\section{Introduction}
The origin of the diffuse flux of astrophysical neutrinos from $\sim$10 TeV to a few PeV, detected by IceCube Observatory, is a long-standing problem in astroparticle physics \citep{Aartsen_2013, Aartsen_2014}. After almost ten years of their discovery, their sources are still concealed from the eyes of multiwavelength telescopes. Since neutrinos are produced only in hadronic processes, it is a smoking gun evidence of cosmic-ray (CR) acceleration inside the source \citep[see for eg.,][]{Aartsen_2018}. CRs consist mainly of protons and
nuclei and can interact with the ambient matter and radiation fields at the production site and/or during their propagation to Earth. Astrophysical neutrinos are produced along with high-energy photons via photohadronic ($p\gamma$) or hadronuclear ($pp$) interactions in the source emission region. The magnetic rigidity is insufficient for retaining directional signatures of cosmic rays over cosmological distances. The high-energy universe is opaque to photons beyond a few hundred TeV due to $e^+e^-$ pair production with cosmic background photons. Neutrinos are ideal messengers of cosmic-ray acceleration because they are weakly interacting, and thus unattenuated by intervening matter or radiation and undeflected by magnetic fields. Thus the discovery of the neutrino emitters will evidently unveil the sources of extragalactic cosmic rays, and even the ultrahigh-energy cosmic rays \citep[UHECRs; $E>10^{17}$ eV, see, for e.g.,][for a recent review]{Anchordoqui:2018qom, AlvesBatista:2019tlv}.

The real-time alert system at the IceCube neutrino Observatory, located in the South Pole ice, allows rapid follow-up by multiwavelength detectors to search for the source of a high-energy muon-neutrino event \citep{IceCube:2016cqr}. Neutrinos of astrophysical origin at $\gtrsim1$ TeV and at declination $\delta \gtrsim 5^\circ$ can be efficiently discriminated from the atmospheric muon contamination with an angular resolution of $<1^\circ$ for track-like events. Active Galactic Nuclei (AGNs), powered by accretion onto a supermassive black hole, are considered as prominent candidates of IceCube astrophysical neutrino signal. The $3\sigma$ detection of IC-170922A by IceCube led to the first spatial and temporal correlation of a high-energy $\nu_\mu$ event with a flaring $\gamma$-ray blazar, detected by Fermi-LAT, and followed up by several detectors in an intensive multiwavelength campaign \citep{IceCube:2018dnn, IceCube:2018cha}. Several other neutrino events are identified thereafter having positional coincidence with blazars but with lower statistical significance \citep{Franckowiak:2020qrq, Giommi:2020hbx}.

Blazars are a class of AGN with their collimated beam of outflow pointing toward the observer's line of sight (\citealt{1995PASP..107..803U}). These powerful luminous jets are relativistic in nature, extending from pc to kpc scale distances. Blazar jets can provide a suitable environment for cosmic-ray acceleration and also neutrino production \citep{IceCube:2022zbd}. The blazars with weak or no emission lines are called BL Lacs; they are usually highly variable sources. Based on the frequency of their synchrotron emission peaks ($\nu_{peak}$) in broadband SED, BL Lacs are classified into low energy-peaked (LBL) ($\nu_{peak}\approx 10^{13-14}$ Hz), intermediate energy-peaked (IBL) ($\nu_{peak}\approx 10^{15-16}$ Hz) and high energy-peaked (HBL) ($\nu_{peak}\approx 10^{17-18}$ Hz) sources. More recently, the LBLs with peak frequency above $10^{13.5}$ Hz have been named intermediate high energy-peaked BL Lacs (IHBLs) \citep{giommi2021astrophysical}. Earlier analyses have shown, however, that a maximum of 30\% of the cumulative neutrino background can originate from blazars \citep{IceCube:2016qvd}.

It is possible that along with electrons, protons are also accelerated to very high energy inside the jet. However, the direct evidence of their presence and their maximum energy can not be identified by modeling the multiwavelength electromagnetic radiation. In the lepton model, the broadband spectral energy distribution can be well explained by the synchrotron and inverse-Compton emission. It has been noted earlier that in some cases, where the leptonic model fails to explain the broadband SED, lepto-hadronic modeling is preferred (\citealt{Bottcher2013}).  In hadronic interactions, the shock-accelerated protons or heavy nuclei interact with the cold protons in the ambient medium (p-p interaction) or low-energy photons (p-$\gamma$ interaction) and produce the high-energy gamma-rays as well as neutrinos (\citep{Bottcher2013, Reimer:2018vvw}). Although both $\gamma$-rays and neutrinos maintain their directionality, due to their non-interacting nature, high-energy neutrinos are the best candidates to probe the acceleration site of cosmic rays and their maximum energy \citep{Sikora_1987, Eichler_1979}.

