⚛️ nuclear theory

Representation dependence of the neutron-proton kinetic energy in adiabatic deuteron stripping

This paper demonstrates that the significant sensitivity of adiabatic distorted-wave approximation (ADWA) calculations for deuteron stripping to the choice of neutron-proton interaction stems from the representation dependence of the short-range kinetic energy rather than physical observables, a limitation that is largely resolved by more exact theoretical treatments.

Original authors: Jin Lei

Published 2026-10-16

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CC BY 4.0

Original authors: Jin Lei

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the heart of every atom lies a nucleus, a dense cluster of protons and neutrons held together by forces so powerful they defy everyday intuition. To understand how these nuclei are built, scientists often fire beams of deuterons—simple atoms made of just one proton and one neutron—at target materials. When a deuteron strikes a nucleus, it can strip away its neutron, leaving the proton behind. By watching where the proton goes, researchers can map out the internal structure of the target nucleus, revealing how its own neutrons and protons are arranged. This technique, known as a stripping reaction, is a standard tool in nuclear physics. However, to make sense of the data, scientists must rely on complex mathematical models to describe how the deuteron behaves during the collision. One of the most widely used models is called the adiabatic distorted-wave approximation. It is a powerful shortcut that allows physicists to calculate reaction outcomes without solving the full, incredibly difficult three-body problem of the neutron, the proton, and the target nucleus all at once. But like any shortcut, it relies on assumptions, and the question remains: how much do the details of those assumptions change the final answer?

A recent study by Jin Lei at Tongji University and the Chinese Academy of Sciences investigates a specific weakness in this popular model. The research focuses on a particular number that the model uses to describe the energy of the neutron and proton as they interact inside the deuteron. In the standard model, this energy is not just a simple average; it is a weighted average that pays extra attention to the moments when the two particles are very close together. Previous calculations using different, realistic descriptions of the force between neutrons and protons showed that this number could vary wildly, sometimes changing the predicted reaction rate by as much as fifty percent. This huge spread led some to wonder if the reaction itself was acting as a microscope, revealing new, hidden details about how neutrons and protons behave at extremely short distances. If true, these reactions could help scientists refine our understanding of the fundamental forces that hold matter together.

The new paper challenges this exciting possibility. The author asks a simpler, more fundamental question: is this energy number a fixed property of nature, or does it depend on how we choose to write down the mathematics of the neutron-proton force? To find out, the researcher performed a precise experiment, not with physical particles, but with the mathematical equations themselves. Starting with a well-established description of the neutron-proton force, the author applied a mathematical transformation that changed the force only at very short distances, deep inside the deuteron. Crucially, this change was designed to leave all observable properties of the deuteron completely untouched. The binding energy, the way the deuteron scatters off other particles, and the shape of its outer edge remained exactly the same. In the world of physics, if two descriptions produce the exact same observable results, they are considered equivalent representations of the same reality.

The result of this transformation was striking. Even though the physical properties of the deuteron did not change at all, the specific energy number used in the adiabatic model jumped dramatically. It rose from 218 MeV to 314 MeV, a shift that accounted for more than half of the entire range of variation seen among different realistic models. Meanwhile, other parts of the model that describe the low-energy behavior of the deuteron changed by less than one percent. This finding demonstrates that the large variations seen in previous calculations are not caused by differences in the physical structure of the deuteron or by new short-range physics. Instead, they are an artifact of the mathematical representation chosen by the scientist. The number is not a direct measurement of nature; it is a feature of the specific equation used to describe it.

The study explains why the adiabatic model is so sensitive to this issue. The model makes two key approximations that amplify the effect of short-range details. First, it weights the calculation heavily toward the moments when the neutron and proton are closest together, exactly where the mathematical transformation was applied. Second, it treats the internal motion of the deuteron as if it were frozen at a single energy level, ignoring the fact that the particles are constantly moving and exchanging energy. When these two features are combined, the model becomes highly responsive to how the force is written down at short distances. In contrast, more exact calculations that do not freeze the energy or use the same weighting scheme show much smaller variations, suggesting that the large spread in the adiabatic results is a flaw in the approximation, not a window into new physics.

The paper concludes that the adiabatic model cannot, on its own, be used to constrain or discover new details about the short-range structure of the neutron-proton interaction. The variations it produces are not a signal of different physical realities but rather a reflection of the mathematical freedom scientists have in choosing how to describe those realities. To truly understand the short-range behavior of nuclear forces, researchers must use methods that treat the neutron, proton, and target nucleus with equal rigor, avoiding the shortcuts that introduce these artificial dependencies. The study serves as a reminder that in the quest to understand the building blocks of the universe, the tools we use to measure them must be as carefully scrutinized as the objects themselves. Without this care, what looks like a discovery of new physics may simply be a shadow cast by the way we chose to look.

