KEK-TH-1616

# What can we learn from the 126 GeV Higgs boson
for the Planck scale physics ?
\- Hierarchy problem and the stability of the vacuum - 111Contribution to
SCGT12 ”KMI-GCOE Workshop on Strong Coupling Gauge Theories in the LHC
Perspective”, 4-7 Dec. 2012, Nagoya University and HPNP2013 ”Toyama
International Workshop on Higgs as a Probe of New Physics 2013”, 13–16, Feb.
2013, Toyama University

Satoshi Iso  Theory Center, Institute of Particles and Nuclear Studies
High Energy Accelerator Research Organization (KEK)
1-1 Oho, Tsukuba, Ibaraki, Japan 305-0801

###### Abstract

The discovery of the Higgs particle at around 126 GeV has given us a big hint
towards the origin of the Higgs potential. The running quartic self-coupling
decreases and crosses zero somewhere in the very high energy scale. It is
usually considered as a signal of the instability of the standard model (SM)
vacuum, but it can also indicate a link between the physics in the electroweak
scale and the Planck scale. Furthermore, the LHC experiments as well as the
flavor physics experiments give strong constraints on the physics beyond the
SM. It urges us to reconsider the widely taken approach to the physics beyond
the SM (BSM), namely the approach based on the gauge unification below the
Planck scale and the resulting hierarchy problem. Motivated by the recent
experiments, we first revisit the hierarchy problem and consider an
alternative appoach based on a classical conformality of the SM without the
Higgs mass term.

In this talk, I review our recent proposal of a B-L extension of the SM with a
flat Higgs potential at the Planck scale IsoOrikasa ; IOO . This model can be
an alternative solution to the hierarchy problem as well as being
phenomenologically viable to explain the neutrino oscillations and the baryon
asymmetry of the universe. With an assumption that the Higgs has a flat
potential at the Planck scale, we show that the B-L symmetry is radiatively
broken at the TeV scale via the Coleman-Weinberg mechanism, and it triggers
the electroweak symmetry breaking through a radiatively generated scalar
mixing. The ratio of these two breaking scales is dynamically determined by
the B-L gauge coupling.

##  I Central dogma of particle physics

In the LHC era, we acquired various hints towards the physics beyond the SM.
The first hint is of course the mass of the recently discovered Higgs-like
particle. The value of 126 GeV is quite interesting because it is close to the
boarder of the stability bound. Given the vev of the Higgs at 246 GeV, its
mass gives an information of the curvature of the potential at the minimum. As
the mass 126 GeV is smaller than 246 GeV, the Higgs potential is rather
shallow and unstable against the radiative corrections. The (in)stability of
the SM vacuum can be investigated explicitly by looking at the running
bahaviour of the quartic coupling. The beta function of the quartic Higgs
coupling $\lambda_{H}$ is given by

$\displaystyle\beta_{H}=\frac{1}{16\pi^{2}}\left(24\lambda_{H}^{2}-6Y_{t}^{4}+\frac{9}{8}g^{4}+\frac{3}{8}g_{Y}^{4}\right).$
(1)

$Y_{t}$ is the top Yukawa coupling and $g,g_{Y}$ are $SU(2)_{L},U(1)_{Y}$
gauge couplings. It is either positive or negative whether the negative
contribution by the large top Yukawa coupling is compensated by the gauge
couplings and the Higgs quartic coupling. The corresponding quartic coupling
to the 126 GeV Higgs boson does not suffice to compensate it and the beta
function is negative. So the running quartic coupling crosses zero somewhere
at the UV energy scale. It is very suggestive that the observed value of the
Higgs mass is close to the stability bound up to the Planck scale Espinosa

$\displaystyle
M_{h}[\mbox{GeV}]>129.2+1.8\left(\frac{M_{t}[\mbox{GeV}]-173.2}{0.9}\right)-0.5\left(\frac{\alpha_{s}(M_{Z})-0.1184}{0.0007}\right)\pm
1.0_{th}.$ (2)

When the Higgs mass is lighter than the above bound, new physcis must appear
below the Planck scale. But if it lies just on the border of the stability
bound, it gives a big hint to the origin of the Higgs potential at the Planck
scaleFP1 .

