# The variability of the broad-line Balmer decrement for quasars from the
Sloan Digital Sky Survey Reverberation Mapping

Yan-Song Ma1, Shao-Jun Li1, Chen-Sheng Gu1, Jian-Xia Jiang1, Kai-Li Hou1, Shu-
Hao Qin1, Wei-Hao Bian1
1School of Physics and Technology, Nanjing Normal University, Nanjing 210023,
People’s Republic of China
E-mail<EMAIL_ADDRESS>
###### Abstract

Based on the spectral decomposition through a code of PrepSpec, the light
curves (spanning 6.5 years in the observed frame) of the broad-line Balmer
decrement, i.e., the flux ratio of the broad H$\alpha$ to the broad H$\beta$
line, are calculated for a sample of 44 Sloan Digital Sky Survey
reverberation-mapped quasars ($z<0.53$). It is found that the logarithm of the
mean broad-line Balmer decrement is 0.62 with a standard deviation of 0.15
dex. The relations between the mean Balmer decrement and the SMBH accretion
properties (the luminosity, black hole mass, Eddington ratio, accretion rate)
are investigated and no obvious correlations are found. It is found that there
are 27 quasars ($61\%$) showing strong negative correlations between the
Balmer decrement variance and the continuum variance, i.e., the Balmer
decrement would be smaller with larger continuum flux. Assuming that the dust
obscuration leads to the variance in the Balmer decrement and the continuum,
an expected slope is $-1/3$, which is not consistent with most of measured
slopes. Using the interpolated cross-correlation function, the time delays
between the inverse Balmer decrement and the continuum are measured for 14
quasars with the maximum correlation coefficient larger the 0.6. It suggests
that the size corresponding to the Balmer decrement lag extends from the BLR
size to the torus size.

###### keywords:

galaxies: active – quasars: emission lines – quasars: general – quasars:
supermassive black holes

##  1 INTRODUCTION

Active galactic nuclei (AGN) are accreting supermassive black holes (SMBHs) at
the centres of galaxies. The standard paradigm for AGN assumes that the
central SMBH is surrounded by an accretion disc (AD) and broad-line region
(BLR), which are surrounded by a dusty torus, and a farther narrow-line region
(NLR). The NLR/BLR Balmer decrement ($\rm H\alpha/H\beta$ ) is usually used as
a tool to investigate the properties of gas and dust in AGN (e.g., Brinchmann
et al., 2004; Osterbrock & Ferland, 2006; Lu et al., 2019b).

Predicted by Case B recombination (at a typical low electron density of $\rm
10^{2}~{}cm^{-3}$ and temperature of $\rm 10^{4}K$ ), an intrinsic Balmer
decrement value of 2.86 is suggested for H II region photoionized by a hot
star. For Balmer decrement of AGN, a larger value for the intrinsic decrement
is predicted from CLOUDY code, where is due to the presence of NLR/BLR gas of
higher densities , much harder ionizing continuum, lower ionization parameter
(Osterbrock & Ferland, 2006; Netzer, 2013; Heard & Gaskell, 2016).

The distributions of Balmer decrement of NLR/BLR were reported for different
selected AGN samples (e.g., Dong et al., 2005, 2008; Heard & Gaskell, 2016;
Baron et al., 2016; Lu et al., 2019b). For a sample of blue AGN defined by the
spectral slopes in order to minimize the effect of dust extinction, the
distributions of the intrinsic BLR Balmer decrement was investigated (Dong et
al., 2008). Based on this sample of extremely blue AGN, Gaskell (2017)
suggested that the intrinsic Balmer is $2.72\pm 0.041$, which is consistent
with the Case B value of 2.74 (Osterbrock & Ferland, 2006). The small scatter
of 0.04 dex suggested the BLR decrement can be used a reddening indicator for
the BLR, as well as for the NLR decrement (e.g., Dong et al., 2008; Bian et
al., 2010a; Heard & Gaskell, 2016; Lu et al., 2019b). For a compiled sample of
AGN with reliable decomposition of the narrow/broad $\rm H\alpha/H\beta$ lines
(Osterbrock, 1977; Cohen, 1983; Dong et al., 2005, 2008), it was also found
that the average BLR Balmer decrement is larger than NLR (Heard & Gaskell,
2016), suggesting an additional dust reddening the BLR is interior to the NLR,
which is possibly due to the dust in BLR and/or in torus. However, this is
opposite to the results obtained by others, (de Zotti & Gaskell, 1985; Dong et
al., 2008; Baron et al., 2016; Lu et al., 2019b). The comparison between the
NLR and BLR Balmer decrement depends on the method to de-composite the BLR/
NLR contribution in H$\alpha$ and H$\beta$ lines.

For quasars from the Sloan Digital Sky Survey (SDSS), it was found that BLR
Balmer decrement depends on the optical continuum slope, although with a
larger scatter (see Fig. 7 in Baron et al., 2016). It was also found that
there is no correlation between the decrement and the luminosity (Dong et al.,
2008; Lu et al., 2019b). Recently the variability of the BLR Balmer decrement
is investigated for individual AGN (e.g., Siram et al., 2021; Pandas &
Seiegowska, 2022; Li et al., 2022; Selwood et al., 2023). The Balmer decrement
variability for a sample with spectral monitoring would provide clues to these
problems.

It was suggested that the diversity of AGN can be unified by the orientation
and the Eddington ratio (Shen & Ho, 2014). It is necessary to investigate the
relation between the BLR/NLR decrement and the accretion process. For broad-
line AGN, the BLR clouds (size $R_{\rm BLR}$ and velocity $\Delta V$) can be
used as a probe to calculate its virial mass as follows (e.g., Peterson et
al., 2004; Yu et al., 2020b):

$M_{\rm BH}=f\times\frac{R_{\rm BLR}~{}(\Delta V)^{2}}{G},$ (1)

where G is the gravitational constant, $f$ is a virial factor. From the
standard disk model of Shakura & Sunyaev (1973), the dimensionless accretion
rate $\dot{\mathscr{M}}$ can be calculated from $L_{5100}$ and $M_{\rm BH}$ as
follows (e.g., Du et al., 2016):

$\dot{\mathscr{M}}\equiv\dot{M}/\dot{M}_{Edd}=20.1\left(\frac{\it{l}\mathrm{{}_{44}}}{\cos\,\it{i}}\right)^{3/2}m_{7}^{-2},$
(2)

where $\dot{M}$ is the accretion rate, the Eddington accretion rate
$\dot{M}_{Edd}=L_{\rm Edd}/c^{2}$, $\l_{44}=L_{5100}/10^{44}{\rm erg\ s}^{-1}$
is the 5100 Å luminosity in units of $10^{44}$ erg s-1 , $L_{\rm Edd}$ is the
Eddington luminosity, $m_{7}=M_{\rm BH}/10^{7}M_{\odot}$ is the SMBH mass in
units of $10^{7}M_{\odot}$, $i$ is the accretion disk inclination relative to
the observer and $cosi=0.75$. The Eddington ratio $L_{\rm Bol}$ /$L_{\rm Edd}$
and the dimensionless accretion rate $\dot{\mathscr{M}}$ are two important
parameters describing the SMBH accretion process, where $L_{\rm Bol}$ is the
bolometric luminosity. Considering $L_{\rm Bol}=\eta\dot{M}c^{2}$, where
$\eta$ is the accretion efficiency, $L_{\rm Bol}/L_{\rm
Edd}=\frac{\eta\dot{M}c^{2}}{\dot{M}_{\rm Edd}c^{2}}=\eta\dot{\mathscr{M}}$.

In this paper, for a sample of 44 SDSS-RM quasars, we use light curves
(spanning 6.5 years) of H$\beta$ , H$\alpha$ and the continuum at 5100 Å
derived from the spectral decomposition through PrepSpec to investigate the
variability of BLR Balmer decrement ($\rm H\alpha/H\beta$ ). In Section 2, we
describe the sample and light curve data. The mean value of the Balmer
decrement and it connection with the SMBH accretion, the relative variance of
the Balmer decrement, the relation of $\rm H\alpha/H\beta$ variability with
the continuum, and the lag of the Balmer decrement with respect to the
continuum are presented in Section 3. Section 4 is our results.

##  2 Sample

SDSS-RM Project monitored a flux-limited ($i_{\rm psf}<21.7$) sample of 849
quasars with redshifts of $0.1<z<4.5$ on the 2.5 m SDSS telescope with the
BOSS spectrograph (Shen et al., 2016; Grier et al., 2017). The wavelength
coverage of BOSS spectra is $\sim$ 3650–10400Å with a spectral resolution of
$R\sim 2000$. Additional photometric data were acquired with the 3.6 m
Canada–France–Hawaii Telescope (CFHT) and the Steward Observatory 2.3 m Bok
telescope.

A code called PrepSpec 111PrepSpec: https://github.com/Alymantara/pyPrepSpec
is used to improve the relative flux calibrations and produce line/continuum
light curves (Shen et al., 2016; Grier et al., 2017; Khatu et al., 2023). What
PrepSpec does is similar to the decomposition approach used in other studies
(e.g., Bian et al., 2010b; Barth et al., 2015; Hu et al., 2015, 2021). Because
we use the light curves derived from PrepSpec, the code is introduced as
follows. For each SDSS-RM spectrum for an RM quasar, the PrepSpec model is:
$\mu(\lambda,t)=p(t)[A(\lambda,t)+B(\lambda,t)+C(\lambda,t)]$, where $p(t)$
are time-dependent photometric corrections, $A(\lambda)$ is the average
spectrum, and $B(\lambda,t)$ and $C(\lambda,t)$ are variations in the BLR
spectrum and continuum, respectively.
$A(\lambda)=\bar{F}(\lambda)+N(\lambda)$, where $\bar{F}(\lambda)$ includes
both the continuum and broad-line components, as well as the $\rm Fe\ \sc{II}$
pseudo-continuum (host-galaxy light is not included ). The narrow-line
component, $N(\lambda)$, is isolated using a piecewise cubic spline fit to
$A(\lambda)$ as a high-pass filter and then multiplying the result by a window
function that is 0 outside and 1 inside a defined velocity range around a
specified list of narrow emission lines. The BLR variations $B(\lambda,t)$ are
represented by a separable function for each line,
$B(\lambda,t)=\Sigma_{l=1}^{N_{l}}B_{l}(\lambda)L_{l}(t)$, where the light
curve $L_{l}(t)$ is normalized to a mean of 0 and rms of 1, and
$B_{l}(\lambda)$ is the rms spectrum of line $l$. A key feature of PrepSpec is
the inclusion of a time-dependent flux correction calculated by assuming that
there is no intrinsic variability of the narrow emission-line fluxes. PrepSpec
minimizes the apparent variability of the narrow lines by fitting a model to
the spectra that includes intrinsic variations in both the continuum and
different broad emission lines. The narrow-line and broad-line window widths
are determined through iterating until convergence in PreSpec. It is different
to other decomposition methods using multi-Gaussians for one by one spectrum
(e.g., Bian et al., 2010b; Barth et al., 2015; Hu et al., 2015, 2021).

The light curves of H$\beta$ , H$\alpha$ , and the continuum derived from
PrepSpec model for 44 quasars ($z<0.53$) are given in the website of Keith
Horne 222http://star-www.st-and.ac.uk/$\sim$kdh1/pub/sdss/2020/sdssz.html.
Because the long coverage of light curves (MJD from 2456660 to 2459029, about
6.5 years ) and the updated spectral analysis, we use the data provided by the
website of Keith Horne instead of the data shown in Table 2 in Grier et al.
(2017) 333https://cdsarc.cds.unistra.fr/ftp/J/ApJ/851/21/ . The observation
cadence ( average sampling interval of the objects) is large , about 5 days in
the years of 2014, about 13 days in years of 2015-2017, and about 20 days in
years of 2018-2020. The sample of these 44 SDSS-RM quasars is used to
investigate the Balmer decrement $\rm H\alpha/H\beta$ .

Figure 1: Light curves of the H$\alpha$ , H$\beta$ , $\rm H\alpha/H\beta$ ,
continuum at 5100 Å (from top to bottom) for an example of quasar, SDSS
J141123.42+521331.7 (RMID 0371, z = 0.472). The relative variance and its
error for each light curve are shown in the upper middle corner of each panel.

Figure 1 shows an example of the light curve of the Balmer decrement, as well
as the H$\alpha$ , H$\beta$ and the continuum at 5100 Å for one quasar (RMID
0371, z = 0.472). For the light curve of $\rm H\alpha/H\beta$ , the ratio of
H$\alpha$ to H$\beta$ emission line flux is calculated from the website of
Keith Horne. Through the error propagation formula, errors of H$\alpha$ and
H$\beta$ flux are used to calculate the error of $\rm H\alpha/H\beta$ ,

$\sigma=\frac{f_{\rm H\alpha}}{f_{\rm H\beta}}\sqrt{(\sigma_{\rm
H\beta}/f_{\rm H\beta})^{2}+(\sigma_{\rm H\alpha}/f_{\rm H\alpha})^{2}}$ (3)

where $\sigma$ is the error of Balmer decrement, $f_{\rm H\alpha}$ and $f_{\rm
H\beta}$ are the emission fluxes of H$\alpha$ and H$\beta$ , $\sigma_{\rm
H\alpha}$ and $\sigma_{\rm H\beta}$ are the corresponding errors of H$\alpha$
and H$\beta$ .

