Bethe–Heitler Cascades and Hard Gamma-Ray Spectra in Flaring TeV Blazars: 1ES 0414009 and 1ES 1959650

Abstract

In this work, we present updated models of the spectral energy distributions (SEDs) for two high-frequency-peaked BL Lac objects (HBLs), that is, 1ES 0414+009 and 1ES 1959+650. The hard gamma-ray spectra observed during their flaring states suggest the presence of an additional emission component beyond the standard synchrotron self-Compton (SSC) scenario. We explore the possibility that this hard gamma-ray emission arises from inverse Compton (IC) scattering by Bethe–Heitler pairs produced along the line of sight, pointing to a more complex high-energy emission mechanism in these sources.

1. Introduction

Active galactic nuclei (AGN) constitute a prominent population of energetic extragalactic sources, with blazars representing an extreme subclass of radio-loud AGN. Blazars, a subclass of jetted AGN that includes flat-spectrum radio quasars (FSRQs) and BL Lac objects (BLOs), exhibit non-thermal continuum emission originating from relativistic jets aligned close to our line of sight [1]. Moreover, they are characterized by distinctive characteristics such as rapid variability, high luminosity, and strong polarization [2,3,4,5,6]. FSRQs are distinguished by their strong, broad emission lines, whereas BLOs exhibit weak or nearly absent emission lines [7]. Their multiwavelength spectral energy distributions (SEDs) typically feature a characteristic double-hump structure [3,4,8,9,10,11]. The low-energy hump, spanning from the radio to the X-ray bands, is well explained by synchrotron radiation. However, the origin of the high-energy hump, which falls within the MeV-TeV energy range, remains an open question. Two theoretical models describe the high-energy photon emission in blazars: the leptonic model and the hadronic model. In the leptonic model scenarios, the high-energy component may result from inverse Compton (IC) scattering of relativistic electrons, either on synchrotron photons (Synchrotron Self-Compton, SSC [12,13]) or on external photon fields (External Compton, EC [4,14,15]), such as the broad-line region (BLR), accretion disk, or the external torus. In contrast, the hadronic model suggests that high-energy gamma-rays are generated either through proton synchrotron radiation in sufficiently strong magnetic fields [16,17,18] or via meson and lepton production in cascades triggered by proton-proton or proton-photon interactions [19,20,21,22].

The position of the first peak in SEDs, ${\nu }_p^S$ (synchrotron peak frequency), classifies the sources as low-frequency peaked BL Lacs (LBLs; e.g., ${\nu }_p^S$ < 10¹⁴ Hz), intermediate-frequency peaked BL Lacs (IBLs; e.g., 10¹⁴ Hz < ${\nu }_p^S$ < 10¹⁵ Hz), and high-frequency peaked BL Lacs (HBLs; e.g., ${\nu }_p^S$ > 10¹⁵ Hz) [8]. Several studies [12,23,24] have shown that the SEDs of BL Lac objects, particularly HBLs, are well described by a pure SSC model. The formation of relativistic jets in AGN remains an open question, with various models proposed to explain their origin. Two of the most well-established theories are the Blandford-Znajek mechanism [25], in which the jet derives energy from the black hole’s rotation, and the Blandford-Payne mechanism [26], where the jet primarily extracts rotational energy from the accretion disk. In both scenarios, magnetic fields play a crucial role in channeling energy from the black hole or disk into the jet [27]. Initially, the jet’s energy is primarily carried by Poynting flux, which progressively converts into the kinetic energy of the plasma as the flow accelerates [28]. Theoretical studies on energy dissipation suggest that, in this scenario, the emitting electrons and the electromagnetic field may share an equal distribution of energy [29]. Understanding the strength of the magnetic field within the jet is essential for unraveling its formation and energy distribution. By analyzing the frequency-dependent position of the optically thick jet core [30], Zamaninasab et al. (2014) [31] found that the jet’s magnetic flux on parsec scales correlates with the power of the corresponding accretion current, aligning with predictions from magnetohydrodynamics. Another reliable approach for estimating the magnetic field in the innermost emission region of the jet is through modeling the SEDs of blazars [3,32,33,34].

