# Study of quantum decoherence at Protvino to ORCA experiment

Chinmay Bera<EMAIL_ADDRESS>Department of
Physics, École Centrale School of Engineering - Mahindra University,
Hyderabad, Telangana, 500043, India  K. N. Deepthi
<EMAIL_ADDRESS>Department of Physics, École
Centrale School of Engineering - Mahindra University, Hyderabad, Telangana,
500043, India

###### Abstract

Protvino to ORCA (Oscillation Research with Cosmics in the Abyss) (P2O) is an
upcoming neutrino oscillation experiment with a very long baseline of 2595 km.
Due to the substantial baseline, this experiment provides a unique opportunity
to study the earth matter effects over very large distances. This makes it a
suitable experiment to investigate the environmental decoherence in neutrino
oscillations, where the neutrino system could interact with a stochastic
environment and lead to a loss in the coherence of neutrino states. In this
work, we consider an open quantum system framework to simulate the neutrino
oscillations in P2O experiment and obtain bounds on the decoherence parameters
in different phenomenological models. We assume that the decoherence parameter
$\Gamma$ depends on neutrino energy $E_{\nu}$ as
$\Gamma_{ij}(E_{\nu})=\Gamma_{0}(\frac{E_{\nu}}{E_{0}})^{n}$. Further, we use
these bounds to study the effect on the neutrino mass ordering sensitivity and
CP violation sensitivity of P2O experiment.

Lindblad dynamics; quantum decoherence; neutrino oscillation; P2O; mass
ordering; CP violation

##  I Introduction

Over the past few years, leading neutrino experiments have been diligently
collecting data to estimate the neutrino oscillation parameters precisely.
Till date, the majority of the data from these experiments aligns with the
conventional three flavour neutrino oscillation framework, characterized by
the three mixing angles, two mass square differences, and a Dirac Charge-
Parity (CP) phase. Nevertheless, the forthcoming high precision neutrino
oscillation experiments provide a significant opportunity to probe sub-leading
new physics (NP) phenomenon. One such interesting phenomenon is the
environmentally induced decoherence among the neutrino states passing through
the earth matter.

The standard three flavour neutrino oscillation phenomenology considers
neutrinos to be isolated from the environment and the coherence of the
neutrino states is preserved over very long distances. However several
theories Hawking:1978sf ; Amati:1987string ; Saurya:2018se ; Stuttard:2020mik
; Luciano:2021qg  propose the possibility of interaction of neutrinos with a
stochastic environment, leading to the loss of coherence in the neutrino
states. This phenomenon is called quantum neutrino decoherence Liu:1997km ;
Feng:1998hu ; Chang:1999dai ; Benatti:2000ni ; Kalpdor:2000pas ; Gago:2001mo ;
Benatti:2001fa ; Fogli:2003li ; Ashie:2004sk ; Luis:2005rg ; Barenboim:2006os
; Farzan:2008zv ; Oliveira:2010zzd ; Oliveira:2013nua ; Oliveira:2016asf ;
Coelho:2017byq ; Carpio:2017nui ; Carrasco:2018sca ; Buoninfante:2020iyr ;
deHolanda:2019tuf ; JUNO:2021wang ; cheng:2022man ; carrasco:2022dia  and it
is different in its origin from the quantum mechanical wave packet decoherence
Giunti:1998kim ; Blennow:2005ohl ; Akhmedov:2012her ; Chang:2016chu ;
Gouvea:2021rom . In this paper, we consider only the former case.

In the recent years, significant research efforts have been carried out in
constraining the decoherence parameters. In ref. Lisi:2000zt , the authors
have analysed Super-Kamiokande atmospheric neutrino experiment data and
obtained bounds on the decoherence parameter $\Gamma<3.5\times 10^{-23}~{}GeV$
(90% CL). In ref. deOliveira:2013dia  the bounds on the decoherence parameters
were obtained using MINOS data while in ref. gomesalg:2023 , data from MINOS
and T2K experiments has been analysed to update the bounds on the decoherence
parameter by assuming power law dependency on the neutrino energy. The bounds
obtained from the analysis of solar and KamLAND data have been presented in
ref. fogligl:2007 . In a further analysis, in ref. Gomes:2016ixi  the authors
included the recent KamLAND data. In a seminal paper ref. Coelho:2017zes
authors pointed out that the $2\sigma$ tension between the T2K and NO$\nu$A
data could be improved by considering environmental decoherence of strength
$\Gamma=(2.3\pm 1.1)\times 10^{-23}~{}GeV$. The authors of ref. Coloma:2018ice
have analyzed the IceCube/DeepCore data on atmospheric neutrinos and reported
the bounds on the $\Gamma$ considering $n=0,~{}\pm 1,~{}\pm 2$. Recently, in
ref. Collab:2024icecube  the IceCube collaboration reported bounds on $\Gamma$
with atmospheric neutrinos for $n=0,1,2,3$. References Carpio:2019mass ;
Gomes:2019for  present the sensitivity studies of DUNE to the decoherence
parameters. Recently, authors in ref. Romeri:2023cgt  have analysed several
reactor (KamLAND, Daya Bay and RENO) and accelerator (T2K, NOvA, MINOS/MINOS+)
data as well as future (JUNO, DUNE) experiments and reported the updated
bounds on the decoherence parameters. In a most recent study, in ref.
Collab:2024essnusb  authors have constrained $\Gamma$ parameter for a future
long baseline ESSnuSB experiment.

In an open quantum system framework the neutrino system interacts with the
environment, leading to the loss of quantum coherence among the neutrino
states. This decoherence effect manifests as a damping term $e^{-\Gamma L}$ in
the neutrino oscillation probabilities, where $\Gamma$ is the decoherence
parameter and L is the distance travelled by the neutrinos. $\Gamma$ can be
parameterized using power-law dependency i.e., $\Gamma\propto E_{\nu}^{n}$
where $E_{\nu}$ is the neutrino energy and the value of $n$ depends on the
origin of the decoherence model. In this context, without delving into the
origin of the decoherence effect, we perform a phenomenological study of the
effect of quantum neutrino decoherence in the Protvino to ORCA (P2O)
experiment. P2O is an upcoming neutrino oscillation experiment with a
substantial baseline of 2595 km Akindinov:2019p2o ; KM3Net:2016zxf . This very
large baseline of P2O experiment, makes it an ideal candidate to study the
impact of environmentally induced decoherence on the neutrino oscillations.

This paper is structured as follows. In section II we present a concise
overview of the density matrix formalism which is employed to derive the
neutrino oscillation probabilities. We discuss the experimental and simulation
details of our analysis in III. In section IV we examine the impact of
environmental decoherence on various oscillation channels (baseline 2595 km),
taking into account the power law dependencies of decoherence on neutrino
energy. Finally, we conclude our results in section V.

##  II Theoretical formulation

In an open quantum system framework, neutrino sub system can interact very
weakly with the stochastic environment. Keeping a phenomenological approach,
we discuss the dissipative effects of the decoherence on neutrino oscillations
in a model-independent way. To achieve this we consider density matrix
formalism and represent the time evolution of the density matrix using the
Lindblad master equation Lindblad:1976g ; GKS:1976vit ,

$\frac{d\rho_{m}(t)}{dt}=-i\left[H,\rho_{m}(t)\right]+\mathcal{D}\left[\rho_{m}(t)\right]~{},$
(1)

where $\rho_{m}$ is the density matrix of neutrino in the mass basis and H is
the Hamiltonian of the neutrino system. The decoherence effect is introduced
through a dissipative term $\mathcal{D}[\rho_{m}(t)]$. Parameterization of the
decoherence matrix is performed by imposing mathematical properties of the
density matrix such as complete positivity and preserving trace normalization
(probability must be positive and conserved w.r.t time). Imposing complete
positivity gives the Lindblad form of the dissipator as GKS:1978g

$\displaystyle\mathcal{D}\left[\rho_{m}(t)\right]$
$\displaystyle=\frac{1}{2}\sum_{n=1}^{N^{2}-1}\left\\{[\mathcal{V}_{n},\rho_{m}\mathcal{V}_{n}^{\dagger}]+[\mathcal{V}_{n}\rho_{m},\mathcal{V}_{n}^{\dagger}]\right\\}~{},$
(2)

where $N$ is the dimension of the Hilbert space and $\mathcal{V}_{n}$ are the
operators arising from the interaction of the subsystem with the environment.
In addition, we impose an increase in von Neumann entropy
$S=-Tr(\rho_{m}\ln\rho_{m})$ Banks:1984bsp ; Benatti:1988nar  and conservation
of average energy $Tr(\rho_{m}H)$ of the neutrino system. These conditions
lead to $\mathcal{V}_{n}=\mathcal{V}_{n}^{\dagger}$ , $[\mathcal{V}_{n},H]=0$
and imply that $\mathcal{V}_{n}$ and $H$ are simultaneously diagonalisable.
For a three-flavor neutrino system ($N=3$), operators $\mathcal{V}_{n}$
($n=1~{}\text{to}~{}8$ ) are the linear combinations of the Gell-Mann
matrices. Substituting for $\mathcal{V}_{n}$ in eq. (2) and using the above
mentioned conditions simplifies the dissipator term to

