Magnetic Field Properties inside the Jet of Mrk 421 Multiwavelength Polarimetry Including the Imaging X-ray Polarimetry Explorer

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Introduction
Relativistic jets from active galactic nuclei (AGN) are the most luminous long-lived phenomena across the entire electromagnetic spectrum in the universe.This characteristic of jets makes them natural laboratories that can be studied through multiwavelength and multimessenger observations from radio to γ-rays.
Blazars are a subclass of AGN in which a relativistic plasma jet, propelled from the vicinity of a supermassive black hole, is aligned closely with our line of sight (e.g.Hovatta & Lindfors 2019).The emission from the jet is relativistically boosted toward us, and therefore dominates the spectral energy distribution (SED).The SED typically displays two broad nonthermal radiation components.The lower-frequency component is generally ascribed to be synchrotron emission from relativistic electrons.On the other hand, interpretation of the higher-frequency component falls into two different scenarios: leptonic and hadronic models.In the case of the leptonic model, the high-energy emission is attributed to Compton scattering of synchrotron photons (synchrotron self-Compton, or SSC; e.g., Jones et al. 1974;Maraschi et al. 1992) or photons from outside the jet (e.g., Dermer & Schlickeiser 1993;Sikora et al. 1994).The hadronic scenario explains the high-energy emission as proton synchrotron radiation (e.g., Aharonian 2000;Böttcher et al. 2013) or photopion production (e.g., Mannheim 1993).
Blazars are sub-classified into three different types, depending on the frequency of the synchrotron peak of the SED.Lowsynchrotron-peaked blazars (LSP) have a peak frequency below 10 14 Hz, intermediate-synchrotron peaked blazars (ISP) between 10 14 Hz and 10 15 Hz, and high-synchrotron peaked blazars (HSPs) above 10 15 Hz (Abdo et al. 2010).Thus, among these subclasses, HSPs feature a synchrotron spectrum that smoothly connects from radio to X-ray bands (Fossati et al. 1998).Studies of HSPs therefore provide an opportunity to probe the geometry and dynamics of the magnetic field components present in the synchrotron emission features of the jet.
In this respect, multiwavelength polarimetry can be a prominent diagnostic tool to investigate the particle acceleration processes and the geometrical characteristics of the magnetic field inside the jet (e.g., Rybicki & Lightman 1979;Zhang 2019;Böttcher 2019;Tavecchio 2021).Until recently, polarimetric observations of blazars have been limited to optical and radio bands.However, the successful launch of the Imaging X-ray Polarimetry Explorer in late 2021 (IXPE; Weisskopf et al. 2022) has extended multiwavelength polarimetry up to the X-ray energy band.IXPE is a joint mission of NASA and the Italian Space Agency (Agenzia Spaziale Italiana, ASI).It carries three identical X-ray telescope systems (detector units; DUs) corresponding to three gas pixel detectors (GPDs, Costa et al. 2001) and three mirror module assemblies (MMAs).IXPE measures linear polarization in the 2-8 keV band from the photoelectric track produced by the interaction of X-ray photons and gas inside each GPD.
Mrk 421 (redshift, z = 0.030) is the brightest HSP blazar at X-ray energies, and therefore a prime target for measuring polarization with IXPE (Liodakis et al. 2019).Indeed, IXPE securely detected linear polarization at a level exceeding 10% during a two-day pointing of Mrk 421 in May 2022 (Di Gesu et al. 2022).In addition, during subsequent observations in June 2022, smooth rotation of the electric-vector position angle ψ X was discovered (Di Gesu et al. 2023), in contrast to the relatively stable polarization degree and direction found for a few other HSP blazars (e.g., Liodakis et al. 2022;Middei et al. 2023).This find-⋆ corresponding author, dawoon.kim@inaf.iting can be interpreted as evidence for a helical magnetic field inside the X-ray emitting portion of the jet.Furthermore, if the X-ray emission, and therefore helical field, occurs mainly in the innermost jet, then it is expected that such rotations of ψ X should be routinely observed.If so, X-ray observations can probe the section of the jet where the flow is accelerated and collimated (Marscher et al. 2008(Marscher et al. , 2010)).
Here we report the results of a new IXPE observation of Mrk 421, conducted as part of a multiwavelength campaign.In §2, we present the analysis of the time-averaged data and we search for variability of the polarization properties as a function of time and photon energy.To do this, we compare the finding with the results of the previous three IXPE observations, and we investigate the polarization variability on long time scales.Next, in §3, we report the multiwavelength polarimetry results obtained from simultaneous and quasi-simultaneous measurements.Based on these results, we propose possible geometrical and physical interpretations of the magnetic field geometry inside the jet of Mrk 421 in §4.
We also observed Mrk 421 with XMM-Newton on 2022 December 7 for 7.5 ks.In addition, the Neil Gehrels Swift Observatory (hereby referred to as Swift-XRT) monitored the blazar to improve the spectral and temporal coverage.Table 1 presents a summary of the X-ray observations.Detailed information on the data reduction procedures is presented in Appendix A. Throughout the manuscript, errors are given at the standard 1σ (68%) confidence level.