##  3 RESULTS AND DISCUSSIONS

###  3.1 The relative variance of the Balmer decrement $\rm H\alpha/H\beta$

Figure 2: Distribution of $F_{\rm var}$ for $\rm H\alpha/H\beta$ (filled with
red lines), the continuum at 5100 Å (filled with blue lines), H$\alpha$
(orange lines) and H$\beta$ (green lines). The inserted panel in the upper
right corner shows the corresponding cumulative distribution of $F_{\rm var}$.

Accounting for the measurement uncertainties, the relative intrinsic
variability amplitude $F_{\rm var}$ is used to show the variance in the light
curve. $F_{\rm var}$ and its uncertainty $\sigma_{\rm Fvar}$ of a light curve
are as commonly defined (e.g., Rodriguez-Pascual et al., 1997; Edelson et al.,
2002; Fausnaugh et al., 2016; Lu et al., 2019a; Zhao et al., 2020):

$F_{\rm var}=\frac{(\sigma^{2}-\Delta^{2})^{1/2}}{\bar{f}},\sigma_{\rm
Fvar}=\frac{1}{F_{\rm var}}(\frac{1}{2N})^{1/2}\frac{\sigma^{2}}{\bar{f}},$
(4)

where $\bar{f}$ is the average flux of $f_{i}$, $\sigma$ is the standard
deviation of flux $f_{i}$, $N$ is the number of observations, $\Delta^{2}$ is
the mean square of uncertainty $\Delta_{i}$ on the flux $f_{i}$. $F_{\rm var}$
gives an estimate of the relative intrinsic variability amplitude by
accounting for the measurement uncertainties.

For each light curve, we calculate the relative intrinsic variability
amplitude $F_{\rm var}$ and its error, where are presented in Table 1. There
are 7 quasars without values of $F_{\rm var}$ because of $\Delta^{2}$ larger
than $\sigma^{2}$. Figure 2 is the $F_{\rm var}$ distribution of the broad-
line Balmer decrement $\rm H\alpha/H\beta$ , as well as H$\alpha$ , H$\beta$
and continuum at 5100 Å for our sample. The mean $F_{\rm var}$ of $\rm
H\alpha/H\beta$ is 0.11$\pm$0.019, the mean $F_{\rm var}$ of H$\alpha$ of all
sample is 0.11$\pm$0.009, the mean $F_{\rm var}$ of H$\beta$ is
0.22$\pm$0.023, and the mean $F_{\rm var}$ of continuum is 0.10$\pm$0.008. The
mean $F_{\rm var}$ of $\rm H\alpha/H\beta$ is similar to H$\alpha$ and the
continuum, but smaller than H$\beta$ .

Kolmogorov–Smirnov (K–S) test 444We used python (scipy.stats.ks_2samp,
scipy.stats.spearmanr) to do our following analysis for the K-S test and the
Spearman correlation test. is performed to investigate the distribution
differences among them. For the distributions of the decrement $\rm
H\alpha/H\beta$ and H$\alpha$ , K-S test shows the statistic d and the
significant level probability for the null hypothesis ($p$) are 0.23 and 0.20.
It shows that there are no obvious difference between their distributions. It
is the case for the distributions of $\rm H\alpha/H\beta$ and continuum at
5100 Å with d=0.24 and $p=0.17$ . For the distributions of $\rm
H\alpha/H\beta$ and H$\beta$ , d=0.46 and $p=2\times 10^{-4}$, which showing
the difference between their distributions.

###  3.2 The mean value of $\rm H\alpha/H\beta$ and the relation with the SMBH
accretion

Figure 3: The distribution of the mean Balmer decrement ($\log\rm
H\alpha/H\beta$) showing a large range from 0.3 to 1.

Figure 4: The relations of the mean Balmer decrement with the the continuum
$L_{5100}$ (top left), the SMBH mass $M_{\rm BH}$ (top right), the Eddington
ratio $L_{\rm Bol}/L_{\rm Edd}$ (bottom left) and the dimensionless accretion
rate $\dot{\mathscr{M}}$ (bottom right). The Spearman test result is shown in
the top corner in each panel.

For each quasar, we use the square of the measurement error as the weight to
calculate the mean value of $\rm H\alpha/H\beta$ . The error of the mean $\rm
H\alpha/H\beta$ is calculated from the weighted mean of the measurement errors
and the weighted standard deviation. Figure 3 shows the distribution of the
mean Balmer decrement $\log(\rm H\alpha/H\beta)$. The mean value of $\log(\rm
H\alpha/H\beta)$ is 0.62 with the standard deviation of 0.15 dex. For $\rm
H\alpha/H\beta$, the mean value is 4.17 with the standard deviation of 1.42.
We notice that our measured BLR $\rm H\alpha/H\beta$ have a larger range
extending from 2 to 10. It is found that the BLR $\rm H\alpha/H\beta$ has a
strong relation with the equivalent width (EW) of the broad H$\beta$ line
($r_{s}=-0.43,p_{\rm null}=4.0\times 10^{-3}$), while no obvious relation
relation with H$\alpha$ EW ($r_{s}=0.073,p_{\rm null}=0.64$). It is possible
that the large $\rm H\alpha/H\beta$ is due to weak H$\beta$ line. Therefore,
if we excluding quasars with $\rm H\beta(\rm EW)<20$, the mean $\log\rm
H\alpha/H\beta$ is 0.57 with the standard deviation of 0.11 dex, i.e.,
$3.71\pm 0.90$ for the mean $\rm H\alpha/H\beta$. For BLR $\rm H\alpha/H\beta$
, its value changes from one sample to another. Dong et al. (2008) found it is
3.06, Gaskell (2017) suggested it is 2.72 and Lu et al. (2019b) found it is
3.16. However, Lakicevic et al. (2017) found it is 3.76, as well as 3.38 found
by Siram et al. (2021). Our result is similar to that by Lakicevic et al.
(2017).

For quasars with reliable lag detection (Grier et al., 2017), $M_{\rm BH}$ is
adopted from col. (11) in Table 1 from Yu et al. (2020b), as well as its
error, which is calculated from the broad H$\beta$ line dispersion from the
rms spectrum and the lag suggested by (Grier et al., 2017). For other quasars,
considering the extended empirical $R_{\rm BLR}({\rm H\beta})-L_{\rm 5100}$
relation including $R_{\rm Fe}$ (Du & Wang, 2019; Yu et al., 2020a), $M_{\rm
BH}$ is calculated as follows: $\log\frac{M_{\rm
BH}}{M_{\odot}}=7.01+2\log\frac{\rm FWHM_{H\beta}}{1000~{}\rm km/s}+0.48\log
l_{44}-0.38R_{\rm Fe}$ (Liu & Bian, 2022), where $l_{44}$, $\rm FWHM_{H\beta}$
and optical $R_{\rm Fe}$ are measured from the mean spectra (Shen et al.,
2019). Based on the error transfer formula, the error of $M_{\rm BH}$ is
calculated from the errors of $\rm FWHM_{\rm H\beta}$, $L_{5100}$ , $R_{\rm
Fe}$ , including a systematic error of 0.2 dex for $R_{\rm BLR}(\rm H\beta)$.
The error of $\dot{\mathscr{M}}$ is about double $M_{\rm BH}$ error. The error
of $L_{\rm Bol}/L_{\rm Edd}$ is about 0.33 dex (e.g., Liu & Bian, 2022; Chen
et al., 2022).

In Table 1, $L_{5100}$ , $\rm FWHM_{\rm H\beta}$, $R_{\rm Fe}$ , $M_{\rm BH}$
are shown in cols. (14-17). Sulentic et al. (2000) proposed two main
populations on the plane of $\rm FWHM_{H\beta}$ versus $R_{\rm Fe}$ (Shen &
Ho, 2014): Population A (Pop. A) for quasars with $\rm
FWHM_{H\beta}<4000~{}{\rm km\ s}^{-1}$ and Population B (Pop. B) for those
with $\rm FWHM_{H\beta}>4000~{}{\rm km\ s}^{-1}$. For our sample, there are 23
quasars belong to Pop. B, 21 quasars belong to Pop. A and 5 quasars belong to
extreme Pop. A with $R_{\rm Fe}>1.2$.

In Figure 4, we show the relations of the mean value of $\rm H\alpha/H\beta$
with $L_{5100}$ , $M_{\rm BH}$ , $L_{\rm Bol}/L_{\rm Edd}$ and
$\dot{\mathscr{M}}$ . There is a large $M_{\rm BH}$ error bar for RMID 0767,
which is due to its large error of $\rm FWHM_{H\beta}$, i.e., $2003\pm
3868~{}{\rm km\ s}^{-1}$. It seems that there exists a trend of smaller $\rm
H\alpha/H\beta$ with larger $L_{5100}$ and $L_{\rm Bol}/L_{\rm Edd}$ or
$\dot{\mathscr{M}}$ . Using the Spearman correlation test to investigate the
correlations between the mean Balmer decrement and $L_{5100}$ , $M_{\rm BH}$ ,
$L_{\rm Bol}/L_{\rm Edd}$ , $\dot{\mathscr{M}}$ . The test coefficient $r_{s}$
and the probability of the null hypothesis $p_{\rm null}$ are 0.03, 0.85 for
$L_{5100}$ , 0.15 and 0.34 for $M_{\rm BH}$ , -0.1 and 0.53 for $L_{\rm
Bol}/L_{\rm Edd}$ , -0.11 and 0.47 for $\dot{\mathscr{M}}$ . No obvious
correlations are found between the mean Balmer decrement and the $L_{5100}$ ,
$M_{\rm BH}$ , $L_{\rm Bol}/L_{\rm Edd}$ , $\dot{\mathscr{M}}$ .

###  3.3 The relation between the Balmer decrement variance and the continuum
variance at 5100 Å

Figure 5: Top: the relationship between $\rm H\alpha/H\beta$ and continuous
spectral flux for an example of quasar, SDSS J141123.42+521331.7 (RMID 0371, z
= 0.472). The red solid line is the best linear fitting and the blue region is
the confidence band. Bottom: the continuum at 5100 Å versus the continuum flux
at g+i band for the spectra at the year of 2014 (Grier et al., 2017) for this
quasar. The red line and the blue region are the same as in the top panel.
Figure 6: The distribution of the Spearman correlation coefficient $r_{\rm s}$
versus the distribution of probability of the null hypothesis $p_{\rm null}$.
The orange shaded region shows $p_{\rm null}$ $\geq$ 0.01, indicating no
strong correlation.  Figure 7: The slope versus $r_{s}$ for the relation
between the Balmer decrement and continuum flux at 5100 Å. The color bar shows
the $p_{\rm null}$ from the Spearman test . The red solid line shows the
theoretical value of slope ($k=-1/3$) predicted from the extinction and the
constant continuum/line flux.

It was suggested that there existed a correlation between $\rm H\alpha/H\beta$
and the continuum for a single AGN (e.g., Siram et al., 2021; Pandas &
Seiegowska, 2022; Li et al., 2022). Figure 5 shows an example of the relation
between the variance of the Balmer decrement $\rm H\alpha/H\beta$ and the
variance of the continuum at 5100 Å for one quasar (RMID 0371). There exists a
strong correlation with $r_{s}=-0.65$ and $p_{\rm null}=5.8\times 10^{-12}$.
Its continuum changes about 0.2 dex, and its decrement changes about 0.18 dex.
Considering the errors in both coordinates, through kmpfit
555https://www.astro.rug.nl/software/kapteyn/kmpfittutorial.html., the best
linear fit is $\log\rm H\alpha/H\beta=-(0.68\pm 0.03)logf_{con}+(0.84\pm
0.01)$. The fitting slope is $-0.68\pm 0.03$, which would be test by the dust
obscuration (see next paragraph). In the bottom panel in Figure 5, we also
show the continuum at 5100 Å versus the continuum flux at g+i band for the
spectra at the year of 2014 (Grier et al., 2017). The relation between them is
strong with $r_{s}=0.92$. The linear fitting slope by kmpfit is $1.038\pm
0.037$. Although existing line-contribution at g+i band, the slope of 1.038
shows the spectral continuum fluxes are consistent well with the photometric
continuum fluxes at g+i band .

For all 44 quasars in our sample, the correlation coefficient $r_{s}$, $p_{\rm
null}$ and the fitting slope $k$ are given in Cols. (11-13) in Table 1. For
the $r_{s}$ distribution, the mean value of the total sample is -0.45 and the
standard deviation is 0.78. In Figure 6, we show $p_{\rm null}$ versus
$r_{s}$, as well their histograms in the top and right panel. The orange
shaded region shows the $p_{\rm null}\geq 0.01$. Considering $p_{\rm
null}<0.01$, there are about 61% ( 27/44) quasars showing obvious anti-
correlation. There is only one quasar with $r_{s}>0$ with $p_{\rm null}<0.01$.
For the subsample with $p_{\rm null}<0.01$, the mean value of $r_{s}$ is -0.55
with a standard deviation of 0.16.

It is generally believed that Balmer decrements can be used for dust
extinction. The extinction at the wavelength of $\lambda$ can be calculated
using the formula: $<A(\lambda)/A(V)>=a(x)+b(x)/R_{V}$, where
$x{\equiv}1/{\lambda}$. According to the definition of extinction,
$A(\lambda)=m^{\rm obs}_{\lambda}-m^{\rm int}_{\lambda}=-2.5\log\frac{f^{\rm
obs}_{\lambda}}{f^{\rm int}_{\lambda}}$, where $m^{\rm
obs}_{\lambda},m^{int}_{\lambda}$ are the observed and intrinsic magnitude,
$f^{\rm obs}_{\lambda},f^{int}_{\lambda}$ are the observed and intrinsic flux.
Therefore, at the wavelength of $\lambda=5100~{}\AA$:

$f^{int}_{5100}=f^{obs}_{5100}\left[{\rm\frac{({\frac{H\alpha}{H\beta}})^{obs}}{({\rm\frac{H\alpha}{H\beta}})^{\rm
int}}}\right]^{\frac{A(5100\AA)}{A(\rm H\beta)-A(\rm H\alpha)}}$ (5)

Using the extinction law of the Milky Way with $R_{V}=3.1$, it is found that
$A(5100\AA)=1.11/A_{V},A(\rm H\alpha)=1.18/A_{V},A(\rm H\beta)=0.81/A_{V}$. We
obtain the index in Eq. 5 is 3.