Comparing multiwavelength emission from blazars with numerical models that simulate radiative emission and transfer under specific assumptions about particle content and emission region characteristics is a key method for investigating the source’s microphysics and physical parameters. Data from the very high-energy (VHE) segment of the SED is essential for constraining model parameters of high-frequency peaked BL Lac objects, which emit a significant portion of their radiation in this energy range. Additionally, VHE observations are valuable for softer sources, as they probe the extreme limits of particle acceleration and are particularly sensitive to photon-photon absorption within internal or external radiation fields [35]. An essential element for understanding the high-energy gamma-ray emission is the Bethe-Heitler pair production (also called the proton-photon ($p\gamma$) pair production process), which occurs when relativistic protons interact with soft photons (ambient synchrotron or external photons), leading to the production of electron-positron pairs [36,37]. The generated electron-positron pairs then undergo synchrotron and IC processes, contributing to the broadband gamma-ray emission observed from blazars. It is worth mentioning that this process is typically subdominant compared to proton synchrotron or pion decay (which predict correlated neutrino and gamma-ray emissions) mechanisms in hadronic models but can still play a role in shaping the observed SED of blazars.

In this work, we investigate the SED of two blazars detected at TeV energies: 1ES 0414+009 and 1ES 1959+650. These high-energy sources provide valuable insights into particle acceleration mechanisms and emission processes. By modeling their multiwavelength emission, we aim to constrain the physical properties of their jets and explore the contribution of leptonic and hadronic processes to their observed radiation. In Section 2, we introduce our sample. In Section 3 and Section 4, we present the fitting tools and the fitting process. Section 5 describes the results and provides a discussion. Finally, we present the conclusion of this work in Section 5. Throughout the paper, we assume the Hubble constant $H_0$ = 67.77 km s⁻¹ Mpc⁻¹, the matter energy density ${\Omega }_M$ = 0.27, the radiation energy density ${\Omega }_r$ = 0, and the dimensionless cosmological constant ${\Omega }_{\Lambda }$ = 0.73.

2. TeV Blazars: 1ES 0414+009 and 1ES 1959+650

In this section, we are going to present a brief description of the two TeV blazars studied in this work and the set of data used in our analysis.

2.1. 1ES 0414+009

The BL Lac object 1ES 0414+009 was first detected by the HEAO 1 satellite [38] in the 0.9–13.3 keV energy range and later identified in X-ray images from the Einstein Observatory [39]. Situated at a redshift of z = 0.287 [40], it is powered by a supermassive black hole (SMBH) with a mass of approximately 2 × 10⁹ M_⊙ [41]. According to the scheme proposed by [42], 1ES 0414+009 belongs to the class of HBLs, which exhibit a synchrotron-emission peak at UV/soft X-ray frequencies. In such sources, X-ray emission is primarily dominated by synchrotron radiation. In the VHE gamma-ray domain, data from the HEGRA experiment were used to establish an upper limit on the integral flux of 1ES 0414+009, corresponding to 13.5 × 10⁻¹² cm⁻² s⁻¹ above the threshold energy of 910 GeV [43]. Costamante & Ghisellini [44] identified 1ES 0414+009 as a strong candidate for VHE emission based on its high X-ray and radio flux. Its detection became even more probable following blazar gamma-ray spectrum analyses, which suggested a low intensity of the diffuse extragalactic background light (EBL) [45].

The H.E.S.S. array of Cherenkov telescopes has detected significant VHE gamma-ray emission from 1ES 0414+009. With an average flux of ∼0.6% of the Crab Nebula flux above E > 200 GeV, this blazar is among the faintest extragalactic sources observed in the TeV range. Additionally, 1ES 0414+009 was detected by the Fermi-LAT instrument during its first 20 months of operation (2008–2010), exhibiting very faint emission in the high-energy (HE) domain as well [46]. The HE and VHE spectra of 1ES 0414+009, corrected for absorption using an EBL model close to the lower limits, exhibit a best-fit power law with an index harder than 2, classifying it as a hard-TeV BL Lac object. The overall SED is averaged over five years, though there is no strict simultaneity between Swift and H.E.S.S. observations. With this limitation, the SED properties, particularly an IC peak energy above 1–2 TeV, are challenging to explain within the framework of a pure one-zone SSC model, unless unusual parameter values are assumed [47].

The study of 1ES 0414+009 has implications beyond its characterization, extending to the broader context of AGN and their role in multi-messenger astronomy [48]. This blazar, along with others, has been proposed as a potential source of ultra-high-energy cosmic rays [49]. These investigations collectively enhance our understanding of the extreme environments surrounding supermassive black holes and their relativistic jets, contributing to advancements in high-energy astrophysics [50,51].

2.2. 1ES 1959+650

The object 1ES 1959+650 [52], with a redshift of z = 0.047, was classified as a BL Lac in 1993 using a specialized radio/optical/X-ray technique [53]. Its first detection at TeV energies was reported by the Utah Seven Telescope Array collaboration during the 1998 observational season [54], revealing an excess with a statistical significance of 3.9$\sigma$ above 600 GeV after 57 h of observation. In May 2002, 1ES 1959+650 experienced a powerful TeV outburst, observed by the VERITAS [55], HEGRA [56] collaborations. Significant flux variations were recorded, reaching levels up to three times the Crab Nebula flux [57].

