Limited impact of jet induced feedback in the multi-phase nuclear interstellar medium of 4C12.50

Although the ultraluminous infrared radio galaxy 4C12.50 at z=0.12 is a promising candidate to reveal how radio induced feedback may regulate star formation in galaxies, we find no solid evidence for current or past impact of this mechanism on the evolution of this system, neither by clearing out the dusty central cocoon efficiently, nor by suppressing star formation. We study in detail for the first time the hot (>~1500 K) molecular gas in this object. The potential impact of the radio jet on this gas phase, as well as on the star formation activity, are investigated. 4C12.50 hosts (2.1+/-0.4)x1e4 Msun of hot molecular gas. An unusually high rotational temperature T =3020+/-160 K is inferred. The molecular gas mass obeys a power law temperature distribution d(M(H2))/dT ~ T^-5 from T~300 K and up to ~3000 K. Both results support that shocks (probably induced by the radio jet) contribute to the heating and excitation of the hot molecular gas. A molecular outflow is not detected. The coupling of the outflowing ionized and neutral outflows with the hot molecular gas is poor. We find no evidence for star formation supression. NIR and MIR integral field spectroscopy at very high spatial resolution (for instance, with the JWST) would be of key value to further investigate these issues.


Introduction
Gigahertz-Peaked Spectrum (GPS) sources are compact and often powerful radio sources.They are estimated to make up around 10% of the bright radio-source population (O'Dea et al. 1998;2021, Sadler 2016)).There are three main hypotheses for their nature, which can vary from source to source.They might be: (1) very young radio galaxies which will evolve into large radio galaxies.(2) compact due to the confinement by interactions with dense gas in their environments; (3) transient or intermittent sources (O'Dea & Saikia 2021).
The GPS sources are entirely contained within the extent of the narrow-line region (NLR, < ∼ 1 kpc).Because of the similar spatial scales, the feedback effects that result from the interaction between the radio source and the dense circumnuclear interstellar medium (ISM) can be very strong.Thus, they are interesting targets to investigate whether and how radio induced feedback may affect the evolution of galaxies in the early phases of radio activity (O'Dea 1998;Holt et al. 2003Holt et al. , 2011;;Morganti et al. 2013;Santoro et al. 2020).
GPS sources (P 5GHz ∼10 33 W Hz −1 ; O'Dea 1998; Holt et al. 2003).Radio emission stretching outside the host galaxy provides evidence of a previous radio outburst, that occured ∼ 10 7−8 yr ago (Stanghellini et al. 2005).The jet may have restarted only recently (<10 5 yr) after a long period of inactivity or be a central component in a continuous supply of energy from the core to the extended lobes (Lister et al. 2003;O'Dea et al. 2000;Stanghellini et al. 2005;Morganti et al. 2013).Jet frustration appears to be working at some level, but the amount of mass seems to be insufficient to confine the jet completely (Morganti et al. 2004(Morganti et al. , 2013)).
4C12.50 is very rich in dust and molecular gas with a mass of cold ( < ∼ 25 K) molecular gas ∼10 10 M (Dasyra & Combes 2012), in the range of other ULIRGs (Mirabel 1989;Evans et al. 1999;Solomon & Vanden Bout 2005).It is the most molecular gas-rich radio galaxy known in the nearby Universe (e.g. Ocaña Flaquer et al. 2010;Smolčić & Riechers 2011).Most of this gas is highly concentrated within few kpc of the western active nucleus, including a small, ∼4 kpc wide disk (Fotopoulou et al. 2019).
This primary nucleus hosts fast outflows (up to ∼2000 km s −1 ) which have been detected in emission in the ionized phase and in absorption in the cold HI circumnuclear gas (e.g.Holt et al. 2003Holt et al. , 2011;;Spoon & Holt 2009a, Morganti et al. 2005;Rose et al. 2018; see also Rupke et al. 2005a).The ionized and the neutral HI outflows in 4C12.50 are driven by the radio plasma.The compact jet seems to be fighting its way out and emerging from the dense cocoon of gas and dust in the western nucleus, clearing it out via these kinematically extreme outflows (Morganti et al. 2003;2013).
It is of great interest to investigate the potential impact of this feedback mechanism in the molecular phase, since this is the fuel used by galaxies to form stars.The evidence for a cold molecular outflow in 4C12.50 is controversial.Blueshifted CO(3-2) absorption (shift relative to the systemic velocity ∆V ∼-950 km s −1 ) and tentative CO(1-0) absorption (∆V ∼-1100 km s −1 ) (Dasyra & Combes 2012) have been reported at similar velocity as the neutral outflow.On the other hand, posterior studies (some making use of the Atacama Large Millimeter Array, ALMA) could not corroborate this outflow, neither in absorption or emission using different CO lines (Dasyra et al. 2014;Fotopoulou et al. 2019;Lamperti et al. 2022).The detection of outflow signatures in the warm (∼400 K) molecular gas is also con- troversial.The high velocity blueshifted mid-infrared (MIR) H 2 emission detected by Dasyra & Combes (2011), could not be confirmed by Guillard et al. (2012).
The outflow imprint is yet to be searched for in the hot (>1500 K) molecular phase.In fact, this gas, which emits the strongest lines in the near-infrared (NIR), has been barely studied in 4C12.50 (Veilleux et al. 1997;Rose et al. 2018).Our goal is to fill this gap in the knowledge of this otherwise widely studied system.We investigate the general hot H 2 properties (mass, temperature, reddening, excitation mechanisms) and whether the outflow has affected it.Moreover, we present new results on the ionized phase, including the high ionization coronal gas.All the results are discussed in the context of prior studies of 4C12.50.This work is based on Gran Telescopio Canaria (GTC) Kband and Very Large Telescope (VLT) J, H, K bands long slit spectroscopy of the primary western active nucleus.It is also based on ALMA 12-m array CO(2-1) and 220 GHz continuum observations described in Lamperti et al. (2022) (program 2018.1.00699.S; PI: M. Pereira-Santaella).See also Pereira-Santaella et al. 2021).
The paper is organized as follows.We describe the observations and data in §2.The spectral fitting techniques are explained in §3.The results of the study of the ionized (including coronal) and the molecular gas components in 4C12.50 are presented in §4 and discussed in §5.The summary and conclusions are in §6.
In order to find a compromise between spectral resolution and flux coverage, the slit width used during the observations was 0.8 , adapted to the K-band seeing size (FWHM∼0.8).The instrumental profile measured from the arc lines is FWHM IP =6.32±0.44Å (85.7±6 km s −1 at ∼2.2 µm).
The slit position angle PA 104 • N to E was chosen to align the slit with the two nuclei (Fig. 1).The only line detected from the secondary, Eastern nucleus is Paα , which is ∼90 times fainter than that from the primary nucleus.For this reason, we will focus our study on the primary nucleus.
Eight spectra were obtained in 4 different nights (Feb 12 th , April 8 th , 9 th , and 12 th of 2022).The total exposure time on source was 8×3360 sec = 26880 sec or 7.5 hours.A typical ABBA nodding pattern was applied.
The spectra were reduced using several Python routines customised by GTC staff for EMIR spectroscopic data.The sky background was first subtracted using consecutive A-B pairs.They were subsequently flat-fielded, calibrated in wavelength, and combined to obtain the final spectrum.To correct for telluric absorption, we observed a telluric standard star (HR 5238) with the same observing set-up as the science target, immediately after the 4C12.50observations and at similar airmass.To apply the correction we used a version of Xtellcor (Vacca 2003) specifically modified to account for the atmospheric conditions of Roque de los Muchachos observatory in La Palma (Ramos Almeida et al. 2009).Flux calibration was applied using the spectrum of the standard star obtained with a wide 5 slit.
The K-band EMIR spectrum is shown in Fig. 2 (left).A 0.8 ×1.4 aperture centered at the continuum centroid of the western nucleus was chosen.This optimizes the extraction of the maximum line fluxes and the signal to noise for several important faint lines so that the kinematic parameters can be constrained more accurately.The strange continuum bump marked with a red arrow in Fig. 2 is an artifact.It suggests a problem with the relative flux calibration in that region of the spectrum.At bluer λ the continuum shape is very similar in the eight spectra and the accuracy of the absolute flux calibration is estimated ∼10%.The shape varies more towards the red.The comparison of the line fluxes in the individual spectra suggests an additional ∼15% uncertainty on the flux calibration in that spectral window (H 2 1-0 S(2) and redder).This will not affect our conclusions.Moreover, we will also have the Xshooter NIR spectrum for comparison.
The Xshooter spectrum was described and shown in Rose et al. (2018).It covers the J+H+K spectral range so that very valuable information can be obtained from additional emission lines (the K band range is shown in Fig. 2, right).The spectral resolution values for the 1.2 slit were 69.2±1.3,70.1±0.9and 72.3±1.3 km s −1 for the J, H and K bands respectively.The slit was placed at PA 20 • , the paralactic angle during the observations (Fig. 1).The pixel scale for the NIR arm is 0.2 .The spectrum was extracted from a 1.2 ×4.0 aperture, also centered at the western nucleus continuum centroid.The seeing was in the range 0.88 -0.95 (g band).The authors reported a < ∼ 8% relative flux calibration accuracy.
The Xshooter data has the advantage of covering a much wider spectral range, including very valuable molecular lines in the J band, as well as 1-0 S(1) (which lies on the edge of the EMIR spectrum).The advantage of the EMIR spectrum is its higher S/N.The r.m.s values in different continuum windows are in the range ∼(1.3-2.6)×10−18 erg s −1 cm −2 Å −1 , ∼1.7-3.5 times lower than for the Xshooter data in the same regions.This spectrum is also more severely affected in some spectral windows by artifacts, including a glitch in the 2.272-2.283µm range (see Rose et al. (2018) for a detailed explanation).Both spectra are therefore valuable.
Galactic extinction is very low (A K =0.01) and no correction was applied for this effect.
In spite of the different aperture sizes, we measure almost identical Paα fluxes in the EMIR (F Paα =(1.6±0.2)×10−14 erg s −1 cm −2 ) and the Xshooter spectra (1.5±0.1)×10−14 erg s −1 cm −2 ).The same can be said about most (if not all) K-band emission lines (see next section).This suggests that the Paα and other line emission is very compact and strongly concentrated in the primary nucleus and there is little contamination by more extended gas in both apertures.