Assuming the variance in $\rm H\alpha/H\beta$ and the continuum is coming from
the dust extinction, i.e., the intrinsic $\rm H\alpha/H\beta$ and the
continuum are stable, from Eq. 5, a relation can be found: $\Delta(\log\rm
H\alpha/H\beta)\propto-\frac{1}{3}\Delta(\log L_{\rm 5100})$, i.e, a slope of
$-1/3$. Figure 7 shows the slope $k$ versus $r_{s}$ ,as well as the color bar
based on $p_{\rm null}$. Red solid line shows the line of $k=-\frac{1}{3}$
predicted from the obscuration and the pink shaded area shows the 10%
uncertainty. Considering error of 10%, there are only 4 quasars consistent
with the slope of -1/3. Most of our measured slopes are not consistent with
the expected value of $-1/3$ derived from the dust obscuration. It implies
that there exists intrinsic variance of the continuum and the line flux.

###  3.4 The lag of the Balmer decrement

Figure 8: Light curves and CCF model for the the inverse of $\rm
H\alpha/H\beta$ analysis for an example of a quasar, SDSS J141041.25+531849.0
(RMID 0160, z = 0.359). The light curves of the continuum ($\log f_{\rm con}$)
and the inverse of the broad-line Balmer decrement ($\log\rm H\beta/H\alpha$)
(vertically shifted to assure positive values) are presented in the top and
middle panels. For display purposes, we show the weighted mean of all epochs
observed within a single night. The bottom three panels show the results of
the time-series analysis. The bottom left panel shows the CCF. The other two
panels present the lag distributions for the two different methods, normalized
to the tallest peak in the distribution. The bottom middle panel shows the
CCCD, the bottom right panel shows the CCPD lag distribution. For RMID 0160,
the lag is $49.8^{+9.4}_{-9.9}$ light days. Figure 9:  Lag of the Balmer
decrement $\tau_{\rm cent}(\rm H\beta/H\alpha)$ versus the maximum correlation
coefficient $r_{\rm max}$ for light curves obtained in years of 2014-2020. The
red line shows $r_{\rm max}=0.6$.

Figure 10: Comparison between the $\rm H\alpha/H\beta$ lag and the H$\beta$
lag (top), the H$\alpha$ lag (bottom). The dot line is 1:1, and the solid line
is 1:4 for the size ratio of BLR to the torus. The red circles denote quasars
with $r_{\rm max}>0.6$. The labels near locations of data points are quasars
RMID.

Because of the anti-correlation between $\rm H\alpha/H\beta$ and the
continuum, we calculate the light curve of the inverse $\rm H\alpha/H\beta$ ,
i.e., $\log\rm H\beta/H\alpha$, and use the common methods to detect the lag
with respect to the continuum ($\log f_{\rm con}$). There are mainly three
methods to do that, the interpolated cross-correlation function (ICCF) (e.g.,
Peterson et al. 2004), JAVELIN (Zu et al., 2011, 2013), and CREAM (Starkey et
al., 2016). ICCF is the most common method to measure time lags between two
light curves in AGN (e.g., Peterson et al., 2004; Bian et al., 2010b; Grier et
al., 2017). For 26 quasars with reliable measurement of H$\beta$ or H$\alpha$
lag given by Grier et al. (2017) , from light curves of $\rm H\alpha/H\beta$
and the continuum at 5100 Å, we calculate CCFs and centroid lag $\tau_{\rm
cent}$ using an interpolation grid spacing of 2 days, and a time lag limiting
to a range between -100 and 600 days. The centroid lag is measured using
points surrounding the maximum correlation coefficient $r_{\rm max}$ out to
$r\geq 0.8r_{\rm max}$, as is standard for CCF analysis (e.g., Peterson et
al., 2004; Grier et al., 2017). The lag of the Balmer decrement is listed in
col. 22 in Table 2.

Figure 8 shows an example about the CCF analysis for the light curves of
H$\beta$ /H$\alpha$ and the continuum at 5100 Å for years of 2014-2020 for
SDSS J141041.25+531849.0 (RMID 0160, z = 0.359). The light curves of the
continuum and H$\beta$ /H$\alpha$ are presented in the top and middle panels.
For display purposes, we show the weighted mean of all epochs observed within
a single night. The bottom three panels show the results of the time-series
analysis. The bottom left panel shows the CCF. With the increasing lag, the
correlation coefficient reach the maximum ($r_{\rm max}=0.86$) at about 49.8
light days, and then slow down with weak correlation. The other two panels
present the lag distributions from the two different methods, cross-
correlation centroid distribution ( CCCD) and cross-correlation peak
distribution (CCPD).

Figure 9 shows the lag $\tau_{\rm cent}$ of the Balmer decrement versus the
maximum correlation coefficient $r_{\rm max}$ for light curves obtained in
years of 2014-2020. The Balmer decrement lags are removed for quasars with
$r_{\rm max}<0.3$ as not significant. The maximum ICCF correlation
coefficient, $r_{\rm max}$, was suggested to be greater than 0.45. This
ensures that the behaviour in the two light curves is well correlated (Grier
et al., 2017). Adopting $r_{\rm max}>0.6$, there are 14 out 26 quasars showing
a reliable measurement of $\tau_{\rm cent}$.

For SDSS RM quasars, the observation numbers are about 32, 12, 13, 12, 9, 8, 4
during seven years from 2014 to 2020. Considering the larger gaps between
adjacent years (see Figure 1), we also do the ICCF analysis for other two
cases, one is for the spectra observed during the year of 2014, and the other
is during years of 2014-2015. For our sample, the results on $\tau_{\rm cent}$
and $r_{\rm max}$ for the Balmer decrement, as well as H$\beta$ , H$\alpha$
from ICCF are given in cols. (6-11). and cols. (12-17), cols. (18-23) in Table
2 for the year of 2014, for years of 2014-2015 and for years of 2014-2020,
respectively. Comparing the decrement lag in three cases, i. e., 2014,
2014-2015, 2014-2020, it is found that, for the light curves obtained in the
year of 2014, the decrement lag has large uncertainty. With all data in the
years of 2014 to 2020, the decrement lag has small uncertainty. It is possible
due to small amount of data for ICCF (about 30 data points). Therefore, we use
the decrement lag measurement from the data during seven years from 2014 to
2020 (about 90 data points).

Balmer decrement is obtained as the ratio of fluxes of H$\alpha$ and H$\beta$
, which have different lags as the response to the continuum light curve. The
curve obtained from the ratio of H$\alpha$ and H$\beta$ may represent the
curve of the one with the larger amplitudes. It is possible that the Balmer
decrement lag is false if the decrement lag is similar to the lag of the
H$\beta$ or H$\alpha$ . Comparing our measured H$\beta$ lags with the
decrement lags through ICCF from light curves obtained during the years of
2014-2020, it is found that there are ten quasars showing obvious lag
difference (considering lag errors). For the case of our measured H$\alpha$
lags and the decrement lags through ICCF, there are 12 quasars showing obvious
lag difference (considering lag errors). The subsample of these 12 quasars is
used to investigate the relation between the decrement lag and the BLR lag.

The RM program gave empirical BLR $R_{\rm BLR}({\rm H\beta})-L_{\rm 5100}$
relation (Peterson et al., 2004; Bentz et al., 2013; Du & Wang, 2019; Yu et
al., 2020a). Yang et al. (2020) used infrared WISE W1 data to measure the
torus size and gave the torus R-L relation. The torus size is about 4 times of
the BLR size. Considering our smaller numbers of the continuum data points and
large gaps between adjacent years, we use the lag results for H$\beta$ and
H$\alpha$ from Grier et al. (2017). In Figure 10, we compare the Balmer
decrement lag and the H$\beta$ lag (top), or the H$\alpha$ lag (bottom). Red
circles denote quasars with $r_{\rm max}>0.6$. The solid line is 1:4, i.e.,
$\tau_{\rm H\alpha/H\beta}=4\tau_{\rm H\beta}(or~{}\tau_{\rm H\alpha})$ and
the dash line is 1:1. There are larger uncertainties in the decrement lags,
H$\beta$ , or H$\alpha$ lags. For the top panel in Figure 10 , i.e., the
decrement lag versus the H$\beta$ lag, there are 3 quasars located above the
line of 1:4, 6 quasars located between two lines of 1:1 and 1:4, and one
quasar located below the line of 1:1. For the bottom panel in Figure 10, i.e.,
the decrement lag versus the H$\alpha$ lag, there is 4 quasars located between
two lines of 1:1 and 1:4, three quasars located below the line of 1:1. From
our results, the size corresponding to the Balmer decrement lag extends from
the BLR size to the torus size.

We use the data from the spectrum decomposition through PreSpec, the host
contribution is not included. It would influence the measurements of the
contribution of the AGN continuum, as well as $\rm Fe\ \sc{II}$ quasi-
continuum (e.g., Bon et al., 2020). This limitations in our data may influence
the final results.

##  4 Conclusions

For a sample of 44 SDSS RM quasars ($z<0.53$) with spectral monitoring of
H$\beta$ and H$\alpha$ observed from 2014 to 2020 ( 6.5 years), using the
light curves of H$\alpha$ , H$\beta$ and the continuum at 5100 Å derived from
the spectral decomposition through PrepSpec, the variance of H$\beta$
/H$\alpha$ is investigated. The main conclusions can be summarized as follows:

  * •

The logarithm of the mean broad-line Balmer decrement is 0.62 with a standard
deviation of 0.15 dex.

  * •

No obvious correlations are found between the mean Balmer decrement and the
SMBH accretion properties, such as the continuum luminosity, the black hole
mass, the Eddington ratio, the dimensionless accretion rate.

  * •

There are 27 quasars (61%) showing strong negative correlations between the
Balmer decrement and the continuum, i.e., the Balmer decrement would be
smaller with larger continuum flux. Assuming that the dust obscuration leads
to the variance in the Balmer decrement and the continuum, the expected slope
is -1/3, which is not consistent with most of our measured slopes.

  * •

For 26 quasars with reliable measurement of the H$\beta$ and/or H$\alpha$
delay , through ICCF, the time delays between the inverse Balmer decrement and
the continuum are measured. Using the maximum correlation coefficient larger
the 0.6, reliable decrement lags are detected for 14 quasars . It suggests
that the size of the Balmer decrement lag extends from the BLR size to the
torus size.

## Acknowledgements

We are also very grateful to the anonymous referee for her/his instructive
comments which significantly improved the content of the paper. This work is
supported by the National Key Research and Development Program of China (No.
2017YFA0402703). This work has been supported by the National Science
Foundation of China (No. 11973029).

## Data Availability

The data underlying this article are available at http://star-www.st-
and.ac.uk/ kdh1/pub/sdss/2020/sdssz.html.

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Table 1: The relative variability amplitude for $\rm H\alpha/H\beta$ ,
H$\beta$ , H$\alpha$ , $L_{5100}$ , as well as the properties for the relation
between $\rm H\alpha/H\beta$ and the continuum at 5100 Å and the SMBH masses.
Col. (1): RMID, col. (2): redshift; col. (3): the broad Balmer decrement, col.
(4): $F_{\rm var}$ of H$\alpha$ /H$\beta$ , col. (5): the flux of the broad
H$\alpha$ in units of $\rm 10^{-17}~{}erg~{}s^{-1}~{}cm^{-2}$, col. (6):
$F_{\rm var}$ of H$\alpha$ , col. (7): the flux of the broad H$\beta$ in units
of $\rm 10^{-17}~{}erg~{}s^{-1}~{}cm^{-2}$, col. (8): $F_{\rm var}$ of
H$\beta$ , col. (9): the continuum flux at 5100 Å in $\rm
10^{-17}~{}erg~{}s^{-1}~{}cm^{-2}~{}\AA^{-1}$, col. (10): $F_{\rm var}$ of the
continuum at 5100 Å, cols. (11-13): $r_{s},p_{\rm null},k$ for the relation
between the $\rm H\alpha/H\beta$ and the continuum, cols. (14-17) : the
luminosity in 5100 Å in units of erg s-1 , FWHM of broad H$\beta$ line in
units of km s-1 , $R_{\rm Fe}$ and $\log M_{\rm BH}/M_{\odot}$.