$\mathcal{D}\left[\rho_{m}(t)\right]=\begin{pmatrix}0&-\Gamma_{21}\rho_{12}(t)&-\Gamma_{31}\rho_{13}(t)\\\
-\Gamma_{21}\rho_{21}(t)&0&-\Gamma_{32}\rho_{23}(t)\\\
-\Gamma_{31}\rho_{31}(t)&-\Gamma_{32}\rho_{32}(t)&0\end{pmatrix},$ (3)

with

$\Gamma_{ij}=\Gamma_{ji}=\frac{1}{2}\sum_{n=1}^{8}(d_{n,i}-d_{n,j})^{2}~{},$
(4)

where, $d_{n,i}$, $d_{n,j}$ are the diagonal elements of $\mathcal{V}_{n}$
operator and $i,j$ take the values $1,2,3$.

Considering vacuum Hamiltonian in the mass basis
$H=diag(0,~{}\Delta_{21},~{}\Delta_{31})$, where $\Delta_{ij}=\Delta
m_{ij}^{2}/2E$, $\Delta m_{ij}^{2}$ being the neutrino mass-square difference,
one can obtain $\commutator{H}{\rho_{m}(t)}$ to be

$\displaystyle\commutator{H}{\rho_{m}(t)}=\begin{pmatrix}0&-\rho_{12}(t)\Delta_{21}&-\rho_{13}(t)\Delta_{31}\\\
\rho_{21}(t)\Delta_{21}&0&-\rho_{23}(t)\Delta_{32}\\\
\rho_{31}(t)\Delta_{31}&\rho_{32}(t)\Delta_{32}&0\end{pmatrix}~{}.$ (5)

The density matrix (mass basis) at time t is obtained by substituting eq. (3),
eq. (5) in eq. (1) as

$\rho_{m}(t)=\begin{pmatrix}\rho_{11}(0)&\rho_{12}(0)\exp-(\Gamma_{21}+i\Delta_{21})^{*}t&\rho_{13}(0)\exp-(\Gamma_{31}+i\Delta_{31})^{*}t\\\
\rho_{12}(0)\exp-(\Gamma_{21}+i\Delta_{21})t&\rho_{22}(0)&\rho_{23}(0)\exp-(\Gamma_{32}+i\Delta_{32})^{*}t\\\
\rho_{31}(0)\exp-(\Gamma_{31}+i\Delta_{31})t&\rho_{32}(0)\exp-(\Gamma_{32}+i\Delta_{32})t&\rho_{33}(0)\end{pmatrix}~{}.$
(6)

Eq. (6) can be converted into the flavour basis using the modified PMNS mixing
matrix $\tilde{U}$, as we include the MSW effect. We consider the modified-
mixing matrix ($\tilde{U}$) up to the 1st order approximation, from ref.
Denton:2018dmp  as

$\tilde{\rho}_{\alpha}=\tilde{U}~{}\tilde{\rho}_{m}~{}\tilde{U}^{\dagger}~{}.$
(7)

The neutrino transition probability from initial flavor ’$\nu_{\alpha}$’ to
final flavor ’$\nu_{\beta}$’ in terms of density matrix is

$\displaystyle P_{\alpha\beta}(t)$
$\displaystyle=Tr[\tilde{\rho}_{\alpha}(t)\tilde{\rho}_{\beta}(0)]~{}.$ (8)

The explicit form of the transition probability assuming ultra-relativistic
neutrinos ($t\approx L$) is given by Gomes:2019for ; Coloma:2018ice

$\displaystyle P_{\alpha\beta}(L)$
$\displaystyle=\delta_{\alpha\beta}-2\sum_{j>k}Re\left(\tilde{U}_{\beta
j}\tilde{U}_{\alpha j}^{*}\tilde{U}_{\alpha k}\tilde{V}_{\beta
k}^{*}\right)+2\sum_{j>k}Re\left(\tilde{U}_{\beta j}\tilde{U}_{\alpha
j}^{*}\tilde{U}_{\alpha k}\tilde{U}_{\beta
k}^{*}\right)\exp(-\Gamma_{jk}L)\cos(\frac{\tilde{\Delta}m_{jk}^{2}}{2E}L)$
(9) $\displaystyle+2\sum_{j>k}Im\left(\tilde{U}_{\beta j}\tilde{U}_{\alpha
j}^{*}\tilde{U}_{\alpha k}\tilde{U}_{\beta
k}^{*}\right)\exp(-\Gamma_{jk}L)\sin(\frac{\tilde{\Delta}m_{jk}^{2}}{2E}L)~{}.$

Here in eq. (9), $\Gamma_{ij}$ appear in the dissipative term
$e^{-\Gamma_{ij}L}$, which alters with respect to the baseline. Note that in
the absence of the decoherence effect i.e., $\Gamma_{ij}=0$ the probability
expression reduces to the standard oscillation probability. We consider a
general power law dependency of $\Gamma_{ij}$ on the neutrino energy given by

$\Gamma_{ij}(E_{\nu})=\Gamma_{0}\left(\frac{E_{\nu}}{E_{0}}\right)^{n}~{},$
(10)

where, $\Gamma_{0}$ is constant, $E_{0}$ is the reference energy taken as 1
GeV and $n=0,\pm 1,\pm 2$. Different physical origins explain the decoherence
phenomena leading to different integral power law dependencies Romeri:2023cgt
.

From eq. (4) it is clear that the $\Gamma_{ij}$’s are not independent of each
other. In our analysis, we consider the cases mentioned in table-1 where at
least two $\Gamma_{ij}$’s are activated simultaneously. Each case has a unique
decoherence matrix, eq. (3), and thus damps different neutrino oscillation
channels.

Cases | Assumptions on $\Gamma_{ij}$
---|---
Case-I | $\Gamma_{21}=\Gamma_{31}=\Gamma_{32}\neq 0$
Case-II | $\Gamma_{21}=\Gamma_{31},~{}\Gamma_{32}=0$
Case-III | $\Gamma_{21}=\Gamma_{32},~{}\Gamma_{31}=0$
Case-IV | $\Gamma_{31}=\Gamma_{32},~{}\Gamma_{21}=0$
Table 1: The decoherence models considered in this work.

In this work, we study the effect of the different power law dependencies
given in eq. (10), for each of the cases in table-1.

##  III Experimental and Simulation details

Protvino to ORCA (P2O) is an upcoming long baseline neutrino oscillation
experiment. The neutrino beam travels for about 2595 km from Protvino before
it reaches the 8 MT water cherenkov KM3NeT/ORCA detector. The initial plan is
to have a 90 kW proton beam resulting in $0.8\times 10^{20}$ protons on target
(POT) per year. A later upgrade targets a 450 kW proton beam. The resultant
muon neutrino beam of $2-7$ GeV passes through the upper mantle of the earth
with matter density $\sim~{}3.3~{}gm/cm^{3}$ before it reaches the far
detector. The oscillation channels of utmost interest are
$\nu_{\mu}\rightarrow\nu_{e},~{}\nu_{\mu}\rightarrow\nu_{\mu}$ channels and
the corresponding anti-neutrinos channels.

We have used the GLoBES software Huber:2004ka ; Huber:2007ji  to simulate the
P2O experiment and perform our analysis. Considering the standard three
flavour oscillation picture we reproduced the event spectra w.r.t the true
neutrino energy that is reported in ref. Akindinov:2019p2o . In this work, we
simulated the data corresponding to 90 kW proton beam resulting in $0.8\times
10^{20}$ POT/year and also 450 kW beam corresponding to the $4.0\times
10^{20}$ POT/year and total run time 6 years (3 years for $\nu$ and 3 years
for $\bar{\nu}$ mode) in each case.