Time-averaged polarization properties
We have derived the linear polarization parameters from Obs. 4 by four different methods: (1) the Kislat et al. (2015) method that is implemented in the PCUBE algorithm in ixpeobssim (Baldini et al. 2022), (2) an event-based maximum likelihood method (Marshall 2021b, ML) that includes simultaneous background estimation, (3) the spectropolarimetric analysis described in Strohmayer (2017) using XSPEC, and (4) a maximum likelihood spectropolarimetric (MLS) fit implemented by the multi nest algorithm.Table 1 summarizes the Π X and ψ X values obtained from these methods.
First, using the Kislat et al. (2015) method, we estimated Π X =13±2% and ψ X =107±3 • in the 2-8 keV band after background subtraction from the three combined DUs.The PCUBE measurement includes the spectral-model-independent polarization properties over a given energy and time range (Kislat et al. 2015).We derived Π X and ψ X from the normalized Stokes parameters 2  and ψ X =1/2 tan −1 (U/Q).Error estimation was calculated based on   Kislat et al. (2015) and Muleri (2022).Figure 1 indicates the detection significance of this measurement in the form of a polarization contour plot.
The second method (ML) uses an event-based maximum likelihood method to determine Q and U (Marshall 2021b).The method accounts for background using data from an annulus about the target by including a term in the likelihood, as used in the analysis of the IXPE data of Cen A (Ehlert et al. 2022).Source events were selected from a region of 60 ′′ radius about the source centroid, while the background events were selected from an annulus with inner and outer radii of 200 ′′ and 300 ′′ .In this case, Π X = 13±1% and ψ X =107±3 • .
Next, the spectropolarimetric analysis examined the X-ray spectra and polarization properties based on spectral modeling with XSPEC (ver.12.13.0c;Arnaud 1996).With this method, we obtained Π X =14±1% and ψ X =107±3 • .The detection significance of this measurement was ≳11σ.The flux was estimated as 2.25 (±0.04) ×10 −10 erg s −1 cm −2 over 2-8 keV, the same band over which the IXPE I, Q, and U spectra were obtained.In this analysis, we additionally included the simultaneous XMM-Newton data in order to refine the constraints on the spectral shape over the 0.3-10 keV energy range.The crosscalibration factors among all the spectra are accounted for using the CONSTANT model, normalized to the XMM-Newton spectrum while the other spectra were varied.In all the fits, we considered the Galactic absorption along the line of sight of Mrk 421.For this, we used the TBABS model with weighted average column density values from HI4PI Collaboration et al. (2016) (N H = 1.34 × 10 20 cm −2 ).The WILM model was applied to take into account metal abundance (Wilms et al. 2000).For the spectral modeling, we first applied a simple power law model (POWERLAW in XSPEC) to reproduce the I synchrotron spectrum, but the best-fit result was poor, with χ 2 /d.o.f.=3308/651.Hence, we employed the log-parabolic model (LOGPAR), in which the photon index varies with energy following a log parabola function (Massaro et al. 2004): where the pivot energy E pivot is a scaling factor, α describes the slope of the photon spectrum at E pivot , β expresses the spectral curvature, and K denotes a normalization constant.This spectral model generally describes a typical HSP spectrum well, including that of Mrk 421, both in quiescence and in flaring states (Donnarumma et al. 2009;Baloković et al. 2016).As the photon index varies with energy in this model, the choice of reference energy E pivot changes the determined value of α.In our spectral fit, we fixed the pivot energy to 5.0 keV (e.g., Baloković et al. 2016;Middei et al. 2022).In this case, the α parameter approximately corresponds to the photon index over 3.0-7.0keV.In our spectral fitting, we also allowed the values of α, β, and K to vary.These free parameters are coupled with the reference spectrum, which here is the XMM-Newton PN spectrum  A.1 present the parameter values of our best-fit results.We note that in the Stokes-spectra-decoupled case, which only fits I spectra without involving POLCONST, the best-fit results obtained the same values as we derived from the simultaneous spectropolarimetric fit.Finally, the MLS method was also applied to determine the polarization properties for the POWERLAW spectral model.With this procedure, we derived the polarization and spectral properties to be Π X =13±1% and ψ X =109±3 • .
We have thus derived that the X-ray polarization properties by four different methods are consistent within the uncertainties (see Table 1).The small differences can be explained by the fact that the PCUBE analysis estimated spectral-modelindependent polarization properties, while the XSPEC methods take into account the best fit from the spectral modeling.Furthermore, among the current choices, only the XSPEC method can improve sensitivity by applying event weight methods introduced by Di Marco et al. (2022).Overall, we find that the Π X and ψ X results are robust, with modest statistical errors (see Figure 1).

