[@a11] \_[m\_1m\_2]{}\^[\_h]{} \~(-t\_1)\^[|m\_1|/2]{} (-t\_2)\^[|m\_2|/2]{}, m\_1+m\_2 = \_h. \[eq:mouse\] Note that all values of $m_2$ ($m_1$) enter [(\[eq:jackal\])]{}, but, due to [(\[eq:mouse\])]{}, for small $t_1,t_2$ it is enough to consider the lowest values of $m_2$ ($m_1$) consistent with [(\[eq:koala\])]{}.

It is often convenient to write the spin structure of the amplitudes in terms of the characteristic 3-vectors of the problem. Such a representation is closely related to the helicity amplitudes discussed above [@a13], but the formulas become more transparent. In this case the central vertex $g_{ik}^h$ is written as a scalar (or pseudoscalar) function (depending on the product $\eta_h(-1)^{S_h}\sigma_iP_i\sigma_kP_k$) of the vectors $\vec
p_{3\perp},\vec p_{4\perp}$ and the spin vectors (tensors) of particle $h$.

Let us consider particular examples for the spin-parity $J^P$ of $h$, in each case taking$\sigma_iP_i\sigma_kP_k = +1 $ as for double-Pomeron exchange.

-   $J^P(h) = 0^+$

    For a scalar particle $h$, the vertex coupling is simply g\_[ik]{}\^h = f\_[0\^+]{}(p\_[3]{}\^2, p\_[4]{}\^2, p\_[3]{}p\_[4]{}), \[eq:nightingale\] where $f_{0^+}$ is a function of the scalar variables which can be formed from the transverse momenta $\vec
    p_{3\perp}$ and $£\vec p_{4\perp}$ of the outgoing protons. When $\vec p_{3\perp}$ or $\vec p_{4\perp}\to 0$, this function in general tends to some constant $f(0,0,0)$. In order to obtain 