PKS~0735+178 is an IHBL located at a redshift of $z=0.45\pm0.06$ as determined from the detection of its host galaxy \citep{2012A&A...547A...1N}. The redshift value of 0.65 has been suggested in a more recent study by \citet{2021ATel15132....1F}, which has been used in this work. During its largest flaring state in December 2021, it was found to be associated with neutrino events detected by IceCube \citep{2021GCN.31191....1I}, Baikal \citep{2021ATel15112....1D}, Baksan \citep{2021ATel15143....1P}, and KM3NeT \citep{2022ATel15290....1F}. We have analyzed the observational data from {\it Swift-XRT}, {\it UVOT}, {\it NuSTAR} and {\it Fermi-LAT} to build the multi-wavelength spectral energy distribution (SED) during the flaring state. We initially modeled the SED with a leptonic model, which can explain the SED satisfactorily, and subsequently with a lepto-hadronic model to determine whether PKS~0735+178 could be the source of astrophysical neutrinos. An external radiation field, which may originate from the Broad Line Region (BLR), is needed to enhance the neutrino flux, without violating the luminosity budget. We have discussed the neutrino event detected by the IceCube collaboration in section 2 and the analysis of the UV, X-ray, and $\gamma$-ray data in section 3. Our results are presented in section 4, they are discussed in section 5, and the conclusions from our results are given in section 6.


\section{Neutrino event and its follow-up}
A GCN circular 31191 was posted confirming the detection of high energy neutrino candidate with a track-like event (IceCube-211208A; \citealt{2021GCN.31191....1I}). In IceCube, the detection was recorded on 2021-12-08 at 20:02:51.1 UT with a moderate probability of being of astrophysical origin. It was found that within the 90$\%$ error region, there are two gamma-ray sources as reported by the Fermi 4FGL-DR2 catalog. The blazar PKS 0735+178 (4FGL J0738.4+1539) was found to be just 0.1 degrees away from the event constructed location by the IceCube, which triggered a massive search of its counterparts in different wavebands across the globe. Soon after, many Atels were reported (\citealt{2021ATel15021....1S, 2021ATel15113....1F, 2021ATel15108....1H, 2021ATel15105....1K, 2021ATel15143....1P, 2021ATel15136....1L, 2021ATel15148....1C}). The first broadband confirmation came from the optical MASTER telescope which reported a brightening in the optical band (\citealt{2021ATel15098....1Z}) with magnitude 14.1 and soon after the source was observed in a high flux state in $\gamma$-ray (\citealt{2021ATel15099....1G}), X-ray (\citealt{2021ATel15102....1S}), and radio bands (\citealt{2021ATel15105....1K}). The simultaneous brightening in almost all the electromagnetic regime during the IceCube detection clearly suggest that this blazar could be the possible source of the neutrino.
This motivates us to do a broadband study to understand the physical mechanism responsible for the broadband emission and explore the neutrino connection with the blazar.

\section{Broadband observations and data analysis }
We collected the archival data that is available for this source during this important event and provide a detailed broadband study including the temporal and spectral properties.
\subsection*{Fermi-LAT:}
We analyzed the Pass8 Fermi-LAT dataset within energy 100 MeV to 350 GeV of the source 4FGL J0904.9-5734. We use the {\it fermipy} package developed by Wood et al.(2012) for analyzing the \textit{Fermi}-LAT data. A $10^{\circ}$ region of interest width (ROI) was selected around the source for the photons extraction. The events with zenith angle $\theta <  90^\circ$ have been rejected to avoid contamination from the earth's limb $\gamma$-ray. The filter "gtmktime"(DATA\_QUAL>0 \&\& LAT\_CONFIG==1) is used to select the good time intervals (GTIs). While analyzing the data both the front and back-type events were considered. The above analysis has been performed with the latest instrument response function (IRF) ``P8R3\_SOURCE\_V3", isotropic background model ``iso\_P8R3\_SOURCE\_V3\_v01'' and the galactic diffuse emission model ``gll\_iem\_v07''. These models can be found on Fermi Science Support Center \footnote{\url{https://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html}}.

We generated the light curves of the source by fixing the model parameters of all other sources spanning the entire 10 degrees of the region width.
The upper panel of Figure \ref{MWL} shows the weekly binned light curve. The source was analyzed from 1 Jan 2021 to 10 April 2022.
\begin{figure}
   
    \includegraphics[width=0.49\textwidth]{MWL-lc_Sep4.pdf}
    \caption{Lightcurve for PKS 0735+17. The shaded region is chosen for SED modeling. The $\gamma$-ray data are in units of 10$^{-7}$ ph cm$^{-2}$ s$^{-1}$, XRT are in 10$^{-12}$ erg cm$^{-2}$ s$^{-1}$, UVOT are in 10$^{-11}$ erg cm$^{-2}$ s$^{-1}$, and ASAS-SN are in units of mJy. }
    \label{MWL}
\end{figure}

In order to generate the spectral energy distribution (SED) of the sources for modeling, we keep the parameters of the galactic and isotropic diffuse components free and initially, the parameters of all the other sources present within a radius of 5 degrees from the source of interest are also kept free. We impose another restriction on the background sources based on their TS values. The parameters of the sources having a TS value less than 10 are kept fixed. We calculate flux upper limits in energy bins if the source TS is found to be less than 9. 