Another important information is that the LHC results are almost consistent
with the SM. Furhtermore the precision experiments of the flavor physics,
Babar, Belle and LHCb, gave stringent constraints on the physics beyond the
SM. Of course, in spite of the above rather unexpectedly good agreement with
the SM, there exist phenomena which cannot be explained within the SM.
Nuetrino oscillation requires the dimension 5 operator $l\phi l\phi$ and a new
scale beyond the SM must be introduced. The baryon asymmetry of the universe
also requires an additional source of the CP violation. The SM anyway needs to
be extended to explain these phenomana.

The most common appoach to go beyond the SM is based on a unification of the
gauge couplings below the Planck scale, i.e. the GUT scale. Then we need a
natural explanation why the electroweak scale is much smaller than the GUT
scale. In order to solve the hierarchical structure of the scales, the
supersymmetry is introduced. Here I call the sequence of ideas from GUT to the
hierarchy problem and the low energy supersymmetry the central dogma of
particle physics. In addition to solving the hierarchy problem, it can improve
the gauge coupling unification as well as providing candidates of the dark
matter particles. But as the bonus we get or as the price we pay, it predicts
many new particles at the TeV scale and the recent experiments have given
strong constraints on the models with low energy supersymmetry.

In such circumstances, it may be a good time to reconsider the central dogma
of particle physics. In this note, we take an approach to the hierarchy
problem suggested by Bardeen. In the next section, we interpret the Bardeen’s
argument in terms of the renormalization group. If we adopt the argument, the
most natural mechanism to break the electreweak symmetry is the Coleman-
Weinberg (CW) mechanism. But we know that the CW mechanism does not work
within the SM because of the large top Yukawa coupling, so we need to extend
the SM. In section 3, we introduce our model, a classically conformal $B-L$
extension of the SM and then discuss the dynamics of the model.

##  II Bardeen’s argument of the hierarchy problem

We pay a special attention to the almost scale invariance of the SM. At the
classical level, the SM Lagrangian is conformal invariant except for the Higgs
mass term. Bardeen argued Bardeen  that once the classical conformal
invariance and its minimal violation by quantum anomalies are imposed on the
SM, it may be free from the quadratic divergences.

Bardeen’s argument on the hierarchy problem may be interpreted as follows
AokiIso . We classify divergences of the scalar mass term in the SM into the
following 3 classes,

  * •

quadratic divergences: $\Lambda^{2}$

  * •

logarithmic divergences with a small coefficient: $m^{2}\log(\Lambda/\mu)$

  * •

logarithmic divergences with a large coefficient: $M^{2}\log(\Lambda/\mu)$

The logarithmic divergences are operative both in the UV and the IR. In
particular, it controls a running of coupling constants and is observable. On
the other hand, the quadratic divergence can be always removed by a
subtraction. Once subtracted, it no longer appears in observable quantities.
In this sense, it gives a boundary condition of a quantity in the IR theory at
the UV energy scale where the IR theory is connected with a UV completion
theory. Indeed, the RGE of a Higgs mass term $m^{2}$ in the SM

$\displaystyle V(H)=-m^{2}H^{\dagger}H+\lambda_{H}(H^{\dagger}H)^{2}$ (3)

is approximately given by

$\displaystyle\frac{dm^{2}}{dt}=\frac{m^{2}}{16\pi^{2}}\left(12\lambda_{H}+6Y_{t}^{2}-\frac{9}{2}g^{2}-\frac{3}{2}g_{Y}^{2}\right).$
(4)

The quadratic divergence is subtracted by a boundary condition either at the
IR or UV scale. Once the initial condition of the RGE is given at the UV
scale, it is no longer operative in the IR. The RGE shows that the mass term
$m^{2}$ is multiplicatively renormalized. If it is zero at a UV scale
$M_{UV}$, it continues to be zero at lower energy scales. In this sense, the
quadratic divergence is not the issue in the IR effective theory, but the
issue in the UV completion theory. Hence if the SM (and its extension at the
TeV scale) is directly connected with a UV completion theory at the Planck
scale physics, the hierarchy problem turns out to be a problem of the boundary
condition at the UV scale. If the UV completion theory is an ordinary field
theory, it will be difficult to protect the masslessness of the scalar
particle against radiative corrections by massive particles of the UV scale
unless we introduce, e.g. the low-energy supersymmetry. But in the string
theory, symmetry is sometimes enhanced on a moduli space and massless scalars
can survive even without supersymmetry. Also discrete symmetry like T-duality,
which is invisible in the low energy effective theory, may prohibit a
generation of potential at the string scale.

