The intermediate-ionization lines as virial broadening estimators for Population A quasars

The identification of a virial broadening estimator in the quasar UV rest frame suitable for black hole mass computation at high redshift has become an important issue. We compare the HI Balmer H-beta line width to the ones of two intermediate ionization lines: the Aliii 1860 doublet and the Ciii] 1909 line, over a wide interval of redshift and luminosity (0<z<3.5; 43<logL<48.5 [erg a/]), for 48 sources belonging to the quasar population characterized by mid-to-high values of the Eddington ratio (Population A). The present analysis indicates that the line width of Aliii 1860 and H-beta are highly correlated, and can be considered equivalent for most Population A quasars over five orders of magnitude in luminosity; forCiii] 1909, multiplication by a constant correction factor xi ~ 1.25 is sufficient to bring the FWHM of Ciii] in agreement with the one of H-betaa. The statistical concordance between low-ionization and intermediate-ionization lines suggests that they predominantly arise from the same virialized part of the broad line region. However, blueshifts of modest amplitude (few hundred km /s) with respect to the quasar rest frame and an excess (~ 1.1) Aliii broadening with respect to H-beta are found in a fraction of our sample. Scaling laws to estimate M_BH of high redshift quasar using the Aliii and the Ciii] line widths have rms scatter ~ 0.3 dex. The Aliii scaling law takes the form log M_BH ~ 0.58 log L1700,44 + 2 log FWHM + 0.49 [solar masses].


INTRODUCTION
The energetics of all accretion-related phenomena occurring in active galactic nuclei (AGN) can be tied down to the mass of the central black hole. The mass (M BH ) of the black holes at the origin of the AGN phenomenon is now reputed a key parameter in the evolution of galaxies and in cosmology as well (e.g., Kormendy & Ho 2013;Vogelsberger et al. 2014;Heckman & Best 2014), and its estimation has become an important branch of extragalactic research. Black hole mass estimates on large type-1 AGN samples are carried out employing a deceptively-simple formulation of the virial theorem, under the assumption that all the mass of the system is concentrated in the center of gravity provided by the black hole (see e.g., Shen 2013;Peterson 2014, for reviews): M BH = f S r BLR (δv) 2 /G, where f S is a structure factor (a.k.a. virial or form factor) dependent on the emitting region geometry and dynamics, the radius r BLR the distance of the line emitting region from the continuum source, and δv a suitable measure of the line broadening (e.g., FWHM or dispersion σ, Vestergaard & Peterson 2006;Peterson et al. 2004). The main underlying assumptions are that the broadening is due to Doppler effect because of the line emitting gas motion, and that the velocity field is such that the emitting gas remains gravitationally bound to the black hole.
Early UV and optical inter-line shift analysis provided evidence that not all the line emitting gas is bound to the black hole (e.g., Gaskell 1982;Tytler & Fan 1992;Brotherton et al. 1994;Marziani et al. 1996;Leighly & Moore 2004). The scenario emerging from more recent studies is that outflows are ubiquitous in active galactic nuclei. They occur under a wide range of physical conditions, and are detected in almost every band of the electromagnetic spectrum and on a wide range of spatial scales, from a few gravitational radii to tens of kpc (e.g., Capetti et al. 1996;Colbert et al. 1998;Everett 2007;Carniani et al. 2015;Bischetti et al. 2017;Komossa et al. 2018;Kakkad et al. 2020;Vietri et al. 2020;Laurenti et al. 2021). At high luminosity, massive outflows provide feedback effects to the host galaxy (e.g., Fabian 2012; King & Pounds 2015;King & Muldrew 2016;Barai et al. 2018), and are invoked to account for the M BH -bulge velocity dispersion correlation (e.g., Kormendy & Ho 2013, and references therein). For z 4, M BH estimates rely on the Civλ1549 high-ionization line, and the highest-z sources appear to be almost always high-accretors (Bañados et al. 2018;Nardini et al. 2019). Two studies pointed out 20 years ago the similarity between X-ray and UV properties of high-z quasars and local quasars accreting at high rates (e.g., Mathur 2000;Sulentic et al. 2000a). The source of concerns is that high-ionization lines such as Civλ1549 are subject to a considerable broadening and blueshifts associated with outflow motions already at low redshift (Coatman et al. 2016;Sulentic et al. 2017;Marinello et al. 2020b, see Marinello et al. 2020a for a detailed study of the prototypical source PHL 1092). Overestimates of the virial broadening by a factor as large as ≈5 (Netzer et al. 2007;Sulentic et al. 2007;Mejía-Restrepo et al. 2016, 2018b for supermassive black holes at high z may even pose a spurious challenge to concordance cosmology (e.g., Trakhtenbrot et al. 2015) and lead to erroneous inferences on the properties of the seed black holes believed to be fledgling precursors of massive black holes. This paper is focused on the measurement of the line width of the UV intermediate-ionization lines at ≈ 1900Å and on their use for black hole mass measurements for large quasar samples, over a wide interval of luminosity and of redshift. The blend at λ1900Å is due, at least in part, to the Aliiiλ1860 doublet and to the Siiii]λ1892 and Ciii]λ1909 lines. Aliii is a resonant doublet ( 2 P o 3 2 , 1 2 → 2 S1 2 ) while Siiii] and Ciii] are due to inter-combination transitions ( 3 P o 1 → 1 S 0 ) with widely different critical densities (≈ 2 · 10 11 cm −3 and ≈ 3 · 10 9 cm −3 , respectively; Zheng 1988;Negrete et al. 2012). The parent ionic species imply ionization potentials 15 χ i 30 eV, intermediate between the ones of low-ionization lines (LILs), and the ones of high-ionization lines (HILs; χ i 40 − 50 eV). The intermediate-ionization lines (IILs) at 1900Å are well-placed to provide a high redshift estimator; they can be observed with optical spectrometers up to z ∼ 4. Observations can be extended in the NIR (13,500Å) up to z ∼ 5.7 without solution of continuity, thereby sampling a redshift domain that is crucial for understanding the primordial growth of massive black holes and galaxy formation. In principle, observations could be extended to the H band to cover the as yet mostly uncharged range 6.5 z 8, a feat that may well become possible with the advent of James Webb Space Telescope (Gardner et al. 2006), of the ESO Extremely Large Telescope (Gilmozzi & Spyromilio 2007), and of the next-generation large-aperture telescopes (see e.g., D'Onofrio & Marziani 2018, for a review of foreseeable technological developments).
The quasar main sequence provides much needed discerning abilities for the exploitation of the IILs (e.g., Sulentic et al. 2000a;Bachev et al. 2004;Marziani et al. 2001;Shen & Ho 2014;Panda et al. 2018), as line profiles and intensities of individual sources are not considered as isolated entities, but interpreted as part of consolidated trends in the main sequence context. Broad line measurements involving Hβ line width and Feii strength are not randomly distributed but instead define a sequence that has become known as the quasar "main sequence" (MS; e.g., Sulentic et al. 2000a;Shen & Ho 2014;Panda et al. 2019;Wildy et al. 2019). The Feii strength is parameterized by the intensity ratio involving the Feii blue blend at 4570Å and broad Hβ i.e., R FeII = I(Feiiλ4570)/I(Hβ), and the Hydrogen Hβ line width by its FWHM. MS sources with higher R FeII show narrower broad Hβ (Population A, FWHM(Hβ) 4000 km s −1 ), and sources with broader Hβ profiles tendentially show low R FeII (Pop. B with FWHM(Hβ) 4000 km s −1 , Sulentic et al. 2000aSulentic et al. , 2011. It is also known that optical and UV observational properties are correlated (e.g., Sulentic et al. 2000b;Bachev et al. 2004;Sulentic et al. 2007;Du et al. 2016;Śniegowska et al. 2018). In this paper, the attention is restricted to sources radiating at relatively high Eddington ratio (L/L Edd 0.1 − 0.2) i.e., to Population A that accounts for the large majority of sources discovered at high z. In the course of our analysis we realised that sources radiating at lower Eddington ratio (0.01 L/L Edd 0.1 − 0.2) show a different behaviour of the 1900 blend and will be considered elsewhere.
The coverage of the Hβ spectral range greatly easies the determination of the redshift as well as the positional classification of sources along the MS. In addition, FWHM Hβ has been employed as a virial broadening estimator of M BH since the earliest single-epoch observations of large samples of quasars, and in more recent times as well (e.g., McLure & Jarvis 2002;McLure & Dunlop 2004;Vestergaard & Peterson 2006;Assef et al. 2011;Trakhtenbrot & Netzer 2012;Shen & Liu 2012). The Hβ line is likely to be still the most widely used line for M BH computations for low redshift quasars (z 1). Our analysis relies on the availability of both Hβ and the 1900 blend lines, as we will consider FWHM Hβ as the reference "virial broadening estimator" (VBE).
Section 2 introduces the samples used in the present work, covering a wide range in redshift and luminosity, 0 z 3.5; 44 log L 48.5 [erg s −1 ]. The data were obtained with instruments operating in widely different spectral ranges (UV, optical, IR); as a consequence, S/N ratio values vary widely and the uncertainty assessment requires a dedicated approach (Section 3, and Appendix A). The Section 4 introduces paired fits to Hβ and the 1900 blend (an atlas is provided in Appendix B), along with several line width measures, and the relation between Hβ and Aliii measurements. A scaling law for M BH determination equivalent to the one based on Hβ but based on the IIL broadening is discussed in Section 5.

