  In this paper, we consider the existence and asymptotic behavior on mass of
the positive solutions to the following system:
\begin{equation}\label{eqA0.1}\nonumber \begin{cases} -\Delta
u+\lambda_1u=\mu_1u^3+\alpha_1|u|^{p-2}u+\beta v^2u\quad&\hbox{in}~\R^4,\\
-\Delta v+\lambda_2v=\mu_2v^3+\alpha_2|v|^{p-2}v+\beta
u^2v\quad&\hbox{in}~\R^4,\\ \end{cases} \end{equation} under the mass
constraint $$\int_{\R^4}u^2=a_1^2\quad\text{and}\quad\int_{\R^4}v^2=a_2^2,$$
where $a_1,a_2$ are prescribed, $\mu_1,\mu_2,\beta>0$; $\alpha_1,\alpha_2\in
\R$, $p\!\in\! (2,4)$ and $\lambda_1,\lambda_2\!\in\!\R$ appear as Lagrange
multipliers. Firstly, we establish a non-existence result for the repulsive
interaction case, i.e., $\alpha_i<0(i=1,2)$. Then turning to the case of
$\alpha_i>0 (i=1,2)$, if $2<p<3$, we show that the problem admits a ground
state and an excited state, which are characterized respectively by a local
minimizer and a mountain-pass critical point of the corresponding energy
functional. Moreover, we give a precise asymptotic behavior of these two
solutions as $(a_1,a_2)\to (0,0)$ and $a_1\sim a_2$. This seems to be the first
contribution regarding the multiplicity as well as the synchronized mass
collapse behavior of the normalized solutions to Schr\"{o}dinger systems with
Sobolev critical exponent. When $3\leq p<4$, we prove an existence as well as
non-existence ($p=3$) results of the ground states, which are characterized by
constrained mountain-pass critical points of the corresponding energy
functional. Furthermore, precise asymptotic behaviors of the ground states are
obtained when the masses of whose two components vanish and cluster to a upper
bound (or infinity), respectively.
