Both leptonic and hadronic
decays of the $\tau$ lepton are exploited, with the lepton from the Higgs and
the $\tau$ of a different flavor to reject the strong di-lepton background
from Dell-Yan processes. The analysis exploits boosted decision tree
algorithms for the separation of signal from background events, being
$Z\to\tau^{+}\tau^{-}$ and pairs of top quarks the most significant background
components. In the absence of a significant excess, the upper limits

$\displaystyle{\cal{B}}(H\to\tau^{\pm}e^{\mp})$ $\displaystyle<4.7\times
10^{-3},$ (7) $\displaystyle{\cal{B}}(H\to\tau^{\pm}\mu^{\mp})$
$\displaystyle<2.5\times 10^{-3},$

at 95% C.L. are set for a Higgs boson with $m_{H}=125~{}$GeV/c2 Aad et al.
(2020b). The upper limits are computed assuming no cross-feed contamination
between the channels.

##  VI Summary

A brief experimental overview of LFV searches has been presented, with
emphasis on the strongest limits set to the date. References have been
provided for the reader, where the details of each result can be consulted.

Other possibilities not discussed during this talk, but not less relevant,
include searches of LFV in decays of kaons: $K_{L}^{0}\to\mu^{\pm}e^{\mp}$
Ambrose et al. (1998), $K^{+}\to\pi^{+}\mu^{+}e^{-}$ Appel et al. (2000);
$J/\psi$ decays: $J/\psi\to\mu^{\pm}e^{\mp}$ Ablikim et al. (2022),
$J/\psi\to\tau^{\pm}e^{\mp}$ Ablikim et al. (2021),
$J/\psi\to\tau^{\mp}\mu^{\mp}$ Ablikim et al. (2004); decays with BSM
particles involved: $\ell^{-}\to\ell^{\prime-}\alpha$ Albrecht et al. (1995);
Ibarra et al. (2022), $Z^{\prime}\to\ell\ell^{\prime}$ Adachi et al. (2020),
and several more not covered due to time limitations.

In all the sectors presented, an improvement of 1-2 orders of magnitude is
expected with the data collected at the future muon experiment facilities,
Belle II at KEK, and the upgrades of ATLAS, CMS, and LHCb. The experiments in
the next decade will not only rely on larger statistics but also on improved
hardware and tools that will increase the signal efficiency, and in a better
understanding of the background contributions.

## References

  * Bernstein and Cooper (2013) R. H. Bernstein and P. S. Cooper, Phys. Rept. 532, 27 (2013), eprint 1307.5787.
  * Cei and Nicolo (2014) F. Cei and D. Nicolo, Adv. High Energy Phys. 2014, 282915 (2014).
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$\alpha$-Attractors and Reheating in a Non-Minimal Inflationary Model
R<EMAIL_ADDRESS>K<EMAIL_ADDRESS>and
F<EMAIL_ADDRESS>
aDepartment of Physics, Azarbaijan Shahid Madani University,
P. O. Box 53714-161, Tabriz, Iran
bDepartment of Theoretical Physics, Faculty of Basic Sciences,
University of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran

###### Abstract

We study $\alpha$-attractor models with both E-model and T-model potential in
an extended Non-Minimal Derivative (NMD) inflation where a canonical scalar
field and its derivatives are non-minimally coupled to gravity. We calculate
the evolution of perturbations during this regime. Then by adopting inflation
potentials of the model we show that in the large $N$ and small $\alpha$
limit, value of the scalar spectral index $n_{s}$ and tensor to scalar ratio
$r$ are universal. Next, we study reheating after inflation in this formalism.
We obtain some constraints on the model’s parameter space by adopting the
results with Planck 2018.
PACS: 98.80.Cq , 98.80.Es
Key Words: Non-Minimal Derivative Inflation, $\alpha$-Attractor, E-Model,
T-Model.

##  1 Introduction

The big bang theory had a number of problems. Inflationary paradigm has
emerged as candidate for solve this problems that we can see today [1-9].
Recently, the Planck measurements of temperature fluctuation of cosmic
microwave background (CMB) shows that spectrum of primordial scalar
perturbations has small deviation from scale invariance. It is encoded in
scalar spectral index in form $n_{s}=0.9649\pm 0.0042$. Furthermore, the
Planck has placed a constraint on the tensor to scalar ratio $r<0.056$. New
inflationary models provide compelling explanation of such perturbations.
During the last five years, a broad class of inflationary models have been
proposed. So that in the context of completely different theories, all of
which share the same predictions providing an excellent match to the WMAP and
Planck data [10-13]. These models are called ’Cosmological Attractors’. To see
more details, one can refer to Refs. [14-22].

There are several types of attractor models which among them we study
$\alpha$-attractors model [16-19]. $\alpha$-attractor scenario has been
proposed in the framework of supergravity, so that $\alpha$ describe moduli
space curvature given by $R_{K}=\frac{2}{3\alpha}$. This model provide a very
good fit to the observational data. There are two types of $\alpha$-attractor
models called E-model and T-model according to its potentials. In the large
e-folds number and small $\alpha$ limit, predicted for scalar spectral index
and tensor-to-scalar ratio, measured by CMB experiments, have universal form
as

$n_{s}\approx 1-\frac{2}{N}\,\,\,\,,\,\,\,\,r\approx\frac{12\alpha}{N^{2}}.$
(1)

For $\alpha=1$ is coincide with predictions of Higgs inflation and Starobinsky
model and for large $\alpha$ theory asymptotes to the power-law inflation.
(for more details on the issue of $\alpha$-attractors refer to Refs. [23-25])

On the other hand, consideration of process by which the universe reheats,
provide additional constraints to inflationary models [26-30]. Universe after
inflation without reheating, would be cold and empty. After inflation ends,
inflaton begins to oscillate around the minimum of its potential. When
inflaton oscillates, energy stored in inflaton field is transferred to the
plasma of relativistic particles.