As a HBL source, 1ES 1959+650 is generally faint in the Fermi-LAT energy range (20 MeV–300 GeV) compared to nearby LBLs and exhibits weaker variability in this band than in the X-ray and VHE ranges [58]. Notably, the 0.3–10 GeV flux did not display significant long-term flares, with the photon flux derived from two-week binned data rarely exceeding 4 × 10⁻⁸ photons cm⁻² s⁻¹ [59,60,61]. However, the source remained mostly above this threshold between August 2015 and August 2016, during which two strong, long-term HE flares were observed [62,63].

During intense X-ray flares in 2016–2017, the source exhibited very hard X-ray spectra, with the 0.3–300 GeV photon index also remaining hard during the same periods [63]. While reproducing such hard gamma-ray spectra can be challenging for standard leptonic models, certain hadronic scenarios may offer a more natural explanation under specific conditions [64]. For example, the proton blazar model introduced by Mannheim [20] predicts X-ray spectra with photon indices in the range 1.5–1.7 and can account for uncorrelated X-ray–TeV variability, a feature reported for this source [58]. Nonetheless, leptonic models remain widely favored in the literature (e.g., [12]), and further multiwavelength studies are needed to distinguish between competing scenarios.

3. Multi-Wavelength SED Fitting of Jetted AGNs

Modeling the SEDs of jetted AGN across the electromagnetic spectrum, from radio to gamma-rays, is essential for understanding the physical mechanisms governing relativistic jets. Through multiwavelength SED fitting, we can investigate particle acceleration, energy dissipation, and radiative processes responsible for the observed emission.

3.1. SED Leptonic Modeling

We employed the JetSeT v1.3.0 software package [65,66,67] for leptonic SED modeling. JetSeT is an open-source C/Python framework designed to simulate radiative and particle acceleration processes in relativistic jets and galactic sources, both beamed and unbeamed. It supports various leptonic emission mechanisms, including synchrotron radiation, SSC, and external Compton (EC) scattering of photons from the accretion disk, BLR, dusty torus (DT), and the cosmic microwave background (CMB). The framework also includes $\gamma \gamma$ absorption based on established EBL models [68,69,70].

To ensure consistency across sources, we adopted a one-zone synchrotron+SSC scenario, where the emission originates from a spherical region of radius $R=c{t}_{\mathrm{var}}\delta /(1+z)$ [71], with $t_{\mathrm{var}}$ set to one day [72,73]. The region moves relativistically with a Doppler factor $\delta ={[\Gamma (1-\beta cos\theta )]}^{-1}$, where $\Gamma$ is the bulk Lorentz factor and $\theta$ the viewing angle. Electrons are assumed to follow a broken power-law distribution [12,23,28]:

$$N\left(\gamma \right)=N_0\left\{\begin{array}{cc}{\gamma }^{-p_1},\hfill & {\gamma }_{\min }\le \gamma \le {\gamma }_{\mathrm{b}},\hfill \\ {\gamma }_b^{p_2-p_1}{\gamma }^{-p_2},\hfill & {\gamma }_{\mathrm{b}}<\gamma \le {\gamma }_{\max },\hfill \end{array}\right.$$

(1)

where ${\gamma }_{\min }$, ${\gamma }_{\mathrm{b}}$, and ${\gamma }_{\max }$ are the minimum, break, and maximum electron Lorentz factors, and $N_0$ is the normalization constant.

JetSeT employs a two-stage fitting procedure. First, a phenomenological characterization of the SED is performed using the SEDShape module, which applies power-law and log-parabolic fits to binned data spanning the radio to TeV range. This step provides initial constraints on the synchrotron and SSC components, helping to define parameter boundaries. Next, the ObsConstrain module is used to derive input parameters for the physical modeling, where we adopt the broken power-law electron distribution defined in Equation (1). A Bayesian approach is adopted for the final model fitting, incorporating prior constraints to ensure that all parameters remain within physically meaningful bounds.

Model parameters are implemented as Astropy quantities, allowing seamless integration with other Python-based astrophysical tools. JetSeT supports both frequentist and Bayesian fitting approaches. For frequentist optimization, plugins are available for iminuit [74] and SciPy’s bounded least squares method [75]. Bayesian inference is carried out using a Markov chain Monte Carlo (MCMC) sampler through integration with the emcee package [76]. In all cases, the redshift is fixed, and we assume a cold proton to relativistic electron ratio of 0.1 [77].

