# Nucleon Polarisabilities from Deuteron Compton Scattering, and Its Lessons for Chiral Power Counting

\bodymatterPolarisabilities measure the global stiffness of the nucleon’s constituents
against displacement in an external electro-magnetic field. We examined them
in elastic deuteron Compton scattering for photon
energies between zero and in Chiral Effective Field Theory EFT with explicit degrees of freedom, see
Refs. [1, 2] for details and better
references. An excellent tool to identify the active low-energy degrees of
freedom are the *dynamical polarisabilities*, defined by a multipole
decomposition of the structure part of the Compton amplitude at fixed energy.
Unique signals allow one to study the temporal response of each constituent.

For example, the strong, para-magnetic -to- transition induces a strong energy-dependence which is pivotal to resolve the “SAL-puzzle”, see Fig. 1: While all previous analyses of the SAL-data at extracted vastly varying nucleon polarisabilities, EFT with an explicit captures correctly both normalisation and angular dependence of the data without altering the static (namely zero-energy) polarisabilities.

A consistent description must also give the correct Thomson limit, i.e. the exact low-energy theorem which is a consequence of gauge invariance. Its verification is straight-forward in the 1-nucleon sector, where the amplitude is perturbative. In contradistinction, all terms of the leading-order Lippmann-Schwinger equation of -scattering, including the potential (and hence one-pion exchange), must be of order when all nucleons are close to their non-relativistic mass-shell, to accomodate the shallow bound-state [3]. In a consistent power-counting, all -rescattering processes between photon-absorption and emission must thus be included. Our Green’s function approach embeds these diagrams to guarantee the Thomson limit, which is however statistically significant only below , see Fig. 1. Up to next-to-leading order, the only unknowns are contributions to the polarisabilities from short-distance Physics, leading to two energy-independent parameters. The iso-scalar Baldin sum rule is in excellent agreement with our 2-parameter fit, serving as input to model-independently determine the iso-scalar, spin-independent dipole polarisabilities of the nucleon at zero energy from all Compton data below :

(in ). We estimate the theoretical uncertainty to be from typical higher-order contributions in the 1- and 2-nucleon sector. Dependence on the -potential or deuteron wave-function used is virtually eliminated with the correct Thomson limit. Comparing this with our analysis of all proton Compton data below by the same method,

we conclude that the proton and neutron polarisabilities are to this leading order identical within (predominantly statistical) errors, as predicted by EFT. More and better data from MAXlab (Lund) will lead to a more precise extraction, allowing one to zoom in on the proton-neutron differences.

## References

- [1] R.P. Hildebrandt, H.W. Grießhammer, T.R. Hemmert: [nucl-th/0512063], subm. Phys. Rev. C .
- [2] R.P. Hildebrandt, PhD thesis TU München Dec. 2005 [nucl-th/0512064].
- [3] H.W. Grießhammer, forthcoming.
- [4] S.R. Beane, M. Malheiro, J.A. McGovern, D.R. Phillips, U. van Kolck: Phys. Lett. B567 (2003), 200; err. B607 (2005), 320; Nucl. Phys. A747 (2005), 311.