# Implication of photon-ALP interactions in sub-PeV neutrino source TXS
0506+056

Bhanu Prakash Pant<EMAIL_ADDRESS>Sunanda  Sarathykannan S  Reetanjali
Moharana<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Jodhpur, Karwar 342037, India.

###### Abstract

We investigate the effect of photon-ALP oscillations on the gamma-ray spectra
of TXS 0506+056 using observations by Fermi Large Area Telescope (FERMI-LAT)
and Major Atmospheric Gamma Imaging Cherenkov Telescopes (MAGIC) data. We
individually used both the quiescent and the two VHE flaring phases around
IC170922-A and additionally, the 2014 neutrino flare phase for the source to
study the ALP effect at sub-PeV energies.

##  I Introduction

Axion-like particles (ALPs) are pseudo-scalar (spin 0) bosons with very light
mass and weak coupling to standard model (SM) particles. They are proposed as
an extension of physics beyond the standard model (BSM) [1, 2], similar to QCD
axions that are proposed originally by Peccei and Quinn to solve the strong CP
problem in QCD [3, 4].

In the presence of an external magnetic field, ALPs can couple to photons via
two coupling vertices which leads to the photon-ALP oscillation effect in
which the free photon and ALP state may convert into each other. In
astrophysical environments, photon-ALP conversion drastically reduces the
absorption of very high energy (VHE) photons by extragalactic background light
(EBL) through pair production above 100 GeV. This increased transparency can
modulate and enhance the observed $\gamma$-ray spectra of the TeV photons
originating from higher redshift sources. In addition to the modification of
VHE $\gamma$-ray spectra, they are also considered as a potential dark matter
(DM) candidate and thus may account for a significant fraction of cold dark
matter in the Universe [5, 6, 7, 8, 9, 10].

Many studies have been performed in the lab, for instance CAST [11, 12], and
using observations of $\gamma$-ray spectra from VHE sources [13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40] to set stringent constraints on ALP mass and coupling
constant.

In this work, we investigate the implication of photon-ALP oscillation on the
BL Lac-type blazar TXS 0506+056 [41]. It was first discovered in 1983 as a
radio source [42] and later by space missions, like Energetic Gamma Ray
Experiment Telescope (EGRET) and Fermi Gamma-ray Space Telescope [43, 44, 45].
It is located at a redshift of $z_{0}=0.3365$ [46], which is about 1 GPc from
the Earth. It is also the first-ever known source associated with sub-PeV
neutrinos of astrophysical origin [47]. On 2017 September 22, the IceCube
Neutrino Observatory detected a very high energy $\sim$ 290 TeV muon neutrino
coinciding with the direction of TXS 0506+056 [48]. Soon, follow up
observations were performed at various energy bands like FERMI-LAT
($\gamma$-rays) [49], NuSTAR (X-rays) [50], Swift (X-rays, UV, optical) [51]
including VHE $\gamma$-ray observations by many instruments like MAGIC [52],
H.E.S.S [53], HAWC [54] and VERITAS [55]. Thus, confirming the evidence of
associated VHE $\gamma$-ray emissions from this blazar.

This paper is organized as follows. In Section II, we briefly describe the
propagation of photon-ALP beam in an external magnetic field. Section III,
describes the various magnetic field models used for the analysis. In Section
IV, we discuss the methodology used for data fitting and predicting the
expected $\gamma$-ray flux. In Section V, we describe the significance of the
ALP effect in TXS 0506+056 and its Fermi-LAT analysis. In Section VI, we
discusses the result.

##  II Photon-ALP Oscillation and Propagation in Magnetic Fields

The minimal interaction coupling $g_{a\gamma}$, between photons of energy
$E^{\prime}_{\gamma}$ and ALP, in the presence of an external magnetic field B
and electric field E has been proposed in literatures [56, 17].