Parameters | True values |  $3\sigma$ ranges
---|---|---
$\sin^{2}{\theta_{12}}$ | $0.304$ | Fixed
$\sin^{2}{\theta_{13}}$ | $0.0222$ | Fixed
$\sin^{2}{\theta_{23}}$ | $0.573$ | $[0.405:0.620]$
$\delta_{CP}$ | $194\degree$ | $[0:360\degree]$
$\frac{\Delta m^{2}_{21}}{10^{-5}~{}eV^{2}}$ | $7.42$ | Fixed
$\frac{\Delta m^{2}_{31}}{10^{-3}~{}eV^{2}}$ (NH) | $2.515$ | $[2.431:2.599]$
$\frac{\Delta m^{2}_{31}}{10^{-3}~{}eV^{2}}$ (IH) | $-2.498$ | $[-2.584:-2.413]$
Table 2: True oscillation parameters and $3\sigma$ ranges have been considered
in our analysis are taken from NuFIT Esteban:2020nufit .

Later we incorporated a new oscillation probability engine into GLoBES by
taking into account the effect of decoherence on neutrino propagation. The
best-fit values of the standard oscillation parameters and their $3\sigma$
ranges used in the analysis are given in table-2. For the statistical
analysis, the $\chi^{2}$ function is calculated using Poisson chi-square
function given by

$\chi^{2}=\min_{\alpha_{s},\alpha_{b}}\sum_{\text{channels
}}2\sum_{i}\left[N_{i}^{\mathrm{test}}-N_{i}^{\mathrm{true}}+N_{i}^{\mathrm{true}}\log\left(\frac{N_{i}^{\mathrm{true}}}{N_{i}^{\mathrm{test}}}\right)\right]+\alpha_{s}^{2}+\alpha_{b}^{2}~{},$
(11)

where, $N_{i}^{\mathrm{test}}$ and $N_{i}^{\mathrm{true}}$ are the number of
test and true events (signal + background) in $i$-th bin, respectively.
$\alpha_{s}$ and $\alpha_{b}$ are the signal and background normalisation
errors treated using the pull method Fogli:2002pt ; Huber:2002mx . The
analysis window of the neutrino energy is considered from 2 to 12 GeV. We
marginalize over the $\delta_{CP}$, $\theta_{23}$, $\Delta m_{31}^{2}$ in
their $3\sigma$ ranges without assuming any priors.

##  IV Results and discussion

In this segment, we outline the findings of our analysis across four
subsections, each focusing on the cases mentioned in table-1. The sensitivity
studies have been performed for both 90 kW proton beam and 450 kW proton beams
in all the subsections and the corresponding bounds of $\Gamma_{ij}$ have been
reported in table 3 and table 4 respectively. Henceforth, we will refer to the
simulations with 90 kW proton beam as P2O and that with 450 kW beam as
P2O-upgrade.

In each subsection, firstly we show the effect of non-zero decoherence on the
three oscillation channels $\nu_{\mu}\rightarrow\nu_{e}$ (left),
$\nu_{\mu}\rightarrow\nu_{\mu}$ (middle) and $\nu_{\mu}\rightarrow\nu_{\tau}$
(right) for P2O baseline. To achieve this, we have considered the true values
of the standard oscillation parameters given in the second column of table 2
allowing NH. However, for the sensitivity studies we have taken into account
only the relevant oscillation channels $\nu_{\mu}\rightarrow\nu_{e}$,
$\nu_{\mu}\rightarrow\nu_{\mu}$, $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$,
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\mu}$ . Each figure contains the
standard neutrino propagation in matter (SM) and the non-zero decoherence
assuming energy dependency index in eq. (10) as $n=-2,~{}-1,~{}0,~{}1,~{}2$.
In each plot, we show five graphs where green, magenta, blue, dark green and
red line represent $n=-2,~{}-1,~{}0,~{}1~{}\text{and}~{}2$ respectively.

Secondly, to quantify the sensitivity of P2O experiment to the decoherence
parameter $\Gamma_{ij}$, we plot $\chi^{2}$ as a function of
$\Gamma_{ij}$(test). We simulate the data by taking $\Gamma_{ij}=0$(true) and
the test spectrum by considering $\Gamma_{ij}\neq 0$. We calculate $\chi^{2}$
as described in sec III, and obtain

$\Delta\chi_{\Gamma}^{2}=\chi^{2}(\Gamma(true)=0,\Gamma(test)\neq 0)~{},$ (12)

after marginalizing over $\theta_{23},~{}\delta_{CP}$ and $(\Delta
m_{31}^{2})_{NH}$ in the test spectrum. Further, we provide the expected
bounds on $\Gamma_{ij}$ obtained for P2O and P2O-upgrade in table 3 and table
4 respectively.

Later, assuming the presence of decoherence in nature, we study the effect of
non-zero $\Gamma_{ij}$ on the mass hierarchy (MH) and CP violation (CPV)
sensitivity of the P2O and P2O-upgrade experiment. For this, we consider the
$3\sigma$ value of $\Gamma_{ij}$ obtained from the table 3 and table 4 as the
true values in each subsection. We estimate the MH and CPV sensitivity as a
function of $\delta_{CP}$ (true).

The $\Delta\chi_{MH}^{2}$ is defined as

$\Delta\chi_{MH}^{2}=\chi_{true}^{2}(\Gamma\neq 0,\Delta
m_{31}^{2}>0)-\chi_{test}^{2}(\Gamma=0,\Delta m_{31}^{2}<0)~{}.$ (13)

Here, we marginalize over the test parameters $\theta_{23}$ and $\delta_{CP}$.

For the CPV sensitivity, the $\Delta\chi_{CPV}^{2}$ is given by

$\displaystyle\Delta\chi_{0}^{2}=\chi_{true}^{2}(\delta_{CP}(true),\Gamma\neq
0)-\chi_{test}^{2}(\delta_{CP}=0,\Gamma=0)~{},$ (14)
$\displaystyle\Delta\chi_{\pi}^{2}=\chi_{true}^{2}(\delta_{CP}(true),\Gamma\neq
0)-\chi_{test}^{2}(\delta_{CP}=\pi,\Gamma=0)~{},$
$\displaystyle\Delta\chi_{CPV}^{2}=min[\Delta\chi_{0}^{2},\Delta\chi_{\pi}^{2}]~{}.$

We marginalize over the test parameters $\theta_{23}$ and $(\Delta
m_{31}^{2})_{NH}$. The significance ($\sigma$) of the MH and CPV sensitivity
is obtained by using $\sigma_{MH/CPV}=\sqrt{\Delta\chi_{MH/CPV}^{2}}$ .

###  IV.1 Case-I: $\Gamma_{21}=\Gamma_{31}=\Gamma_{32}\neq 0$

In this case, we assume $\Gamma_{21}$, $\Gamma_{31}$ and $\Gamma_{32}$ to be
equal to $2.3\times 10^{-23}$ GeV and non-zero. Fig. (1) shows the probability
of the three oscillation channels, $\nu_{\mu}\rightarrow\nu_{e}$,
$\nu_{\mu}\rightarrow\nu_{\mu}$ and $\nu_{\mu}\rightarrow\nu_{\tau}$ in the
left, middle and right panels, respectively. Legends corresponding to each
plot are mentioned in the figure. It can be seen that all the three
oscillation probabilities are significantly modified for the positive powers
of $n\geq 0$ when compared to the SM case (black curve). Additionally, in the
left plot, when $n\geq 0$, there is an extra peak around 10 GeV in the $P_{\mu
e}$ channel. The probability $P_{\mu\mu}$ ($P_{\mu\tau}$) is higher (lower)
than the SM probability in the energy window $E_{\nu}\sim 4-9~{}GeV$ and lower
(higher) in the ranges $E_{\nu}\sim 2-4~{}GeV$ and $E_{\nu}>9~{}GeV$. The
first oscillation peak (dip) in $\nu_{e}$ appearance ( $\nu_{\mu}$
disappearance) channel is effected by the decoherence for the chosen set of
decoherence parameters $\mathcal{O}(10^{-23}GeV)$. However, we noticed that
this effect is not significant when the reference value of decoherence
parameters is chosen to be $\mathcal{O}(10^{-24}GeV)$ and the extra peak
around 10 GeV remains in place with lesser amplitude.

In the left plots of fig. (2) and fig. (3), we constrain $\Gamma_{ij}$
presuming the conditions on decoherence parameters as given in case I. We
calculate the test statistics using eq. (12). The horizontal dashed and dotted
lines correspond to $90\%$ ($\sim 1.64\sigma$) and $3\sigma$ confidence level
(CL) respectively. The corresponding bounds of these parameters for different
$n$ values are tabulated in the first column of table-3 and table-4
respectively. From left most plots of fig. (2) and fig. (3), it can be seen
that the sensitivity to the decoherence parameter has slightly improved after
the proton beam upgrade to 450 kW. It can be verified from the first column of
table-3 and -4, that the $3\sigma$ bounds on $\Gamma_{ij}$ have improved by
one order of magnitude for the cases with $n\geq 0$ for the P2O-upgrade (450
kW).

