4.1. Comparison with other Doppler factors in the literature

The Doppler factor ( $\delta$ ), an important parameter to reveal the relativistic beaming effect and explain the observed extreme properties of blazars, is proverbially unmanageable to measure since there is no straight method at present. Many indirect methods are proposed to estimate the beaming Doppler factor: (i) it can be deduced by a synchrotron self-Compton (SSC) model thus denoted as $\delta_{\textrm{ssc}}$ (e.g. Ghisellini et al. Reference Ghisellini, Padovani, Celotti and Maraschi1993); (ii) to be derived from adopting single-epoch radio data by assuming that the sources hold an equipartition of energy between radiating particles and magnetic field as denoted as $\delta_{\textrm{eq}}$ (Readhead Reference Readhead1994); (iii) to be estimated using the radio flux density variations or brightness temperature denoted as $\delta_{\textrm{var}}$ (Lähteenmäki & Valtaoja Reference Lähteenmäki and Valtaoja1999; Hovatta et al. Reference Hovatta, Valtaoja, Tornikoski and Lähteenmäki2009). (iv) One can calculate it based on the broadband SED (e.g. Chen Reference Chen2018). However, due to these different assumptions, each method will render discrepant results.

The relativistic beaming effect plays a crucial role in the $\gamma$ -ray emission, and particularly, for Fermi-detected blazars. Fan et al. (Reference Fan, Huang, He, Yang, Hua, Liu and Wang2009) found that the $\gamma$ -ray luminosity of Fermi-detected blazars correlates tightly with the radio Doppler factor $\log L_{\gamma} \sim 0.47\log \delta^{3+\alpha}_{\textrm{R}}$ . Kovalev (Reference Kovalev2009) pointed out that the sources detected by Fermi/LAT have higher brightness temperature with respect to those not detected by Fermi. Savolainen et al. (Reference Savolainen2010) compiled 62 AGNs with apparent superluminal motion and adopted their Doppler factors from Hovatta et al. (Reference Hovatta, Valtaoja, Tornikoski and Lähteenmäki2009) and found that the Fermi blazars have, on average, higher Doppler factor than non- Fermi-detected blazars. Xiao et al. (Reference Xiao2019) collected 291 sources with superluminal motions, in which 189 are $\gamma$ -ray sources detected by Fermi, and reported that the Fermi-detected sources show higher proper motion, apparent velocity, Doppler factor, Lorentz factor, and smaller viewing angles than non- Fermi-detected sources, also suggesting the strong Doppler effect lies on those $\gamma$ -ray sources. Pei et al. (Reference Pei, Fan, Bastieri, Yang and Xiao2020b) obtained that the $\gamma$ -ray luminosity increases with radio core-dominance parameter for Fermi AGNs.

Beaming effect is mostly studied using the radio emission, which yields that the radio variability Doppler factor ( $\delta_{\textrm{var}}$ ) method should be perhaps an appropriate way to describe the blazars population and beaming effect (Fan et al. Reference Fan, Huang, He, Yang, Hua, Liu and Wang2009, Reference Fan, Yang, Zhang, Hua, Liu, Qin and Huang2013b; Liodakis & Pavlidou Reference Liodakis and Pavlidou2015; Liodakis et al. Reference Liodakis2017a,b).

By modelling the radio light curves of 1 029 sources as a series of flares characterised by an exponential rise and decay, Liodakis et al. (Reference Liodakis, Hovatta, Huppenkothen, Kiehlmann, Max-Moerbeck and Readhead2018) estimated the variability Doppler factor ( $\delta_{\textrm{var}}$ ) for 837 blazars, which included 670 FSRQs and 167 BL Lacs. They calculated the variability brightness temperature ( $T_{\textrm{var}}$ ) using

(7) \begin{equation}T_{\textrm{var}} = 1.47\times10^{13}\displaystyle\frac{d^{2}_{\textrm{L}}\Delta S_{\textrm{ob}}(\nu)}{\nu^{2}t^{2}_{\textrm{var}}(1+z)^{4}}{\rm K},\end{equation}

here, $S_{\textrm{ob}}(\nu)$ the amplitude of the flare in Jy, $\nu$ the observed frequency in GHz, and $t_{\textrm{var}}$ the rise time of a flare in days. Then, the variability Doppler factor ( $\delta_{\textrm{var}}$ ) can be defined as:

(8) \begin{equation}\delta_{\textrm{var}}=(1+z)\sqrt[3]{\frac{T_{\textrm{var}}}{T_{\textrm{eq}}}},\end{equation}

where $T_{\textrm{eq}}$ is the equipartition brightness temperature, and $T_{\textrm{eq}}=2.78\times10^{11}$ K was adopted. After cross-checking with our sample, there are 285 common sources, which contains 210 FSRQs and 75 BL Lacs. When we compared our results with theirs, it was found that $\log \delta_{\gamma}=(0.22\pm0.03)\log \delta_{\textrm{L18}}+(0.49\pm0.04)$ with a correlation coefficient $r=0.37$ and a chance probability of $P<10^{-10}$ . We show this plot in the left panel of Figure 2.