Reheating era is an important part of inflationary models for several reasons.
Firstly, It can be explains how inflation is connected to hot big bang phase.
Also, micro physics of this era depends on the interaction between inflaton
and other fundamental fields, then by constraining reheating, we can learn
about coupling this fields.

Recently, show that the Planck satellite measurements of CMB anisotropies
constrain kinematic properties of the reheating era. By studying reheating
process we are able to additional constrains on inflationary models [31].
Reheating epoch characterized by three major parameters: e-fold number
$N_{re}$, reheating temperature $T_{re}$ and effective equation of state
parameter $\omega_{re}$. Once inflaton field decays, radiation dominated epoch
starts and the universe obtains a temperature. Measuring this temperature is
important for understanding thermal history of our universe. Furthermore, in
standard reheating scenario effective equation of state parameter is zero. But
numerical experiments of thermalization phase suggest a possibly wide range of
values $0\leq\omega\leq 0.25$ [32]. Though, $\omega_{re}>-\frac{1}{3}$ is is
required at end of inflation but $\omega_{re}<\frac{1}{3}$ is needed for
beginning of the radiation-dominated phase. This parameters also provides some
constraints to the models.

In this work we extend the nonminimal inflationary models to the case that a
canonical scalar field is coupled nonminimally to the gravity and in the same
time its derivatives are also coupled to the Einstein s tensor. We study
cosmological inflation to find possible $\alpha$-attractor then we consider in
detail reheating phase in this nonminimal inflationary model. We compare
results with latest observational data to see the viability of this extended
model. We obtain some constraints on the model’s parameter space.

##  2 The Model

In this letter, we study models with a canonical scalar field so that gravity
is coupled non-minimally to both the scalar field and its derivatives [33-34].
The general action for such model described by the following expression

$S=\frac{1}{2}\int
d^{4}x\sqrt{-g}\Bigg{[}M_{p}^{2}f(\phi)R+\frac{1}{\widetilde{M}^{2}}G_{\mu\nu}\partial^{\mu}\phi\partial^{\nu}\phi-2V(\phi)\Bigg{]}\,,$
(2)

where $\phi$ is a scalar field, $f(\phi)$ is a general function of $\phi$,
$M_{p}$ is a reduced Planck mass, and $\widetilde{M}$ is a mass parameter.

###  2.1 Field Equations

First let us assume that $f(\phi)=\frac{\phi^{2}}{2}$. In a spatially flat
FLRW geometry, the equations of motion following from action (2) are [27]

$H^{2}=\frac{1}{3M_{p}^{2}}\Bigg{[}-\frac{3}{2}M_{p}^{2}H\phi(2\dot{\phi}+H\phi)+\frac{9H^{2}}{2\widetilde{M}^{2}}\dot{\phi}^{2}+V(\phi)\Bigg{]}\,,$
(3)

$\displaystyle\dot{H}$ $\displaystyle=$
$\displaystyle-\frac{1}{2M_{p}^{2}}\Bigg{[}\dot{\phi}^{2}\bigg{(}\frac{3H^{2}}{\widetilde{M}^{2}}-\frac{\dot{H}}{\widetilde{M}^{2}}\bigg{)}-\frac{2H}{\widetilde{M}^{2}}\dot{\phi}\ddot{\phi}-\frac{3}{2}M_{p}^{2}H\phi(2\dot{\phi}+H\phi)$
(4)
$\displaystyle+\frac{1}{2}M_{p}^{2}\bigg{(}(2\dot{H}+3H^{2})\phi^{2}+4H\phi\dot{\phi}+2\phi\ddot{\phi}+2\dot{\phi}^{2}\bigg{)}\Bigg{]}\,,$

$-3M_{p}^{2}(2H^{2}+\dot{H})\phi+\frac{3H^{2}}{\widetilde{M}^{2}}\ddot{\phi}+3H\bigg{(}\frac{3H^{2}}{\widetilde{M}^{2}}+\frac{2\dot{H}}{\widetilde{M}^{2}}\bigg{)}\dot{\phi}+V^{\prime}(\phi)=0\,,$
(5)

where a dot denotes derivative with respect to the cosmic time and prime
refers to derivative with respect to the scalar field. The slow-roll
parameters are defined as

$\epsilon\equiv-\frac{\dot{H}}{H^{2}}\,\,\,\,,\,\,\,\,\eta\equiv-\frac{1}{H}\frac{\ddot{H}}{\dot{H}}\,.$
(6)

