Star formation traced by optical and millimeter hydrogen recombination lines and free-free emissions in the dusty merging galaxy NGC 3256 -- MUSE/VLT and ALMA synergy

A galaxy-galaxy merger and the subsequent triggering of starburst activity are fundamental processes linked to the morphological transformation of galaxies and the evolution of star formation across the history of the Universe. Both nuclear and disk-wide starbursts are assumed to occur during the merger process. However, quantifying both nuclear and disk-wide star formation activity is non-trivial because the nuclear starburst is dusty in the most active merging starburst galaxies. This paper presents a new approach to this problem: combining hydrogen recombination lines in optical, millimeter, and free-free emission. Using NGC~3256 as a case study, H$\beta$, H40$\alpha$, and free-free emissions are investigated using the Multi Unit Spectroscopic Explorer at the Very Large Telescope of the European Southern Observatory (MUSE/VLT) and the Atacama Large Millimeter/submillimeter Array (ALMA). The H$\beta$ image obtained by MUSE identifies star-forming regions outside the nuclear regions, suggesting a disk-wide starburst. In contrast, the H40$\alpha$ image obtained by ALMA identifies a nuclear starburst where optical lines are undetected due to dust extinction ($A_{\rm V}\sim25$). Combining both MUSE and ALMA observations, we conclude that the total SFR is $49\pm2~M_{\odot}$~yr$^{-1}$ and the contributions from nuclear and disk-wide starbursts are $\sim34~\%$ and $\sim66~\%$, respectively. This suggests the dominance of disk-wide star formation in NGC~3256. In addition, pixel-by-pixel analyses for disk-wide star-forming regions suggest that shock gas tracers (e.g., CH$_3$OH) are enhanced where gas depletion time ($\tau_{\rm gas}$=$M_{\rm gas}/SFR$) is long. This possibly means that merger-induced shocks regulate disk-wide star formation activities.

respectively. This suggests the dominance of disk-wide star formation in NGC 3256. In

