Anomaly-induced Quadrupole Moment of the Neutron in Magnetic Field
aa r X i v : . [ h e p - ph ] M a y Anomaly-induced Quadrupole Momentof the Neutron in Magnetic Field
Dmitri E. Kharzeev,
1, 2
Ho-Ung Yee, and Ismail Zahed Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA Department of Physics, Brookhaven National Laboratory, Upton, New York 11973-5000, USA (Dated: November 2, 2018)The neutrons cannot possess a quadrupole moment in the vacuum. Nevertheless, we show thatin the presence of an external magnetic field the neutrons acquire a new type of quadrupole mo-ment Q ij = χ σ i B j involving the components of spin and magnetic field. This “chiral magnetic”quadrupole moment arises from the interplay of the chiral anomaly and the magnetic field; we esti-mate its value for the neutron in the static limit, and find χ ≃ . · − fm . The detection of thequadrupole moment of the neutron would provide a novel test of the role of the chiral anomaly inlow-energy QCD and can be possible in the presence of both magnetic and inhomogeneous electricfields. The quadrupole moment of the neutron may affect e.g. the properties of neutron stars andmagnetars. PACS numbers: 12.39.Fe,12.38.Qk,13.40.Em Quantum anomalies are known to play an im-portant role in QCD. For example, the Abelian chiralanomaly [1, 2] accounts for the π → γγ decay and forceschiral Skyrmions to be baryons. In this short letter wewill explore the effect of the chiral anomaly on the defor-mation of the neutron in an external magnetic field.The electric dipole moment of the neutron has at-tracted a lot of attention as it allows to put a strictbound on the amount of CP violation in QCD and be-yond, see [3] for a recent result. The electric quadrupolemoment defined as the expectation value of the opera-tor ˆ Q ijE = P e (3 x i x j − δ ij x ) is not possible for a spin1 / Q ijE can contain only the components of the angular momen-tum of the system, and for a spin 1 / Q ijM ∼ B i B j (see e.g. [5] for the case of hydrogen atom).We will now argue that the chiral anomaly induces anew type of quadrupole deformation of the nucleons inthe presence of a magnetic field - a “chiral magnetic”quadrupole moment Q ij ∼ σ i B j that involves both thecomponents of spin and magnetic field. Our analysisis motivated in part by a recent study [6] of the ef-fects of chiral anomaly on the electric charge distribu-tion inside the nucleons, and in part by the finding of theanomaly-induced quadrupole deformation of the quark-gluon plasma in a magnetic field [7]. In QCD the Abelian anomaly is given by [1, 2] L = N c e π π f π ~E · ~B, (1)where π is the neutral pion field, f π is the pion decayconstant, and N c is the number of colors. The corre- sponding effective Hamiltonian can be obtained by tak-ing the expectation value of (1) over a neutron state inthe static limit: H = N c e π ×× lim p ′ → p Z d x ~E ( ~x ) · ~B (cid:28) N ( p ) (cid:12)(cid:12)(cid:12)(cid:12) π ( ~x ) f π (cid:12)(cid:12)(cid:12)(cid:12) N ( p ′ ) (cid:29) . (2)In this limit, the pion field is sourced by the neutronthrough (cid:0) ∇ − m π (cid:1) π ( x ) = − ig πNN Nγ τ N ( x ) . (3)Inserting the static solution to (3) in (2) and using thenon-relativistic reduction for the neutron wave functionyields H = − N c e π g πNN M N f π m π ×× Z d x ( ∇ i E j ) B j N + σ i τ N. (4)It is easy to see that the Hamiltonian (4) gives rise to theneutron’s chiral magnetic quadrupole moment Q ij : Q ij = δH δ ∇ i E j = − N c e π g πNN M N f π m π N + σ i τ N B j (5)which can be re-written as Q ij = − N c α π g A ( f π m π ) N + σ i τ N B j (6)after the use of the Goldberger-Treiman relation. Notethat the chiral magnetic quadrupole moment involvesthe correlation between different components of spin andmagnetic field, and is thus different from the “conven-tional” electric and magnetic quadrupole moments.Numerically, using g A = 1 . f π = 93 MeV and m π =135 MeV, we get for the chiral magnetic quadrupole mo-ment of the neutron from (6) Q ij ≈ (cid:0) . · − fm (cid:1) σ i B j . (7) The chiral corrections to (7) mostly contribute tothe pion nucleon coupling g πNN or to the axial charge g A and renormalize them to their physical values. The non-static corrections to the pion field are not expected to beimportant for the static quadrupole moment. The quan-tum chiral loops would induce higher-order contributionsto the induced quadrupole moment; however these cor-rections appear to be numerically small (see below). If aprecise measurement of the chiral magnetic quadrupolemoment is possible, these corrections would have to beevaluated. Such a measurement would open a possibilityof a precision determination of the nucleon axial charge g A . It is evident from the arguments presented abovethat the Abelian chiral anomaly affects solely the mag-netically induced neutron quadrupole moment and notits electric charge. Measuring this quadrupole moment isof fundamental interest and would provide a novel test ofthe role of the chiral anomaly in low energy QCD. Suchmeasurements would require the presence of very strongmagnetic and inhomogeneous electric fields, e.g. the fieldof an intense laser. The operator (6) also appears in the low energyexpansion of the amplitude of polarized Compton scat-tering at order ω ( ω is the energy of the photon)[16], seee.g. [8–10]. Our evaluation of the quadrupole moment Q ij corresponds to the pion pole part of the ∼ σ i B j termin the spin-polarizability of the neutron. The pole contri-bution dominates over the higher order chiral loops and∆ π contributions [11, 12]. In the case of a static externalmagnetic field that we consider in this note, the effect ofthis term can be easily understood. Indeed, an externalmagnetic field ~B due to the interaction with magneticmoment of the neutron will align its spin so that ~σ k ~B .In contrast, the quadrupole moment ∼ σ i B j would tendto create a misalignment of the spin and magnetic field,causing the precession of the neutron’s spin around thedirection of magnetic field.It is now established that neutron stars and magne-tars can develop very strong magnetic fields eB ∼ G [13]. We speculate that if the neutron stars containa P anisotropic neutron superfluid (see [14] and refer-ences therein), the precession of the neutron spins may translate into a macroscopic precession of the angularmomentum of the star. Indeed, in the presence of astrong magnetic field ~B the angular momentum l = 1locked to the spin S = 1 of the Cooper pair would precessaround the axis of ~B . Since the Cooper pairs in a super-fluid form a Bose condensate and are in the same quan-tum state, this precession would translate into a macro-scopic precession of the system. This effect is similar tothe coherent enhancement of microscopic parity violationin anisotropic superfluids predicted by Leggett [15].We thank Martin Savage and Brian Tiburzi for bring-ing our attention to spin polarizabilities in Compton scat-tering, and to Yannis Semertzidis for the discussion ofthe prospects for experimental detection. 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