Nucleon Magnetic Moments and Electric Polarizabilities Page: 3 of 7
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Nucleon moments and polarizabilities
in the effective action are constrained by the underlying symmetries: gauge invariance, Lorentz
invariance, W, iY, and J. The leading operator is the magnetic moment term, NugvFvN, and
the corresponding effective Hamiltonian is given by: H = -pnK -F, where the matrices K are
generators of boosts in the spin-1/2 representation of the Lorentz group. Operators at the next
order contain the s-wave and d-wave couplings of the neutron to two photons, NNFvFPV, and
i(N ydyN- ayNyN)FPPFpV}. While the latter term would naively be supressed by a power of
the neutron mass, a field redefinition is required to arrive at a canonically normalized kinetic term,
and both operators end up being of the same order. A linear combination of these two operators
gives rise to the second-order effective Hamiltonian: H = -4aEE2.
Both effective interactions, H1 and H(2), give rise to electric dipole moments (EDMs). The
magnetic moment interaction generates a motional EDM [5], d i) = pn x v, where v is a small
neutron velocity. The interaction energy of the dipole and the electric field is merely -P) - = -
-p - B, where B = v x E is the magnetic field in the neutron's rest frame. The induced EDM,
J(2) = -aEF, is proportional to the strength of the applied field with a coefficient that is the
electric polarizability.
The electric polarizability lowers the neutron's energy. For a neutron at rest, the energy shift is
merely AE = - aEE2. Less obvious, however, is the effect of the magnetic moment. The leading- t
order contribution clearly vanishes for a neutron at rest. One must treat the magnetic moment t
operator to second order to account for all terms in the neutron energy at 6(E2). For a small
velocity v, the second order contribution has the form
1n(v x E) Pn(i x )(2.1)
and survives the v -> 0 limit. In this way, one avoids the neutron pole. Beyond this schematic
discussion, one can make the result exact by calculating the neutron propagator to all orders in H
and H . The result is a neutron propagator with an energy shift of the form
1 t p 2
AE=-- IaE- iF. (2.2)
2 \MUJ
As a consequence, studying the electric-field dependence of unpolarized neutron correlation func-
tions is not enough to determine the electric polarizability without knowledge of the magnetic
moment.
In order to determine aE from two-point functions, one must perform the analogue of a Born
subtraction. For the case of a magnetic field, the magnetic moment can be isolated by considering
spin-projected correlation functions. One way to access magnetic moments in an electric field is, by
analogy, to use boost-projected correlation functions. We use this terminology to refer to the spinor
structure; the neutron remains at rest throughout. With projection matrices, _iY0+ = j(1 K3), we
observe that the boost-projected two-point functions have the form
Tr [ +G(t)] = Z(1 pnE)e-it(M1+AE), (2.3)
for the electric field F = Ez, with the energy shift AE given in Eq. (2.2). Thus a simultaneous mea-
surement of both boost-projected correlation functions will allow one to determine the magnetic3
B. C. Tiburzi
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Detmold, W. & Walker-Loud, A. Nucleon Magnetic Moments and Electric Polarizabilities, article, June 1, 2010; Newport News, Virginia. (https://digital.library.unt.edu/ark:/67531/metadc832863/m1/3/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.