In this regard, we find these parameters as follows

$\epsilon=\bigg{[}1+\frac{\phi^{2}}{2}-\frac{\dot{\phi^{2}}}{2\widetilde{M}^{2}M_{p}^{2}}\bigg{]}^{-1}\bigg{[}\frac{3\dot{\phi}^{2}}{2\widetilde{M}^{2}M_{p}^{2}}+\frac{\phi\dot{\phi}}{2H}+\frac{\ddot{\phi}}{H\dot{\phi}}\bigg{(}\frac{\phi\dot{\phi}}{2H}-\frac{\dot{\phi^{2}}}{\widetilde{M}^{2}M_{p}^{2}}\bigg{)}\bigg{]}\,,$
(7)

$\eta=-2\epsilon-\frac{\dot{\epsilon}}{H\epsilon}\,.$ (8)

To have inflation phase, should satisfy the slow-roll conditions ($\epsilon\ll
1$ , $\eta\ll 1$). In our setup within the slow-roll approximation, the
equations of motion (3-5) is obtained as follows respectively

$H^{2}\simeq\frac{1}{3M_{p}^{2}}\Bigg{[}-\frac{3}{2}M_{p}^{2}H^{2}\phi^{2}+V(\phi)\Bigg{]}\,,$
(9)

$\dot{H}\simeq-\frac{1}{2M_{p}^{2}}\Bigg{[}\frac{3H^{2}\dot{\phi}^{2}}{\widetilde{M}^{2}}-M_{p}^{2}H\phi\dot{\phi}+M_{p}^{2}\dot{H}\phi^{2}\Bigg{]}\,,$
(10)

$-6M_{p}^{2}H^{2}\phi+\frac{9H^{3}\dot{\phi}}{\widetilde{M}^{2}}+V^{\prime}(\phi)\simeq
0\,.$ (11)

In this respect, within the slow-roll conditions, (7) become

$\epsilon\simeq\frac{\widetilde{M}^{2}M_{p}^{4}}{2V}\Bigg{[}[(1+\frac{\phi^{2}}{2})(\frac{V^{\prime}}{V})]^{2}-\frac{3\phi}{M_{p}^{2}}(1+\frac{\phi^{2}}{2})(\frac{V^{\prime}}{V})+2\phi^{2}\Bigg{]}\,.$
(12)

The e-folds number during inflation is defined as

${\cal N}=\int_{t_{hc}}^{t_{e}}H\,dt\,,$ (13)

where $t_{hc}$ and $t_{e}$ are defined as the time of horizon crossing and end
of inflation, respectively ($\epsilon(t_{hc})=\epsilon(t_{e})=1$). In our
setup, the e-fold number in the slow-roll approximation can be written as
follows

${\cal
N}\simeq\int_{\phi_{hc}}^{\phi_{e}}\frac{V(\phi)d\phi}{M_{p}^{2}\bigg{(}1+\frac{1}{2}\phi^{2}\bigg{)}\Bigg{[}2M_{p}^{2}\widetilde{M}^{2}\phi-
M_{p}^{2}\widetilde{M}^{2}\frac{V^{\prime}(\phi)}{V(\phi)}\bigg{(}1+\frac{1}{2}\phi^{2}\bigg{)}\Bigg{]}}\,.$
(14)

###  2.2 Linear Perturbations

Now, we study linear perturbations of the model around the homogeneous
background solution. In the first step, we should expand the action (2) up to
the second order in small fluctuations of the space-time background metric. It
is convenient to work in Arnowitt-Deser-Misner (ADM) formalism in which one
can eliminate one extra degree of freedom of the perturbations from the
beginning. In this formalism the space-time metric is given by [35]

$ds^{2}=-N^{2}dt^{2}+h_{ij}(N^{i}dt+dx^{i})(N^{j}dt+dx^{j})\,,$ (15)

where $N^{i}$ and $N$ are the shift vector and lapse function, respectively.
We can obtain general perturbed form of the ADM metric by expanding $N^{i}$ as
$N^{i}=\delta^{ij}\partial_{j}\Upsilon$ and $N$ as $N=1+2\Phi$ in which $\Phi$
and $\Upsilon$ are three-scalars. Also, the coefficient $h_{ij}$ in the second
term of the above metric can be written as
$h_{ij}=a^{2}(t)[(1+2\Psi)\delta_{ij}+\gamma_{ij}]$, where $\Psi$ is spatial
curvature perturbation and $\gamma_{ij}$ is symmetric and traceless shear
three-tensors. In the rest, we choose $\gamma_{ij}=0$ and $\delta\Phi=0$.
Finally, we can write the perturbed metric up to the linear level as [36]

$ds^{2}=-(1+2\Phi)dt^{2}+2\partial_{i}\Upsilon
dtdx^{i}+a^{2}(t)(1+2\Psi)\delta_{ij}dx^{i}dx^{j}\,.$ (16)

By replacing the perturbed metric in action (2) and expanding it up to the
second order in small perturbations, the second-order action can be rewritten
as follows [37]

