Write the superpotential so that the only combination of X and Φ2 that talks to Φ1 is manifest: W = 1/2 Φ1^2 (h1 X + h2 Φ2) + f X.

Make a linear, unitary change of variables in the X–Φ2 plane:

• S ≡ (h1 X + h2 Φ2)/h,
• T ≡ (−h2 X + h1 Φ2)/h, with h ≡ √(|h1|^2 + |h2|^2).

In this basis, W = 1/2 h S Φ1^2 + f (h1 S − h2 T)/h.

Two immediate consequences follow:

• SUSY breaking (tree level): Along the classical pseudomoduli space Φ1 = 0, one has F_S = f h1/h and F_T = f h2/h, so V = |f|^2 and SUSY is broken (since f ≠ 0) with two classically flat complex directions S and T.

• What one loop can and cannot do (without computing the Coleman–Weinberg potential explicitly): • Only the multiplet Φ1 has masses that depend on the background pseudomoduli, and those masses depend solely on S through m_f = h S and m_b^2 = |h S|^2 ± |h F_S|. Therefore any one-loop effective potential can depend only on S, not on T. Consequently, the T direction remains exactly flat to all orders in perturbation theory (it decouples from any interaction except for the linear term). • If h1 ≠ 0, then F_S ≠ 0 along the pseudomoduli space, so the Φ1 bosons are split relative to the Φ1 fermion. This necessarily generates an S-dependent Coleman–Weinberg potential (equivalently, an S-dependent one-loop correction to the Kähler metric Z_S(S†S), giving V_eff ≈ |F_S|^2/Z_S). Thus the S direction is lifted at one loop, just as in the standard O’Raifeartaigh model W = κ S + 1/2 λ S φ^2 with κ = f h1/h and λ = h.

Special case:

• If h1 = 0, then S = Φ2 and F_S = 0 on the pseudomoduli space; the Φ1 bosons and fermion are degenerate, so there is no S-dependent one-loop potential at all. In that limit the model is simply Polonyi (X with linear term) plus a decoupled free chiral Φ2 coupled quadratically to Φ1, and the entire pseudomoduli space is flat at one loop.

So, without computing the Coleman–Weinberg potential explicitly, a simple field redefinition shows:

• The orthogonal combination T is not lifted (it decouples).
• The combination S is generically lifted at one loop (unless h1 = 0). Therefore the statement that “one-loop corrections do not lift the classical pseudo-moduli space” is not true in general; it holds only in the special limit h1 = 0.