SAMPLE
Low-luminosity 1900 and Hβ data (FOS sample) -We considered a Faint Object Spectrograph (FOS) sample from Sulentic et al. (2007, hereafter S07) as a low-L and low-z sample. For the sake of the present paper, we restrict the S07 sample to 28 sources covering the 1900 blend spectral range and with previous measurements for the Hβ profile and R FeII (FOS sample). The spectra covering the Hβ spectral range come from Marziani et al. (2003, hereafter M03), as well as from the SDSS (York et al. 2000) and the 6dF (Jones et al. 2004; Table 1 provides information on the provenience of of individual spectra). The FOS high-resolution grisms yielded an inverse resolution λ/δλ ∼ 1000, equivalent to typical resolution of the data of M03 and of the SDSS. The S/N is above 20 for both the optical and UV low-redshift data. The FOS sample has a typical bolometric luminosity log L ∼ 45.6 [erg s −1 ] and a redshift z 0.5.
High-luminosity VLT and TNG data for Hamburg-ESO quasars (HE sample) -The sample of high-L quasars includes 10 sources identified in the Hamburg-ESO survey (Wisotzki et al. 2000) in the redshift range 1.4 z 2.6. All HE quasars satisfy the condition on the bolometric luminosity log L 10 47.5 erg s −1 and are discussed in detail by Sulentic et al. (2017, hereafter S17), where Civλ1549 and Hβ were analyzed. The sample used in this paper is restricted to the 9 Population A sources with VLT/FORS1 spectra and 1 TNG/DOLORES (HE1347-2457) spectrum that cover the 1900Å blend. The spectral resolutions at FWHM are 300 km s −1 and 600 km s −1 for the spectrographs FORS1 and DOLORES, respectively. The resolution of the ISAAC spectra covering Hβ is ≈ 300 km s −1 ). Typical S/N values are 50.
Additional high-luminosity sources (ISAAC sample) -Additional ISAAC spectra were obtained under programme 083.B-0273(A), for three targets SDSS J005700.18+143737.7, SDSS-J132012.33+142037.1, SDSS J161458.33+144836.9. They have been reduced following the same procedures employed for the HE quasars. The data will be presented in a forthcoming paper (Deconto Machado et al., in preparation). Matching rest-frame UV spectra were collected from the SDSS and BOSS (Smee et al. 2013), with a resolving power R = λ/FWHM ∼ 2000.
High-luminosity sources from the WISSH (WISSH sample) -We included near-infrared (NIR) spectroscopic observations of 7 WISSH Population A quasars QSOs (Vietri et al. 2018), obtained with LUCI at the Large Binocular Telescope and in one case with SINFONI at VLT. Basic information on this sample is provided in Table 1 of Vietri et al. (2018). The matching rest-frame UV spectra are from the SDSS. The higher resolution implies a somewhat lower S/N with respect to the ISAAC spectra; we restrict our analysis to the spectra above a minimum S/N ≈ 15. Redshifts measured for this paper agree very well with the values reported by Vietri et al. (2018) if the Hβ profile is sharp; they are lower by 300-400 km s −1 in four cases with relatively shallow profiles due to the different fitting techniques.
Joint sample - Table 1 lists in the following order source identification, redshift, specific rest-frame flux in the UV at 1700Å f λ,1700 , S/N at 1700Å reference to the origin of the spectrum, specific flux in the optical at 5100Å (f λ,5100 ), S/N at 5100Å, and reference to the origin of the optical spectrum. Table footnotes list references to the flux scale origin, in case the spectrum had uncertain of no absolute spectrophotometric flux calibration. Notes include the radio loudness classification (Zamfir et al. 2008;Ganci et al. 2019): radio-loud (RL), radio-intermediate (RI), and radio-quiet (RQ). Only two sources (HE 0043-2300 and 3C 57) are "jetted" in the sense of having a powerful relativistic jet (Padovani 2017). HE 0043-2300 is listed as a flat-spectrum radio quasar with dominant blazar characteristics in the Roma-BZCAT (Massaro et al. 2009), and 3C 57 is a compact-steep source (CSS;O'Dea 1998, Sulentic et al. 2015. Two other sources qualify as radio-intermediate , and are briefly discussed in Appendix C.

The quasar main sequence as an interpretative aid
In the following the framework of the quasar MS to make assumptions on line shapes, both in the optical and in the UV spectral ranges. There are several papers that provide a description of the main trends associated with the MS. Fraix-Burnet et al. (2017) reviews the main multifrequency trends. Sulentic et al. (2000aSulentic et al. ( , 2011) review the case for two different quasar Populations: Population A (at low z, FWHM Hβ 4000 km s −1 ), and Population B (FWHM Hβ 4000 km s −1 ). The limit is luminosity-dependent (S17), and reaches FWHM 5500 km s −1 at high luminosity log L ∼ 48). In the optical plane of the MS defined by FWHM Hβ vs R FeII Population A has been subdivided into 4 spectral types (STs) according to Feii prominence: A1, with R FeII 0.5; A2, with 0.5 R FeII 1.0; A3, with 1.0 R FeII 1.5; A4, with 1.5 R FeII 2.0 (Sulentic et al. 2002, see also Shen & Ho 2014 for an analogous approach). The condition R FeII 1.0 restricts the MS to the tip of high R FeII values, and encompasses 10% of objects (referred to as extreme Population A). At low-z they are mostly narrow-line Seyfert-1 (NLSy1s) driving the MS correlations (Boroson & Green 1992;Sulentic et al. 2000a;Du et al. 2016). Sources with R FeII 2 do exist (Lipari et al. 1993;Graham et al. 1996) but they are exceedingly rare (less than 1%) in optically-selected samples (Marziani et al. 2013a, hereafter M13a). We therefore group all sources with R FeII 1.5 in A4.