The photon index of the target was left free to vary (along with all the other parameters) but that of the other sources was kept fixed. For the background sources which satisfied the TS$\geq$10 limit, only the normalization or prefactor was left free to vary (depending on the model being Power law or LogParabola) whereas their spectral indices were kept fixed. The effect of energy dispersion was acknowledged during the analysis with the exception of 'galdiff' and 'isodiff' for which the effect of energy dispersion was ignored.

\subsection*{Swift-XRT:}
Soon after the detection of the neutrino event and the high source activity in $\gamma$-ray, the source was proposed for monitoring with \texttt{Swift} telescope in X-ray and optical-UV wavebands. We have collected and analyzed the long-term data and also focused on the period close to the neutrino event. The analysis is done following the standard Swift-XRT procedure. A source and background region of 15$"$ and 30$"$ is chosen around the source and away from the source to extract the spectrum. The ancillary response file (ARFs) corresponding to each observation is created using the tool \texttt{XRTMKARF} and the corresponding re-distribution matrix file (RMFs) is obtained from the latest version of the CALDB. The RMFs and ARFs file along with the source and background spectrum is loaded into the tool \texttt{GRPPHA} to combine and grouped them together. The final output spectrum of the \texttt{GRPPHA} is binned with 20 counts per bin to achieve the minimum of 20 counts in each bin. The final spectra are loaded into \texttt{XSPEC} and modeled with a simple unabsorbed power law. The X-ray photons in the soft energy band are accounted for the galactic correction using the galactic column density, n$_H$ = 4.5$\times$10$^{20}$ cm$^{-2}$ (\citealt{2013MNRAS.431..394W}).

\subsection*{Swift-UVOT:}
The advantage of having \texttt{Swift} telescope is that it can provide simultaneous observations in optical-UV along with the X-ray. The Swift has an ultraviolet optical telescope (UVOT) onboard with six filters namely, UVV, UVU, UVB, UVW1, UVM2, and UVW2 where the first three are optical filters and the latter three are ultraviolet. The raw data were analyzed following the standard procedure where the source and background region were selected with a circular region of radius 5$"$ and 15$"$ around the source and away from the source location, respectively. The magnitude was extracted using the tool \texttt{UVOTSOURCE} from filters and later was corrected for galactic extinction using the reddening, E(B-V) = 0.0298 (\citealt{2011ApJ...737..103S}) and extinction ratio corresponding to each wavelength, A$\lambda$/E(B-V) from \citealt{2006A&A...456..911G}. Further, the corrected magnitude was converted to fluxes using the zero point (\citealt{2011AIPC.1358..373B}) and the conversion factors (\citealt{2008MNRAS.383..627P}).

\subsection*{NuSTAR:}
Following the IceCube event the source was proposed for ToO observation in NuSTAR and two observations were done on 11 December 2021 and 13 December 2021 with $\sim$22 \& $\sim$21 ks exposure  \citep{2021ATel15113....1F}. We have analyzed these two observations using the standard NuSTAR data analysis tool \texttt{NuSTARDAS} 1.9.2 provided by \texttt{HEASOFT}. \texttt{Nupipeline} was run on both observations to produce the cleaned event file. A circular source and background region were chosen of radius 20$"$ and 40$"$, respectively. A tool called \texttt{nuproduct} was run to create the source and background spectra along with corresponding \texttt{rmf} and \texttt{arf} files. Further, the spectra were processed in \texttt{xspec}. 