A polarized, monoenergetic photon beam propagating along the
$\hat{\textbf{z}}$ direction in a cold plasma medium with homogeneous B-field
along $\hat{\textbf{y}}$ axis, have the equation of motion,

$\left(i\frac{d}{dz}+E^{{}^{\prime}}_{\gamma}+\mathcal{M}_{0}\right)\psi(z)=0\,\Big{|}_{E^{\prime}_{\gamma}\gg
m_{a}}\,,$ (1)

where $\mathcal{M}_{0}$ represents the photon-ALP mixing matrix. On further
simplifications like weak magnetic fields that neglect the QED vacuum
polarization and Faraday rotation as $E^{\prime}_{\gamma}$ is in high
energies, the mixing matrix becomes,

$\mathcal{M}_{0}=\begin{pmatrix}\Delta^{xx}&0&0\\\
0&\Delta^{yy}&\Delta^{y}_{a\gamma}\\\
0&\Delta^{y}_{a\gamma}&\Delta^{zz}_{a}\end{pmatrix}\,,$ (2)

where $\Delta^{xx}=\Delta^{yy}=-\omega^{2}_{pl}/2E$ and $\omega^{2}_{pl}$ is
the plasma frequency resulted from charge screening effect. The propagation
region of the photon-ALP beam is divided into N-sub regions. In each region,
the probability for photon survival is calculated. Considering the initial
beam state as

$\rho(0)=\frac{1}{2}\,diag.(1,1,0).$ (3)

The final photon survival probability can be written as:

$P_{\gamma\gamma}=Tr\left[(\rho_{11}+\rho_{22})T(s)\rho(0)T^{\dagger}(s))\right],$
(4)

where $\rho_{11}$ $=$ diag(1, 0, 0) and $\rho_{22}$ $=$ diag(0, 1, 0) denotes
the polarization along the x and y-axis, and $T(s)=T(s_{3})_{Gal}\times
T(s_{2})_{Ext}\times T(s_{1})_{Source}$ is the whole propagation transfer
matrix. Here, the sub-regions are: at the source, the extragalactic medium,
and the Milky Way region.

##  III Magnetic field models

In this section, we give a brief overview of the magnetic field models used in
our calculation.

###  III.1 Blazar jet region

We consider the photon-ALP oscillation at the source in the presence of blazar
jet magnetic field (BJMF). The magnetic field in the jet region of BL Lac
objects can be modelled with a poloidal (along the jet axis, $B\propto
r^{-2}$) and a toroidal (transverse to the jet axis, $B\propto r^{-1}$)
components. At distances large enough from the central black hole, the
toroidal component dominates over the poloidal and thus the latter can be
neglected. We adopt the toroidal magnetic field strength $B^{jet}(r)$ given by
[57, 58]:

$B^{jet}(r)=B^{jet}_{0}\left(\frac{r}{r_{VHE}}\right)^{\eta},$ (5)

where $B^{jet}_{0}$ is the magnetic field strength at the core and $r_{VHE}$
is the distance between VHE $\gamma$-ray emitting region and the central black
hole. We assume $r_{VHE}\sim R_{VHE}/\theta_{j}$, where $R_{VHE}$ is the blob
radius of the VHE emitting region and $\theta_{j}$ is the angle between the
jet axis and the line of sight.

Assuming equipartition between the magnetic field and particle energies, the
electron density profile $n_{el}(r)$ can be modelled as a power law given by
[59]:

$n_{el}(r)=n_{0}\left(\frac{r}{r_{VHE}}\right)^{\xi},$ (6)

where $n_{0}$ is the electron density at $r_{VHE}$. In Ref. [60], a more
realistic model is provided which takes into account the fact that electron
distribution is non-thermal in a relativistic AGN Jet.

The photon energy $E^{{}^{\prime}}_{\gamma}$ in the jet frame is related to
the lab-frame energy $E_{\gamma}$ by
$E^{{}^{\prime}}_{\gamma}=E_{\gamma}/\delta_{D}$, where
$\delta_{D}=\left[\Gamma_{L}(1-\beta_{j}^{2}cos\theta_{j})\right]^{-1}$ is the
Doppler factor with $\Gamma_{L}$ and $\beta_{j}$ as the bulk Lorentz and beta
factor, respectively. We assume at $r>1$ kpc, the BJMF strength is negligible.
Further details of the BJMF model can be found in Ref. [19, 27].