The multiplicative renormalization of the Higgs mass term is violated by a
mixing with a massive field in the loop. If the massive field aquires its mass
in a different mechanism with the EWSB, the Higgs mass has a logarithmic
divergence

$\displaystyle\delta
m^{2}\sim\frac{\lambda^{2}M^{2}}{16\pi^{2}}\log(\Lambda^{2}/m^{2})$ (5)

which modifies the RGE as

$\displaystyle\frac{dm^{2}}{dt}=\frac{m^{2}}{16\pi^{2}}\left(12\lambda_{H}+6Y_{t}^{2}-\frac{9}{2}g^{2}-\frac{3}{2}g_{Y}^{2}\right)+\frac{M^{2}}{8\pi^{2}}\lambda^{2}.$
(6)

The last term corresponds to the logarithmic divergence with a large
coefficient. The coefficient $M^{2}$ has nothing to do with the mass of the
Higgs $m^{2}$, and it violates the multiplicativity of the Higgs mass. Thus
the hierarchy problem, namely the stability of the EWSB scale, is caused by
such a mixing of relevant operators (mass terms) with hierarchical energy
scales $m\ll M$. In the Bardeen’s argument, he also imposes an absence of
intermediate scales above the EW scale. The logarithmic divergence with a
large coefficient (5) is sometimes confused with the quadratic divergence, but
if the UV completion theory is something like a string theory, they should be
distinguished.

From the above considerations, the hierarchy problem can be solved by imposing
the following two different conditions;

  * •

Correct boundary condition at the UV (Planck) scale $M_{pl}$

  * •

Absence of mixings in intermediate scales below $M_{pl}$

The first condition subtracts the quadratic divergence at the Planck scale. It
must be solved in the UV completion theory such as the string theory. The most
natural boundary condition is that scalar fields which appear in the low
energy physics are massless at the Planck scale. On the other hand, the second
condition assures the absence of logarithmic divergences with large
coefficients. Even if the scalars are massless at the Planck scale, they
receive large radiative corrections from the mixing with other relevant
operators. Without a cancellation mechanism like the supersymmetry, we need to
impose an absence of intermediate scales between EW (or TeV) and Planck
scales. Hence all symmetries are broken either at the Planck scale or near the
EW scale. Especially, the breakings of the supersymmetry or the grand
unification of gauge coupling should occur at the Planck scale. This second
condition is also emphasized in the Bardeen’s argument Bardeen . In such a
scenario, Planck scale physics is directly connected with the electroweak
physics shapo .

Hence a natural boundary condition of the mass term at the UV cut-off scale,
e.g. $M_{Pl}$, is

$\displaystyle m^{2}(M_{Pl})=0.$ (7)

This is the condition of the classical conformality of the BSM. The condition
(7) must be justified in the UV completion theory, and from the low energy
effective theory point of view, it is just imposed as a boundary condition
222The condition (7) may look similar to the Veltman condtion Veltman , but
they are conceptually different at all. In the Veltman condition, the
quadratic divergence is considered to be cancelled between various
contributions of bosons and fermions. Such a cancellation occurs in a very
special situation of the IR physics. On the contrary, the condition (7) is
independent of the matter content in the IR, and robust against a change of
scales. .

##  III Flat potential at the Planck scale

The condition (7) restricts the form of the Higgs potential as

$\displaystyle V(H)=\lambda_{H}(H^{\dagger}H)^{2}.$ (8)

Here $\lambda_{H}$ is the running coupling and the RG improved effecitve
potential is given by making the coupling $\lambda_{H}(H)$ field dependent.
The mass term is not generated even in the IR as discussed in the previous
section once the boundary condtion (7) is imposed at the boundary with the UV
completion theory. The mass of the Higgs at 126 GeV suggests that the running
coupling becomes asymptotically vanishing near the Planck scale. The current
bounds (2) is a bit heavier than the experimental data, but in this note, we
assume that the Higgs quartic coupling vanishes at the Planck scale. Hence

$\displaystyle V(H)=0\mbox{ at the Planck scale}.$ (9)

The condition may connect the SM in the IR with the string theory in the UV.
Now we have to solve two problems. The first is whether we can construct a
phenomenologically viable model starting from the condtion of the flatness of
the Higgs potential (9), and the second is to derive such a boundary condtion
from the UV completion theory such as a string theory. Supersymmetry or grand
unification, if exists, are broken at the Planck scale. In the following we
focus on the first problem by proposing a B-L extension of the SM with a flat
potential at the Planck scale. The second issue is left for future
investigations.