Based on the SED fitting, Chen (Reference Chen2018) estimated the jet physical parameters of 1 392 $\gamma$ -ray loud blazars taking from Fan et al. (Reference Fan2016), and particularly, they calculated the Doppler factor ( $\delta_{\textrm{SED}}$ ) and obtained that the median values of the Doppler factors of FSRQs, BL Lacs, and total blazars were 10.7, 22.3, and 13.1, respectively. It is usually assumed in SED modelling that the $\gamma$ -ray emission is produced closer to the supermassive black hole than the radio core of the jet where most of the radio emission originates. For comparison, we investigate the relation between the derived $\delta_{\gamma}$ in the present paper and those in Chen (Reference Chen2018) for 597 sources are in common in both papers (in fact, we found 682 sources to be in common after cross-checking; however, there are 85 sources obtained an extremely large or small value of $\delta_{\textrm{SED}}$ in Chen (Reference Chen2018), thus we excluded those 85 sources). The best fitting is $\log \delta_{\gamma}=(0.21\pm0.02)\log \delta_{\textrm{C18}}+(0.45\pm0.03)$ with $r=0.34$ and $P\sim0$ for the 597 sources (see the right panel of Figure 2).

However, according to equation (5), the estimation of lower limit on $\delta_{\gamma}$ would be affected by redshift, the above correlations then have a redshift dependence, thus the redshift effect needs to be removed. To do so, we adopt a partial correlation analysis (see e.g. Padovani Reference Padovani1992):

(9) \begin{equation}r_{12,3} = \displaystyle\frac{r_{12}-r_{13}r_{23}}{\sqrt{1-r^{2}_{13}}\sqrt{1-r^{2}_{23}}},\end{equation}

where $r_{ij}$ denotes the correlation coefficient between $x_{i}$ and $x_{j}$ , while $r_{ij,k}$ denotes the partial correlation coefficient between $x_{i}$ and $x_{j}$ with $x_{k}$ dependence excluded ( $i, j, k=1,2,3$ ). In our case, we let $x_{1}=\log\delta_{\gamma}$ , $x_{2}=\log\delta_{\textrm{L18}}$ or $\log\delta_{\textrm{C18}}$ , and $x_{3}=z$ . For the left panel, we have $r_{12}=0.37$ , $r_{1z}=0.74$ , and $r_{2z}=0.31$ , which yields $r_{12,z}=0.21$ . Using the similar calculation, we obtain $r_{12,z}=0.19$ for the right panel. Their P-values are both $<10^{-4}$ . It still rendered a statistically correlation between our derived values of $\log\delta_{\gamma}$ and those from other methods after removing the redshift effect, implying that they are truly correlated. This result indicates that our derived lower limit on $\gamma$ -ray Doppler factors are reasonable.

We draw an equality line in both panels of Figure 2 (labelled in blue solid) and note that some points are quite disperse and most importantly, the derived values of $\log \delta_{\gamma}$ are fairly small than that of $\log \delta_{\textrm{var}}$ or $\log \delta_{\textrm{SED}}$ and the range is relatively compact, ranging from 0.95 to 28.38 and the average value is 5.39. We should point out that, firstly, because our estimation is the lower limit on $\delta_{\gamma}$ . Hovatta et al. (Reference Hovatta, Valtaoja, Tornikoski and Lähteenmäki2009) had shown the median value of $\delta$ was 12.02 ranging from 0.30 to 35.50; Liodakis et al. (Reference Liodakis, Hovatta, Huppenkothen, Kiehlmann, Max-Moerbeck and Readhead2018) reported the average value to be 14.35 in the range from 0.08 to 88.44; Chen (Reference Chen2018) on average, 14.30 was obtained and spanning from 1.00 to 99.50. The typical values of Doppler factor in blazars should be in the range from a few to ${\sim}$ 50 based on various methods as mentioned above. However, since our derived results are the lower limit on $\delta_{\gamma}$ , thus the distribution of $\delta_{\gamma}$ should be smaller than those from the literature. This is consistent with Dondi & Ghisellini (Reference Dondi and Ghisellini1995), they found the average value $\langle\delta_{\gamma}\rangle\sim4.4$ , ranging from 1.3 to 11 for a sample of EGRET blazars. Fan et al. (Reference Fan, Yang, Liu and Zhang2013a) obtained an average value $\langle\delta_{\gamma}\rangle\sim7.22$ for 138 $\gamma$ -ray blazars and Fan et al. (Reference Fan, Bastieri, Yang, Liu, Hua, Yuan and Wu2014) also found the average value was ${\sim}$ 7.00 with regard to their sample.