Analysis
Relevant H and H 2 lines detected in the NIR spectrum of 4C12.50 are listed in Tables 1 and 2. Except for Paα and the blend of Brδ, H 2 (1-0) S(3) and [SiVI]λ1.9630(see below), each line flux was measured by integrating the area underneath its spectral profile and above the continuum level.The FWHM were obtained from single Gaussian fits.The splot task in iraf was used.A single Gaussian provides acceptable fits in general, even when slight asymmetries are hinted.
For complex line profiles and blends (see below), we applied multiple-Gaussian fits using the Starlink package dipso.We used the minimum number of components required to produce an adequate fit, without leaving significant features in the residuals.dipso is based on the optimization of fit coefficients, in the sense of minimizing the sum of the squares of the deviations of the fit from the spectrum data.The output from a complete fit consists of the optimized parameters (FWHM, central λ, peak and integrated fluxes) and their errors (calculated in the linear approximation, from the error matrix).
All FWHM values were corrected for instrumental broadening by subtracting the instrumental profile in quadrature.

Paα
Paα shows a very prominent blue broad wing (Fig. 3).Three kinematic components are identified in the multi-Gaussian fit of the EMIR spectrum with FWHM and velocity shifts relative to the narrow component shown in Table 1.The results obtained with the Xshooter spectrum are in good agreement (Table 1).The only difference is that an additional very narrow component (FWHM=101±12 km s −1 and redshifted by +167±9 km s −1 ) is isolated in these data  set.This can be due to the higher spectral resolution or the different position angle and width of the Xshooter slit picking up an additional narrow component.It only contributes ∼2% of the total line flux.Holt et al. (2003, see also Holt et al. 2011;Rodríguez Zaurín et al. 2013;Rose et al. 2018) found relatively similar complex kinematics for the strongest optical emission lines from the western nucleus, including forbidden transitions such as [NeV]λ3426, [OIII]λλ4959,5007, [OI]λ6300 and [SII]λ6716,6731 ([NeV], [OIII], [OI] and [SII] hereafter).The three components isolated in the [OIII] dou-blet have FWHM=340±23 (narrow), 1255±12 (intermediate) and 1944±65 km s −1 (broad), with these last two shifted by -402±9 and -1980±36 km s −1 relative to the narrow, systemic component.Rose et al. (2018) identified an additional, even more extreme component in the [OIII] lines using the Xshooter spectrum mentioned here.It has FWHM=3100±200 km s −1 and ∆V =-331±60 km s −1 .This 4th component is not apparent in the Paα fit of either the EMIR or the Xshooter spectra and thus we will not consider this option.The Paα narrow component has z=0.12175±0.00008.This is consistent with the z of the narrow [OIII]λ5007 (z sys =0.12174±0.00002).According to Holt et al. (2003), this component is at the systemic z implied by the stars.It is also consistent within the errors with the CO(2-1) redshift z= 0.121680±0.00004(Lamperti et al. 2022).In what follows, we will also assume that the narrow Paα component is at z sys .

The Brδ+H 2 S(3)+[SiVI] blend
A broad pedestal is apparent underneath Brδ (1.9451µm), H 2 (1-0) S(3) and [SiVI]λ1.9630(Fig. 4).Our aim is to discern its nature, since it reveals very high velocity gas that may trace a coronal and/or molecular outflow.It is also necessary to constrain the flux of the S(3) line.
Due to the complex blend of lines, it is not possible to apply a multiple Gaussian fit avoiding degeneracies.We have applied a different method consisting of creating an artificial Brδ that we subtracted from the original data, as explained below.We then fitted the residual spectrum, which should retain the contribution of S(3) and [SiVI].This method was applied to both the EMIR and the Xshooter spectra.
To create the expected artificial Brδ profile for a given spectrum, we assumed the same spectral shape as Paα.The differential reddening of the three kinematic components (Holt et al. 2003(Holt et al. , 2011) ) does not affect the line profile significantly.For E B−V = 0.59±0.11(as derived by Hα/Hβ, Paα/Hβ and Paβ/Hβ, Rose et al. 2018), the expected flux ratio Paα/Brδ ∼18.1 is very close to the case B value, 18.29 (Osterbrock & Ferland 2006).Thus, we used the fitted Paα profile (see previous section), shifted it to the Brδ wavelength and divided its flux by 18.1.The expected Brδ contribution is shown in Fig. 4, together with the con-tinuum fitted by interpolating between the blue and red sides of the line blend.The results of subtracting Brδ from the original spectra are.A broad pedestal is still obvious in both cases.
We then fitted this new spectrum with the smallest number of components (three) that provides a reasonable fit to the residual blend.The results are shown in Table 3 and Fig. 5.The two narrower components correspond to H 2 S(3) and the core of [SiVI].
The pedestal is consistent with a very broad component with FWHM=3500±375 km s −1 for the EMIR spectrum (values for Xshooter will be given in brackets: 3355±247 km s −1 ).One possibility is that it is dominated by a molecular outflow, blueshifted by -425±83 (-478±136) km s −1 relative to the narrow S(3) component.The fact that it is not detected in other molecular lines is not in contradiction.If its contribution relative to the narrow component was similar in all H 2 lines (F broad /F narrow ∼1.15 (1.07), as in S(3)), the expected fluxes would be below or just close to the 3σ detection limits in all cases.
While we cannot unambiguously rule out a broad H 2 S(3) component, the very turbulent kinematics of the ionized gas (Paα, [OIII], etc) suggests that the broad pedestal may be dominated by a coronal [SiVI] outflow.An intermediate situation, with contribution from both broad S(3) and [SiVI] or even contamination by [Si XI]λ1.9359 on the blue side of the blend could also be possible.Overall, this just reflects the difficulty to deblend the Brδ+H 2 S(3)+[SiVI] lines.
In spite of this, several arguments support that the pedestal is due to a coronal outflow.In this case, the line would consist of two components, both blueshifted relative to z sys (Table 3).The blueshift of both components may indicate that the whole [SiVI] emitting gas (and not only the broad component) is outflowing, although asym-