###  III.2 Inter-cluster magnetic fields

The inter-cluster medium (ICM) magnetic field, $B^{ICMF}$ can be modelled as,

$B^{ICMF}(r)=B_{0}^{ICMF}\left(\frac{n_{el}(r)}{n_{el}(r_{0})}\right)^{\eta},$
(7)

with $0.5\leq\eta\leq 1.0$, where $B_{0}^{ICMF}$ and $n_{el}(r_{0})$ are the
magnetic field strength and the electron density at the cluster center,
respectively. The electron density distribution $n_{el}(r)$ at a distance $r$
from the cluster center is

$n_{el}(r)=n_{0}^{ICMF}\left(1+\frac{r}{r_{core}}\right)^{\zeta},$ (8)

with $r_{core}$ as the core radius and $\zeta=-1$. The typical order of the
electron density and the core radius are $\sim\mathcal{O}(10^{-3})$ cm-3, and
$\sim\mathcal{O}(100)$ kpc, respectively.

In a cluster rich environment, the turbulent magnetic field is of $\sim$
$\mathcal{O}$(1)$\mu$G [61, 62, 63]. Such cluster can have a significant
effect on the conversion between photons and ALPs [21, 64].

###  III.3 Extragalactic magnetic fields

The actual strength of the extragalactic magnetic field on the cosmological
scale $\sim\mathcal{O}$(1) Mpc, is still unknown but the current accepted
limit is $\sim\mathcal{O}$(1) nG [65, 66]. In this work, we neglect this
magnetic field and consider only the absorption effect due to EBL.

The optical depth of EBL attenuation can be written as [16]:

$\displaystyle\tau(E_{\gamma},z_{0})=c\,\int_{0}^{z_{0}}\frac{dz}{(1+z)H(z)}\,\int_{E_{th}}^{\infty}dE^{BG}_{\gamma}\,\frac{dn(z)}{dE^{BG}_{\gamma}}\,$
$\displaystyle\times\,\tilde{\sigma}(E_{\gamma},E^{BG}_{\gamma})\,,$ (9)

with

$\displaystyle\tilde{\sigma}(E_{\gamma},E^{BG}_{\gamma})=\int_{-1}^{1-\frac{2(m_{e}c^{2})^{2}}{E^{BG}_{\gamma}E_{\gamma}}}\,dcos\theta\,\frac{(1-cos\theta)}{2}\,$
$\displaystyle\times\,\sigma_{\gamma\gamma}(E_{\gamma},E^{BG}_{\gamma},\theta)\,,$
(10)

and

$E_{th}=\frac{2(m_{e}c^{2})^{2}}{E_{\gamma}(1-cos\theta)},$ (11)

where $z_{0}$ is the redshift of the source, $H(z)$ is the rate of Hubble
expansion, $E_{th}$ is the threshold energy for pair production,
$\frac{dn(z)}{dE^{BG}_{\gamma}}$ is the proper number density of the EBL,
$\sigma_{\gamma\gamma}(E_{\gamma},z,E^{BG}_{\gamma})$ is the pair production
cross section, $\theta$ is the angle between the projectile and target
photons, $E_{\gamma}$ is the projectile photon energy, and $E^{BG}_{\gamma}$
is the target background photon energy.

###  III.4 Milky way region

Finally, the effect of photon-ALP oscillation in the presence of a galactic
magnetic field (GMF), having a large-scale regular and a small-scale random
component is considered. Due to the smaller coherence length than the
oscillation length, we neglect the random component in our analysis and
consider only the regular GMF component model given in [67], and the latest
model can be found in [68, 69].

##  IV Methodology

The photon beam of the blazar can be considered as the intrinsic photons
generated by the accelerated leptons or hadrons. In general, one can consider
the intrinsic spectrum to follow the superexponential cut-off power law
(SEPWL) to fit the observed data points under the null hypothesis,

$\Phi_{int}(E)=N_{0}\left(\frac{E}{E_{0}}\right)^{-\alpha}exp\left[-\left(\frac{E}{E_{cutoff}}\right)^{\beta}\right],$
(12)

where $E_{0}$ is taken to be 1 GeV, $N_{0}$, $\alpha$, $E_{cutoff}$, and
$\beta$ are treated as free parameters. It is to be noted that we also test
other forms of the intrinsic spectrum like exponential cut-off and log
parabola. We find that the SEPWL, gives the smallest value of the best-fit
$\chi^{2}$ under the null hypothesis.

We used an open-source python-based package
gammaALPs111https://gammaalps.readthedocs.io/en/latest/index.html [70] to
compute the photon-ALP oscillation probability $P^{ALP}_{\gamma\gamma}$ in
different astrophysical environments as described in Section III.