Secondly, since we could not ascertain the variability timescale for each source in our large sample, thus we adopted $\Delta T=24$ h (1 d) in our present calculation. However, this operation was examined by Fan et al. (Reference Fan, Yang, Liu and Zhang2013a) and Fan et al. (Reference Fan, Bastieri, Yang, Liu, Hua, Yuan and Wu2014), who also used $\Delta T=1$ d for calculation and obtained reliable results. For instance, those authors found a tendency for the $\gamma$ -ray Doppler factors to increase with the radio Doppler factors and $\delta_{\gamma}$ are also correlated with the superluminal velocity. This supports the fact that the $\gamma$ -rays are strongly beamed, and likewise suggesting that the radio Doppler factors estimated from the variability can be used to discuss the beaming effect in Fermi loud blazars. Dondi & Ghisellini (Reference Dondi and Ghisellini1995) used a similar method as this paper to estimate the lower limit on $\delta_{\gamma}$ for 46 $\gamma$ -ray loud blazars. They could not find enough information of variability timescale for nearly half of their sample either, and they also adopted $\Delta T=1$ d for calculation. However, different synchrotron peaked sources may have different timescales. Abdo et al. (Reference Abdo2011), Bonnoli et al. (Reference Bonnoli, Ghisellini, Foschini, Tavecchio and Ghirlanda2011), Hu et al. (Reference Hu, Chen, Guo, Jiang and Li2014), and Prince (Reference Prince2020) processed systematic well studied on Mrk 421, 3C 454.3, S5 0716 + 714, and 3C279, respectively, although Mrk 421 is a HSP blazar and the other three are LSP blazars (Fan et al. Reference Fan2016), those authors had shown that a typical variability timescale in the source frame for Fermi/LAT blazars is $\approx$ 1 d (see also Nalewajko Reference Nalewajko2013; Zhang et al. Reference Zhang, Xue, He, Liang and Zhang2015). Chen (Reference Chen2018) also adopted this idea for simplicity in the constraint of Doppler factor from SED modelling for a larger sample of blazars (see also Kang, Chen, & Wu Reference Kang, Chen and Wu2014). From this point of view, we consider that our choice of variability timescale that $\Delta T=1$ d is also reliable.

On the other hand, therefore, different values of $\Delta T$ can be also considered. We compute the $\gamma$ -ray Doppler factor with $\Delta T=6$ h (denoted in $\delta_{\gamma}^{\rm6\,h}$ ) and $\Delta T=48$ h (denoted in $\delta_{\gamma}^{\rm48\,h}$ ), then a relation is found $\delta_{\gamma}^{\rm6\,h}\sim1.32\delta_{\gamma}^{\rm24\,h}$ and $\delta_{\gamma}^{\rm48\,h}\sim0.87\delta_{\gamma}^{\rm24\,h}$ . A slight higher than our present result if we choose $\Delta T=6$ h and smaller while $\Delta T=48$ h is considered (see also Fan et al. Reference Fan, Bastieri, Yang, Liu, Hua, Yuan and Wu2014). For the sake of simplicity, we adopt $\Delta T=24$ h in this paper for calculation as did in Dondi & Ghisellini (Reference Dondi and Ghisellini1995), Fan et al. (Reference Fan, Yang, Liu and Zhang2013a), and Fan et al. (Reference Fan, Bastieri, Yang, Liu, Hua, Yuan and Wu2014). Besides, the non-simultaneous observations will also result in some discrepancies.

For comparison, we draw a half proportion dividing line that is parallel to the equality line in two panel (labelled in dashed pink), which means there are 50/50 distribution of sources on two sides. When we consider this line in the plot of $\log \delta_{\gamma}$ versus $\log \delta_{\textrm{L18}}$ , it corresponds to a variability timescale $\Delta T \sim$ 2 h. For the case of $\log \delta_{\gamma}$ versus $\log \delta_{\textrm{C18}}$ , we found $\Delta T \sim$ 1.5 h. This could suggest that the on average, variability timescale for Fermi-detected blazars is around 1.5 $\,\sim\,$ 2 h; however, we cannot reach firm conclusions.

Based on the SSC limit, a so-called ‘classical’ method to estimate the Doppler factor, Ghisellini et al. (Reference Ghisellini, Padovani, Celotti and Maraschi1993) constrained a relation between the $\delta$ derived from moving sphere and a continuous jet:

(10) \begin{equation}\delta_{3+\alpha}^{(4+2\alpha)/(3+2\alpha)}=\delta_{2+\alpha}.\end{equation}

We also assume that the emission region from the jet is spherical, which means $p=3+\alpha$ has been taken into account. Note that, if the source is a continuous jet, $p=2+\alpha$ , then the equation (5) of derived $\delta_{\gamma}$ should be regenerated as:

(11) \begin{equation}\begin{aligned}\delta_{\gamma}\ge\left[{1.54\times10^{-3}\displaystyle\left(1+z\right)^{3+2\alpha}\left(\frac{d_{L}}{\rm Mpc}\right)^{2}\left(\frac{\Delta T}{\rm h}\right)^{-1}} \right. \\\left. \left(\frac{F_{\textrm{1\,keV}}}{\mu \rm Jy}\right){\left(\frac{E_{\gamma}}{\rm GeV}\right)^{\alpha}}\right]^{\frac{1}{3+2\alpha}}.\end{aligned}\end{equation}

For comparison, we also estimate the lower limit on $\gamma$ -ray Doppler factor for this case and obtain $\langle\delta_{\gamma}|_{\textrm{FSRQ}}\rangle=8.93\pm6.78$ and $\langle\delta_{\gamma}|_{\textrm{BL\,Lac}}\rangle=5.42\pm4.29$ with P-value of K–S test indicating the two distributions being from the same apparent distribution is $P=4.58\times10^{-35}$ . Our results show that $\log\delta_{\gamma}^{2+\alpha}\sim1.14\log\delta_{\gamma}^{3+\alpha}$ for FSRQs and $\log\delta_{\gamma}^{2+\alpha}\sim1.13\log\delta_{\gamma}^{3+\alpha}$ for BL Lacs. We can find the estimated value of $\delta_{\gamma}$ for the case of continuous jet to be larger than that of moving blob sphere, and FSRQs also have on average higher $\gamma$ -ray Doppler factor than BL Lacs.

From our calculation, FSRQs have on average significantly higher $\delta_{\gamma}$ than BL Lacs (P-value is $2.54\times10^{-27}$ from K–S test), which is consistent with estimations from other literatures (Fan et al. Reference Fan, Huang, He, Yang, Hua, Liu and Wang2009; Hovatta et al. Reference Hovatta, Valtaoja, Tornikoski and Lähteenmäki2009; Fan et al. Reference Fan, Yang, Liu and Zhang2013a, Reference Fan, Bastieri, Yang, Liu, Hua, Yuan and Wu2014; Liodakis et al. Reference Liodakis, Hovatta, Huppenkothen, Kiehlmann, Max-Moerbeck and Readhead2018). In the standard model of AGNs, FSRQs should have a smaller viewing angle $\theta$ with regard to BL Lacs, thus showing stronger Doppler-boosted effect and resulting in a larger $\delta$ due to the definition of Doppler factor $\delta=[\Gamma(1-\beta\cos\theta)]^{-1}$ . In the cross-checked sample with Liodakis et al. (Reference Liodakis, Hovatta, Huppenkothen, Kiehlmann, Max-Moerbeck and Readhead2018), 157 out of 285 sources were given the estimation of viewing angle, which included 127 FSRQs with average around $\langle\theta\rangle|_{\textrm{FSRQ}}\sim4.61$ deg and 30 BL Lacs with $\langle\theta\rangle|_{\textrm{BL\,Lac}}\sim10.20$ deg, leading FSRQs have higher $\delta$ than BL Lacs.

As for the weaker emission line features in BL Lacs is mainly ascribable to the fact that the isotropic emission component in BL Lacs is intrinsically weaker than in FSRQs and directs to different parent populations for the two subclasses of blazars (Madau, Ghisellini, & Persic Reference Madau, Ghisellini and Persic1987; Ghisellini et al. Reference Ghisellini, Padovani, Celotti and Maraschi1993). For example, Padovani (Reference Padovani1992) pointed out that [O $III$ ] line luminosity and radio-extended luminosity for BL Lacs are lower than for FSRQs with two orders of magnitude. It is possible that the different emission line property is from the factor that the ratio of the core emission to the extend one in the co-moving frame is higher in BL Lacs than in FSRQs (Fan Reference Fan2003).

4.2. Correlation analysis

The beaming model anticipates that more core-dominated sources should be more beamed and consequently have larger Doppler-boosted factors. Dondi & Ghisellini (Reference Dondi and Ghisellini1995) had probed the correlation between the core-dominance parameter $\log \, R$ and the $\gamma$ -ray Doppler factor $\log \, \delta_{\gamma}$ , a correlation coefficient $r=0.38$ and a chance probability of $P=8.0\times10^{-2}$ were found for 28 sources after excluding BL Lacs since their core-dominance parameters were quite large. The relation $R-\delta_{\gamma}$ existed but their sample is small. We also explore the correlation between $\log \, R$ and the derived $\log \, \delta_{\gamma}$ as the scatter plot in Figure 3. The best fitting for FSRQs is

\[\log R=(0.81\pm0.24)\log\delta_{\gamma}+(0.25\pm0.19),\]

with a correlation coefficient $r=0.22$ and a chance probability of $P<10^{-4}$ :

\[\log R=(0.59\pm0.24)\log\delta_{\gamma}+(0.35\pm0.14),\]

with $r=0.16$ and $P=0.01$ for BL Lacs. The $R-\delta_{\gamma}$ relation shows that a $\gamma$ -ray source with more core-dominated to be more Doppler-beamed.