Flux(norm)
Predicted Brδ Original spectrum -Brδ Original spectrum metries in the spatial and/or velocity distributions of the systemic coronal gas cannot be discarded.The broad component (the pedestal) has a blueshift of ∆V sys =-1296±133 km s −1 (EMIR) or -1398±136 km s −1 (Xshooter) and it contributes 83±11% of the total [SiVI] flux.This interpretation is supported by the similar kinematics of the coronal [FeVII]λ6087 line (Fig. 6, Table 3).The profile is very broad and asymmetric and clearly blueshifted relative to z sys .Two components are isolated, both broad and blueshifted.
We therefore propose that the H 2 S(3) emission is traced by the narrow S(3) above the pedestal, while this (possibly the whole [SiVI] flux) is dominated by a coronal outflow.

The warm ionized and coronal gas
The extreme Paα kinematics are roughly consistent with those seen in the optical lines, including the forbidden ones (Sect.3.1).This implies that the broad Paα (FWHM∼2750 km s −1 and ∆V =-1400±43 km s −1 (see Table 2) is not emitted by the broad line region (see also Rupke et al. 2005a).It is instead emitted by the kinematically extreme ionized compact outflow, whose radial size ∼69 pc was measured by Tadhunter et al. (2018)   +1.66  −1.01 ) × 10 5 cm −3 , compared with n = (2.94+0.71  −1.03 )×10 3 , (1.47 +0.60 −0.47 )×10 4 cm −3 for the narrow and the intermediate components respectively (see also Rose et al. 2018).With these values, and using the reddening corrected Paα luminosities we calculate the mass of each kinematic component as: where L Hβ is the the Hβ luminosity, inferred from the reddening corrected L Paα and assuming case B Paα Hβ =0.332 (Osterbrock & Ferland 2006), m p is the mass of the proton, α eff Hβ =3.03×10 −14 cm −3 s −1 is effective Case B recombination coefficient of Hβ for T =10,000 K and n =10 4 cm −3 (Osterbrock & Ferland 2006), h is Planck's constant, ν Hβ is the frequency of Hβ, and n is the electron density.The total mass is M HII ∼9.0×10 5 M , of which ∼6% corresponds to the broadest component and ∼28% to the total outflowing gas (broad+intermediate components).
We plot in Fig. 7 the observed relative contribution of the broadest component to the total line fluxes F br Ftot against the critical density n crit for all the forbidden lines with this information available.Although with a large scatter, Line  1984).The authors proposed that the outflow in MRK477 has been triggered at < ∼ 220 pc, possibly at < ∼ 30 pc, from the AGN (also by the radio jet) and the correlation shows how its emission weakens as it propagates outwards from the inner denser coronal region (e.g.Müller Sánchez et al. 2006;Rose et al. 2011) outwards in the NLR, following the decreasing density gradient.We propose a similar scenario for 4C12.50.
This also suggests that a single mechanism (the radio jet in this case) is responsible for the outflow identified in all emission lines emitted by the ionized phase, from the coronal to the lowest ionization species.This is also supported by the similar, unusually high values of V max = |∆V sys − F W HM/2| (a frequent definition of the maximum outflow velocity, computed for the most blueshifted component; Rupke et al. 2005b) for most lines (Table 4).
The narrow Paα component is at the systemic redshift (Sect.3.1).The prominent blue excess and the almost total lack of redshifted emission (Fig. 3) could be due to an asymmetric spatial distribution of the outflowing gas.This is expected in 4C12.50, since the interaction between the radio source and the ambient gas at both sides of the AGN is indeed very asymmetric (Morganti et al. 2004).
Another possibility is that the receding side of the outflow is almost completely extinguished by dust.Let us consider the broadest, most turbulent component.If the gas moving away on the far side of the nucleus has the same kinematics and emits the same intrinsic Paα flux as the approaching side (∼2.1×10 −14 erg s −1 cm −2 , corrected for reddening with E B−V =1.44 mag, Holt et al. 2003) comparison with the 3σ upper limit < ∼ 9.4×10 −16 erg s −1 cm −2 , implies E B−V >5.8 or A V >23.5 mag.Assuming the same gas-to-extinction ratio as in our Galaxy (Zhu et al. 2017), this corresponds to a column density of HI, N HI ∼2.08×10 21 ×A V cm −2 > ∼ 4.9×10 22 cm −3 , which is consistent with N HI measurements in the central region (∼200 pc) of 4C12.50 (Morganti et al. 2013).
Therefore, it is possible that the receding ioinized outflow is completely extinguished.This would not be surprising, given the dusty circumnuclear environment of this and other ULIRGs and the presence of a circumnuclear torus related to the AGN.The outflow, therefore, could be trully kinematically extreme in this case, with FWHM∼2×FWHM broad ∼5400 km s −1 (∼7000 km s −1 for the coronal gas, Table 3), only comparable to those seen in a handful of high z extremely luminous quasars (Perrotta et al. 2019;Villar Martín et al. 2020).The outflow mass, M ∼2.5×10 5 M (broad and intermediate Paα components, see above), the mass outflow rate, Ṁ and the kinetic power Ėkin would still be moderate.These are calculated as (e.g. Rose et al. 2018): where V is the average velocity of the outflowing gas and r is the outflow radius.We assume V = V max (Rupke et al. 2005b).For a Gaussian of a given FWHM centered at V sys , V max =FWHM/2.In the current scenario, V max ∼2783 km s −1 and∼1029 km s −1 for the broadest and intermediate components respectively (Table 1).Assuming r=69 pc (Tadhunter et al. 2018), then Ṁ ∼3.3 M yr −1 and Ė ∼1.1×10 43 erg s −1 for the broad component; Ṁ ∼4.4 M yr −1 and Ė ∼2.0×10 42 erg s −1 for the intermediate component.In total, Ṁ ∼7.7 M yr −1 ( SFR∼100 M yr −1 , Rupke et al. 2005a) and Ė ∼1.3×10 43 erg s −1 , which is ∼0.14% of the bolometric luminosity L bol ∼9×10 45 ergs.The values are still moderate (see also Holt et al. 2011;Rose et al. 2018).Given, in addition, the small size (much smaller than the effective radius of the western bulge component, 2.59±0.58kpc, Dasyra et al. 2006) and volume apparently affected by the ionized outflow, it is not clear whether it will affect the evolution of the host galaxy.