3.2. SED Lepto-Hadronic Modeling

To explore scenarios involving hadronic contributions, we used the AM³ (astrophysical multi-messenger modeling) framework [78], an open-source software package designed to simulate time-dependent lepto-hadronic interactions in astrophysical sources. AM³ computes the coupled evolution of photon, electron, positron, proton, neutron, and neutrino populations, along with intermediate products, within an isotropic magnetic field. It includes non-linear processes such as electromagnetic cascades and secondary particle feedback, providing a self-consistent description of particle interactions.

The emission region is modeled as a spherical blob of radius $R^{\prime }$ moving with bulk Lorentz factor $\Gamma$ along the jet. Primary electrons and protons are injected isotropically into this region. Electrons follow a broken power-law energy distribution, while protons follow a single power-law extending up to ${\gamma }_{\mathrm{p},\max }^{\prime }$. These high-energy protons interact with ambient photon fields following the framework of Hümmer et al. (2010) [79], producing charged and neutral pions. The decay of pions gives rise to secondary gamma-rays, neutrinos, electrons, and positrons, which in turn participate in electromagnetic cascades.

AM³ incorporates several key processes: synchrotron emission and self-absorption, IC scattering by both electrons and protons, Bethe–Heitler pair production ($p+\gamma \to p+e^{-}+e^{+}$), photon-photon pair production and annihilation, and the evolution of secondary particles. The magnetic field $B^{\prime }$ is assumed to be randomly oriented within the blob, and its turbulence plays a significant role in shaping the resulting SED and multi-messenger signatures.

This approach allows us to assess the hadronic contribution to the SED and explore scenarios where neutrino and gamma-ray emission arise from the same physical region, providing insight into possible associations between high-energy astrophysical neutrinos and blazar flares.

4. Results and Discussion

Based on the methods presented in Section 3.1 and Section 3.2, it was possible to perform the non-thermal multi-wavelength modeling of the blazars 1ES 0414+009 and 1ES 1959+650. For this, data from space and ground-based observatories, ranging from radio to VHE gamma-rays, were extracted from the Space Science Data Center (SSDC)1 and Firmamento2. First, the multi-wavelength observations were fitted using the MCMC method in JetSeT, and the best-fitting parameters were then used as input for the lepto-hadronic modeling with AM³.

Table 1. Observatory catalogs used in the modeling of the broadband SED of the source 1ES 0414+009, listed with their respective energy ranges and references.

Observatory/Instrument	Catalog(s)	Energy/Wavelength Range	Catalog Reference
NRAO 91-m Telescope	NORTH20CM	Radio (1.4 GHz)	[80]
VLA	NVSS, CRATES, CLASSSCAT	Radio (1.4–8.4 GHz)	[81,82,83]
Parkes Telescope	PMN	Radio (4.85 GHz)	[84]
Green Bank Telescope	GBE	Radio (4.85 GHz)	[85]
EXOSAT CMA	CMA	X-ray (0.05–2.0 keV)	[86]
ROSAT	RASS, RXS2CAT, WGACAT2	X-ray (0.1–2.4 keV)	[87,88,89]
Einstein IPC	IPC	X-ray (0.15–4.5 keV)	[39,52,90]
Swift-XRT	1SWXRT, 1SXPS	X-ray (0.3–10 keV)	[91,92]
BeppoSAX	BeppoSAX Spectra	X-ray/Gamma-ray (0.1–300 keV)	[93]
NASA/IPAC Database (Multi)	NED	Multi-wavelength (Radio to Gamma-ray)	[94]
Fermi-LAT	Fermi1FGL, FERMI2FGL, FERMI3FGL, 2FHL, FERMI 4FGL-DR3, BAND P5	Gamma-Ray (50 MeV–2 TeV)	[95,96,97,98,99,100]
ARGO-YBJ	ARGO2LAC	Gamma-ray (0.3–10 TeV)	[101]
VERITAS	VERITAS	VHE Gamma-ray (85 GeV–30 TeV)	[102]
H.E.S.S.	HESS	VHE Gamma-ray (100 GeV–100 TeV)	[47]

and

Table 2. Observatory catalogs used in the modeling of the broadband SED of the source 1ES 1959+650, listed with their respective energy ranges and references.