Figure 3. Plot of the core-dominance parameter $\log \, R$ against the $\gamma$ -ray Doppler factor $\log \, \delta_{\gamma}$ for FSRQs (left panel), and BL Lacs (right panel).

Hovatta et al. (Reference Hovatta, Valtaoja, Tornikoski and Lähteenmäki2009) collected 80 sources with valid core-dominance parameters to investigate the correlation between $\log R$ and Doppler factor they derived and found a positive correlation with $r=0.37$ and $P=4.0\times10^{-4}$ . Pei et al. (Reference Pei, Fan, Bastieri, Yang and Xiao2020b) studied $R-\delta$ relation by cross-checking the Doppler factor from Liodakis et al. (Reference Liodakis, Hovatta, Huppenkothen, Kiehlmann, Max-Moerbeck and Readhead2018) and a significant correlation was also obtained. This demonstrates that the indication of sources with more core-dominated are also more boosted. Ghisellini et al. (Reference Ghisellini, Padovani, Celotti and Maraschi1993) expressed the core-dominance parameter as $R = f\delta^{3+\alpha}$ for the sources with the jets directly pointing to the observer. Therefore, the core-dominance parameter is statistically a good indicator for beaming effect (Urry & Padovani Reference Urry and Padovani1995; Fan Reference Fan2003; Pei et al. Reference Pei2016, Reference Pei, Fan, Bastieri, Sawangwit and Yang2019).

Since the emission from the jet is strongly boosted by a factor $\delta^{p}$ , thus one can assume that the $\gamma$ -ray luminosity should be proportional to the $\gamma$ -ray Doppler factor, that is, $L_{\gamma}\sim\delta_{\gamma}^{3+\alpha_{\gamma}}$ for stationary continuous jets or $L_{\gamma}\sim\delta_{\gamma}^{4+\alpha_{\gamma}}$ for moving sphere blob (Fan et al. Reference Fan, Yang, Zhang, Hua, Liu, Qin and Huang2013b). We probe this relation in Figure 4. Within the case for continuous jets, we ascertain that the best fittings are

\[\log L_{\gamma}({\rm erg\,s^{-1}})=(0.71\pm0.03)\log\delta_{\gamma}^{3+\alpha_{\gamma}}+(43.91\pm0.13),\]

with $r=0.75$ and $P\sim0$ for FSRQ and

\[\log L_{\gamma}({\rm erg\,s^{-1}})=(0.75\pm0.02)\log\delta_{\gamma}^{3+\alpha_{\gamma}}+(43.38\pm0.07),\]

with $r=0.82$ and $P\sim0$ for BL Lacs; besides, for a moving sphere,

\[\log L_{\gamma}({\rm erg\,s^{-1}})=(0.62\pm0.03)\log\delta_{\gamma}^{4+\alpha_{\gamma}}+(43.96\pm0.12),\]

with $r=0.75$ and $P\sim0$ for FSRQ and

\[\log L_{\gamma}({\rm erg\,s^{-1}})=(0.68\pm0.02)\log\delta_{\gamma}^{4+\alpha_{\gamma}}+(43.40\pm0.07),\]

with $r=0.82$ and $P\sim0$ for BL Lacs are also obtained. The values of $\gamma$ -ray Doppler factor in $\delta_{\gamma}^{3+\alpha_{\gamma}}$ and $\delta_{\gamma}^{4+\alpha_{\gamma}}$ from the relation above are derived by equation (11) and (5), respectively. We state parenthetically that although our derived $\delta_{\gamma}$ here is a lower limit, the above scaling relations should be valid for the cases when the estimated $\gamma$ -ray Doppler factor $\ge$ this lower limit and also different variability timescales are taken into account.

Figure 4. Plot of the correlation of $\log L_{\gamma}$ against $\log\delta_{\gamma}^{3+\alpha_{\gamma}}$ and $\log\delta_{\gamma}^{4+\alpha_{\gamma}}$ . The best-fit relations of FSRQs signify that $\log L_{\gamma}=(0.71\pm0.03)\log\delta_{\gamma}^{3+\alpha_{\gamma}}+(43.91\pm0.13)$ and $\log L_{\gamma}=(0.62\pm0.02)\log\delta_{\gamma}^{4+\alpha_{\gamma}}+(43.96\pm0.12)$ (left panel). On the other hand, for BL Lacs, $\log L_{\gamma}=(0.75\pm0.02)\log\delta_{\gamma}^{3+\alpha_{\gamma}}+(43.38\pm0.07)$ and $\log L_{\gamma}=(0.68\pm0.02)\log\delta_{\gamma}^{4+\alpha_{\gamma}}+(43.40\pm0.07)$ (right panel), respectively.