\section{Results}
\subsection{Broadband Light curves}
After the IceCube alert, the source was rigorously monitored across the wavebands. We have shown the temporal behavior of the source during the event and a few weeks before and after the events to have a clear understanding of its variability.
In Figure \ref{MWL}, we present the broadband light curve during the neutrino event. We noticed that during the neutrino event, the source was in a slightly alleviated state in $\gamma$-ray where the flux rises from $\sim$2$\times$10$^{-7}$ ph cm$^{-2}$ s$^{-1}$ to $\sim$6.3$\times$10$^{-7}$ ph cm$^{-2}$ s$^{-1}$ within a day. Then it goes down again reaching the flux level $\sim$2$\times$10$^{-7}$ ph cm$^{-2}$ s$^{-1}$ within 2 days and again started rising. This confirms a short-term variability in the $\gamma$-ray light curve. Though the variability does not coincide exactly with the neutrino event (vertical red line in Figure \ref{MWL}). Similar behavior was also seen in the X-ray band where we observed that the source was in a high state during the IceCube event but the observation was missed in Swift. Just after the event, the X-ray observation was done and it shows a clear fast-flux variability. The flux goes down by a factor of three from 7.3$\times$10$^{-12}$ to 2.7$\times$10$^{-12}$ ergs cm$^{-2}$ s$^{-1}$ just within 3 days after the event and again rises to the high flux state to 6.76$\times$10$^{-12}$ ergs cm$^{-2}$ s$^{-1}$ within 3 days and then went down to a low flux state. During this variability, we noticed that the photon spectral index does not change much but we see a slight trend of the "softer-when-brighter". This trend is more clear when plotting all the observations together in Figure \ref{flux-index}. The trend is more like an FSRQ-type blazar rather than a BL Lac type. BL Lac types sources generally show the opposite trend in X-rays.

Using the 1-day binned $\gamma$-ray light curve, we have estimated the flux variability time and the size of the emission region. For that, we followed the expression,
\begin{equation}
    F(t_2) = F(t_1) \times 2^{(t_2 - t_1)/T_d}
\end{equation}
Here $F(t_1$) and $F(t_2$) are the observed fluxes at time $t_1$ and $t_2$, respectively and $T_d$ is the flux doubling/halving time scale. The fastest doubling/halving time in $\gamma$-ray was found to be 0.5 days. The value for $t_{var}$ can be given by $t_{var} = ln(2) \times T_f$ which is $\sim$0.35 days or 8.3 hours. The hour scale variability is very common in blazars suggesting a compact emitting region close to the central supermassive black hole. 
The size of the emission region (R) can be estimated by using the equation,
\begin{equation} \label{eq:3}
    R = c \delta t_{var}/(1+z),
\end{equation}
where R is estimated to be 1.8$\times$10$^{16}$cm, using the $\delta$=30 and t$_{var}$ 0.35 days.

Simultaneous to $\gamma$-ray and X-ray this source appears to be in a high state also in optical-UV. In UVOT, we do not have any optical-UV observation on the exact date of neutrino detection but the observation done after the event suggests a brightening. We also have access to the long-term ASAS-SN  optical data as shown in panel 4, which confirms the optical brightening during the flaring and the IceCube event. In the lower panel, we show the detection of high-energy photons with probability $>$99$\%$. Two high-energy photons of energy above 30 GeV were detected after the IceCube event.





\begin{figure}
\includegraphics[scale=0.35]{flux-index.pdf}

\caption{The Flux-Index variation of X-ray emission during the long period. The "softer-when-brighter" trend more resembles the FSRQ-type nature rather than a BL Lac-type source.}
\label{flux-index}
\end{figure}

\subsection{Multiwavelength SED and neutrino production}

\begin{figure*}
\includegraphics[width=0.49\textwidth]{ssc_with_redshift_correction.pdf}
\includegraphics[width=0.49\textwidth]{ssc_neu_case2.pdf}
\caption{Broadband SED obtained for only synchrotron and synchrotron self-Compton emission with no external photon field. Red data points are UVOT, blue are XRT, green and orange are NuSTAR (FPMA \& FPMB), and magenta is GeV data from Fermi-LAT. The grey data points are the archival data. The left panel shows the purely leptonic emission case. Here the different line styles (shown in the legend for the synchrotron spectrum) correspond to the time evolution of the SED over a period of 10 days. The right panel shows the SED including $p\gamma$ interactions on the steady-state leptonic SED. The red dashed curve shows the secondary cascade radiation. The final spectrum is corrected for $\gamma\gamma$ pair production inside the jet and attenuation in the EBL (gray-shaded region). The magenta line is the muon neutrino flux obtained in this model for the best-fit proton maximum energy.}
\label{fig:hadronic}
\end{figure*}
\begin{figure*}
\centering
   
\includegraphics[width=0.49\textwidth]{ext_ic_with_redshift_correction.pdf}
\includegraphics[width=0.49\textwidth]{ext_ic_hadronic.pdf}
\caption{Broadband SED modeled using synchrotron and inverse Compton emission off an external radiation field. The colors are the same as in Fig \ref{fig:hadronic}. The left panel shows the purely leptonic emission case. The different line styles (shown in the legend for the synchrotron spectrum) correspond to the time evolution of the SED over a period of 10 days. The right panel shows the SED including $p\gamma$ interactions on the steady-state leptonic SED and also on the external blackbody radiation field. The red dashed curve shows the secondary cascade radiation. The final spectrum is corrected for $\gamma\gamma$ pair production over all the available radiation fields in the jet emission region and attenuation in the EBL (gray-shaded region). The magenta line is the muon neutrino flux obtained in this model for the best-fit proton maximum energy.}
\label{fig:with-External}
\end{figure*}

Broadband SED modeling is important to unveil the physical mechanism responsible for broadband emission and the simultaneous detection of the astrophysical neutrino by IceCube. From the temporal study, it is clear that during the time of neutrino detection, the source was in an active state in $\gamma$-ray, X-ray, and optical UV. We have combined the observations from all these wavebands and produced a broadband SED which we have modeled with a possible lepto-hadronic model.