Observatory/Instrument	Catalog(s)	Energy/Wavelength Range	Catalog Reference
2MASS Survey	2MASS	Near-Infrared (1.25–2.17 µm)	[103]
WISE	WISE	Mid-Infrared (3.4–22 µm)	[104]
Planck	PCCS2143	Microwave (30–857 GHz)	[105]
GMRT	TGSS150	Radio (150 MHz)	[106]
Green Bank Telescope (GB)	GB6, GB87	Radio (4.85 GHz)	[85,107]
VLA	CRATES, NVSS, VLASSQL	Radio (1.4–8.4 GHz)	[81,82,108]
NRAO 91-m Telescope	NORTH20	Radio (1.4 GHz)	[80]
Pan-STARRS	PanSTARRS	Optical (400–1000 nm)	[109]
Gaia	GAIA	Optical (330–1050 nm)	[110]
HST, GSC	HSTGSC	Optical (300–1000 nm)	[111]
Compiled from multiple optical surveys	DEBL	Optical (350–950 nm)	[112]
Swift-UVOT	UVOT	UV/Optical (170–650 nm)	[113]
GALEX	GALEX	UV (135–280 nm)	[114]
XMM-Newton Optical Monitor	XMMOM	UV/Optical (180–600 nm)	[115]
Owens Valley + others	OUNBLZ, OUSXB	Optical (400–800 nm), X-ray (0.3–10 keV)	[116,117]
Einstein IPC	IPCSL	X-ray (0.16–3.5 keV)	[52]
ROSAT	RASS	X-ray (0.1–2.4 keV)	[88]
Swift-XRT	2SXPS, XRTSPEC	X-ray (0.3–10 keV)	[118,119]
XMM-Newton	4XMM-DR10, XMMSL2	X-ray (0.2–12 keV)	[120,121]
Swift-BAT	BAT105m	Hard X-ray (14–195 keV)	[122]
BeppoSAX	BeppoSAX	X-ray/Gamma-ray (0.1–300 keV)	[93]
Oulu Neutron Monitor	OULC	Cosmic-ray flux (secondary particles, ∼GeV range)	[123]
Fermi-LAT	3FGL, 4FGL-DR2, 3FHL, 2FHL, 2BIGB, FMonLC	Gamma-ray (50 MeV–2 TeV)	[95,96,97,98,99]
MAGIC	MAGIC	VHE Gamma-ray (50 GeV–50 TeV)	[124]
VERITAS	VERITAS	VHE Gamma-ray (85 GeV–30 TeV)	[59]
Whipple Telescope	WHIPPLE	VHE Gamma-ray (300 GeV–10 TeV)	[125]

summarize the observatories employed in the modeling of the sources 1ES 0414+009 and 1ES 1959+650, respectively, together with the corresponding energy/wavelength ranges covered by the data collected at each facility.

4.1. SSC SED Fitting

In the fitting method, each free parameter has a specific physical boundary. We can fix specific parameters and set the fitting range of the remaining parameters to speed up the convergence of the fitting process. Given that all our samples are HBLs and we adopt the one-zone lepton model Syn+SSC, we fix the redshift and the distance of the radiation region from the central black hole, $R_{\mathrm{H}}$ - 10¹⁷ cm (default value for JetSeT). To prevent biased results, ensure that output parameters remain within physically meaningful ranges (based on the literature [102,126]), and improve convergence, we define fitting ranges for the radius of the emission region R, magnetic field strength B, ${\gamma }_{\min }$, ${\gamma }_{\mathrm{b}}$, ${\gamma }_{\max }$, and the spectral indexes (p and $p_1$). The best-fitting values are described in

Table 3. SSC model parameters for 1ES 0414+009 and 1ES 1959+650.

Symbol	Description	1ES 0414+009	1ES 1959+650
${\gamma }_{min}$	Minimum electron Lorentz factor	$6.35\times {10}^1$	$9.60\times {10}^2$
${\gamma }_{\mathrm{break}}$	Break electron Lorentz factor	$8.11\times {10}^4$	$1.96\times {10}^5$
${\gamma }_{max}$	Maximum electron Lorentz factor	$1.70\times {10}^6$	$2.40\times {10}^6$
B [G]	Magnetic field strength	$3.62\times {10}^{-1}$	$1.05\times {10}^{-1}$
R [cm]	Radius of the emitting region (blob)	$1.02\times {10}^{16}$	$3.15\times {10}^{15}$
${\theta }_{\mathrm{obs}}$ [deg]	Viewing angle	$1.61$	$2.9$
N [cm−3]	Particle number density	$45.11$	$12.47$
p	Spectral index below ${\gamma }_{\mathrm{break}}$	$2.29$	$2.41$
$p_1$	Spectral index above ${\gamma }_{\mathrm{break}}$	$4.45$	$4.37$
$\Gamma$	Bulk Lorentz factor	$15.02$	$37.25$

.