The tight relations verify that $L_{\gamma}\sim\delta_{\gamma}^{3+\alpha_{\gamma}}$ or $\sim\delta_{\gamma}^{4+\alpha_{\gamma}}$ . The high-energy $\gamma$ -rays detected from blazars imply that the presence of the beaming effect in those sources, otherwise, the $\gamma$ -rays should have been absorbed due to pair production on collision with the lower-energy photons populating the region (Mattox et al. Reference Mattox1993). Thus, from the point of view of the present considerations and our correlation analysis, the $\gamma$ -ray emission is strongly beamed. It also manifests that we can utilise $\delta_{\gamma}$ to flourish our understanding of Doppler beaming effect. In addition, we can see that the slope corresponding to the continuous is 0.71 and 0.75 for FSRQs and BL Lacs, respectively. This provides hints that the jet is possibly continuous in the $\gamma$ -ray bands.

4.3. More implications

The assumption we take for the estimation of $\delta_{\gamma}$ , primarily based upon that the X-ray emission is originated from the same region as $\gamma$ -ray emission, and the intensities of X-ray and $\gamma$ -ray are similar when $\gamma$ -ray is observed. We do not have direct evidence that the $\gamma$ -ray and the X-ray emission regions are cospatial is true in general but have indirect evidence for some sources. For example, Prince (Reference Prince2020) presented a multiwavelength temporal and spectral analysis of flares of 3C 279 during the period from 2017 November to 2018 July, and three bright $\gamma$ -ray flares were observed simultaneously in the X-ray and optical/UV band. A so-called ‘harder-when-brighter’ tendency has been observed in both $\gamma$ -rays and X-rays during the flaring period. The cross-correlation study of the emission from different wavebands has been performed and shown a strong correlation between them without any time lags. This is the first time that 3C 279 has shown a strong correlation between $\gamma$ -ray and X-ray emission with zero time lag, indicating that their origins are cospatial. Besides, Larionov et al. (Reference Larionov2020) also reported the results of multiwavelength decade-long monitoring (starting from 2008 and ending at 2018) of 3C 279. The $\gamma$ -ray and X-ray light curves correlate fairly with no delay $\gtrsim$ 3 h, suggesting the cospatiality of the $\gamma$ -ray and X-ray emission regions. Recently, Vercellone et al. (Reference Vercellone2019) studied a high-redshift blazar 4C + 71.07 from the multiwavelength simultaneous observations, and one of those prominent results substantiated that the $\gamma$ -ray emission and X-ray emission are perhaps produced within the same region.

The blazar TXS 0506 + 056, that is, 4FGL J0509.4 + 0542 ( $z=0.3365$ , Paiano et al. Reference Paiano, Falomo, Treves and Scarpa2018), the first detected neutrino emitter, was reported by the IceCube Neutrino Observatory on 2017 September 22 (IceCube Collaboration et al. 2018a,b), becoming a target of multiwavelength monitoring due to the detection of a 290-TeV muon-track neutrino event IceCube-170922A coincident with its direction and arrival time during a $\gamma$ -ray flare, opening up the possibility of an association between blazars and very high energy neutrinos, and providing the evidence for the blazar jet acting as an accelerator of cosmic ray particles. Liodakis et al. (Reference Liodakis, Hovatta, Huppenkothen, Kiehlmann, Max-Moerbeck and Readhead2018) obtained the Doppler factor for TXS 0506 + 056 is 14.67 and Chen (Reference Chen2018) reported 14.3. The above two values are quite close to the averages of their sample. However, Li et al. (Reference Li, An, Mohan and Giroletti2020) has reported on the radio properties of the parsec-scale jet in TXS 0506 + 056 derived from the analysis of multiepoch and multifrequency archive very long baseline interferometry (VLBI) data. They obtained $\delta=2.65$ . We derived that the $\gamma$ -ray Doppler factor for this source is 4.25 in this paper, with adopting $F_{\textrm{1\,keV}}=0.042$ $\mu$ Jy, $E_{\gamma}=4.21$ GeV, and $\alpha_{X}=1.89$ . Our result is lower than that from Chen (Reference Chen2018) or Liodakis et al. (Reference Liodakis, Hovatta, Huppenkothen, Kiehlmann, Max-Moerbeck and Readhead2018) but higher than that from Li et al. (Reference Li, An, Mohan and Giroletti2020). Considering the core-dominance parameter of this source to be $\log R=-0.13$ (Pei et al. Reference Pei, Fan, Bastieri, Yang and Xiao2020b) and also take into account the fact that the estimated values of Doppler factor in our work and from other literature are not large, it suggests that a moderately relativistic jet originated in TXS 0506 + 056.