Figure 1: Probability versus neutrino energy for $\nu_{e}$ appearance (left),
$\nu_{\mu}$ disappearance (middle), $\nu_{\tau}$ appearance (right)
considering case-I ( $\Gamma_{21}=\Gamma_{31}=\Gamma_{32}=2.3\times 10^{-23}$
GeV). The different colors correspond to the probabilities for the standard
case and various $n$ dependencies are as mentioned in the legend.

Figure 2: Case-I for P2O experiment (90kW beam). In the left, we present
$\Delta\chi^{2}$ as a function of $\Gamma_{0}$ (test) based on power law index
$n=0,\pm 1,\pm 2$. Horizontal dashed and dotted lines stands for 90% and
$3\sigma$ CL respectively. In the middle and right panels we show mass
hierarchy sensitivity and CP-violation sensitivity.

Figure 3: Case-I for P2O-upgrade (450 kW beam power). In the left, we present
$\Delta\chi^{2}$ as a function of $\Gamma_{0}$ (test) based on power law index
$n=0,\pm 1,\pm 2$. Horizontal dashed and dotted lines stands for 90% and
$3\sigma$ CL respectively. In the middle and right panels we show mass
hierarchy sensitivity and CP-violation sensitivity.

The MH and CPV sensitivity are calculated at $3\sigma$ value of $\Gamma_{ij}$
using eq. 13 and eq. 14. The MH sensitivity (middle panel) and CPV sensitivity
(right panel) are plotted for the standard case and for non-zero decoherence
(with $n=0,\pm 1,\pm 2$) in fig.(2) and fig.(3) respectively. For the 90 kW
beam, we can see the MH sensitivity (middle plot) for $n=-2$ is higher than
that for $n=2$. The significance $\sigma_{MH}$ is maximum around
$\delta_{CP}\text{(true)}=270\degree$. In the case of 450 kW beam, the over
all significance to the mass hierarchy is higher ($\sim 12\sigma$) and the
order of decoherence curves corresponding to different values of n obeys a
similar trend to that in 90 kW beam. However, since the MH sensitivity of P2O
(for both 90 kW and 450 kW) is very high, one can note that the effect of
decoherence is not prominent on the determination of MH. The same conclusions
have been observed in the plots where true hierarchy is assumed to be IH.

In the right most plots of fig. (2) and fig. (3), we observe that there is
non-zero significance corresponding to CP conserving values of
$\delta_{CP}=0,\pm\pi$ for all values of n. It implies that in the presence of
decoherence a non-zero CP value could be reported even for the true
$\delta_{CP}=0,\pm\pi$.

###  IV.2 Case-II (Solar limit I) : $\Gamma_{21}=\Gamma_{31},~{}\Gamma_{32}=0$

The transition probabilities in this case are shown in fig. (4). From the left
plot, one can see no notable change in the $\nu_{e}$-appearance probability
compared to the SM for the chosen set of decoherence parameters and also no
extra peak is observed at $E_{\nu}\sim 10~{}GeV$. In the middle plot, the
$\nu_{\mu}$ disappearance probabilities are presented for different $n$.
Clearly, the probability values significantly vary for the cases with $n\geq
0$. The same can be noted from the right most plot showing
$\nu_{\mu}\rightarrow\nu_{\tau}$ probability.

To obtain the bounds on the decoherence parameters, we plot $\chi^{2}$ vs
$\Gamma$ in the left panel of fig. (5) and fig. (6). The corresponding bounds
of $90\%$ and $3\sigma$ CL on $\Gamma$ are listed in the second column of
table-3 and table-4. In the middle and right plots of figures, we present MH
and CPV sensitivity using $3\sigma$ bounds obtained from the left plot. We can
see the MH and CPV sensitivities are almost equal for all $n$ considered in
our analysis and higher than the SM sensitivities. The maximum MH sensitivity
($\sim 20\sigma$) and maximum CPV sensitivity ($\sim 5\sigma$) are obtained
around $\delta_{CP}\text{(true)}=270\degree$.

Figure 4: Probability versus neutrino energy for $\nu_{e}$ appearance (left),
$\nu_{\mu}$ disappearance (middle), $\nu_{\tau}$ appearance (right)
considering case-II, $\Gamma_{21}=\Gamma_{31}=2.3\times
10^{-23}~{}GeV,~{}\Gamma_{32}=0$. The different colors correspond to the
probabilities for the standard case and various $n$ dependencies are as
mentioned in the legend.

Figure 5: Case-II for P2O experiment (90kW beam). In the left, we present
$\Delta\chi^{2}$ as a function of $\Gamma_{0}$ (test) based on power law index
$n=0,\pm 1,\pm 2$. Horizontal dashed and dotted lines stands for 90% and
$3\sigma$ CL respectively. In the middle and right panels we show mass
hierarchy sensitivity and CP-violation sensitivity..

Figure 6: Case-II for P2O-upgrade (450 kW beam power). In the left, we present
$\Delta\chi^{2}$ as a function of $\Gamma_{0}$ (test) based on power law index
$n=0,\pm 1,\pm 2$. Horizontal dashed and dotted lines stands for 90% and
$3\sigma$ CL respectively. In the middle and right panels we show mass
hierarchy sensitivity and CP-violation sensitivity.

###  IV.3 Case-III (Solar limit II) :
$\Gamma_{21}=\Gamma_{32},~{}\Gamma_{31}=0$

Figure 7: Probability versus neutrino energy for $\nu_{e}$ appearance (left),
$\nu_{\mu}$ disappearance (middle), $\nu_{\tau}$ appearance (right)
considering case-III, $\Gamma_{21}=\Gamma_{32}=2.3\times
10^{-23}~{}GeV,~{}\Gamma_{31}=0$

Figure 8: Case-III for P2O experiment (90kW beam). In the left, we present
$\Delta\chi^{2}$ as a function of $\Gamma_{0}$ (test) based on power law index
$n=0,\pm 1,\pm 2$. Horizontal dashed and dotted lines stands for 90% and
$3\sigma$ CL respectively. In the middle and right panels we show mass
hierarchy sensitivity and CP-violation sensitivity.

Figure 9: Case-III for P2O-upgrade (450 kW beam power). In the left, we
present $\Delta\chi^{2}$ as a function of $\Gamma_{0}$ (test) based on power
law index $n=0,\pm 1,\pm 2$. Horizontal dashed and dotted lines stands for 90%
and $3\sigma$ CL respectively. In the middle and right panels we show mass
hierarchy sensitivity and CP-violation sensitivity.

We plot $\nu_{\mu}\rightarrow\nu_{e}$ transition in the left,
$\nu_{\mu}\rightarrow\nu_{\mu}$ in the middle and
$\nu_{\mu}\rightarrow\nu_{\tau}$ in the right considering case-III in fig.
(7). Looking at the left plot one can state the $\nu_{e}$-appearance
probability varies similar to case-I. The $\nu_{\mu}$-disappearance
probability decreases for $E_{\nu}\geq 6~{}GeV$ and $\nu_{\tau}$-appearance
probability increases for $E_{\nu}\geq 11~{}GeV$.

The bounds on the decoherence parameters in this case are shown in the left
plots of fig. (5) and fig. (6) and are tabulated in the third column of
table-3 and table-4. In the middle panel, we plot the MH sensitivities where
we can see the sensitivities for all $n$ are lesser than the sensitivities
corresponding to the SM and the maximum sensitivity ($\sim 20\sigma$) visible
around $270\degree$. In the right panel, we plot the CPV sensitivities. We
notice the sensitivities for all $n$ are better than the sensitivity
corresponding to the SM and maximum ($>5\sigma$) is obaserved around
$90\degree$.

###  IV.4 Case-IV (Atmospheric limit) :
$\Gamma_{31}=\Gamma_{32},~{}\Gamma_{21}=0$

We present the oscillation probabilities in fig. (10) with $\Gamma_{31}$ and
$\Gamma_{32}$ are nonzero and equal. The $P_{\mu e}$ in the left plot is
similar to the corresponding plot of case-I and case-III but $P_{\mu\mu}$ (in
the middle) and $P_{\mu\tau}$ (in the right) differ. $P_{\mu\mu}$
($P_{\mu\tau}$) increases (decreases) significantly w.r.t SM for $n\geq 0$ in
the energy range $4-9~{}GeV$. Beyond that both the $P_{\mu\mu}$ and
$P_{\mu\tau}$ are lesser than the corresponding SM probability.