Figure 1: Final Photon survival probability for the TXS 0506+056 blazar for
six typical sets of parameters. The dot-dashed line and the solid lines
represent the photon survival probability without and with the ALP effect,
respectively. The notation used to represent the parameters are mneV $\equiv$
ma/1 neV and g11 $\equiv$ gaγ/10-11 GeV-1.

The modulated $\gamma$-ray spectrum obtained after the photon-ALP oscillation
is given by

$\Phi_{wALP}(E)=P^{ALP}_{\gamma\gamma}\Phi_{int}(E)\,.$ (13)

In order to get the final model expected $\gamma$-ray spectrum observed in the
detector, we must take into account their energy resolution. Considering the
energy dispersion function $D(E_{t},E^{i}_{\gamma},E^{j}_{\gamma})$ to be a
Gaussian with $\sigma$ being the energy resolution, the expected flux between
energy bins $E^{i}_{\gamma}$ and $E^{j}_{\gamma}$ is given by [38]

$\Phi^{exp}(E_{\gamma})=\frac{\int_{0}^{\infty}D(E_{t},E^{i}_{\gamma},E^{j}_{\gamma})\,\Phi_{wALP}(E_{t})\,dE_{t}}{E^{i}_{\gamma}-E^{j}_{\gamma}},$
(14)

where $E_{t}$ is the true energy of $\gamma$-rays. The energy resolution of
FERMI-LAT and MAGIC are taken as $15\%$
222https://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/Cicerone_Introduction/LAT_overview.html
and $16\%$ [71], respectively.

One can obtain the best-fit photon-ALP oscillation probability by fitting the
observed flux with the expected flux. We define $\chi^{2}$ as,

$\chi^{2}=\sum_{i}^{N_{bins}}\left(\frac{\Phi^{exp}_{i}(E_{\gamma})-\phi^{obs}_{i}(E_{\gamma})}{\sigma_{i}}\right)^{2}\,,$
(15)

where $\phi^{obs}_{i}$ is the observed $\gamma$-ray flux, with $\sigma_{i}$ as
the corresponding uncertainty. The best-fit $m_{a}$ and $g_{a\gamma}$ are
calculated by minimizing the $\chi^{2}$.

##  V Significance of ALP effect in TXS 0506+056

The first extra-galactic TeV-PeV neutrino source TXS 0506+056 is the best
candidate to study high-energy radiation. The follow-up observations of
IC170922-A show that the source has multi-wavelength emissions. Additionally,
the source showed an excess of $13\pm 5$ in the time window of 158 days, from
2014 September (MJD 56937.81) to 2015 March (MJD 57096.21), known as the
neutrino flare phase [72]. Recently, a $\sim$ 41 hr and $\sim 74$ hr
observation of VHE ( $>90$ GeV) events by Major Atmospheric Gamma Imaging
Cherenkov Telescopes (MAGIC) between 2017 September to 2018 December showed
three activities [73, 74]. Hence, the blazar TXS 0560+056 would be the best
probe to study the photon-ALP interaction from MeV to the sub-PeV range. We
studied the VHE activity phases as well as the neutrino-flare phase of the
source.

###  V.1 Fermi-LAT analysis of TXS 0506+056

We analysed the Fermi-LAT data for TXS 0506+056 in four phases:

  * •

VHE Flare 1 : 2017 September 04 (MJD 58000) to November 03 (MJD 58060).

  * •

VHE Quiescent : 2017 November 04 (MJD 58061) to 2018 October 10 (MJD 58422).

  * •

VHE Flare 2 : 2018 September 01 (MJD 58423) to 2019 January 02 (MJD 58485).

  * •

Neutrino Flare : 2014 September (MJD 56937.81) to 2015 March (MJD 57096.21).

We select above mentioned time period of FERMI-LAT (Pass 8) processed data
from Fermi Science Data Center (FSDC)
333https://fermi.gsfc.nasa.gov/ssc/data/access/. We choose the spacecraft
files and the instrument response functions (IRFs) of P8R3_SOURCE_V2 to match
the extracted data. We select the SOURCE event class (evclass=128 and
evtype=3) in the energy range of 100 MeV to 300 GeV. The region of interest
(ROI) is selected to be 2∘ centred on the target source. We consider all the
4FGL sources around 10∘ of the TXS 0506+056 as background sources together
with pre-processed templates of Galactic diffuse emission, gll_iem_v08.fits ,
and the extragalactic isotropic diffuse emission, iso_P8R3_SOURCE_V2_v2.fits.