Rotational temperature, T rot
We have calculated the H 2 rotational excitation temperature, T rot , using the extinction corrected fluxes and upper limits of the NIR H 2 lines and following Pereira-Santaella et al. (2014).The molecular lines are not necessarily affected by the same extinction as the ionized gas emission.To estimate E mol B−V , we have used H2 2−0 S(3)λ1.1175H2 2−1 S(3)λ2.0735, which has a theoretical value of 0.83.Both lines are detected in the Xshooter spectrum.Because they are faint and noisy, different aperture sizes were attempted to maximize the S/N.We infer a value of 0.58±0.05.This implies E mol B−V =0.35±0.08,which is not significantly different in comparison with the ionized gas total extinction E(B-V)=0.59±0.11(Holt et al. 2003;Rose et al. 2018).
We show in Fig. 8 the result of modelling the relative population levels of the NIR H 2 lines using single excitationtemperature LTE models (see Pereira-Santaella et al. 2014).We infer T rot =3020±160 K.This is quite high in comparison with typical values in nearby AGN and ULIRGs.As an example, Riffel et al. (2021) obtained T rot in the range ∼760-2075 K in a sample of 36 nearby Sy1 and Sy2 (0.001 < ∼ z < ∼ 0.056) selected among the hard X-ray (14-195 keV) sources in the Swift Burst Alert Telescope (BAT) survey (Oh et al. 2018).The ULIRGs analysed by Davies et al. (2003), which also host very large M hot (see next section), have maximum temperatures of ∼2400 K.In general, T rot <2500 K in galaxies, including U/LIRGs and AGN, (Murphy et al. (2001); Davies et al. (2003); Rodríguez-Ardila et al. (2004, 2005); Ramos Almeida et al. (2009); Mazzalay et al. (2013); Pereira-Santaella et al. (2014).Such high temperature is not found either in hot molecular outflows (T rot ∼1900-2300 K), although there is only a handful of objects where it has been possible to isolate the NIR H 2 outflow emission (Emonts et al. 2014;Tadhunter et al. 2014;Ramos Almeida et al. 2019).
The NIR H 2 lines extend the gradient found by Guillard et al. (2012) in 4C2.50 to higher temperatures.They fitted three components using the MIR H 2 lines with T ∼100, 275 and 1500 K respectively.A temperature gradient, described with a power law, exists in the molecular gas of numerous galaxies (Davies et al. 2003;Ogle et al. 2010;Guillard et al. 2012;Pereira-Santaella et al. 2014;Togi & Smith 2016).

Mass
The mass of hot molecular gas M hot H2 can be estimated from the extinction corrected S(1) flux, F H21−0S(1) , under the assumptions of local thermal equilibrium and an excitation temperature of 3020±160 K (e.g.Scoville et al. 1982;Riffel et al. 2014).We infer F H21−0S(1) =(2.44±0.41)×10−15 erg s −1 cm −2 (we use the Xshooter line flux in this calculation because it is more accurate), assuming E B−V =0.35±0.08 (see Sect. 4.2.1).If the gas was not thermalized, the mass would be underestimated.
We obtain M hot H2 = (2.10±0.44)×10 4 M .We show in Fig. 9 Mezcua et al. 2015;Riffel et al. 2021).It is similar to other nearby ULIRGs and consistent with the value expected from the observed M hot H2 vs. L IR correlation.If slit losses were significant, the intrinsic mass would be higher, although this situation is unlikely.The hot molecular gas in luminous AGN is usually mostly concentrated within < ∼ several×100 pc (Mezcua et al. 2015;Riffel et al. 2021).At the z of 4C12.50, such physical size is not resolved spatially in our data.Moreover, since the K-band seeing (see Sect. 2), was significantly narrower than the Xshooter 1.2 slit, losses are expected to be low in the direction perpendicular to the slit also.The ratio of hot to cold H 2 masses,
The dependence of M H2 with the excitation temperature, T , is shown in Fig. 10.The mass at T ∼3000 K follows the trend of this and other galaxies, where the bulk of the molecular mass is concentrated at the lowest temperature (Guillard et al. 2012).The mass-temperature function at T > ∼ 300 K is well described in 4C12.50 with a power law: M H2 (M ) = 9×10 17 T −3.9 (coefficient of determination, R 2 =0.937) or dMH2 dT ∝ T −n with n ∼5.Aperture effects do not have a significant impact in this plot.Although the warm masses (Guillard et al. 2012) were obtained with Spitzer IRS data (which provides much larger aperture sizes (3.6 × 57 to 11 × 22 , depending on the line), the gas is expected to be highly concentrated within a spatial region smaller than the physical region covered by the Xshooter aperture (2.6×8.8 kpc 2 ).A mass-temperature power law distribution of the molecular gas at T > ∼ 100 K is frequently observed (or assumed) for galaxies independently of the excitation mechanism.Togi & Smith (2016) found that a continuous powerlaw distribution of rotational temperatures, with dMH2 dT ∝ T −n , reproduces well the H 2 excitation from a wide range of galaxy types using a single parameter, the power-law slope n (see also Pereira-Santaella et al. 2014).This model, can recover the mass at T > ∼ 100 K, with n in the range 3.79-6.4and average 4.84±0.61.n gives information on the relative importance of gas heating by shocks, photoelectric heating, UV pumping, etc.According to Neufeld & Yuan (2008) n ∼4-5 is consistent with the predictions of simple models for paraboloidal bow shocks.
For 4C12.50, the high T rot and the n ∼5 power law mass-temperature distribution suggest that shocks play an important role on the excitation of the molecular gas at T > ∼ 300 K.

Excitation mechanism
We have seen that the high T rot and the power law masstemperature distribution suggest that shocks play an important role on the excitation of the molecular gas at T > ∼ 300 K. We now check whether the influence of shocks is apparent in the diagnostic diagram [Fe II] λ1.257µm/Paβ vs. H 2 1-0 S(1)/Brγ (Larkin et al. 1998;Rodríguez-Ardila et al. 2005;Riffel et al. 2021; see also Colina et al. 2015).[Fe II] λ1.257µm/Paβ=0.73±0.08 for 4C12.50 is obtained from the Xshooter spectrum.
This, however, does not imply that shocks are not present and a more likely scenario is that a combination of excitation mechanisms exists.4C12.50 is part of the MO-HEG (molecular hydrogen emission galaxy) sample of radio galaxies hosting fast ionized and HI jet-driven outflows studied by Guillard et al. (2012).They discarded AGN Xray heating as the dominant source of excitation of the MIR H 2 in these systems, based on the large ratio of the H 2 line luminosity (summed over the MIR S(0) to S(3) rotational transitions) to the unabsorbed 2-10 keV nuclear X-ray luminosity (see also Ogle et al. 2010).Instead, shocks are proposed as the main excitation mechanism.Using magnetic shock models, they showed that the dissipation of a small fraction (<10%) of the kinetic energy of the radio jet heats the gas to a range of temperatures (Sect.4.2.1) and is enough to power the observed mid-IR H 2 emission.
An important difference between 4C12.50 and most MOHEGs is that it hosts a very luminous AGN (Ogle et al. 2010).Spoon et al. (2009b) and Guillard et al. (2012) show that, while the lower ionization lines in the MIR are consistent with shocks, the high ionization lines (in particular [NeV]), arise primarily from photoionization of the gas by the AGN.The presence of strong coronal lines (Sect.3.2) indicates that it must contribute to the excitation of at least the ionized gas.This could explain why this system, which is expected to host strong jet-induced shocks affecting the molecular gas (see also previous section), is located in the AGN area of the diagnostic diagram.