Multicomponent χ 2 -minimization
Resolution and S/N of the available spectra are adequate for a multicomponent nonlinear fitting analysis using the IRAF routine specfit (Kriss 1994), involving an accurate deconvolution of Hβ, [Oiii]λλ4959,5007, Feii, Heiiλ4686 in the optical, and of Aliii, Ciii] and Siiii] in the UV. A χ 2 minimization analysis is necessary in all cases, since the strongest lines are heavily blended together, and the blend involves also features extended over a broad wavelength range, due to Feii (mainly optical, and UV to a lesser extent) and Feiii (UV only).

Hβ line
The Hβ Balmer emission line is a reliable estimator of the "virial" broadening in samples of moderate-to-high luminosity (Wang et al. 2009;Trakhtenbrot & Netzer 2012;Shen & Liu 2012). Typically, the Hβ line profiles are fairly symmetric, and are thought to be dominated by a virialized component (Peterson & Wandel 1999;Peterson et al. 2004, S17). Several previous works noted that Hβ shows a Lorentzian-like profile in sources belonging to Population A (e.g., Véron-Cetty et al. 2001;Sulentic et al. 2002;Cracco et al. 2016, this is also seen in Mgiiλ2800, Marziani et al. 2013b;Popović et al. 2019). However, the Hβ profiles can be affected by slight asymmetries and small centroid shifts. In Population A they are mostly due to blueshifted excess, often modeled with a blueward asymmetric Gaussian component (BLUE), strongly affecting the Civλ1549 line profiles, and related to outflows (e.g., S17, and references therein, Negrete et al. 2018). In Hβ, BLUE is detected as a faint excess on the blue side of the symmetric profile assumed as the virialized component of Hβ, almost only in extreme Population A (several examples are shown in the Figures of the atlas of Appendix B). Even when the BLUE component is detected, its influence on the Hβ FWHM is modest, leading at most to an increase of the broadening ≈ 10% over the FWHM of the symmetric broad profile (Negrete et al. 2018).
• the Hβ BC , modeled with an (almost) unshifted Lorentzian profile; • a blueshifted excess (BLUE) modeled with a blueshifted Gaussian with free skew parameter (Azzalini & Regoli 2012). The skewed Gaussian function has no more outliers than the normal distribution, and retains the shape of the normal distribution on the skewed side. It is consistent with the suppression of the receding side of an optically thin flow obscured by an optically thick structure (i.e., the accretion disk); • the Hβ NC , modeled with a Gaussian, unshifted with respect to rest frame; • Feii emission, modeled with a scaled and broadened template (e.g., Boroson & Green 1992), as defined by Negrete et al. (2018); • [Oiii]λλ4959,5007, modeled by a core-component (assumed Gaussian and symmetric) and a semi-broad component (assumed Gaussian but with the possibility of being skewed). This approach has been followed in several previous work (e.g., Zhang et al. 2011); • Heiiλ4686, broad and narrow component. Heiiλ4686 is not always detected in the spectra, especially in the case of strong Feii emission, but the line was included in the fits.

The 1900 blend
The range 1700 -2000 is dominated by the 1900Å blend which includes Aliii, Siiii], Ciii], as well as Feii and Feiii lines. On the blue side of the blend Siiiλ1816 and Niii]λ1750 are also detected. The relative intensity of these lines (apart from Niii]λ1750 that is not affecting the blend and for which further observations are needed) is known to be a function of the location along the quasar main sequence (Bachev et al. 2004). The line profiles and relative intensities are systematically different not only between Population A and B, but also within Pop. A there is a systematic trend of increasing Aliii and decreasing Ciii] prominence with increasing Feii emission (Bachev et al. 2004).
Our interpretation of the 1900 blend for Population A sources closely follows previous analyses (Baldwin et al. 1996;Wills et al. 1999;Baldwin et al. 2004;Richards et al. 2011;Negrete et al. 2012;Martínez-Aldama et al. 2018). The fits include the following components (described in detail by Martínez-Aldama et al. 2018): • Aliii, Siiii], and Ciii], modeled with a Lorentzian profile. We assume that the shapes of the strongest lines are consistent with the ones considered for the Hβ broad components (Lorentzian for Pop. A), and that FWHM Aliii = FWHM Siiii] (Negrete et al. 2013). The fitting routine may introduce a systematic blueshift to minimize χ 2 , in the case the profile is significantly affected by an unresolved blue shifted component, as observed in the case of Mgiiλ2800 (M13a); • Feiii emission, very intense in extreme Population A spectra, modeled with an empirical template (Vestergaard & Wilkes 2001 (2008) template. Considering the difficult assessment of the Feii UV emission, no measurements are reported in the present paper.
• a blueshifted excess (BLUE) modeled with a blueshifted skew Gaussian. At high luminosity in the HE sample, there are 2 cases (HE0359-3959 and HE1347-2457) where a strong blueshifted component is obviously affecting the profile of the 1900 blend. Other cases are also detected in the WISSH sample (see §5.3 for the interpretation of the 1900 blend profiles involving a blue shifted excess). For two objects, the BLUE emission is overwhelming and masking the emission of the individual Aliii, Siiii], Ciii] broad components (Section 5.3). Otherwise, the appearance of the blend is not suggesting, even at the highest luminosity, the presence of an outflow component spectroscopically resolved (i.e., of significant blueshifted emission as detected in Civλ1549). Small in Aliii blueshifts do occur, but with amplitude than their FWHM.

Full profile measurements
We assume that the symmetric and unshifted Hβ BC is the representative line components of the virialized part of the BLR. It is expedient to define a parameter ξ as follows: where the FWHM vir is the FWHM of the "virialized" component, in the following assumed to be Hβ BC , and the FWHM is the FWHM measured on the full profile (i.e., without correction for asymmetry and shifts) of any line. In the case of Hβ, FWHM Hβ ≈ FWHM Hβ BC , and ξ Hβ ≈ 1 (Section 4.1). For the sake of this work, Hβ and Hβ BC can be considered almost equivalent, so that we will rely on the Hβ BC -Hβ decomposition obtained with specfit only in a few instances. The blue excess is usually faint with respect to Hβ BC and no empirical correction has been applied.
A goal of the present paper is to derive ξ for Aliii and Ciii]. Similarly as for Hβ, the Aliii lines are fit by symmetric functions. This approach has been applied in all cases and appears appropriate for the wide majority of spectra (≈ 90%), where there is no evidence of a strong BLUE in Aliii and the Aliii peak position is left free to vary to account for small shifts that might be due to a spectroscopically-unresolved outflowing component. A few cases for which there is evidence of contamination by a strong blue shifted excess are discussed in Sect. 5.3.

Error estimates
The data used in this paper come from an array of instruments yielding spectra with widely different S/N. In addition, the comparison is between two emission lines, one of which is relatively strong (W (Hβ)∼ 100Å), and one faint (W (Aliii) 10Å in most cases). To make things worse, at low z the Aliii line is also recorded on lower S/N spectra. These and other systematic differences have to be quantitatively taken into account in the error estimates. A quality parameter Q has been defined for Aliii, Hβ, and Ciii] as the ratio between the line equivalent width and its FWHM multiplied by the S/N ratio measured on the continuum. The Q values can be computed using the parameters reported in Tables 1 and 2. The systematic differences in the spectra covering Aliii and Hβ are reflected in the distribution of Q: Hβ and Aliii occupy two different domains (Figures in Appendix  A). The corresponding fractional uncertainties in FWHM computed from dedicated Markov Chain Monte Carlo (MCMC) simulations or by defining a relation with the Q parameter as detailed in Appendix A are significantly different for the two lines, being just a few percent in the FWHM of the narrowest sources with strong and sharp Hβ and at worst ≈ 10%, but in the range ≈ 10% -50 % for Aliii.