INTRODUCTION
It has been known for decades that mergers of two disk galaxies can induce nuclear starbursts in the central ∼ 1 kpc region during the coalescence stage (e.g., Keel et al. 1985), triggered by massive gas inflows. In a more early stage of a merger, "disk-wide starbursts" ( 1 kpc) are seen both in theoretical models (Barnes 2004) e.g., the Antennae galaxy (Wang et al. 2004), Arp 140 (Cullen et al. 2007), and NGC 2207+IC 2163 (Elmegreen & Elmegreen 2005). In particular, Cortijo-Ferrero et al. (2017) investigated the star formation history of merging galaxies using optical integral field unit (IFU) observations, showing that disk-wide starbursts arise in the early stages whereas nuclear starbursts occur in the more advanced stages of a merger process. Theoretical models predict that such disk-wide starbursts can be explained by interstellar medium (ISM) turbulence and fragmentation into dense clouds in the disk region (Teyssier et al. 2010;Bournaud 2011). In addition, Saitoh et al. (2009) suggest that shock-induced star formation may be efficient during a merger process. Observationally, it is difficult to quantify both nuclear and disk-wide starbursts in a consistent manner. For example, the mapping of hydrogen recombination lines (i.e., Hα and Hβ) by optical IFUs enables us to investigate the spatial distributions of star formation activities in regions where dust extinction is insignificant (e.g., Thorp et al. 2019;Pan et al. 2019) such as the disk component of galaxies. However, optical observation is hampered by extinction from thick layers of interstellar dust clouds, and correct quantification of the star formation activity in dusty regions such as the central nucleus of a merging galaxy is highly non-trivial. One of the best methods of investigating the properties of star formation activities in such extremely dusty regions is hydrogen recombination lines in the millimeter (mm) range (Scoville & Murchikova 2013). Recently, Atacama Large Millimeter/-submillimeter Array (ALMA) has detected recombination lines from nearby galaxies; e.g., NGC 253 (Bendo et al. 2015), NGC 4945 (Bendo et al. 2016), and NGC 5253 (Bendo et al. 2017). By cross-checking star formation rate (SFR) measurements from the other wavelengths, Bendo et al. (2015Bendo et al. ( , 2016Bendo et al. ( , 2017 demonstrated that ALMA is effective to study the starburst activity in dusty regions (A V 10). In this paper, we apply this method to investigate dusty starbursts in a merging galaxy.
The SFR estimated from the hydrogen recombination line luminosity (hereafter, SFR RL ) allows us to estimate the calibration constant between SFR and total infrared (TIR) luminosity. The calibration constant changes depending on the duration of the currently observed starbursts (Calzetti 2013), because both high-mass short-lived stars and low-mass long-lived stars heat the dust and contribute to the TIR emission. If a young stellar population is the predominant energy source within a system, the TIR emission is mainly produced by dust heated by these young stars. However, the recombination line mainly traces the current starbursts because only stars more massive than ∼ 20 M ⊙ produce a measurable ionizing photon flux. As such, the ratio between SFR RL and recombination line luminosity is constant (when the age of starburst is longer than ∼ 6 Myr.), allowing us to estimate the age of the starburst by comparing SFR RL with the TIR luminosity. Hence, if the age of the starburst is shorter than the age of the galaxy merger, it is likely that the starburst was triggered by the galaxy interaction.
Little has been reported on the observations of mm recombination lines in merging galaxies. For example, in the case of Arp 220, Anantharamaiah et al. (2000) detected H42α, H40α, and H31α using IRAM 30m telescopes, and the results suggest multiple starbursts. Scoville et al. (2015) searched for H26α emission from Arp 220, but detection was unclear due to the contamination of a nearby HCN(4-3) line. In order to investigate optical and millimeter hydrogen recombination lines, we focus on one specific merging galaxy, NGC 3256. In this galaxy, H40α and H42α were detected by ALMA (Harada et al. 2018) and Erroz-Ferrer et al. (2019);den Brok et al. (2020) mapped the Hα and Hβ emissions with the Multi Unit Spectroscopic Explorer at the Very Large Telescope of the European Southern Observatory (MUSE/VLT) (Bacon et al. 2010).
NGC 3256 (redshift z = 0.00935 1 ) is a merging galaxy with a TIR luminosity (5-1100µm, L TIR ) of 4.8 × 10 11 L ⊙ (see SECTION 3.2.1 in detail) 2 . This system is at a distance of D ∼ 41.7 Mpc, which translates to 1 ′′ ∼198 pc. There are two nuclei (northern and southern) separated by ∼ 970 pc in NGC 3256. The systematic velocity of the merger is assumed to be cz ∼ 2800 km s −1 (c is light speed). Lira et al. (2002Lira et al. ( , 2008 derive normalization of the extinction curve (A V = 5.5 and 16 for the northern and southern nuclei, respectively) from the NICMOS 3 H − K color. The large A V makes it impossible to investigate the southern nuclear starburst activity using optical hydrogen recombination lines. The southern nucleus is an ideal laboratory to quantify how much the Hα and Hβ emission miss the SFR using H40α emission. In SECTION 2, the VLT and ALMA observations are explained. In SECTION 3, the formula to calculate the SFR is introduced. In SECTION 4, we investigate the nuclear starbursts, disk-wide starbursts, starburst timescale, and electron temperature. Finally, we summarize this project in SECTION 5.
2. DATA 2.1. MUSE NGC 3256 was observed by MUSE as one of the targets for the MUSE Atlas of Disks (MAD) project (Erroz-Ferrer et al. 2019). The processed MUSE 3D data cube of NGC 3256 can be downloaded from the ESO science archive portal 4 , and has a field of view (FoV) of 1 arcmin 2 , with spatial sampling of 0. ′′ 2, the full width half maximum (FWHM) of the effective spatial resolution is ∼ 0. ′′ 6, spectral sampling of 1.25Å, and an observation date of April 6, 2016. 5 Figures 1 (a) and (b) show the extinction map and extinction-corrected Hβ map processed by Erroz-Ferrer et al. (2019). The 2D maps in Figure 1 are downloaded from the MAD project web-page 6 . We use these 2D maps for the main analysis (i.e., measurements of SFR). We use the 3D data cube only for measuring line profiles (see section 2.3). The errors for the emission line flux are about 10% for the low signal-to-noise-ratio (S/N< 3) regions, and 2% for the higher S/N regions (Erroz-Ferrer et al. 2019). A conservative overall photometric error of 5% is adopted for the analysis using the Hβ map. In order to compare the MUSE/VLT and ALMA images, the Hβ peak position at the non-dusty northern nucleus is assumed to be same as the H40α peak position.

