3C 264 is one of the few FRI radio galaxies with detected TeV emission. It is a lowluminosity active galactic nucleus LLAGN and is generally associated with a radiatively inefficient accretion flow RIAF. Earlier multiwavelength studies suggest that the Xray emission originates from a jet. However the possibility that the RIAF can significantly contribute to the Xrays cannot be ruled out. In particular hard Xray emission 10 keV has never been detected making it challenging to distinguish between Xray models. Here we report a NuSTAR detection up to 25 keV from 3C 264. We also present subpixel deconvolved Chandra images to resolve jet emission down to 0.2 from the center of the unresolved Xray core. Together with a simultaneous Swift observation we have constrained the dominant hard Xray emission to be from its unresolved Xray core presumably in its quiescent state. We found evidence of a cutoff in the energy around 20 keV indicating that at least some of the Xrays from the core can be attributed to the RIAF. The Comptonization model suggests an electron temperature of about 15 keV and an optical depth ranging between 4 and 7 following the universality of coronal properties of black hole accretion. The cutoff energy or electron temperature of 3C 264 is the lowest among those of other LLAGNs. The detected hard Xray emission is at least an order of magnitude higher than that predicted by synchrotron selfCompton models introduced to explain ray and TeV emission suggesting that the synchrotron electrons might be accelerated to higher energies than previously thought.
['High energy astrophysics', 'Low-luminosity active galactic nuclei', 'Active galactic nuclei', 'Radio active galactic nuclei', 'X-ray active galactic nuclei', 'Jets', 'Astrophysical black holes', 'Galaxy jets', 'Accretion', 'X-ray astronomy', 'Relativistic jets', 'Gamma-ray sources', '739', '2033', '16', '2134', '2035', '870', '98', '601', '14', '1810', '1390', '633', 'Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Astrophysics of Galaxies']
NuSTAR Observation of the TeVdetected Radio Galaxy 3C 264 Core Emission and the Hot Accretion Flow Contribution
{'NuSTAR Observation of the TeV-Detected Radio Galaxy 3C 264: Core Emission and the Hot Accretion Flow Contribution': "Ka-Wah Wong, 1 Colin M. Steiner, 1 Allison M. Blum, 1 Dacheng Lin, 2 Rodrigo Nemmen, 3, 4 Jimmy A. Irwin, 5 and Daniel R. Wik 6  \n1 Department of Physics, SUNY Brockport, Brockport, NY 14420, USA  \n2  \nDepartment of Physics, Northeastern University, Boston, MA 02115-5000, USA 3 Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA 4 Instituto de Astronomia, Geof'ısica e Ciˆencias Atmosf'ericas, Universidade de S˜ao Paulo, S˜ao Paulo, SP 05508-090, Brazil 5 Department of Physics and Astronomy, University of Alabama, Box 870324, Tuscaloosa, AL 35487, USA 6 Department of Physics & Astronomy, The University of Utah, 115 South 1400 East, Salt Lake City, UT 84112, USA", 'ABSTRACT': "3C 264 is one of the few FRI radio galaxies with detected TeV emission. It is a low-luminosity AGN (LLAGN) and is generally associated with a radiatively inefficient accretion flow (RIAF). Earlier multiwavelength studies suggest that the X-ray emission originates from a jet. However, the possibility that the RIAF can significantly contribute to the X-rays cannot be ruled out. In particular, hard X-ray emission ≳ 10 keV has never been detected, making it challenging to distinguish between X-ray models. Here we report a NuSTAR detection up to 25 keV from 3C 264. We also present subpixel deconvolved Chandra images to resolve jet emission down to ∼ 0 . '' 2 from the center of the unresolved X-ray core. Together with a simultaneous Swift observation, we have constrained the dominant hard X-ray emission to be from its unresolved X-ray core, presumably in its quiescent state. We found evidence of a cutoff in the energy around 20 keV, indicating that at least some of the X-rays from the core can be attributed to the RIAF. The Comptonization model suggests an electron temperature of about 15 keV and an optical depth ranging between 4 and 7, following the universality of coronal properties of black hole accretion. The cutoff energy or electron temperature of 3C 264 is the lowest among those of other LLAGNs. The detected hard X-ray emission is at least an order of magnitude higher than that predicted by synchrotron self-Compton models introduced to explain γ -ray and TeV emission, suggesting that the synchrotron electrons might be accelerated to higher energies than previously thought.", '1. INTRODUCTION': "Studying active galactic nuclei (AGNs) and their relativistic outflows or jets remains one of the most vibrant areas of research in astrophysics today. It is critical not only for probing the properties of black holes but also for understanding their impact on the formation of larger-scale structures. Some AGNs are identified as very high energy (VHE) TeV sources, indicating that particles within them are accelerated to equivalent energy scales. Therefore, studying TeV-emitting AGNs provides crucial insights into the mechanisms of VHE radiation and acceleration (e.g. Madejski & Sikora 2016).  \nAmong the VHE sources discovered, blazars are the most common type of extragalactic sources, with their jets oriented closely to our line of sight, and are thought  \nto be responsible for the TeV emission. A number of radio galaxies with misaligned jets have also been identified as TeV sources (Rieger & Levinson 2018; Rani 2019; Rulten 2022). In particular, four of these belong to the FRI type (NGC 1275, 3C 264, M87, and Cen A) and seem to form a subclass of VHE sources 1 . Specifically, FRI radio galaxies typically host low-luminosity AGNs (LLAGNs; Wu et al. 2007), which are believed to accrete in a radiatively inefficient accretion flow (RIAF) or a hot mode (Narayan & Yi 1994; Yuan & Narayan 2014). Studying these TeV-emitting FRI sources will undoubtedly provide a unique scientific perspective on the study of VHE particle acceleration in AGNs, whether it be from a misaligned jet (Abdo et al. 2010) or from the hot accretion flow.  \n3C 264 is a FRI radio galaxy located at a redshift of z = 0 . 0217 (Baum et al. 1990). TeV emission was detected with VERITAS in 2018 (Mukherjee 2018). The mass of the supermassive black hole is about 2 . 6 × 10 8 M ⊙ (van den Bosch 2016). The X-ray jet has been resolved with Chandra , alongside an unresolved X-ray core (Perlman et al. 2010, hereafter, core means the central source unresolved by Chandra at the subarcsecond scale). The X-ray emission is dominated by the core with an X-ray luminosity of L 0 . 5 -10 keV = 2 × 10 42 ergs s -1 (Sun et al. 2007). Soft X-ray emission below ∼ 2 keV has been detected from the host galaxy NGC 3862, extending up to 6 '' , using Chandra (Sun et al. 2007). The coronal temperature has been measured to be 0 . 65 +0 . 29 -0 . 09 keV and the measured coronal luminosity is L 0 . 5 -2 keV = 1 . 4 × 10 40 ergs s -1 . The contribution of X-ray emission from LMXBs to the total X-ray emission is estimated to be at most 4% (Sun & Murray 2002).  \nDue to its low Eddington ratio of L bol /L Edd ≈ 7 × 10 -5 , 3C 264 is a LLAGN (Donato et al. 2004), with the black hole accreting in the RIAF mode. Donato et al. (2004) and Evans et al. (2006) found significant correlations between the X-ray core luminosity and the radio/optical luminosities, suggesting that at least some of the X-rays come from a jet, although an origin from RIAF cannot be completely ruled out. By measuring the X-ray, optical, and radio spectral slopes, Perlman et al. (2010) argue that the X-ray emission is likely due to the synchrotron process, which is consistent with a jet model. These observations suggest that jets can be associated with RIAF as expected (Yuan & Narayan 2014). However, the contribution of X-rays from the RIAF itself remains a possibility in these observations. For instance, the photon index Γ of the RIAF model, especially at very low accretion rates, can be similar to that of a jet model (Nemmen et al. 2014). Similarly, in another TeV FRI galaxy, Cen A, the source of hard X-ray emission could be attributed to the RIAF, a jet, or a combination of both (Furst et al. 2016).  \n3C 264 is distinct from the other three VHE FRI galaxies in a number of aspects. For example, Cen A and NGC 1275 have been detected with fluorescent Fe-K line, indicating that X-ray emission can be dominated by accretion flows or coronae illuminating accretion disks (e.g., Fukazawa et al. 2015), while none was detected from 3C 264 or M87, suggesting that X-rays may be dominated by jets instead (see, e.g., Wong et al. 2017, for the discussion of M87). Additionally, M87 is located in a hot gas rich environment (center of the Virgo Cluster), whereas the host galaxy of 3C 264 is offset from its host galaxy cluster (Perlman et al. 2010). Thus, comparing 3C 264 with other VHE FRI galaxies will provide  \ninsights into the impact of the accretion flow/jet and the environment on the nature of the X-ray and TeV emissions.  \n3C 264 has been observed from radio to TeV energy (Kagaya et al. 2017; Boccardi et al. 2019; Archer et al. 2020). The most favored model to explain the γ -ray and TeV emission is the jet scenario, with synchrotron emission from a jet plus the inverse Compton emission (synchrotron self-Compton, SSC). The full spectral energy distribution (SED) from radio to γ -ray appears to be well explained by the SSC model. However, prior to our study, hard X-ray emission above 10 keV had not been detected. In these SSC models, the hard X-ray region represents the transition between synchrotron and inverse Compton emissions. Therefore, modeling the full SED, including the hard X-ray spectrum, is crucial to identifying the emission origin.  \nIn this paper, we report the detection of hard X-ray emission from 3C 264 above 10 keV, using simultaneous NuSTAR and Swift observations. Additionally, we analyzed archival Chandra data to provide better constraints on the origin of the X-ray emission. We describe the X-ray observations in Section 2. In Section 3, we present the image analysis of NuSTAR data, including deconvolved Chandra images, as well as an analysis of surface brightness and hardness profiles. Section 4 describes the spectral analysis results, focusing on the soft excess discovered with Swift , the overall hard Xray emission from the AGN, and the decomposition the hard X-ray emission from the core and the jet. We discuss the origin of the X-ray emission and compare our results with SSC models in Section 5 and conclude in Section 6.  \nWeassume a flat universe with H 0 =70kms -1 Mpc -1 , Ω M =0.3, and Ω Λ =0.7. At the redshift of z = 0 . 0217 (Baum et al. 1990), this gives an angular scale of 0.439 kpc arcsec -1 and a luminosity distance of 94.5 Mpc to 3C 264. Errors are given at 1 σ confidence level unless otherwise specified.", '2. X-RAY OBSERVATIONS': "3C 264 was observed with NuSTAR on 2019 July 14, for 55ks (ObsID: 60501016002). All data were reduced using the HEASoft v6.31.1 and CALDB version 20230530. We reprocessed the data using the nupipeline script of the NuSTAR Data Analysis Software package with standard screening criteria, except that we set the filter mode using the parameter SAAMODE=optimized in order to reduce the slightly elevated background caused by the South Atlantic Anomaly.  \nA Swift snapshot was taken during the NuSTAR observation on 2019 July 15 for 1.9 ks (ObsID: 00088885001).  \nThe Swift-XRT spectrum was extracted using online tools 2 provided by the UK Swift Science Data Centre (Evans et al. 2009).  \nWe also included a deep Chandra observation taken with the ACIS-S detector on 2004 January 24 (ObsID: 4916 for 38 ks). All the data were reprocessed using the Chandra Interactive Analysis of Observations ( CIAO ) software version 4.13 and the Chandra CALDB version 4.9.6. The default subpixel event-repositioning algorithm ' EDSER ' was used. After removing the flares using the CIAO deflare script, the final cleaned exposure time came to 34 ks.", '3.1. Spatial Analysis of NuSTAR Image': "With the NuSTAR observation, hard X-rays above ∼ 10 keV are clearly detected from 3C 264 for the first time. We explored images at different energy bands to determine the highest energy that could be detected. Figure 1 shows the NuSTAR images in 3-7 keV and 1520 keV. Each image was created by combining the observations from the two detectors. Relative astrometry was corrected by matching the centroids of the 3-7 keV emission. Hard X-ray emission in 15-20 keV is clearly detected at 4 σ . In the higher energy band of 20-25 keV, the detection significance of the source is at about 2 σ . No emission is detected at higher energy.  \nTo quantify the spatial structure of the soft (3-7 keV) and hard (15-20 keV) X-ray emission, and to test if the emissions are point-like or extended, we fit the images using CIAO 's modeling and fitting package, Sherpa . More specifically, we compared the spatial model with a known bright point source, Cyg X-1, as observed with NuSTAR (ObsID: 10002003001). The data for Cyg X-1 were processed in the same way as 3C 264 with standard screening criteria. We fitted the NuSTAR soft and hard images of Cyg X-1 with a 2D Gaussian model and a 2D β -model to represent the core and the wing of the PSF, respectively. The central coordinates of the models are tied. A constant background model was also included in the fittings. The fitting region is limited to a circular region of radius ∼ 3 . ' 5 centered near the peak of the emission. This model fits the point source, Cyg X-1, very well (Figure 2). We use the best-fit 2D Gaussian model + 2D β -model as a point source template, along with a background model, to fit the images of 3C 264. The relative normalizations and all other parameters, except the central coordinates and the normalization of the 2D Gaussian model, are fixed.  \nFigure 2 shows that the point source model fits the images of 3C 264 very well. We conclude that both the soft and hard X-ray emissions are unresolved by NuSTAR . The size of the X-ray jet resolved with Chandra is ≲ 2 '' , which is much smaller than the 18 '' FWHM of the NuSTAR PSF. Thus, both the soft and hard X-ray emissions detected with NuSTAR are consistent with an origin that is similarly compact, presumably coming from the Chandra core and jet regions.  \nThe locations of the NuSTAR soft and hard X-ray peaks can be determined with about 2 . '' 5 and 10 . '' 4 of precision at 3 σ for two parameters of interest, respectively. The confidence regions are shown in the righthand panel of Figure 1. Due to the 8 '' absolute astrometric uncertainty of NuSTAR , there is an offset of about 10 '' between the Chandra and NuSTAR soft peak. In the figure, a relative astrometric correction has been made so that the peak of the NuSTAR 3-7 keV aligns with that of the Chandra emission. The locations of the NuSTAR soft and hard X-ray peaks are consistent with each other. Thus, the origin of the hard X-ray emission is consistent with that of the soft emission (i.e., core/jet emission).", '3.2. Chandra Image Deconvolution': "Using a maximum entropy deconvolution algorithm, the X-ray jet has already been resolved with Chandra with the size of the jet of about 2 '' pointing toward the northeast direct (Perlman et al. 2010).  \nTo resolve the jet structure down to subarcsecond scale, we applied the Lucy-Richardson deconvolution algorithm to the Chandra subpixel resolution images (Richardson 1972; Lucy 1974). For each energy band used, we first performed a raytracing simulation using the Chandra Ray Tracer (ChaRT) 3 . The Chandra spectrum of 3C 264, fitted as an absorbed power-law model, and the aspect solution file from the Chandra observation were used for these raytracing simulations. For each binning size of the image to be deconvoluted, a simulated event file was created by projecting the rays onto the detector plane using the CIAO simulate psf script and the MARX software 4 . This script also created an image of the PSF. Finally, the CIAO arestore script was used, along with the PSF images, to perform the deconvolution.  \nTo test the convergence of the deconvolution, we experimented with various image binning sizes (0.0625, 0.1, 0.125, 0.25, and 1 physical pixel), as well as different numbers of iterations (50, 100, and 150). All of  \n<!-- image -->  \n<!-- image -->  \nFigure 1. Smoothed 3-7keV (left-hand panel) and 15-20 keV (middle panel) NuSTAR images of 3C 264. The circle at the center of the image indicates the 30 '' radius spectral extraction region. The circle on top is the background spectral extraction region with the same radius of 30 '' . Using a larger background region gives essentially the same results. A 0.3-6 keV Chandra subpixel resolution image is shown in the right-hand panel, with one image pixel size = 0 . '' 123. The two circles indicate the 3 σ position errors of the emission peaks measured with NuSTAR in different energy bands. Relative astrometric correction has been made so that the peak of the NuSTAR 3-7 keV aligns with that of the Chandra emission (see the text). The arc feature at about 0 . '' 8 southeast from the center is an artifact of the Chandra PSF (Section 3.2).  \n<!-- image -->  \n<!-- image -->  \nFigure 2. Radial surface brightness profiles in the soft (3-7 keV, left-hand panel) and hard (15-20 keV, right-hand panel) bands. NuSTAR data of 3C 264 are shown in blue circles. The Cyg X-1 data, represented by purple circles with much smaller error bars, are normalized to match the surface brightness model of 3C 264. The green and red lines represent the point source model and the background used to fit the 3C 264 data, respectively. The total emission profiles are shown in orange. The very good fitting indicates that both the soft and hard X-ray emissions are unresolved by NuSTAR .  \n<!-- image -->  \nthese trials yielded similar results, with the same notable features appearing in each image. As a result, we selected a binning size of 0.125 physical pixels and 100 iterations as our nominal choice.  \nThe upper panels in Figure 3 show the high-resolution Chandra subpixel resolution image in 0.3-6 keV and the corresponding deconvolved image. The green region in the upper left-hand panel, created by the CIAO make psf asymmetry region script, indicates the artifact of the PSF 5 . Aside from the artifact, the X-ray emission is elongated in the direction of the jet to the northeast. Most of the emission is confined to an area of approximately 2 '' . The deconvolved image reveals more details of the X-ray jet, showcasing a very narrow linear structure. The PSF artifact shows up in the same region and has been excluded on the image.  \nThe jet is clearly one-sided with the core located to the southwest. Most of the X-ray emission originates from the core, which has a radius of ∼ 0 . '' 2. The jet is most luminous within ∼ 0 . '' 8, with the surface brightness decreasing progressively from the core. Beyond ∼ 0 . '' 8, the emission has diminished significantly and the jet appears to fade away at about 2 '' . The general spatial structure has been discussed by Perlman et al. (2010) using the maximum entropy deconvolution algorithm. However, the image deconvolved by the Lucy-Richardson method we used is noticeably sharper. For instance, our deconvolved image shows two X-ray tails beyond ∼ 0 . '' 8, one stronger to the northwest and the weaker one appears to turn west. Our deconvolved image is also sharper than that deconvolved with the Chandra HRC-I image observed in 2018 (Archer et al. 2020).  \nAll these features are remarkably similar to the HST and VLA images, with much higher angular resolutions presented in Perlman et al. (2010). This suggests that the deconvolution method is robust and that the jet emission originates from the same source across the radio to X-ray spectrum.  \nThe lower three panels in Figure 3 show the deconvolved images in three energy bands: the wide band in 0.5-6 keV (lower left-hand panel), the soft band in 0.5-1 keV (lower middle panel), and the hard band in 1-6 keV (lower right-hand panel). These figures were created for the analysis of surface brightness and hardness ratio profiles that follows. The general structures of the jets across the three bands are similar. However, the soft emission appears to be more extended, while the hard emission is narrower.", '3.3. Surface Brightness and Hardness Ratio Profiles of the Jet': "The upper left-hand panel in Figure 4 displays the background-subtracted, exposure-corrected surface brightness profiles of the core and jet within 4 '' from the core center in three energy bands (wide, soft, and hard). The source regions used are depicted in the lower lefthand panel of Figure 3. An angular width of 50 · is chosen to encompass most of the X-ray emission observed within 0 . '' 8. The position angle of the central jet axis is set at 40 . · 43 from north to east (Meyer et al. 2015). We selected a partial annulus background region with an angular width of 270 · extending from 5 '' to 10 '' in the direction opposite the jet. Vignetting correction has been applied using the corresponding exposure maps. The energy bands have been chosen to optimize the hardness ratio analysis that follows. Errors are estimated using the Gehrels prescription 6 (Gehrels 1986).  \nBeyond the unresolved core, approximately 0 . '' 2 in radius, the surface brightness rapidly decreases out to about 0 . '' 8, where the most obvious part of the jet lies. Past this point, the jet becomes quite faint and the surface brightness decreases more gradually up to about 2 . '' 4. Any further out, the emission is consistent with background.  \nThe deconvolved images from the soft and hard bands enable us to investigate the rough spectral properties at a spatial resolution higher than that achievable through spectral analysis. The latter is limited by PSF blurring. To accomplish this, we performed a hardness ratio analysis, a technique that has been widely used to characterize various X-ray astrophysical sources (see, e.g., Sarazin et al. 2001; Park et al. 2006; Wong et al. 2008; Thimmappa et al. 2022).  \nWe experimented to determine the optimal energy bands for the hardness ratio analysis, and two energy bands were chosen: the soft (0.5-1 keV) and hard (16 keV) bands. The lower limit of 0.5 keV was chosen to minimize uncertainty in the Galactic column density, which more significantly impacts soft photons. Additionally, when fitting the Chandra spectrum within the central 2 '' region, we observed residuals around 0.3 keV. This makes the interpretation of energies lower than ∼ 0.5 keV challenging. The upper limit at 6 keV was set to maximize the signal-to-noise ratio. The energy value of 1 keV, which serves as the midpoint between our selected energy bands, is chosen to ensure that the signal-to-noise ratio in each region is greater than three within a 2 '' radius.  \nFigure 3. Upper left-hand panel: 0.3-6 keV Chandra subpixel resolution image. The arc feature highlighted in the green region is an artifact of the Chandra PSF. The emission is elongated in direction of the jet to the northeast. Upper right-hand panel: Deconvolved Chandra image in 0.3-6 keV. Lower panels: Deconvolved Chandra image in the wide band (0.5-6 keV; left-hand panel), soft band (0.5-1 keV; middle panel), and hard band (1-6 keV; right-hand panel). The regions shown on the left are used for the analysis of surface brightness and hardness ratio profiles. In all panels, the color bar units are counts per image pixel, and one image pixel size = 0 . '' 0615. For each deconvolved image, the PSF artifact, located within the box region, has been excluded.  \n<!-- image -->  \n0.0  \n0.1  \n0.3  \n0.7  \n1.6  \n3.3  \n6.6  \n13.2  \n26.6  \n53.1 105.9  \n<!-- image -->  \n0  \n1  \n3  \n7  \n14  \n29  \n59  \n119  \n239  \n478  \n952  \n<!-- image -->  \n<!-- image -->  \n0.0  \n0.4  \n1.2  \n2.8  \n5.9  \n12.3  \n24.8  \n49.9  \n100.4 200.2 399.1  \n<!