Since the IR theory is assumed to have the boundary condition (9), the
electroweak symmetry breaking should occur radiatively, namely the Coleman-
Weinberg mechanism. However, it is now well-known that the CW mechanism cannot
occur within the SM because of the large top-Yukawa coupling. Indeed, the CW
mechanism is realized only when the beta-function of the quartic scalar
coupling is positive and the running quartic coupling crosses zero somewhere
in the IR. But as we saw, the beta function of the quartic Higgs coupling is
positive in the SM and its behavior is opposite to the CW mechanism. Hence, in
order to realize the EWSB, we need an additional sector in which the symmetry
is broken radiatively by the CW mechanism and whose symmetry breaking triggers
the EWSB. In the next section, we introduce our model, namely a B-L extension
of the SM with a flat potential at the Planck scale.

##  IV B-L extension of the SM with flat potential at Planck

The idea to utilize the CW mechanism to solve the hierarchy problem was first
modelled by Meissner and Nicolai MaNi  (see also Dias ). In addition to the SM
particles, they introduced right-handed neutrinos and a SM singlet scalar
$\Phi$. Inspired by the work, we proposed a minimal phenomenologically viable
model IOO . It is the minimal B-L model B-L , but with a classical
conformality. The model is similar to the one proposed by Meissner and Nicolai
MaNi , but the difference is whether the B-L symmetry is gauged or not. In a
recent paper we further showed that by imposing the flatness (9) of the Higgs
potential at the Planck scale the B-L breaking scale is related with the EWSB
scale. The ratio of two scales is dynamically determined by the B-L gauge
coupling and the B-L breaking scale is naturally constrained to be around TeV
scale B-L2  for a not so small B-L gauge coupling.

Besides the SM particles the model consists of the B-L gauge field with the
gauge coupling $g_{B-L}$, right-handed nuetrinos $\nu_{R}^{i}$ ($i=1,2,3$
denotes the generation index) and a SM singlet complex scalar field $\Phi$
with two units of the B-L charge. The model is anomaly free. The Lagrangian
contains Majorana Yukawa coupling $\sim
Y_{N}^{i}\Phi\bar{\nu}_{R}^{ic}\nu_{R}^{i}$, and the see-saw mechanism gives
masses to the left-handed neutrinos once the scalar $\Phi$ acquires vev.

##  V Symmetry breakings of B-L and EW

Since the B-L gauge symmetry is broken by the CW mechanism, the breaking scale
is correlated with the quartic coupling $\lambda_{\Phi}$ at the UV scale. Its
running is described by

$\frac{d\lambda_{\Phi}}{dt}=\frac{1}{16\pi^{2}}\left(20\lambda_{\Phi}^{2}-\frac{1}{2}Tr\left[Y_{N}^{4}\right]+96g_{B-L}^{4}+\cdots\right).$
(10)

If the Mayorana Yukawa coupling is not so large, the beta function is
positive. The typical behavior of the running $\lambda_{\phi}$ is drawn in
Fig. 1. It crosses zero at a lower energy scale $M_{0}$, then the B-L symmetry
is broken at $M_{B-L}\sim M_{0}\exp(-1/4)$ through the CW mechanismIsoOrikasa
.

Figure 1:  RG evolution of the self-coupling $\lambda_{\phi}$ of a SM singlet
scalar $\phi$. Since the $\beta$ function is positive, the running coupling
crosses zero at a lower energy scale.

As shown in the paper IOO , the ratio of the scalar boson mass to the B-L
gauge boson mass is given by

$\displaystyle\left(\frac{m_{\Phi}}{m_{Z^{\prime}}}\right)^{2}\sim\frac{6}{\pi}\alpha_{B-L}\ll
1.$ (11)

The condition that the B-L gauge coupling does not diverge up to the Planck
scale requires $\alpha_{B-L}<0.015$ at $M_{B-L}$. Hence the scalar boson
becomes lighter than the B-L gauge boson, $m^{2}_{\Phi}<0.03\
m^{2}_{Z^{\prime}}$. Such a very light scalar boson is a general prediction of
the CW mechanism.

