How much can we stretch the proton and the neutron?

How does a nucleon respond to an electromagnetic field? Professors Haiyan Gao and Calvin Howell of Duke University outline the Experimental Programme using Photons to Stretch the Nucleon

More than 99% of visible matter in the Universe is from atomic nuclei with building blocks of protons and neutrons, known together as nucleons. Atomic nuclei are interesting laboratories that nature provides in which all four fundamental forces are present. They include the strong nuclear force that binds nucleons into the nucleus, nuclear beta decay driven by the weak force, electromagnetic force between the nucleons associated with charge and magnetic dipole moments, and the gravitational force between any two nucleons.

While the theory of strong force, quantum chromodynamics (QCD), has been well established for more than half a century, our understanding of how the theory works is rather limited, especially in the energy region where ordinary matter exists. As such, the study of the nucleon structure that started in the 1950s continues to be a vibrant field due to the richness and internal dynamics of the nucleon as a relativistic many-body system with quarks and gluons as the underlying degrees of freedom. They are an important laboratory for advancing our knowledge about how QCD works. One aspect regarding the nucleon is its electromagnetic (em) polarizabilities, also known as the scalar dipole polarizabilities α_E1, β_M1, two quantities characterising how nucleons respond to an external electric or magnetic field. The induced electric or magnetic dipole moment of the nucleon is defined as the product of the applied field strength and its corresponding polarizability, just like those of a piece of microscopic material in an electromagnetic field. Such quantities have been demonstrated to be calculable by simulating QCD on the lattice (known as LQCD),^1,2 as shown with excellent precision in recent years for the scalar electric polarizability.

Compton scattering has proven to be a powerful tool to probe the em polarizabilities of the nucleon. In such a process, the incoming photon scatters off a target such as a proton at rest in the laboratory frame, and there are only the scattered photon and the recoiling proton in the final state, as illustrated in Fig. 1. The photon in this process provides an external electromagnetic field that is strong enough to induce a measurable effect. The scalar polarizabilities and spin polarizabilities of the nucleon appear in the Compton amplitude as structure constants which manifest in experimental observables such as differential cross section and spin-dependent asymmetries. At low energies, below the threshold of producing a pion from the nucleon, one can extract the scalar dipole polarizabilities from Compton scattering using state-of-the-art effective field theory calculations of the corresponding experimental observables.³ Therefore, these polarizabilities provide an important bridge connecting EFT and LQCD calculations.

Currently, the best knowledge we have about the nucleon scalar dipole polarizabilities is based on the recommended values by the Particle Data Group (PDG) in 2024:⁴

αp_E1= (11.5±0.4) ×10^-4 fm³, β^p_M1 = (2.31±0.29) ×10^-4 fm³

αn_E1 = (11.8±1.1) ×10^-4 fm³, βⁿ_M1 = (3.7±1.1) ×10^-4 fm³

Compared with the proton, where physicists have used liquid hydrogen targets for Compton scattering measurements for decades, the knowledge about the scalar dipole polarizabilities of the neutron is much less well known due to two important aspects. One is that the neutron decays; therefore, one needs to use an effective neutron target that is stable. Secondly, the Compton scattering cross section from the neutron is significantly smaller than that from a proton because it is a charge-neutral particle. Regarding effective neutron targets, deuterium has been used the most, as have lithium and helium-4. Most recently at the High Intensity γ-ray Source Facility (HIγS) at the Triangle Universities Nuclear Laboratory (TUNL), a cryogenic liquid ³He target was used for the first time for elastic Compton scattering measurements to probe the neutron polarizabilities αⁿ_E1, βⁿ_M1. Below we will describe the experimental capability of the HIγS facility and the Compton programme.

High-intensity γ-ray source

The HIγS facility at TUNL is the only Compton γ-ray source that is based on a storage ring Free Electron Laser (FEL). The γ-ray beam is produced by Compton scattering of the photons off circulating electron bunches inside the FEL optical cavity. The implementation of intracavity Compton backscattering along with the high current of the circulating electron beam bunches in the storage ring (up to more than 100 mA) make HIγS the most intense Compton-scattering γ-ray source in the world, delivering up to about 10⁴ γ/s/keV to targets.⁵

The floor layout of HIγS is shown in Fig. 2. HIγS consists of three accelerator assemblies: an electron linac, a booster synchrotron, and a storage ring. The linac injects electron bunches into the booster; the booster accelerates the bunches up to the energy corresponding to the magnet settings in the storage ring and injects the electron bunches into the ring to keep the electron current in the ring constant.