We start with a pure leptonic scenario, where a distribution of leptons was injected inside the jet. The particle distribution evolved with time and later the synchrotron and synchrotron self-Compton (SSC) emission was derived. We used a publicly available code, called \textsc{GAMERA}, which solves the time-dependent transport equation given below,
\begin{equation}
\frac{\partial N(E,t)}{\partial t}= Q(E,t)-\frac{\partial}{\partial E}(b(E,t)N(E,t))-\frac{N(E,t)}{\tau\textsubscript{esc}}
\end{equation}
where, $Q(E,t)$ is the injected lepton distribution and $b(E,t)$ corresponds to the energy loss rate by a synchrotron, SSC, and external Compton (EC) if there is an external radiation field and can be defined as $-{dE}/{dt}$\big. In the last term, we also included the constant escape time $\tau_{\rm esc} \sim R'/c$, where $R'$ denotes the size of the emitting region and $c$ is the speed of light. This code uses the `Klein-Nishina' cross-section to compute Inverse Compton emission \citep{RevModPhys.42.237}.

Since the source is active simultaneously in all the wavebands and emitting neutrinos, we fit the SED using a one-zone emission model. A log-parabola injection spectrum is assumed for electrons which gives a better fit to the optical data in our case. The functional form of the LP distribution is given below,
\begin{equation}
Q(E)=L\textsubscript{0}(E/E_0)^{-(\alpha+\beta \log(E/E_0)}
\end{equation}
where $L_{0}$ is the normalization constant, $E_{0}$ is the reference energy, and $\alpha$, $\beta$ are the spectral index and the curvature parameter. 

The cosmic ray protons accelerated inside the jet can interact with the soft photons produced in synchrotron radiation of leptons, thus undergoing various photohadronic processes. Neutrinos and high-energy photons are produced from the decay of charged and neutral pions produced in the photo-pion interactions ($p\gamma$ $\rightarrow$ $p + \pi^0$ or $n + \pi^+$). The decay of $\pi^+$ gives secondary leptons ($Q'_{e, \pi}$) and also results in a flux of $\nu_e$ and $\nu_\mu$. The high energy $\gamma$-rays are attenuated beyond TeV energies due to $\gamma\gamma$ pair production with the soft photons produced by the leptons, as a result of these interactions secondary $e^\pm$-s are produced. These secondary leptons along with those from $\pi^+$ decay and the $e^\pm$ pair produced in Bethe-Heitler (BH) interactions ($p\gamma$ $\rightarrow$ $p + e^{-}e^{+}$) generate secondary cascade emission. The spectrum of high-energy $\gamma$-rays from $\pi^\circ$ decay, $e^+$ and neutrinos from $\pi^+$ decay in the $p\gamma$ process and the spectrum of secondary $e^{-}e^{+}$ in BH process are estimated following the parametrization in  \citet{Kelner:2008ke}. 

The steady-state spectrum of secondary leptons is calculated using the method given in \cite{Boettcher:2013wxa}. At such high energies, the emission of secondary leptons is synchrotron dominated, due to the suppression of IC efficiency by virtue of the Klein-Nishina effect. The flux of escaping high-energy $\gamma$-rays from pion decays is given as
\begin{equation}
    Q'_{\gamma, \rm esc}(\epsilon_\gamma') = Q'_{\gamma,\pi}(\epsilon'_\gamma) (1-\exp(-\tau_{\gamma\gamma}))/{\tau_{\gamma\gamma}}
\end{equation}
where, $\tau_{\gamma\gamma}$ is the absorption probability per unit path length assuming an isotropic photon field, calculated using the formalism given in \cite{Gould_1967}. To increase the efficiency of neutrino production, we consider the escape timescale of protons to be much higher than the photohadronic interaction timescales. Otherwise, the Eddington luminosity budget is exceeded to obtain a substantial flux of neutrinos \citep[see, for e.g.,][]{Das:2022nyp}. Hence the energy loss due to proton escape is ignored. The relative weight of the rate of the corresponding process ($\pi$ production or BH interaction) is applied to the output fluxes. We assume a power-law distribution of protons, extending upto $E_{p, \rm max}$, given by $dN/dE_p\propto E_p^{-\alpha}$, where we consider the same injection spectral index $\alpha$ as for electrons. The normalization of the spectrum is fixed on the kinetic power required in protons to explain the observed SED. The muon neutrino flux at Earth is given as,
\begin{align}
    E_\nu^2J_\nu = \dfrac{1}{3} \dfrac{V'\delta_D^2\Gamma^2}{4\pi d_L^2}E'^2_\nu Q'_{\nu, p\gamma}
\end{align}
where $Q'_{\nu, p\gamma}$ is the total all-flavor neutrino flux produced in the comoving jet frame. The factor 1/3 approximately takes into account the effect of neutrino oscillations. The $\nu_\mu+\overline{\nu}_\mu$ event rate at IceCube in a given operation time  $\Delta T$ is calculated using the expression
\begin{equation}
    \mathcal{N}_{\nu_\mu} =\Delta T \int_{\epsilon_{\nu, min}}^{\epsilon_{\nu,max}} d\epsilon_\nu \dfrac{d\Phi_\nu}{d\epsilon_\nu} \langle A_{\rm eff} \rangle_\theta
\end{equation}
where $\langle A_{\rm eff} \rangle_\theta$ is the effective detector area averaged over the zenith angle bin concerned \citep{IceCube:2018ndw}. We use the effective area for event selection as a function of neutrino energy from \cite{Stettner:2019tok}. The all-flavor neutrino flux can be approximated as three times the muon neutrino flux.