Polarization variability
We have investigated whether the polarization varies as a function of time or energy.We tested the time dependence using two methods.In the first one, we determined the null-hypothesis probability with the χ 2 test using the combined PCUBE and XSPEC analysis (e.g., Di Gesu et al. 2022;Gianolli et al. 2023;  Di Gesu et al. 2023).For the second method, we used the unbinned event-based maximum likelihood analysis implemented in Marshall (2021a).With this method, we can avoid the error that can be caused by subjective selection bias resulting from the binning criteria.
The first method measured each normalized Stokes parameter value by dividing the data into identical time spans that depended on the selected number of bins (e.g., 2 bins= 100 ks/2 bins = 50 ks/bin).In particular, we split Obs. 4 into 2 to 20 time bins.We then compared Q and U, which followed a Gaussian error distribution, with the results from fitting each parameter treated as constant over time (i.e., Q(t) = Q 0 and U(t) = U 0 ).The result of fitting the constant model was calibrated for each number of bins.We then calculated the χ 2 and the null-hypothesis probability (for the corresponding degrees of freedom) for each case.Figure 2 indicates the null-hypothesis probability as a function of the number of bins.In this figure, the green-shaded area (P Null > 1%) corresponds to cases where the data are statistically consistent with values of Q and U that are constant in time.Conversely, if the points lie outside the region (P Null < 1%), it implies that the polarization varies with time.As seen in Figure 2, we found that splitting the Q(t) light curves with 5,7,8,9,11,12, and 14 time bins does not pro- duce a good fit with the constant model (P Null < 1%).The case of 7 time bins has the smallest null-hypothesis probability, with χ 2 /d.o.f.=25.26/6,beyond the 3σ confidence level.In the polarization light curve of Obs. 4 with seven identical time bins (Figure 3), we can see that ψ X varies from the second to the fourth bin, after which it returns to a more stable value near that of the first three bins.We estimate the rotation rate of this variation as ψ ∼ 92 • /day (change by 61 • over ∼57 ks from the second to the fourth bin).This result is comparable to the rotation rate of ψ = 85 • /day reported in Di Gesu et al. (2023).
Additionally, in order to examine the possibility of continuous change of ψ X , we have tested whether a sinusoidal function (ψ X (t)=A sin(Ct − D) + B) provides a better fit of ψ X (t) with respect to a constant function corresponding to the mean value of ψ X over 7 time bins (blue line in Figure 3).This approach yielded a smaller χ 2 /d.o.f.value of 4.75/3, with a Bayesian information criterion (BIC) of 12.53, than the constant model (χ 2 /d.o.f.=16.66/6,BIC=18.60).
As a second approach, we employed a maximum likelihood method that allows for ψ X rotation in each interval, as described elsewhere (Di Gesu et al. 2023) and shown in Figure 4. Briefly, the ML method is used (see §2.1) for eight equal time intervals but also allowing for ψ X rotation during the time interval, as an "uninteresting" parameter.This approach prevents ψ X rotations during a time interval from reducing the average polarization of that interval, which is important when there are large ψ X changes.For example, Figure 4 shows that there appear to be large, nearly 90 • ψ X shifts between intervals one and two and between intervals three and four.As a result, we confirmed that ψ X varied over time during Obs. 4 with ∼ 4.5σ confidence.
In addition, with a maximum likelihood method, we tested a more complex model fit -the multicomponent model (Pacciani et al. in preparation), which involves the convolution of constant and rotating polarization components -to check the possibility of continuous variation within the Obs. 4 period.The details of this model are described in Appendix B. From this test, we computed the significance of the multicomponent model relative to the constant polarization model based on the delta likelihood, We obtained a decrease in the likelihood estimator, ∆L = −32.4,which follows a chi-square distribution with three degrees of freedom.Hence, Table 2: Parameters for the multicomponent model Therefore, based on these different methods, we conclude that we have found an episodic ψ X variation over time during Obs. 4. Furthermore, the results of sinusoidal and multicomponent models suggest that the variations may be attributed to continuous changes in stable components.However, due to differences in the analysis methods employed, these two models were not compared in this study.Moreover, since these model fit results were derived as relative outcomes compared to the constant model, we cannot rule out the possibility of encountering more complex components and potential stochastic variations.
We also tested whether the X-ray polarization depends on energy by applying the same null-hypothesis probability test discussed above, but with the IXPE energy band (2-8 keV) divided into smaller ranges.We did so for two energy bins (2-4 and 4-8 keV), as well as three energy bins (2-4, 4-6, and 6-8 keV), etc., up to 12 energy bins.We found no statistically significant differences with a constant fit, as P Null ≥ 13% from both Q and U, hence the data are consistent with energy-independent X-ray polarization.

X-ray polarization and spectral variability across multiple IXPE observations
Significant X-ray polarization from Mrk 421 has been detected at four epochs with IXPE (see Table 1).Figure 5 shows error contours for both time-averaged and time-resolved data of each observation.Over the seven months from IXPE Obs. 1 to Obs. 4, the value of ψ X varied widely, with a continuous rotation over 180 • observed during Obs. 2 and Obs. 3. In contrast, measurements of Π X for all events were similar within a range of 10-15%, even though the time-averaged Π X of Obs. 2 and Obs. 3 appears to be clearly lower than in the other cases due to dilution by the changing of ψ X .However, the mean value of Π X during periods of nonrotation of ψ X (Obs. 1 and Obs.4) was 14 ± 2%, a factor of 1.4 ± 0.  creased number of measurements from future X-ray polarimetry monitoring observations.The radio electric-vector position angle ψ 43GHz obtained from the 43 GHz Very Long Baseline Array (VLBA) observations (black line in Figure 5) differs from ψ X and which also changes by 70 • over the 7 months from Obs. 1 to Obs. 4. The pronounced variation of ψ X and ψ 43GHz contrasts with the steadier X-ray polarization observed in other sources of the same subclass of blazars, whose synchrotron SED peaks at X-ray frequencies (i.e., Mrk 501; Liodakis et al. 2022).
We have also investigated the spectral properties and X-ray activity of Mrk 421 through Swift-XRT monitoring observations.Figure 6 presents the α and β spectral parameters obtained from the LOGPAR model fit.We also derived the X-ray flux in the soft (0.3-2 keV), F soft , and hard (2-10 keV), F hard , energy bands, and define the hardness ratio as (F hard − F soft ) / (F hard + F soft ).We have found that the α parameter, representing the slope of the spectrum at the pivot energy, maintained similar values in Obs. 1 and Obs. 4.However, during Obs. 2 and Obs. 3, α decreased with time.In addition, the hardness ratio also varied during the ψ X rotation, while its value in Obs. 1 was consistent with that in Obs.The X-ray fluxes of Mrk 421 during all epochs of IXPE observations were within 1σ of the median historical value (e.g., Liodakis et al. 2019).Nonetheless, we found significant fluctuations during Obs. 2 and Obs. 3.Although the flux variations occurred during all of the IXPE epochs (see, e.g., Figure 3), the discrepancy between the minimum and maximum counting rates was larger in Obs. 2 and Obs. 3. Therefore, we suggest that the smooth ψ X rotation behavior may accompany more pronounced spectral and flux fluctuations.

