Analysis of $M1$ capture in the $α(d,γ)^6$Li reaction

Technical Summary: Analysis of M1 capture in the $\alpha(d, \gamma)^6$Li reaction

Problem Statement
The radiative capture reaction $\alpha(d, \gamma)^6$Li is a critical process in nuclear astrophysics, particularly regarding the "lithium problem," which involves a discrepancy between observed and calculated $^6$Li abundances in the universe. While the origin of this discrepancy is likely astrophysical, the reaction rate remains a subject of intense theoretical scrutiny. A specific point of contention is the relative importance of the magnetic dipole (M1) contribution to the total S-factor at low energies. Existing literature presents contradictory results: some models (e.g., Refs. [8, 12]) suggest the M1 contribution is negligible compared to Electric Dipole (E1) and Electric Quadrupole (E2) transitions, while an ab initio calculation using the no-core shell model with continuum (Ref. [13]) reports a large M1 S-factor that increases at very small energies. This paper aims to resolve this controversy by analyzing the structure of the M1 component using a specific effective operator.

Methodology
The authors employ a two-pronged approach combining analytical operator theory with numerical calculations in a three-body model.

1. Effective M1 Operator:
   The study introduces an effective M1 transition operator, $\hat{M}^{M1}_{\mu}$, which is exactly equivalent to the standard long-wavelength approximation (LWA) magnetic dipole operator in radiative capture matrix elements, provided the initial and final states are orthogonal.

   • The standard operator is decomposed into isoscalar and isovector parts.
   • The effective operator is derived by subtracting a term proportional to the total angular momentum operator ($J_\mu$), which vanishes between orthogonal states of different energies or quantum numbers.
   • The resulting effective operator (Eq. 5) has a simplified isoscalar part proportional only to the total spin operator ($S_\mu$), while the isovector part remains unchanged. This structure allows for a clearer analytical understanding of selection rules.

2. Three-Body Model ($p + n + \alpha$):
   Numerical evaluations are performed using a three-body model where the $^6$Li nucleus is treated as a system of a proton, neutron, and a structureless $\alpha$ particle interacting via effective forces.

   • Wave Functions: The final $^6$Li($1^+$) ground state is calculated using a hyperspherical coordinate expansion with a three-body Hamiltonian. The initial $\alpha + d$ scattering states are described as coupled products of the deuteron ground state and $\alpha+d$ scattering waves, neglecting distortion of the $p+n$ part in the entrance channel.
   • Isospin Mixing: The model simulates isospin mixing by enforcing antisymmetry in the $p+n$ wave functions. Components with $l+S$ odd correspond to isospin $T=0$, while $l+S$ even corresponds to $T=1$.
   • Orthogonality: Since the initial and final states arise from different Hamiltonians, the initial wave function is explicitly orthogonalized to the final ground state.

Key Contributions and Analytical Findings
The primary theoretical contribution is the derivation of strict selection rules based on the structure of the effective operator:

• Forbidden Isoscalar Transitions: The analysis demonstrates that isoscalar M1 transitions from an initial S-wave are strictly forbidden if the initial state is an eigenstate of the spin operator (i.e., $S_z \Psi = M \Psi$) and distortion is neglected. This is a consequence of the orthogonality between the initial and final states and the fact that the isoscalar part of the effective operator depends solely on the total spin.
• Allowed Isovector Transitions: Transitions from an initial S-wave remain possible only via the isovector part of the operator. These transitions depend entirely on the presence of small isospin $T=1$ components in the wave functions.
• D-Wave Suppression: Transitions from initial D-waves are shown to be suppressed at low energies due to centrifugal barriers.

Results
The numerical application of the three-body model yields the following results:

• Negligible M1 S-Factor: The calculated M1 S-factor is extremely small at astrophysical energies. The dominant contribution arises from isovector transitions from the initial S-wave to the $T=1$ component of the $^6$Li ground state.
• Comparison with Other Multipoles: The M1 contribution is orders of magnitude smaller than the E2 contribution and remains negligible compared to the total S-factor.
• Agreement with Specific Models: These results align with the findings of Refs. [8] and [12], which also found M1 contributions to be negligible.
• Contradiction with Ab Initio Results: The results stand in direct contradiction to Ref. [13], which reported a large M1 S-factor. The authors note that the ab initio model in Ref. [13] uses a different wave function description (no-core shell model) and does not employ the effective operator analysis presented here.

Significance and Claims
The paper claims that the use of the effective M1 operator clarifies the physical origin of M1 transitions and explains why they are generally suppressed in standard cluster models. The authors suggest that the large M1 S-factor found in Ref. [13] may stem from specific features of that model's wave function or the treatment of the operator.

The authors modestly conclude that while their three-body model confirms the negligible nature of the M1 contribution in that framework, the discrepancy with the ab initio calculation remains unexplained. They propose that applying the effective M1 operator (Eq. 5) to the ab initio framework of Ref. [13] would be a valuable next step to identify the physical origin of the large M1 value in that approach and potentially resolve the controversy. The paper does not propose new experimental measurements but emphasizes the need for theoretical consistency checks using the derived effective operator.