Kinematics: a molecular outflow cannot be confirmed
Dasyra & Combes (2011) identified prominent blue wings in two out of the three H 2 MIR lines detected in the Spitzer spectrum of 4C12.50,H 2 (0-0) S(1) at 17.04 µm and S(2) at 12.28 µm.The main component, which they consider to trace the systemic velocity, is spectrally unresolved with FWHM < ∼ 550 km s −1 .The blue wing is ∼2.6 times fainter, it is shifted by ∼-640 km s −1 and has an instrumentallycorrected FWHM∼521 km s −1 (errors unavailable).They propose this is produced by an AGN jet or wind-driven outflow.The outflow mass, 5.2×10 7 M , is a very high fraction, ∼27%, of the total warm (∼400 K) H 2 mass.Guillard et al. (2012) also studied the MIR H 2 lines based on Spitzer data.They could not confirm the outflow and report the tentative detection of a blue wing in H 2 (0-0) S(1) only.
We investigate in this section whether the molecular outflow is detected in the NIR H 2 lines.
The hot molecular gas has much simpler kinematics than the ionized gas, whose motions are strongly defined by the outflow triggered by the radio source (Sect.3.1).The NIR H 2 line profiles are well reproduced by single Gaussians and have instrumentally-corrected FWHM with median values ∼446±13 and 455±10 km s −1 for the EMIR and Xshooter spectra respectively (Table 2).No broad components that may be indicative of an outflow are detected for any line (see below).
Information on the potential presence of gas turbulence can be obtained by comparing with the stellar FWHM * .As in many interacting systems, complex, non-ordered stel-  12. Observed emission (black) of H2 S(5) (left), S(2) (middle) and S(1) (right) and the expected profiles (red), for which we assume the presence of a molecular outflow with similar kinematic properties and relative flux contribution as in the MIR H2 lines.The troughs at the peak of the S(2) and S(1) are due to the noise.lar motions have been identified in 4C12.50 (Perna et al. 2021).Dasyra et al. (2006) measured FWHM * =392±112 km s −1 , which is affected by a large uncertainty.The narrow (systemic) component of the emission lines from the ionized gas has FWHM=340±23 km s −1 for the optical lines (Holt et al. 2003; 319±6 km s −1 according to Rose et al. 2018) and FWHM=419±18 km s −1 or 392±15 km s −1 for Paα according to the EMIR and the X-shooter spectra respectively; Table 2).All are consistent with FWHM * within the errors, although the FWHM of the narrow Paα is somewhat broader (3.2σ significance) in comparison with the optical lines.This is not surprising, given the rich dust content of 4C12.50.Slightly larger stellar velocity widths are often inferred for galaxies using NIR stellar features compared with the optical values.This suggests that the NIR features probe more deeply embedded (and therefore higher velocity dispersion) stellar populations than the optical ones (Caglar et al. 2020).
Depending on which value we use (Fig. 11), the molecular lines are all broader than the narrow optical component or similar to the narrow Paα.This is not likely to be affected by aperture effects, given the diversity of aperture sizes for the data and the consistency of the result for most lines.Based on this comparison, therefore, it is not possible to confirm whether the hot (neither the warm) H 2 gas shows turbulent motions in relation to the systemic motions.
An indication of turbulence of the warm and hot molecular gas is suggested by the fact that all lines appear to be broader than CO(2-1) and possibily CO(1-0).The origin is however unknown.Although the dominant kinematic component of the hot molecular gas in ULIRGs and Seyferts is rotation, non rotational components are often also identified (e.g.Bianchin et al. 2021).They could be gas elements out of dynamical equilibrium, such as gas streams related to galaxy interactions, inflows or outflows Dasyra & Combes (2011); Guillard et al. (2012); Fotopoulou et al. (2019).
Another result of the kinematic analysis is that a counterpart of the prominent MIR H 2 molecular outflow identified by Dasyra & Combes (2011) is not confirmed.We show next that if the NIR H 2 lines had the same relative contribution of the outflow as in the MIR, we should have detected it.
For this, we have created the expected spectral profiles of H 2 S(5), S(2) and S(1) assuming the same kinematic substructure in km s −1 as the MIR lines (instrumental broadening has been taken into account).We then compared these with the data.For S(1) we have used the Xshooter spectrum because the line is on the edge of the EMIR data.The results are shown in Fig. 12.The red lines are the expected line profiles.The molecular outflow should have been detected for the three lines as a clear blue excess, but this is not the case.
A faint blue excess may be hinted on the blue wing of S(2) in the EMIR spectrum (Fig. 12).However, it is not clear this is real based on the non detection in the Xshooter spectrum, the structure of the noise in adjacent spectral regions and the absence of the wing in other H 2 lines, both in the Xshooter and EMIR spectra.
In summary, we find no evidence for a hot molecular outflow in the NIR H 2 lines.Kinematic turbulence is suggested by somewhat broader line widths in comparison with CO(2-1) and CO(1-0).The origin of this turbulence can be diverse.Given the clear role of the jet induced shocks in heating the H 2 gas, it seems natural that they may also affect the kinematics inducing some turbulence (see alsoGuillard et al. 2012).
It is possible that faint spectroscopic features related to the feedback induced by the jet are lost in the overwhelming glare of the bright nuclear line emission in the spatially integrated spectra analyzed in this work.NIR and MIR integral field spectroscopy at very high spatial resolution (for instance with NIRSPEC and/or MIRI on the JWST) would be of key value to map in two spatial dimensions the impact of the interaction between the radio jet and the ambient hot and warm molecular gas with spatial resolution of FWHM∼several×100 pc.

Discussion
Observations of ionized and neutral gas outflows in radio galaxies suggest that AGN radio jet feedback has the potential to affect the gaseous environment of their hosts from nuclear to galactic scales, and out into the circumgalactic medium.This feedback mechanism may also be relevant in systems hosting moderate power radio sources (e.g.Villar Martín et al. 2017;2021, Jarvis et al. 2019;;Girdhar et al. 2022).To determine whether and how the radio sources can regulate the star formation in their host galaxies it is necessary to understand how the molecular gas is affected (e.g.Tadhunter et al. 2014;Morganti et al. 2021).
4C12.50 is a very relevant system in this context.If radio induced feedback can regulate the star formation ac-tivity in galaxies, it is a promising candidate to reveal this phenomenon in action.The compact twin jet is still within the region where a huge accumulation of molecular gas and dust formed during the course of a gas-rich merger, prior to the coalescence with the companion secondary nucleus.This process has favoured intense star formation (see Sect. 1).The large mechanical energy of the powerful radio source and the high concentration of gas have resulted on a strong jet-gas interaction that has triggered kinematically extreme ionized and neutral outflows (Sect.4.1 and references therein).
The regulation of the system's evolution by the jet may occur, on one hand, by expelling the dusty nuclear cocoon.The system may transition in this way from a ULIRG to an optically less obscured radio galaxy (Sect.1).On the other hand, the radio source could act on the molecular reservoir by heating (or cooling) gas, which may result in quenching (or triggering) star formation and thus halting (or triggering) the growth of the galaxy.