$\rm RMID$ $z$ H$\alpha$ /H$\beta$ $F_{\rm var}(\rm H\alpha/H\beta)$ H$\alpha$
$F_{\rm var}(\rm H\alpha)$ H$\beta$ $F_{\rm var}(\rm H\beta)$ $f_{5100}$
$F_{\rm var}(f_{5100})$ $r_{s}$ $p_{\rm null}$ $k$ $L_{5100}$ $\rm FWHM_{\rm
H\beta}$ $R_{\rm Fe}$ $M_{\rm BH}$ (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
(11) (12) (13) (14) (15) (16) (17) 17 0.46 3.58$\pm$0.54 - 343.73$\pm$36.09
0.25$\pm$0.02 54.77$\pm$6.33 0.68$\pm$0.05 5.28$\pm$0.9 0.38$\pm$0.03 -0.81
0.0 -0.78$\pm$0.02 43.91$\pm$0.0 6075.62$\pm$121.48 0.67$\pm$0.01
$9.09^{+0.28}_{-0.15}$ 85 0.24 3.0$\pm$0.33 0.07$\pm$0.02 1188.08$\pm$129.16
0.12$\pm$0.01 384.49$\pm$42.82 0.13$\pm$0.01 6.29$\pm$0.76 0.16$\pm$0.01 -0.01
0.91 -0.04$\pm$0.01 43.27$\pm$0.0 13797.44$\pm$363.93 0.26$\pm$0.02
$8.84^{+0.22}_{-0.22}$ 88 0.52 4.56$\pm$0.51 0.04$\pm$0.03 497.95$\pm$44.34
0.06$\pm$0.01 107.71$\pm$12.12 0.09$\pm$0.01 2.6$\pm$0.31 0.09$\pm$0.01 -0.37
0.0 -0.3$\pm$0.04 44.14$\pm$0.0 5492.04$\pm$82.07 0.37$\pm$0.03
$8.44^{+0.22}_{-0.22}$ 101 0.46 3.08$\pm$0.34 0.03$\pm$0.01 1100.52$\pm$120.82
0.04$\pm$0.0 353.74$\pm$39.09 0.04$\pm$0.0 4.8$\pm$0.58 0.09$\pm$0.01 -0.27
0.01 -0.09$\pm$0.01 44.36$\pm$0.0 2651.7$\pm$302.18 0.43$\pm$0.04
$7.42^{+0.14}_{-0.2}$ 126 0.19 4.02$\pm$0.45 0.06$\pm$0.02 461.65$\pm$50.66
0.1$\pm$0.01 113.84$\pm$12.72 0.09$\pm$0.01 7.61$\pm$0.88 0.06$\pm$0.0 -0.05
0.64 -0.04$\pm$0.06 43.3$\pm$0.0 1879.01$\pm$155.38 1.27$\pm$0.03
$6.94^{+0.27}_{-0.27}$ 160 0.36 5.09$\pm$0.57 0.1$\pm$0.04 844.94$\pm$89.94
0.16$\pm$0.01 153.99$\pm$17.03 0.25$\pm$0.02 2.15$\pm$0.25 0.15$\pm$0.01 -0.71
0.0 -0.62$\pm$0.02 43.79$\pm$0.0 5034.4$\pm$34.91 0.3$\pm$0.01
$8.01^{+0.13}_{-0.07}$ 177 0.48 3.71$\pm$0.44 0.08$\pm$0.03 436.59$\pm$46.5
0.08$\pm$0.01 116.55$\pm$13.31 0.14$\pm$0.01 2.13$\pm$0.27 0.14$\pm$0.01 -0.7
0.0 -0.41$\pm$0.02 43.96$\pm$0.0 5229.7$\pm$76.04 0.47$\pm$0.02
$7.73^{+0.81}_{-0.18}$ 184 0.19 2.02$\pm$0.22 0.06$\pm$0.01 830.7$\pm$90.81
0.07$\pm$0.01 405.1$\pm$45.05 0.06$\pm$0.0 12.56$\pm$1.44 0.08$\pm$0.01 -0.13
0.23 -0.04$\pm$0.02 43.68$\pm$0.0 1837.28$\pm$11.06 0.9$\pm$0.01
$7.17^{+0.21}_{-0.21}$ 191 0.44 4.58$\pm$0.58 0.23$\pm$0.12 133.34$\pm$15.04
0.22$\pm$0.02 27.61$\pm$3.29 0.33$\pm$0.03 1.59$\pm$0.19 0.1$\pm$0.01 -0.21
0.05 -0.28$\pm$0.08 43.65$\pm$0.01 2177.65$\pm$43.98 0.65$\pm$0.04
$7.07^{+0.19}_{-0.11}$ 229 0.47 4.72$\pm$0.56 0.13$\pm$0.06 198.06$\pm$21.75
0.11$\pm$0.01 38.47$\pm$4.3 0.24$\pm$0.02 1.21$\pm$0.14 0.1$\pm$0.01 -0.59 0.0
-0.69$\pm$0.04 43.55$\pm$0.0 4328.84$\pm$297.53 0.6$\pm$0.03
$7.82^{+0.12}_{-0.18}$ 252 0.28 3.14$\pm$0.36 0.08$\pm$0.03 352.08$\pm$39.12
0.06$\pm$0.0 109.28$\pm$12.36 0.09$\pm$0.01 2.56$\pm$0.29 0.05$\pm$0.0 -0.29
0.01 -0.49$\pm$0.09 42.74$\pm$0.01 10565.13$\pm$692.7 0.44$\pm$0.03
$8.32^{+0.26}_{-0.26}$ 270 0.42 5.84$\pm$0.72 - 217.7$\pm$25.92 0.09$\pm$0.01
37.04$\pm$4.46 0.13$\pm$0.01 1.52$\pm$0.18 0.12$\pm$0.01 -0.17 0.13
-0.17$\pm$0.06 43.48$\pm$0.0 6470.68$\pm$181.01 0.89$\pm$0.05
$8.17^{+0.23}_{-0.23}$ 272 0.26 3.75$\pm$0.41 0.06$\pm$0.02 2368.22$\pm$258.11
0.07$\pm$0.0 620.05$\pm$68.34 0.09$\pm$0.01 7.74$\pm$0.92 0.12$\pm$0.01 -0.71
0.0 -0.33$\pm$0.01 43.93$\pm$0.02 3514.91$\pm$17.43 0.33$\pm$0.01
$7.75^{+0.14}_{-0.2}$ 305 0.53 4.89$\pm$0.55 0.06$\pm$0.03 504.81$\pm$56.88
0.07$\pm$0.01 102.54$\pm$11.64 0.08$\pm$0.01 2.59$\pm$0.32 0.08$\pm$0.01 0.01
0.91 -0.03$\pm$0.03 44.19$\pm$0.0 2917.6$\pm$61.7 0.47$\pm$0.04
$8.49^{+0.06}_{-0.05}$ 320 0.26 3.41$\pm$0.43 0.14$\pm$0.04 509.16$\pm$57.2
0.11$\pm$0.01 145.64$\pm$16.99 0.22$\pm$0.02 3.69$\pm$0.43 0.1$\pm$0.01 -0.7
0.0 -0.97$\pm$0.04 43.42$\pm$0.0 4700.09$\pm$54.96 0.67$\pm$0.02
$7.84^{+0.12}_{-0.15}$ 338 0.42 6.83$\pm$1.06 0.08$\pm$2.53 138.23$\pm$16.67
0.11$\pm$0.01 15.45$\pm$1.93 0.37$\pm$0.04 1.59$\pm$0.19 0.07$\pm$0.01 -0.47
0.0 -1.91$\pm$0.23 43.36$\pm$0.0 6789.45$\pm$579.67 0.0$\pm$0.0
$7.86^{+0.34}_{-0.27}$ 341 0.42 3.47$\pm$0.4 0.04$\pm$0.01 1199.25$\pm$137.62
0.08$\pm$0.01 347.85$\pm$39.71 0.1$\pm$0.01 5.53$\pm$0.69 0.1$\pm$0.01 -0.43
0.0 -0.19$\pm$0.02 44.42$\pm$0.0 3471.49$\pm$24.98 0.57$\pm$0.01
$8.14^{+0.21}_{-0.21}$ 371 0.47 4.07$\pm$0.47 0.08$\pm$0.03 506.45$\pm$56.22
0.08$\pm$0.01 122.14$\pm$13.75 0.14$\pm$0.01 2.13$\pm$0.24 0.08$\pm$0.01 -0.65
0.0 -0.68$\pm$0.03 44.1$\pm$0.0 4105.97$\pm$37.6 0.75$\pm$0.02
$7.54^{+0.07}_{-0.04}$ 377 0.34 4.49$\pm$0.63 0.22$\pm$0.46 148.29$\pm$16.57
0.17$\pm$0.01 30.46$\pm$3.71 0.34$\pm$0.03 2.44$\pm$0.28 0.05$\pm$0.0 -0.39
0.0 -1.26$\pm$0.18 43.37$\pm$0.0 4429.12$\pm$105.45 0.95$\pm$0.05
$7.39^{+0.05}_{-0.07}$ 497 0.51 3.24$\pm$0.37 0.06$\pm$0.02 612.98$\pm$69.21
0.07$\pm$0.01 186.55$\pm$21.43 0.03$\pm$0.0 3.88$\pm$0.46 0.08$\pm$0.01 0.0
0.96 -0.03$\pm$0.03 44.21$\pm$0.0 11162.28$\pm$264.83 0.21$\pm$0.02
$9.12^{+0.22}_{-0.22}$ 518 0.46 5.68$\pm$0.64 0.04$\pm$0.04 266.66$\pm$29.53
0.05$\pm$0.0 46.56$\pm$5.24 0.07$\pm$0.01 1.24$\pm$0.14 0.09$\pm$0.01 -0.18
0.08 -0.08$\pm$0.05 43.91$\pm$0.0 3321.98$\pm$900.62 0.0$\pm$0.0
$7.96^{+0.44}_{-0.44}$ 541 0.44 3.21$\pm$0.37 0.03$\pm$0.05 122.05$\pm$13.74
0.11$\pm$0.01 37.49$\pm$4.28 0.11$\pm$0.01 1.08$\pm$0.13 0.12$\pm$0.01 0.19
0.07 0.21$\pm$0.08 43.54$\pm$0.0 3245.29$\pm$111.17 1.52$\pm$0.07
$7.49^{+0.24}_{-0.24}$ 645 0.47 8.94$\pm$1.44 0.52$\pm$0.76 453.81$\pm$50.61
0.06$\pm$0.0 37.09$\pm$4.6 0.47$\pm$0.04 2.28$\pm$0.26 0.1$\pm$0.01 -0.8 0.0
-3.18$\pm$0.08 44.11$\pm$0.0 4259.19$\pm$90.47 0.4$\pm$0.01
$7.74^{+0.03}_{-0.1}$ 720 0.47 5.58$\pm$0.66 0.11$\pm$0.05 677.02$\pm$77.4
0.08$\pm$0.01 119.07$\pm$13.81 0.14$\pm$0.01 3.68$\pm$0.45 0.08$\pm$0.01 -0.4
0.0 -0.42$\pm$0.03 44.3$\pm$0.0 3405.89$\pm$22.33 0.52$\pm$0.01
$7.91^{+0.23}_{-0.13}$ 733 0.46 4.77$\pm$0.55 0.07$\pm$0.04 389.41$\pm$44.72
0.1$\pm$0.01 80.97$\pm$9.26 0.1$\pm$0.01 1.96$\pm$0.23 0.11$\pm$0.01 -0.02
0.83 0.05$\pm$0.03 43.86$\pm$0.0 4689.79$\pm$44.47 0.6$\pm$0.02
$8.13^{+0.21}_{-0.21}$ 766 0.16 2.44$\pm$0.28 0.08$\pm$0.02 1494.08$\pm$165.3
0.07$\pm$0.01 601.35$\pm$67.42 0.09$\pm$0.01 21.19$\pm$2.75 0.07$\pm$0.01 0.12
0.27 0.06$\pm$0.02 43.73$\pm$0.0 1936.26$\pm$7.28 1.45$\pm$0.01
$7.14^{+0.2}_{-0.2}$ 767 0.53 8.81$\pm$2.73 - 110.39$\pm$25.11 0.02$\pm$0.0
8.26$\pm$2.13 0.5$\pm$0.12 1.02$\pm$0.23 0.07$\pm$0.01 -0.65 0.0
-1.52$\pm$1.36 43.93$\pm$0.0 2003.55$\pm$3868.39 0.0$\pm$0.0
$7.53^{+1.88}_{-1.88}$ 768 0.26 3.95$\pm$0.47 0.09$\pm$0.03 959.57$\pm$105.6
0.17$\pm$0.01 233.07$\pm$26.53 0.18$\pm$0.01 6.21$\pm$0.81 0.1$\pm$0.01 -0.49
0.0 -0.42$\pm$0.02 43.31$\pm$0.0 9093.06$\pm$611.77 0.53$\pm$0.02
$8.45^{+0.26}_{-0.26}$ 769 0.19 5.25$\pm$0.85 - 249.68$\pm$28.19 0.16$\pm$0.01
37.75$\pm$4.82 0.5$\pm$0.04 5.51$\pm$0.72 0.09$\pm$0.01 -0.71 0.0
-1.12$\pm$0.12 42.97$\pm$0.0 6250.34$\pm$124.45 1.05$\pm$0.02
$7.87^{+0.22}_{-0.22}$ 772 0.25 4.52$\pm$0.66 0.42$\pm$0.23 454.49$\pm$49.76
0.12$\pm$0.01 74.52$\pm$8.66 0.46$\pm$0.03 5.48$\pm$0.63 0.07$\pm$0.01 -0.54
0.0 -2.53$\pm$0.06 43.44$\pm$0.0 3089.67$\pm$65.88 0.67$\pm$0.02
$6.72^{+0.15}_{-0.15}$ 775 0.17 3.5$\pm$0.39 0.08$\pm$0.02 1489.45$\pm$166.07
0.08$\pm$0.01 411.82$\pm$45.64 0.13$\pm$0.01 13.54$\pm$1.66 0.06$\pm$0.0 -0.19
0.08 -0.27$\pm$0.03 43.54$\pm$0.0 3662.02$\pm$26.73 0.73$\pm$0.01
$7.84^{+0.53}_{-0.27}$ 776 0.12 4.39$\pm$0.56 0.28$\pm$0.11 1275.48$\pm$140.8
0.11$\pm$0.01 265.29$\pm$30.33 0.29$\pm$0.02 12.09$\pm$1.48 0.12$\pm$0.01
-0.64 0.0 -0.9$\pm$0.03 43.12$\pm$0.0 4344.84$\pm$557.6 0.78$\pm$0.04
$7.43^{+0.06}_{-0.14}$ 779 0.15 3.53$\pm$0.41 0.07$\pm$0.02 767.31$\pm$86.9
0.1$\pm$0.01 213.23$\pm$24.35 0.09$\pm$0.01 4.81$\pm$0.57 0.08$\pm$0.01 -0.05
0.64 -0.14$\pm$0.02 43.09$\pm$0.0 2944.79$\pm$20.28 0.57$\pm$0.01
$7.34^{+0.04}_{-0.08}$ 781 0.26 3.19$\pm$0.36 0.05$\pm$0.01 570.41$\pm$63.34
0.06$\pm$0.0 178.32$\pm$20.09 0.08$\pm$0.01 4.59$\pm$0.56 0.09$\pm$0.01 -0.43
0.0 -0.26$\pm$0.02 43.6$\pm$0.02 2989.96$\pm$48.32 0.59$\pm$0.04
$8.06^{+0.04}_{-0.04}$ 782 0.36 3.54$\pm$0.41 0.03$\pm$0.07 444.06$\pm$52.26
0.18$\pm$0.01 124.07$\pm$14.35 0.21$\pm$0.02 4.4$\pm$0.63 0.17$\pm$0.01 -0.56
0.0 -0.41$\pm$0.02 43.91$\pm$0.0 3428.09$\pm$36.56 1.03$\pm$0.02
$7.67^{+0.04}_{-0.1}$ 789 0.42 3.14$\pm$0.36 0.09$\pm$0.03 263.38$\pm$30.0
0.07$\pm$0.01 82.37$\pm$9.38 0.12$\pm$0.01 1.55$\pm$0.18 0.08$\pm$0.01 -0.33
0.0 -0.5$\pm$0.05 43.68$\pm$0.0 4148.68$\pm$70.17 0.55$\pm$0.02
$7.94^{+0.22}_{-0.22}$ 790 0.24 3.29$\pm$0.38 0.08$\pm$0.02 770.02$\pm$88.43
0.11$\pm$0.01 235.63$\pm$26.97 0.12$\pm$0.01 6.57$\pm$0.82 0.07$\pm$0.01 -0.08
0.47 -0.09$\pm$0.03 43.26$\pm$0.0 9207.55$\pm$268.85 0.66$\pm$0.04
$8.45^{+0.68}_{-0.25}$ 792 0.53 6.54$\pm$0.27 - 50.53$\pm$9.13 0.09$\pm$0.02
3.82$\pm$4.2 0.47$\pm$0.09 0.87$\pm$0.25 0.09$\pm$0.01 -0.05 0.7 -1.2$\pm$0.47
43.0$\pm$0.01 5111.22$\pm$823.91 0.81$\pm$0.09 $7.75^{+0.34}_{-0.34}$ 797 0.24
2.21$\pm$0.36 0.09$\pm$0.02 79.38$\pm$47.06 0.08$\pm$0.01 35.47$\pm$15.43
0.14$\pm$0.01 2.11$\pm$0.48 0.05$\pm$0.0 -0.1 0.36 -0.18$\pm$0.12
43.14$\pm$0.0 1474.1$\pm$47.1 1.7$\pm$0.03 $6.58^{+0.23}_{-0.23}$ 798 0.42
3.08$\pm$0.26 0.04$\pm$0.01 407.92$\pm$18.47 0.06$\pm$0.01 131.24$\pm$8.19
0.05$\pm$0.0 3.69$\pm$0.53 0.08$\pm$0.01 0.07 0.51 0.07$\pm$0.03 44.05$\pm$0.0
2311.19$\pm$62.02 1.13$\pm$0.06 $7.5^{+0.23}_{-0.23}$ 822 0.29 2.32$\pm$0.38
0.1$\pm$0.03 185.84$\pm$50.08 0.27$\pm$0.02 74.47$\pm$17.02 0.26$\pm$0.02
4.46$\pm$0.66 0.16$\pm$0.01 0.03 0.81 -0.02$\pm$0.07 43.6$\pm$0.0
1847.81$\pm$63.39 1.47$\pm$0.05 $7.03^{+0.23}_{-0.23}$ 840 0.24 3.37$\pm$1.48
0.09$\pm$0.03 534.45$\pm$11.65 0.19$\pm$0.01 152.26$\pm$1.24 0.17$\pm$0.01
5.21$\pm$0.41 0.12$\pm$0.01 0.37 0.0 0.18$\pm$0.02 43.18$\pm$0.0
7409.47$\pm$113.24 1.05$\pm$0.02 $8.11^{+0.2}_{-0.18}$ 845 0.27 5.57$\pm$1.5 -
83.07$\pm$6.54 0.17$\pm$0.02 7.06$\pm$0.57 0.72$\pm$0.11 2.63$\pm$0.11
0.1$\pm$0.01 0.2 0.15 1.94$\pm$0.62 42.73$\pm$0.01 9023.24$\pm$930.44
0.87$\pm$0.07 $8.1^{+0.29}_{-0.29}$ 846 0.23 3.48$\pm$0.93 - 13.72$\pm$2.17
0.18$\pm$0.03 1.85$\pm$0.34 0.49$\pm$0.12 0.34$\pm$0.05 0.19$\pm$0.02 -0.23
0.13 0.31$\pm$0.34 41.83$\pm$0.01 3464.89$\pm$1038.81 0.0$\pm$0.0
$7.0^{+0.46}_{-0.46}$