Figure 1 shows the modeling via the SSC physical process for the 1ES 0414+009 source. The first peak in the figure represents the synchrotron emission, which occurs at lower frequencies, peaking at $\sim {10}^{17}$ Hz, thus indicating that 1ES 0414+009 is a HSP BL Lac. The second peak, at $\sim {10}^{25}$ Hz, represents the emission due to IC scattering. According to the residual plot at the bottom of the figure, the model fits the data well by plotting the difference between actual and predicted values, with only one discrepancy in the TeV region of the energy spectrum.

In JetSeT, jet power can be readily estimated by fitting the spectral energy distributions. The jet kinetic power is carried by electrons, magnetic fields, and cold protons [28,71,127], and is expressed as $L_i=\pi R^2{\Gamma }^{2}c{U}_i$, where $U_i$ represents the energy density of each component (i = e, p, B). Consequently, the total jet kinetic power is given by $L_{\mathrm{kin}}=L_{\mathrm{e}}+L_{\mathrm{B}}+L_{\mathrm{p}}$. The derived energy densities and luminosities for the source 1ES 0414+009 are summarized in

Table 4. Derived physical quantities for 1ES 0414+009 and 1ES 1959+650.

Symbol	Description	1ES 0414+009	1ES 1959+650
$U_e$ [erg cm−3]	Electron energy density	$9.19\times {10}^{-3}$	$3.04\times {10}^{-2}$
$U_B$ [erg cm−3]	Magnetic energy density	$5.22\times {10}^{-3}$	$4.37\times {10}^{-4}$
$U_{\mathrm{sync}}$ [erg cm−3]	Synchrotron photon energy density	$1.22\times {10}^{-3}$	$3.54\times {10}^{-4}$
$L_{\mathrm{syn}}$ [erg s−1]	Synchrotron radiative power	$3.33\times {10}^{42}$	$3.98\times {10}^{41}$
$L_{\mathrm{SSC}}$ [erg s−1]	SSC radiative power	$5.83\times {10}^{41}$	$1.81\times {10}^{41}$
$L_{\mathrm{rad}}$ [erg s−1]	Total radiated power	$3.91\times {10}^{42}$	$5.79\times {10}^{41}$
$L_{\mathrm{kin}}$ [erg s−1]	Jet kinetic power	$2.09\times {10}^{44}$	$5.52\times {10}^{43}$

. The total radiative power is calculated as $L_{\mathrm{rad}}\simeq L_{\mathrm{syn}}+L_{\mathrm{SSC}}$, where $L_{\mathrm{syn}}$ and $L_{\mathrm{SSC}}$ are the powers emitted through synchrotron and synchrotron self-Compton processes, respectively [13].

The comparison between the parameters of the SSC model for 1ES 0414+009 obtained in this work and those reported by Aliu et al. (2012) [102] reveals important similarities despite differences in the modeling approaches and data selections. Both studies find a comparable size for the emitting region, with $R\sim {10}^{16-17}$ cm, consistent with a compact emission zone typical of high-frequency peaked BL Lac objects. In addition, the viewing angle (${\theta }_{\mathrm{obs}}\sim 1.{61}^{\circ }$) and the general SSC framework adopted for the SED modeling are similar, indicating a shared assumption of a relativistic jet closely aligned with the observer’s line of sight. Both works also conclude that purely leptonic models face challenges in fully reproducing the broadband SED, particularly in explaining the hard TeV spectra, thus motivating the consideration of more complex scenarios, such as lepto-hadronic contributions. Although there are differences in specific parameter values, such as the minimum Lorentz factor, magnetic field strength, and bulk Lorentz factor, these arise naturally from the focus of each study: Aliu et al. (2012) [102] emphasized modeling the TeV emission with extreme parameters, while our work provides a global fit to a broader, multiwavelength dataset. Overall, both studies reinforce the notion that 1ES 0414+009 exhibits characteristics requiring detailed modeling beyond the simplest SSC assumptions.

Similarly to the previous source, Figure 2 shows the modeling of 1ES 1959+650 using the SSC physical process, considering also the presence of the host galaxy component (emission peak at approximately ${10}^{15}$ Hz). As indicated by the figure, the synchrotron emission peaks at ∼${10}^{18}$ Hz, classifying the source as an extreme high-frequency peaked BL Lac (EHBL) with ${\nu }_{\mathrm{sync}}^{\mathrm{peak}}>{10}^{16}$ Hz. The second peak, located at higher energies, is attributed to IC scattering. The best-fitting parameters are summarized in Table 3. For this source, as found for most HBLs, the kinetic powers carried by electrons and protons ($L_{\mathrm{e}}$ and $L_{\mathrm{p}}$) dominate over the magnetic contribution $L_{\mathrm{B}}$ [28].

