⚛️ nuclear theory

Spurious center-of-mass mode as an $S$ -matrix pole within the Jost-random-phase approximation

This paper demonstrates that within the Jost-random-phase approximation, the spurious center-of-mass mode manifests as a second-order pole of the $S$ -matrix at zero excitation energy, a property that holds regardless of self-consistency and allows for the mode's principal part to be subtracted from the response without altering physical strength at non-zero energies.

Original author: Kazuhito Mizuyama

Published 2026-11-05

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CC BY 4.0

Original author: Kazuhito Mizuyama

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

Inside the heart of every atom lies a nucleus, a dense cluster of protons and neutrons held together by powerful forces. To understand how these tiny systems behave, physicists often treat the nucleus not as a collection of individual particles, but as a fluid that can ripple and vibrate. When an external force, such as a passing particle, nudges this fluid, it creates waves. Scientists use mathematical tools to predict how these waves move and interact, hoping to distinguish between genuine vibrations of the nucleus itself and artifacts that arise simply because the entire nucleus is moving through space. One such artifact is the "center-of-mass mode," a motion where the whole nucleus shifts position rather than changing its internal shape. While this shift is not a true internal vibration, it often appears in calculations as if it were one, creating a confusing signal that researchers must learn to identify and separate from the real physics.

For decades, a specific question has lingered in the theoretical study of these nuclear vibrations: does this spurious shift appear as a distinct, singular point in the mathematical map that describes how particles scatter off the nucleus? This map, known as the scattering matrix, acts like a ledger of all possible outcomes when particles collide. If the center-of-mass motion is a true feature of this ledger, it should show up as a specific type of mathematical singularity, a point where the values blow up in a predictable way. Previous work had suggested that this motion might not appear as a pole in this map at all, or that it might be hidden in a more complex form. However, a new study by Kazuhito Mizuyama has revisited this question using a refined mathematical approach called the Jost–random-phase approximation. By applying this method, which carefully tracks the boundaries of the nuclear fluid, the researcher has demonstrated that the spurious center-of-mass mode is indeed a pole in the scattering matrix, and specifically, it is a "double pole," a point of unique mathematical intensity that occurs exactly when the energy of the system is zero.

The study focuses on two nuclei, oxygen-16 and lead-208, using a standard model of nuclear forces known as the Woods-Saxon potential. In these simulations, the researcher found that when the model is not perfectly tuned to match the real world, the spurious mode does not sit at zero energy. Instead, it appears as a pair of points floating on the imaginary axis of the energy map, a location that corresponds to a specific type of unstable or non-physical vibration. However, when the model is calibrated so that the residual interactions between particles are adjusted to force the mode to zero energy, the behavior changes dramatically. In this calibrated state, and also in a fully self-consistent model where the nuclear forces are derived directly from the density of the particles themselves, the two points merge into a single, powerful double pole at zero energy. This finding confirms that the spurious motion is not just a background noise but a fundamental mathematical feature of the scattering process, carrying a specific weight that accounts for nearly all of the energy associated with the external force used to probe the nucleus.

The research also clarifies how to handle this feature when calculating the actual strength of nuclear vibrations. Because the spurious mode creates a massive spike in the mathematical description at zero energy, it can distort the results for real, physical energies. The study shows that by mathematically removing the specific contribution of this double pole, the remaining signal for real energies changes very little, except for a slight smoothing effect. In fact, in a self-consistent model, removing this pole is mathematically identical to projecting out the ideal spurious state, a technique physicists have used for years to clean up their data. This means that the new mathematical proof does not require a change in how experimentalists analyze their data, but it does provide a rigorous foundation for why those methods work. The study further reveals that in the self-consistent models, every physical measurement of the scattering process, when traced back to zero energy, carries this double-pole signature, correcting an earlier conclusion that had suggested the scattering matrix was free of this feature at zero energy.

Ultimately, the paper establishes that the spurious center-of-mass mode is a double pole of the scattering matrix, a result that holds true whether the zero energy is achieved through self-consistency or through calibration of the interaction strength. The state associated with this double pole is a zero-energy solution that represents the translation of the entire nucleus, normalized by its inertia, which is a measure of its resistance to acceleration. The researchers found that in the uncalibrated models of oxygen-16 and lead-208, the mode exists as a pair of poles on the imaginary axis at energies of 10.46 i MeV and 8.07 i MeV respectively. When calibrated, these merge into a double pole at zero energy. The study concludes that while the translation of the nucleus is not an intrinsic excitation, its mathematical footprint is undeniable and precise, appearing as a second-order pole that carries between 90 and 95 percent of the energy-weighted sum rule for the probing field. This work resolves a long-standing theoretical ambiguity, confirming that the scattering matrix does indeed contain this pole, and providing a clear, rigorous method for separating the motion of the whole nucleus from the vibrations of its parts.