$S^{(2)}=\int
dtdx^{3}a^{3}\Bigg{[}-\frac{3}{2}(M_{p}^{2}\phi^{2}-\frac{\dot{\phi}^{2}}{\widetilde{M}^{2}})\dot{\Psi}^{2}+\frac{1}{a^{2}}((M_{p}^{2}\phi^{2}-\frac{\dot{\phi}^{2}}{\widetilde{M}^{2}})\dot{\Psi}$
$-(M_{p}^{2}H\phi^{2}+M_{p}^{2}\phi\dot{\phi}-\frac{3H\dot{\phi}^{2}}{\widetilde{M}^{2}})\Phi)\partial^{2}\Upsilon-\frac{1}{a^{2}}(M_{p}^{2}\phi^{2}-\frac{\dot{\phi}^{2}}{\widetilde{M}^{2}})\Phi\partial^{2}\Psi$
$+3(M_{p}^{2}H\phi^{2}+M_{p}^{2}\phi\dot{\phi}-\frac{3H\dot{\phi}^{2}}{\widetilde{M}^{2}})\Phi\dot{\Psi}+3H(-\frac{1}{2}M_{p}^{2}H\phi^{2}-M_{p}^{2}\phi\dot{\phi}$
$+\frac{3H\dot{\phi}^{2}}{\widetilde{M}^{2}})\Phi^{2}+\frac{1}{2a^{2}}(M_{p}^{2}\phi^{2}+\frac{\dot{\phi}^{2}}{\widetilde{M}^{2}})(\partial\Psi)^{2}\Bigg{]}\,.$
(17)

By variation of the above action with respect to $N^{i}$ and $N$ we obtain

$\Phi=\frac{M_{p}^{2}\phi^{2}-\frac{\dot{\phi}^{2}}{\widetilde{M}^{2}}}{M_{p}^{2}H\phi^{2}+M_{p}^{2}\phi\dot{\phi}-\frac{3H\dot{\phi}^{2}}{\widetilde{M}^{2}}}\dot{\Psi}\,,$
(18)

$\partial^{2}\Upsilon=\frac{2a^{2}}{3}\frac{(-\frac{9}{2}M_{p}^{2}H^{2}\phi^{2}-9M_{p}^{2}H\phi\dot{\phi}+\frac{27H^{2}\dot{\phi}^{2}}{\widetilde{M}^{2}})}{(M_{p}^{2}H\phi^{2}+M_{p}^{2}\phi\dot{\phi}-\frac{3H\dot{\phi}^{2}}{\widetilde{M}^{2}})}$
$+3\dot{\Psi}a^{2}-\frac{M_{p}^{2}\phi^{2}-\frac{\dot{\phi}^{2}}{\widetilde{M}^{2}}}{M_{p}^{2}H\phi^{2}+M_{p}^{2}\phi\dot{\phi}-\frac{3H\dot{\phi}^{2}}{\widetilde{M}^{2}}}\dot{\Psi}\,.$
(19)

By substituting the equation (18) in equation (17) and taking some integration
by part, the quadratic action can be rewritten as following expression

$S^{(2)}=\int
dtdx^{3}a^{3}\vartheta_{s}\bigg{[}\dot{\Psi}^{2}-\frac{c_{s}^{2}}{a^{2}}(\partial\Psi)^{2}\bigg{]}$
(20)

where the parameters $\vartheta_{s}$ and $c_{s}^{2}$ are expressed as

$\vartheta_{s}\equiv
6\frac{(M_{p}^{2}\phi^{2}-\frac{\dot{\phi}^{2}}{\widetilde{M}^{2}})^{2}(-\frac{1}{2}M_{p}^{2}H^{2}\phi^{2}-M_{p}^{2}H\phi\dot{\phi}+\frac{3}{\widetilde{M}^{2}}H^{2}\dot{\phi}^{2})}{(M_{p}^{2}H\phi^{2}+M_{p}^{2}\phi\dot{\phi}-\frac{3}{\widetilde{M}^{2}}H\dot{\phi}^{2})^{2}}+3\bigg{(}\frac{1}{2}M_{p}^{2}\phi^{2}-\frac{1}{2\widetilde{M}}\dot{\phi}^{2}\bigg{)}\,,$
(21)

and

$\displaystyle c_{s}^{2}$ $\displaystyle\equiv$
$\displaystyle\frac{3}{2}\bigg{\\{}\bigg{(}M_{p}^{2}\phi^{2}-\frac{\dot{\phi}^{2}}{\widetilde{M}^{2}}\bigg{)}^{2}\bigg{(}M_{p}^{2}H\phi^{2}+M_{p}^{2}\phi\dot{\phi}-\frac{3H\dot{\phi}^{2}}{\widetilde{M}^{2}}\bigg{)}H-4\bigg{(}M_{p}^{2}H\phi^{2}+M_{p}^{2}\phi\dot{\phi}-\frac{3H\dot{\phi}^{2}}{\widetilde{M}^{2}}\bigg{)}^{2}$
(22)
$\displaystyle\bigg{(}M_{p}^{2}\phi^{2}-\frac{\dot{\phi}^{2}}{\widetilde{M}^{2}}\bigg{)}\bigg{(}M_{p}^{2}\phi^{2}-\frac{\dot{\phi}^{2}}{\widetilde{M}^{2}}\bigg{)}\bigg{(}M_{p}^{2}\phi\dot{\phi}-\frac{\dot{\phi}\ddot{\phi}}{\widetilde{M}^{2}}\bigg{)}\bigg{(}M_{p}^{2}H\phi^{2}+M_{p}^{2}\phi\dot{\phi}-\frac{3H\dot{\phi}^{2}}{\widetilde{M}^{2}}\bigg{)}$
$\displaystyle-\bigg{(}M_{p}^{2}-\frac{\dot{\phi}^{2}}{\widetilde{M}^{2}}\bigg{)}^{2}\bigg{(}M_{p}^{2}\dot{H}\phi^{2}+2M_{p}^{2}H\phi\dot{\phi}M_{p}^{2}\dot{\phi}^{2}+M_{p}^{2}\phi\ddot{\phi}-\frac{3\dot{H}\dot{\phi}^{2}}{\widetilde{M}^{2}}-\frac{6}{\widetilde{M}^{2}}H\dot{\phi}\ddot{\phi}\bigg{)}\bigg{\\}}$
$\displaystyle\bigg{\\{}9\bigg{[}\frac{1}{2}M_{p}^{2}\phi^{2}-\frac{\dot{\phi}^{2}}{2\widetilde{M}^{2}}\bigg{]}\bigg{[}4\bigg{(}\frac{1}{2}M_{p}^{2}\phi^{2}-\frac{\dot{\phi}^{2}}{2\widetilde{M}^{2}}\bigg{)}\bigg{(}-\frac{1}{2}M_{p}^{2}H^{2}\phi^{2}-M_{p}^{2}H\phi\dot{\phi}+\frac{3}{\widetilde{M}^{2}H^{2}\dot{\phi}^{2}}\bigg{)}$
$\displaystyle+\bigg{(}M_{p}^{2}H\phi^{2}+M_{p}^{2}\phi\dot{\phi}-\frac{3H\dot{\phi}^{2}}{\widetilde{M}^{2}}\bigg{)}^{2}\bigg{]}\bigg{\\}}^{-1}\,.$