In PKS 0735+178 IR-optical-UV and soft-X-ray emissions are expected to be produced by the synchrotron emission of  relativistic leptons in a strong magnetic field. However, the hard X-ray and $\gamma$-ray are expected to be produced by the SSC process. In our analysis, although the X-ray data corresponds to 10 days around the time of flaring, we assume a constant proton injection unchanged by the flaring activity, and hence the electromagnetic cascade and neutrino fluxes correspond to that produced in the flaring state once a steady state is reached as shown on the left panels of Fig.~\ref{fig:hadronic} and Fig.~\ref{fig:with-External}. The multiwavelength SED fits obtained by using a pure leptonic and a lepto-hadronic emission model is shown in the left and right panels of Fig.~\ref{fig:hadronic}. The X-ray flux at keV energies provides stringent constraints on the secondary cascade emission and hence, limits the neutrino flux. We vary the proton maximum energy $E'_{p,max}$ in logarithmic intervals between 0.1 and 10 PeV to find the best fit to SED and simultaneously maximize the neutrino flux. A steep cutoff in the proton spectrum is required to explain the observed neutrino event. The neutrino flux peaks at $\sim0.1$ PeV. However, the luminosity requirement, in this case, is extremely high, exceeding the Eddington limit. 



\subsection{With external photon field}

Since the source is argued to be a "masquerading blazar" that means it can also have an external radiation field such as a broad line region and the dusty molecular torus. In the modeling, we consider that the emission region is within the boundary of the broad line region and hence radiation field from BLR will dominate the emission in $\gamma$-ray and the neutrino event flux.
As argued in \citet{Ghisellini2011} the transition line between BL Lac and FSRQ type object can be given by the ratio of BLR luminosity to Eddington luminosity, $L_{\rm BLR}$/$L_{\rm Edd}$ $\sim$ 5$\times$10$^{-4}$. Considering the black hole mass as $\sim 10^9 M_{\odot}$, the Eddington luminosity is estimated as 1.26$\times$10$^{47}$ erg/s which lead to the $L_{\rm BLR}$ = 6.3$\times$10$^{43}$ erg/s. Considering the fact that a very small portion of the disk photon gets reprocessed in BLR (2$\%$) the disk luminosity can be derived as $L_{\rm disk}$ = 3.15$\times$10$^{45}$ erg/s. The BLR energy density in the co-moving frame can be estimated using the expression, 
\begin{equation}
    U'_{\rm BLR} = \frac{\Gamma^2 L_{\rm BLR}}{4 \pi R_{\rm BLR}^2 c}
\end{equation}
Where, $R_{\rm BLR}$ $\sim$ 10$^{17}$ $L_{\rm disk, 45}^{0.5}$. Using the above values of $L_{\rm BLR}$ and $L_{\rm disk}$, the size of the broad line region and BLR energy density is estimated as, $R_{\rm BLR} = 1.77\times10^{17}$ cm and $U'_{\rm BLR} = 2.0 $ erg/cm$^3$, respectively. The important thing to note here is that these values are an approximation.
We also derive the location of the emission region down the jet using an expression, $d\sim 2 c \Gamma^2$ $t_{\rm var}/(1+z) \sim 1.86\times10^{17}$ cm, which suggests that the emission region is located at the boundary of the BLR.