-- image -->  \n0.0  \n0.5  \n1.4  \n3.3  \n7.1  \n14.6  \n29.6  \n59.4 119.5  \n238.5 475.3  \nThe upper right-hand panel in Figure 4 displays the hardness ratio profile, which is derived by dividing the net photon counts in the soft band by those in the hard band for each radial region, or equivalently dividing the soft surface brightness profile by the hard surface brightness profile. The larger the hardness ratio, the softer the emission. The general trend suggests that the hardness ratio remains flat within ∼ 1 . '' 2 and softens from there to ∼ 2 '' , with an anomalous data point at 0 . '' 7 displaying  \nparticularly hard emission. This anomalous data point is due to the abrupt decrease in soft photons in that region (see the upper left-hand panel in Figure 4). The surface brightness profiles change significantly around that radius, which could introduce a large systematic uncertainty in that region. The hardness ratio beyond ∼ 1 . '' 2 seems excessively soft for jet emission, which could be due to soft emission from the galaxy or systematic  \n<!-- image -->  \n<!-- image -->  \n1  \n<!-- image -->  \nFigure 4. Upper left-hand panel: Background-subtracted, exposure-corrected surface brightness profiles of the jet in different energy bands. The error bars are at 1 σ confidence level. For clarity, the profile in 0.5-6 keV is multiplied by a factor of 10. Upper right-hand panel: Hardness ratio profile defined as the ratio of the count in the soft energy band to the count in the hard energy band (Park et al. 2006). The larger the hardness ratio, the softer the emission. Errors are calculated using standard error propagation procedures. The red data point, calculated using a large region, represents the average value across the brighter part of the jet. The jet emission appears to be softer beyond 1 . '' 2, but this could be due to soft emission from the galaxy or subject to systematic uncertainties. Bottom left-hand panel: Photon index vs. hardness ratio corresponding to the Chandra observation, assuming a power-law model absorbed by the Galactic column density. Bottom right-hand panel: Similar to the upper right-hand panel, but the hardness ratio has been converted to photon index (thin black and red crosses) using the figure in the bottom left-hand panel. The low photon index at 0 . '' 7 appears to be an outlier (see text). The photon indices determined using the spectral fitting method are also shown (thick gray and brown crosses).  \n<!-- image -->  \nuncertainties. More data is needed to confirm whether this rising trend and the anomalous point are real.  \nUpon closer inspection of the hardness ratio profile, the unresolved core within 0 . '' 2 is significantly harder than the jet region beyond that at 0 . '' 2-0 . '' 4. This does not seem to be due to the systematic error related to the PSF's dependence on energy because a larger PSF at higher energy should only make the outer regions harder, not the core. Spectral fitting also shows that the central core is harder than the jet on average, although the spa-  \nion of the spectra is limited by the PSF. In principle pileup can make the central spectrum harder, but we have checked this with a PILEUP model in XSPEC 7 and found that it can bias the photon index (Γ) by at most 0.6%, which is negligible.  \n7 The X-ray Spectral Fitting Package ( XSPEC ), https://heasarc. gsfc.nasa.gov/xanadu/xspec/  \nTo convert the hardness ratio into the photon index of the absorbed power-law model, we calculate the ratio of model-predicted rates in the soft and hard bands with different photon indices using XSPEC . Galactic absorption was assumed and the corresponding response files were used. The relationship between the photon index and the hardness ratio is shown in the lower left-hand panel of Figure 4. The converted photon index profile within the brighter part of the jet ( < 1 . '' 2) is shown as black crosses in the lower right-hand panel of Figure 4. The photon index at the < 0 . '' 2 core is Γ = 2 . 16 +0 . 03 -0 . 03 , which is significantly harder than that in the jet region of 0 . '' 2-0 . '' 4, where Γ = 2 . 41 +0 . 07 -0 . 08 . We also measured the average hardness ratio of the jet between 0 . '' 2 and 1 . '' 2, and found that the average photon index of the jet is Γ = 2 . 26 +0 . 05 -0 . 05 (red cross in the right-hand panels of Figure 4). Including the core from 0 '' to 1 . '' 2, the average photon index is Γ = 2 . 19 +0 . 03 -0 . 03 (not shown on the figure for clarity).  \nWe also created spectra in the corresponding surface brightness regions and performed fitting to obtain the best-fit photon indices (thick crosses in the lower righthand panel of Figure 4). Since the region sizes are smaller than the PSF, the spectra across regions are mixed. Nevertheless, the general trend of the photon indices obtained by spectral fitting and the magnitude of the uncertainties are consistent with those obtained by the hardness ratio method. In particular, the average spectral index of the jet between 0 . '' 2 and 1 . '' 2 determined by the two methods are almost the same, suggesting that the hardness ratio analysis is reliable.  \nNote that the innermost two regions are harder according to the spectral fitting, while only the innermost region appears harder in the hardness ratio analysis. This discrepancy might be due to spectral mixing in the spectral analysis, whereas for the hardness ratio analysis deconvolved images are used. The spectral analysis does not confirm the particularly hard region at 0 . '' 7, which could possibly be attributed to spectral mixing.", '4. SPECTRAL ANALYSIS': "We extracted NuSTAR spectra from the central region of 3C 264 for the observation and created the corresponding response files for the unresolved source (Figure 5, and see the discussion below for further details). The extraction region is circular with a 30 '' radius, which is close to the 29 '' half-power radius, centered at the NuSTAR 3-7 keV peak (Figure 1). The background cir-  \ngion with the same radius was chosen to be far enough from the center and also located on the same detector chip as the source. The latter criterion is important to minimize instrumental background variations from chip to chip (Wik et al. 2014a). During the spectral analysis with XSPEC , the two NuSTAR spectra from both detectors were joint-fitted during the spectral analysis with XSPEC . To constrain the softer emission of the spectrum, we also included the Swift data taken during the NuSTAR observation (Section 2).  \nThe unresolved NuSTAR X-ray emission is overwhelmingly dominated by the AGN activities, specifically the X-ray core and the jet, with only minimal contributions from the diffuse emission from hot gas in the galaxy, the ICM, or low-mass X-ray binaries (LMXBs).  \nTo detect fainter sources unrelated to the AGN activities, we incorporated a Chandra observation taken in 2004 (Section 2; see Wong et al. (2014, 2017) for the procedure to detect and analyze Chandra point sources). Only one source is detected in the NuSTAR extraction region with a luminosity of ∼ 10 40 erg s -1 in 0.510 keV if it is at the distance of 3C 264, which is more than two orders of magnitudes fainter than the emission from 3C 264. Sun & Murray (2002) estimated that the contribution from unresolved low-mass X-ray binaries (LMXBs) is at most 4% using the scaling relation from the LMXB X-ray-to-optical ratios in Sarazin et al. (2001). The corresponding luminosity is approximately 1 . 