Let us now calculate quantum perturbations of $\Psi$. To this end, we can
obtain equation of motion of the curvature perturbation by variation of action
(20) which follows

$\ddot{\Psi}+\bigg{(}3H+\frac{\dot{\vartheta_{s}}}{\vartheta_{s}}\bigg{)}\dot{\Psi}+\frac{c_{s}^{2}k^{2}}{a^{2}}\Psi=0\,.$
(23)

By solving the equation of motion up to the lowest order of the slow-roll
approximation, we obtain

$\Psi=\frac{iH\exp(-ic_{s}k\tau)}{2c_{s}^{\frac{3}{2}}\sqrt{k^{3}}\vartheta_{s}}(1+ic_{s}k\tau)\,.$
(24)

Two-point correlation function of curvature perturbations can be found by
obtaining vacuum expectation value at the end of inflation. By using this
function, we can study power spectrum of the curvature perturbation in our
model. We describe the power spectrum $P_{s}$ as following expression

$\langle
0|\Psi(0,\textbf{k}_{1})\Psi(0,\textbf{k}_{2})|0\rangle=\frac{2\pi^{2}}{k^{3}}P_{s}(2\pi)^{3}\delta^{3}(\textbf{k}_{1}+\textbf{k}_{2})\,,$
(25)

where by definition

$P_{s}=\frac{H^{2}}{8\pi^{2}\vartheta_{s}c_{s}^{3}}\,.$ (26)

The spectral index is given by [38-42]

$n_{s}-1=\frac{d\ln P_{s}}{d\ln
k}|_{c_{s}k=aH}=-2\epsilon-\delta_{F}-\eta_{s}-S,$ (27)

by definition

$\delta_{F}=\frac{\dot{f}}{H(1+f)}\,\,\,\,,\,\,\,\,\eta_{s}=\frac{\dot{\epsilon_{s}}}{H\epsilon_{s}}\,\,\,\,,\,\,\,\,S=\frac{\dot{c_{s}}}{Hc_{s}},$
(28)

where

$\epsilon_{s}=\frac{\vartheta_{s}c_{s}^{2}}{M_{p}^{2}(1+f)},$ (29)

which gives finally

$n_{s}-1=-2\epsilon-\frac{1}{H}\frac{d\ln
c_{s}}{dt}-\frac{1}{H}\frac{d\ln[2H(1+\frac{\phi^{2}}{2})\epsilon+\phi\dot{\phi}]}{dt}\,,$
(30)

which shows the scale dependence of perturbations due to deviation of $n_{s}$
from $1$.

Let us now study the amplitude of tensor perturbation and its spectral index.
In this regard, we consider the metric as follows

$ds^{2}=-dt^{2}+a(t)^{2}(\delta_{ij}+T_{ij})dx^{i}dx^{j}\,,$ (31)

where $T_{ij}$ is transverse and traceless shear three-tensor. We set $T_{ij}$
in terms of the two polarization tensors, as follows

$T_{ij}=T_{+}e^{+}_{ij}+T^{\times}e^{\times}_{ij}\,,$ (32)

where $e^{+}_{ij}$ and $e^{\times}_{ij}$ are symmetric, transverse and
traceless. In this case, the second-order action for the tensor mode of the
perturbations (known as gravitational waves) can be written as

$S_{T}=\int
dtdx^{3}a^{3}\vartheta_{T}\bigg{[}\dot{T}_{(+,\times)}^{2}-\frac{c_{T}^{2}}{a^{2}}(\partial
T_{(+,\times)})^{2}\bigg{]}\,,$ (33)

by definition

$\vartheta_{T}\equiv\frac{1}{8}(M_{p}^{2}\phi^{2}-\frac{\dot{\phi}^{2}}{\widetilde{M}^{2}})$
(34)

and

$c_{T}^{2}\equiv\frac{\widetilde{M}^{2}M_{p}^{2}\phi^{2}+\dot{\phi}^{2}}{\widetilde{M}^{2}M_{p}^{2}\phi^{2}-\dot{\phi}^{2}}\,.$
(35)