The FEL is located on the front straight section of the ring. The optical cavity is defined by two high-reflectivity concave spherical mirrors, one at each end of the cavity. The light inside the FEL optical cavity is generated by the acceleration of the electron bunches as they pass through the array of periodic dipole magnets inside each Optical Klystron (OK). The OKs time-compress the electron bunches, thereby amplifying stimulation of coherent light radiation. For lasing to occur, the round-trip time of a light pulse inside the optical cavity must be exactly equal to the time for an electron bunch to travel once around the ring. This requirement is mainly achieved by making the circumference of the storage ring (108m) equal to twice the length of the optical cavity (54m). The light pulse generated by an electron bunch during its previous trips through the OKs will overlap with that electron bunch in all subsequent passes through the OKs. The interaction of the electromagnetic fields of the light pulse with the electron bunch inside the OK field causes the stimulated light emission to be in phase with the previously emitted light, i.e., coherent light amplification.

There are two types of optical klystrons at HIγS, the OK-4 and OK-5 undulators. Inside the OK-4, the electrons accelerate in the horizontal plane in a sinusoidal pattern, thereby radiating linearly polarised light with the electric field vector in the horizontal plane. The OK-5 undulators have a three-dimensional magnetic field created by an array of dipole magnets, with the magnetic field alternating from the vertical to the horizontal direction. The electrons moving through this field follow a helical path and radiate circularly polarised light. The wavelength of the light emitted by the electrons moving through the OKs is adjustable by changing the strength of the magnetic field in the OKs and by the electron energy.

For γ-ray production, two electron bunches are loaded into the ring separated by half the ring’s circumference. Each bunch is the source of FEL light and the target for Compton scattering. With this beam bunch configuration, the electron bunch collides with the FEL photons in a field-free region midway along the straight section. Because the electron bunches are the sources of the photon pulses inside the FEL cavity, the γ-ray beam position and intensity on target are highly stable due to their common beam axis. The Compton-scattering process preserves the polarisation of the optical photons, making the polarisation of the HIγS γ-ray beam greater than 95%. Linearly or circularly polarised beams can be delivered to experiments by using either the OK-4 (with planar undulators) or the OK-5 (with helical undulators) FEL. The high intensity of HIγS relative to other Compton γ-ray sources is due mostly to the combination of the high intra-cavity optical power relative to external laser light sources and the high average beam current in the electron storage ring. The γ-ray beam energy range is from 1 to 100 MeV, and the energy spread of the γ-ray beam is selectable down to about 2% by collimation.

The Compton-scattering Experimental Programme at HIγS

At energies below about 300 MeV, the response of a nucleon to an electromagnetic wave can be parameterised in terms of six frequency (ω) dependent structure functions, two spin-independent and four spin-dependent functions.⁶ The values of these dynamical functions extrapolated to ω = 0 are the static polarizabilities. In elastic Compton scattering from a nucleon with real photons, referred to as Real Compton Scattering (RCS), the values of these fundamental parameters are encoded in the angular differential cross section.

Substantial progress has been made during the past decade on determining the proton polarizabilities. The uncertainties in the dipole electric and magnetic polarizabilities of the proton are 3.5% and 12.6%, respectively.^4,7,8 Also, the first double-polarised RCS asymmetries on the proton were measured by the Mainz A2 Collaboration using circularly polarised incident photons on either a transversely or longitudinally polarised proton target at the Mainz Microtron (MAMI).⁹

However, the situation is not so bright with the neutron. The experimental uncertainties on the determination of the neutron’s α_E1 and β_M1 are about 10% and 34%, respectively.⁴ The overarching goal of the HIγS Compton-scattering programme is to reduce the experimental uncertainty of αⁿ_E1 and βⁿ_M1 by a factor of two or more, thereby putting their precision on a similar footing as the proton. This work is carried out by the HIγS Compton-scattering Collaboration (Compton@HIγS), which consists of experimental and theoretical physicists from 12 institutions. The challenges in extracting the em polarizabilities of the neutron arise due to the absence of free neutron targets. Elastic Compton scattering from a light nucleus is an established method to determine the isospin-averaged nucleon polarizabilities, αs_E1 and βs_M1. A challenge in applying this technique to determine αs_E1 and βs_M1 is that it requires EFT calculations of the differential cross section for Compton scattering from the light nucleus that serves as a surrogate neutron target. In this approach, the isoscalar nucleon polarizabilities are determined by adjusting their values in the theoretical calculations to fit the differential cross-section data. The αn_E1 and βn_M1 are solved for using the extracted αs_E1 and βs_M1 and the PDG values for the proton polarizabilities.