Immediate results
The specfit analysis results are provided in form of an atlas (Appendix B) for the FOS , HE, ISAAC, and WISSH samples. The Aliii and Hβ spectral range are shown after continuum subtraction, on a normalized flux scale (at 1700 and 5100Å). The parameters measured with the specfit analysis or on the full profiles for Hβ, and Aliii are reported in Table 2. Table 2 lists, in the following order: identification by IAU code name (Col. 1), rest-frame flux and equivalent width of the Hβ line (Cols. 2-3). The following columns (Cols. 4-6) report the Hβ profile parameters: FWHM Hβ, FWHM Hβ BC , and shift. Here for shift s we intend the radial velocity of the line peak with respect to the rest frame as defined from the redshift measured in the Hβ spectral range; parameter R FeII and spectral type (Cols. 7-8); rest-frame flux, equivalent width, FWHM and shift of the Aliii line (Cols. 9-12). The FWHM refers to the individual component of the doublet, whereas flux and equivalent width W are measured over the full doublet; flux of Siiii] (Col. 13); Ciii] flux and FWHM (Cols. 14-15). The Feiii flux measurement (Col. 16) was obtained by integrating the template over the range 1800 − 2150Å. The upper limit of the wavelength range set at 2150Å allows the inclusion of a broad feature peaking at ≈ 2050 − 2080Å and mostly ascribed to Feiii emission (Martínez-Aldama et al. 2018). Further information on the reported parameters are given in the Table footnotes. Errors on line widths have been computed from the numerical simulations described in Appendix A or from the data listed in Tables 1 and 2 that yield Q. The same approach has been followed for errors on line intensities and line shifts.
The values of the Hβ FWHM for the WISSH quasars are fully consistent with those reported by Vietri et al. (2018) for all but two targets, namely SDSS J152156.48+520238.5 and SDSS J115747.99+272459.6, for which a discrepancy can be explained in terms of a different fitting technique. Intensity ratios computed between lines in the UV and the optical should be viewed with extreme care. The observations are not synoptical and were not collected with the aim of photometric accuracy. Fig. 2 shows the FWHM Aliii vs FWHM Hβ full profile. The overall consistency in the FWHM of the two lines is rather obvious from the plot. In the case of Aliii and Hβ, the Pearson's correlation coefficient is ≈ 0.785 (P ∼ 5 · 10 −8 of a chance correlation). A best fit with the ordinary least-squares (OLS) bisector yields FWHM AlIII ≈ (273 ± 216) + (0.933 ± 0.059)FWHM Hβ.

FWHM Hβ vs. FWHM Aliii
(2) The two lines are, on average, unbiased estimators of each other, with a 0 point offset that reflects the tendency of the Aliii lines to be somewhat broader than Hβ but is not statistically significant (the offset by 250 km s −1 is at less than 1 σ confidence level). An orthogonal LSQ fit yields slope and offset consistent with the OLS. The normalized χ 2 ν ≈ 1 also indicates that the ratio between the FWHM of the two lines is 1 within the uncertainties. The maximum FWHM ≈ 6000 km s −1 is observed for the sources of the highest luminosity (Section 4.4) and is below the luminosity-dependent FWHM limit of Population A. A sources Civλ1549 is systematically broader than Hβ, apart from in two cases in the HE sample, and FWHM(Civλ1549) shows a broad range of values for similar FWHM Hβ i.e., FWHM(Civλ1549) is almost degenerate with respect to Hβ. The Civλ1549 line FWHM values are so much larger than the ones of Hβ making it possible that the M BH derived from FWHM Civλ1549 might be higher by even more than one order of magnitude than the one derived from the Hβ FWHM, as pointed out in several past works (Sulentic et al. 2007;Netzer et al. 2007;Mejía-Restrepo et al. 2016). We remark again that the Aliii line may show a blueshifted excess in 6 sources in our sample, with convincing evidence in only two cases (Sect. 5.3) but that the line profile is otherwise well represented by a symmetric Lorentzian. In the case of a blueshifted excess, the good agreement between FWHM Aliii and FWHM Hβ is in part due to the Siiiλ1816 emission that, if no blueshifted Aliii emission is allowed, becomes very strong in the fit of the 6 sources, and compensate for the blueshifted excess. Siiiλ1816 is expected to be enhanced in the physical conditions of extreme sources (Negrete et al. 2012, Section 5.3). At the same time, including the Siiiλ1816 line in the fits allows for a standard procedure that does not require identification and a screening for the sources with a strong blueshifted excess, which apparently follow the correlation between Aliii and Hβ full profile in Fig. 3. 4.3. Dependence on spectral type and R FeII 4.3.1. FWHM In Fig. 2 the data points are color coded according to their original subsamples. Fig. 3 shows the joint sample FWHM Aliii vs FWHM Hβ full profile color-coded according to ST. There are systematic differences between the various STs, in the sense that A1 sources have Aliii narrower than Hβ (at the relatively high confidence level of 2σ), and Aliii and Hβ FWHM are almost equal for ST A2. The Aliii and Hβ FWHM ratio is reversed, in the sense that Hβ is narrower than Aliii, for STs A3 and A4 grouped together. The difference between STs is reinforced if only the BC of Hβ is considered (Fig. 4), since the FWHM Hβ BC is slightly lower than the FWHM Hβ of the full Hβ broad profile, with ξ Hβ ≈ 0.97 ± 0.05 on average, but ξ Hβ ≈ 0.79 for the spectral types A3 and A4. If we define ξ AlIII = FWHM(Hβ BC )/FWHM(Aliii), we have the following median values (± semi-interquartile range): A2 is the highest-prevalence ST in Population A, with ξ ≈ 1. However, across Population A there is a significant trend that implies differences of ±20 − 25% with respect to unity. The A3-A4 result is after all not surprising, considering that quasars belonging to these spectral types with the strongest Feii emission are the sources most affected by the high-ionization outflows detected in the Civλ1549 line. The A1 result i.e., Aliii lines narrower than Hβ by ≈ 20% comes more as a surprise, and it is intriguing that it is consistent that also for the B Population Aliii is narrower than Hβ (del Olmo et al., in preparation; Marziani et al. 2017). This result may hint at a small but systematic extra broadening not associated with virial motions in A2.