ALMA
The H40α, H42α, 13 CO (1-0), and CH 3 OH (2 k -1 k ) data cubes were obtained as part of the 85-110 GHz range line search ALMA project for NGC 3256 (ID: 2015.1.00993.S). In addition, the data from two other ALMA projects (ID: 2015.1.00412.S and 2016.1.00965.S) (Harada et al. 2018) were combined during data processing in order to produce higher quality H40α and H42α images ( Table 1). The calibrated visibility data were obtained by the calibration scripts that were provided by ALMA east Asian Regional Centerand processed using Common Astronomy Software Applications (CASA) (McMullin et al. 2007). We manually applied band-edge flagging and flux scaling for the data obtained in one specific Execution Block (uid A002 Xb00ce7 X47b4). Eight channels were flagged at the band-edge, whereas the original script flags 15 channels which included channels near the H40α emission. In addition, the absolute flux was corrected by a factor of 1.115 since the continuum flux for this Execution Block was systematically lower than the others. The continuum emissions were subtracted using the uvcontsub task in CASA. The data cubes were produced by using the tclean task in CASA with the Briggs weighting (robust = 2.0; Natural waiting), the velocity resolution of 50 km s −1 , and the pixel size of 0. ′′ 125. The clean masks were selected by the automatic masking loop (sidelobethreshold=2.0, noisethreshold=2.5, lownoisethreshold=1.5, minbeamfrac=0.3, growiterations=75, and negativethreshold=0.0). The FoV of the ALMA map is 59. ′′ 4 at the sky frequency of H40α emission. For 13 CO (1-0) imaging, we applied robust = 0.5 since the signal to noise ratio is high enough. The continuum map was produced using the line-free channels beside the H40α emission line. Table 2 is a summary of the achieved angular resolution and sensitivity for each line. Figure 2 shows the Hubble Space Telescope (HST) optical color image 7 and the integrated intensity map of H40α and H42α. Channel maps are shown in Figure 3 and the spectra are shown in Figure 4 for each region. We use H40α line flux to derive physical parameters, because the image quality (i.e., angular resolution and sensitivity) is better than that of H42α. In order to identify the H II regions probed by the H40α line, we use the imfit task in CASA to fit elliptical Gaussian components on the integrated intensity map. H40α is detected at the northern nucleus, southern nucleus, and northeastern (NE) peak with S/N of > 10, > 8, and > 4, respectively. Table 3 is a summary of the coordinates, line flux, and source size (FWHM of major and minor axes) of the detected regions. Figure 5 shows the spatial distribution of 13 CO (1-0) and 99 GHz continuum emission. Disk-wide distributions are seen in both the 13 CO (1-0) and rest-frame 99 GHz continuum. Figure 6 shows the line profiles for each line, and the results of Gaussian fittings are shown in Table 4. We use 3D data cube (without extinction correction) obtained by ESO archive that is not processed by MAD project. The velocity range is consistent among each line. The peak velocity of H40α emission at the southern nucleus is blue-shifted compared with the Hα and Hβ lines, while the three lines have similar velocities at the northern nucleus and NE peak. This may mean that dusty star formation activities that H40α can trace (but optical lines cannot) have different velocity components, yielding variation of the derived SFR between H40α and Hβ lines. The velocity width is larger in the optical lines than in the mm ones. This is likely due to the lower S/N of the H40α detection than optical line detections.