Technical Summary: Spurious Center-of-Mass Mode as an S-Matrix Pole within the Jost–Random-Phase Approximation

Problem Statement
In the random-phase approximation (RPA) applied to a self-consistent mean field, the translation of the nucleus manifests as a solution at zero excitation energy ( $E=0$ ), known as the spurious center-of-mass (c.m.) mode. While this mode is not an intrinsic excitation, it contaminates the response to external fields with angular momentum $L=1$ . Previous treatments have yielded conflicting results regarding the nature of this mode within the scattering matrix ( $S$ -matrix). Specifically, Ginocchio, Schucan, and Weidenmüller [20] concluded that the $S$ -matrix possesses no pole at $E=0$ . Conversely, discrete RPA and certain continuum models suggest the mode forms a Jordan pair or a double pole in the response function, contributing an $E^{-2}$ term. The central question addressed is whether the spurious mode constitutes a pole of the physical $S$ -matrix, and if so, what is its order and behavior in both self-consistent and non-self-consistent (calibrated) models.

Methodology
The paper employs the Jost–random-phase approximation (Jost-RPA), a framework that maintains scattering boundary conditions for all channels and identifies $S$ -matrix poles as zeros of the Jost determinant.

Formalism: The study utilizes a $2M \times 2M$ Green function formalism involving forward and backward particle-hole channels. The Jost matrices $J^{(\pm)}(E)$ are defined via the asymptotic behavior of regular and Jost solutions. The $S$ -matrix is derived from these matrices, and its poles are analyzed by examining the zeros of $\det J^{(+)}(E)$ .
Models: Two distinct scenarios are investigated:
1. Self-consistent models: Where the mean field and residual interaction are derived from a single energy density functional of neutron and proton densities.
2. Woods–Saxon models: Uncalibrated and calibrated versions for $^{16}\text{O}$ and $^{208}\text{Pb}$ . In the calibrated case, the residual interaction is rescaled to force the Jost determinant to vanish at $E=0$ .
Analytical Tools: The analysis relies on the Autonne–Takagi factorization to define the state of a pole, Wronskian relations to establish symmetry properties, and pair arguments to determine the reality of eigenvalues. The study also examines the removal of the spurious mode via the subtraction of principal parts (Mittag-Leffler terms) from the response function.

Key Contributions and Results

Nature of the Spurious Mode as an S-Matrix Pole:
The paper demonstrates that within the Jost-RPA, the spurious center-of-mass mode is indeed a zero of the Jost determinant and, consequently, a pole of the $S$ -matrix. This finding directly contradicts the earlier conclusion by Ginocchio et al. [20], which claimed no such pole exists. The discrepancy is resolved by showing that while the integrand in previous analyses appeared regular, the integral itself diverges as $E^{-2}$ in the self-consistent limit.
Order of the Pole:
Under specific premises—natural parity, every occupied orbital being a hole, every unoccupied bound orbital lying above the holes (S1), and configurations containing every allowed partial wave (S2)—the mode at $E=0$ is proven to be a pole of exactly second order (a double pole) when there is a single independent solution.
- This second-order nature arises from the position of the zero ( $E=0$ ) and the structure of the solutions, not merely from self-consistency.
- In self-consistent models, every physical $S$ -matrix element continued to $E=0$ carries this $E^{-2}$ term.
- In uncalibrated Woods–Saxon models, the mode appears as a pair of poles on the imaginary axis (e.g., at $E = \pm 10.46i$ MeV for $^{16}\text{O}$ ).
State of the Double Pole:
The state associated with the double pole is identified via the Autonne–Takagi factorization of the coefficient matrix of the $E^{-2}$ term. This state corresponds to the zero-energy solution normalized by the c.m. inertia. In the self-consistent limit, this solution represents the rigid translation of the nucleus.
Removal of the Spurious Mode:
The paper analyzes the effect of subtracting the principal part of the response function (the Mittag-Leffler terms) associated with these poles.
- For real excitation energies $E \neq 0$ , subtracting the principal part changes the strength only through the smoothing parameter $\eta$ ; as $\eta \to 0$ , the strength at real $E$ remains unchanged.
- In self-consistent models, this subtraction procedure is shown to be mathematically equivalent to the projection of the ideal spurious state out of the response [18].
- For the double pole at $E=0$ , the subtraction removes the $E^{-2}$ term, leaving a regular response function $S^{\text{reg}}_F(E)$ .
Sum Rules:
The study confirms that if the Jost determinant has no zeros in the upper half of the complex energy plane ( $\text{Im } E > 0$ ), the energy-weighted sum rule holds for the regularized response $S^{\text{reg}}_F(E)$ . Numerical results for $^{16}\text{O}$ and $^{208}\text{Pb}$ show that the spurious mode (or the pair of imaginary poles) carries 90–95% of the energy-weighted sum rule for the field $r$ in the $L=1$ channel.

Significance
The paper claims to resolve a long-standing question regarding the analytic structure of the $S$ -matrix in the presence of spurious modes. It establishes that the spurious c.m. mode is a double pole of the $S$ -matrix at $E=0$ in self-consistent models, correcting the analytic conclusion of Ref. [20] while affirming its physical statement that translation is not an intrinsic excitation. The work provides a rigorous theoretical basis for handling spurious modes in continuum RPA calculations, demonstrating that subtracting the principal part of the response is equivalent to projecting out the spurious state and that the remaining strength obeys standard sum rules. The findings apply to both self-consistent theories and calibrated mean-field models, offering a unified description of the spurious mode's pole structure.

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