Shifts
In addition to shifts with respect to the rest frame, we consider also the shift s defined as the radial velocity difference between the peak position of the Lorentzian function describing the individual components of the Aliii doublet with respect to the quasars rest frame and the peak position of Hβ and (reported in Table 2) i.e., s (AlIII) = s(AlIII) − s(Hβ). We adopt this definition because the spectral resolution and the intrinsic line width make it difficult to resolve the outflow in the Aliii profile. In the Civλ1549 and Hβ profiles, and even in the ones of Mgiiλ2800, it is possible to isolate a blue-shifted excess that contributes most of the Civλ1549 flux in several cases, superimposed to a symmetric profile originating in the low ionization part of the BLR. In the case of Aliii, this approach is more difficult, in part because of the low S/N (a blueshifted contribution as faint as the one of Hβ would be undetectable for Aliii, especially in the FOS spectra), in part because of the intrinsic rarity of sources with a strong blue-shifted excess. In addition, the Hβ NC is often overwhelmed by the Hβ BC , making it difficult to accurately measure its width and shift. The shifts of Hβ broad profile with respect to the rest frame set by the Hβ NC are usually modest, and the median is consistent with 0, µ(s(Hβ)) ≈ 0 ± 65 km s −1 . The shift estimators s and s therefore yield results that are in close agreement. Fig. 5 shows that the s amplitude is relatively modest for spectral type A3. ST A4 shows a larger shift, associated with an increase in the FWHM Aliii with respect to Hβ, although with a large scatter. This is due to an outstanding case, HE0132-4313, a NLSy1 with FWHM Aliii / FWHM Hβ ≈ 5, a behaviour frequent for the Civλ1549 in strong Feii emitters but apparently rarer for Aliii.
Taken together, the FWHM and shift data suggest that major discrepancies are more likely for the relatively rare, stronger Feii emitters, ST A3 and A4. The left panel of Fig. 5 shows that only A4 has a median shift that is significantly different from 0. A4 sources are relatively rare sources (4 in our sample, and ≈ 3 % in M13a). In addition to the shift amplitude in km s −1 , the line shift normalized by line width may be a better description of the "dynamical relevance" of the outflowing component (Marziani et al. 2013b). The parameters δ( i 4 ) = c( i 4 )/FW i 4 M, i = 1, 2, 3, yield the centroids c( i 4 ) at fractional peak intensity i 4 normalized by the full width at the same fractional       (1/ξ AlIII ) and the SIQR are reported for spectral types A1 and A2, and for the union of A3 and A4, as in Fig. 3. An important clue to the interpretation of the Aliii broadening is provided by the trends with luminosity. Fig. 6 shows FWHM Aliii vs. FWHM Hβ with data points color-coded according to luminosity. There is no significant deviation from equality for the FWHM of Hβ and Aliii. At higher luminosity, both the Aliii and the Hβ line become broader, and the largest line widths are measured on the ESO, ISAAC and WISSH samples. The ratio r = FWHM Aliii / FWHM Hβ ≈ 1/ξ also does not depend on luminosity: dividing the sample by about one half at log L 1700 = 46 [erg s −1 ] yields median r values for the subsamples above and below this limit that are very close to 1 (lower panel of Fig. 6).
The statistical equality between FWHM Hβ and FWHM Aliii is not breaking down at any luminosity, at least within the limit sets by our sample and data quality. The explanation resides in the fact that both line FWHM increase in the same way with luminosity, as shown by the left, top panel of Fig. 7. The trends for Hβ, Aliii, and Ciii] alike (Ciii] is discussed in Section 4.5) can be explained if the broadening of the line is basically set by the virial velocity at the luminosity-dependent radius of the emitting region, r BLR ∝ L While L/L Edd is changing in a narrow range (0.2 -1) and f S is also changing by a factor a few, M BH is instead spanning around 4 orders of magnitude. Over such a broad ranges of masses or, alternatively, luminosity, we might expect that the dominant effect in the FWHM increase is associated right with mass or luminosity. In Fig. 7 the trend expected for FWHM(Hβ) ∝ L 1 4 1700 is overlaid to the data points, and is consistent with the data in the luminosity range 44.5 log L 1700 47.5 [erg s −1 ], where a fivefold increase of the FWHM of both Hβ and Aliii is seen. The blueshifts involve radial velocities that are relatively modest (right panel of Fig. 7). Aliii shifts exceed 1000 km s −1 only in two cases of extreme luminosity. Even if we see some increase toward higher s values in the highest luminosity range, several high-luminosity sources remain unshifted within the uncertainties. If we consider the dependence of shifts on luminosity, at high L there is an increase in the range of blueshifts involving values that are relatively large (several hundred km s −1 ;  Fig. 7). The parameter δ as a function of L has a more erratic behaviour (Fig. 8), but only at the highest L δ ≈ −0.1, and the effect of the line shift is at a 10% level. Fig. 8 is consistent with Aliii blueshifts becoming more frequent and of increasing amplitude with luminosity.

FWHM Hβ and FWHM Ciii]
The Ciii] line has been considered as a possible virial broadening estimator, and has been a target of reverberation mapping monitoring (Trevese et al. 2007(Trevese et al. , 2014Lira et al. 2018;Kaspi et al. 2021).
The problem in Population A is that Ciii] shows a strong gradient in its intensity. In spectral type A1, Ciii] is by far the strongest line in the 1900Å blend, but its prominence decreases with increasing R FeII , i.e., going toward spectral type A3 (Bachev et al. 2004). In spectral types A3 and A4 Ciii] is affected by the severe blending with Siiii] and Feiii emission (much more severe than for Aliii), and may become so weak to the point of being barely detectable or even undetectable (Vestergaard & Wilkes 2001;Negrete et al. 2014;Temple et al. 2020;Bachev et al. 2004;Martínez-Aldama et al. 2018). In addition, the Ciii] line has a rather low critical density, and its intensity is exposed to the vagaries of density and ionization fluctuations much more than Aliii, whose emission is highly efficient in dense gas over a broad range of ionization levels (Marziani et al. 2020, and Sect. 5.3). The measurements of the Ciii] width might be inaccurate in extreme Population A if Feiii contamination is strong. It is therefore legitimate to expect a greater dispersion in the width relation of Ciii] with Hβ.
The FWHM of Ciii] is shown against the FWHM of Hβ full profile in Fig. 9, upper panel. Error bars show uncertainties computed following the prescription of Appendix A. Not surprisingly, the scatter in the FWHM ratio between Ciii] and Hβ is larger than in the case of Aliii. The top and middle panels of Fig. 9 show that there is a significant deviation from unity, although for relatively narrow profiles around 2000 km s −1 the FWHM Ciii] is close to the 1:1 line. The χ 2 ν is much higher than 1. The bottom panel shows the ratio between FWHM Ciii] and FWHM Hβ color-coded according to Ciii] strength. The limit was set at normalized intensity (roughly equal to equivalent width) I = 10. The trend for sources above this limit implies FWHM Ciii]= ξ CIII] FWHM ≈ 0.77 FWHM Hβ.
We didn't detect any strong difference in the trend with luminosity of Hβ and Ciii] FWHM, as it has been the case for Hβ and Aliii. The two lines follow a similar trend with luminosity at 1700Å (Fig. 10). No shift analysis was carried out for Ciii] due to the severe blending with Siiii] and Feiii.
The narrower profile of Ciii] indicates a higher distance from the central continuum source than the one obtained from Hβ, if the velocity field is predominantly virial (Peterson & Wandel 2000). This result is also consistent with the findings of Negrete et al. (2013) who, using Ciii] intensity ratios in a photoionization estimate of the emitting region radius, obtained much larger radii than the ones obtained from reverberation mappings of Hβ.

