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The Role of $Δ(1232)$ in Two-pion Exchange Three-nucleon Potential

In this paper we have studied the two-pion exchange three-nucleon potential $(2πE-3NP)$ using an approximate $SU(2) \times SU(2)$ chiral symmetry of the strong interaction. The off-shell pion-nucleon scattering amplitudes obtained from the Weinberg Lagangian are supplemented with contributions from the well-known $σ$-term and the $Δ(1232)$ exchange. It is the role of the $Δ$-resonance in $2πE-3NP$, which we have investigated in detail in the framework of the Lagrangian field theory. The $Δ$-contribution is quite appreciable and, more significantly, it is dependent on a parameter Z which is arbitrary but has the empirical bounds $|Z| \leq 1/2$. We find that the $Δ$-contribution to the important parameters of the $2πE-3NP$ depends on the choice of a value for Z, although the correction to the binding energy of triton is not expected to be very sensitive to the variation of Z within its bounds.