Analysis of p-Wave Sommerfeld Resonance Behavior in Finite-Size Dark Matter

The gist: The analysis investigates how finite-size effects modify p-wave Sommerfeld resonance behavior for dark matter, finding that these effects weaken resonances compared to point-like dark matter while leaving the velocity dependence largely unchanged.

Size Effects on Resonance Behavior

Finite size introduces an additional parameter and may relax constraints on dark matter and mediator masses and their coupling The study examines the effect of finite size on the p-wave Sommerfeld enhancement factor, using a simple top-hat profile for the charge-density distribution Both s-wave and p-wave resonance peaks decrease once the particles acquire a finite size This suppression becomes more evident at larger values of bf(y).

P-Wave Resonance Analysis for Finite-Size Dark Matter

The analysis characterizes the p-wave Sommerfeld resonances of finite-size dark matter using two parameters, (bf(y), a√y/f(y)) For generic finite-size dark matter without a specified internal structure, the resonance boundaries remain relatively clear while the internal structure of each resonance region becomes blurred. The results show that the p-wave contribution to the Sommerfeld enhancement is larger than the s-wave contribution However, finite-size effects reduce the magnitude of both factors.

Nugget Dark Matter Case

When considering nugget dark matter composed of a small number of constituents, the analysis shows that after the internal structure is specified, the resonant behavior becomes similar to that of a point particle. Specifically, for nugget dark matter with a specified internal structure, distinct p-wave resonance peaks emerge Furthermore, both panels of Fig. 5 show a smaller region of Sommerfeld enhancement compared to point-like dark matter in the upper-left panel of Fig. 3.

Implications for Dark Matter Models

The findings have important phenomenological implications, as finite-size effects may relax some constraints imposed by previous analyses of the point-particle p-wave Sommerfeld enhancement These changes in resonant behavior may help reconcile the different annihilation cross sections required at different stages of cosmic evolution. Conversely, constraints on the annihilation cross section can help probe the internal structure of finite-size dark matter, such as constraining the constituent number and radius of nugget dark matter.

Key Findings Summary

** As in the point-particle case, the p-wave contribution to the Sommerfeld enhancement is larger than the s-wave contribution. **

** Finite-size effects reduce the magnitude of both swave and p-wave Sommerfeld resonances. **

** Finite-size objects without a specified internal structure exhibit p-wave resonance regions with sharp boundaries but smeared internal structures. **

** Nugget dark matter with a specified internal structure exhibits distinct p-wave resonance peaks similar to point-like dark matter. **

** Higher partial waves can provide the dominant contribution to the Sommerfeld resonances. **

This behavior suggests that finite-size effects on the p-wave Sommerfeld enhancement may help reconcile the different annihilation cross-section requirements at different stages of cosmic evolution.

--- Page 1 ---

Prepared for submission to JHEP Analysis of p-Wave Sommerfeld Resonance Behavior for Finite-Size Dark Matter Wu-Long Xua, Jin Min Yangb, Rui Zhud, School of Integrated Circuits, Chengdu Technological University, Chengdu 611730, P. R. China Center for Theoretical Physics, Henan Normal University, Xinxiang 453007, P. R. China Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, P. R. China State Key Laboratory of Dark Matter Physics, Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, P. R. China Key Laboratory for Particle Astrophysics and Cosmology (MOE) & Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai 200240 E-mail: 396440567@qq.com(corresponding author), jmyang@itp.ac.cn, zhu.rui@sjtu.edu.cn(corresponding author) Abstract: Different stages of cosmic evolution impose different requirements on the dark matter annihilation cross section Sommerfeld enhancement provides a possible way to accommodate them. However, in the point-like dark matter scenario, satisfying these requirements with either s-wave or p-wave Sommerfeld enhancement can strongly constrain the dark matter and mediator masses and their coupling A finite dark matter size introduces an additional parameter and may relax these constraints. Motivated by this possibility, in this work we study the p-wave Sommerfeld resonances of finite-size dark matter (like proton or neutron) without specifying its internal structure We find that the p-wave contribution can exceed the s-wave contribution. Nevertheless, the finite-size p-wave resonances are weaker than those in the point-like case. Despite this suppression, their velocity dependence remains similar to that of point-like dark matter. We then extend our analysis to nugget dark matter composed of a small number of constituents In this case, the p-wave Sommerfeld enhancement exhibits behavior similar to that of point-like dark matter. arXiv:2610.11166v1 [hep-ph] 8 Oct 2026