Table 2: The lags of the broad-line Balmer decrement H$\beta$ /H$\alpha$ ,
H$\beta$ , H$\alpha$ for light curves obtained during the year of 2014, years
of 2014-2015, years of 2014-2020, as well as the H$\beta$ , H$\alpha$ lags
during the year of 2014 given by Grier et al. (2017). Col. (1): RMID, cols.
(2-5): H$\beta$ lag, quality, H$\alpha$ lag, quality, cols. (6-11): lag and
$r_{\rm max}$ obtained from the data during the year of 2014 for H$\beta$ ,
H$\alpha$ , H$\beta$ /H$\alpha$ , respectively, cols. (12-17): lag and $r_{\rm
max}$ obtained from the data during the years of 2014-2015 for H$\beta$ ,
H$\alpha$ , H$\beta$ /H$\alpha$ , respectively, col. (18-23): lag and $r_{\rm
max}$ obtained from the data during the years of 2014-2020 for H$\beta$ ,
H$\alpha$ , H$\beta$ /H$\alpha$ , respectively. The lag is in units of light
days in the observed frame. The lines in bold denote quasars shown in Figure
10.

$\rm RMID$ $\tau^{a}(\rm H\beta)$ $q^{a}(\rm H\beta)$ $\tau^{a}(\rm H\alpha)$
$q^{a}(\rm H\alpha)$ $\tau^{b}(\rm H\beta)$ $r^{b}(\rm H\beta)$ $\tau^{b}(\rm
H\alpha)$ $r^{b}(\rm H\alpha)$ $\tau^{b}(\rm H\beta/H\alpha)$ $r^{b}(\rm
H\beta/H\alpha)$ $\tau^{c}(\rm H\beta)$ $r^{c}(\rm H\beta)$ $\tau^{c}(\rm
H\alpha)$ $r^{c}(\rm H\alpha)$ $\tau^{c}(\rm H\beta/H\alpha)$ $r^{c}(\rm
H\beta/H\alpha)$ $\tau^{d}(\rm H\beta)$ $r^{d}(\rm H\beta)$ $\tau^{d}(\rm
H\alpha)$ $r^{d}(\rm H\alpha)$ $\tau^{d}(\rm H\beta/H\alpha)$ $r^{d}(\rm
H\beta/H\alpha)$ Grier 2014 2014-2015 2014-2020 (1) (2) (3) (4) (5) (6) (7)
(8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23)
17 $32.7^{+15.4}_{-15.9}$ 4.0 $65.3^{+20.0}_{-15.4}$ 5.0
$36.0^{+22.9}_{-34.1}$ 0.61 $53.9^{+24.4}_{-24.7}$ 0.69 $24.3^{+30.7}_{-23.4}$
0.62 $47.9^{+21.8}_{-16.3}$ 0.81 $54.0^{+18.7}_{-23.5}$ 0.8
$51.0^{+15.9}_{-21.6}$ 0.78 $85.8^{+28.3}_{-25.5}$ 0.81
$111.3^{+29.0}_{-25.0}$ 0.8 $85.4^{+21.9}_{-23.1}$ 0.79 88 - -
$84.0^{+5.7}_{-8.3}$ 3.0 $76.1^{+6.9}_{-158.2}$ 0.8 $0.0^{+86.0}_{-82.9}$ 0.81
$26.9^{+20.1}_{-85.9}$ 0.67 $19.1^{+68.7}_{-20.0}$ 0.65 $0.9^{+119.9}_{-1.9}$
0.8 $44.1^{+226.9}_{-17.2}$ 0.5 $56.9^{+54.6}_{-54.7}$ 0.79
$70.2^{+96.0}_{-70.1}$ 0.69 $62.0^{+59.5}_{-31.0}$ 0.71 101 $\bf
31.1^{+5.9}_{-11.4}$ 5.0 - - $\bf 85.9^{+5.2}_{-65.0}$ 0.72 $\bf
91.0^{+3.1}_{-168.0}$ 0.6 $\bf 65.0^{+7.1}_{-46.0}$ 0.64 $\bf
71.3^{+11.6}_{-19.4}$ 0.9 $\bf 165.1^{+112.9}_{-35.7}$ 0.7 $\bf
26.8^{+44.3}_{-20.3}$ 0.66 $\bf 152.9^{+27.3}_{-18.3}$ 0.8 $\bf
194.4^{+36.1}_{-37.7}$ 0.71 $\bf 523.9^{+14.4}_{-152.2}$ 0.55 160 $\bf
14.6^{+8.9}_{-9.6}$ 3.0 $\bf 16.5^{+10.3}_{-9.2}$ 4.0
$\bf-68.0^{+77.3}_{-7.5}$ 0.94 $\bf 8.7^{+17.2}_{-75.1}$ 0.82
$\bf-68.1^{+85.0}_{-8.8}$ 0.93 $\bf 71.2^{+127.6}_{-17.4}$ 0.75 $\bf
146.3^{+7.9}_{-37.4}$ 0.83 $\bf 49.1^{+144.3}_{-20.0}$ 0.82 $\bf
100.3^{+11.5}_{-9.7}$ 0.81 $\bf 130.3^{+23.0}_{-20.0}$ 0.65 $\bf
49.8^{+9.4}_{-9.9}$ 0.86 177 $\bf 8.3^{+9.1}_{-6.6}$ 4.0 - -
$\bf-0.9^{+6.0}_{-87.1}$ 0.71 $\bf 2.1^{+42.9}_{-83.1}$ 0.65
$\bf-14.9^{+36.3}_{-64.1}$ 0.58 $\bf 16.6^{+16.3}_{-10.0}$ 0.94 $\bf
22.7^{+17.5}_{-9.4}$ 0.88 $\bf 12.2^{+22.9}_{-21.2}$ 0.86 $\bf
28.4^{+11.6}_{-8.5}$ 0.9 $\bf 40.8^{+15.0}_{-11.0}$ 0.79 $\bf\bf
9.0^{+12.4}_{-19.4}$ 0.77 191 $14.0^{+5.7}_{-5.8}$ 5.0 $26.0^{+9.6}_{-11.8}$
4.0 $34.9^{+3.2}_{-80.0}$ 0.64 $37.4^{+19.7}_{-39.3}$ 0.43
$34.0^{+16.7}_{-84.1}$ 0.58 $117.1^{+10.1}_{-11.7}$ 0.71
$41.3^{+97.5}_{-40.3}$ 0.55 $120.2^{+7.9}_{-12.4}$ 0.64
$106.0^{+27.2}_{-29.4}$ 0.77 $110.5^{+28.3}_{-29.2}$ 0.76
$101.4^{+56.7}_{-65.2}$ 0.64 229 $21.0^{+6.3}_{-8.7}$ 5.0
$34.0^{+19.1}_{-11.5}$ 3.0 $28.1^{+14.0}_{-34.2}$ 0.57 $44.8^{+10.2}_{-67.9}$
0.54 $15.0^{+22.9}_{-83.0}$ 0.52 $111.1^{+19.1}_{-95.0}$ 0.46
$134.0^{+111.8}_{-94.9}$ 0.55 $112.9^{+15.2}_{-98.7}$ 0.51
$23.2^{+26.3}_{-7.5}$ 0.68 $28.7^{+126.8}_{-25.7}$ 0.67 $25.7^{+11.8}_{-5.2}$
0.66 252 - - $14.1^{+8.1}_{-6.7}$ 5.0 $69.0^{+21.9}_{-142.7}$ 0.54
$24.0^{+11.9}_{-64.2}$ 0.55 $70.0^{+17.1}_{-144.1}$ 0.55
$86.1^{+7.0}_{-145.4}$ 0.51 $-6.4^{+102.4}_{-37.5}$ 0.34
$86.0^{+59.0}_{-157.2}$ 0.42 $15.0^{+288.1}_{-5.2}$ 0.42
$460.6^{+46.3}_{-246.0}$ 0.26 $16.1^{+284.0}_{-19.2}$ 0.36 272 $\bf
21.1^{+7.5}_{-9.0}$ 5.0 $\bf 42.2^{+24.0}_{-21.1}$ 3.0 $\bf
26.3^{+6.0}_{-4.1}$ 0.82 $\bf 50.0^{+13.9}_{-36.0}$ 0.59 $\bf
21.0^{+12.7}_{-11.0}$ 0.79 $\bf 27.4^{+4.6}_{-5.6}$ 0.9 $\bf
47.7^{+21.3}_{-21.8}$ 0.59 $\bf 17.8^{+7.3}_{-6.8}$ 0.83 $\bf
32.2^{+9.4}_{-9.9}$ 0.84 $49.0^{+11.2}_{-10.1}$ 0.65 $\bf
13.6^{+15.7}_{-14.1}$ 0.71 305 $-74.0^{+22.2}_{-12.8}$ 2.0 - -
$38.8^{+17.8}_{-133.4}$ 0.68 $-67.0^{+101.2}_{-8.1}$ 0.48
$41.0^{+29.0}_{-132.0}$ 0.57 $131.7^{+105.1}_{-75.9}$ 0.36
$0.0^{+138.0}_{-75.0}$ 0.44 $155.0^{+119.9}_{-92.2}$ 0.39
$179.1^{+23.0}_{-27.9}$ 0.73 $38.5^{+179.6}_{-30.0}$ 0.61
$152.8^{+80.6}_{-22.5}$ 0.59 320 $\bf 33.9^{+10.1}_{-17.4}$ 4.0 $\bf
25.9^{+18.7}_{-18.7}$ 4.0 $\bf 37.0^{+2.9}_{-32.1}$ 0.72 $\bf
17.1^{+21.8}_{-16.1}$ 0.56 $\bf 39.2^{+37.8}_{-94.5}$ 0.69 $\bf
39.0^{+118.1}_{-27.0}$ 0.7 $\bf 8.8^{+40.3}_{-18.7}$ 0.52 $\bf
40.1^{+113.0}_{-39.9}$ 0.56 $\bf 66.2^{+32.7}_{-36.9}$ 0.86 $\bf
97.4^{+24.9}_{-38.5}$ 0.78 $\bf 39.1^{+22.5}_{-17.7}$ 0.79 338
$10.9^{+14.2}_{-12.8}$ 4.0 - - $24.9^{+69.0}_{-92.0}$ 0.73
$88.0^{+10.0}_{-163.5}$ 0.83 $19.6^{+72.4}_{-86.7}$ 0.71
$-8.9^{+35.0}_{-47.1}$ 0.54 $23.1^{+77.8}_{-55.1}$ 0.45 $-8.8^{+35.7}_{-47.2}$
0.53 $-5.1^{+13.7}_{-47.8}$ 0.51 $15.4^{+8.9}_{-12.9}$ 0.52
$-12.2^{+23.1}_{-44.8}$ 0.51 371 $\bf 9.5^{+12.9}_{-8.0}$ 3.0 $\bf
19.4^{+14.8}_{-20.8}$ 3.0 $\bf 28.5^{+8.0}_{-9.3}$ 0.89 $\bf
29.8^{+10.7}_{-9.6}$ 0.89 $\bf 29.9^{+18.1}_{-18.3}$ 0.9 $\bf
