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Locations of optical and $γ$-ray emitting regions in the jet of PMN J2345-1555

We collect long term $γ$-ray, optical and radio $15$ GHz light curves of quasar object PMN J2345-1555. The correlation analyses between them are performed via the local cross-correlation function (LCCF). We found that all the optical $V$, $R$ band and the infrared $J$ band are correlated with the radio 15 GHz at beyond $3σ$ significance level, and the lag times are $-221.81^{+6.26}_{-6.72}$, $-201.38^{+6.42}_{-6.02}$ and $-192.27^{+8.26}_{-7.37}$ days, respectively. The $γ$-ray is strongly correlated with optical, but weakly correlated with the radio. We present that time lags between different frequencies can be used as an alternative parameter to derive the core-shift measurement. For this target, the magnetic field and particle density at 1 parsec in jet are derived to be $0.61$ Gauss and $1533/γ_{\rm min}$ cm$^{-3}$, respectively. The black hole mass and the 15 GHz core position in jet are estimated to be $10^{8.44} {\rm M}_{\odot}$ and $30$ parsec, respectively. The lag times enable us to derive that the optical and the $γ$-ray emitting regions coincide, which are located at $4.26^{+0.83}_{-0.79}$ pc away from 15 GHz core position in jet and beyond the broad line region (BLR). We found that a $3σ$ correlation between the color index and the radio light curve, which indicates that opacity may play an important role in the variation. The $δV-δR$ behaviors are complex, while the $R-J$ shows a bluer when brighter trend. As hinted from radio images, we proposed a positional dependent spectral index model to explain the color index behaviors, which is complementary for the shock in jet model. The curvature effects and contribution from accretion disk may also affect variables of blazars in many aspects.