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Perturbative expansions, the Born series, of the scattering length and theamplitude of $S$-wave neutron-deuteron scattering for spin quartet channelbelow deuteron breakup threshold are studied in pionless effective field theoryat small cutoff values. A three-body contact interaction is introduced when theintegral equation is solved using the small cutoffs. After renormalizing thethree-body interaction by using the scattering length, we expand the integralequation as the ordinary and inverted Born series. We find that the scatteringlength and the phase shift are considerably well reproduced with a few terms ofthe inverted Born series at relatively large cutoff values, $\Lambda\simeq 100$MeV.

The Born series for $S$-wave quartet $nd$ scattering at small cutoff values