The likelihood analysis, spectral energy distribution (SED), and the light
curve are obtained by using the python based fermipy package
444https://fermipy.readthedocs.io/en/latest/index.html [75].

##  VI Results & Discussions

We calculated the photon-ALP oscillation probability for TXS0506+056 following
section IV.

###  VI.1 VHE activity around IC170922-A

In this section, we discuss the results obtained for VHE Flare 1, VHE
Quiescent and VHE Flare 2 using the Fermi-LAT and MAGIC events. The VHE events
observed by MAGIC are collected from [73, 74]. We used the modelling
parameters for these phases as listed in [74]. The parameters used in the BJMF
model in gammaALPs package are listed in Table 1. Note that for this study, we
have not considered ICMF model.

Figure 1 shows the photon survival probability as a result of EBL and ALP
effect for several typical sets of mneV $\equiv$ ma/1 neV and g11 $\equiv$
gaγ/10-11 GeV-1.

Table 1: Summary of the BJMF model parameters in the quiescent and flaring states of TXS0506+056 taken from Ref. [74]. Parameter Name | Quiescent | Flaring
---|---|---
R.A.(J2000) | 05 09 25 (hh mm ss) | ”
Dec.(J2000) | +05 42 09 (dd mm ss) | ”
z | 0.337 | ”
$\theta_{view}$ [deg] | 0.8 | ”
$\delta$ | 40 | ”
$\Gamma$ | 22 | ”
$B^{Jet}_{0}$ [G] | 1 | ”
$n^{Jet}_{0}$ [cm-3] | $10^{3}$ | 520
$r_{VHE}$ [pc] | 10 | ”
$R^{{}^{\prime}}_{blob}$ [$10^{16}$ cm] | 1.1 | ”
$\eta$ | -1 | ”
$\xi$ | -2 | ”
$\gamma_{e,min}$ | 800 | ”
$\gamma_{e,max}$ | $10^{4}$ | $2\times 10^{4}$

Figure 2 shows the distribution of $\chi^{2}_{ALP}$ in the $m_{neV}-g_{11}$
parameter space for all three VHE activity phases. In Table 2, we summarize
the best-fit $\chi^{2}_{w/oALP}$ and $\chi^{2}_{ALP}$ values along with the
$m_{neV}-g_{11}$ parameters obtained under the null and ALP hypothesis. And to
summarize, we show the $\gamma$-ray flux for the best-fit $\chi^{2}$ values
along with the final photon survival probability in figure 3.

Table 2: Summary of the best-fit $\chi^{2}$ values and ALP parameters under the null and ALP effect hypothesis for all the phases. Phase | $\chi^{2}_{w/oALP}$ | $\chi^{2}_{ALP}$ | $m_{neV}$ | $g_{11}$
---|---|---|---|---
VHE Flare 1 | 4.67 | 3.43 | 17.78 | 35.48
VHE Quiescent | 2.94 | 6.12 | 11.22 | 94.41
VHE Flare 2 | 3.36 | 5.75 | 1.99 | 14.12
Neutrino Flare 2014 | 4.67 | 3.43 | 4.47 | 39.81
Figure 2: Distribution of $\chi^{2}_{ALP}$ in the $m_{neV}$-$g_{11}$ parameter
space for all three phases. The ”$\star$” symbol in black represents the best-
fit parameter point. The black horizontal line represents the upper limit set
by the CAST experiment of g${}_{a\gamma}<$ 6.6$\times$10-11 GeV-1 [76]. Figure
3: Top panel: Best-fit $\gamma$-ray spectra of TXS 0506+056 for all three VHE
phases. The dotted black and the solid magenta lines represent the spectra
under the null and ALP hypothesis. The corresponding best-fit $\chi^{2}$
values are listed in Table 2. The experimental data are from Fermi-LAT 666see
footnote 3 and MAGIC [74]. The dot-dashed red line represents the intrinsic
$\gamma$-rays and the dashed red line represents the flux expected by
considering the ALP effect (see text). The sensitivity of LHAASO for one-year
exposure is shown by green dash-dot-dot line. Bottom panel: Final photon
survival probability for best-fit $\chi^{2}_{ALP}$ parameter values under the
ALP hypothesis.