In our setup, the amplitude of the tensor perturbations is given by

$P_{T}=\frac{H^{2}}{2\pi^{2}\vartheta_{T}c_{T}^{3}}\,,$ (36)

we have defined the tensor spectral index of the gravitational waves as

$n_{T}\equiv\frac{d\ln P_{T}}{d\ln k}|_{c_{T}k=aH}\,=-2\epsilon-\delta_{F}.$
(37)

leads to the following expression

$n_{T}=-2\epsilon-\frac{\phi\dot{\phi}}{H(1+\frac{\phi^{2}}{2})}\,.$ (38)

Finally, the ratio between the amplitudes of the tensor and scalar
perturbations (tensor-to-scalar ratio) in our setup is given by

$r=\frac{P_{T}}{P_{s}}=16c_{s}\bigg{(}\epsilon+\frac{\phi\dot{\phi}}{2H(1+\phi^{2}/2)}+o(\epsilon^{2})\bigg{)}\simeq-8c_{s}n_{T}.$
(39)

As we can see, NMD inflation yields the standard consistency relation.

##  3 $\alpha$-Attractors

We consider a class of NMD inflationary models of $\alpha$-attractors in a
unified manner. As shown in Ref. [12] $\alpha$-attractors are consistent with
observations of the CMB anisotropies. For consistency with observed $r$, the
parameter $\alpha$ should be less than $O(100)$. In this section, we focus on
the E-model and T-model as generalised models of $\alpha$-attractors so that
specified by the potentials as (for more details refer to Refs.[14,17])

$V=V_{0}\bigg{[}1-exp(-\sqrt{\frac{2}{3M_{pl}^{2}\alpha}}\phi)\bigg{]}^{2n},$
(40)

$V=V_{0}\tanh^{2n}(\frac{\phi}{\sqrt{6M_{pl}^{2}\alpha}}),$ (41)

respectively. $V_{0}$ and $n$ are the parameters.

Now, by using the potentials defined in Eqs. (40) and (41) and solving
integral (14), we find the value of the field at the horizon crossing. We can
therefore substitute this value in Eqs. (12), (30) and (39). Finally, when
$n=1$ and $\alpha\ll 1$, also using the slow-roll approximation, we can find
the values of the scalar spectral index and the tensor-to-scalar ratio for
each model. Under these conditions, the values of $n_{s}$ and $r$ for E-model
potential are given by

$n_{s}=1-\frac{2}{1+N+3\alpha\ln(1+N)/4},$ (42)

$r=\frac{12\alpha}{(1+N)^{2}+3\alpha(1+N)\ln(1+N)/2},$ (43)

where in the large e-folds number and small $\alpha$ limit become

$n_{s}\approx 1-\frac{2}{N},$ (44)

$r\approx\frac{12\alpha}{N^{2}}.$ (45)

As we can see, predicted form of the scalar spectral index and tensor-to-
scalar ratio have universal form.

Also, Under above conditions, the values of $n_{s}$ and $r$ for T-model
potential can be written as follows

$n_{s}=1-\frac{2}{N}+\frac{2}{N^{2}+N},$ (46)

$r=\frac{12\alpha}{(1+N)^{2}}.$ (47)

Now, we study the perturbation parameters numerically. To this end, we plot
tensor to scalar ratio versus the scalar spectral index and compare them with
the observations. The results are shown in Figure (1). As we shown, the models
for $N=50,60$ and $70$ are consistent with the Planck measurements.

Figure 1: Tensor to scalar ratio versus the scalar spectral index for NMD
model with the E-model (left panel) and T-model (right panel) potential.

##  4 Reheating

After inflation ends, the process of reheating take places. By consider this
process we can investigate some constraints on the model’s parameter space. In
this section, we obtain expressions for e-fold number and temperature of
reheating era.

The e-fold number between the time of horizon crossing and end of inflation is
written as

$N_{hc}=\ln\bigg{(}\frac{a_{e}}{a_{hc}}\bigg{)},$ (48)

where $a_{hc}$ and $a_{e}$ are scale factor at the horizon crossing and end of
inflation, respectively.

We assume that at the end of reheating epoch, energy density of the universe
is given by

$\rho_{re}=\frac{\pi^{2}g_{re}}{30}T_{re}^{4},$ (49)

where $g_{re}$ is effective number of relativistic particles at the end of
reheating.

In this respect, the length of reheating is written as

$N_{re}=\ln\bigg{(}\frac{a_{re}}{a_{e}}\bigg{)}=-\frac{1}{3(1+\omega_{eff})}\ln\bigg{(}\frac{\rho_{re}}{\rho_{e}}\bigg{)},$
(50)

so that we substituted the relation $\rho\sim a^{-3(1+\omega_{eff})}$ at above
equation. Where $\rho$ and $\omega_{eff}$ are energy density and effective
equation of state, respectively.

