Manifest orthogonality of dispersive-bound contributions in the proposed parametrisation

Establish a representation of the dispersive bound for the hadronic form factor parametrisation F(q^2) = [W(z)/phi_F(z)] · f(z)/([z − z_r][z − z_r^*]) with z = z(q^2) that maintains manifest orthogonality of the contributions to the saturation; specifically, derive the bound explicitly as a sum of positive-definite terms so that the saturation can be exhibited in a manifestly orthogonal form.

Background

Dispersive-bounded z-expansions of hadronic form factors traditionally lead to bounds expressed as sums of squared coefficients due to orthogonality on the unit circle, yielding a manifestly positive-definite saturation (e.g., ∑|a_n|^2 < 1).

The paper proposes an extension to cover the timelike region by including a weight function W(z) to control the outer function’s behaviour near z → ±1 and explicit factors for above-threshold poles ([z − z_r] [z − z_r^*]). In this extended ansatz, the authors note that they cannot maintain a manifestly orthogonal decomposition of the contributions to the bound, so the saturation is not transparently a sum of positive-definite quantities.

Clarifying or restoring manifest orthogonality within this framework would strengthen theoretical control of the bound and its interpretation, aligning the extended parametrisation with the traditional orthogonal structure of dispersive bounds.