Extending the best-fit survival probability to sub-PeV energy, we can estimate
the residual photon flux of the observed neutrino’s counterpart. Assuming
IC170922-A resulted from proton-proton interaction, as calculated in [77], the
counterpart $\gamma$-rays is [78],

$\left(E^{2}_{\gamma}.\frac{dN_{\gamma}}{dE_{\gamma}}\right)=\frac{2}{3}\left(E^{2}_{\nu}.\frac{dN_{\nu}}{dE_{\nu}}\right)\,,$
(16)

where $E_{\gamma}\approx 2E_{\nu}$ is the energy of the photons produced from
$\pi^{0}$ decay. Subsequently, these VHE photons attenuate by interacting with
the synchrotron and synchrotron self-Compton (SSC) resulting from relativistic
electrons.

The fraction of VHE photons that can escape from the blob is estimated as in
[79],

$\mathcal{F}^{esc}_{\gamma\gamma}=\frac{1-\exp{(-\tau_{\gamma\gamma}(\epsilon^{\prime}_{\gamma})})}{\tau_{\gamma\gamma}}\,,$
(17)

where the optical depth due to this interaction is,

$\tau_{\gamma\gamma}(\epsilon^{\prime}_{\gamma})=R^{\prime}_{blob}\int_{\epsilon_{thr}}\sigma_{\gamma\gamma}(\epsilon^{\prime}_{\gamma},\epsilon^{\prime}_{k})\,n^{\prime}_{k}(\epsilon^{\prime}_{k})\,d\epsilon^{\prime}_{k}\,,$
(18)

with $n^{\prime}_{k}(\epsilon^{\prime}_{k})$ being the number density of the
photons of energy $\epsilon^{\prime}_{k}$ (in $m_{e}c^{2}$) in the comoving
frame,

$n^{\prime}_{k}(\epsilon^{\prime}_{k})=\frac{2D_{L}^{2}}{cR^{\prime
2}_{blob}\delta^{2}\Gamma^{2}_{k}}\,\frac{F_{k}(\epsilon_{k})}{m_{e}c^{2}\epsilon^{\prime
2}_{k}}\,,$ (19)

where $\sigma_{\gamma\gamma}$ is the pair production cross section [80],
$R^{{}^{\prime}}_{blob}$ is the radius of the blob, $D^{2}_{L}$ is the
luminosity distance of the source, $c$ is the velocity of light, $\delta$ is
the Doppler factor, $\Gamma_{k}$ is the bulk Lorentz factor, $m_{e}$ is the
mass of the electron, and $F_{k}(\epsilon_{k})$ is the photon flux in the
observer frame.

Hence, the intrinsic $\gamma$-ray spectrum can be obtained by multiplying the
escape fraction to Equation 16 and is shown by a dot-dashed line in the top
panel of Figure 3. This flux generally gets exhausted due to interaction with
the EBL. On multiplying the intrinsic flux with the photon-ALP oscillation
probability, the photon flux gets lower but not exhausted. The $\gamma$-ray
due to the ALP effect at the sub-PeV energy is shown with a dashed line (red)
line. We show also the differential sensitivity of LHAASO for one-year
exposure by dash-dot-dot line (green) for comparison [81].

###  VI.2  Neutrino Flare (2014-2015)

We also investigated the ALP effect for the neutrino flare phase of the blazar
TXS 0506+056. For the neutrino flare, there are no VHE observations, hence we
calculated the best fit oscillation probability using only the Fermi-LAT
events and following the method as in section IV. The survived $\gamma$-rays
are shown with the cross-hatched region while the intrinsic photon flux is
shown with the vertical-hatched lines in figure 4. Interestingly, the survival
photon with ALP effect comes out to be above the one-year detection
sensitivity of LHAASO [81].

Figure 4: Left panel: Best-fit $\gamma$-ray flux for the neutrino flare 2014
under the null and ALP hypothesis. The vertical-hatched region represents the
intrinsic flux and the cross-hatched region represents the flux after
considering the ALP effect. Right panel: Final photon survival probability for
best-fit $\chi^{2}_{ALP}$ parameter values under the ALP hypothesis.

###### Acknowledgements.

The authors would like to thank Science and Engineering Research Board (SERB)
Grant No. SRG/2020/001932 for the support.

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