Based on the assumption in case-IV, in fig. (11) and fig. (12) we have shown
the bounds on $\Gamma$ in the left panel, the MH and CPV sensitivity in the
middle and the right panels respectively. These plots are more or less similar
to the case-I, although the assumptions do not match.

From fig. (1), fig. (4), fig. (7), fig. (10), we observe that varying values
of $n$ result in distinct spectral distortions of oscillation probabilities in
all the scenarios. It is evident from these figures, that in all the cases the
neutrino oscillation probabilities are sensitive to positive powers of $n$.
This arises due to the high energy range ($E_{\nu}\geq E_{0}$) in eq. (10) of
the neutrino beam w.r.t reference value ($E_{0}=1$ GeV) considered. This
result aligns with conclusions drawn in ref. Romeri:2023cgt  for the
accelerator data. From case-I, case-III and case-IV, it can be seen that
$P_{\mu e}$ has the same spectral distortion for all the values of $n$. In
addition, for $n\geq 0$ there is a peak appearing around 10 GeV for these
cases. This can be attributed to the non-zero value of $\Gamma_{32}$ in these
cases. The same conclusion that $\Gamma_{32}$ only effects the $P_{\mu e}$
channel can be verified from the left plot of fig. (4) belonging to case-II.
On the contrary, $\Gamma_{32}$ does not play a significant role in the
$P_{\mu\mu}$ channel for the neutrino energy $2GeV<E_{\nu}<12GeV$. This can be
seen from the middle panels of fig. (4) and fig. (7), where $\Gamma_{32}=0$ in
the former and $\Gamma_{32}\neq 0$ in the latter.

Figure 10: Probability versus neutrino energy for $\nu_{e}$ appearance (left),
$\nu_{\mu}$ disappearance (middle), $\nu_{\tau}$ appearance (right)
considering case-IV, $\Gamma_{31}=\Gamma_{32}=2.3\times
10^{-23}~{}GeV,~{}\Gamma_{21}=0$

Figure 11: Case-IV for P2O experiment (90kW beam). In the left, we present
$\Delta\chi^{2}$ as a function of $\Gamma_{0}$ (test) based on power law index
$n=0,\pm 1,\pm 2$. Horizontal dashed and dotted lines stands for 90% and
$3\sigma$ CL respectively. In the middle and right panels we show mass
hierarchy sensitivity and CP-violation sensitivity.

Figure 12: Case-IV for P2O-upgrade (450 kW beam power). In the left, we present $\Delta\chi^{2}$ as a function of $\Gamma_{0}$ (test) based on power law index $n=0,\pm 1,\pm 2$. Horizontal dashed and dotted lines stands for 90% and $3\sigma$ CL respectively. In the middle and right panels we show mass hierarchy sensitivity and CP-violation sensitivity. $\mathrm{n}$ | $\mathrm{CL}$ | Case-I | Case-II | Case-III | Case-IV
---|---|---|---|---|---
$\Gamma_{21}=\Gamma_{31}=\Gamma_{32}$ | $\Gamma_{21}=\Gamma_{31}$ | $\Gamma_{21}=\Gamma_{32}$ | $\Gamma_{31}=\Gamma_{32}$
$n=-2$ | $90\%$ | $8.06\times 10^{-23}$ | $6.0\times 10^{-23}$ | $1.008\times 10^{-22}$ | $8.0\times 10^{-23}$
$3\sigma$ | $1.15\times 10^{-22}$ | $1.11\times 10^{-22}$ | $1.56\times 10^{-22}$ | $1.15\times 10^{-22}$
$n=-1$ | $90\%$ | $1.8\times 10^{-23}$ | $1.36\times 10^{-23}$ | $2.19\times 10^{-23}$ | $1.75\times 10^{-23}$
$3\sigma$ | $2.55\times 10^{-23}$ | $2.46\times 10^{-23}$ | $3.37\times 10^{-23}$ | $2.53\times 10^{-23}$
$n=0$ | $90\%$ | $3.76\times 10^{-24}$ | $2.76\times 10^{-24}$ | $4.26\times 10^{-24}$ | $3.76\times 10^{-24}$
$3\sigma$ | $5.3\times 10^{-24}$ | $5.18\times 10^{-24}$ | $6.7\times 10^{-24}$ | $5.23\times 10^{-24}$
$n=1$ | $90\%$ | $7.4\times 10^{-25}$ | $5.4\times 10^{-25}$ | $7.31\times 10^{-25}$ | $7.3\times 10^{-25}$
$3\sigma$ | $1.04\times 10^{-24}$ | $1.02\times 10^{-24}$ | $1.23\times 10^{-24}$ | $1.008\times 10^{-24}$
$n=2$ | $90\%$ | $1.24\times 10^{-25}$ | $9.07\times 10^{-26}$ | $1.007\times 10^{-25}$ | $1.2\times 10^{-25}$
$3\sigma$ | $1.79\times 10^{-25}$ | $1.87\times 10^{-25}$ | $1.95\times 10^{-25}$ | $1.74\times 10^{-25}$
Table 3: Bounds obtained in different cases for 90 kW beam power. $\mathrm{n}$ | $\mathrm{CL}$ | Case-I | Case-II | Case-III | Case-IV
---|---|---|---|---|---
$\Gamma_{21}=\Gamma_{31}=\Gamma_{32}$ | $\Gamma_{21}=\Gamma_{31}$ | $\Gamma_{21}=\Gamma_{32}$ | $\Gamma_{31}=\Gamma_{32}$
$n=-2$ | $90\%$ | $6.4\times 10^{-23}$ | $3.9\times 10^{-23}$ | $6.5\times 10^{-23}$ | $6.4\times 10^{-23}$
$3\sigma$ | $9.0\times 10^{-23}$ | $7.6\times 10^{-23}$ | $1.1\times 10^{-22}$ | $9.3\times 10^{-23}$
$n=-1$ | $90\%$ | $1.4\times 10^{-23}$ | $8.5\times 10^{-24}$ | $1.5\times 10^{-23}$ | $1.46\times 10^{-23}$
$3\sigma$ | $2.1\times 10^{-23}$ | $1.75\times 10^{-23}$ | $2.5\times 10^{-23}$ | $2.0\times 10^{-23}$
$n=0$ | $90\%$ | $3.1\times 10^{-24}$ | $1.89\times 10^{-24}$ | $2.5\times 10^{-24}$ | $3.1\times 10^{-24}$
$3\sigma$ | $4.2\times 10^{-24}$ | $4.0\times 10^{-24}$ | $5.0\times 10^{-24}$ | $4.7\times 10^{-24}$
$n=1$ | $90\%$ | $6.2\times 10^{-25}$ | $3.6\times 10^{-25}$ | $4.2\times 10^{-25}$ | $6.3\times 10^{-25}$
$3\sigma$ | $8.5\times 10^{-25}$ | $7.7\times 10^{-25}$ | $8.5\times 10^{-25}$ | $8.1\times 10^{-25}$
$n=2$ | $90\%$ | $7.8\times 10^{-26}$ | $5.87\times 10^{-26}$ | $6.0\times 10^{-26}$ | $7.5\times 10^{-26}$
$3\sigma$ | $1.5\times 10^{-25}$ | $1.4\times 10^{-25}$ | $1.1\times 10^{-25}$ | $1.3\times 10^{-25}$
Table 4: Bounds obtained in different cases for 450 kW beam power.

From table-3 and table-4, it can be observed that case-I and case-IV give rise
to similar bounds on $\Gamma$ parameter. Firstly, $\Gamma_{21}\neq 0$ does not
effect the $P_{\mu e}$ channel as can be seen from the left panel of fig. (1)
and fig. (10). Additionally, from the middle panel of fig. (1) and fig. (10)
it can be seen that the difference in $P_{\mu\mu}$ values dominant after
$E_{\nu}\geq 8$ GeV. However, the P2O flux gradually falls after $E_{\nu}\sim
7$ GeV and the analysis window for the $\chi^{2}$ plots are considered as
$2GeV<E_{\nu}<12GeV$. Consequently, the bounds on $\Gamma$ for these cases
(case-I and case-IV) are similar in both P2O and P2O-upgrade experiments. For
the case of $n=0$ (3rd row) in table-3 and table-4 one can conclude that the
bounds obtained for P2O and P2O-upgrade are better than the bounds reported in
ref. Romeri:2023cgt . Specifically, case-II poses strong constraints on
$\Gamma_{ij}\leq 2.76\times 10^{-24}$ GeV for P2O and $\Gamma_{ij}\leq
1.89\times 10^{-24}$ GeV for P2O-upgrade, which are found to be better than
the bounds reported by T2K and MINOS joint analysis in ref. gomesalg:2023 ,
upcoming DUNE experiment in ref. Gomes:2019for  and competitive with the
recent bounds obtained from neutrinos detected by IceCube in ref.
Coloma:2018ice ; Collab:2024icecube . For $n=-1,-2$ the bounds on
$\Gamma_{ij}$ are $\mathcal{O}(10^{-23})$ at 90% CL. These bounds are less
constrained when compared to the bounds from reactor data of ref.
Romeri:2023cgt  as expected.