20.1^{+6.1}_{-4.0}$ 0.92 $\bf 41.4^{+9.9}_{-6.2}$ 0.86
$\bf-2.8^{+11.3}_{-11.9}$ 0.82 $\bf 25.1^{+4.0}_{-3.2}$ 0.83 $\bf
38.7^{+8.0}_{-6.4}$ 0.77 $\bf 13.7^{+7.9}_{-7.8}$ 0.69 377
$12.0^{+16.0}_{-15.5}$ 3.0 $7.1^{+14.3}_{-44.0}$ 3.0 $21.1^{+69.0}_{-60.0}$
0.5 $16.9^{+11.3}_{-54.9}$ 0.66 $24.9^{+68.1}_{-105.9}$ 0.51
$13.9^{+10.5}_{-30.3}$ 0.57 $19.1^{+15.9}_{-21.9}$ 0.68 $9.1^{+115.9}_{-50.1}$
0.46 $339.1^{+60.1}_{-317.0}$ 0.51 $350.9^{+53.4}_{-337.2}$ 0.51
$332.1^{+69.8}_{-163.5}$ 0.48 645 $7.5^{+9.5}_{-12.5}$ 4.0
$26.7^{+9.4}_{-22.9}$ 5.0 $28.5^{+12.8}_{-16.2}$ 0.93 $29.0^{+15.6}_{-20.3}$
0.86 $32.4^{+10.6}_{-18.1}$ 0.93 $41.7^{+10.5}_{-9.4}$ 0.91
$31.7^{+7.9}_{-8.1}$ 0.88 $43.6^{+10.3}_{-10.2}$ 0.91 $44.3^{+7.1}_{-7.6}$
0.85 $39.3^{+7.2}_{-6.2}$ 0.83 $44.9^{+7.2}_{-8.7}$ 0.84 720 $\bf
66.0^{+11.9}_{-15.4}$ 2.0 - - $\bf 0.0^{+4.9}_{-2.0}$ 0.79 $\bf
24.4^{+68.6}_{-80.6}$ 0.51 $\bf 2.9^{+55.0}_{-10.4}$ 0.64 $\bf
0.0^{+3.8}_{-2.8}$ 0.81 $\bf 0.0^{+10.0}_{-2.1}$ 0.65 $\bf
4.2^{+56.8}_{-10.0}$ 0.56 $\bf 3.0^{+381.0}_{-4.0}$ 0.75 $\bf
1.0^{+371.7}_{-2.9}$ 0.7 $384.7^{+47.9}_{-179.8}$ 0.55 733 - - $\bf
74.0^{+13.9}_{-21.8}$ 2.0 $\bf-0.9^{+46.0}_{-53.0}$ 0.4 $\bf
44.0^{+14.9}_{-97.0}$ 0.42 $\bf 10.1^{+79.9}_{-28.3}$ 0.57 $\bf
119.8^{+122.2}_{-73.7}$ 0.61 $\bf 102.2^{+47.1}_{-51.1}$ 0.55 $\bf
103.0^{+182.0}_{-127.7}$ 0.34 $\bf 44.1^{+103.8}_{-37.1}$ 0.53 $\bf
11.4^{+109.8}_{-11.4}$ 0.48 $\bf 122.7^{+132.5}_{-111.7}$ 0.3 768 - - $\bf
42.0^{+17.8}_{-13.0}$ 5.0 $\bf 28.0^{+5.0}_{-5.0}$ 0.59 $\bf
36.5^{+23.5}_{-5.4}$ 0.35 $\bf 20.9^{+5.1}_{-27.8}$ 0.63 $\bf
27.4^{+6.0}_{-5.6}$ 0.83 $\bf 30.8^{+26.1}_{-9.7}$ 0.69 $\bf
27.4^{+77.4}_{-15.1}$ 0.8 $\bf 36.2^{+13.0}_{-9.1}$ 0.88 $\bf
67.8^{+31.0}_{-14.7}$ 0.76 $\bf 2.1^{+15.9}_{-10.5}$ 0.62 772 $\bf
1.3^{+4.6}_{-5.2}$ 5.0 $\bf 9.0^{+4.3}_{-7.0}$ 5.0 $\bf-59.0^{+70.9}_{-3.9}$
0.52 $\bf 14.0^{+25.0}_{-59.9}$ 0.35 $\bf-60.0^{+69.9}_{-3.1}$ 0.6 $\bf
190.1^{+97.9}_{-175.5}$ 0.53 $\bf 189.9^{+96.0}_{-151.6}$ 0.5 $\bf
189.9^{+97.9}_{-178.9}$ 0.53 $\bf 38.8^{+9.5}_{-11.1}$ 0.69 $\bf
60.6^{+19.0}_{-10.2}$ 0.7 $\bf 30.1^{+12.3}_{-11.8}$ 0.69 775
$22.1^{+10.2}_{-12.9}$ 4.0 - - $31.1^{+4.8}_{-4.9}$ 0.86 $48.0^{+7.1}_{-12.9}$
0.75 $23.2^{+6.8}_{-15.3}$ 0.73 $28.1^{+6.9}_{-4.8}$ 0.82
$43.0^{+149.9}_{-13.4}$ 0.74 $24.9^{+167.0}_{-5.7}$ 0.74 $26.9^{+9.6}_{-8.0}$
0.17 $35.1^{+12.2}_{-35.5}$ 0.0 $25.0^{+51.6}_{-9.1}$ 0.29 776
$10.0^{+6.3}_{-4.2}$ 4.0 $8.0^{+5.4}_{-6.3}$ 4.0 $2.9^{+4.3}_{-20.1}$ 0.6
$-26.0^{+25.1}_{-30.0}$ 0.6 $7.4^{+4.6}_{-9.2}$ 0.6 $154.1^{+39.4}_{-94.1}$
0.51 $68.2^{+158.6}_{-11.3}$ 0.54 $150.0^{+59.0}_{-28.1}$ 0.49
$137.7^{+67.9}_{-22.9}$ 0.8 $130.7^{+21.6}_{-19.9}$ 0.85
$155.9^{+73.0}_{-42.7}$ 0.73 779 $12.1^{+9.5}_{-10.1}$ 4.0
$16.9^{+21.6}_{-13.7}$ 2.0 $19.7^{+7.4}_{-7.5}$ 0.89 $14.3^{+25.5}_{-15.3}$
0.71 $66.0^{+19.0}_{-51.5}$ 0.75 $29.6^{+11.0}_{-5.7}$ 0.78
$41.9^{+141.1}_{-32.0}$ 0.52 $55.7^{+13.3}_{-27.4}$ 0.65 $23.0^{+7.3}_{-4.6}$
0.69 $21.7^{+6.8}_{-8.8}$ 0.6 $326.0^{+136.1}_{-38.7}$ 0.28 781 $\bf
93.0^{+3.1}_{-10.5}$ 2.0 - - $\bf 1.0^{+85.1}_{-80.0}$ 0.51 $\bf
64.1^{+10.9}_{-147.1}$ 0.6 $\bf 0.0^{+19.0}_{-75.2}$ 0.46 $\bf
38.3^{+23.7}_{-28.6}$ 0.89 $\bf 46.7^{+25.5}_{-35.6}$ 0.77 $\bf
22.1^{+63.8}_{-18.9}$ 0.76 $\bf 116.9^{+40.8}_{-39.4}$ 0.72 $\bf
38.2^{+46.6}_{-18.7}$ 0.58 $\bf 185.3^{+41.8}_{-37.3}$ 0.64 782 $\bf
15.0^{+14.4}_{-6.8}$ 4.0 - - $\bf-73.1^{+165.2}_{-11.9}$ 0.55
$\bf-76.0^{+163.1}_{-10.0}$ 0.32 $\bf 88.9^{+5.0}_{-161.6}$ 0.63 $\bf
3.9^{+112.5}_{-7.9}$ 0.97 $\bf 2.0^{+140.4}_{-7.0}$ 0.97 $\bf
104.8^{+174.8}_{-96.8}$ 0.43 $\bf 5.1^{+80.6}_{-8.4}$ 0.96 $\bf
3.8^{+128.1}_{-9.6}$ 0.96 $\bf 20.9^{+85.8}_{-18.9}$ 0.49 790
$11.0^{+6.1}_{-6.5}$ 3.0 $0.9^{+11.9}_{-5.6}$ 2.0 $-1.0^{+7.2}_{-19.3}$ 0.58
$-0.9^{+41.0}_{-19.0}$ 0.59 $0.1^{+74.8}_{-32.1}$ 0.42 $0.0^{+246.2}_{-15.9}$
0.54 $1.9^{+265.3}_{-2.9}$ 0.49 $26.0^{+78.2}_{-50.1}$ 0.52
$127.2^{+77.1}_{-104.5}$ 0.75 $182.6^{+40.7}_{-25.1}$ 0.78
$28.9^{+56.3}_{-47.2}$ 0.11 840 $8.1^{+3.4}_{-2.3}$ 5.0 $11.9^{+5.6}_{-3.7}$
5.0 $36.2^{+10.8}_{-27.3}$ 0.7 $2.9^{+21.0}_{-92.0}$ 0.76
$86.1^{+4.8}_{-101.1}$ 0.62 $39.6^{+11.5}_{-14.5}$ 0.89 $32.0^{+12.7}_{-15.5}$
0.91 $164.1^{+85.8}_{-28.1}$ 0.1 $17.1^{+16.6}_{-6.1}$ 0.84
$25.9^{+16.3}_{-10.9}$ 0.88 $353.9^{+146.9}_{-187.6}$ 0.1


# Dynamics of particle production in Pb–Pb collisions at $\sqrt{s_{NN}}$ =
2.76 TeV using PYTHIA8 Angantyr model

Ravindra Singh  Yoshini Bailung  Ankhi Roy  Discipline of Physics, Indian
Institute of Technology Indore, Simrol, Indore 453552, India

###### Abstract

We study the dynamics of identified, strange, and multi-strange particle
production in Pb–Pb collisions at $\sqrt{s_{{}_{NN}}}$ = 2.76 TeV using the
recently developed Angantyr model, incorporated within PYTHIA8. We show the
interplay between multi-parton interactions (MPI) and color reconnection (CR)
on the experimentally measured quantities. The charged-particle multiplicity
($N_{\rm ch}$) and mean transverse momentum ($\langle$$p_{\rm{T}}$$\rangle$)
distributions are well explained by PYTHIA8 Angantyr with proper tuning, as
presented in this paper. Predictions of $p_{\rm{T}}$ spectra,
$\langle$$p_{\rm{T}}$$\rangle$, $p_{\rm{T}}$ integrated yields of the
identified, strange and multi-strange particles are studied. To provide
insight into the collective nature of the produced particles, we look into the
ratio of particle yields to pions and kaons. PYTHIA8 Angantyr with CR and MPI
mimic signs of collectivity and is possibly one of the suitable candidates to
study ultra-relativistic heavy-ion collisions.