6 × 10 41 erg s -1 in 0.5-10 keV. To assess the impact of all these point sources, we conservatively incorporated a LMXB model with the aforementioned luminosity (Wong et al. 2017; Wik et al. 2014b). The changes in the best-fit photon index and cutoff energy of the LLAGN are negligible, being at most 0.05% and 0.4%, respectively. Thus, we ignore the contributions from these point sources in the spectral analysis.", '4.1. Soft Excess in the Swift Spectrum': "In the Swift spectrum, there is an evident excess of Xray emission around ∼ 0.7-1 keV beyond what a powerlaw model predicts. This excess can be described by an APEC model absorbed by the Galactic absorption. We fixed the redshift to that of 3C 264 and set the metallicity to 0.8 solar, following the value used by Sun et al. (2007) when fitting the X-ray spectrum of the hot gas of the host galaxy. The best-fit temperature is approximately 0.7-0.8 keV. The unabsorbed X-ray flux between 0.5 and 2 keV is about 3 × 10 -13 erg cm -2 s -1 (see Table 1 below). Assuming the distance to 3C 264, the corresponding X-ray luminosity is about 3 × 10 41 erg s -1 . To determine the significance of this excess and ascertain the necessity of the extra APEC model, we sim-  \nFigure 5. X-ray spectra with different models as listed in Table 1. POWERLAW models were used on the left-hand panels while CUTOFFPL models were included on the corresponding right-hand panels. Upper left-hand panel: The absorbed POWERLAW model of the NuSTAR spectra (black and red), taken from within a 30 '' circular region centered on the X-ray peak, was jointly fitted with the Swift spectrum (blue). This approach was used to constrain the overall hard X-ray emission originating from AGN activities (Core+Jet). The dotted lines represent the individual components ( POWERLAW + APEC ) as discussed in the text. Upper right-hand panel: Similar to the upper left-hand panel, but with the CUTOFFPL model replacing the POWERLAW model. Middle lefthand panel: Similar to the upper left-hand panel, but including the Chandra jet spectrum (cyan) modeled with POWERLAW . The flux corrected jet models for the NuSTAR and Swift spectra are shown as dotted lines. The unresolved core is also modeled as a POWERLAW (dashed lines). Middle right-hand panel: Similar to the middle left-hand panel, but with the CUTOFFPL model replacing the POWERLAW model for the unresolved core. Lower left-hand panel: Similar to the middle left-hand panel, but including the Chandra core spectrum (magenta) modeled with POWERLAW . Middle right-hand panel: Similar to the lower left-hand panel, but with the CUTOFFPL model replacing the POWERLAW model for the unresolved core.  \n<!-- image -->", 'Wong, Steiner et al.': "Table 1. Best-fit Spectral Results using Phenomenological Models  \nTable 2. Best-fit Spectral Results using Comptonization ( compPS ) Model  \nulated 1000 spectra using the single POWERLAW model and then compared the likelihood ratio for the single POWERLAW model ralative to the APEC+POWERLAW model (likelihood ratio test lrt, in XSPEC ). We found that 957 of the simulated likelihood ratios are smaller than the observed ratio ( p -value = 4.3%), strongly suggesting that the APEC+POWERLAW model is preferred.  \nHowever, such a pronounced excess was not observed in the Chandra data, which was taken 15 yr prior to the NuSTAR and Swift observations. Although Sun et al. (2007) identified hot gas in the host galaxy with a temperature roughly around 0.65 keV from the Chandra observation, the flux of the excess in the Swift data is 20 times higher than the hot gas flux they reported. Thus, it is unlikely that the soft excess detected with Swift originates from the diffuse hot gas of the host galaxy. Instead, it might be attributed to an increase in soft flux related to AGN activities, such as an additional Comptonization component (Dewangan et al. 2007), ionized reflection (Crummy et al. 2006), or complex absorption (Gierli'nski & Done 2004; Done et al. 2007). Alternatively, it could also be attributed to another unresolved transient event, such as an off-nuclear hyperluminous X-ray source (HLX) occurring during the NuSTAR and Swift observations. Fortunately, this excess is soft and sufficiently faint in comparison to the LLAGN and jet spectra, ensuring that it does not significantly influence the spectral fitting results. These results are robustly anchored by the broader energy range of the spectra. When excluding the APEC component from the fitting, the deviations in the best-fit photon index and cutoff energy are less than the statistical uncertainties. Nevertheless, the soft excess is taken into account by the absorbed APEC model, with both temperature and normalization being free parameters tied across all spectra.", '4.2. Overall Hard X-ray Emission from AGN Activities': 'To constrain the overall hard X-ray emission from the AGN activities, which includes the unresolved X-ray core and the jets, we fitted the two 3-30 keV NuSTAR spectra jointly with the 0.3-10 keV Swift spectra. All the spectra were grouped with a minimum of one count per bin and were fitted using the C -statistic in XSPEC . Errors of spectral parameters were determined by using ∆ C = 1 (68% confidence) for one parameter of interest.  \nAfter accounting for the soft excess (Section 4.1), we modeled the Swift and NuSTAR spectra using a single POWERLAW model to represent the combined emission from both the core and the jet. To investigate the potential presence of a spectral curvature, which could suggest emission from a hot accretion flow or an X-ray corona proximate to the black hole, we also employed  \na cutoff power-law ( CUTOFFPL ) model for the fit. No intrinsic absorption was found in the spectra. Thus, the same Galactic absorption model was applied to all components, which was fixed at the Galactic value of N H = 2 . 45 × 10 20 cm -2 (COLDEN program from the Chandra X-ray Center using the data from Dickey & Lockman (1990); see also, Evans et al. (2006) and Perlman et al. (2010)). All the parameters were tied together across all the spectra, with the exception of the normalizations for each spectrum, to account for the crosscalibration uncertainties between the two NuSTAR and the Swift detectors. The best-fit parameters for the overall AGN emission as well as the core and jet components (Section 4.3 below) are listed in Table 1.  \nThe overall spectrum, including the unresolved Chandra core and the jet, can be sufficiently characterized by a single POWERLAW model with a photon index of Γ = 2 . 26 +0 . 07 -0 . 07 , absorbed by the Galactic column density (upper left-hand panel in Figure 5). This is consistent with the value of 2 . 19 +0 . 03 -0 . 03 determined using the hardness ratio method described in Section 3.3. This is also consistent with the value of 2 . 24 ± 0 . 05 measured by Perlman et al. (2010). The overall unabsorbed 20-30 keV flux, which is completely dominated by the central regime of the AGN activities (core and jet), is 1 . 69 +0 . 13 -0 . 13 × 10 -13 erg cm -2 s -1 . This translates into a luminosity of 1 . 81 +0 . 14 -0 . 13 × 10 41 erg s -1 .  \nFor the absorbed CUTOFFPL model, the best-fit photon index is Γ = 1 . 93 +0 . 18 -0 . 19 (upper right-hand panel in Figure 5). The cutoff energy is E cutoff = 20 +21 -7 keV with a 3 σ lower limit of 6 keV. The unabsorbed 2030 keV flux of the overall AGN-related activities is 1 . 13 +0 . 21 -0 . 18 × 10 -13 erg cm -2 s -1 , giving a luminosity of 1 . 21 +0 . 22 -0 . 20 × 10 41 erg s -1 .  \nIn the soft X-ray band of 0.5-10 keV measured with Swift in 2019, both the POWERLAW and CUTOFFPL models estimate a luminosity of about 2 × 10 42 erg s -1 . This is similar to the luminosity of the unresolved core, 1 . 6 × 10 42 erg s -1 , observed with Chandra in 2004 (Perlman et al. 2010), and the power-law luminosity (core) of 2 . 26 × 10 42 erg s -1 observed with XMM-Newton in 2001 (Donato et al. 2004). These fluxes are about three to four times lower than the flux observed during a 2018 outburst observed with Chandra HRC-I (Archer et al. 2020), and it aligns with its low state observed in 2004 and in the preceding years (Boccardi et al. 2019). These observations suggest that the 3C 264 AGN, as observed by NuSTAR and Swift in 2019, is in a quiescent phase.', '4.2.1. Supporting Evidence of a Cutoff in Spectrum': 'The CUTOFFPL model offers a slightly better fit based on the face values of the C -statistics, but this improve-  \nment is not statistically significant. The likelihood ratio test ( lrt in XSPEC ) also does not suggest a statistical preference for the CUTOFFPL model. We have calculated the Akaike information criterion (AIC: Akaike 1974) to distinguish between models. The preferred model for the data is the one that minimizes the AIC (Liddle 2007; Medvedev et al. 2021). Table 1 shows that the CUTOFFPL model has a slightly smaller AIC than the POWERLAW model, suggesting that the CUTOFFPL model is slightly preferred. There is also additional evidence supporting the CUTOFFPL model, which is detailed below.  \nFirst, a significant discrepancy is observed between the normalizations of the POWERLAW model for Swift and NuSTAR , despite the Swift observation being conducted simultaneously with the NuSTAR observation. No time variation was detected in the NuSTAR light curve, suggesting that this discrepancy is not attributable to variability. The discrepancy between model normalizations, as measured with NuSTAR and Swift , is approximately 20% for the CUTOFFPL model and 45% for the POWERLAW model, respectively. While a cross-calibration uncertainty of 10-15% between different X-ray observatories is within expected limits, and a 20% difference is on the high end but not alarming (Madsen et al. 2017; Molina et al. 2019; Abdelmaguid et al. 2023), a discrepancy exceeding 40% is inconsistent with the current calibration standards 8 . The discrepancy observed in the single POWERLAW model could potentially be explained by adjusting the normalizations to account for spectral curvature.  \nMore specifically, the fitting is biased toward the NuSTAR data because there are more photon counts in the NuSTAR data. When fitting a single POWERLAW model to both the NuSTAR and Swift datasets and tying their photon indices, if there is a cutoff in energy, the POWERLAW slope will try to fit the steeper NuSTAR spectra at higher energy. Therefore, if the normalization of Swift is the same as the NuSTAR normalization, the model will overestimate the softer photons measured by Swift . To compensate for this curvature in the fitting, the Swift normalization needs to be lower than the NuSTAR normalization.  \nWe simulated the overall spectra with the absorbed CUTOFFPL + APEC model using 1000 times longer exposures for each detector. The model parameters used are listed in Table 1, except that we set all the normalizations of the CUTOFFPL model to those determined by the NuSTAR FPMA detector. The best-fit models to  \nFigure 6. Photon index fitted at various energy bins for different dataset. NuSTAR and Chandra data were fitted with absorbed POWERLAW model. For Swift data, since there is a clear soft excess, an absorbed APEC + POWERLAW model was used.  \n<!-- image -->  \nthe simulated datasets largely reproduce this normalization discrepancy, with the NuSTAR normalizations being more than 30% higher than that of Swift 9 .  \nSecond, we divided the NuSTAR spectra into different energy bands and fit the NuSTAR data separately for each band using a POWERLAW model. For comparison, we also fitted the Chandra and Swift data individually. The spectral slope steepens (i.e., the photon index increases) at higher energies (see Figure 6), indicating the presence of an energy cutoff.  \nThird, when decomposing the core and jet spectra with Chandra data (Section 4.3.2), the CUTOFFPL model is statistically favored for the core emission, suggesting that a cutoff is needed at least for the core.  \nThese points serve as supporting evidence for the CUTOFFPL model. However, to definitively confirm the presence of an energy cutoff, deeper simultaneous observations spanning a broad energy range (e.g., 0.5-40 keV) would be necessary.', '4.2.2. Comptonization Model with Reflection': 'We replace the phenomenological POWERLAW or the CUTOFFPL models with the compPS model (Poutanen & Svensson 1996), which is a physical model that includes the thermal Comptonization emission from hot  \nplasma and reflection from a standard accretion disk. Following Ursini et al. (2015), we set the geometry of the hot plasma to spherical ( GEOM = -4). The disk is assumed to have a multicolor temperature, with the inner disk temperature set to 10 eV (Beckmann et al. 2011; Ursini et al. 2015) and the inclination angle set to i = 50 · (Donato et al. 2004). The free parameters for the compPS model include the hot plasma temperature T e , the Compton parameter y , reflection strength R , and normalizations.  \nWhile the reflection strength cannot be precisely constrained, the best-fit value is effectively zero ( R = 4 × 10 -14 ), and the 1σ upper limit is R ≤ 0 . 68. Therefore, since there is no evidence of a reflection hump, we fixed R = 0 in the fitting process.  \nThe best-fit parameters for the overall AGN emission as well as the core and jet components (Section 4.3 below) are listed in Table 2. The best-fit hot plasma temperature is T e = 15 +3 -2 keV, and the Compton parameter is y = 0 . 56 +0 . 77 -0 . 09 . The optical depth, τ , is related to the Compton parameter and temperature by y = 4 τ ( kT e /m e c 2 ). Therefore, the optical depth falls within the range of τ = 3-12. Using different geometrical settings for the hot plasma (e.g., slab, cylinder, or hemisphere) gives similar results.', '4.3.1. With Chandra Jet Spectrum': "We further constrained the X-ray emission from the individual components of the unresolved X-ray core and the jet by jointly fitting their corresponding Chandra spectra 10 . We first included the Chandra jet spectrum, extracted from the partial annular regions between 0 . '' 2 and 1 . '' 2, as shown in the lower left-hand panel of Figure 3. This jet spectrum was modeled as an extra absorbed POWERLAW . The power-law indices for the NuSTAR , Swift , and Chandra spectra were tied together. Given that the Chandra PSF size scale is comparable to the jet spectral extraction region, some photon leakage from this region is anticipated. However, for the NuS-  \nTAR and Swift spectra, the spectral extraction regions are expected to encompass most of the jet emission. By counting the total number of photons in the deconvolved image within a rectangular region that includes most of the jet's photons and comparing this count to that observed in the Chandra jet spectrum, we estimated that the jet spectrum accounts for 59.6% of the total counts. We corrected for this flux loss by tying the NuSTAR and Chandra normalizations to the ratio of this fraction. The Swift and NuSTAR normalizations were also tied together. Based on the deconvolved image, we assessed that the emission beyond 1 . '' 2 contributes less than 5% of the total emission. However, for the emission inside 0 . '' 2, we lack a precise estimate of the unresolved jet's contribution to the X-ray emission. Thus, we did not apply such corrections in the spectral analysis.  \nFor the NuSTAR and Swift spectra, after accounting for the jet emission, the remaining emission predominantly originates from the unresolved Chandra core. Similar to Section 4.2, we modeled this emission with either an absorbed POWERLAW model or an absorbed CUTOFFPL model. Additionally, we kept the normalizations for NuSTAR and Swift separate (untied).  \nThe best-fit spectra with all the model components are displayed in the middle panels of Figure 5. The best-fit photon indices for the POWERLAW jet component, when jointly fitted with either the single POWERLAW or the CUTOFFPL models of the core, are 2 . 25 +0 . 04 -0 . 04 and 2 . 26 +0 . 04 -0 . 04 , respectively. For the core, the single POWERLAW model yields a photon index of 2 . 27 +0 . 10 -0 . 10 , which is essentially the same as the jet's index. The CUTOFFPL model for the core has a significantly harder photon index of 1 . 68 +0 . 31 -0 . 35 , and the cutoff energy is determined to be E cutoff = 12 +12 -4 keV. The 2030 keV flux of the core, as measured with NuSTAR , is 1 . 25 +0 . 14 -0 . 13 × 10 -13 erg cm -2 s -1 for the POWERLAW model and 0 . 66 +0 . 19 -0 . 16 × 10 -13 erg cm -2 s -1 for the CUTOFFPL model, accounting for about 74% and 58% of the overall AGN emission, respectively.  \nIn the case of the POWERLAW model for the core component, the discrepancy between the NuSTAR and Swift normalizations is larger compared to the analysis presented in Section 4.2. The NuSTAR normalizations are twice as high as those from Swift , and they are inconsistent at a level beyond 90% confidence statistically. For the CUTOFFPL model, however, the NuSTAR normalizations are about 25% higher than the Swift normalization, and they are consistent within 1 σ statistically. This further supports the hypothesis of a necessary cutoff in the power-law model for the core.  \nUsing the compPS model discussed in Section 4.2.2 for the core, the best-fit hot plasma temperature is deter-  \nmined to be T e = 14 +3 -1 keV, and the Compton parameter is y = 0 . 9 +5 . 0 -0 . 4 . The optical depth is within the range of τ = 4-58.", '4.3.2. With Chandra Jet and Core Spectra': "Finally, we extracted a Chandra core spectrum within a radius of 0 . '' 2. PSF correction was applied when generating the response file. As discussed in Section 4.3.1, the amount of emission originating from the unresolved jet within 0 . '' 2 is unknown, and therefore we did not include any corrections for this.  \nWe additionally included the Chandra core spectrum and jointly fitted it with the NuSTAR and Swift spectra. We modeled the core component using either a single POWERLAW or a CUTOFFPL model. Given the potential for variability, the spectral shapes of the Chandra and NuSTAR / Swift observations might differ. To investigate the potential spectral variation, we separately fitted the Chandra and NuSTAR spectra in the overlapping 3-8 keV energy band using an absorbed POWERLAW model. For the Swift spectrum, we extended the fitting range to 0.3-8 keV to increase the statistics and included an APEC model, as described in Section 4.1, to account for the soft excess. The photon indices from all three instruments are consistent within a 1 σ level, indicating no evidence of spectral variation. Consequently, we tied the spectral indices of the core components across all observations during the fitting process. The normalizations for NuSTAR , Swift , and Chandra were kept separate (untied) to account for calibration uncertainties or potential flux variations.  \nThe best-fit spectra with all model components are shown in the lower panels of Figure 5. The best-fit photon indices of the jet component, when jointly fitted with the single POWERLAW and the CUTOFFPL core models, are 2 . 26 +0 . 04 -0 . 04 and 2 . 25 +0 . 04 -0 . 04 , respectively. These values are essentially the same as those obtained from the analysis without the Chandra core spectrum.  \nFor the core, the single POWERLAW model yields a photon index of 2 . 05 +0 . 04 -0 . 04 , which is significantly harder than the jet's spectrum. The photon index for the CUTOFFPL model is 1 . 93 +0 . 06 -0 . 06 , which is also notably harder. The cutoff energy is determined to be E cutoff = 19 +10 -5 keV. The associated 20-30keV flux measured with NuSTAR is 1 . 65 +0 . 10 -0 . 10 × 10 -13 erg cm -2 s -1 and 0 . 81 +0 . 17 -0 . 15 × 10 -13 erg cm -2 s -1 for the POWERLAW and CUTOFFPL models, respectively. These values account for approximately 98% and 72% of the overall AGN emission, respectively.  \nIn the case of both the POWERLAW and CUTOFFPL models for the core component, while the Swift data normalizations are 40-50% lower than those of NuSTAR or  \nChandra , they remain statistically consistent at the 90% confidence level. Notably, the NuSTAR and Chandra normalizations are consistent within 1 σ . This supports the assumption that there is no significant spectral variation.  \nWe conducted a likelihood ratio test ( lrt ) in XSPEC with 1000 simulations to differentiate between the POWERLAW and CUTOFFPL models for the core component. The test results suggest that the POWERLAW model is rejected at a confidence level of 5.4%, strongly indicating the need for a cutoff in the spectrum.  \nThe compPS model for the core, as discussed in section 4.2.2, yields a best-fit hot plasma temperature of T e = 14 . 9 +1 . 7 -1 . 3 keV and a Compton parameter of y = 0 . 61 +0 . 15 -0 . 06 . The optical depth falls within the range of τ = 4-7.", '5. DISCUSSION': "We have detected hard X-ray emission ≳ 10 keV from 3C 264. However, both the soft and hard X-ray emissions are unresolved by NuSTAR . The locations of the soft and hard X-ray peaks are consistent with each other, indicating that the origin of the hard X-ray emission aligns with that of the soft emission, likely from the core and the jet resolved by Chandra and by other optical and radio observations.  \nWe have presented the highest spatial resolution Xray Chandra image using the Lucy-Richardson deconvolution method. The X-ray jet can be distinctly resolved down to approximately 0 . '' 2 from the center of the unresolved core. The X-ray morphology bears a remarkable similarity to the radio and optical emissions. The Xray jet is most luminous within about 0 . '' 8 from the core and appears linear, indicating that the jet's momentum is dominant. This region, at or less than 0 . '' 8 from the core, is also where four knobs have been resolved with HST , ranging from 0 . '' 15 to 0 . '' 6 away from the core center (Perlman et al. 2010; Meyer et al. 2015). Beyond that, the jet significantly diminishes in brightness and splits into two directions. The stronger path continues northeast, while the weaker one turns to the east. X-ray emissions from the jet are undetectable beyond ∼ 2 '' .  \nThe deconvolved Chandra images in the 0.5-1 keV and 1-6 keV bands have been utilized to create a hardness ratio map, allowing us to investigate the spectral properties at a spatial resolution surpassing that achievable through spectral analysis. The average photon index of the jet, determined by this hardness ratio map within the soft X-ray emission range (0.5-6 keV), is Γ = 2 . 26 +0 . 05 -0 . 05 . This finding is in excellent agreement with the spectral analysis, which yields a Γ ranging between 2.21 and 2.30 (see Table 1). The hardness  \nratio map also indicates that the photon index of the unresolved Chandra core (within < 0 . '' 2) is significantly harder than in the adjacent regions of the jet.  \nWhen jointly fitting the NuSTAR spectra with the Swift spectrum taken during the NuSTAR observation, the overall AGN spectrum (core+jet) can be adequately modeled with either the POWERLAW or CUTOFFPL models. However, there is evidence supporting the presence of a cutoff energy at E cutoff = 20 +21 -7 keV. Including the resolved Chandra core and jet spectra in the soft Xray, taken 15 yr prior to the NuSTAR and Swift observations, further reinforces the likelihood of a cutoff at E cutoff = 19 +10 -5 keV. The CUTOFFPL model also indicates a significantly harder photon index for the core, aligning with the hardness ratio analysis. Such a cutoff in energy implies that the X-ray emission from the core can be at least partially contributed by the RIAF. Notably, the cutoff energy for 3C 264 is the lowest among other LLAGNs measured in X-ray (Chakraborty et al. 2023; Jana et al. 2023).  \nWhen fitted with the Comptonization model ( compPS ), the best-fit electron temperature is approximately T e ≈ 15 keV, and the optical depth is more precisely constrained within the range of τ = 4-7, especially when combined with Chandra data. This aligns excellently with the anticorrelation observed between optical depth and electron temperature in a sample of 16 LLAGNs (Chakraborty et al. 2023). Such an anticorrelation has also been previously noted in more luminous Seyfert galaxies (Tortosa et al. 2018) and in the hard state of black hole binaries (Banerjee et al. 2020), suggesting the universality of coronal properties across different black hole masses and accretion rates. This has important implications for a departure from a fixed disk-corona configuration in radiative balance (see Tortosa et al. 2018; Chakraborty et al. 2023, for further discussions).  \nThe overall X-ray spectrum is consistent with the X-ray properties of LLAGNs discussed by Ho (2008). The photon index between 0.5 and 10 keV is approximately 2.1, which is within the range of 1 . 4-2 . 