##  V Conclusions

In this work, we have investigate the effect of environmentally induced
decoherence on the neutrino oscillations at the upcoming P2O experiment.
Considering three flavor oscillation framework, we study the viable
phenomenological models of decoherence where we assume all the decoherence
parameters are non-zero (or) two of them are non-zero. In both the scenarios,
we consider a power law based relation between the decoherence parameter
$\Gamma$ and the neutrino energy $E_{\nu}$. After a thorough review of the
oscillation probabilities we note that $\Gamma_{32}$ plays a significant role
on the $\nu_{e}$ appearance channel. Given that in P2O, the neutrino beam
energy $E_{\nu}>E_{0}$ (reference $E_{0}=1$ GeV), we observe that when $n\geq
0$ the effect decoherence on the oscillation probabilities is prominent.

Assuming that there is no decoherence in the true neutrino spectra, we obtain
bounds on $\Gamma_{ij}$ parameter for both 90kW (P2O) and 450kW proton beam
(P2O-upgrade). For the case of $n=0$ the bounds obtained on $\Gamma_{ij}$ in
both P2O and P2O-upgrade across all the cases are found to be better than the
bounds reported in ref. Romeri:2023cgt . For case-II, we obtain
$\Gamma_{ij}\leq 2.76\times 10^{-24}$ GeV for P2O and $\Gamma_{ij}\leq
1.89\times 10^{-24}$ GeV for P2O-upgrade, which are found to be the most
stringent bounds among the bounds reported from other experiments.

Further, we also explore how the MH and CPV sensitivity of P2O and P2O-upgrade
are effected if one assumes the presence of non-zero decoherence in nature. We
note that the comprehensive effect of $P_{\mu e}~{},~{}P_{\mu\mu}$ and the
corresponding anti-neutrino probabilities has not significantly changed the MH
sensitivity of the experiment. However, the CPV sensitivity of P2O and
P2O-upgrade is observed to be non-zero even for CP-conserving values of
$\delta_{CP}$. In conclusion, based on our analysis, it is evident that P2O
experiment shows significant promise in constraining the environmental
decoherence parameters.

Acknowledgements: We thank Anatoly Sokolov for fruitful discussions. We
acknowledge Dmitry Zaborov for useful information related to P2O flux. One of
the authors (K. N. Deepthi) would like to thank DST-SERB SIRE program for the
financial support.

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# New criteria for starlikeness in the unit disc

Milutin Obradović  Department of Mathematics, Faculty of Civil Engineering,
University of Belgrade, Bulevar Kralja Aleksandra 73, 11000, Belgrade, Serbia
<EMAIL_ADDRESS>and  Nikola Tuneski  Department of Mathematics and
Informatics, Faculty of Mechanical Engineering, Ss. Cyril and Methodius
University in Skopje, Karpoš II b.b., 1000 Skopje, Republic of North
Macedonia<EMAIL_ADDRESS>
###### Abstract.

It is well-known that the condition
$\operatorname{Re}\left[1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right]>0$,
$z\in{\mathbb{D}}$, implies that $f$ is starlike function (i.e. convexity
implies starlikeness). If the previous condition is not satisfied for every
$z\in{\mathbb{D}}$, then it is possible to get new criteria for starlikeness
by using
$\left|\arg\left[\alpha+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right]\right|$,
$z\in{\mathbb{D}}$, where $\alpha>1.$

###### Key words and phrases:

starlike functions, convex functions, criteria

###### 2010 Mathematics Subject Classification:

30C45, 30C50

##  1\. Introduction and definitions

Let $\mathcal{A}$ be the class of functions $f$ which are analytic in the open
unit disc ${\mathbb{D}}=\\{z:|z|<1\\}$ of the form
$f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$, and let $\mathcal{S}$ be the subclass
of $\mathcal{A}$ consisting of functions that are univalent in ${\mathbb{D}}$.

Also, let

$\begin{split}\mathcal{C}&=\left[f\in\mathcal{A}:\operatorname{Re}\left[1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right]>0,\,(z\in{\mathbb{D}})\right],\\\
\mathcal{S}^{\star}&=\left[f\in\mathcal{A}:\operatorname{Re}\left[\frac{zf^{\prime}(z)}{f(z)}\right]>0,\,(z\in{\mathbb{D}})\right],\end{split}$

denote the classes of convex and starlike functions, respectively. It is well-
known that

$f\in\mathcal{C}\quad\Rightarrow\quad f\in\mathcal{S}^{\star},$

(see Duren [1]). If the condition for convexity,
$\operatorname{Re}\left[1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right]>0,$
is not satisfied for every $z\in{\mathbb{D}}$, then we cannot conclude that
$f\in\mathcal{C}$, and consequently, $f\in\mathcal{S}^{\star}$.

In this paper $f\in\mathcal{S}^{\star}$ in the terms of the expression

$\left|\arg\left[\alpha+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right]\right|,\quad(z\in{\mathbb{D}})$

where $\alpha>1$.

##  2\. Maine results

For our consideration we need the next lemma given by Nunokawa in [3].

Lemma A. Let $p(z)=1+p_{1}z+p_{2}z^{2}+\ldots$ be analytic in the unit disc
${\mathbb{D}}$ and $p(z)\neq 0$ for $z\in{\mathbb{D}}$. Also, let suppose that
there exists a point $z_{0}\in{\mathbb{D}}$ such that
$\operatorname{Re}\,p(z)>0$ for $|z|<|z_{0}|$ and
$\operatorname{Re}\,p(z_{0})=0$, where $p(z_{0})\neq 0$, i.e. $p(z_{0})=ia$,
$a$ is real and $a\neq 0$. Then we have

$\frac{z_{0}p^{\prime}(z_{0})}{p(z_{0})}=ik,$

where $k\geq\frac{1}{2}\left(a+\frac{1}{a}\right)$, when $a>0$, and
$k\leq-\frac{1}{2}\left(|a|+\frac{1}{|a|}\right)$, when $a<0$.

######  Theorem 1.

Let $f\in\mathcal{A}$, $\alpha>1$, and let

(1)
$\left|\arg\left[\alpha+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right]\right|<{\operatorname{arctg}}\frac{\sqrt{3}}{\alpha-1}\quad(z\in{\mathbb{D}}).$

Then $f\in\mathcal{S}^{\star}$.

###### Proof.

If we put that $p(z)=\frac{zf^{\prime}(z)}{f(z)}$, then $p(0)=1$ and
$\frac{zp^{\prime}(z)}{p(z)}+p(z)-1=\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}$.
So, we would need to prove that the following implication holds:

(2)
$\begin{split}\left|\arg\left[\frac{zp^{\prime}(z)}{p(z)}+p(z)+\alpha-1\right]\right|&<\operatorname{arctg}\frac{\sqrt{3}}{\alpha-1}\quad(z\in{\mathbb{D}})\\\\[5.69054pt]
\Rightarrow\quad\operatorname{Re}p(z)&>0\quad(z\in{\mathbb{D}}).\end{split}$

First we will prove that $p(z)\neq 0,z\in{\mathbb{D}}$. On a contrary, if
there exists $z_{1}\in{\mathbb{D}}$ such that $z_{1}$ is the zero of order $m$
of the function $p$, then $p(z)=(z-z_{1})^{m}p_{1}(z)$, where $m$ is positive
integer, $p_{1}$ is analytic in ${\mathbb{D}}$ with $p_{1}(z_{1})\neq 0$, and
further

$\frac{zp^{\prime}(z)}{p(z)}=\frac{mz}{z-z_{1}}+\frac{zp_{1}^{\prime}(z)}{p_{1}(z)}.$

This means that the real part of the right hand side can tend to $-\infty$
when $z\rightarrow z_{1}$, which is a contradiction to the assumption of the
theorem regarding the argument. Thus $p(z)\neq 0$, $z\in{\mathbb{D}}.$

Now, let suppose that the implication (2) does not hold in the unit disc. It
means that there exists a point $z_{0}\in{\mathbb{D}}$ such that
$\operatorname{Re}p(z)>0$ for $|z|<|z_{0}|$ and $\operatorname{Re}p(z_{0})=0$,
where $p(z_{0})\neq 0$. If we put $p(z_{0})=ia$, $a$ is real and $a\neq 0$,
then by Lemma A we have

$\frac{z_{0}p^{\prime}(z_{0})}{p(z_{0})}=ik,$

where $k\geq\frac{1}{2}\left(a+\frac{1}{a}\right)$, when $a>0$, and
$k\leq-\frac{1}{2}\left(|a|+\frac{1}{|a|}\right)$, when $a<0$.