A comparison between Table 3 and Table 4 from this work and Table 5 from Tavecchio et al. (2010) [12] highlights similarities in the physical parameters derived for 1ES 1959+650. Both studies model the broadband emission using a one-zone leptonic scenario dominated by synchrotron and SSC processes, adopting similar assumptions for the jet composition and emission region structure. The magnetic field strength ($B\sim 0.1$ G) and the size of the emission region ($R\sim 3\times {10}^{15}$ cm) are consistent across the studies, suggesting comparable estimates for the energy balance within the jet. The inferred jet kinetic powers are also similar, with both analyses concluding that the power carried by particles (electrons and cold protons) largely exceeds the magnetic contribution, a common characteristic of high-frequency-peaked BL Lac objects. These similarities reinforce the model of the one-zone SSC interpretation for the quiescent state of 1ES 1959+650, as presented in both studies.

4.2. Lepto-Hadronic SED Modeling

The lepto-hadronic SED modelling was performed using the open-source AM³ software. The non-thermal lepton distribution in the source came from the JetSeT fitting process (section above), where the electron luminosity in the jet is $L_{\mathrm{e}}\sim 2.50\times {10}^{43}$ erg/s for 1ES 0414+009 and $L_{\mathrm{e}}\sim 3.93\times {10}^{43}$ erg/s for 1ES 1959+650. For the hadronic interaction process, the proton luminosity $L_{\mathrm{p}}$ was estimated to be less than or equal to the Eddington luminosity ($L_{\mathrm{p}}\le L_{\mathrm{Edd}}$) [128], where $L_{\mathrm{Edd}}=1.3\times {10}^{38}(M_{\mathrm{BH}}/M_{\odot })$ erg/s. Assuming that $M_{\mathrm{BH}}\sim 2\times {10}^9{\mathrm{M}}_{\odot }$ for 1ES 0414+009 [47], $L_{\mathrm{Edd}}\sim 2.52\times {10}^{47}$ erg/s. For 1ES 1959+650 we also assumed $L_{\mathrm{p}}\le L_{\mathrm{Edd}}$ [128], where $M_{\mathrm{BH}}\sim 3.16\times {10}^8{\mathrm{M}}_{\odot }$ [129] and the resulting Eddington luminosity is $L_{\mathrm{Edd}}\sim 3.98\times {10}^{46}$ erg/s. Although jet loading at super-Eddington rates may occur during brief episodes of flaring activity, it is unrealistic to expect such conditions to persist during extended periods of steady, quiescent emission [128]. For both sources, the injection of protons follows a simple power-law distribution with ${\gamma }_{\min }=100$, ${\gamma }_{\max }={10}^6$ and ${\alpha }_{\mathrm{p}}=1$ (spectral index).

Figure 3 and Figure 4 display the multiwavelength SED for the blazars 1ES 0414+009 and 1ES 1959+650 along with observational data from several catalogs, including BeppoSAX, VERITAS, Fermi-LAT and others, covering frequencies from radio to gamma-rays. The dark blue and red lines correspond to synchrotron emission and IC scattering of primary electrons (best fitting parameters from JetSeT). The black line represents synchrotron emission and IC scattering of protons. The light blue line represents Bethe-Heitler pair production ($p\gamma \to p{e}^{+}{e}^{-}$) from ${10}^{-3}$ eV to ${10}^{15}$ eV (in the case of 1ES 0414+009) and ${10}^{16}$ eV (in the case of 1ES 1959+650). The green line refers to the synchrotron emission and IC scattering of electron-positron pair production ($\gamma \gamma \to e^{+}{e}^{-}$), which is observed in the energy range from ${10}^{-3}$ eV to ${10}^{15}$ eV (in the case of 1ES 0414+009) and ${10}^{17}$ eV (in the case of 1ES 1959+650). The yellow line corresponds to the multi-wavelength emission from proton-photon interaction generating charged pions ($p\gamma \to {\pi }^{\pm }\to {\mu }^{\pm }\to e^{\pm }$) in the range from ${10}^{-1}$ eV to ${10}^{15}$ eV (in the case of 1ES 0414+009) and ${10}^{17}$ eV (in the case of 1ES 1959+650). The purple dotted curve illustrates the gamma-ray emission resulting from the decay of neutral pions produced in proton–photon interactions ($p\gamma \to {\pi }^0\to \gamma \gamma$) observed in the range from ${10}^{11}$ eV to ${10}^{16}$ eV (in the case of 1ES 0414+009) and ${10}^{10}$ eV to ${10}^{17}$ eV (in the case of 1ES 1959+650). The gamma-ray emission from the decay of neutral pions produced in proton–proton interactions ($pp\to {\pi }^0\to \gamma \gamma$) is illustrated by the dark blue dashed line, which is observed in the range ${10}^{11}$ eV to ${10}^{15}$ eV (in the case of 1ES 0414+009) and ${10}^9$ eV to ${10}^{17}$ eV (in the case of 1ES 1959+650).