Multiwavelength polarization analysis
Multiwavelength campaigns for the first three observations of Mrk 421 are reported in Di Gesu et al. (2022, 2023).During Obs. 4, Mrk 421 was observed with the VLBA, the Effelsberg 100-m Radio Telescope, the Korean VLBI Network (KVN), the Submillimeter Array (SMA), the Kanata telescope, the Perkins telescope, and the Sierra Nevada Observatory (SNO, T90 telescope).Details of the observations and observatories can be found in Appendix A.3.
We have analyzed VLBA data obtained for Mrk 421 within the BEAM-ME (Blazars Entering the Astrophysical Multimessenger Era) program1 during the period from MJD 59616 (2022 February 5) to MJD 59986 (2023 February 11) to investigate the parsec-scale jet behavior during the IXPE observations.The data include total and polarized intensity images at 43 GHz at 12 epochs.The data reduction and modeling are described in §A.4. Figure A.2 exhibits the evolution of polarization properties of Mrk 421 obtained from the contemporaneous multiwavelength polarimetry campaign from just before IXPE Obs. 1 to shortly after Obs. 4. Throughout this time period, we find a similar strong chromatic behavior, with Π 2-3 times higher at X-ray rather than at longer wavelengths.In contrast, the infrared, optical, and radio degrees of polarization were similar.Meanwhile, the polarization position angle exhibited marked changes, with different values during the various IXPE pointings.The range of ψ observed in Obs. 1 across millimeter to X-ray wavelengths was ∼30 • , whereas a larger range of ∼90 • was evident in Obs. 4.Moreover, (Di Gesu et al. 2023) have reported that, during the rotation of ψ X of Obs. 2 and Obs. 3, ψ values at other wavelengths were consistent with each other, with a weak temporal variation.Hence, we conclude that the region where the polarized X-rays are emitted is mostly or completely distinct from that at longer wavelengths.
Furthermore, Mrk 421 exhibited a clockwise ψ O rotation of approximately 120 • over ∼40 days during the first three IXPE observation periods.Similar behavior has been reported in previous optical polarimetry monitoring studies, where the polarization angle changed by ∼180 • over ∼50 days (Blinov et al. 2015;Jermak et al. 2016;Blinov et al. 2016a;Fraija et al. 2017;Blinov et al. 2018).The direction of the ψ O rotation was opposite to the counter-clockwise ψ X rotation observed by IXPE during Obs. 2 and Obs. 3.