The EWSB is triggered by the B-L breaking. The flatness condition (9) of the
Higgs potential predicts an absence of the scalar mixing at the Planck scale.
Hence B-L and EW sectors are decoupled each other in the UV. But since the
matter fields are coupled to both $U(1)_{Y}$ and $U(1)_{B-L}$, these two
sectors become mixed through the $U(1)$-mixing. As a result, the scalar mixing
term $\lambda_{mix}(H^{\dagger}H)(\Phi^{\dagger}\Phi)$ appears in the IR. It
is interesting that a very small negative mixing $\lambda_{mix}$ is always
generated irrespective of the details of other parameters once we assume the
flatness condition $\lambda_{mix}(M_{pl})=0$. By solving the RGE IsoOrikasa
we showed that the scalar mixing term is dynamically generated around
$\lambda_{mix}\sim-4\times 10^{-4}$. If the $\Phi$ field acquires a VEV
$\langle\Phi\rangle=M_{B-L}$, the mixing term
$\lambda_{mix}(H^{\dagger}H)(\Phi^{\dagger}\Phi)$ gives an effective mass term
of the Higgs field. Since the coefficient $\lambda_{mix}$ is negative, the
EWSB is triggered and the Higgs VEV is given by

$\displaystyle v=\langle
H\rangle=\sqrt{\frac{|\lambda_{mix}|}{\lambda_{H}}}M_{B-L}$ (12)

This gives a ratio between the EWSB scale to the B-L symmetry breaking scale.
The scalar mixing is determined in terms of the gauge couplings, so the ratio
of two breaking scales is also determined dynacamilly in terms of the gauge
coupling $g_{B-L}$.

##  VI Model predictions

The dyanamics of the model is controlled by two parameters, $g_{B-L}$ and
$\lambda_{\Phi}$, which determines the two breaking scales of B-L and EW.

Figure 2: Model prediction is drawn in the black line (from top left to down
right). The $B-L$ gauge coupling $\alpha_{B-L}$ and the gauge boson mass
$m_{Z^{\prime}}$ are related because of the flat potential assumption at the
Planck scale. The left side of the most left solid line in blue has been
already excluded by the LEP experiment. The left of the dashed line can be
explored in the 5-$\sigma$ significance at the LHC with $\sqrt{s}$=14 TeV and
an integrated luminosity 100 fb-1. The left of the most right solid line (in
red) can be explored at the ILC with $\sqrt{s}$=1 TeV, assuming 1% accuracy.

The experimental input $v=246$ GeV gives a relation between these two and the
dynamics of the model is essentially described by a single parameter. The
figure 2 shows the prediction of our model. The vertical axis is the strength
of $\alpha_{B-L}$ and the horizontal axis is the mass of the B-L gauge boson.
The black line (from top left to down right) shows the prediction of our
model. If an extra $U(1)$ gauge boson and a SM singlet scalar are found in the
future, the prediction of our model is the mass relation (11), e.g.,
$m_{\phi}\sim 0.1\ m_{Z^{\prime}}$ for $\alpha_{B-L}\sim 0.005$. The CW
mechanism in the B-L sector predicts a much lighter SM singlet Higgs boson
than the extra $U(1)$ gauge boson. It is different from the ordinary TeV scale
B-L model where the symmetry is broken by a negative squared mass term.

Nuetrino oscillation is realized by the type I see-saw mechanism with small
neutrino Yukawa couplings. Baryon number asymmetry of the universe may be
generated through the TeV scale leptogenesis with almost degenerate Majorana
massesIOO3 . Furhter phenomenological issues such as $U(1)$ mixing or the
lepton number violation at the TeV scale are discussed in a separate paper.

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Center at Stockholm University, the University of Maryland, the University of
Washington, Deutsches Elektronen-Synchrotron and Humboldt University, Los
Alamos National Laboratories, the TANGO Consortium of Taiwan, the University
of Wisconsin at Milwaukee, and Lawrence Berkeley National Laboratories.
Operations are conducted by COO, IPAC, and UW.

##  Appendix A Format of Data Products

The final SDSS-RM data set and related data products are provided via
anonymous ftp and https
services555ftp://quasar.astro.illinois.edu/public/sdssrm/final_result/,
https://ariel.astro.illinois.edu/sdssrm/final_result/ (or contact Y. Shen for
latest archival info). We organize the data products by the object ID (RMID).
In each object directory (e.g., ./rm000/), we provide the original spectra,
continuum and emission-line light curves, PrepSpec outputs, and lag
measurements. Additional information about the SDSS-RM sample is provided in
the sample characterization paper Shen et al. (2019b). Table 5 provides a
brief description of the data products available. Future updates of these
products will also be distributed via this site. Below we describe the main
data products released with this paper.