SFR diagnostic for hydrogen recombination lines
The relation between ionizing photon rate Q [s −1 ] and SFR [M ⊙ yr −1 ] depends on the initial mass function (IMF), mass range of stellar IMF, and timescale (τ ) over which star formation needs to remain constant, and on stellar rotation effects. According to Bendo et al. (2016), SFR can be calculated using the relation of We note that the coefficient in this equation can vary by a factor of two depending on the adopted assumption (e.g. the coefficient increases without stellar rotation effects). The details are explained in Bendo et al. (2015Bendo et al. ( , 2016. The emission measure (EM = n e n p V , where n e , n p , and V are ionized electron volume density, proton volume density, and volume of ionized H II region, respectively) is described using the total recombination coefficient (α B ). (2) Using the specific emissivity (ǫ) of each recombination line, the recombination line luminosity can be calculated by Here, the emissivity is given per unit n e n p . From equation (1)- (3), The luminosity can be calculated from the observed total line flux (Solomon & Vanden Bout 2005): The calibration constant between SFR RL and recombination line luminosity is applicable to the case where SFR is constant over >6 Myr. There is no dependency on long timescales, unlike the calibration constant between SFR and TIR luminosity (Calzetti 2013). Finally, the relation between SFR and total line flux (Bendo et al. 2016) is The α B terms depend on electron temperature (T e ) and electron density (n e ), assuming case-B recombination. The α B values are listed in Storey & Hummer (1995). We fixed the n e to 10 3 cm −3 , as the dependence is negligible in the range of 10 2 -10 5 cm −3 (Storey & Hummer 1995;Bendo et al. 2015). We use an interpolated relation between α B and T e (500-30000 K) for hydrogen recombination: The ǫ values are also listed by Storey & Hummer (1995), and we fixed the n e of 10 3 cm −3 to use the interpolated relation between ǫ and T e (500-30000 K). For example, in the case of optical, infrared, and mm recombination lines, In order to estimate the electron temperature, 99 GHz flux density can be used. The free-free (bremsstrahlung) continuum emission can also be used to probe the ionized gas EM. Therefore, it is possible to calculate SFR from the 99 GHz flux density (Draine 2011;Scoville & Murchikova 2013;Bendo et al. 2016): Here, we assume an ionic charge of Z = 1. From equation 6 and 9, the ratio of the line flux density integrated over velocity v to the free-free flux density can be written as The 99 GHz continuum emission is dominated by free-free emission in most cases (Saito et al. 2016). However, there is a possible contribution from non-thermal radio emissions and dust emissions. In order to check this contribution, we use the 5.0, 8.3, and 15 GHz continuum flux density measured by Very Large Array (VLA) from the literature (Neff et al. 2003) and 200 GHz Band6 data from archival ALMA data (Harada et al. 2018). Figure 7 shows the spectral energy density (SED) of the northern and southern nuclei. Three components can explain 1-300 GHz SED. The first component is the power law from non-thermal emission using the slope as a free parameter. The second is the freefree emission, which is scaled by the Gaunt factor (equation 10). The third component is dust emission with a slope of 4.0. Assuming T e = 5000 K, the SED fittings show that the contribution of free-free emission at 99 GHz continuum flux density (frac-FF) is ∼ 76% and ∼ 90% at the northern and southern nuclei, respectively. We note that the values of frac-FF obtained by SED fitting are not significantly sensitive to the assumption of electron temperature. Therefore, the variation in electron temperature can be investigated by equation 11. Subsequently, α B , ǫ, and SFR can be derived.

Molecular gas mass
Assuming optically thin emission and local thermodynamic equilibrium (LTE) conditions, the molecular gas mass associated with H40α detected regions can be estimated from 13 CO (1-0). It is better to use 13 CO (1-0) than 12 CO (1-0) when investigating very dusty regions in LIRGs, because the 12 CO (1-0) line is most likely optically thick. It is assumed that the excitation temperature of 10 K (Harada et al. 2018) and the 12 CO/ 13 CO ratios (R 12/13 ) of ∼ 100 (Henkel et al. 2014) are constant. Finally, we use the equation when we derive molecular gas mass from 13 CO luminosity (Battisti & Heyer 2014). Table 5 shows the information of gas mass in each region.