Another neutrino event IceCube-190730A was reported to be in spatial coincidence with the bright $\gamma$ -ray FSRQ PKS 1502 + 106, that is, 4FGL J1504.4 + 1029, and the neutrino was reported with a signalness of 67% and an energy of 300 TeV (IceCube Collaboration et al. 2018a; Franckowiak et al. Reference Franckowiak2020). This source is also included in our present sample and $\delta_{\gamma}=13.41$ , which is much higher than the average value for our subsample of FSRQs. Chen (Reference Chen2018) and Liodakis et al. (Reference Liodakis, Hovatta, Huppenkothen, Kiehlmann, Max-Moerbeck and Readhead2018) obtained a quite high Doppler factor for this source as well, 23.8 and 13.77 were reported, respectively. It shows that this neutrino emitter is strongly Doppler-boosted. We calculated the $\gamma$ -ray luminosity of $\log L_{\gamma}=48.49$ erg $\text{s}^{-1}$ , which indicates that PKS 1502 + 106 is a very bright source in the 4FGL catalogue in terms of $\gamma$ -ray energy flux at >100 MeV, even though it has a large redshift of 1.838 (Hewett & Wild Reference Hewett and Wild2010), indicating an extremely high intrinsic luminosity.

From the multiwavelength characterisation of SED, Righi, Tavecchio, & Pacciani (Reference Righi, Tavecchio and Pacciani2019) studied some neutrino blazars candidates with their physical composition and the emission processes, in which a HSP source RXJ1022.7 $-$ 0112, that is, 4FGL J1022.7 $-$ 0112 is also listed in our sample. We report an extremely high lower limit on $\gamma$ -ray Doppler factor for this source of $\delta_{\gamma}=21.22$ . For another HSP source listing in both papers, TXS 0628 $-$ 240, that is, 4FGL J0630.9 $-$ 2406, we obtained $\delta_{\gamma}=10.43$ . Chen (Reference Chen2018) also reported a very high Doppler factor for this neutrino candidate of 51.

In addition, we found some more blazars candidates of neutrino emitters in our present sample after cross-checking with the literature. For example, $\delta_{\gamma}=11.40$ for PKS B1424 $-$ 418, that is. 4FGL J1427.9 $-$ 4206, which was in coincidence with the arrival time of a PeV neutrino (Kadler et al. Reference Kadler2016; Franckowiak et al. Reference Franckowiak2020); $\delta_{\gamma}=3.67$ for a distant BL Lac object MG3 J225517 + 2409, that is, 4FGL J2255.2 + 2411 with a redshift of 1.37 (Franckowiak et al. Reference Franckowiak2020; Abdollahi et al. Reference Abdollahi2020); $\delta_{\gamma}=6.48$ for FSRQ S4 1716 + 68, that is, 4FGL J1716.1 + 6836, which the duration of the neutrino flare is short (Franckowiak et al. Reference Franckowiak2020); $\delta_{\gamma}=6.71$ for OJ 508, that is, 4FGL J1022.7 $-$ 0112 with statistically significant potential for neutrino emission (Smith, Hooper, & Vieregg Reference Smith, Hooper and Vieregg2020); $\delta_{\gamma}=5.19$ for HSP PMNJ0953 $-$ 0840, that is, 4FGL J0953.0 $-$ 0840 (Righi et al. Reference Righi, Tavecchio and Pacciani2019). Therefore, from our comparison, it reaches to an interesting finding that the lower limit on $\gamma$ -ray Doppler factor for these neutrino emitter candidates are relatively quite high, suggesting that these sources are also possibly strongly Doppler-boosted.

For the well-studied FSRQ 3C 279 (i.e. 4FGL J1256.1 $-$ 0547), $\delta_{\gamma}=7.21$ are ascertained in this paper. Fan et al. (Reference Fan, Yang, Liu and Zhang2013a, Reference Fan, Bastieri, Yang, Liu, Hua, Yuan and Wu2014) also calculated the $\gamma$ -ray Doppler factor for this source with adopting the data from 2FGL and obtained 5.99 and 5.62, respectively, both by setting $\Delta T$ = 1 d. Our estimation here is higher than the results from Fan et al. (Reference Fan, Yang, Zhang, Hua, Liu, Qin and Huang2013b) or Fan et al. (Reference Fan, Bastieri, Yang, Liu, Hua, Yuan and Wu2014), since some input parameters for calculation are quite different from there, $F_{\textrm{1\,keV}}=4.338$ $\mu$ Jy, $E_{\gamma}=3.13$ GeV, and $\alpha_{X}=0.84$ are adopted. Fan et al. (Reference Fan, Yang, Zhang, Hua, Liu, Qin and Huang2013b) and Fan et al. (Reference Fan, Bastieri, Yang, Liu, Hua, Yuan and Wu2014) both reported $E_{\gamma}=3.54$ which are similar with this paper but quite low $F_{\textrm{1\,keV}}$ were used, 1.340 $\mu$ Jy and 0.961 $\mu$ Jy presented in Fan et al. (Reference Fan, Yang, Zhang, Hua, Liu, Qin and Huang2013b) and Fan et al. (Reference Fan, Bastieri, Yang, Liu, Hua, Yuan and Wu2014), respectively, and thus led their low estimation of $\delta_{\gamma}$ . From constraining the brightness temperature, Liodakis et al. (Reference Liodakis, Hovatta, Huppenkothen, Kiehlmann, Max-Moerbeck and Readhead2018) found the Doppler factor of 11.64 for 3C 279, this value is lower than the average value of their sample. Chen (Reference Chen2018) obtained $\delta=27.7$ that is higher than their average from SED modelling. Our derived result is also higher than our average value ( $\langle\delta_{\gamma}\rangle\sim6.87$ for FSRQ). This could possibly suggest that the $\gamma$ -ray and radio regions share the same relativistic effects.