SDSS spectra Customarily reprocessed (e.g., Shen et al., 2015a) optical
spectroscopy from SDSS is provided in the ./spec/ directory in ascii format. A
total of 90 epochs are available for each object. These are the original
spectra before the PrepSpec run. However, these spectra were customarily
reprocessed as described in Shen et al. (2015a), thus are slightly different
from the spectra included in the SDSS-III/IV public data releases.

Continuum and emission-line light curves The merged continuum light curve and
emission-line light curves (after processed by PrepSpec) are provided in plain
ascii files with the _lc.txt affix in each object directory (e.g., ./rm000/).
These light curves are used in the lag measurements. The emission-line light
curve files only include H$\alpha$, H$\beta$, Mg ii, and C iv, with prefixes
ha, hb, mg2 and c4, respectively. Additional emission-line light curves, if
covered by spectroscopy, will be included in the ./prepspec/ directory in the
future. We are aware that in rare occasions, the merging of the continuum
light curves failed, with discrepant fluxes from different facilities. These
corrupted light curves usually do not produce successful lags, and we did not
attempt to resolve these flux discrepancies in this work.

PrepSpec outputs Model light curves, flux calibration scaling factor
(_p0_t.dat), mean and rms spectra, and line width measurements from PrepSpec
run are provided in the ./prepspec/ directory. These outputs include all
available lines covered in SDSS spectra. The flux calibration scaling factor
p0_t is used to scale the SDSS spectra to improve spectrophotometry. A subset
of the PrepSpec outputs are also compiled in the summary Table 2.

Lag measurements Outputs from our lag analyses are provided in the ./pypetal/
directory in each object directory. These outputs include the DRW fits to the
continuum light curves and outlier masks, ICCFs, Javelin/PyROA results, and
weights applied to the lag calculations. The essential subset of these
measurements are compiled in the summary Table 2. For convenience, we provide
a lag summary plot in each object directory (e.g., ./rm000/lag_summary.pdf) to
display the lag measurements for all available lines.

Software Scripts of running lag detection and post-processing are provided in
the ./scripts/ directory.

Table 5: Content of Data Products

File Path | Description
---|---
./ | Top directory
./spec/ | Directory for SDSS spectra (before PrepSpec)
./prepspec/ | Directory for PrepSpec outputs
./rm*/ | Object directory for each SDSS-RM quasar
./scripts/ | Batch scripts and additional software tools
./summary.fits | Summary table (Table 2)

Content of ./prepspec/
File Path Description ./rm*/ Object directory – ./rm*_p0_t.dat Scaling factor
of flux calibration (tabulated in $\ln p_{0}(t)$) – ./rm*_avg_w.dat Average
spectrum – ./rm*_rms_w.dat RMS spectrum; column 2 is the ‘RMSx’ estimate (Shen
et al., 2016) – ./rm*_$cont$.dat Continuum light curve (e.g., ‘c3000’,
‘c5100’) – ./rm*_$cont$_t_stats.dat Statistics of the continuum light curve –
./rm*_vblr.dat Line width measurements from the average spectrum –
./rm*_vvblr.dat Line width measurements from the rms spectrum –
./rm*_$line$_t.dat Line light curve – ./rm*_$line$_w.dat RMS line profile –
./rm*_$line$_t_stats.dat Statistics of the line light curve

Content of ./rm*/
File Path Description ./lag_summary.pdf Summary plot of lag measurements
./cont_lc.txt Merged continuum light curve ./$line$_lc.txt Original line light
curve (first three columns in PrepSpec output rm*_$line$_t.dat) ./pypetal/ Lag
measurement directory – ./light_curves/ Copied original light curves with
flags for outliers added – ./processed_lcs/ Outlier-rejected light curves and
light curves formatted for Javelin and PyROA – ./cont/drw_rej/ Continuum DRW
fitting result and outlier mask – ./$line$/ Lag measurements for the line –
./pyccf/ ICCF results – ./javelin/ Javelin results – ./pyroa/ PyROA results –
./weights/ Weights for lag posteriors