Is the radio jet clearing out the cocoon?
Like many local ULIRGs, 4C12.50 hosts an obscured and very compact nucleus.Based on the CO(2-1) ALMA data described in Lamperti et al. (2022), we measure a deconvolved half light radius of the CO(2-1) emisssion of r CO =358±2 pc (see also Evans et al. 2002).A molecular gas mass log(M H2 ) = 9.36±0.05M (∼23% of the total cold molecular gas mass) is enclosed within.Most of the remaining cold molecular gas is in a ∼4 kpc wide disk (Fotopoulou et al. 2019).
For comparison, compact obscured nuclei with r CO ∼200-400 pc and M H2 ∼(0.1-several)×10 9 M are common in local ULIRGs (Condon et al. 1991;Soifer et al. 2000;Pereira-Santaella et al. 2021).The small sizes of the twin jet in 4C12.50 (total size ∼220 pc) and of the ionized (radial size r ∼69 pc, Tadhunter et al. 2018) and neutral (r ∼100 pc, Morganti et al. 2013) outflow triggered by it indicate that they are well within the dusty cocoon.Therefore, the clearing up process may be at work.
Currently, the radio plasma appears to be dragging a small fraction of the total cocoon mass, with ∼8×10 5 M of ionized gas and 1.6×10 4 M of neutral gas at a total mass outflow rate of ∼10 M yr −1 at most (Holt et al. 2011;Morganti et al. 2013;Rose et al. 2018).Considering all studies of the molecular gas component, there is no solid evidence for a molecular outflow in 4C12.50, neither in emission, nor in absorption (Sect. 1 and 4.2.4), not at any temperature, hot (this work), warm or cold.In spite of the clear and dramatic impact of the jet on the ionized and neutral gas kinematics, it has no obvious effects on the kinematics of the hot (neither warm) molecular gas except, possibly, some enhanced turbulence in comparison with the cold molecular gas.This is not enough to remove significant amounts of molecular gas and clear out the central cocoon (Guillard et al. 2012).As discussed by the later authors, this implies that dynamical coupling between the molecular gas and the ionized and neutral outflowing gas is weak.
At the total observed Ṁ , and assuming the unlikely case that all the gas is successfully removed from the central region, the radio source would have to be trapped and removing the gas within the cocoon for an unrealistically long time ∼2.3×10 8 yr.This is as long as the maximum radio source dynamical ages in radio galaxies in general.These are typically < ∼ several×10 8 yr, with the longest ages measured in giant radio galaxies (e.g.Machalski et al. 2007Machalski et al. , 2009)).
At the moment, it appears that the radio jet is displacing mass at a too slow rate to efficiently clear up the dusty cocoon.It is unclear whether the removal of a significantly lower amount of mass would suffice to promote the transtion into an optically less obscured radio galaxy.5.2.Has star formation been suppressed by jet-induced feedback in 4C12.50?
It is not clear whether the outflow can affect substantially the star formation activity in 4C12.50, given the moderate mass outflow rate ( Ṁ < SF R), kinetic power and small volume (see Rose et al. 2018 for a detailed dicussion).The radio source size is ∼220 pc.The affected volume is much smaller than the galactic bulge (the effective radius of the galaxy hosting the western nucleus is r eff =2.59±0.58kpc, Dasyra et al. 2006).Star formation may be affected within the inner r < ∼ 100 pc, the maximum estimated outflow size, but the impact on larger scales is lacking evidence.As argued by Rose et al. (2018), although the presence of a lower density outflow component that has a high mass and contributes relatively little to the emission line fluxes cannot be ruled out, there is currently no observational evidence for it.
Even if a powerful molecular outflow is not triggered, supression of star formation may occur as a consequence of other processes related to the jet, such as molecular gas heating and/or the injection of turbulence.Evidence for this process lies in the unusually high temperature (T rot =3020±160 K) of the hot component, the power law Tmass function relation (Sect.4.2.1 and 4.2.2) and the fact that shocks are needed to explain the MIR H 2 emission (Sect.4.2.3;Ogle et al. 2010;Guillard et al. 2012).Shocks can be generated in different ways.In ULIRGs in particular, interactions with nearby galaxies can excite large-scale shocks that will cool by means of H 2 emission (Zakamska 2010;Rich et al. 2015).In 4C12.50, a natural scenario is that the compact, powerful jet plays a major role on this regard.The clear impact of the interaction with the ionized and neutral components suggests that shocks are indeed present.If the jet encounters molecular gas on its path, it may inject mechanical energy capable of heating it, even if the kinematics is not significantly affected.
We investigate next whether there is evidence for star formation supression in 4C12.50.Lanz et al. (2016) found that MOHEGs (including 4C12.50)fall below the K-S relation of galaxies, log(Σ SFR ) vs. log(Σ CO ) (Kennicutt 1998).These are the surface density of star formation and the surface density of molecular gas respectively.According to the authors, this suggests that the star formation rate (SFR) is suppressed by a factor of 3-6, depending on how the molecular gas mass is estimated.Approximately 25% of their sample shows a suppression by more than a factor of 10.For 4C12.50, they found a factor ∼10 in comparison with normal galaxies and a factor ∼100 in comparison with ULIRGs.They suggested that the shocks driven by the radio jets are responsible for the suppression, by injecting turbulence into the interstellar medium (ISM).They also found that the degree of SFR suppression does not correlate with indicators of jet feedback including jet power, diffuse X-ray emission, or intensity of warm molecular H 2 emission.
Using the ALMA data previously mentioned, we have revised the location of 4C12.50 in the K-S relation of different galaxy types.For the purpose of comparison with Lanz et al. (2016) , where r SF and r H2 are the radial sizes of the star forming region and of the molecular gas distribution respectively.
The main sources of uncertainty to determine both Σ values come from the uncertain SFR and the areas of the star forming region and the molecular gas distributions.
The main difficulty to estimate the SFR is the uncertain fraction of AGN contribution to L IR , f IR = 100× L AGN IR LIR .Veilleux et al. (2009) applied six different methods to infer the fraction of AGN contribution to the bolometric luminosity L bol , f bol = 100 × L AGN bol L bol and obtained values in the range ∼28-84%, with an average of 57 %.Perna et al. (2021) constrained the range further to f bol ∼60-82% depending on the method.Since ∼90% of L bol is emitted in the infrared for 4C12.50,we assume f bol ∼ f IR .f IR ∼60-82% is a reasonable range of values also based on the the ratio of the mid to far infrared continuum fluxes of 4C12.50, log( F5−25µm 40−122µm )=-0.19, which is intermediate between starburst dominated ULIRGs (∼-1.25) and AGN dominated ULIRGs (∼0.35) (Veilleux et al. 2009).This suggestas a significant but not total contamination of a MIR bump due to the AGN (Lanz et al. 2016).
This implies SFR∼63-141 M yr −1 (Kennicutt 1998).For comparison Rupke et al. (2005b) quote 101 M yr −1 .These SFR are in the range of other ULIRGs (e.g.Daddi et al. 2010a;De Looze et al. 2014;Perna et al. 2021), SFR > ∼ 100 M yr −1 is consistent with the value expected for its gas mass (see Fig. 1 in Daddi et al. 2010a).Lanz et al. 2016 fitted the spectral energy distribution (SED) of 4C12.50 with both an AGN and a SB component, which is essential to obtain a more accurate f IR .They inferred SFR∼24 M yr −1 for 4C12.50 and, thus, f IR ∼93%.This is indeed bellow the value expected for its gas mass.Notice, however, that the authors warn that a combination of dust temperatures not implemented in their method would be more adequate to fit the IR SED.Given all uncertainties, we will consider three possibilities: SFR=24, 63 and 141 M yr −1 .
Regarding the areas of the star forming region and the molecular gas distributions, we have considered several possibilities summarized in Table 5.We measure a half light radius of the CO(2-1) emitting region r CO =358±2 pc (see previous section), which is very similar to the median value, 320 pc, inferred by Pereira-Santaella et al. (2021) for their sample of nearby ULIRGs.We consider this r CO a reasonable upper limit for r SF , since r SF < r CO invariably in ULIRGs (Pereira-Santaella et al. 2021).This is also consistent with MIR observations which indicate that most of the star formation in ULIRGs occurs in very compact regions (<1 kpc; e.g.Soifer et al. 2000;Díaz-Santos et al. 2010;Alonso-Herrero et al. 2016).his implies lower limits for log(Σ SF ) of > ∼ 1.76 or > ∼ 2.19 and 2.54 depending of the three assumed SFR values for 4C12.50.For comparison, most ULIRGs in Pereira-Santaella et al. (2021) sample are in the range 2.8-4.33 .
Based on the CO(1-0), (3-2), and (4-3) line obsevations with ALMA), Fotopoulou et al. (2019) discovered that most of the total molecular gas of 4C12.50 is within r ∼2 kpc (this is consistent with our CO(2-1) ALMA data), including a disk of radius r disk =2 kpc, associated with the western nucleus.Although they detected more extended molecular gas, its mass is relatively low.In case (B), we thus assume r H2 =2 kpc and log(M H2 /M )=10.00, which is the total cold molecular gas of the system.For the sake of comparison with Lanz et al. (2016), we assume r SF =r H2 , although this r SF is likely to be unrealistically large (see above).
Finally, we will assume the values adopted by Lanz et al. (2016), r SF =r H2 =4.2 kpc (case C).This is the extent of the main CO(1-0) spatial component, which Dasyra et al. (2014) found to be marginally resolved within the beam size based on IRAM Plateau de Bure Interferometer data.We emphasize that these authors used α CO =4.3 (K km −1 s −1 pc 2 ) −1 (instead of 0.78), leading to a significantly higher total molecular gas mass of log(M H2 /M )=10.73.
Σ MH 2 and Σ SFR for cases (A) to (C) are shown in Table 5 and their location on the the K-S relation in Fig. 13.As argued by Daddi et al. (2010a), this diagram suggests the existence of two different SF regimes: a longlasting mode for disks and a more rapid mode for starbursts (including U/LIRGs and submillimeter galaxies (SMG)), the latter probably occurring during major mergers or in dense nuclear SF regions.They considered the total neutral and molecular gas.We use only the cold molecular gas in 4C12.50, because the neutral gas mass in ULIRGs is in likely negligible in comparison (Daddi et al. 2010b).
Being a ULIRG undergoing a major merger, we would expect 4C12.50 to lie near the starburst sequence.On the contrary, Lanz et al. (2016) found that the object is not only well below it, but well below the "normal" disks sequence as well (see black diamond in Fig. 13).
Based on our revised location of 4C12.50 in the K-S diagram we cannot confirm this conclusion.In fact, the location based on the new ALMA measurements (Case (A) lower limits represented as triangles in the figure) in Table 5), all lie close or above the starburst region.Given that r CO is most probably a lower limit of r SF , we can claim that we find no evidence for star formation supression in this object.Positive feedback, in the sense of an enhacement of the star formation activity induced by the jet, may be present.To investigate this more accurate determinations of both the SFR and, specially, r SF would be very valuable.
The main reason for the discrepancy with Lanz et al. (2016) is their assumption of unrealistically large r H2 ∼ r SF ∼4.2 kpc.Another important difference is the lower SFR∼24 M yr −1 and the assumed α CO =4.3.While this conversion factor is appropriate for galaxies in the disk sequence, α CO =0.78 is more appropriate for U/LIRGs.This difference was taken into account by Daddi et al. (2010a) 2016), we have assumed the area corresping to each radial size, r, as A = πr 2 .M enc H 2 is the assumed cold molecular gas mass enclosed within rH 2 .SFR in units of M yr −1 .ΣM H 2 in M pc −2 .rCO=358 pc is the CO(2-1) half light radius inferred with ALMA data (see text).r disk =2 kpc is the radius of the molecular disk discovered by Fotopoulou et al. (2019).Case (C) corresponds to Lanz et al. (2016).All molecular masses were calculated with αCO=0.78,except in case (C), for which αCO=4.3 as in Lanz et al. (2016).The last column specifies the symbol used in Fig. 13.  5).
The black diamond comes from Lanz et al. 2016 (case (C)).The triangles represent the values obtained from recent high angular resolution ALMA CO(2-1) data.

