Helium-3 relativistic wave function in light-front dynamics

The relativistic wave function of $^3$He nucleus is calculated in the framework of Light-Front Dynamics. It is determined by 32 spin-isospin components, each of which depends on five scalar variables. For NN interaction, the one-boson exchange model is assumed, but without a potential approximation. The relativistic effects manifest themselves in deviation of the relativistic components from the non-relativistic input, in the appearance of the components absent in the non-relativistic limit, and in dependence of solutions on specific variables that don’t exist in the non-relativistic wave function.

💡 Research Summary

The paper presents a comprehensive calculation of the relativistic wave function of the helium‑3 (³He) nucleus within the framework of Light‑Front Dynamics (LFD), employing an explicitly covariant formulation. Recognizing that traditional non‑relativistic many‑body Schrödinger approaches fail to describe the high‑momentum tails of nuclear wave functions—where nucleon momenta become comparable to their rest mass—the authors turn to LFD, which naturally incorporates relativistic kinematics and spin dynamics.

A central result is that the ³He wave function is decomposed into 32 spin‑isospin components. Each component depends on five independent scalar variables: two transverse Jacobi momenta (|R⊥|) and three longitudinal momentum fractions (x) associated with the three nucleons. This is a substantial increase in complexity compared to the deuteron, whose relativistic LF wave function involves six components depending on only two variables (|k⊥| and x). The authors emphasize that the additional variables have no analogue in the non‑relativistic description and are responsible for genuinely relativistic effects.

The nucleon‑nucleon interaction is modeled by a one‑boson‑exchange (OBE) kernel without invoking a potential approximation. By treating the exchange diagrams directly, the calculation retains full relativistic covariance and avoids the ambiguities associated with constructing effective potentials in a relativistic setting. The OBE kernel is inserted into the three‑body Faddeev equations formulated on the light front, leading to a set of coupled integral equations for the 32 amplitudes.

Two distinct bases are introduced to handle the spin‑isospin structure. The first basis, denoted V₍ij₎ (i, j = 1…4), is built from Dirac spinors, charge‑conjugation matrices, and four‑by‑four matrices Tᵢ and Sⱼ that depend on the light‑front fraction x₃ and the scalar product ω·p (ω being the light‑front direction vector). This basis is orthonormal by construction (½ V†₍i′j′₎ V₍ij₎ = δᵢ′ᵢ δⱼ′ⱼ) and simplifies the projection of the Faddeev kernel. The second basis, χₙ (n = 1…16), is designed to reduce smoothly to the familiar non‑relativistic components (S‑ and D‑waves) in the limit where the light‑front variables collapse to the relative momentum q. While χₙ is not as straightforward to orthogonalize, it provides a direct link to traditional nuclear‑structure language. The two bases are linearly related through an explicit transformation (Eq. 83), allowing the authors to compute the amplitudes in the V‑basis, transform them to the χ‑basis, and compare with non‑relativistic wave functions.

The three‑body wave function is expressed as a sum over cyclic permutations of a Faddeev component Φ₁₂(1,2,3). Antisymmetry under exchange of the first pair ensures overall antisymmetry when combined with the appropriate isospin functions ξ^(t) (t = 0, 1). Consequently, each isospin channel contributes 16 amplitudes, yielding the total of 32 independent functions g⁽ᵗ⁾₍ij₎(k₁,k₂,k₃).

Numerically, the coupled integral equations are solved using high‑performance parallel computing. The momentum space is discretized on a grid adapted to the light‑front variables, and convergence is verified by varying grid density and employing different quadrature schemes. The resulting amplitudes display clear deviations from the input non‑relativistic wave function: in addition to the dominant S‑ and D‑like components, sizable P‑ and F‑like relativistic components appear, especially at large transverse momenta. These new components are absent in any non‑relativistic treatment and are directly linked to the dependence on the extra light‑front variables and on the ω‑direction.

The authors analyze the impact of these relativistic components on observable quantities. They find that the high‑momentum behavior of the electromagnetic form factors of ³He is significantly altered, with the new components contributing to the tail of the charge and magnetic form factors. Moreover, the ω‑dependence introduces a subtle breaking of rotational symmetry that is restored only after integrating over the full light‑front phase space, illustrating the delicate interplay between boost invariance and rotational invariance in LFD.

In conclusion, the work demonstrates that Light‑Front Dynamics, equipped with an explicitly covariant formulation and a realistic OBE kernel, can successfully handle the full relativistic three‑nucleon problem. The emergence of 32 spin‑isospin amplitudes depending on five scalar variables underscores the richness of the relativistic structure of ³He. The study provides a benchmark for future investigations of few‑body nuclei at high momentum transfer, and it paves the way for applying the same methodology to more complex systems, to the calculation of deep‑inelastic scattering off light nuclei, and to the exploration of short‑range correlations in a fully relativistic framework.