##  Appendix B Informing future MOS-RM Programs

The lag yields from the final SDSS-RM data are roughly in line with the
simulation forecast in Shen et al. (2015a), despite the slightly different
total baseline and lag detection criteria assumed in the forecast, and the
fact that the predictions were based on simulated light curves. In Fig. 16, we
show the final lag detection fraction as a function of the line variability
SNR2 parameter described in §3.3 (also see Shen et al., 2019b). For a future
MOS-RM program, it is necessary to have a sufficient number of epochs to
sample variability patterns over a sufficiently long (e.g., multi-years)
spectroscopic baseline. However, in order to detect the line variability
properly, each spectroscopic epoch must also achieve a significant S/N for the
line flux measurement. A value of ${\rm SNR2}=20$ with the benchmark SDSS-RM
cadence/baseline (with $N_{\rm epoch}\approx 100$) approximately corresponds
to ${\rm line\ RMS}/\left<\sigma_{i}\right>\approx 2$, where line RMS is the
rms line flux variability (absolute not fractional), and
$\left<\sigma_{i}\right>$ is the average per-epoch flux uncertainty. Thus for
a $20\%$ fractional line variability, i.e., ${\rm RMS/AVG}=0.2$, we require a
fractional line flux measurement uncertainty of $\left<\sigma_{i}\right>/{\rm
AVG}=10\%$ in order to reach ${\rm SNR2=20}$. Deeper epoch spectra, e.g., with
5% flux measurements, will increase SNR2 to $\sim 40$ and subsequently boost
the probability of lag detection (Fig. 16).

For MOS spectroscopy, there is also a systematic floor of flux uncertainties
limited by the spectrophotometry accuracy of the MOS program. For example, in
SDSS-RM, we achieved a typical systematic flux uncertainty floor of $\sim
5\%$. Using PrepSpec and for quasars with strong narrow lines (such as [O iii]
5007) at low redshift, we were able to further improve the spectrophotometry
to $\sim 2-3\%$. This additional improvement is likely partially responsible
for the higher lag detection rate for H$\beta$ at fixed SNR2 (computed using
statistical flux uncertainties only), compared with Mg ii and C iv for which
the PrepSpec improvement on spectrophotometry is marginal. Moreover, the
systematic uncertainty of spectrophotometry increases toward the blue end of
SDSS spectra (e.g., Shen et al., 2015a), resulting in a larger impact on lag
detection for Mg ii and C iv in high-redshift quasars than for H$\beta$ in the
low-redshift subset.

The outcome from the SDSS-RM final data set thus provides useful guidance for
future MOS-RM programs with similar baseline and cadence. Given the typical
$\sim 10\%$ RMS variability of broad lines, a low systematic floor of
spectrophotometry will greatly improve the lag detection rate. At the same
time, we recommend to reach a statistical (per epoch) line flux uncertainty of
better than $\sim 10\%$ at the flux limit of the sample. For SDSS-RM, the
latter goal is achieved with 2 hr integration per epoch at a limiting
magnitude of $i_{\rm psf}<21.7$, for the general quasar population.

Figure 16: Lag detection fraction as a function of line variability parameter
SNR2, for H$\beta$, Mg ii and C iv, respectively. Error bars are Poisson
counting uncertainties.

##  Appendix C Additional Quality Cuts on Lags

As discussed in Sec 4.2, we impose a set of quantitative cuts for lag
detection. Our choice of the $r_{\rm max}>0.4$ cut is more lenient than those
used in other high-quality RM studies. However, while imposing this less
stringent cut inevitably introduces more false positives in the lag sample, it
also ensures that we are not removing lags near the detection boundary that
could potentially bias the $R-L$ relation measurements.

We test the effects of imposing more stringent quality cuts on lag detection.
Fig. 17 shows the comparison in the $R-L$ plane with our fiducial lag results
for: (1) a higher $r_{\rm max}>0.6$ cut; (2) a ${\rm SNR2>35}$ cut; (3) high-
quality lags from manual inspection as detailed in §4.3. These additional
quality cuts ($r_{\rm max}$, SNR2, grade) roughly reduce the fiducial lag
sample by half for each line.

As shown in Fig. 17, for the quantitative cuts with $r_{\rm max}$ and SNR2, we
generally found consistent results on the best-fit $R-L$ relation in terms of
the slope, normalization and intrinsic scatter. For the visual inspection
grade cut, however, the Mg ii and C iv $R-L$ relations are notably more
flattened and shifted to lower normalizations. This is because long lags with
limited light curve overlap are generally less likely to be perceived as a
good-quality detection. As a result, manual inspection preferentially removes
long lags and bias the measurement of the $R-L$ relation. This highlights the
danger of cherrypicking the best-quality lags in the $R-L$ relation
measurements from a systematic RM program such as ours.

For full disclosure, we provide the best-fit $R-L$ relation for these higher-
quality lag subsets in Table 6, but we strongly caution that these relations
are not intended for any practical uses for the reasons discussed above.