A M BH scaling law based on Aliii
The goal is to obtain an M BH estimator based on Aliii that is consistent with the scaling law derived for Hβ. In this context, the process is much simpler than in the case of Civλ1549, where large blueshifts introduced a significant correction and a second-order dependence on luminosity of FWHM Civλ1549 could not be bypassed. The Hβ and Aliii widths of the two lines grow in a similar trend with L (Fig. 7). The M BH scaling law can be derived in the form log M BH = α·log L+2·log FWHM+γ by minimizing the scatter and any systematic deviation of M BH estimated from Aliii with respect to the Hβ-derived masses from the Vestergaard & Peterson (2006) scaling law: log M BH (Hβ) ≈ 0.5 log L 5100,44 + 2 log (FWHM(Hβ)) + 0.91, where L 5100,44 is the rest frame luminosity λL λ at 5100Å in units of 10 44 erg s −1 , and the FWHM Hβ is in km s −1 , considering that no correction is needed to FWHM Aliii (i.e., ξ AlIII ≈ 1). The Vestergaard & Peterson (2006) scaling law is a landmark that has been applied in hundreds of quasar studies. However, the Vestergaard & Peterson (2006) Hβ scaling law provides individual estimates with large error bars in relative estimates (≈0.5 dex at 1σ, see the discussion in their paper). This is a limit to the precision of any scaling law based on the match with the one based on Hβ. The large error bars of individual mass estimates can be mainly explained on the basis of orientation effects (M19) and of trends along the spectral types of the main sequence (Du & Wang 2019). If this condition is enforced, the relation between M BH from Aliii and from Hβ (Fig. 11) can be written as: log M BH (AlIII) ≈ (1.000 ± 0.043) log M BH (Hβ) − (0.001 ± 0.367). (4) The Aliii scaling law takes the form, with the FWHM in km s −1 : log M BH (AlIII) ≈ (0.579 +0.031 −0.029 ) log L 1700,44 + (5) 2 log (FWHM(AlIII)) + (0.490 +0.110 −0.060 ), with an rms scatter σ ≈ 0.29. Figure 11 suggests the presence of a well-behaved distribution with a few outlying points. The relation of Eq. 5 considers the FWHM of 47 sources. One data point has been excluded applying a σ clipping algorithm (the one A4 outlier, HE0132-4313). This selective procedure is justified by the fact that only some of the most extreme sources of Population A (not all of them) show large blueshifts and only one (HE0132-4313) a FWHM in excess to Hβ by a large factor, deviating at more than 3 times the sample rms. Removal of HE0132-4313 provides however only a minor, not significant change in the fitting parameters. The scaling law parameter uncertainties have been estimated computing the coefficients α and γ over a wide range of values and defining the interval where agreement between M BH from Hβ and Aliii (Eq. 4) is satisfied within 1.00±1σ for the best-fitting slope and 0.00 ± 1σ for the intercepts (respectively ≈0.043 and ≈0.367, as per Eq. 4). Due to some scatter in FWHM relation at FWHM Hβ BC ∼ 1000 km s −1 and the possibility of predominantly face-on orientation (Mejía-Restrepo et al. 2018a) the M BH estimates at M BH 10 7 M should be viewed with care due to the paucity of data points.
It is important to remark that this result, unlike the one on Civλ1549 M BH estimates, is obtained without any correction on the measured FWHM. The Aliii and Hβ relation should be considered equivalent with respect to M BH estimates in large samples of Population A sources. No scaling law r BLR − L has been ever derived from reverberation mapping on the Aliii lines. However, it is reassuring that the luminosity exponent (≈ 0.579 +0.031 −0.029 ) deviates by about 1σ from the one entering the scaling law r BLR − L derived by Bentz et al. (2013) for Hβ.

A M BH scaling law based on Ciii]
An approach analogous to the one adopted for Aliii was also applied to Ciii]. The goal is to obtain an M BH estimator based on Ciii] that is consistent with the scaling law derived for Hβ. The process is again much simpler than in the case of Civλ1549, where large blueshift introduced a significant correction and a second-order dependence on luminosity of FWHM Civλ1549 could not be bypassed. Considering that only a very simple correction is needed to FWHM Ciii], ξ CIII] ≈ 1.25, the M BH scaling law can be derived in the form log M BH = α · log L + 2 · log ξ CIII] FWHM + γ by minimizing the scatter and any systematic deviation of M BH estimated from Ciii] with respect to the Hβ-derived masses. Figure 12 suggests the presence of a well-behaved distribution with a few outlying points. The condition is satisfied if the Ciii] scaling law takes the form: log M BH CIII] ≈ (0.6765 ± 0.0450) log L 1700,44 + (7) 2 log ξ CIII] FWHM(CIII]) + (0.332 ± 0.120), with an rms scatter σ ≈ 0.35 (excluding 1 outlier, for 44 objects). The scaling law parameter uncertainties have been estimated varying the coefficients α and γ as for the Aliii case.
The M BH Ciii] scaling law is derived with a simple correction on the measured FWHM Ciii]. The Ciii] and Hβ relation should be considered equivalent with respect to M BH estimates in large samples of Population A sources. Care should however be taken to consider sources in which Ciii] can be measured (W 10Å) and to identify extreme objects, as discussed in Section 5.3. In addition, the heterogeneity of the sample and the possibility of different trends in different FWHM domains (FWHM (Ciii]) ≈ FWHM(Hβ) if FWHM 3000 km s −1 ) makes the scaling law with Aliii more reliable.

DISCUSSION
In the following we first compare the scaling laws obtained for Aliii and Ciii](Sect. 5.1). The present work detects systematic shifts of Aliii toward the blue. Even if they are on average slightly above the uncertainties, it is interesting to analyze them in the context of systematic shifts affecting the prominent high and low-ionization lines of Civλ1549 and Mgiiλ2800 (Section 5.2), and to consider more in detail sources in the spectral type where Aliii are broader than the Hβ ones (Section 5.3).

Consistency of scaling laws
The scaling laws derived for Aliii and Ciii] are mutually consistent. Black hole mass estimates are shown in Fig. 13. An unweighted lsq fit yields slope 0.975±0.043, consistent with unity, and intercept 0.246 ± 0.371, with an rms scatter ≈ 0.302. The median difference of the mass values obtained with the two scaling laws is µ(log M BH (CIII]) − log M BH (AlIII)) ≈ 0.06 ± 0.20 (SIQR).
The r BLR − L implicit in Eq. 6 is consistent with the r BLR − L relation derived for Civλ1549 in a previous study (Trevese et al. 2014), and only slightly higher than the one derived in the early study of Kaspi et al. (2007, α ≈ 0.52 − 0.55, c.f. Kaspi et al. 2021). More high S/N spectroscopic observations, including monitoring, should be carried out to explore the full potential of the 1900Å blend lines and especially of Ciii] as VBEs.  Vestergaard & Peterson (2006) based on the Hβ spectral range vs. the one computed from the Ciii] data following Eq. 7, with spectral types color-coded as in Fig. 3. As for Fig. 11, the filled line traces Eq. 6, while the dot-dashed line is the equality line. The lower panel shows residuals as a function of M BH derived from Hβ. Average, standard deviation, and χ 2 ν of the log M BH difference are reported on the top-row of the inside caption; the bottom row reports median and SIQR. The scaling law derived from the Aliii line is consistent with the Civλ1549-based scaling law (M19): Fig. 14 indicates only a slight bias (less than the SIQR ≈ 0.2 dex, and the rms ≈ 0.3 dex), as the Civλ1549 scaling law apparently overestimates the M BH by ≈ 20% and ≈ 40% with respect to the M BH estimates based on Aliii and Hβ, respectively. Considering that the Civλ1549-based scaling law requires a large correction (ξ CIV can be as low as ≈ 0.2) to the Civλ1549 FWHM dependent on both line shift and luminosity, the Aliii scaling law should be preferred in case observations of both Civλ1549 and Aliii are available.