Horizon crossing occurs at $a_{hc}H_{hc}=k$, where $k$ denotes wave number and
$H_{hc}$ is Hubble parameter at horizon crossing during inflation. Then we can
write

$0=\ln\bigg{(}\frac{k}{a_{hc}H_{hc}}\bigg{)}=\ln\bigg{(}\frac{a_{e}}{a_{hc}}\frac{a_{re}}{a_{e}}\frac{a_{0}}{a_{re}}\frac{k}{a_{0}H_{hc}}\bigg{)},$
(51)

where $a_{0}$ is scale factor of present epoch. Now, from Eqs. (48) and (50)
we obtain

$N_{hc}+N_{re}+\ln\bigg{(}\frac{a_{0}}{a_{re}}\bigg{)}+\ln\bigg{(}\frac{k}{a_{0}H_{hc}}\bigg{)}=0.$
(52)

From conservation of entropy we can write

$\frac{a_{0}}{a_{re}}=\bigg{(}\frac{43}{11g_{re}}\bigg{)}^{-\frac{1}{3}}\frac{T_{re}}{T_{0}},$
(53)

where $T_{0}$ is the current temperature. From Eqs. (49) and (53) we obtain

$\frac{a_{0}}{a_{re}}=\bigg{(}\frac{43}{11g_{re}}\bigg{)}^{-\frac{1}{3}}\bigg{(}\frac{\pi^{2}g_{re}}{30\rho_{re}}\bigg{)}^{-\frac{1}{4}}\frac{1}{T_{0}}.$
(54)

In our framework, the energy density during the inflation epoch can be
rewritten as follows

$\rho=\frac{9H^{2}}{2\widetilde{M}^{2}}\dot{\phi}^{2}-\frac{3}{2}M_{p}^{2}H\phi(2\dot{\phi}+H\phi)+V(\phi).$
(55)

Now by setting $\epsilon=1$, the energy density at the end of inflation phase
is the following form

$\rho_{e}=\bigg{(}\frac{2}{2+\phi_{e}^{2}}\bigg{)}V_{e},$ (56)

where $\phi_{e}$ and $V_{e}$ are scalar field and potential at the end of
inflation, respectively. Then from Eqs. (50) and (56) we have

$\rho_{re}=\bigg{(}\frac{2}{2+\phi_{e}^{2}}\bigg{)}V_{e}\exp[-3(1+\omega_{eff})N_{re}].$
(57)

$H_{hc}$ can be find from Eq. (26). Now by substituting $H_{hc}$ and Eqs. (54)
and (57) in the Eq. (52) we can write the following expression for e-fold
number of reheating era

$\displaystyle N_{re}$ $\displaystyle=$
$\displaystyle\frac{4}{(1-3\omega_{eff})}\bigg{[}-N_{hc}-\ln\bigg{(}\frac{k_{hc}}{a_{0}}\bigg{)}+\frac{1}{2}\ln(8\pi^{2}P_{s}\vartheta_{s}c_{s}^{3})+\frac{1}{3}\ln\bigg{(}\frac{43}{11g_{re}}\bigg{)}$
(58) $\displaystyle+\frac{1}{4}\ln\bigg{(}\frac{\pi^{2}g_{re}}{30}\bigg{)}+\ln
T_{0}-\frac{1}{4}\ln\bigg{(}\frac{2V_{e}}{2+\phi_{e}^{2}}\bigg{)}\bigg{]}.$

and from equations (49) and (57) we obtain the temperature during reheating

$T_{re}=\bigg{(}\frac{60V_{e}}{\pi^{2}g_{re}(2+\phi_{e}^{2})}\bigg{)}^{\frac{1}{4}}\exp\bigg{[}-\frac{3}{4}(1+\omega_{eff})N_{re}\bigg{]}.$
(59)

The scalar spectral index and wave number are related through $\phi$ with Eqs.
(30), (48) and $a_{hc}H_{hc}=k$, therefore one can investigate $N_{re}$ and
$T_{re}$ in terms of the scalar spectral index for each model.

###  4.1 E-model

First, we consider E-model potential defined in Eq. (40). As we know, in the
small $\alpha$ and large $N$ limit, one can rewrite the potential as

$V=V_{0}\bigg{[}1-2n\exp(-\sqrt{\frac{2}{3M_{pl}^{2}\alpha}}\phi)\bigg{]},$
(60)

where we assume that $n=1$. We also need to the e-fold number that happen
after a mode with wave number $k$. By substituting the above equation in the
Eq. (14) we can obtain e-folding as function of $\phi$ which defined as
following expression

$\displaystyle N_{hc}$ $\displaystyle\approx$
$\displaystyle\frac{V_{0}}{M_{pl}^{4}\widetilde{M}^{2}}\bigg{[}-\frac{1}{4}\phi^{2}(\phi^{2}-4)+\frac{1}{2}(\phi^{4}+4\sqrt{6M_{pl}^{2}\alpha}\phi$
(61)
$\displaystyle+12M_{pl}^{2}\alpha-4)\exp\bigg{(}-\sqrt{\frac{2}{3M_{pl}^{2}\alpha}}\phi\bigg{)}\bigg{]}_{\phi_{hc}}^{\phi_{e}}.$

Now, by plotting the e-fold number and temperature of reheating era versus the
scalar spectral index we can study the parameters numerically. We consider the
case with $\alpha=1$ for some values of effective equation of state. The
results are shown in Figure (2).

Figure 2: $N_{re}$ (left panel) and $T_{re}$ (right panel) as function of the
scalar spectral index for E-model when $n=1$ with $\alpha=1$. In each panel,
the red line corresponds to $\omega_{re}=-\frac{1}{3}$, the purple line to
$\omega_{re}=0$, the green line to $\omega_{re}=\frac{2}{3}$ and the blue line
to $\omega_{re}=1$. The yellow shaded region indicate the Planck 2018
constraint.