The connection between $\gamma$ -ray and parsec-scale radio flares has been claimed by previous studies (e.g. Pushkarev, Kovalev, & Lister Reference Pushkarev, Kovalev and Lister2010; Agudo et al. Reference Agudo2011a; Jorstad et al. Reference Jorstad2013). However, the location of $\gamma$ -ray emission originating is still debated. Two scenarios have been proposed: on subparsec scales or on a few parsec scales. The former scenario indicates that the $\gamma$ -ray emission zone is located close to the true base of the jet and the black hole (e.g. Pushkarev et al. Reference Pushkarev, Kovalev and Lister2010; Tavecchio et al. Reference Tavecchio, Ghisellini, Bonnoli and Ghirlanda2010; Rani et al. Reference Rani, Krichbaum, Marscher, Jorstad, Hodgson, Fuhrmann and Zensus2014). The latter one implies that $\gamma$ -rays are produced from many parsecs away from the jet apex (e.g. Marscher et al. Reference Marscher2010; Agudo et al. Reference Agudo2011b; Schinzel et al. Reference Schinzel, Lobanov, Taylor, Jorstad, Marscher and Zensus2012). If a $\gamma$ -ray event occurs in a part of a relativistic jet, that is, optically thin for radio emission, then radio band variability is expected to be simultaneous at different radio frequencies and be simultaneous or even precede the $\gamma$ -ray variability (e.g. Ghisellini & Tavecchio Reference Ghisellini and Tavecchio2008; Agudo et al. Reference Agudo2011a; Jorstad et al. Reference Jorstad2013).

Lisakov et al. (Reference Lisakov, Kovalev, Savolainen, Hovatta and Kutkin2017) performed a comprehensive VLBA study of the blazar 3C 273 associated with a strong $\gamma$ -ray flare and found that the $\gamma$ -ray emission region in 3C 273 is close to the true base of the jet, located $2-7$ pc upstream from the 7-mm radio core, implying that the location of $\gamma$ -ray emission and radio emission is in the same jet in 3C 273 but different sites.

Relativistic beaming in blazars is evidenced by the rapid variability property from radio to $\gamma$ -ray wavelengths (Aller & Aller Reference Aller and Aller1996; Lähteenmäki & Valtaoja Reference Lähteenmäki and Valtaoja2003; Kovalev Reference Kovalev2009), and it is originated from within the AGN jet and strongly beamed along the line of sight. This yields an interesting fact that all $\gamma$ -ray-loud AGNs are also radio-loud, while not all radio-loud AGNs are $\gamma$ -ray loud. This could be explained by two ways: the $\gamma$ -ray emission is in a Doppler-boosted state inside a narrow conoid if the $\gamma$ -ray-emitting regions are moving faster than the radio-emitting regions (Salamon & Stecker Reference Salamon and Stecker1994). Then outside the $\gamma$ -ray beaming conoid and within the radio beaming conoid, the $\gamma$ -ray radiation would be in a Doppler dimmed state, displaying that $\gamma$ -ray-quiet but radio-loud AGNs.

Another scenario for explicating this fact should be related to the AGN jet bending (von Montigny et al. Reference von Montigny1995). Figure 2 shows that around one-third (left panel) and one-fourth of sources (right panel) locate at the upper left of the equality line, implying that the derived lower limit on $\delta_{\gamma}$ for those sources are higher than that from the radio estimation. This could be possibly explained by the jet bending within those blazars. If a bend in the jet takes place at upstream of the $\gamma$ -ray emission region and downstream of extended radio-emitting region, then the radiation would be Doppler-boosted in alignment to their respective sections of the jet. If so, the $\gamma$ -ray emission is beamed towards our line of sight (resulting in a high $\delta_{\gamma}$ ), while the radio emission is beamed away from us (resulting in a low $\delta_{\gamma}$ ) ascribable to a better alignment of the $\gamma$ -ray-emitting jet section with our line of sight. In this case, we would probably detect a $\gamma$ -ray-loud and radio-loud AGNs.