ΔΔ intermediate state in1S0NNscattering from effective field theory

We examine the role of the ΔΔ intermediate state in $\mathrm{NN}$ scattering in the ${}^1{S}_0$ channel. The computation is performed at lowest order in an effective-field theory involving local four-fermion operators and one-pion exchange using dimensional regularization with minimal subtraction ($\overline{\mathrm{MS}}$). As first discussed by Weinberg, in the theory with only nucleons, the large-scattering length in this channel requires a small scale for the local $N^4$ operators. When Δ's are included (but without pions) a large-scattering length can be obtained from operators with a scale $\sqrt{{2M}_N{(M}_{\Delta }-M_N)},$ but fine-tuning is required. The coefficients of the contact terms involving the Δ fields are not uniquely determined but for reasonable values one finds that, in general, $\mathrm{NN}$ scattering computed in the theory with Δ's looks like that computed in the theory without Δ's. The leading effect of the Δ's is to change the coefficients of the four-nucleon contact terms between the theories with and without Δ's. Further, the decoupling of the Δ's in the limit of large mass and strong coupling is clearly demonstrated. When pions are included, the typical scale for the contact terms is $\sim 100\mathrm{MeV},$ both with and without Δ's and is not set by $\sqrt{{2M}_N{(M}_{\Delta }-M_N)}.$ For reasonable values of contact terms that reproduce the scattering length and effective range (at lowest order) the phase shift is not well reproduced over a larger momentum range as is found in the theory without Δ's at lowest order.