\begin{table}
\centering
\caption{Multiwavelength SED modeling results with the values of the best-fit parameters. The input injected electron distribution is LogParabola with reference energy 60 MeV. The Doppler factor and the Lorentz factor are fixed at 30.0 considering $\delta$ $\sim$ $\Gamma$. }
 \begin{tabular}{l c c}
 \hline \noalign{\smallskip}
Parameters & SSC & SSC+EC \\
\noalign{\smallskip}  \hline  \noalign{\smallskip}
    $\delta_D$ & 30 & 30\\
    $B'$ [Gauss] & 0.55 & 4.5 \\
    $R'_{\rm blob}$ [cm] & $1.0\times 10^{16}$ & $1.0\times 10^{16}$\\
    $\alpha$ ($e/p$ spectral index) & 1.90 & 1.90 \\
    $\beta$ (log parabola index) & 0.04 & 0.01\\
    $E_0$ [GeV] & 0.09 & 0.1\\
    $\gamma'_{e,\rm min}$& 20 & 7 \\
    $\gamma'_{e,\rm max}$& 2.5$\times$10$^{4}$ & 1.0$\times$10$^{4}$\\
    $U'_{\rm BLR}$ [erg cm$^{-3}$]& -& 2.0 \\
    $T'_{\rm BLR}$ [K] & - & $2.5\times10^{5}$ \\
    $E'_{p, \rm min}$ [TeV] & 0.01 & 0.01 \\ 
    $E'_{p, \rm max}$ [TeV] & $1.6\times10^{2}$ & $4.0\times10^{3}$ \\
   \noalign{\smallskip}  \hline  \noalign{\smallskip}
    $L_{j,e}$ [erg s$^{-1}$] & $0.36\times10^{46}$ & $0.14\times 10^{46}$ \\
    $L_{j,B}$ [erg s$^{-1}$] & $0.01\times10^{46}$ & $0.68\times10^{46}$ \\
    $L_{j,p}$ [erg s$^{-1}$] & $3.60\times10^{50}$ & $1.35\times10^{46}$\\
    


 \noalign{\smallskip} \hline   \noalign{\smallskip}
 \end{tabular}
 \label{tab:sed_param}

\end{table}


\section{Discussions}

Neutrinos from astrophysical phenomenon is a smoking-gun evidence of cosmic-ray acceleration. The identification of the sources of IceCube detected diffuse astrophysical neutrino flux is of paramount importance to unveil extreme high-energy processes. Blazars are a potential candidate, among others, for neutrino emission through hadronic processes in their relativistic jets. Several sources have been identified in spatial coincidence with IceCube muon tracks. TXS~0506+056 is the only source correlated spatially and temporally during the flaring activity in the Fermi-LAT waveband, with a $\nu_\mu$ event at the IceCube. However, the blazar PKS~0735+178 provides a testbed of one more such association. This BL Lac object lies within $90\%$ error region of one track-like event at IceCube of energy $E_\nu=172$ TeV. It can also be associated with a larger error region of a cascade event at the Baikal observatory of energy 43 TeV \citep{Dzhilkibaev_2022}, and an 18 TeV neutrino detected by KM3NeT \citep{Filippini_2022}.

In this article, we have studied the long-term $\gamma$-ray, X-ray, and UV optical data of the source. The SED reveals an active state during the IceCube detection epoch. The $\gamma$-ray and X-ray lightcurve exhibits a day-scale variability where the flux varies by a factor of $\sim3$ in the high state in comparison to the low state.
In our one-zone lepto-hadronic modeling, we consider two cases viz., with and without external photon field. The absence of spectral emission lines suggests the source to be a BL Lac. However, the SSC spectrum provides insufficient target photons for neutrino production. The neutrino flux peaks at $\approx0.1$ PeV and the corresponding flux value is $3.9\times10^{-13}$ erg cm$^{2}$ s$^{-1}$ For higher values of $E'_{p, \rm max}$, the Bethe-Heitler cascade increases near the X-ray energies, hindering increase in neutrino flux due to the restriction from Swift data. It should be noted, that the required kinetic power in protons exceeds the Eddington luminosity of the source. We have checked that for protons to interact dominantly with the synchrotron photons, $E'_{p,\rm max}$ must be $\gtrsim 10^{17}$ eV, and hence $\Gamma E'_{p, \rm max}$ corresponds to ultrahigh energies, which will lead to signatures of the extragalactic cascade at multi-TeV energies \citep{Das:2021cdf}, which is not detected for this source. Hence, for more reasonable results, PKS~0735+178 can be considered a ``masquerading BL Lac'' similar to TXS~0506+056, with hidden emission lines \citep{Padovani:2019xcv}. The external photons from BLR region are then the primary target for $p\gamma$ interactions, producing a detectable neutrino flux.

