In the SU(3) limit the masses of all the hadrons belonging to a given SU(3) multiplet have the same value. By direct product of monoparticle states of particles belonging to different SU(3) multiplets we have multiparticle scattering states. In particular, for the two-body interaction process involving such particles we would have only to distinguish between the common masses of the particles in the SU(3) representation involved. For instance, for the lightest pseudoscalar–pseudoscalar scattering we would have only one mass since all the particles belong to the same octet SU(3) representation. Other two-body states of interest for our purposes is the one made by a baryon and one of the lightest pseudoscalars, all of them belonging to octet representations.

It is clear that because of the Wigner–Eckart theorem the matrix element of SU(3) operators transforming within a given SU(3) multiplet between states belonging to definite representations are independent of the hypercharge and third component of isospin, that characterize the different states and operators in a given irreducible representation [46]. The T matrix is an SU(3) singlet and therefore the scattering matrix is diagonal in a basis of states with definite transformation properties under SU(3). Denoting these states by \(|R,\lambda \rangle \), with R corresponding to the SU(3) irreducible representation and \(\lambda \) including the other quantum numbers needed to distinguish between states within R (e.g., third component of isospin and hypercharge). The momenta and spin indices are not indicated in the following since they do not play any active role in the next considerations. We then have for the T matrix,

$$\begin{aligned} \langle R',\lambda '|T|R,\lambda \rangle&=T_R \delta _{RR'}\\&=\frac{1}{\mathcal{N}_{R}^{-1}+g_R(s)}\delta _{RR'}~. \nonumber \end{aligned}$$
(9.1)

Here we have used the general parameterization of Eq. (7.2) with the unitarity function \(g_R(s)\) containing the subtraction constant \(a_R\).

Let us further denote by \(|i\rangle \) the physical states in the charged basis and the Clebsch–Gordan coefficients connecting both bases by \(\langle i, R\lambda \rangle \). These real coefficients satisfy the orthogonality relations

$$\begin{aligned} \sum _i\langle i,R\lambda \rangle \langle i,R'\gamma \rangle&=\delta _{RR'}\delta _{\lambda \gamma }~,\\ \sum _{R,\lambda }\langle i,R\lambda \rangle \langle j, R \lambda \rangle&=\delta _{ij}~.\nonumber \end{aligned}$$
(9.2)

Notice that since \(\langle i, R\lambda \rangle =\langle i|R,\lambda \rangle \) and is real, then it also follows that \(\langle i, R\lambda \rangle =\langle R,\lambda |i\rangle \).

In the physical basis the T-matrix elements \(T_{ij}(s)\) also obeys Eq. (7.2), with \(\mathcal{N}(s)\) calculated in the charged basis and the functions \(g_i(s)\) involving the subtraction constants \(a_i\). Let us show that all the \(a_R\) and \(a_i\) have the same value in the SU(3) limit, as derived in Ref. [47]. For that we proceed with the change of basis of a singlet SU(3) matrix A, from the physical basis to the SU(3) one. Then,

$$\begin{aligned} \sum _{ij}\langle i,R\lambda \rangle A_{ij}\langle j, R'\gamma \rangle&=A_{R}\delta _{RR'} \delta _{\lambda \gamma }~. \end{aligned}$$
(9.3)

For instance, this is case for the QCD Hamiltonian, and therefore for the T matrix, as well as for the unitarity loop function \(g_i(s)\), Eq. (8.5). The latter function, contrary to the T matrix, is also diagonal in the physical basis. This is a key distinctive feature that allows us to perform the following manipulations. By inverting Eq. (9.3) with g(s) instead of A, we have that

$$\begin{aligned} g_{i}(s)\delta _{ij}&=\sum _{R,\lambda }\langle i,R\lambda \rangle g_R(s)\langle j, R\lambda \rangle ~. \end{aligned}$$
(9.4)

Next, we multiply by \(\langle j, R'\gamma \rangle \) and sum over j,

$$\begin{aligned} g_i(s) \langle i,R'\gamma \rangle&= \sum _{R,\lambda }\sum _j \langle i,R\lambda \rangle g_R(s) \underbrace{\langle j,R\lambda \rangle \langle j,R'\gamma \rangle }_{\delta _{RR'}\delta _{\lambda \gamma }} = \langle i, R'\gamma \rangle g_{R'}(s)~. \end{aligned}$$
(9.5)

From this relation it is sufficient to take physical states with components in different irreducible representation to conclude that for any state \(|i\rangle \) in the charged basis and for any irreducible SU(3) representation R involved in the decomposition of the charged basis in SU(3) multiplets, one has

$$\begin{aligned} g_i(s)&=g_R(s)=g(s)~. \end{aligned}$$
(9.6)

As a result it follows the equality in the SU(3) limit of all the subtraction constants for the two-particles states \(|AB\rangle \), with A and B belonging to the irreducible SU(3) representations \(R_A\) and \(R_B\), in order.

This result is manifestly evident when every subtraction constant \(a_i(\mu )\) is given by its natural value, Eq. (8.8), because then the masses \(m_1\) and \(m_2\) (as well as the three-momentum cut-off \(\Lambda \)) are common to all the two-particle states in the SU(3) limit.