##  I Introduction

The study of how matter behaves at extreme temperature and energy densities is
vital to understand the nature of phase transition in Quantum Chromodynamics
(QCD)(Gao:2020qsj, ; Tawfik:2016cot, ; Satz:2013xja, ). Experiments like
Relativistic Heavy Ion Collider (RHIC) at BNL, USA, and Large Hadron Collider
(LHC) at CERN, Geneva, Switzerland, are of prime importance to unravel the
properties of the hot QCD matter produced(Busza:2018rrf, ; Teaney:2000cw, ;
Cassing:1999es, ; PHOBOS:2004zne, ). A de-confined state of quarks and gluons,
also known as Quark-Gluon Plasma (QGP), is believed to be created for a very
short lifetime in these heavy-ion collisions experiments. In this QGP phase,
the relevant degrees of freedom are quarks and gluons rather than mesons and
baryons, confined to color-neutral states Becattini:2014rea ; Busza:2018rrf ;
Aoki:2006we .

An initial and important observable is the multiplicity distribution ($\rm
d\it{N}_{\rm ch}/d\eta$) of the charged particles, which is essential to
extract the properties of produced particles and their interactions
PHOBOS:2004zne . Such distributions in a particular pseudorapidity range were
measured in the CERN proton anti-proton ($\rm p\overline{p}$) collider
experiments in 1980’s (UA5:1988gup, ; UA1:1989bou, ; UA5:1986yef, ;
UA1:1982fux, ). These measurements provide information on the energy density
and centrality of the colliding system. The centrality is directly related to
the initial overlap region geometry, which corrrespond to the number of
participating nucleons and binary collisions(Miller:2007ri, ).

For the final-state charged particles, the rapidity ($\rm{y}$) or
pseudorapidity ($\eta$) and transverse momentum ($p_{\rm{T}}$) spectra are
known to reflect the degrees of longitudinal extension and transverse
excitation of the interacting system, respectively(PHENIX:2004vcz, ; Braun-
Munzinger:2003pwq, ). Distributions in low $p_{\rm{T}}$ ranges let us inspect
the transverse excitation and soft processes, whereas higher $p_{\rm{T}}$
correspond to hard scattering processes. In low energy collisions, one can
neglect hard processes, while most of the contribution comes from soft
processes. At high center of mass energies, hard processes have finite
contribution albeit, the soft processes are predominant Liu:2013uba . From the
final state charged particle $p_{\rm{T}}$ spectra, one can extract information
about the thermal nature of the interacting system (Cleymans:2012ya, ;
Cleymans:2011in, ), and can comment on the formation and characteristic
properties of the formed matter. According to the Maxwell-Boltzmann
distribution law, the $p_{\rm{T}}$ spectra are related to the temperature of
the system formed in these collisions. The $N_{\rm ch}$ of charged particles
formed in ultra-relativistic collisions also depends on the system’s
temperature and density. Since most of the final state charged particles are
part of a locally thermalized medium, the mean transverse momentum
$\langle$$p_{\rm{T}}$$\rangle$ distribution vs. $N_{\rm ch}$ is expected to be
more or less flat in heavy-ion systems like Pb–Pb at high $N_{\rm ch}$. The
contributions at lower $N_{\rm ch}$ are mostly from the peripheral collisions
where a QGP is less likely to be produced.

The ratios of yields of identified hadrons are important to understand to the
mechanism of hadron production. The ratio of proton to pion ($p/\pi$) and kaon
to pion ($K/\pi$) characterize the relative baryon and meson production,
respectively. Additionally, $K/\pi$, $\Lambda/\pi$, $\Sigma^{0}/\pi$,
$\Xi^{0}/\pi$, and $\Omega/\pi$ ratios represent the strangeness production at
higher multiplicities, indicating a universal underlying dynamics in hadron
production for different quark-gluon final states. Strangeness enhancement is
proposed as a signature of QGP formation in heavy-ion collisions (Koch:1986ud,
; Soff:1999et, ; Capella:1994pr, ) because of the faster equilibration of
strangeness production processes in a QGP than any other process in a hadron
gas Cuautle:2016huw ; Torrieri:2009mq . This is observed to be more prominent
for multi-strange hadrons ALICE:2016fzo . The production mechanism of strange
hadrons provides a way to investigate the properties of the hot QCD matter.

Another essential medium characteristic is anisotropic flow, considered the
proof of collective behavior of partons and hadrons(Huovinen:2001cy, ;
ALICE:2010suc, ; STAR:2004jwm, ). In a heavy-ion collision, the hydrodynamic
expansion is a consequence of the transverse pressure gradient. This
transverse flow shifts the produced particles to higher momenta, and due to
higher gain in momenta of heavy particles from flow velocity, the increment is
more for heavier particles. This effect is seen commonly for heavy-ion systems
and even high multiplicity p–p and p–Pb collisions ALICE:2016fzo .

In this work, we have explored all the above-mentioned aspects of particle
production in Pb–Pb collisions at $\sqrt{s_{{}_{NN}}}$ = 2.76 TeV using the
Angantyr model, which is the heavy-ion extension of the PYTHIA8, extensively
used for pp collisions. The paper is organized as follows. In section II, the
details of event generation and analysis methodology are discussed. section
III presents the results and discussions. Finally, in section IV, we summarize
our findings of this investigation.

##  II Event generation and Analysis methodology

PYTHIA is an event generator that is extensively and successfully used for the
study of proton-proton and proton-lepton collisions. Recent advancement in
PYTHIA8 enables the study of heavy nuclei collisions, namely proton-nuclei
(pA) and nuclei-nuclei (AA). In this work, PYTHIA8 event generator is used to
simulate ultra-relativistic Pb–Pb collisions with Angantyr Bierlich:2018xfw .
PYTHIA8 natively does not support heavy-ion systems; however, the Angantyr
model combines several nucleon-nucleon collisions into one heavy-ion
collision. It is a combination of many-body physics (theoretical) models
suitable for producing hard and soft interactions, initial and final-state
parton showers, particle fragmentations, multi-partonic interactions, color
reconnection mechanisms, and decay topologies Sjostrand:2006za . We use 8.235
version of PYTHIA8 Sjostrand:2008za , which includes multi-parton interactions
(MPI). MPI is vital to expostulate the underlying events, multiplicity
distributions, and charmonia production. In general, an event generator at
high colliding energies produces around four to ten partonic interactions,
which depend on the overlapping region of colliding particles Weber:2018ddv .
The perturbative scattering processes are implemented by Initial State
Radiation (ISR) and Final State Radiation (FSR)(BaBar:2015onb, ;
Martinez:2014cfa, ).

Hadronization in PYTHIA8 is done using the Lund string fragmentation model. In
this model, the probability of creation of hadrons from an initial state of
partons is described by the Lund area law Andersson:2001yu . The beam remnants
and the produced partons are interconnected via color fluxtubes or strings
storing potential energy. New quark-antiquark pairs are formed as strings
break when the partons move away from each other. Until the strings reduce to
small pieces, this process continues. These small string pieces are recognized
as shell hadrons. In this scheme, strings are reconnected between the partons
in such a way that strings length decreases; particle production decreases and
consequently multiplicity reduces. In literature, flow-like feature in pp
collisions is well mimicked by Color Reconnection (CR) OrtizVelasquez:2013ofg
tune of PYTHIA8.

In the current version of Angantyr included in PYTHIA8, hadronization is done
using the string fragmentation model; a thermalized medium of partons (QGP) is
not a part of this model Bierlich:2018xfw . In Angantyr, positions and the
number of the interacting nucleons and binary nucleon-nucleon collisions are
performed by Glauber model-based eikonal approximation in impact-parameter
space. Furthermore, Gribov’s corrections are implemented in order to include
diffractive excitation, which appears due to the fluctuations in the nucleon
substructure. Angantyr is the first model in which individual fluctuations for
both projectile and target are included. It is required in order to
extrapolate into AA systems Bierlich:2018xfw . The contribution to the final
state from each participating nucleon is induced from the Fritiof model with
the concept of wounded nucleons (diffractive and non-diffractive). The hard
partonic sub-collisions, normalized by nucleon-nucleon sub-collisions, play a
crucial role at high energies. The model treats the projectile and target
nucleons via two interaction scenarios. In one case, the interactions between
the species are considered as pp like non-diffractive (ND) processes. This is
entirely driven by PYTHIA8. However, in the second scenario, a wounded
projectile nucleon can have ND interactions with many target nucleons, which
are termed secondary ND (SD) collisions. Subevents are generated solely
through PYTHIA8, where these SD collisions are put into play as modified SD
processes Behera:2019cds . Depending on the interaction probability,
interactions between wounded nucleons are classified as elastic, non-
diffractive, secondary non-diffractive, single diffractive, and double-
diffractive.

We have generated around 2 million events in Pb–Pb at $\sqrt{s_{{}_{NN}}}$ =
2.76 TeV. The inelastic, non-difractive component of the total cross-section
for all soft QCD processes is used with the switch SoftQCD:all = on with MPI
based scheme of color reconnection (ColorReconnection:mode(0)). The classes
based on charged particle multiplicities ($N_{\rm ch}$) have been chosen in
the acceptance of TPC detector with pseudorapidity range of ($-0.8<\eta<0.8$)
ALICE:2010suc  to match with experimental conditions and centrality classes in
ALICE at the LHC. The events generated using these cuts are divided into nine
multiplicity classes, each class containing 10% of total events except first
two classes which contain 5% of total events as used in ALICE:2010suc . The
$N_{\rm ch}$ classes corresponding to different centralities are tabulated in
TABLE 1. Reconstruction of heavy strange particles are done via specific decay
channels or their weak decay topologies, namely
$\Xi^{\mp}\xrightarrow{}\pi^{\mp}+\Lambda$,
$\Omega^{\mp}\xrightarrow{}K^{\mp}+\Lambda$,
${\Phi\xrightarrow{}\mu^{-}+\mu^{+}}$,
${\Sigma^{0}\xrightarrow{}\Lambda+\gamma}$ , and
${\Lambda\xrightarrow{}\pi^{-}+p}$ and for the anti-particles we used
corresponding charge conjugate decay channels. We use -0.8 $<\eta<0.8$ and
-0.9 $<y<0.9$ kinematics range for reconstruction these particles to account
for ALICE experimental acceptances.

Table 1: Centrality classes and the corresponding charged particle
multiplicities.

S.No. | Centrality (%) | $N_{\rm ch}$
---|---|---
I | 0-5 | 2314-3050
II | 5-10 | 1947-2314
III | 10-20 | 1387-1947
IV | 20-30 | 967-1387
V | 30-40 | 644-967
VI | 40-50 | 399-644
VII | 50-60 | 224-399
VIII | 60-70 | 108-224
IX | 70-80 | 43-108

Table 2: Mean and RMS of charge particle multiplicity in different PYTHIA8
tunes.

| MPI + CR | CR off | MPI off | MPI off + CR off
---|---|---|---|---
Mean | 704.1 | 882.8 | 276.6 | 276.6
RMS | 759.5 | 961.2 | 274.3 | 274.3

##  III Results and Discussion

The charged-particle multiplicity distributions for different PYTHIA8 tunes
within $|\eta|<0.8$) is shown in the FIG. 1. To see the effect of different
PYTHIA tunes, we consider the following configurations: MPI with/without CR,
No MPI, and both MPI and CR off. It is observed that MPI+CR is compatible with
ALICE data. MPI without CR overestimates, whereas the tune without MPI is seen
to underestimate our results. We also observe that there is no effect of CR if
MPI is off. The particle production increases with MPI, due to inter-partonic
interactions; on the other hand, while turning CR off, particle production
increases. In the color reconnection (CR) scheme, the string lengths are
reduced; in consequence, when CR is kept on, particle production lessens
OrtizVelasquez:2013ofg . Turning MPI off removes the strings between the
partons. As a result, we do not observe any effect of CR. To quantify the
effect of MPI and CR for the different PYTHIA tunes, the mean and RMS of the
multiplicity distributions are measured and reported in TABLE 2. The mean for
MPI+CR is around 2.5 times larger without MPI and around 3.2 times larger
without CR compared without MPI. A similar comparison can be made for the RMS
values between these settings. For MPI+CR turned off, we report similar values
for mean and RMS, which confirms our statement made earlier.