Results
The dust-extinction-corrected Hβ map (Figure 1b) shows disk-wide starbursts. The total SFR based on the Hβ map is SFR total Hβ ∼ 40 ± 2 M ⊙ yr −1 , assuming T e = 5000 K. The SFR measured by Hβ is insensitive to T e , as the relations of α B -T e and ǫ-T e have similar indexes of ∼ 0.8 (equations 6, 7, and 8). However, this value likely underestimates the total SFR due to dust extinction. Table 6 shows the SFR at star-forming regions where H40α is detected. A conservative overall photometric error of 5% is adopted 8 . The SFRs of the three detected regions measured by H40α emissions are SFR N H40α = 9.8 ± 0.5, SFR S H40α = 6.8 ± 0.3, and SFR NE H40α = 0.98 ± 0.05 M ⊙ yr −1 . In contrast, the SFRs of these regions measured by extinction-corrected Hβ data are SFR N Hβ ∼ 6.8 ± 0.3, SFR S Hβ ∼ 1.7 ± 0.1, and SFR NE Hβ ∼ 0.47 ± 0.02 M ⊙ yr −1 . The systematically lower SFR derived from the Hβ line suggests the presence of intervening dust, especially in the southern nucleus. Finally, the total SFR (SFR total Hβ+H40α ) is calculated as SFR total Hβ −(SFR N (Hβ) + SFR S (Hβ) + SFR NE (Hβ) )+(SFR N H40α + SFR S H40α + SFR NE H40α ) = 40 − (6.8 + 1.7 + 0.47) + (9.8 + 6.8 + 0.98) ∼ 48 ± 2 M ⊙ yr −1 . We note that the total SFR derived here is estimated assuming all the Hβ and Hα emissions originate from H II regions. As mentioned by Rich et al. (2011), shocks could also contribute to the line emissions. In order to estimate the regions ionized by pure H II regions, we use Baldwin, Phillips & Terlevich (BPT) cuts for each pixel derived by Erroz-Ferrer et al. (2019). The total SFR from pure H II regions is calculated as ∼ 40 M ⊙ yr −1 , which is consistent with SFR total Hβ derived by this project. The total SFR from TIR luminosity (L TIR = (4.8 ± 0.2) × 10 11 L ⊙ ) is 51.5 ± 2.6 M ⊙ yr −1 , assuming a young starburst (100 Myr), Kroupa IMF, and a mass range of 0.1 − 100 M ⊙ (Calzetti 2013). The comparison between hydrogen recombination lines and TIR luminosity is investigated in SECTION 4.4 in terms of the starburst age. Figure 8 shows the relation between SFR derived by Hβ and free-free emission. The SFR traced by free-free emission is systematically higher than SFR from Hβ, which suggests the contamination from synchrotron and/or dust in the 99 GHz continuum flux density. The typical frac-FF can be roughly estimated from the ratio of the SFRs derived by the Hβ and free-free emission. The mean value of the ratio is ∼ 0.7, indicating the typical frac-FF of ∼ 70 %. This fraction is consistent with the frac-FF of typical starburst galaxies such as NGC 253 (Bendo et al. 2015). While uncertainties in the dust extinction correction exist, we adopt the SFR derived using the Hβ line in the following sections because the S/N is higher than the 99GHz continuum map. In SECTION 4.3, we investigate the possible regions where Hβ may underestimate the SFR outside the southern nucleus.

DISCUSSION
The key questions we endeavor to answer are: (i) " What is the fraction of star formation missed by optical and infrared observations (e.g., Hα, Hβ, and Brγ)?"; (ii) " What is the fraction of the nuclear starburst that contribute to the total SFR?; (iii) " Can the variation of gas depletion time be seen within NGC 3256?"; and (iv) " How long is the starburst timescale in NGC 3256?". Finally, we investigate the properties of H II regions (i.e., electron temperature) in H40α detected regions.

Northern nucleus
The northern nucleus contains the largest (area = 0.38 ± 0.01 kpc 2 ) H40α nebula of the three identified (Table 3). The derived SFR is SFR N H40α = 9.8±0.5 M ⊙ yr −1 , which is ∼ 20% of the total SFR 9 . The SFR derived from Hβ is SFR N Hβ = 6.8 ± 0.3 M ⊙ yr −1 , and this is ∼ 70% of the SFR derived from H40α (Table 6). This difference may be explained by insufficient dust extinction correction which was performed using optical lines alone (i.e., the conversion from Hα/Hβ ratio to A V ). The star formation rate surface density (Σ inner SF R N H40α ) is 32.9±1.6 M ⊙ yr −1 kpc −2 , and the molecular gas mass surface density (Σ inner M H 2 ) around the northern nucleus is 2772 ± 139 M ⊙ pc −2 , which is a typical disk-averaged surface densities for starburst galaxies (Kennicutt 1998). This suggests that the characteristics of the H II regions near the northern nucleus are consistent with regions in typical starburst galaxies.