According to the picture, we can obtain constraints on the model s parameter
space by adopting our results with the Planck 2018. The constrains on the
E-model parameters are shown in Table 1.

Table 1: The ranges of e-fold number and temperature during reheating epoch for E-model potential which are consistent with the Planck 2018. $n=1,\alpha=1$ | $\omega_{eff}=-\frac{1}{3}$ | $\omega_{eff}=0$ | $\omega_{eff}=\frac{2}{3}$ | $\omega_{eff}=1$
---|---|---|---|---
$N_{re}$ | $<31$ | $<39$ | $<15$ | $<8$
$\log(T_{re})$ | $>15$ | $>4$ | $>22$ | $>23$

###  4.2 T-model

Now, we consider T-model potential defined in Eq. (41). By substituting the
potential in the Eq. (14) we can obtain e-folding as function of $\phi$ which
defined as following expression

$\displaystyle N_{hc}$ $\displaystyle\approx$ $\displaystyle
V_{0}\bigg{[}12M_{pl}^{2}\alpha\log(\cosh(\frac{\phi}{\sqrt{6M_{pl}^{2}\alpha}}))+\frac{1}{4}(\phi^{4}-4)\cosh(\frac{\phi}{\sqrt{6M_{pl}^{2}\alpha}})^{-2}$
(62)
$\displaystyle-8\sqrt{6M_{pl}^{2}\alpha}\phi\tanh(\frac{\phi}{\sqrt{6M_{pl}^{2}\alpha}})-\phi^{4}+4\phi^{2}\bigg{]}_{\phi_{hc}}^{\phi_{e}}.$

We consider $\alpha=1$ and plot the e-fold number and temperature of reheating
era versus the scalar spectral index. The results are shown in Figure (3).
Therefore, we obtain some constraints on the model s parameter space. The
constrains on the T-model parameters are shown in Table 2.

Figure 3: $N_{re}$ (left panel) and $T_{re}$ (right panel) as function of scalar spectral index for T-model when $n=1$ with$\alpha=1$. In each panel, the red line corresponds to $\omega_{re}=-\frac{1}{3}$, the purple line to $\omega_{re}=0$, the green line to $\omega_{re}=\frac{2}{3}$ and the blue line to $\omega_{re}=1$. The yellow shaded region indicate the Planck 2018 constraint. Table 2: The ranges of e-fold number and temperature during reheating epoch for T-model potential which are consistent with the Planck 2018. $n=1,\alpha=1$ | $\omega_{eff}=-\frac{1}{3}$ | $\omega_{eff}=0$ | $\omega_{eff}=\frac{2}{3}$ | $\omega_{eff}=1$
---|---|---|---|---
$N_{re}$ | $<8$ | $<20$ | $<52$ | $<27$
$\log(T_{re})$ | $>20$ | $>16$ | $>7.5$ | $>12$

###  4.3 Post-Inflationary era

The reheating evolution of the Horndeski models is much different than the
usual minimal coupling theory. The scalar field (inflaton) oscillates very
fast and coherently around the minimum of its potential. The averaged
expansion of the universe is given by (for more details refer to Refs.
[34,43,44])

$\bar{H}(t,\omega_{re})=\frac{1}{\frac{3}{2}(1+\bar{\omega}_{re})(t-t_{end})+H^{-1}_{end}},$
(63)

for $t_{end}<t<t_{re}$. The averaged equation of state ($\bar{\omega}_{re}$)
after inflationary era is determined by oscillating behavior of the scalar
field about the minimum. The above expression for $\bar{H}$ imply a different
relation for $\bar{\omega}_{re}$. As a result of Ref. [34], for the non-
minimal derivative models, one can obtain different negative values for the
equation of state. In Fig. (4) we plot time evolution of the Hubble parameter
for the possible values of the equation of state.

On the other hand, during the slow-roll phase we have a dual description for
the tensor-to-scalar ratio in the non-minimal derivative and usual minimal
coupling scenario. However, this degeneracy breaks by different value of the
reheating temperature and equation of state parameter when the reheating phase
is taken into account.

Figure 4: Time evolution of the Hubble parameter for the possible values of
the equation of state.

##  5 Summary

In this letter, we have considered a class of non-minimal derivative
inflationary model. We focused on both the E-model and T-model as generalized
models of $\alpha$-attractors. We have obtained the scaler spectral index and
tensor to scaler ratio in the models. As shown above, in the small $\alpha$
and large $N$ limit, the model with E-model potential predicts $n_{s}\approx
1-\frac{2}{N}$ and $r\approx\frac{12\alpha}{N^{2}}$ where there is an
attractor behavior. However, the value of the $n_{s}$ and $r$ predicted by
T-model, has some deviation from the corresponding terms in E-model. We
analyzed the parameters numerically for $n=1$ and compared our results with
observations. We have found that the models for large N are consistent with
the Planck data.

Moreover, we have studied the reheating phase in the NMD model. We obtained
e-fold number and temperature during reheating in terms of scalar spectral
index for both E-model and T-model. We also analyzed the parameters
numerically for $n=1$ with $\alpha=1$ and compared the results with
observational data. As shown in Table (1,2), we have obtained some constraints
on the model’s parameter space by adopting our results with Planck 2018.

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