--- Page 2 ---

Contents 1 Introduction Dark matter remains one of the most mysterious components of the Universe. Its microscopic properties, including its mass, spin, and size, remain largely unknown. Moreover, direct-detection experiments have found no conclusive evidence for dark matter so far. Taken together, these facts motivate us to consider an alternative possibility: present-day dark matter may not consist of elementary point-like particles, instead, it may be composed of spatially extended states formed during cosmic evolution. Such states could resemble protons, glueballs, or even dark atoms. Studies of finite-size dark matter usually begin by assuming a specific internal structure. Typical examples include dark atoms and dark nugget. The resulting scattering or annihilation cross sections are then used to constrain parameters such as the dark matter mass and size. By contrast, recent studies have proposed a more phenomenological approach. Instead of specifying a microscopic model, finite-size dark matter can be characterized by an internal charge distribution. Once this distribution is specified, both the dark matter selfscattering cross section and its scattering cross section with nucleons can be calculated. For example, finite-size effects can generate a velocity-dependent self-scattering cross section. They therefore provide an alternative to light-mediator scenarios and may help alleviate the small-scale structure problems of cosmology. Moreover, the same finite-size effects can lead to distinct regimes in dark matter–nucleon scattering, including the Born, resonant, and classical regimes. Consequently, meaningful constraints may be placed on the dark matter mass, size, and interaction strength without specifying its detailed microscopic structure. In addition to scattering, dark matter annihilation provides another important probe of dark matter physics. Most existing studies focus on point-like dark matter. However, the annihilation rate is tested in cosmological and astrophysical environments with markedly different characteristic velocities. Around recombination, the dark matter velocity can be as low as v ∼ 10−8 c, although its precise value depends on the thermal history. Energy injection from dark matter annihilation can modify the cosmic microwave background (CMB) anisotropies.

1. This system can perform parameter space exploration for dark matter models by using finite-size effects to relax some of these constraints. This allows for a more flexible fitting process when comparing annihilation cross sections probed at different cosmic epochs, as stated in the conclusion: Finite-size effects may relax some of these constraints.

2. The system can distinguish between different dark matter morphologies during analysis by utilizing the predicted spectral features. Specifically, it can identify that nugget dark matter with a specified internal structure exhibits distinct p-wave resonance peaks compared to generic finite-size dark matter where resonances are smeared internal structures.

3. The AI can generate constraints on the physical properties of dark matter constituents by analyzing experimental data against theoretical predictions. For nugget dark matter, the system can infer the constituent number N or the radius of nugget dark matter by examining the observed resonance peak positions, as noted in Section 3.2.

Different stages of cosmic evolution impose different requirements on the dark matter annihilation cross section. Sommerfeld enhancement provides a possible way to accommodate them. However, in the point-like dark matter scenario, satisfying these requirements with either s-wave or p-wave Sommerfeld enhancement can strongly constrain the dark matter and mediator masses and their coupling. A finite dark matter size introduces an additional parameter and may relax these constraints. Motivated by this possibility, in this work we study the p-wave Sommerfeld resonances of finite-size dark matter (like proton or neutron) without specifying its internal structure. We find that the p-wave contribution can exceed the s-wave contribution. Nevertheless, the finite-size p-wave resonances are weaker than those in the point-like case. Despite this suppression, their velocity dependence remains similar to that of point-like dark matter. We then extend our analysis to nugget dark matter composed of a small number of constituents. In this case, the p-wave Sommerfeld enhancement exhibits behavior similar to that of point-like dark matter.