Shifts of Aliii vs shifts of Civλ1549 and Mgiiλ2800
Fig 15 shows the Aliii peak shifts reported in Table 2 vs. the centroid at half maximum of Civλ1549 from S07, S17, Deconto-Machado et al. (2022, in preparation), and the "peak" shift given by Vietri et al. (2018) for the four subsamples of the present work. The Aliii shift is usually very modest, and a factor of ∼ 10 lower than the shift measured for Civλ1549. The Aliii and Civλ1549 shifts are however significantly correlated (Pearson's correlation coefficient ≈ 0.54, with a significance P ≈ 1. − 2 · 10 −3 ): The slope is shallow, but the correlation indicates that the shifts in Aliii and Civλ1549 are likely to be due to the same physical effect. If we ascribe the small displacements observed in the peak shift s of Aliii to the effect of outflows, the outflow prominence is much lower than in the case of Civλ1549, both in terms of radial velocity values and of δ. As mentioned, Civλ1549 shifts and FWHM are correlated, implying that the broader the line, the higher the shift amplitude becomes (Coatman et al. 2016;Sulentic et al. 2017). The Aliii shows a consistent behavior, but apparently masked by the much lower shift amplitudes; the presence of Aliii blueshifts appears to be statistically significant at very high luminosity, and for spectral type A3 and A4. When Civλ1549 shows large blueshifts i.e., for high R FeII or very high luminosity, the Aliii line becomes broader than Hβ (even if the two lines remain in fair agreement). We see a relation between Aliii line shift and widths: in ST A3 and A4, where shifts are larger, the Aliii FWHM exceeds the one of Hβ ( Figure 5).
The Aliii and Civλ1549 results on line shifts are also consistent with the ones obtained for Mgiiλ2800. Small amplitude blueshifts of a few hundreds km s −1 were measured on the full line profile of Mgiiλ2800 (Marziani et al. 2013b). For Mgiiλ2800 the separation of a BLUE component and a symmetric Lorentzian has been possible on median composite spectra because of the high S/N and of the peaky line core of the Mgiiλ2800 line. The same operation is not feasible for in- dividual Aliii profiles that are often significantly affected by noise, and in some cases even barely above noise. The Aliii, Civλ1549 and Mgiiλ2800 are all resonance lines that may be subject to selective line-driven acceleration (Murray & Chiang 1997;Proga 2007;Risaliti & Elvis 2010). The different velocity amplitudes most likely reflect the difference in the line emitting region distance from the continuum source and in physical properties, such as ionization parameter, density and column density.
Rare sources with large shift amplitudes in Aliii are expected to be intrinsically infrequent even at the redshift where luminous quasars were fairly common (z ≈ 1.5 − 2) and, even if over-represented because of a Malmquist-type bias, they are outstanding and pretty easily recognizable, especially in large samples of AGN (Sect. 5.3). The most striking case directly resembling Civλ1549 is the one of HE0132-4313 which is an object of fairly low luminosity and an outlier in the plots FWHM Aliii vs FWHM Hβ. Sources with large shift amplitude may be excluded or flagged if black hole mass estimates are being carried out.

xA quasars
There are 16 sources meeting the criterion R FeII 1 in the joint sample. The wide majority of these sources shows Aliii blueshift with respect to Hβ. The average shift is rather modest, ≈ −250 although for 7 of them, s −250 km s −1 . Six objects show evidence of a strong excess on the blue side of the 1900 blend. Fits to the 1900 blend of these sources following the standard approach are shown in Appendix B. The fits have been repeated by allowing for an extra BLUE in the 1900 blend, represented as a skewed Gaussian (Fig. 16). The blueshifted excess in some cases cannot be distinguished from strong Siiiλ1816 emission. The Siiiλ1816 emission line can be of strength comparable to Aliii in the condition of low ionization and high density derived for the virialized component (Negrete et al. 2012).
In two cases (e.g., HE0359-3959,and SDSS J152156.48+520238.5) the blueshifted excess is so overwhelming that Siiiλ1816 emission cannot account for the excess unless the Siiiλ1816 line itself is significantly blueshifted. These are perhaps the best cases supporting the evidence for a significant BLUE in Aliii. It is reasonable to assume that the blueshifted excess is mainly due to Aliii, being Aliii a resonant line for which BALs are also observed. Broad absorption components are observed in the Aliii profile, even if rarely, and with terminal radial velocity of the absorption through much lower than the one of Civλ1549 (Gibson et al. 2009).
The Eddington ratio values derived from the continuum luminosity at 1700Å (after Galactic extinction correction) multiplied by a bolometric correction factor k bol ≈ 3.5, and from the M BH estimated from Eq. 5 range between L/L Edd ≈ 0.2 and L/L Edd ≈ 1.5. If the luminosity-dependent k bol is applied to the 1700Å continuum, 0.2 L/L Edd 1.1. Both estimates confirm that all quasars of the presented sample are within the range expected for Pop. A sources. The xA sources are at the high end of the L/L Edd distribution, with µ(logL/L Edd )≈ −0.18, and the Aliii BLUE sources are even more extreme with µ(logL/L Edd )≈ −0.105. Extreme radiation forces may make it possible to blow out rather dense/high column density gas from the virialized region associated with the emission of the low-and intermediate ionization lines (Netzer & Marziani 2010). Sources showing a strong BLUE in Aliii could be the most extreme accretors, perhaps in a particular "blow-out" phase of the quasar evolution (D'Onofrio & Marziani 2018).
A related issue is whether sources with a strong blue-shifted component in Aliii can be empirically distinguished from the rest of Pop. A quasars, without resorting to the knowledge of the rest frame. The FWHM of the whole blend (i.e., of the sum of all lines after continuum subtraction) is clearly affected by spectral type: going from A1 to A4 we see an overall decrease of prominence in Ciii], and an increase in Aliii with respect to the other line. The blue-shifted excess should further increase the FWHM of the blend. The parameter normalizes the FWHM of the whole blend FWHM 1900 (i.e., the sum of all line components) because of the increase of the line width with luminosity by a factor L 1 4 . Fig. 17 shows the distribution of the A for the sources of ST A1 and A2, A3 and A4, and the 6 quasars for which the 1900 blend was fit with the addition of a blue-shifted excess. The distributions of A1+A2 and A3+A4 are significantly different at a 3σ confidence level according to a K-S test. However, there is considerable overlap around A ≈ 3000 km s −1 , making it difficult to unambiguously distinguishing between xAs, xAs with blue-shifted excess and other sources.

Consistency of UV and optical classification for extreme Population A
Extreme quasars can be identified by employing selection criteria in the optical and UV (MS14). The consistency of the selection criteria is however little tested, since observations covering both the 1900Å range and the Hβ one are still rare. In the joint sample most sources with R FeII 1 also satisfy the UV intensity-ratio conditions Aliii/Siiii]> 0.5, and Siiii]>Ciii]. Figure 18 shows the location of the data points identified according to spectral type (defined by ranges of the optical parameter R FeII ; A3 and A4 satisfy the condition R FeII 1 by definition) in the plane defined by the UV ratios Ciii]/Siiii] vs. Aliii/Siiii]. There are several borderline cases, but only one in which the criteria are not satisfied: J15591+3501 with intensity ratios Ciii]/Siiii]≈ 2.38±0.45, and Aliii/Siiii]≈ 0.33±0.06. For J14421+3526, the feature at ≈ 1910Å is most likely a blend of Feiii and Ciii]. In this case, only an upper limit can be assigned to the Ciii] intensity, and the UV selection criteria may not have been violated. The reason of the discordance for J15591+3501 is not clear. The majority of objects (≈ 80%) in Figure 18 supports the equivalence between the two xA selection criteria suggested by MS14. Apart from borderline cases, five A2 sources (4 if we exclude J1421+4633 with R FeII ≈ 0.99) out of 19 enter the domain of the xA (the grey shaded area of Figure 18). These sources appear to be genuine xA in terms of the UV intensity ratios, but have lower than expected R FeII (0.5 R FeII 1). It is intriguing that the four sources all belong to the high-z samples. The possibility of systematic differences as a function of redshift in the relative abundance of iron with respect to carbon and α elements should be further investigated (e.g., Martínez-Aldama et al. 2021, and references therein).