Technical Summary: Representation Dependence of the Neutron-Proton Kinetic Energy in Adiabatic Deuteron Stripping

Problem Statement
Deuteron stripping reactions, such as $(d, p)$ , are standard tools for extracting nuclear single-particle structure, typically analyzed using the Adiabatic Distorted-Wave Approximation (ADWA). When employing nonlocal nucleon optical potentials, ADWA cross-sections exhibit a significant sensitivity to the choice of the realistic neutron-proton ( $n$ - $p$ ) interaction, varying by tens of percent to a factor of two. This spread is attributed to the "energy shift" mechanism, where the nonlocality of the nucleon-target potential requires the evaluation of the potential at an energy $E_d/2 + \Delta E$ . The shift $\Delta E$ is proportional to the neutron-proton relative kinetic energy averaged with the weight $V_{np}\phi_d$ , denoted as $\langle T_{np} \rangle_V$ .

Previous studies noted that $\langle T_{np} \rangle_V$ varies by a factor of approximately three among different realistic interactions (e.g., AV18, CD Bonn, N3LO), whereas low-momentum properties like the zero-range constant and the finite-range coefficient of the form factor remain nearly identical. The central question addressed is whether this large variation in $\langle T_{np} \rangle_V$ represents a genuine physical difference in short-range $n$ - $p$ structure that $(d, p)$ data can probe, or if it is merely an artifact of the mathematical representation of the interaction.

Methodology
To distinguish between physical differences and representation artifacts, the author constructs a unitary transformation of the $n$ - $p$ Hamiltonian within the deuteron channel, starting from the AV18 interaction. This transformation:

Acts only at short distances (compact support).
Preserves all two-body observables: the scattering matrix, the deuteron binding energy ( $B_d$ ), and the asymptotic normalization coefficients (ANCs) for both S- and D-waves.
Effectively changes the representation of the interaction without altering the physical fit to two-body data.

The study compares the behavior of two key quantities under this transformation:

$\langle T_{np} \rangle_V$ : The weighted kinetic energy integral that drives the ADWA energy shift.
$c$ : The finite-range coefficient, which determines the low-momentum slope of the form factor.

These results are contrasted with the behavior of continuum-discretized coupled-channels (CDCC) and exact Faddeev calculations, which treat the three-body problem without the adiabatic approximations.

Key Results

Extreme Sensitivity of $\langle T_{np} \rangle_V$ to Representation: A unitary transformation confined to short distances ( $r < 1.5$ fm) with a small rotation angle ( $\theta = 0.05$ ) increases $\langle T_{np} \rangle_V$ from 217.9 MeV (AV18) to 314.0 MeV. This change constitutes 62% of the total spread observed among five different realistic interactions (63–218 MeV).
Stability of Low-Momentum Properties: The same transformation changes the finite-range coefficient $c$ by only 1% of its realistic range. Even with a larger rotation ( $\theta = 0.10$ ), where $\langle T_{np} \rangle_V$ exceeds all realistic values (reaching 518.8 MeV), $c$ shifts by only 3% of its range.
Mechanism of Amplification in ADWA: The paper identifies two specific features of the ADWA projection that amplify the representation dependence:
- The Weight: ADWA weights the deuteron wave function by $V_{np}\phi_d$ , emphasizing short separations where $n$ - $p$ interactions differ significantly. In contrast, CDCC elastic coefficients rely on the overlap of the deuteron with its displaced copy (weighted by density $\rho$ alone), which emphasizes long-range tails where interactions are identical.
- Frozen Energy: ADWA replaces the internal Hamiltonian with $-B_d$ , assigning the deuteron energy to internal propagation. The nonlocality then samples the internal momentum as nucleon energy. The expansion of the nonlocal kernel to lowest order (retaining full $p^2$ ) strongly amplifies the response to short-range changes.
Bounded vs. Unbounded Averages: When the nonlocality is treated with a Gaussian averaging factor (keeping the full overlap $M_0$ ) rather than expanding to lowest order, the sensitivity to representation drops significantly. A transformation that shifts $\langle T_{np} \rangle_V$ by 62% of its realistic span shifts the bounded average $T_\beta$ by only 12% of its own span.

Significance and Claims
The paper concludes that the large variation in $\langle T_{np} \rangle_V$ among realistic interactions is not a fixed observable determined by two-body data. Instead, it reflects how the interaction is written (its representation). Consequently:

The sensitivity of ADWA cross-sections to $\langle T_{np} \rangle_V$ cannot be used in isolation to constrain short-range $n$ - $p$ structure or to discriminate between different nucleon-nucleon forces.
The quantity $\langle T_{np} \rangle_V$ is not an observable; it changes under unitary transformations that leave all physical observables invariant.
The low-momentum content of the form factor (governed by $c$ and the ANC) is robust and consistent across interactions, whereas the high-momentum content (governed by the variance of kinetic energy) is representation-dependent.
For high-momentum probes (e.g., $(p, d)$ at 600 MeV), the form factor shapes and magnitudes vary drastically among interactions, further indicating that interpreting reaction data in terms of short-range structure requires a consistent representation shared by the states and the reaction operator.

The author emphasizes that while the adiabatic projection exposes high-momentum components, the resulting spread in cross-sections is a measure of the theoretical prescription's representation dependence, not a direct measure of physical short-range structure.

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