Southern nucleus
Despite the significant H40α emission, there is no strong emission in the extinction-corrected Hβ map at the southern nucleus ( Figure 1c). Consequently, the SFR derived from H40α (SFR S H40α = 6.8±0.3 M ⊙ yr −1 ) is larger than that derived from the Hβ map (SFR S Hβ ∼1.75 ± 0.09 M ⊙ yr −1 ). This suggests that the optical emission around the southern nucleus is not originated from extremely dust-obscured nebulae emission; rather, it may be contributed from the different components (e.g., the surface of the dusty star-forming region). In addition, the offset in the H40α line profile relative to the Hα and Hβ lines in the southern nucleus (Table 4 and Fig 6) may be the evidence showing that millimeter and optical lines trace different components. This demonstrates the benefits of examining both the spectral line parameters as well as the integrated fluxes when investigating dusty starbursts at the nucleus of U/LIRGs. Emission lines in IR can also be used as an independent proxy of SFR in galaxies. The southern nucleus can be detectable at wavelength 1µm (Lípari et al. 2000). Piqueras López et al. (2012López et al. ( , 2013 detected Brγ emission from the southern nucleus of NGC 3256 (the northern nucleus is not in the FoV) using the Spectrograph for INtegral Field Observations in the Near Infrared (SINFONI) integral field spectroscopy observation with VLT. We use the Brγ data obtained from an online catalog (Piqueras-Lopez et al. 2016) and measured the Brγ to be ∼ 6.3 × 10 −15 erg s −1 cm −2 at the southern nucleus. Assuming A Brg ∼ 1.2 estimated from the Brγ/Brδ ratio (Piqueras López et al. 2013), we find that SFR estimated from Brγ (SF R (Brγ) ) is 1.4 M ⊙ yr −1 . The significantly lower SFR derived from Brγ suggests that it may not be an ideal tracer of SFR in dusty regions, such as the southern nucleus of NGC3256. The comparison between Brγ and H40α flux indicates A Brγ ∼ 2.4 (A V ∼ 25 assuming A Brγ = 0.096A V ).

Disk-wide starburst
The sum of the nuclear starbursts in the northern and southern nuclei derived from the H40α data is 16.6 ± 0.6 M ⊙ yr −1 . Using the total SFR of ∼ 48 ± 2 M ⊙ yr −1 (SECTION 3.3), the contributions of the nuclear and disk-wide starbursts are ∼ 34 % and ∼ 66 %, respectively. In addition, H40α is detected at NE peak ( Figure 2) on the dust lane of the arm that has offset from the two nuclei. Figure 9(a) shows that star-forming regions in NGC 3256 have large scatter in the Σ H 2 -Σ SFR plane, particularly in the regions with τ gas of < 0.1 Gyr −1 as well as regions with τ gas of > 0.4 Gyr outside the nuclear region (the gas depletion time τ gas =M gas /SFR). This large scatter suggests a non-uniform gas depletion time. In addition, from a direct comparison with the results from a broad-band spectral survey of NGC3256 (Harada et al. 2018), we find that shock gas tracers (e.g., CH 3 OH, SiO, HNCO) are coincident with the regions where τ gas is long. Figure 9(b) shows the spatial distribution of τ gas , and the contours show the CH 3 OH (2 k -1 k ) emission. Figure 9(c) shows the relation between CH 3 OH (2 k -1 k )/ 13 CO (1-0) and τ gas . The Spearman's rank correlation coefficient (c-value) is 0.315, possibly suggesting a weak correlation. The probability (p-value) is 0.002, which means that the possibility for rejecting null hypothesis is 2 %. These suggest that merger-induced large-scale shock can possibly suppress the star formation activity in the disk region, although the statistical significance is not very strong. Figures 9(d)-(f) are similar to (a)-(c) but plotted using the SFR measured by 99 GHz continuum after correcting for the contamination from dust and nonthermal emission (assuming 70 %) (see also Wilson et al. 2019). Even after correcting frac-FF, a few regions have τ gas > 0.4 Gyr, suggesting that extinction corrected Hβ map underestimates SFR due to incomplete extinction correction. Alternatively, frac-FF for 99 GHz continuum is much lower than 70 %. It is, however, noteworthy that a possible correlation between CH 3 OH (2 k -1 k )/ 13 CO (1-0) and τ gas exists (Figure 9(f)), with a higher c-value than those shown in Figure 9(c). It is thus necessary to investigate other galaxies for a general conclusion. For example, MUSE/VLT data toward merging starburst galaxies (e.g., VV 114, II ZW 96, IC 214, Arp 256, and NGC 6240) already exist, and future ALMA observations of shocked gas tracers, molecular gas, and ∼100 GHz continuum with the same resolution as MUSE/VLT are important to understand whether shocks can indeed suppress star formation activities.

