A calculational scheme for obtaining the electric polarizability of the neutron in lattice QCD with dynamical quarks is developed, using the background field approach. The scheme differs substantially from methods previously used in the quenched approximation, the physical reason being that the QCD ensemble is no longer independent of the external electromagnetic field in the dynamical quark case. One is led to compute (certain integrals over) four-point functions. Particular emphasis is also placed on the physical role of constant external gauge fields on a finite lattice; the presence of these fields complicates the extraction of polarizabilities, since it gives rise to an additional shift of the neutron mass unrelated to polarizability effects. The method is tested on a SU(3) flavor-symmetric ensemble furnished by the MILC Collaboration, corresponding to a pion mass of m_pi = 759 MeV. Disconnected diagrams are evaluated using stochastic estimation. A small negative electric polarizability of alpha =(-2.0 +/- 0.9) 10^(-4) fm^3 is found for the neutron at this rather large pion mass; this result does not seem implausible in view of the qualitative behavior of alpha as a function of m_pi suggested by Chiral Effective Theory.
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