galaxies. In section 2, we explain the data and sample selection. In section 3, we describe our method to evaluate the $\beta$ value. In section 4, we show the result of the observed $\beta$–$M_{UV}$ relation, and we assess the incompleteness of our sample selection. In section 5, we discuss the intrinsic $\beta$–$M_{UV}$ relation, most active star-forming galaxies, and dust attenuation law for $z \sim 4$ galaxies. In section 6, we give our conclusions to summarize this work. In regard to the cosmological parameters, we assume $\Omega_{m,0} = 0.3$, $\Omega_{\Lambda,0} = 0.7$, $H_{\mathrm{0}} = 70\>\mathrm{km\>s^{-1}\>Mpc^{-1}}$. Finally throughout this work we apply the AB magnitude system ([@OkeGunn]; [@Fkgt96]).

 Data and sample selection  {#S2Dtss}
===========================

 Data  {#S2s1da}
------

In our analysis, we select the Lyman Break Galaxies (LBGs) at $z \sim 4$ in the SXDS field which is partially covered by other surveys (i.e., UDS-UKIDSS/UKIRT, UDS-CANDELS/HST, and SEDS/Spitzer). Figure \[fig1\] shows the field map of each survey and indicates that UDS-CANDELS and SEDS survey do not cover the entire field of SXDS or UDS-UKIDSS.

Our catalog includes *B*, *V*, *R*, *i’*, *z’*, and updated-*z’* from Subaru/Suprime-Cam, *J*, *H*, and *K* from UKIRT/WFCAM, F125W and F160W from HST/WFC3, and 3.6$\>\micron$ and 4.5$\>\micron$ from Spitzer/IRAC. 