5.3.
No solid evidence for a significant impact of radio induced feedback on 4C12.50 evolution Overall, there is no solid evidence for the radio jet to have a significant impact on the evolution of 4C12.50, not by clearing out the dusty central cocoon, nor by suppressing the star formation activity.It is possible that we are observing 4C12.50 in a very early phase of the radio activity.At the moment, the radio source appears to be affecting just a small volume.However, as times passes, unless its advance is frustrated by the rich gaseous medium the jet will propagate through the galaxy and possibly expand beyond it at some point.Signs of a previous radio outburst stretching outside the host shows that this has happened in the past (Lister et al. 2003;Stanghellini et al. 2005).If the radio jet decelerates and inflates into large expanding bubbles, it will affect a much larger galactic volume.Whether it will carry enough energy to expel the cocoon and/or to prevent further gas cooling and/or to inject turbulence, affecting in this way future star formation is unknown (Mukherjee et al. 2016;Morganti et al. 2021).
The presence of other feedback mechanisms acting on different spatial scales should not be forgotten (AGN wind and/or starburst winds).Their action is suggested by Rupke et al. (2005a) results.They detected blueshifted NaID absorption due to a neutral outflow in the Western nucleus with V max = 364 km s −1 , M ∼ 8 × 10 8 M , Ṁ = 0.88 M yr −1 and log( Ė(erg s −1 ))∼41.16 or ∼0.001% of L bol (but see Perna et al. 2021).They locate it up to a radius of ∼10-15 kpc and at a large angle from the radio axis and thus, there is no clear link between this outflow and the radio source.

Summary and conclusions
We present for the first time a detailed analysis of the hot (>1500 K) molecular H 2 gas in the ultraluminous-infrared radio galaxy 4C12.50 at z=0.122 based on GTC EMIR and VLT Xshooter near-infarred long slit spectroscopy of the western primary nucleus.New results on the ionized (including coronal) phase are also presented.
• 4C12.50 hosts a large hot molecular gas content, with M hot H2 =(2.10±0.44)×10 4 M .This is consistent with its high infrared luminosity (in the ULIRG regime) and with the large mass of cold ( < ∼ 25 K) molecular gas.
• An unusually high rotational temperature T rot =3020±160 K is inferred for H 2 , that is at the high end of values measured in galaxies in general.
• The molecular gas mass obeys a power law temperature distribution dMH2 dT ∝ T −5 from T∼300 K and up to ∼3000 K.This is consistent with paraboloidal bow shock model predictions.This, and the high T rot , suggests that, as found by other authors for the warm H 2 MIR emission lines, shocks (probably induced by the radio jet) contribute to the heating and excitation of the hot molecular gas.The jet induced shocks can heat the molecular gas, even it the dynamical coupling is weak.
• The H 2 2-0 S(3)λ1.1175/ 2-1 S(3)λ2.0735ratio implies an extinction of E mol B−V =0.35±0.08 for the hot molecular gas.For comparison, the extinction inferred by other authors for the integrated ionized gas emission is E ion B−V =0.59±0.11.• We find no evidence for a hot molecular outflow in the NIR H 2 lines.The warm (∼400 K) H 2 outflow tentatively identified in the MIR lines by other authors should have been detected, if the relative flux contribution was the same.This is not the case.
• In spite of the dramatic impact of the radio jet induced shocks on the dynamics of the ionized and neutral phases of 4C12.50, the dynamics of the hot molecular gas is not obviously affected.This is consistent with previous works that suggest the poor coupling of the outflowing ionized and neutral outflows with the molecular gas at lower temperatures.
• The extreme Paα kinematics are consistent with those of the optical lines, including several forbidden ones.The prominent broad blueshifted Paα excess is not produced by the broad line region.It is due to the kinematically extreme ionized outflow previously identified in the optical.The outflow dominates (81±4%) the Paα flux.
• The relative contribution of the most turbulent (broadest) outflowing gas component to the total forbidden line fluxes, F br Ftot , correlates with the critical density n crit in log.This suggests that a single mechanism (the radio jet) is responsible for the outflow identified in all lines emitted by the ionized phase, from the coronal to the lowest ionization.
• If the outflowing ionized gas moving away from the observer is completely extinguished, it could be trully kinematically extreme, with FWHM∼5400 km s −1 (∼7000 km s −1 for the coronal gas), only comparable to those seen in a handful of high z extremely luminous quasars.Its mass and energetics would still be moderate.
• Based on ALMA CO(2-1) data and a new estimation of the star forming rate, we revise the location of 4C12.50 in the K-S diagram.Contrary to other studies, we claim that there is no evidence for star formation suppresion in this object.Positive feedback as a consequence of jet induced star formation is not discarded.
• If radio induced feedback can regulate the star formation activity in galaxies, 4C12.50 is a promising candidate to reveal this phenomenon in action.The rich amount of knowledge available for this object, however, has not provided solid evidence so far for this to be the case.We find no solid evidence for current or past impact of this mechanism on the evolution of this system, neither by clearing out the dusty central cocoon efficiently, nor by suppressing the star formation activity.

