Time-delay measurement of MgII broad line response for the
  highly-accreting quasar HE 0413-4031: Implications for the MgII-based
  radius-luminosity relation
  We present the monitoring of the AGN continuum and MgII broad line emission
for the quasar HE 0413-4031 ($z=1.38$) based on the six-year monitoring by the
South African Large Telescope (SALT). We managed to estimate a time-delay of
$302.6^{+28.7}_{-33.1}$ days in the rest frame of the source using seven
different methods: interpolated cross-correlation function (ICCF), discrete
correlation function (DCF), $z$-transformed DCF, JAVELIN, two estimators of
data regularity (Von Neumann, Bartels), and $\chi^2$ method. This time-delay is
below the value expected from the standard radius-luminosity relation. However,
based on the monochromatic luminosity of the source and the SED modelling, we
interpret this departure as the shortening of the time-delay due to the higher
accretion rate of the source, with the inferred Eddington ratio of $\sim 0.4$.
The MgII line luminosity of HE 0413-4031 responds to the continuum variability
as $L_{\rm line}\propto L_{\rm cont}^{0.43\pm 0.10}$, which is consistent with
the light-travel distance of the location of MgII emission at $R_{\rm out} \sim
10^{18}\,{\rm cm}$. Using the data of 10 other quasars, we confirm the
radius-luminosity relation for broad MgII line, which was previously determined
for broad H$\beta$ line for lower-redshift sources. In addition, we detect a
general departure of higher-accreting quasars from this relation in analogy to
H$\beta$ sample. After the accretion-rate correction of the light-travel
distance, the MgII-based radius-luminosity relation has a small scatter of only
$0.10$ dex.