In Figure 3, the model suggests a possible hadronic contribution within the energy range of 10⁸ to $2\times {10}^{12}$ eV. In this scenario, Bethe–Heitler pair production, arising from the interaction of ultra-high-energy protons with ambient photon fields, generates secondary electron–positron pairs that undergo IC scattering. These IC interactions upscatter low-energy ambient photons into the gamma-ray regime, producing the high-energy bump seen in the light blue curve, which matches the gamma-ray flux of $1.78\times {10}^{-11}$ erg cm⁻² s⁻¹ at $1.08\times {10}^{12}$ eV, as reported in the 2FHL catalog by Fermi-LAT [98]. Additionally, a lower energy flux of $5.2\times {10}^{-12}$ erg cm⁻² s⁻¹ at approximately 0.2 GeV, detected by Fermi-LAT (Fermi 1FGL catalog) [95], can be attributed either to synchrotron emission by secondary electrons from muon decay, originating from pion decay in proton-photon interactions, or possibly to inverse Compton scattering by electrons produced in Bethe–Heitler pair production. Both data points are highlighted with dashed-line circles in the figure. Thus, emissions from both Bethe–Heitler pairs and secondary electrons from muon decay contribute significantly to the broadband emission of the source, offering a coherent and self-consistent interpretation of the multiwavelength observations.

In Figure 4, the model indicates a potential hadronic contribution that fills the “gap” between the two characteristic broadband emission features. In this scenario, Bethe–Heitler pair production by ultra-high-energy protons accounts for the X-ray fluxes observed in this intermediate region. These X-ray emissions originate from synchrotron radiation produced by secondary electron–positron pairs generated through the Bethe–Heitler process. Additionally, the IC component, represented by the high-energy bump in the same light blue curve, contributes significantly to explaining the observational data between approximately 10¹¹ eV and 10¹⁴ eV. This energy range corresponds to VHE gamma-rays detected during flaring activity by the Whipple observatory. Notably, on 4 June 2002, the source exhibited a dramatic gamma-ray flare without a simultaneous increase in the X-ray band, marking the first clear detection of an “orphan” gamma-ray flare from a blazar [130]. The analysis of such sources underscores the necessity of adopting a lepto-hadronic framework to fully account for their high-energy emission behavior. This approach has important implications, as it suggests the potential for future neutrino detections and provides strong support for the existence of nuclear acceleration processes in these environments [131,132,133,134].

5. Conclusions

We employed the open-source softwares JetSeT and AM³ to model the SEDs of two high-frequency peaked BL Lac (HBL) sources: 1ES 0414+009 and 1ES 1959+650. We determined the best-fit model parameters by matching the predicted multiwavelength emission to publicly available observational data for both sources. Initial modeling using a purely leptonic scenario yielded fits that could not adequately reproduce the observed hard gamma-ray spectra during flaring states, particularly above 10¹¹ eV. We therefore extended our analysis to a lepto-hadronic model, incorporating proton-proton (pp) and proton-photon (p$\gamma$) interactions within the jet environment. The inclusion of the p$\gamma$ component indicates that hadronic processes may contribute significantly in the high-energy emission of these two HBLs during flaring episodes. For both sources, Bethe–Heitler pair production ($p\gamma \to p+e^{+}+e^{-}$) by ultra-high-energy protons plays a key role in shaping the high-energy emission. In 1ES 1959+650, secondary electron–positron pairs generated through this process produce synchrotron radiation that accounts for the X-ray fluxes observed in the intermediate energy range, while their IC scattering of ambient photons contributes to the VHE gamma-ray emission. In 1ES 0414+009, the VHE gamma-ray flux can likewise be partially attributed to IC scattering by Bethe–Heitler secondaries. Thus, in both cases, Bethe–Heitler pair production injects relativistic leptons into the emission region, where they radiate via synchrotron and/or IC processes. These contributions complement the emission from primary electrons and offer a coherent explanation for the broadband spectral features observed during flaring episodes. In conclusion, the lepto-hadronic framework effectively reproduces the multiwavelength observations across a broad energy spectrum for the two sources, underscoring the importance of incorporating hadronic processes to achieve a comprehensive understanding of BL Lac emission.