Starburst timescale
The calibration constant between SFR and L TIR changes depending on how long the currently observed starbursts have remained constant, because not only the young stellar population but also old, long-lived, low-mass stars contribute to L TIR . If the calibration constant is correct, the SFR estimated from L TIR should be the same as SFR RL . Assuming constant star formation and a Kroupa IMF in the stellar mass range of 0.1-100 M ⊙ , the ratio of SFR RL to L TIR is calculated as below (Calzetti 2013): 1.6 × 10 −44 (τ = 10 Gyr) 2.8 × 10 −44 (τ = 100 Myr) 3.7 × 10 −44 (τ = 10 Myr) The SFR RL of NGC 3256 is 48 ± 2 M ⊙ yr −1 , estimated using the Hβ and H40α maps. Using this SFR RL and L TIR = (4.8 ± 0.2) × 10 11 L⊙ (Sanders et al. 2003), the ratio between SFR RL and L TIR is estimated to be (2.63 ± 0.17) × 10 −44 . This is similar to the theoretical value for τ = 100 Myr, suggesting that the current starburst has continued for ∼100 Myr. This period is shorter than the age of the merger of NGC 3256 (∼500 Myr; Lípari et al. 2000). Thus, it is likely that the current starburst in NGC 3256 was triggered by the galaxy interaction.

Electron temperature variations
The electron temperatures are calculated using equation (11). The electron temperature around the northern nucleus is 5900±400K. This value is consistent with H II regions at the central part (< 4 kpc) of the Milky Way (Shaver et al. 1983) and other starburst galaxies (e.g., NGC 253 and NGC 4945) (Bendo et al. 2015(Bendo et al. , 2016. In contrast, the electron temperature around the southern nucleus is 11500 +800 −700 K, which is consistent with the H II regions in the outer part of the Milky Way (> 10 kpc). The different electron temperatures between the northern and southern nucleus is originally from the different line ratios of R = f νline dν/f νcont .
The freefree emission flux is comparable between the northern and southern nucleus, while the recombination line flux at the southern nucleus is about half of the northern nucleus. An empirical relation between electron temperature and metallicity suggests regions with lower metallicity are higher in electron temperature (Shaver et al. 1983), which is a direct consequence of inefficient cooling in low-metallicity regions (Pagel et al. 1979). Our analysis of NGC 3256 suggest that the metallicity of the extremely dusty (A V ∼ 25) southern nucleus is lower than that of the non-dusty regions where UV and optical emission lines can be detected. Low-metal environments are seen in other galaxies. For example, Kewley et al. (2006);Ellison et al. (2013) show that the metallicity in interacting galaxies tends to be lower than in non-interacting systems of equivalent mass, and later Rupke et al. (2008); Herrera-Camus et al. (2018) find the same trend for U/LIRGs. The low metallicity at the southern nucleus may suggest the past occurrence of a large-scale inflow of metal-poor gas. Other possibilities for the low metallicity include massive outflows (Sakamoto et al. 2014;Michiyama et al. 2018) which can remove gas and metals (e.g., Chisholm et al. 2018).

Possible AGN activity
The origin of recombination line flux may be related to the presence of an AGN, especially at the southern nucleus. The presence of an AGN is suggested from the IRAC 10 color and silicate absorption feature (Ohyama et al. 2015). The possible AGN is categorized as a low-luminosity AGN with the 2-10 keV luminosity of L 2−10keV ∼ 2 × 10 40 erg s −1 (Ohyama et al. 2015;Lehmer et al. 2015). In order to explain the molecular outflows from the southern nucleus, a previously active AGN is needed (Sakamoto et al. 2014;Michiyama et al. 2018). If the AGN ionizes the surrounding gas, the velocity dispersion of hydrogen recombination lines is nominally > 1000 km s −1 . However, the line profile at the southern nucleus has the same line width as that of the northern nucleus (∼ 300 km s −1 ) (Table 4 and Figure 6). In addition, Izumi et al. (2016) show that the expected line flux of mm hydrogen recombination lines is too low to be detected even by ALMA. Therefore, the H40α emission is likely originated from star formation activity at the southern nucleus.
The AGN may enhance the total infrared luminosity independent of star formation activities. In such a case, the expected starburst timescale is shorter than those derived in SECTION 4.4. Finally, higher electron temperature in the southern nucleus could be due to previous AGN activities. For example, Popović (2003) estimated an electron temperature of > 10, 000 K in broad line regions based on the Boltzmann plot method to Balmer lines, which is higher than typical electron temperature at the typical H II regions (e.g., Shaver et al. 1983).