Discussion
Previous IXPE observations of HSPs Mrk 421 (Di Gesu et al. 2022, 2023) and Mrk 501 (Liodakis et al. 2022) suggest that an energy-stratified shock can most readily explain the ∼2-3 times higher degree of polarization at X-ray rather than at longer wavelengths.In this work, we have found consistent multiwavelength measurements from Obs. 4. In this energy-stratified shock model, the relativistic electrons convert their energy to radiation as they move farther from the shock front (Marscher & Gear 1985;Tavecchio et al. 2018).The particles are efficiently accelerated at the shock front, where the magnetic field is relatively well ordered, and hence emit X-ray synchrotron radiation with relatively high polarization.Conversely, electrons lose energy as they propagate away from the shock, causing them to radiate at longer wavelengths, and the degree of polarization decreases as they encounter increasingly turbulent magnetic fields (Di Gesu et al. 2022, 2023;Liodakis et al. 2022).Hence, higher Π is predicted at higher frequencies.This effect ceases at wavelengths long enough that the electrons radiating at those wavelengths can cross the entire shocked region before losing most of their energy (Marscher & Gear 1985).This limitation can explain the similar polarization from millimeter to optical wavelengths observed in Mrk 421.It is also possible that the longer wavelength emission occurred mainly in a relatively slow sheath surrounding a much faster X-ray emitting spine of the jet (e.g., Di Gesu et al. 2023).This is consistent with the subluminal speed we have measured for knot P. In this case, the X-ray and longer wavelength polarization properties may not be related, consistent with their different position angles.
We therefore consider two possible geometries for an energy-stratified jet in Mrk 421: linear and radial models, as suggested by Di Gesu et al. (2022, 2023).In the case of linear geometry, the energy stratification and the ψ rotation can be explained by emission features propagating downstream in the jet, following the helical magnetic field.On the other hand, the radial structure can correspond to a helical, rotating innermost region and a surrounding layer, similar to the spine-sheath jet model (Chhotray et al. 2017).The currently available observational data are insufficient to discern between the linear and radial geometries.Nevertheless, the episodic variation in polarization observed during Obs. 4 offers further insights into the internal geometry of the jet.For instance, it implies alternative perspectives on the geometric structure within the jet, including the intricate interactions between coexisting stable and rotating magnetic field structures, as well as stochastic transitions within the dominant magnetic field structure responsible for particle acceleration.Therefore, future observations of polarization variability are expected to yield further evidence about the geometric structure inside the jet.
On the other hand, despite the comparable multiwavelength results reported for Mrk 421 and Mrk 501, we have found a difference in the behavior of ψ X between them.In the case of Mrk 421, ψ 43GHz and ψ X exhibited significant variations without any consistent alignment with either each other or the direction of the jet axis.However, in the case of Mrk 501, an alignment was observed (within the uncertainties) between measurements of the position angle of the jet axis, ψ X , and ψ 43GHz conducted within a month of each other (Liodakis et al. 2022).Further IXPE and VLBA observations of HSPs are needed to confirm whether, and if so, why, Mrk 421 is different in this regard.
We have found that the X-ray flux and hardness ratio of Mrk 421 were less variable during Obs. 1 and Obs. 4, when ψ X was essentially constant, than during the ψ X rotation of Obs. 2 and Obs. 3.This implies different physical conditions within the jet between the rotating and nonrotating states.Instead, the similar values of Π X across all observations suggest that the basic particle acceleration scenario remained roughly independent of the magnetic field geometry.This can be accommodated within the energy-stratified shock scenario, since the degree of order of the magnetic field could be similar whether the shock moves along a straight or helical trajectory.The flux and spectral variations during the rotation event could have been caused by changes in the Doppler factor as the shock executed helical motion, although the data are too sparsely sampled to test this.Future X-ray spectroscopic and polarimetric observations with improved time resolution can potentially test whether spiral motion along a helical magnetic field causes cyclical Doppler factor variations that lead to observed variations in flux, hardness ratio, and polarization.
Our examination of a sequence of 43 GHz VLBA images has revealed the presence of a prominent, highly polarized knot moving away from the "core" at a speed of 0.7c during a time span that includes Obs. .
This finding suggests a potential connection between the morphological changes near the jet core region and the variability in polarization, as discussed in Di Gesu et al. (2023).The knot could represent a shock containing relativistic electrons accelerated up to Lorentz factor ∼10 6 that radiate at X-ray energies.However, in this scenario, it is difficult to explain the difference in the behavior of ψ X between Obs. 1 and Obs. 2 -Obs.3, unless the geometry of the magnetic field varies across the core, with a tighter helical structure toward the downstream end.However, our analysis does not indicate a direct connection between the X-ray polarization position angle and that of either the core or knot P (see Figure A.5), although the degree of polarization of P, 15-20%, is comparable to Π X .This apparent lack of connection supports the conclusion, drawn above, that emission regions at longer wavelengths (millimeter, IR, and optical) were separate from, or only partially coincident with, that of the X-ray emission.
In the case of Obs. 4, there is no apparent connection between the values of ψ of the polarized features in the jet observed on 2022 December 6 (MJD 59919) and ψ X (Fig. We note that, from Obs. 1 to Obs. 3, the polarization angle at optical, IR, and radio wavelengths rotated in the opposite direction relative to the 5-day rotation of ψ X during Obs. 2 and Obs. 3.This finding supports the conclusion that the X-ray emission region is separate from that at longer wavelengths.The observed similarity in radio to optical polarization properties implies that the emission at these wavelengths originates from spatially interconnected regions.The longer (relative to X-ray) rotation time scale suggests that the X-ray emission region is smaller than that of these other wavelengths.Although the polarization vector rotation at longer wavelengths could be explained by propagation of a larger emission feature down a helical magnetic field (as in Figure 8), the long-term behavior of the optical polarization of Mrk 421 implies stochastic, rather than systematic, variations of both ψ O and Π O (Marscher & Jorstad 2022).