By observing different tunes in FIG. 1 (Left), we can conclude that MPI with
CR is a favorable setting to describe ALICE data ALICE:2010suc . We also
observed similar results after comparing $\langle$$p_{\rm{T}}$$\rangle$
distribution as a function of charged-particle multiplicity using simulated
PYTHIA8 Angantyr and experimental data as shown in FIG. 1 (Right).
Distributions obtained from PYTHIA8 are scaled with a constant (1.138) factor
for better visualization and to compare the slope of different distributions
with data ALICE:2013rdo . The $\langle$$p_{\rm{T}}$$\rangle$ distribution with
MPI and CR describes the data very well, even without hydrodynamics.
$\langle$$p_{\rm{T}}$$\rangle$ with MPI off (or MPI and CR off) describe data
below $N_{\rm ch}=10$ very well but deviates at higher values, becoming almost
flat at high multiplicities. This is probably due to the large production of
low multiplicity events when MPI is kept off. A similar trend is seen for CR
turned off; however, the ratio of $\langle$$p_{\rm{T}}$$\rangle$ over data
decreases as we go to higher values in multiplicity, as reconnection occurs in
such a way that the strings between partons are as small as possible. This
attribute is credited to CR, where a correlation between $N_{\rm ch}$ and
$\langle$$p_{\rm{T}}$$\rangle$ can be seen ALICE:2012aqc . Preceeding
hadronization, strings fuse to form high $p_{\rm{T}}$ hadrons. With CR off,
fewer strings fuse to form hadrons during hard scatterings, explaining the
increment of $\langle$$p_{\rm{T}}$$\rangle$ at higher multiplicities.

To further check the compatibility of simulated data, we compare the
$p_{\rm{T}}$ spectra of final state charged-particles with ALICE measurements
in different centrality classes within experimental kinematic selections,
which is shown in FIG. 2. The $p_{\rm{T}}$ spectra of each centrality class
are scaled to the slope with ALICE measurements for clearer visualization and
comparison. From the lower panel of FIG. 2, it is observed that the
experimental to simulated data is comparable within statistical uncertainties.

Figure 1: (Color Online) (Left) Multiplicity distribution of charged particles
from PYTHIA8 Angantyr with different tunes and ALICE data. (Right)
$\langle$$p_{\rm{T}}$$\rangle$ distribution vs. charged-particle multiplicity
in different PYTHIA8 tunes and ALICE data. The lower panels show the ratio of
PYTHIA Angantyr predictions over Data for the different configurations
considered. Figure 2: (Color Online) Charged-particle $p_{\rm{T}}$ spectra in
nine centrality classes described in TABLE 1 from PYTHIA Angantyr and ALICE
data. The lower panel represents the deviation of PYTHIA Angantyr predictions
from data.

With the assurance of the quality of the simulated data discussed above, we
now move on to study the transverse momentum spectra of identified particles,
$p_{\rm{T}}$ integrated yield of identified and strange particles, and
particle ratios with PYTHIA8.

###  III.1 Transverse momentum spectra of identified particles

FIG. 3. shows the $p_{\rm{T}}$ spectra of identified charge-particles
$\pi^{\pm}$,$K^{\pm}$, and $p(\bar{p})$ in different centrality classes and
for minimum bias (MB). The spectra were obtained using the same selection cuts
in all charged-particles species. To visualize better, we multiplied scale
factors to each $p_{\rm{T}}$ spectra. From the lower panel of FIG. 3., the
$p_{\rm{T}}$ spectra corresponding to (30-40)% centrality is seen to coincide
with the MB spectra. We report a shift in the hardness of the $p_{\rm{T}}$
spectra for each centrality class with respect to MB. For the threshold
centrality class (30-40)%, classes (0-5)%, (5-10)%, (10-20)%, (20-30)% are
harder while classes (40-50)%, (50-60)%, (60-70)% and (70-80)% are softer with
respect to MB. It is to be noted that a similar trend is observed for all the
identified particles.

The $p_{\rm{T}}$ spectra of hadrons obtained at the final state are compared
to measurements from the ALICE Collaboration. In FIG. 4. the PYTHIA
predictions find a good match with the data for pions. For protons and kaons,
we report a slight underestimation by PYTHIA at low $p_{\rm{T}}$. The hump at
low $p_{\rm{T}}$ is probably due to the pQCD implementation of PYTHIA, whereas
we expect NRQCD effects in this regime. We also compare strange baryon
$p_{\rm{T}}$ spectra ($\Lambda$, $\Xi$, $\Omega$) with ALICE in all centrality
ranges considered, as shown in FIG. 5. The ratios show a similar peak at
$p_{\rm{T}}$$\sim$ 3 GeV/$c$. It is also observed that the width of the hump
increases with strangeness and mass, especially for central and semi-central
events. At higher centralities, the strange baryons show good compatibility.
In FIG. 6., we compare the $\phi$ meson and D-meson (D0, D+) $p_{\rm{T}}$
spectra, where the $\phi$ meson spectra are seen to be consistent with ALICE
measurements in all centralities. In the case of D-mesons, we see PYTHIA
predictions depart at low $p_{\rm{T}}$, however in good agreement at
intermediate-higher values. This helps us conclude that the more prominent
peaks observed for strange baryons (FIG. 5.) have a strangeness dependence
rather than mass.

Figure 3: (Color Online) $p_{\rm{T}}$ spectra of identified charged-particles
($\pi^{\pm}$,K±, p$(\bar{\rm p})$) in various centrality classes. The lower
panel show the ratios for each centrality class to MB.

Figure 4: (Color Online) $p_{\rm{T}}$ spectra of identified charge particles
($\pi^{\pm}$,K±, p$(\bar{\rm p})$) compared to ALICE measurements. The lower
panel show the ratios for each centrality class to data.

Figure 5: (Color Online) $p_{\rm{T}}$ spectra of strange and multi-strange
baryons ($\Lambda^{\pm}$, $\Xi^{\pm}$, $\Omega^{\pm}$). The lower panel show
the ratios for different centrality classes to data.

Figure 6: (Color Online) $p_{\rm{T}}$ spectra of $\phi$ and D-mesons. The
lower panel show the ratios for each centrality class to data.

Figure 7: (Color Online) Yield of identified particles (Left) and strange
particles (Right) as a function of centrality.

###  III.2 $p_{\rm{T}}$ integrated yield of identified and strange particles

The $p_{\rm{T}}$ integrated yields of $\pi^{\pm}$,$K^{\pm}$ and $p(\bar{p})$
are shown in FIG. 7 (Left) within rapidity range -0.9 $<y<$ 0.9 normalized by
the total number of events. It is observed that the yields of different
particles are increasing going from peripheral to most central. We can see a
clear mass ordering in yields, with lower mass pions having higher yields,
while protons being heavier have lower yields. This is expected towards
central collisions; the probability for hard scatterings will be higher,
resulting in high particle production. Production of a lighter particle
requires lesser energy as compared to a heavier particle and will be more
dominant in peripheral collisions. This gives further concreteness towards the
use of PYTHIA8 Angantyr with the current configuration for heavy-ion
collisions.

In FIG. 7 (Right), $p_{\rm{T}}$-integrated yields of strange particles are
shown. One can see the same features in strange particles like that of
identified particles observed in FIG. 7 (Left). Production of strange
particles is seen to reduce towards peripheral collisions with a similar trend
in mass, except for $\rm\phi$ mesons.

###  III.3 Particle ratios

By the bare yield distribution, we cannot quantitatively measure the
enhancement or suppression of different particle species. The best way to do
this is to estimate the yield with respect to other particles. We measure the
ratio of proton and kaon yields over pions to inspect the variation over
centrality and $p_{\rm{T}}$. FIG. 8. shows the measured yield ratios vs.
centrality (Left) and $p_{\rm{T}}$(Right). We scale proton over pion ratios
for every centrality class for better comparison with the corresponding
quantity versus $p_{\rm{T}}$ (Left). The scale factor is calculated using the
formula:

$Scale\ factor=\dfrac{K/\pi}{X/\pi}$

Here X refers to different particle species. We can see from this FIG. 8. that
both ratios are increasing towards most central; however proton over pion
($p/\pi$) ratio drops rapidly than kaon over pion ($K/\pi$). $K/\pi$ ratio
increases at lower $p_{\rm{T}}$ but decreases at higher $p_{\rm{T}}$, showing
a bump around 3 GeV/c. In heavy-ion collisions, this is the consequence of
radial flow, but in PYTHIA, this is attributed to color reconnection. We can
argue that CR could be another mechanism of flow where a longitudinal boost is
implemented at the initial state (partonic state), just before hadronization
starts. Understanding this mechanism is important, as it could provide an
explanation of flow-like patterns. At higher $p_{\rm{T}}$, more particles
correspond to jets, in which these particles become insensitive to CR. If one
increases MPI, we see an enhancement in the bump region. We do not observe any
bump in the $K/\pi$ ratio. It increases at lower $p_{\rm{T}}$ and then turns
flat at higher $p_{\rm{T}}$ values. Here, we use the default tune of CR that
is implemented in PYTHIA. By tuning CR, studies report this effect can also be
observed in meson to meson ratio OrtizVelasquez:2013ofg .

Similarly, we show yield ratios of strange particles over pions as a function
of centrality in FIG. 9 (Left). Each yield ratio of different particles is
scaled with a similar method as mentioned before. We observe a clear
strangeness enhancement as we go from most central to peripheral collisions.
Heavy strange particle ratios are showing more enhancement, as reported in
FIG. 9 (Left), slopes of strange particle ratios increase towards heavier
strange particles. This is due to overlapping color strings forming (ropes) at
higher densities Bierlich:2018xfw . In FIG. 9, we show yield ratios of strange
particles as a function of transverse momentum in two different centrality
classes at 0-5% and 70-80%. For all strange particles, yield ratios increase
towards higher $p_{\rm{T}}$. As expected, the ratio of yields is lesser for
strange heavy particles. We also conclude by observing FIG. 8 (Right) and FIG.
9 (Right) that meson to pion ratios are not showing the bump but shows for
baryon to pion ratios.

In FIG. 10. we show the ratio of strange particles over ($K^{+}+K^{-}$)
mesons. The ratio increases as $p_{\rm{T}}$ increases, and after a peak close
to 3-4 GeV/c, it decreases. The position of the peak shift towards higher
$p_{\rm{T}}$ for strange heavy particles. A study reports a similar type of
observation seen in experimental data for Pb–Pb collisions ALICE:2013cdo .
This effect is generally seen in heavy-ion collisions as a consequence of
radial flow Fries:2003fr .

Figure 8: (Color Online) Ratio of yields of identified particles over
$\pi^{+}+\pi^{-}$ as a function of centrality (Left) and as a function of
transverse momentum (Right).

Figure 9: (Color Online) Ratio of yields of strange particles over
$\pi^{+}+\pi^{-}$ as a function of centrality (Left) and as a function of
transverse momentum (Center). Figure 10: (Color Online) Ratio of yields of
strange particles over ($K^{+}+K^{-}$) as a function of transverse momentum.

##  IV Summary

In this work, we attempt to study the dynamics of particle production in Pb–Pb
collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV using the Angantyr model
incorporated in PYTHIA8. This model is an extrapolation of pp collision into
pA and AA collisions without a thermalized medium. Without any collectivity
present, we assume that MPI with CR mimics flow-like features Maldonado-
Cervantes:2014tva . In this regard, the primary observations of this work are
summarised below:

  * $\bullet$

The multiplicity distribution of charged particles with different combinations
of MPI and CR tunes is compared to ALICE measurements. It is observed that
setting CR off, PYTHIA8 Angantyr predictions overestimate ALICE data, whereas
turning MPI off underestimates the data. MPI with CR setting is observed to be
a good tune to match the multiplicity distribution obtained from PYTHIA with
ALICE results.

  * $\bullet$

The $p_{\rm{T}}$ spectra simulated by PYTHIA8 Angantyr at different
centralities are also consistent with experimental data within the
uncertainties. It is also observed that the $\langle$$p_{\rm{T}}$$\rangle$
distribution obtained by using MPI + CR is consistent with ALICE data.

  * $\bullet$

$p_{\rm{T}}$ spectra of identified particles
($\pi^{\pm}$,$K^{\pm}$,$p(\bar{p})$) in different centrality classes along
with minimum bias (MB) is obtained. Hardening of the $p_{\rm{T}}$ spectra can
be seen towards most central and is more pronounced for heavier particles. A
centrality cut-off at (30-40)% is observed and found to coincide with MB,
which implies centralities above this cut-off produce harder particles.

  * $\bullet$

The $p_{\rm{T}}$ integrated yield normalized by total events of identified
charged particles and strange particles show a clear mass ordering. Heavier
particle production is lesser and decreases towards peripheral collisions. The
rate of production is higher for protons compared to kaons towards central
collisions to pions.

  * $\bullet$

It is observed that the ratio of kaon over pion as a function of $p_{\rm{T}}$
increases with increasing $p_{\rm{T}}$ which then saturates toward higher
$p_{\rm{T}}$. In contrast, the ratio of proton over pion shows a peak around 3
GeV/c due to CR with MPI in PYTHIA8, which mimic flow-like patterns. A similar
rise is observed for all the strange baryon over pion ratios, although
unobserved for the meson to meson ratio.

  * $\bullet$

It is observed that the slope of strange particles to pion ratio as a function
of centrality is more significant for strange heavy particles. We report that
the peak of strange baryon to pion ratio and strange baryon to kaon as a
function of $p_{\rm{T}}$ shifts toward higher $p_{\rm{T}}$ for heavier strange
particles. This shows that strangeness enhancement is dominant in strange
heavy particles, which is a consequence of color strings overlapping at higher
densities in accordance with the CR mechanism.

From the present study and the subsequent observations, it can be concluded
that MPI with CR mimics flow-like features, and one can use this tune in
PYTHIA8 Angantyr to carry out heavy-ion studies. Although we have shown
features resembling radial flow, one can also observe higher-order flow
harmonics ($v_{2}$,$v_{3}$) in models that pertain to QGP free collision
systems. daSilva:2020cyn .

##  V Acknowledgement

The author acknowledges the financial support by the DST-INSPIRE program of
the Government of India.

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