  We study the massive scalar field Sorkin-Johnston (SJ) Wightman function
restricted to a flat 2D causal diamond of linear dimension L. Our approach is
two-pronged. In the first, we solve the central SJ eigenvalue problem
explicitly in the small mass regime, upto order (mL)^4. This allows us to
formally construct the SJ Wightman function up to this order. Using a
combination of analytic and numerical methods, we obtain expressions for the SJ
Wightman function both in the center and the corner of the diamond, to leading
order. We find that in the center, it is more like the massless Minkowski
Wightman function than the massive one, while in the corner it corresponds to
that of the massive mirror. In the second part, in order to explore larger
masses, we perform numerical simulations using a causal set approximated by a
flat 2D causal diamond. We find that in the center of the diamond the causal
set SJ Wightman function resembles the massless Minkowski Wightman function for
small masses, as in the continuum, but beyond a critical value it resembles the
massive Minkowski Wightman function as expected. Our calculations suggest that
unlike the massive Minkowski vacuum, the SJ vacuum has a well-defined massless
limit, which mimics the behavior of the Pauli Jordan function underlying the SJ
construction. In the corner of the diamond, moreover, it agrees with the mirror
vacuum for all masses, and not, as might be expected, with the Rindler vacuum.