Conclusions
We have reported X-ray, optical, IR, and radio polarization and flux measurements of the HSP blazar Mrk 421, including four IXPE pointings between 2022 May and December.Such observations probe the magnetic field structure and particle acceleration mechanisms inside the jets of blazars.The combined 7 months of observations sample the time and energy dependence of the X-ray polarization.Over this time span, ψ X varied over the full range of ∼ 180 • , including a 5-day episode of rotation, while the degree of polarization maintained a value between ∼10-15% across all IXPE pointings.The X-ray flux varied by a higher fraction during the rotation than during the two IXPE observations without rotations.
The simultaneous multiwavelength polarimetry results over four IXPE observations provide evidence useful for constraining the physics of the jet.The degree of X-ray polarization was typically ∼2-3 times greater than that at longer wavelengths at all epochs sampled, while the polarization angles fluctuated.The discrepancy between the results of X-ray compared with radio, IR, and optical polarimetry supports the previous conclusion that the X-ray emission region is distinct from that at longer wavelengths in HSP blazars (Liodakis et al. 2022;Di Gesu et al. 2022, 2023).As with these previous studies, we conclude that the observations are consistent with the energy stratified shock model, with the level of turbulence increasing with distance from the shock front.
One difference between Mrk 421 and Mrk 501 is that there is no apparent correlation between the direction of the jet from VLBA and ψ X in the former.While this could be due to the bending of the jet from the X-ray to the radio emitting region, amplified by the narrow angle to the line of sight, the optical polarization angle is also much more highly variable in Mrk 421 than in Mrk 501 (Marscher & Jorstad 2022).This implies an intrinsic difference between the two objects that should be explored with further observations.Following Di Gesu et al. ( 2023), we have proposed a linear and a radial stratification scenarios to explain the rotation behavior of ψ X .The accompanying spectral variation during the ψ X rotation suggests that the physical conditions of the jet, such as the energy distribution of relativistic electrons, differed between the periods of rotation and nonrotation.In addition, we reported rotation in the opposite direction of the ψ between the X-ray and other wavelengths, with the latter occurring over a much longer time scale.This could potentially be interpreted by the presence of multi helical magnetic field structures inside the jet.Morphological changes in the parsec-scale jet, possibly associated with the contemporaneous emergence of a new knot of emission observed to move down the jet in 43 GHz VLBA images, may be linked to the ψ X rotation, although differences in the radio and X-ray polarization angles argue against such a connection.
In conclusion, the present study continues to develop a new perspective on the physical and geometrical features of the magnetic field inside the jets of blazars by employing polarimetry that extends from radio to X-ray wavelengths.The IXPE observations, incorporating data at other wavelengths, have played a significant role in constraining the emission arising from the innermost regions of the jet.The polarization properties, sampled over different time scales and energy regimes, suggest a possible connection between spectral and polarization variations.The connection may include morphological changes in radio images, which coincided with a period of ψ X rotation.However, due to infrequent data sampling, there remain uncertainties regarding apparent correlations, which can be chance coincidences.In addition, thus far the IXPE observations of Mrk 421 have been obtained when the blazar was in average activity states.It is of great interest to determine whether the polarization and physical properties change during strong flaring events.As the IXPE mission continues, further studies of Mrk 421 and other blazars are expected to provide the data needed to improve our understanding of the magnetic field geometry and particle acceleration processes in relativistic jets.and IGN (Spain).Some of the data are based on observations collected at the Observatorio de Sierra Nevada, owned and operated by the Instituto de Astrofísica de Andalucía (IAA-CSIC).Further data are based on observations collected at the Centro Astronómico Hispano en Andalucía (CAHA), operated jointly by Junta de Andalucía and Consejo Superior de Investigaciones Científicas (IAA-CSIC).The Submillimetre Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica.Mauna Kea, the location of the SMA, is a culturally important site for the indigenous Hawaiian people; we are privileged to study the cosmos from its summit.Some of the data reported here are based on observations made with the Nordic Optical Telescope, owned in collaboration with the University of Turku and Aarhus University, and operated jointly by Aarhus University, the University of Turku, and the University of Oslo, representing Denmark, Finland, and Norway, the University of Iceland and Stockholm University at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.E. L. was supported by Academy of Finland projects 317636 and 320045.The data presented here were obtained [in part] with ALFOSC, which is provided by the Instituto de Astrofisica de Andalucia (IAA) under a joint agreement with the University of Copenhagen and NOT.We acknowledge funding to support our NOT observations from the Finnish Centre for Astronomy with ESO (FINCA), University of Turku, Finland (Academy of Finland grant nr 306531).We are grateful to Vittorio Braga, Matteo Monelli, and Manuel Sänchez Benavente for performing the observations at the Nordic Optical Telescope.Part of the French contributions is supported by the Scientific Research National Center (CNRS) and the French spatial agency (CNES).The research at Boston University was supported in part by National Science Foundation grant AST-2108622, NASA Fermi Guest Investigator grants 80NSSC21K1917 and 80NSSC22K1571, and NASA Swift Guest Investigator grant 80NSSC22K0537.This study was based in part on observations conducted using the Perkins Telescope Observatory (PTO) in Arizona, USA, which is owned and operated by Boston University.This research was conducted in part using the Mimir instrument, jointly developed at Boston University and Lowell Observatory and supported by NASA, NSF, and the W.M. Keck Foundation.We thank D. Clemens for guidance in the analysis of the Mimir data.This work was supported by JST, the establishment of university fellowships toward the creation of science and technology innovation, Grant Number JPMJFS2129.This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers JP21H01137.This work was also partially supported by the Optical and Near-Infrared Astronomy Inter-University Cooperation Program from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.We are grateful to the observation and operating members of the Kanata Telescope.The QUIVER observations were performed at several radio bands (depending on receiver availability and weather conditions) from 2.6 GHz to 44 GHz (11 cm to 7 mm wavelength) using six receivers located at the secondary focus of the 100-m Effelsberg Radio Telescope (S110mm, S45mm, S28mm, S20mm, S14mm, and S7mm).The receivers are equipped with two orthogonally polarized feeds (either circular or linear) that can deliver polarimetric parameters using either conventional polarimeters or by connecting the SpecPol spectropolarimetric backend.Instrumental polarization is calibrated via observations of both polarized and unpolarized calibrators performed in each session, and then removed from the data (e.g., Myserlis et al. 2018;Kraus et al. 2003).The polarized intensity, degree, and position angle were derived from the Stokes I, Q, and U cross-scans.The total flux density was successfully recovered at 13 bands between 4.85 GHz and 43.75 GHz.The calibrators 3C 286, 3C 48, and NGC 7027 were used for the total flux and polarization calibration (e.g., Myserlis et al. 2018;Kraus et al. 2003).
Observations were conducted with the KVN simultaneously at 4 frequencies from 22 to 129 GHz in single-dish mode.This was achieved by using the Tamna (22, 43 GHz) and Yonsei (86,129 GHz) antennas with circularly polarized feed horns to conduct two-frequency dual-polarization observations.The polarization angle was calibrated using the Crab nebula (152 • ; Aumont et al. 2010), and the polarization degree using Jupiter (unpolarized) and 3C286 (polarized, Agudo et al. 2012) following Kang et al. (2015).
The SMA observations were taken at 225.538 GHz (1.3 mm) through the SMAPOL program (SMA Monitoring of AGN with POLarization).The SMAPOL observations were taken on 2022 December 7 (MJD 59920) in full polarization mode (Marrone & Rao 2008) using the SWARM correlator (Primiani et al. 2016), and calibrated with the MIR software package 3 .
Observations using HONIR were taken on 2022 December 6 (MJD 59919.7537) in the R and J bands 4 .Each observation comprised four exposures in different positions of the half-wave plate (Kawabata et al. 1999), which were then used to estimate the Stokes parameters and calculate the polarization degree and angle.The host galaxy of Mrk 421 contributes a significant fraction of unpolarized flux to the total emission, thereby reducing the polarization.None of the polarization degree estimates from Kanata have been corrected for the host-galaxy contribution due to the lack of the photometry data needed for such a correction, and hence they should be considered as lower limits to the intrinsic polarization degree of the blazar component.
The optical photometric and polarimetric observations were also obtained at the 1.8 m Perkins telescope (Flagstaff, AZ, USA) with the PRISM camera5 in R band performed before and after Obs. 4. The camera includes a polarimeter with a rotating half-wave plate.Each polarization observation has consisted of four consecutive exposures of 60 s at instrumental position angles 0 • , 90 • , 45 • , and 135 • of the waveplate to calculate the normalized Stokes parameters Q and U.
The SNO observations were performed using polarized filters oriented to represent different positions of a half-wave plate, similar to a conventional polarimeter.The data are then analyzed following standard photometric procedures.Mrk 421 was observed on 2022 December 7 in the R band.In this case, following Nilsson et al. (2007) and Hovatta et al. (2016), we have corrected the host-galaxy contribution.Several observations were taken within the same night.We report the weighted average and uncertainty after considering all of the intra-night observations to account for variations of the system and the observing conditions during the rotation of the filter wheel.