SUMMARY AND CONCLUSION
The present investigation has shown a substantial equivalence of Hβ and Aliii and Ciii] as virial broadening estimators for Population A quasars, thereby providing a tool suitable for M BH estimates up to z 4 from observations obtained with optical spectrometers. More in detail, the salient results of the present investigation can be summarized as follows: • the Aliii and Hβ FWHM are highly correlated and, in the joint sample of 48 Population A sources, can be considered statistically equivalent over 4 orders of magnitude in luminosity (Sections 4.1 and 4.2).
• The FWHM ratio between Aliii and Hβ increases with increasing R FeII or, equivalently, from spectral type A1 to A4 (Sect. 4.3). Extreme Pop. A sources appear to be 20% broader than the sample average, while spectral type A1 20 % narrower than spectral type A2.
• Systematic blueshifts are revealed in Aliii; however, in most cases the amplitude of the blueshifts is modest or smaller than the uncertainties, reaching a sample median for spectral types A3 and A4 of just ≈ −160 km s −1 (Sections 4.3 and 4.4).
• The line FWHM of Hβ, Aliii and Ciii] increases with luminosity as a function of L 1 4 , as expected for a virial velocity field of the line emitting gas (Sect. 4.5).
• The following scaling law between M BH and luminosity and FWHM Aliii: log M BH (AlIII) ≈ (0.579 +0.031 −0.029 ) log L 1700,44 + 2 log (FWHM(AlIII)) + (0.490 +0.110 −0.060 ) (Eq. 5 in Sect. 4.6) provides an estimate of M BH with an rms scatter of ≈ 0.3 dex with respect to the Hβ-derived masses, for M BH 10 7 M . • An analogous scaling law has been defined also for Ciii] (Eq. 7 in Sect. 4.7). The measurement of the Ciii] FWHM is however more strongly affected by the severe blending and by the Ciii] weakness in sources with high R FeII . The Ciii] scaling law requires a constant correction factor to the FWHM of Ciii], ξ CIII] ≈ 1.25. The scaling laws derived from Aliii and Ciii] line width are mutually consistent (Sect. 5.1).
• Although Aliii shift amplitudes are ≈ 1 10 the shifts of Civλ1549 (Section 5.2), it is unclear whether Aliii can be exploited as a virial luminosity estimator for extreme Population A sources (Section 5.3): the Aliii profile is strongly affected by a blueshifted excess in several extreme Pop. A sources (Sect. 5.3). The majority of quasars show consistency between FWHM Aliii and Hβ, and a minority of sources that show FWHM Aliii FWHM Hβ might be easy to recognize in large samples. The extent of systematic effects should however be analyzed by a thorough study of a very large sample of xA sources with full coverage of the optical and UV rest-frame ranges from 1000 to 5500Å.
These results show that the Aliii line is a good UV substitute of Hβ and can be used for black hole mass estimations with the advantage to be at the rest-frame of the source. The results on FWHM Aliii should be compared with the ones obtained for Civλ1549 (M19), where the equivalence was obtained at the expense of corrections that were dependent on the accurate knowledge of the quasar rest frame, and therefore not fully achievable without additional measurements in spectral ranges distinct from the one of Civλ1549: the [Oii]λ3727 line is the narrow low ionization line closest in wavelength to Civλ1549 and offers a reliable rest frame estimator (Bon et al. 2020), but is very rarely covered along with Civλ1549. Uncertainties were estimated following a Bayesian approach, considering the likelihood function where f i are the specific flux values as a function of wavelength (or of pixel number), σ i the uncertainty in f i (in practice from the S/N set constant over the spectrum), m i (Θ) the expectation value for the multicomponent model Θ of the spectrum obtained via a specfit analysis. The Θ can be any set of free parameters employed in the fits: intensity, shift and width of each line, intensity, shift and width of each template. Priors were specified for several parameters in terms of a range of permitted values. The posteriors of the model parameters Θ (for instance, the distributions of FWHM Hβ and Aliii given the data) were obtained by creating a random walk with a modified Metropolis-Hasting algorithm: a new candidate set of model parameters Θ was randomly generated, and screened by an accettance parameter α. The set Θ included model parameters believed to significantly affect the line widths (in practice, most of the parameters included in the specfit analysis). For example, the [Oiii]λλ4959,5007 lines were modeled with two components, a "core" component represented by a symmetric Gaussian, and a semi-broad component modeled by a skew Gaussian. The template Feii emission was scaled, shifted and broadened as done in the specfit procedure. The dispersion of the posterior distribution of each spectral parameter was assumed to yield its uncertainty δ at 1σ confidence level.
A.2. The quality parameter Q The next step was to connect the uncertainty in FWHM, shift and intensity to a quality parameter Q, which may turn useful in case very late samples of quasars are analysed. The quality parameter Q = log 10 W FWHM · S N (A2) defined as the product of the S/N times a line equivalent width W divided by its FWHM, increases with S/N and line prominence over the continuum and decreases with increasing line widths. The signal in each resolution element is proportional to the ratio W /FWHM, which is a measurement of the sharpness of the line, as obviously Q ∝ log 10 I peak ·FWHM FWHMIc · Ic N ∝ I peak N . The quality parameter Q obviates to the inadequacy of the S/N measurement carried out on the continuum. By multiplying it by the ratio W /FWHM we compute a more apt average S/N for a line depending on its strength and width. The parameter Q is larger for sharp lines in spectra with high S/N in the continuum. The large differences in S/N, line width and line strength between Aliii and Hβ is reflected in the distribution of the Q parameter, shown in Fig. 19.
To be of any practical use, the Q parameter needs to be anchored to estimates of the uncertainties. The posterior distributions of the spectral parameters were computed for about 30 sources. Fig.  20 shows a well-defined trend between Q and the fractional uncertainty δFWHM/FWHM for Hβ,  Aliii, and Ciii] derived from the MCMC simulations. Especially for large Q values, the scatter is relatively modest, and the relation between the parameter FWHM, flux and shift and log Q can be written in a linear form, save for the fractional uncertainty of FWHM Aliii that is best fit by δFWHM/FWHM ≈ 1/(a + b log Q). Table 3 provides the coefficients a and b of the best fits along with Q domain. The FWHM relations were obtained by a non-linear fit algorithm implemented in R (R Core Team 2019), and are shown as the thick lines in Fig. 20. Figure 19. Left: distribution of the log of parameter Q parameter for Aliii (blue) and Hβ (red) for the joint sample considered in this paper. Right: distribution of log Q differences between Hβ and Aliii.  Table  3 for the FWHM of Aliii (black) and Ciii] (blue).

B. Hβ AND 1900 BLEND PAIRED COMPARISON
The results of line profile fitting for the 1900 blend and Hβ are shown in the following figures: Fig.  21 for the FOS sample, Fig. 22 for the HE sample, Fig. 23 for the FOS sample, and Fig. 24 for the WISSH sample. All spectra have been continuum subtracted and normalized by the 5100Å (Hβ) and 1700Å (1900 blend) specific flux.