An organising goal in the HIγS programme is to assess the theoretical uncertainty in the technique by determining αn_E1 and βn_M1 using nuclear targets with different combinations of neutrons and protons, e.g., the deuteron (n+p), ³He (2p+n), ⁴He (2p+2n) and ⁶Li (3p+3n) and at different γ-ray beam energies.

The theorists in the collaboration work closely with the experimentalists in motivating and planning experiments and interpreting data in terms of nucleon em polarizabilities. The theory team in the Compton@HIγS Collaboration is leading in the development of EFT calculations for Compton scattering from nuclei. For example, they now have EFT calculations for Compton scattering from the deuteron, ³He, ⁴He¹⁰ and ⁶Li.

Summary

The Compton@HIγS Collaboration is measuring angular differential cross sections of Compton scattering from light nuclei in the photon energy (E_γ) range of 60 to 100 MeV using the monoenergetic beam at HIγS. These data are used to test EFT calculations and to determine the spin-independent em dipole polarizabilities of the neutron. The Collaboration has recently measured cross sections of Compton scattering from the deuteron at E_γ = 61 MeV¹¹ and from ³He at E_γ = 61 and 98 MeV^12,13 using the experiment setup in Fig. 3. This work contributes the first cross-section data for Compton scattering from ³He and high-accuracy data on the deuteron to the global database used in analyses to determine αⁿ_E1 and βⁿ_M1. The estimated impact of these data is a reduction in the uncertainties of αⁿ_E1 and βⁿ_M1 by about 40% relative to the current PDG values. In addition, the most complete angular distribution cross-section measurements for Compton scattering from ⁴He were completed at photon beam energies of 87 and 100 MeV, and data analysis is underway. The next measurements will be on ⁶Li.

The current phase of the HIγS Compton programme is on pace to be completed within the next three years. Turning focus to extracting the spin-dependent em polarizabilities of the nucleons will allow techniques for optimised use of the highly polarised intense monoenergetic photon beam at HIγS. The measurement methods will switch from absolute cross sections to double polarisation (beam and target) asymmetries. A programme in this area requires development of cryogenic polarised target capabilities at TUNL. This work is carried out by the Compton@HIγS Collaboration and supported by the U.S. Department of Energy under grant Nos. DE-FG02-03ER41231 and DE-FG02-97ER41033.

References

1. R. Bignell, W. Kamleh, and D. Leinweber, Phys. Rev. D 101, 094502 (2020).
2. X.-H. Wang et al., Phys. Rev. Lett. 133, 141901 (2024)
3. F. Hagelstein, R. Miskimen, and V. Pascalutsa, Prog. Part. Nucl. Phys. 88, 29 (2016).
4. S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024).
5. C.R. Howell et al., J. Phys. G 49.1 (2022), p. 010502. doi: 10.1088/1361-6471/ac2827. arXiv: 2012.10843 [nucl-ex].
6. Robert P. Hildebrandt, Harald W. Griesshammer, and Thomas R. Hemmert, Eur. Phys. J. A. 20 (2004), pp. 329–344. doi: 10.1140/epja/i2003-10154-7. arXiv: nucl-th/0308054.
7. E. Mornacchi et al., Phys. Rev. Lett. 128 (13 Apr. 2022), p. 132503. doi: 10.1103/PhysRevLett.128.132503. url: https://link.aps.org/doi/10.1103/PhysRevLett.128.132503.
8. X. Li et al., Phys. Rev. Lett. 128 (2022), p. 132502. doi: 10.1103/PhysRevLett.128.132502. arXiv: 2205. 10533 [nucl-ex].
9. D. Paudyal et al., Phys Rev. C 102, 035205 (2020).
10. H.W. Griesshammer et al., arXiv:2401.16995 [nucl-th].
11. D. Godagama, “Elastic and Inelastic Compton Scattering from Deuterium at 61 MeV”, Ph.D. thesis, University of Kentucky (main) (2022).
12. E. Mancil, “Compton Scattering from Liquid 3He with 61.5 MeV Photons”, Ph.D. thesis, Duke U. (main) (2026).
13. J. Zhou, “Compton Scattering on 3He at HIγS and Neutron Polarizabilities”, Ph.D. thesis, Duke U. (main) (2026).

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