A.4. VLBA observations and analysis
The VLBA data were reduced with the Astronomical Image Process System (AIPS) and Difmap software packages in the manner described by Jorstad et al. (2017).The total intensity images are modeled by a number of components with circular Gaussian brightness distributions, with the minimum number of components determined by the best fit between the data and model at each epoch according to a χ 2 test.We identify the brightest feature located at the southeastern end of the jet as the "core," designated as A0, which we assume to be a stationary feature.To obtain polarized intensity images, the data were corrected for the instrumental-polarization "D-terms," calculated by averaging the D-terms obtained individually from a number of sources (usually 15) observed along with Mrk 421 in the BEAM-ME program.The electric-vector position angle (ψ 43GHz ) calibration was obtained by different methods, as discussed in Jorstad et al. (2017).The modeling of total intensity images provides the following parameters of each component: flux density, S , distance from the core, R, position angle with respect to the core, Θ, and angular size of the component, a (FWHM of the best-fit Gaussian).For polarized features in the jet, we have calculated the degree of polarization, Π knot , and ψ knot , by integrating the Q, and U Stokes parameter models obtained with Difmap during imaging within the size of the corresponding feature from the total

Notes. 1 Fig. 1 :
Fig. 1: Polarization contours from Obs. 4. Contours represent the significance of the time-averaged polarization detected with confidence levels of 68.27%, 90.00%, and 99.00%, with two degrees of freedom.The blue contour indicates the values of Π X and ψ X derived from the PCUBE methods, and the red contours show the same properties from simultaneous IXPE and XMM-Newton spectropolarimetric analysis.The radial and angular values represent Π X and ψ X , respectively, with the latter measured from north through east.

Fig. 2 :
Fig. 2: The null-hypothesis probability of the χ 2 test for time variability of the Q (red) and U (blue) Stokes parameters with the constant model for different numbers of time bins for Obs. 4. The left and right vertical axes correspond to the probability values in logarithmic and linear scales, respectively.The green shaded area indicates that the null-hypothesis probability is above the 1% significance level.The black dashed and dotted lines located in the middle of the panel represent 1% and 3σ (99.73%) probability, respectively.

Fig. 3 :
Fig. 3: IXPE polarization and photon counts versus time during Obs. 4. From top to bottom: Π X , ψ X , and count rates.Polarization results are from a time-resolved analysis with seven identical 19 ks time bins.The detection significance for each bin is displayed at the top of the figure.In Π X and ψ X panels, red dashed lines denote a fit to a constant function; the shaded area corresponds to ≤ 3σ uncertainty.In addition, the blue and green solid lines indicate the best-fit result with the sinusoidal and multicomponent model, respectively.The blue dashed line in the light curve indicates the average value during the observation.

Fig. 4 :
Fig. 4: The time variability of the Q and U Stokes parameters during Obs. 4. Each circle indicates the 1σ uncertainties of eight identical time intervals, derived using an event-based maximum likelihood technique as described in Di Gesu et al. (2023).The black circle shows the time-averaged polarization.Each color represents each time interval from the start to the end of the Obs. 4.
day) 12.6±1.6108± 4 6.3 ± 1.8 −96 ± 17 246 ± 23 the constant polarization model is rejected in favor of the multicomponent model, at the ∼ 5σ confidence level that this result occurred by chance.The parameters estimated with the multicomponent model are reported in Table 2.The observed discrepancy between the rotation rate measured from the multicomponent model (i.e., ω = 245 • /day) and the rate estimated directly from the light curve (i.e., ψ ∼92 • /day) can be explained by differences in physical interpretation.The multicomponent model assumes a persistent presence of the rotating component in conjunction with the constant emission component, while the other case assumes that the emission originates from a single dominant component.The multicomponent model is indicated with a green line in Figure 3.