Fig. 1 .
Fig. 1.SDSS colour composite image of 4C12.50 (z=0.122) with the EMIR (position angle, PA=104 • ) and Xshooter (PA=20 • ) slit positions overplotted.The double nuclei and the prominent merger features are appreciated clearly.This work is focussed on the Western active nucleus.

Fig. 3 .
Fig. 3. Fit of Paα using the 0.8 ×1.4 EMIR (top) and 1.2 ×4.0 Xshooter (bottom) rest frame spectra.Left panels: data (black), fit (red) and residuals (blue), shifted on the vertical axis for visualization.The small excess of flux on the blue wing of Paα may be H2 2-1S(7) λ1.8523.Right: Individual components of the fit of Paα, where the values of FWHM and ∆V in km s −1 (velocity shift relative to the narrow or systemic component, in red) refer to the broadest (blue) kinematic component.It is due to a prominent ionized outflow.The spectra in these and all other figures have been normalized by the peak flux of Paα.

Fig. 4 .
Fig. 4. Brδ, H2 (1-0) S(3) and [SiVI] blend (black) in the EMIR (top) and Xshooter (bottom) rest frame spectra.The blue line shows the predicted Brδ profile (+ continuum) (see text).The results of subtracting Brδ from the original spectra are shown in red.The * marks the location of (or near) a possible artifact in the EMIR spectrum, responsible for the unexpectedly narrow [SiVI] peak.

Fig. 8 .
Fig.8.Modelling the relative population levels of the H2 NIR transitions using single excitation-temperature LTE models.The orange circles and diamonds represent de Xshooter and EMIR measurements respectively.Upper limits were obtained with the Xshooter spectrum.They correspond to H2 1-0 S(6) λ1.7880 (orange), the 2-1 transitions (blue) S(2) λ2.1542, S(3) λ2.0729 and S(4) λ2.0041 and the H2 2-0 transitions (light green) S(2) λ1.1382, S(3) λ1.1175 and S(4) λ1.0998 in the J band.The solid red line shows the single-temperature fit based on the fluxes of the lines detected in the Xshooter spectrum and assuming fully thermalized LTE gas conditions.The EMIR data are shown for comparison.
Fig. 9. M hot H 2 in units of M vs. the 8-1000 µm infrared luminosity LIR in units of L for the sample of Sy1 and Sy2 from R21 (Riffel et al. 2021) and ULIRGs from D03 ( Davies et al. 2003) and PL12 (Piqueras López et al. 2012).No extinction correction has been applied, except for L12 ULIRGs (open orange diamonds; PL12 corr.).The same extinction as the ionized gas has been assumed (Piqueras López et al. 2013).4C12.50 is plotted as a red diamond.The observed and extinction corrected masses are very similar.The size of the symbol is similar to the errobar.The large black and green solid circles (in an almost identical location) are the median of the Sy1 (small black circles) and Sy2 (small green circles) values.The dotted black line shows the best fit to all solid data points (this is, the masses not corrected for extinction).

Fig. 10 .
Fig. 10.Dependence of the molecular gas mass in M with excitation temperature for 4C12.50.The color code clarifies the tempertature ranges considered as "cold", "warm" and "hot" in this work.The small red circles indicate the cold H2 masses infered from CO(2-1) ALMA data for different apertures with the radius varying from 0.5 to 2 (see text).The black solid line shows the best fit to the warm (G12) and hot data points, with the equation and coefficient of determination, R 2 .G12: Guillard et al. (2012); D11: Dasyra & Combes (2011); D14: Dasyra et al. (2014); L22: Lamperti et al. (2022).

Fig. 13 .
Fig. 13.Surface density of star formation compared to gas (cold molecular + neutral) density gas (K-S diagrams;Kennicutt 1998).This figure has been adapted from Fig.2inDaddi et  al. (2010a).The coloured areas roughly enclose different galaxy types indicated with labels.The black solid line ("disks sequence") is a fit to local spirals and z ∼1.5 BzK selected galaxies (slope of 1.42).The upper dashed line is the same relation shifted up by 0.9 dex to fit local (U)LIRGs and SMGs.The dotdashed purple line corresponds to a star formation suppression of a factor of 10 relative to the disks sequence.The coloured symbols are different locations of 4C12.50 for the diversity of parameters used in our calculations.The triangles correspond to cases (A) and the circles correspond to cases (B) in Table5).The black diamond comes fromLanz et al. 2016 (case (C)).The triangles represent the values obtained from recent high angular resolution ALMA CO(2-1) data.

Table 2 .
Measurements of the H2 emission lines detected in the EMIR and/or Xshooter spectra of 4C12.50.H2 1-0S(1) is on the red edge of the EMIR spectrum.The S(3) measurements are based on a simple, 1-Gaussian fit of the line profile above the broad pedestal.

Table 3 .
Fit of the H2 (1-0) S(3)+[SiVI] blend using the EMIR spectrum, after removal of the Brδ contribution (Fig.3, right) and assuming the pedestal is dominated by[SiVI]emission.The results of the[FeVII]spectral fit are also shown for comparison.∆V is computed for this line relative to the narrow component of Hα as measured in the optical Xshooter spectrum where[FeVII]is detected.
based on HST narrow band emission line images.The broadest component contributes Fig. 5. Left: Fit of the H2 (1-0) S(3) and[SiVI]blend using the EMIR (top) and Xshooter (bottom) spectra after subracting the continuum and the contribution of Brδ.Color code as in Fig.3.Right: Individual components of the fit.The FWHM and ∆V relative to zsys correspond to the broadest component (blue), which is responsible for the broad pedestal .Xshooter Fig.6.Xshooter rest-frame optical spectrum centered on [FeVII]λ6087.Left: data.The red vertical line marks the expected location of the line for zsys.The line is clearly blueshifted.Middle: data with fit (red) and residuals (blue, shifted vertically for visualization).Right: data with the two kinematic components isolated in the fits.The FWHM and ∆V relative to zsys correspond to the broad most blueshited component.Flux in arbitraty units.

Table 4 .
Bennert et al. 2006)brock 1984;Rodríguez-Ardila et al. 2002;Álvarez-Márquez et al. 2023)ssion lines.Values from a this work, b Holt et al. (2003), c Guillard et al. 2012.The lines below Paα are in order of incresing ncrit.acorrelation is clear 1 .The emission from the most turbulent gas is much stronger (dominant) in lines with high n crit such as the coronal lines than in low n crit lines, specially those with n crit lower than the outflow density (e.g.[OII]λ3727 with n crit =1300 and 4500 cm −3 for the two doublet components and [SII]λλ6716,6731, n crit =1500 and 4000 cm −3 ).This is consistent with the frequent finding that AGN outflows have a more prominent signature in coronal features compared with other lines from both the ionized and specially the molecular phases(De Robertis & Osterbrock 1984;Rodríguez-Ardila et al. 2002;Álvarez-Márquez et al. 2023).A similar correlation was found in MRK477, the nearest QSO2 at z =0.035 by VillarMartín et al. (2015).The NLR density is expected to decrease with distance from the AGN (e.g.Bennert et al. 2006; see also De Robertis & Osterbrock , the 1 The correlation is weaker with ionization potential.It is not shown for simplicity.Relative contribution of the broadest component, F br F tot to the total line fluxes vs. critical density ncrit in log and cm −2 .The dotted line is the linear fit.F br F tot values are from: this work ([SiVI] and [FeVII]); Holt et al. (2003) and Rodríguez Zaurín et al. (2013) (optical lines) andGuillard et al. 2012 ([NeII]12.8µm).Data with no error bars have no errors available.
, we calculate Σ SFR = SF R

Table 5 .
, who used α CO values adequate for each galaxy type.M yr − kpc −2 (A) r H2 =r SF =r CO , α CO =0.Surface density of star formation and of cold molecular gas for different assumptions of SFR and radial sizes of the SF (rSF) and molecular gas (rH 2 ) distributions.For the purpose of comparison withLanz et al. (