Next, if we put $\Phi(z)=\frac{zp^{\prime}(z)}{p(z)}+p(z)+\alpha-1$, then by
the previous facts:

$\operatorname{Re}\Phi(z_{0})=\alpha-1,\quad\operatorname{Im}\Phi(z_{0})=k+a,$

which for $a>0$ implies:

$\begin{split}\arg\Phi(z_{0})&=\operatorname{arctg}\frac{k+a}{\alpha-1}\geq\operatorname{arctg}\frac{\frac{1}{2}\left(a+\frac{1}{a}\right)+a}{\alpha-1}\\\
&=\operatorname{arctg}\frac{3a+\frac{1}{a}}{2(\alpha-1)}\geq\operatorname{arctg}\frac{\sqrt{3}}{\alpha-1},\end{split}$

because the function $\varphi(a)=3a+\frac{1}{a}$ has its minimum value
$\varphi(\frac{1}{\sqrt{3}})=2\sqrt{3}.$ Similarly, for $a<0$:

$\begin{split}\arg\Phi(z_{0})&=\operatorname{arctg}\frac{k+a}{\alpha-1}\leq\operatorname{arctg}\frac{-\frac{1}{2}\left(|a|+\frac{1}{|a|}\right)-|a|}{\alpha-1}\\\
&=-\operatorname{arctg}\frac{3|a|+\frac{1}{|a|}}{2(\alpha-1)}\leq-\operatorname{arctg}\frac{\sqrt{3}}{\alpha-1}.\end{split}$

Combining the cases $a>0$ and $a<0$, we receive

$\left|\arg\Phi(z_{0})\right|\geq\operatorname{arctg}\frac{\sqrt{3}}{\alpha-1},$

which is a contradiction to the relation (1).

This show that
$\operatorname{Re}p(z)=\operatorname{Re}\frac{zf^{\prime}(z)}{f(z)}>0,$ for
all $z\in{\mathbb{D}}$, i.e. that $f\in\mathcal{S}^{\star}$. ∎

######  Example 1.

Let $f\in\mathcal{A}$ is defined by the condition

(3)
$1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}=(\sqrt{3}+1)\left(\frac{1+z}{1-z}\right)^{\frac{1}{2}}-\sqrt{3},$

where we use the principal value of the square root. For real $z$ close to -1
we have
$\operatorname{Re}\left(1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right)<0,$
which means that $f$ is not convex. On the other hand, from (3) we have

$\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}+(\sqrt{3}+1)=(\sqrt{3}+1)\left(\frac{1+z}{1-z}\right)^{\frac{1}{2}}$

and from here

$\begin{split}\left|\arg\left[\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}+(\sqrt{3}+1)\right]\right|&\leq\frac{1}{2}\left|\arg\frac{1+z}{1-z}\right|\\\
&<\frac{\pi}{4}=\operatorname{arctg}\frac{\sqrt{3}}{(\sqrt{3}+1)-1},\end{split}$

which by Theorem 1 (with $\alpha=\sqrt{3}+1$) implies that
$f\in\mathcal{S}^{\star}$.

Letting $\alpha$ tend to 1 in Theorem 1, we have the following well-known
result.

######  Corollary 1.

Let $f\in{\mathcal{A}}$ and

$\left|\arg\left[\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}+1\right]\right|<\frac{\pi}{2}\quad(z\in{\mathbb{D}})$

Then $f\in\mathcal{S}^{\star}.$

Further, it is easy to verify that the disc with center $\alpha$ and radius
$\frac{\alpha\sqrt{3}}{\sqrt{3+(\alpha-1)^{2}}}$ is contained in the angle
$|\arg z|<\operatorname{arctg}\frac{\sqrt{3}}{\alpha-1}$, and so by using the
result of Theorem 1, we have that

$\left|\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right|=\left|\left(\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}+\alpha\right)-\alpha\right|<\frac{\alpha\sqrt{3}}{\sqrt{3+(\alpha-1)^{2}}}\quad(z\in{\mathbb{D}})$

implies $f\in\mathcal{S}^{\star}$. Since the function
$\varphi(\alpha)=:\frac{\alpha\sqrt{3}}{\sqrt{3+(\alpha-1)^{2}}}$, $\alpha\geq
1$, attains its maximal value 2 for $\alpha=4$, we get

######  Corollary 2.

Let $f\in{\mathcal{A}}$ and

$\left|\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right|<2\quad(z\in{\mathbb{D}})$

Then $f\in\mathcal{S}^{\star}.$

In a similar way as in Theorem 1, we can consider starlikeness problem in
connection with the class defined by

$\mathcal{G}=\left[f\in\mathcal{A}:\operatorname{Re}\left[1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right]<\frac{3}{2},\,z\in{\mathbb{D}}\right].$

Ozaki in [4] proved that if $f\in\mathcal{G}$, then $f$ is univalent in
${\mathbb{D}}$. Later, Umezava in [6] showed the functions from $\mathcal{G}$
are convex in one direction. Also, it is shown in the papers [2] and [5] that
$\mathcal{G}$ is subclass of $\mathcal{S}^{\star}.$

If
$\operatorname{Re}\left[1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right]<\frac{3}{2}$
is not satisfied for every $z\in{\mathbb{D}}$, then we can pose a question if
for some $\beta<1$, such that

(4)
$\operatorname{Re}\left[\beta+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right]<\frac{3}{2}\quad(z\in{\mathbb{D}}),$

is it possible obtain sufficient condition for starlikeness, in a similar way
as in Theorem 1. The condition (4) is equivalent to

$\operatorname{Re}\left[\frac{3-2\beta}{2}-\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right]>0\quad(z\in{\mathbb{D}}),$

and we can use similar technique. This lead so

######  Theorem 2.

Let $f\in\mathcal{A}$, $\beta<1$, and let

$\left|\arg\left[\frac{3-2\beta}{2}-\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right]\right|<\operatorname{arctg}\frac{2\sqrt{3}}{5-2\beta}\quad(z\in{\mathbb{D}}).$

Then $f\in\mathcal{S}^{\star}$.

###### Proof.

As in the proof of Theorem 1, we put that $p(z)=\frac{zf^{\prime}(z)}{f(z)}$,
such that $p(0)=1$ and
$\frac{zp^{\prime}(z)}{p(z)}+p(z)-1=\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}$.
Now, our aim is to prove that for some $\beta<1$ the following implication
holds:

(5)
$\begin{split}\left|\arg\left[\frac{5-2\beta}{2}-\frac{zp^{\prime}(z)}{p(z)}-p(z)\right]\right|&<\operatorname{arctg}\frac{2\sqrt{3}}{5-2\beta}\quad(z\in{\mathbb{D}})\\\\[5.69054pt]
\Rightarrow\operatorname{Re}p(z)&>0\quad(z\in{\mathbb{D}}).\end{split}$

First, as in Theorem 1 we conclude that $p(z)\neq 0,z\in{\mathbb{D}}$.

Further, if the implication (5) is not true, there exists a point
$z_{0}\in{\mathbb{D}}$ such that $\operatorname{Re}p(z)>0$ for $|z|<|z_{0}|$
and $\operatorname{Re}p(z_{0})=0$, where $p(z_{0})\neq 0$. If we put
$p(z_{0})=ia$, $a$ is real and $a\neq 0$, then by Lemma A we have

$\frac{z_{0}p^{\prime}(z_{0})}{p(z_{0})}=ik,$

where $k\geq\frac{1}{2}\left(a+\frac{1}{a}\right)$, when $a>0$, and
$k\leq-\frac{1}{2}\left(|a|+\frac{1}{|a|}\right)$, when $a<0$. Also, for

$\Psi(z)=\frac{5-2\beta}{2}-\frac{zp^{\prime}(z)}{p(z)}-p(z),$

using the previous conclusions, we have

$\operatorname{Re}\Psi(z_{0})=\frac{5-2\beta}{2},\quad\operatorname{Im}\Psi(z_{0})=-(k+a).$

Using the same method as in the proof of Theorem 1, we easily conclude that

$\arg\Psi(z_{0})=-\operatorname{arctg}\frac{2(k+a)}{5-2\beta}\leq-\operatorname{arctg}\frac{2\sqrt{3}}{5-2\beta},$

when $a>0,$ and

$\arg\Psi(z_{0})=-\operatorname{arctg}\frac{2(k+a)}{5-2\beta}\geq\operatorname{arctg}\frac{2\sqrt{3}}{5-2\beta},$

when $a<0$. These facts imply contradiction of the assumption in (5).