Fig. 5 :
Fig. 5: Time-averaged and time-resolved polarization contours of four multiple IXPE observations.In the top panel, each colored contour represents the significance of the time-averaged polarization detection for the corresponding observation (Obs.1: red, Obs.2: orange, Obs.3: green, and Obs.4: blue).Contours are shown at confidence levels of 68.27%, 90.00%, and 99.00%, from a χ 2 test with two degrees of freedom.Additionally, the black line and gray shaded areas indicate the jet's electric vector position angle (EVPA) in degrees as observed by the VLBA at 43 GHz (Obs.1: 100±10 • , Obs. 2: 147±7 • , Obs. 3: 147±7 • , and Obs.4: 171±10 • ).In the bottom panel, the 99.00% confidence level contours show the time variation of polarization properties by dividing the data into three identical time intervals for each observation.In the cases of Obs. 2 (orange) and Obs. 3 (green), arrows indicate a rotation of ψ X from the start to the end of the observation.

Fig. 6 :
Fig. 6: SWIFT-XRT X-ray spectral parameters and flux versus time of Mrk 421 from a log-parabolic best-fit model with E pivot =5 keV.From top to bottom: α, β, hardness ratio, and flux in the soft (0.3-2 keV, gray circles) and hard (2-10 keV, black diamonds) energy bands.The time ranges of the IXPE pointings are indicated by the gray-shaded regions.

Fig. 7 :
Fig. 7: VLBA total (contours) and polarized (color scale) intensity images of Mrk 421 at 43 GHz.The peak of the total intensity is 295 mJy/beam; yellow linear segments within the images indicate the direction of polarization; black horizontal line marks the position of the core, A0; colored circles indicate locations of jet features A1 (blue), A2 (gray), and P (red).Images are convolved with the same elliptical Gaussian beam, which is shown in the bottom right corner by a black ellipse.

Fig. 8 :
Fig. 8: Schematic diagram of double helical magnetic field components inside the jet.The downstream direction of the jet is to the left.The arrows indicate each helical magnetic field component involved in the emission at different wavelengths (blue: X-ray, black: longer wavelengths).
A.5). However,Figure A.4  indicates an increase of the core flux density in 2023 February (MJD 59986) that could be a signature of an emerging moving feature that would have been upstream of the 43 GHz VLBI core during IXPE Obs. 4. Further combined IXPE and VLBI monitoring could help to clarify whether there is any relation between the X-ray emission regions and features seen in the jet at millimeter wavelength.

Acknowledgements.
The authors thank the anonymous referee for comments that improved this manuscript.The Imaging X-ray Polarimetry Explorer (IXPE) is a joint US and Italian mission.The US contribution is supported by the National Aeronautics and Space Administration (NASA) and led and managed by its Marshall Space Flight Center (MSFC), with industry partner Ball Aerospace (contract NNM15AA18C).The Italian contribution is supported by the Italian Space Agency (Agenzia SpazialeItaliana, ASI)  through contract ASI-OHBI-2017-12-I.0,agreements ASI-INAF-2017-12-H0 and ASI-INFN-2017.13-H0, and its Space Science Data Center (SSDC), and by the Istituto Nazionale di Astrofisica (INAF) and the Istituto Nazionale di Fisica Nucle-are (INFN) in Italy.This research used data products provided by the IXPE Team (MSFC, SSDC, INAF, and INFN) and distributed with additional software tools by the High-Energy Astrophysics Science Archive Research Center (HEASARC), at NASA Goddard Space Flight Center (GSFC).The IAA-CSIC group acknowledges financial support from the grant CEX2021-001131-S funded by MCIN/AEI/10.13039/501100011033 to the Instituto de Astrofísica de Andalucía-CSIC and through grant PID2019-107847RB-C44.The QUIVER data are based on observations with the 100-m telescope of the MPIfR (Max-Planck-Institut für Radioastronomie) at Effelsberg.Observations with the 100-m radio telescope at Effelsberg have received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 101004719 (ORP).The POLAMI observations were carried out at the IRAM 30m Telescope.IRAM is supported by INSU/CNRS (France), MPG (Germany),

Table 1 :
Results of the X-ray polarimetric and spectral observations of Mrk 421 This research has made use of data from the RoboPol program, a collaboration between Caltech, the University of Crete, IA-FORTH, IUCAA, the MPIfR, and the Nicolaus Copernicus University, which was conducted at Skinakas Observatory in Crete, Greece.D.B., S.K., R.S., N. M., acknowledge support from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program under grant agreement No. 771282.CC acknowledges support from the European Research Council (ERC) under the HORIZON ERC Grants 2021 program under grant agreement No. 101040021.We acknowledge the use of public data from the Swift data archive.Based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.The Very Long Baseline Array is an instrument of the National Radio Astronomy Observatory.The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under a cooperative agreement by Associated Universities, Inc. S. Kang, S.-S.Lee, W. Y. Cheong, S.-H.Kim, and H.-W. Jeong were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MIST) (2020R1A2C2009003).The KVN is a facility operated by the Korea Astronomy and Space Science Institute.The KVN operations are supported by KREONET (Korea Research Environment Open NETwork) which is managed and operated by KISTI (Korea Institute of Science and Technology Information).The VLBA is an instrument of the National Radio Astronomy Observatory.The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.