2 of typical LLAGNs. The power-law component shows very little intrinsic absorption. There is no evidence of Fe K α emission or Compton reflection, suggesting either the absence of an optically thick accretion disk or that it is truncated and replaced by an optically thin RIAF. Note that RIAF models tend to predict harder X-ray spectra than jet models, as can be seen in Figure 17 of Nemmen et al. (2014). As such, the measured photon index of 1.93 from the CUTOFFPL core model agrees better with a RIAF origin. Detailed modeling of the multiwavelength  \nSED will be needed to confirm this (e.g., Nemmen et al. 2014).  \nA similar conclusion regarding the existence of RIAFs has been drawn from studies of the LLAGNs Cen A 11 (Furst et al. 2016), M81 (Young et al. 2018), NGC 7213 (Ursini et al. 2015), NGC 3998 and NGC 4579 (Younes et al. 2019). Although Fe lines have been detected in a number of cases, they are attributed to hot thermal plasma or to broad-line regions (see, e.g., Ursini et al. 2015; Young et al. 2018; Younes et al. 2019). For 3C 264, excess in soft emission below ∼ 2 keV is detected, which can be fitted with a thermal plasma model with a temperature of 0.7-0.8 keV. The origin of the soft excess is not clear. All of these are consistent with the X-ray properties of LLAGNs as described by Ho (2008).", '5.1. Implication of the Low Cutoff Energy': 'The cutoff energy E cutoff ≈ 20 keV of 3C 264 is very low when compared to other AGNs, which have a median E cutoff ∼ 200 keV (Ricci et al. 2017). Lower cutoff values are usually observed at very high Eddington ratios (Ricci et al. 2018). The LLAGN 3C 264 has an Eddington ratio of approximately 7 × 10 -5 . This ratio places it outside the typical correlations observed for AGNs with Eddington ratios higher than about 0.001, such as the photon index-Eddington ratio correlation noted in recent studies (e.g., Chakraborty et al. 2023; Jana et al. 2023). For instance, these cited recent studies do not report higher cutoff energies for LLAGNs with Eddington ratios below 0.001, unlike those expected with higher ratios. Some LLAGNs exhibit very low cutoff energies or coronal temperatures, such as a cutoff energy E cutoff ≈ 80 keV and an electron temperature T e ≈ 30-40 keV in low-accreting Seyferts (Jana et al. 2023) and an electron temperature T e ≲ 10-20 keV in LLAGNs (Chakraborty et al. 2023). Therefore, the notably low cutoff of 3C 264, considering the uncertainties, is not overly surprising.  \nThe low cutoff energy suggests a cooler corona or plasma within the hot accretion flow. If the X-ray emission arises from the thermal Comptonization of seed disk photons by this hot plasma, particularly in a scenario where the disk is truncated in a RIAF, Comptonization could effectively cool the hot plasma. The smaller the truncation radius, the more efficient the cooling becomes (Tortosa et al. 2018). Consequently, the low cutoff energy indicates a small truncation radius. However, there is no evidence of a truncated disk in 3C 264, such as reflection features or a UV bump (Boccardi et al.  \n2019), and the presence of a truncated disk remains unconfirmed. Alternatively, a reduced scale height of the corona or hot accretion flow could also enhance the Compton cooling rate of the hot plasma (Tortosa et al. 2018). The cutoff might also result from bremsstrahlung emission in the outer parts of the RIAF near the Bondi radius. This extended quiescent emission is similarly proposed to explain the low cutoff energy in the X-ray spectrum of Sgr A ∗ (Yuan et al. 2003; Quataert 2002).', '5.2. Testing the Synchrotron Self-Compton Model': 'To account for the γ -ray and TeV emission from 3C 264, the SED has been modeled using the synchrotron self-Compton (SSC) jetted model (Figure 1 in Kagaya et al. 2017). This model adequately fits the softer X-ray emission (up to ≲ 10 keV) and the γ -ray emission observed during the quiescent state of the core. However, it significantly underpredicts the hard X-ray emission (around ∼ 20 keV) observed with NuSTAR by an order of magnitude; our measured SED at 20 keV is around νf ν = 3-4 × 10 -13 erg cm -2 s -1 while the SSC model predicts a value of ∼ 4 × 10 -14 erg cm -2 s -1 . Furthermore, the predicted power-law index of approximately 3 in 1-10 keV is substantially steeper than the value measured. The large discrepancy is likely due to the phenomenological model used to fit the emission below the hard X-ray band. Similarly, recent SSC jetted models of the low state of the 3C 264 core also underestimated the SED at ∼ 20 keV by more than an order of magnitude, and their predicted power-law slope in the 1-10 keV range of ∼ 3-4 is also significantly steeper than our observed values (Boccardi et al. 2019). It is possible that these model fittings underestimated the maximum energy of the electron population in 3C 264. This contrasts with the findings for another TeV radio galaxy, M87, where the SSC model overpredicts the hard X-ray emission at 40 keV by approximately a factor of three during the quiescent state of the core, and the predicted power-law index is flatter than the measured spectral slope (Wong et al. 2017). NuSTAR plays a crucial role in exploring the transition from synchrotron-dominated to (self) inverse Compton-dominated emission around 10 keV, thereby providing key insights into these VHE processes.  \nPresently, our understanding is limited by the statistical uncertainties of our data. Future in-depth observations with NuSTAR , extending detection beyond 30 keV, and simultaneous high angular resolution multiwavelength observations, will offer tighter constraints on accretion models and the VHE emission mechanisms.', '6. CONCLUSION': "For the first time, hard X-ray emission ≳ 10 keV from the TeV radio galaxy 3C 264 has been detected, which is unresolved with NuSTAR . The location of the hard X-ray emission is consistent with that of the soft Xray emission. An excess in soft emission below approximately 2 keV has been detected with Swift , and its origin is unclear. We have generated high-resolution deconvolved Chandra images to more precisely determine the origin of the X-ray emission. These deconvolved Chandra images show that the X-ray jet can be resolved down to approximately 0 . '' 2 from the unresolved Chandra core. The X-ray morphology is similar to that of the radio and optical jet. Furthermore, the X-ray spectrum of the Chandra core is harder than that of the jet. The X-ray spectrum can be adequately modeled with a single power-law model. However, evidence suggests the presence of a cutoff in the energy around 20 keV, which indicates that at least some of the X-ray emission from the core can be attributed to the RIAF. The Comptonization model indicates an electron temperature of about 15 keV and an optical depth ranging between 4 and 7, following the universality of coronal properties of black hole accretion. The cutoff energy or electron temperature of 3C 264 is the lowest among those of other LLAGNs measured in X-rays. Meanwhile, recent SSC models, which were proposed to explain γ -ray and TeV emission, significantly underestimate the hard Xray emission from the low state of the core at 20 keV by an order of magnitude or more. This suggests that the synchrotron electrons might be accelerated to higher energies than previously thought.", 'ACKNOWLEDGMENTS': "The authors would like to thank the referee and the editors for their comments and suggestions, which have improved the manuscript. This work was supported by the following NASA grants: NuSTAR awards 80NSSC21K1855, 80NSSC20K0052, 80NSSC22K0032, and 80NSSC22K0065, Chandra awards GO7-18071B, G09-20111A, and G02-23092X. C.S. and A.B. were also supported by the SUNY Brockport's Summer Undergraduate Research Program (SURP) and the Physics Department's Richard V. Mancuso Summer Research Award and Donald '80 and Diana '81 Hallenbeck Research Scholars Fund. R.N. was supported by NASA through the Fermi Guest Investigator Program (Cycle 16) and a Bolsa de Produtividade from Conselho Nacional de Desenvolvimento Cient'ıfico e Tecnol'ogico. D.L. was supported by the NASA ADAP grant 80NSSC22K0218. This paper employs a list of Chandra datasets, obtained by the Chandra X-ray Observatory, which are contained in doi:10.25574/cdc.263."}