In our work, we consider an external photon field of temperature $2.5\times10^5$ K, i.e., within the visible frequency range. The X-ray data severely restricts the flux of synchrotron radiation from secondary electrons. The cascade flux in turn limits the allowed neutrino flux level. The neutrino event rate in our study is found to be 2.0695$\times$10$^{-8}$ events per second. The expected events rate during the current high state of the source is $\sim0.2$ events per 100 days. Overall, the fit to the observed SED is significantly improved compared to the SSC model. A higher neutrino flux can be achieved if the peak of the external photon spectrum is further blueshifted. However, we have limited the photon field within the optical range, as considered widely in the literature for TXS~0506+056 \citep{Keivani:2018rnh}. Also, we emphasize the fact that an updated and detailed study of the X-ray flux and a fit to the archival radio data provides limited flexibility of the jet parameters. However, it should be noted that both the SSC model and the external-Compton model require the proton spectrum to be cut off steeply at a specific energy, which is difficult to explain.

We find the X-ray -- neutrino correlation is more probable than the $\gamma$ -- neutrino correlation in the case of this source. A similar conclusion can be found for the TXS 0506+056 blazar in the literature, in which case too the pion decay cascade is sub-dominant at the VHE $\gamma$-ray domain. However, the neutrino flux upper limit estimated by IceCube for 0.5-yr or 7.5-yr averaged observation of one event like IC-170922A, is much higher than that obtained here. The IceCube detection potential is not sensitive to the flux level obtained here. Also, for the cascade event detection by Baikal, the angular uncertainty of positional correlation is poor, $\sim5.5^\circ$. In that case, one cannot ignore the possibility that other blazar sources within the $50\%$ containment zone can also be the source. However, for neutrino multiplets from this source, the estimated neutrino flux is more than an order of magnitude higher than that obtained here. Such high neutrino fluxes are extremely difficult to produce in a flare time of 21 days, and one needs to invoke a hidden sector of neutrino production, or multiple zones, or neutrino production in the vicinity of the SMBH, outside the jet region.

The broadband SED of PKS 0735+178 has been modeled in a recent study by \citet{Sahakyan2023} where they have used a lepto-hadronic model and considered the three possible scenarios namely $p\gamma$ with proton-synchrotron, SSC and with external Compton. In $p$-synchrotron the model demands a very high magnetic field of the order of 100 Gauss, which is unrealistic in blazars. Their hybrid model with SSC required a very high proton luminosity exceeding the Eddington luminosity of the source. They also considered the external photon field to explain the high energy gamma-ray and eventually for the p-$\gamma$ interactions to explain the observed neutrino event. They found a neutrino event rate of 0.067 neutrinos per 21 days. Their external photon field has a luminosity of $L_{\rm BLR}=2\times10^{43}$ erg/s, which peaks at $2\times10^{15}$ Hz (i.e. 8.3 eV). 
As mentioned earlier, in our model the external photon field peaks at the optical frequency, and the jet luminosity required is of the order of $10^{46}$ erg/sec, which is comparable to the Eddington luminosity of this source $L_{\rm Edd}=1.2\times 10^{46}(M_{\rm BH}/10^8 M_{\sun})$ erg/sec, where $M_{\sun}$ is the solar mass. The mass of the black hole $M_{BH}\sim 10^{8.8\pm 0.4} M_{\sun}$ has been assumed following \cite{Labita:2006jg}. 


 
\section{Conclusions}

The modeling of multiwavelength emission from jetted AGNs requires the knowledge of apparent jet speeds and their adiabatic evolution, the magnetic field, size and distance of emission region, etc. However, neutrino emission is a direct probe of cosmic-ray acceleration despite the degeneracy in these model parameters and depends dominantly on the available target photons for photopion production, providing important constraints to blazar properties. In the case of BL Lac objects, neutrino production is suppressed in a one-zone photohadronic model due to the lack of sufficient target photons from SSC. In this work, we have studied a recent association of $\gamma$-ray blazar PKS~0735+178 with an IceCube muon track detected on Dec 2021, without invoking any exotic physical phenomenon. We predict the resulting neutrino fluxes, aided by multiwavelength SED fit obtained in various scenarios, with and without external blackbody radiation field. The obtained flux levels are still much lower than the IceCube operation sensitivity. Resolving the uncertainty around the temperature and density of these external seed photons for photohadronic processes is a non-trivial task, and we restrict ourselves to optimal values found in the literature. Interestingly, an association between the X-ray flux and neutrino emission is irrefutable. We conclude that the blazar's contribution to the diffuse neutrino background cannot be neglected. Upcoming detectors with improved sensitivity and angular resolution will be able to pinpoint the neutrino-emitting blazars and constrain our neutrino flux predictions.

\begin{acknowledgements}
R. Prince and B.Z. are grateful for the support of the Polish Funding Agency National Science Centre, project 2017/26/A/ST9/-00756 (MAESTRO 9) and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. [951549]).
The work of S.D. was supported by JSPS KAKENHI Grant Number 20H05852. Numerical computation in this work was partly carried out at the Yukawa Institute Computer Facility.
\end{acknowledgements}





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