So, we have the statement of this theorem. ∎

Letting $\beta$ tend to 1 in Theorem 2, we receive

######  Corollary 3.

Let $f\in{\mathcal{A}}$ and

$\left|\arg\left[\frac{1}{2}-\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right]\right|<\operatorname{arctg}\frac{2}{\sqrt{3}}\approx
49.1^{\circ}\quad(z\in{\mathbb{D}}).$

Then $f\in\mathcal{S}^{\star}.$

As in the case of Theorem 1, we can show that the disc with center
$\frac{3-2\beta}{2}$ and radius
$\frac{(3-2\beta)\sqrt{3}}{\sqrt{12+(5-2\beta)^{2}}}$ is containing in the
angle $|\arg z|\leq\operatorname{arctg}\frac{2\sqrt{3}}{5-2\beta}$. So, if

$\left|\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right|=\left|\left(\frac{3-2\beta}{2}-\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right)-\frac{3-2\beta}{2}\right|<\frac{(3-2\beta)\sqrt{3}}{\sqrt{12+(5-2\beta)^{2}}},$

for all $z\in{\mathbb{D}}$, then Theorem 2 brings that
$f\in\mathcal{S}^{\star}.$ Since the function
$\psi(\beta)=:\frac{(3-2\beta)\sqrt{3}}{\sqrt{12+(5-2\beta)^{2}}}$ is
decreasing on the interval $(-\infty,1]$, and
$\psi(\beta)\rightarrow\sqrt{3}$, when $\beta\rightarrow-\infty,$ we get

######  Corollary 4.

Let $f\in{\mathcal{A}}$ and

$\left|\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right|<\sqrt{3}\quad(z\in{\mathbb{D}}).$

Then $f\in\mathcal{S}^{\star}.$

For a starlike function $f$ it is not necessary that $Re\frac{f(z)}{z}>0,$
$z\in{\mathbb{D}}$. For example, for Koebe function $k(z)=\frac{z}{(1-z)^{2}}$
we have

$\left.\operatorname{Re}\frac{k(z)}{z}\right|_{z=(1+i)/\sqrt{2}}=-(\sqrt{2}+1)<0,$

and this also holds for points in ${\mathbb{D}}$ close enought to
$\frac{1+i}{\sqrt{2}}$.

In the next theorem we give a condition which provides
$\operatorname{Re}\frac{f(z)}{z}>0$, $z\in{\mathbb{D}}$.

######  Theorem 3.

Let $f\in\mathcal{A}$, $\gamma\geq 0$, and let

$\left|\arg\left[\frac{zf^{\prime}(z)}{f(z)}+\gamma\right]\right|<\operatorname{arctg}\frac{1}{1+\gamma}\quad(z\in{\mathbb{D}}).$

Then $\operatorname{Re}\frac{f(z)}{z}>0$, $z\in{\mathbb{D}}.$

###### Proof.

We apply the same method as in the previous theorems. In this case, we use
$p(z)=\frac{f(z)}{z}$, such that $p(0)=1$ and
$\frac{zp^{\prime}(z)}{p(z)}+1=\frac{zf^{\prime}(z)}{f(z)}$. So, the result we
need to prove can be rewritten in the following equivalent form:

(6)
$\begin{split}\left|\arg\left[\frac{zp^{\prime}(z)}{p(z)}+1+\gamma\right]\right|&<\operatorname{arctg}\frac{1}{1+\gamma}\quad(z\in{\mathbb{D}})\\\\[5.69054pt]
\Rightarrow\qquad\operatorname{Re}p(z)&>0\quad(z\in{\mathbb{D}}).\end{split}$

First, $p(z)\neq 0$ for all $z\in{\mathbb{D}}$.

Next, if the implication (6) does not hold in the unit disc, then there exists
a point $z_{0}\in{\mathbb{D}}$ such that $\operatorname{Re}p(z)>0$ for
$|z|<|z_{0}|$ and $\operatorname{Re}p(z_{0})=0$, where $p(z_{0})\neq 0$. If we
put $p(z_{0})=ia$, $a$ is real and $a\neq 0$, then by Lemma A we have

$\frac{z_{0}p^{\prime}(z_{0})}{p(z_{0})}=ik,$

where $k\geq\frac{1}{2}\left(a+\frac{1}{a}\right)$, when $a>0$, and
$k\leq-\frac{1}{2}\left(|a|+\frac{1}{|a|}\right)$, when $a<0$. Now, for $a>0$
we get:

$\begin{split}\arg\left[\frac{z_{0}p^{\prime}(z_{0})}{p(z_{0})}+1+\gamma\right]&=\arg(ik+1+\gamma)=\operatorname{arctg}\frac{k}{1+\gamma}\\\
&\geq\frac{\frac{1}{2}\left(a+\frac{1}{a}\right)}{1+\gamma}\geq\operatorname{arctg}\frac{1}{1+\gamma},\end{split}$

and for $a<0$,

$\arg\left[\frac{z_{0}p^{\prime}(z_{0})}{p(z_{0})}+1+\gamma\right]\leq-\operatorname{arctg}\frac{1}{1+\gamma}.$

By combining the above conclusions, we receive that

$\left|\arg\left(\frac{z_{0}p^{\prime}(z_{0})}{p(z_{0})}+1+\gamma\right)\right|\geq\operatorname{arctg}\frac{1}{1+\gamma},$

which is a contradiction to assumption in (6). ∎

For $\gamma=0$ in previous theorem we have next result.

######  Corollary 5.

Let $f\in\mathcal{A}$ and

$\left|\arg\frac{zf^{\prime}(z)}{f(z)}\right|<\frac{\pi}{4}\quad(z\in{\mathbb{D}}).$

Then $\operatorname{Re}\frac{f(z)}{z}>0$, $z\in{\mathbb{D}}.$

In Theorem 3 the function $f$ from $\mathcal{A}$ need not be starlike as the
next example shows.

######  Example 2.

Let $f\in\mathcal{A}$ is defined by

(7)
$\frac{zf^{\prime}(z)}{f(z)}=\sqrt{3}\left(\frac{1+z}{1-z}\right)^{\frac{1}{3}}+1-\sqrt{3},$

where we use the principal value for square root. For real $z$ close to -1 we
have that $\operatorname{Re}\left[\frac{zf^{\prime}(z)}{f(z)}\right]<0,$ which
means that $f$ is not starlike. But from (7) we have

$\frac{zf^{\prime}(z)}{f(z)}+(\sqrt{3}-1)=\sqrt{3}\left(\frac{1+z}{1-z}\right)^{\frac{1}{3}},$

and from here

$\begin{split}\left|\arg\left[\frac{zf^{\prime}(z)}{f(z)}+(\sqrt{3}-1)\right]\right|&\leq\frac{1}{3}\left|\arg\left(\frac{1+z}{1-z}\right)\right|\\\
&<\frac{\pi}{6}=\operatorname{arctg}\frac{1}{(\sqrt{3}-1)+1},\end{split}$

which by Theorem 3 (with $\gamma=\sqrt{3}-1$) implies that
$\operatorname{Re}\frac{f(z)}{z}>0,$ $z\in{\mathbb{D}}$.

It is possible to show that the disc with center $1+\gamma$ and radius
$\frac{1+\gamma}{\sqrt{1+(1+\gamma)^{2}}}$ lies in the angle $|\arg
z|\leq\operatorname{arctg}\frac{1}{1+\gamma}$, and so, by using the result of
Theorem 3 we have that

$\left|\frac{zf^{\prime}(z)}{f(z)}-1\right|=\left|\left(\frac{zf^{\prime}(z)}{f(z)}+\gamma\right)-(1+\gamma)\right|<\frac{1+\gamma}{\sqrt{1+(1+\gamma)^{2}}}\quad(z\in{\mathbb{D}})$

implies $\operatorname{Re}\frac{f(z)}{z}>0,$ $z\in{\mathbb{D}}.$ When
$\gamma\rightarrow+\infty$ we receive

######  Corollary 6.

Let $f\in{\mathcal{A}}$ and

$\left|\frac{zf^{\prime}(z)}{f(z)}-1\right|<1\quad(z\in{\mathbb{D}}).$

Then $\operatorname{Re}\frac{f(z)}{z}>0,$ $z\in{\mathbb{D}}.$

## References

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