SUMMARY
In order to show evidence of the large contribution of disk-wide starbursts to the total SFR in a merging galaxy NGC 3256, we investigated spatially resolved SFR using optical and mm hydrogen recombination lines. At first, we used optical integral field units (MUSE mounted on VLT) to obtain maps of recombination lines (i.e., Hα and Hβ). We found many star-forming regions outside the nuclear regions. However, it is difficult to investigate star formation activities in dusty nuclear regions using optical observations. ALMA observation of the mm recombination lines H40α and H42α allowed us to the quantify the true star formation activity in these regions. The total SFR obtained by Hβ and H40α line emission is ∼ 48 ± 2 M ⊙ yr −1 . The main findings are as follows: (1) H40α emission is detected at the northern nucleus, southern nucleus, and NE peak. However, there are no bright Hβ emissions at the southern nucleus. This means that there is a dustobscured region at the southern nucleus. The SFR from the southern dusty region is 6.8 ± 0.3 M ⊙ yr −1 , which is ∼ 14% of the total SFR.
(2) The sum of the nuclear starbursts in the northern and southern nuclei is 16.6 ± 0.6 M ⊙ yr −1 , which means that the contributions of the nuclear and disk-wide starbursts are ∼ 34 % and ∼ 66 %, respectively. The disk-wide starbursts are predominant compared to the nuclear starbursts, even considering the very dusty starburst seen in the southern nucleus.
(3) We find that τ gas is not uniform in NGC 3256. There are regions with τ gas < 0.1 Gyr as well as regions with τ gas > 0.4 Gyr outside the nuclear region. One possible explanation is merger-induced large-scale shocks that suppress star formation activities in the disk region.
(4) Recombination lines and total FIR luminosity suggest the current starburst started ∼100 Myr ago. This is shorter than the timescale of a merger process (∼ 500 Myr), and this supports the idea that the current starbursts are triggered by a merger process.
(5) The electron temperature is higher in the dusty southern nucleus (10200 +700 −600 K) than in the non-dusty northern nucleus (5900 +400 −400 K). One possible explanation is the lower metallicity in the southern nucleus than in the northern nucleus, suggesting metal-poor gas inflows or metal-rich gas outflows at the southern nucleus.  (3,4,5,6,7,8,9)    (top) H40α spectrum with photometric beam size of ∼ 1. ′′ 5 at the northern nucleus (red), southern nucleus (green), and NE peak (blue). The black dashed line is the result of Gaussian fitting. The arrows show the sky frequency at the systematic velocity. (bottom) Same figures for H42α. There are three other emission lines in these spectra: c-C 3 H 2 (2 1,2 -1 0,1 ) at 84.547 GHz, CH 3 CCH (5 k -4 k ) at 84.664 GHz, and SO (2 2 -1 1 ) at 85.295 GHz (sky frequency).    Neff et al. (2003), and 224 GHz data points from archival ALMA data (Harada et al. 2018). The gray lines show the three components. We use a synchrotron function with a variable power-law index, a free-free function scaled by g FF , and a modified Rayleigh-Jeans function (dust) with a fixed index of 4. The calculated synchrotron power-law indexes are -0.85 and -1.29 at the northern and southern nucleus, respectively. The free-free contribution on 99 GHz continuum emission is ∼ 76% for the northern nucleus and ∼ 90% for the southern nucleus, assuming an electron temperature of T e = 5000 K. Figure 8. Pixel-by-pixel comparison of SFR measured by Hβ and free-free emission. The blue dashed line corresponds to the 3 sigma limits for free-free emissions. The red line shows where the SFR from Hβ and free-free emission is equal. There is a linear relation; however, the SFR traced by free-free emission is systematically higher than that from Hβ. This means that 99 GHz emission often has contamination from other properties (e.g., dust and synchrotron).
(d)-(f) Same figures as (a)-(c) but with the SFR measured by free-free emission. The SFR from free-free emission is measured by assuming frac-FF = 70 %.  a The source sizes are measured as the half-light radius for 2D Gaussian fitting. The MUSE spectral resolution of 1.25Å corresponds to the velocity resolutions of ∼ 57 km s −1 for Hα and ∼ 77 km s −1 for Hβ. In the case of ALMA observations, the velocity resolution is 50 km s −1 . The flux is measured by the same aperture as in Table 6. If we use the source size in Table 3, the surface density Σ inner M H2 = 2772±139, 2936±147, and 1769±88 M ⊙ pc −2 for the northern nucleus, southern nucleus, and NE peak, respectively.