  This paper considers the single antenna, static Gaussian broadcast channel in
the finite blocklength regime. Second order achievable and converse rate
regions are presented. Both a global reliability requirement and per-user
reliability requirements are considered. The two-user case is analyzed in
detail, and generalizations to the $K$-user case are also discussed. The
largest second order achievable region presented here requires both
superposition and rate splitting in the code construction, as opposed to the
(infinite blocklength, first order) capacity region which does not require rate
splitting. Indeed, the finite blocklength penalty causes superposition alone to
under-perform other coding techniques in some parts of the region. In the
two-user case with per-user reliability requirements, the capacity achieving
superposition coding order (with the codeword of the user with the smallest SNR
as cloud center) does not necessarily gives the largest second order region.
Instead, the message of the user with the smallest point-to-point second order
capacity should be encoded in the cloud center in order to obtain the largest
second order region for the proposed scheme.