Figure 17: Tests of lag detection with more stringent quality cuts. In each
panel, the open symbols represent the fiducial lag sample, and the filled
symbols are for the subset. The left column shows results with a $r_{\rm
max}>0.6$ cut; the middle column shows results with a ${\rm SNR2}>35$ cut; the
right column shows the results with the manual inspection cut. Details
regarding these quality cuts and their impact on the measured $R-L$ relations
are presented in Appendix C.  Table 6: Regression Results for Higher-quality
Lags

$Y=a+bX+{\rm scatter}$
Cut Y$|$X X X range $a$ $b$ $\sigma_{\rm int}$ $N_{\rm fit}$ $r_{\rm max}>0.6$
$\log\tau_{\rm rest,H\beta}$ $\log L_{\rm 5100,host-corr}$ [42.87, 45.22]
$1.432_{-0.047}^{+0.044}$ $0.54_{-0.09}^{+0.08}$ $0.28_{-0.03}^{+0.04}$ 44
${\rm SNR2}>35$ $\log\tau_{\rm rest,H\beta}$ $\log L_{\rm 5100,host-corr}$
[42.87, 45.40] $1.450_{-0.041}^{+0.041}$ $0.43_{-0.07}^{+0.07}$
$0.27_{-0.03}^{+0.03}$ 48 Grade$=4/5$ $\log\tau_{\rm rest,H\beta}$ $\log
L_{\rm 5100,host-corr}$ [43.10, 45.08] $1.418_{-0.048}^{+0.047}$
$0.42_{-0.11}^{+0.10}$ $0.28_{-0.03}^{+0.04}$ 37 $r_{\rm max}>0.6$
$\log\tau_{\rm rest,MgII}$ $\log L_{\rm 3000}$ [43.58, 46.23]
$2.079_{-0.061}^{+0.060}$ $0.21_{-0.09}^{+0.10}$ $0.32_{-0.04}^{+0.05}$ 35
${\rm SNR2}>35$ $\log\tau_{\rm rest,MgII}$ $\log L_{\rm 3000}$ [43.58, 46.28]
$2.111_{-0.053}^{+0.053}$ $0.22_{-0.09}^{+0.09}$ $0.33_{-0.04}^{+0.04}$ 44
Grade$=4/5$ $\log\tau_{\rm rest,MgII}$ $\log L_{\rm 3000}$ [43.69, 45.63]
$1.935_{-0.058}^{+0.056}$ $0.09_{-0.11}^{+0.11}$ $0.27_{-0.04}^{+0.05}$ 32
$r_{\rm max}>0.6$ $\log\tau_{\rm rest,CIV}$ $\log L_{\rm 1350}$ [44.66, 46.95]
$1.752_{-0.119}^{+0.121}$ $0.47_{-0.16}^{+0.16}$ $0.54_{-0.06}^{+0.08}$ 36
${\rm SNR2}>35$ $\log\tau_{\rm rest,CIV}$ $\log L_{\rm 1350}$ [44.69, 46.95]
$1.665_{-0.086}^{+0.085}$ $0.47_{-0.12}^{+0.12}$ $0.43_{-0.04}^{+0.05}$ 57
Grade$=4/5$ $\log\tau_{\rm rest,CIV}$ $\log L_{\rm 1350}$ [44.69, 46.14]
$1.634_{-0.112}^{+0.113}$ $0.005_{-0.219}^{+0.219}$ $0.41_{-0.05}^{+0.07}$ 32

##  Appendix D Lag Examples

We provide several additional examples for H$\beta$, Mg ii and C iv lags in
our fiducial lag sample that represent the range of detection qualities based
on our visual inspection. We caution that our visual inspection is subjective
on individual cases, but on average higher-grade lags are more robustly
detected. Lower-grade lags not necessarily mean the lag is spurious, and may
reflect the complexity of the BLR and the limitations of measuring an average
lag.

Figure 18: Additional examples of H$\beta$ lags (light curves in the left
panel and ICCF in the right panel). Higher (visual inspection) grades
correspond to better measured lags. The continuum and broad-line light curves
are normalized to have unity mean within their respective baseline, and
therefore they do not necessarily line up after shifting the broad-line light
curve by the observed-frame PyROA lag.

Figure 19: Additional examples of Mg ii lags. Higher (visual inspection)
grades correspond to better measured lags.

Figure 20: Additional examples of C iv lags. Higher (visual inspection) grades
correspond to better measured lags.

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