  We study motion of tagged particles in a harmonic chain of active particles.
We consider three models of active particle dynamics - run and tumble particle,
active Ornstein-Uhlenbeck particle and active Brownian particle. We investigate
the variance, auto correlation, covariance and unequal time cross correlation
of two tagged particles. For all three models, we observe that the mean squared
displacement undergoes a crossover from the super-diffusive $\sim t^{\mu}$
scaling for $t \ll \tau_A$ ($\tau_A$ being the time scale arising due to the
activity) to the sub-diffusive $\sim \sqrt{t}$ scaling for $t \gg \tau_A$,
where $\mu = \frac{3}{2}$ for RTP, $\mu = \frac{5}{2}$ for AOUP. For the $x$
and $y$-coordinates of ABPs we get $\mu=\frac{7}{2}$ and $\mu=\frac{5}{2}$
respectively. We show that these crossover behaviours in each case can be
described by appropriate crossover function that connects these two scaling
regimes. We compute these crossover functions explicitly. In addition, we also
find that the equal and unequal time auto and cross correlations obey
interesting scaling forms in the appropriate limits of the observation time
$t$. The associated scaling functions are rigorously derived in all cases. All
our analytical results are well supported by the numerical simulations.
