⚛️ phenomenology

Identification of $D_{s1}(2933)$ as the $2P_{1}$ state in the quark model

This paper identifies the newly observed $D_{s1}(2933)^{+}$ meson as the $2P_1$ excited state of the $c\bar{s}$ system by demonstrating that coupled-channel corrections involving specific intermediate states successfully reproduce its measured mass, predict a partner state, and yield a branching ratio consistent with experimental data.

Original authors: Jun Wang, Qiang Zhao

Published 2026-09-29

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Jun Wang, Qiang Zhao

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

Deep within the heart of matter, protons and neutrons are built from smaller particles called quarks, which are bound together by a powerful force. When these quarks pair up, they form particles known as mesons. For decades, physicists have mapped out the expected "family tree" of these mesons, predicting where they should appear and how heavy they should be based on the rules of the standard model of particle physics. However, nature has a habit of surprising us. Over the last twenty years, experiments have uncovered several charm-strange mesons—particles made of a heavy charm quark and a lighter strange quark—that do not fit neatly into these predicted slots. Their masses are often lower than expected, and their behaviors suggest that the simple picture of two quarks orbiting each other is incomplete. This has led scientists to wonder if these particles are influenced by a more complex environment, perhaps interacting with other particles in a way that shifts their properties, much like a single note sounding different when played in a crowded room versus a quiet one.

Recently, the LHCb collaboration at the Large Hadron Collider discovered a new, heavier member of this family, a particle named Ds1(2933). It has a mass of about 2933 MeV and carries specific quantum numbers that identify it as a type of excited state. The big question was whether this new particle is simply a higher-energy version of the standard two-quark model, or if it is something entirely different, such as a loose cluster of four quarks or a molecule made of two other mesons. In a new study, researchers Jun Wang and Qiang Zhao set out to solve this puzzle by treating the particle as a standard two-quark system but accounting for the messy reality of its surroundings. They proposed that while the particle is fundamentally a charm and strange quark pair, its mass and behavior are significantly altered by "coupled-channel effects." This means the particle briefly fluctuates into other combinations of particles, such as a D meson and a K-star meson, before returning to its original state. These fleeting interactions act like a cloud of virtual particles that pull the mass of the original particle down, making it lighter than the pure theoretical prediction.

To test this idea, the team performed a detailed calculation that started with the standard prediction for this type of particle, which suggested a mass around 3020 MeV. They then added the mathematical corrections for all the possible intermediate particle loops the Ds1(2933) could form. By adjusting a specific parameter that controls how far these virtual interactions reach, they found a setting where the corrected mass matched the experimentally observed 2933 MeV perfectly. This calculation revealed that the particle is indeed the lower-mass version of the expected excited state, but its properties have been heavily reshaped by these interactions. The study also predicted the existence of a "partner" particle, a heavier sibling in the same family, which should exist at a mass of 3012 MeV. Furthermore, the analysis showed that the way these two states mix together is different from what simple theories predict, suggesting that the environment plays a crucial role in defining the particle's identity.

The researchers did not stop at just matching the mass; they also checked if their model could explain how the particle decays. The Ds1(2933) was found by looking at how it breaks apart into other particles, specifically through intermediate steps involving a D-star meson and a K-star meson. The team calculated the ratio of these different decay paths and found their theoretical result to be consistent with the experimental data, despite the large uncertainties in the measurements. This agreement provides strong support for the idea that the Ds1(2933) is a standard two-quark excited state that has been modified by its interactions with nearby particle channels. The study effectively rules out the possibility that this particle is the heavier partner state, as that scenario would require physically unreasonable parameters to fit the data. Instead, the evidence points to a picture where the fundamental structure is simple, but the observed properties are a complex result of the particle constantly borrowing energy and identity from its neighbors. This work helps clarify the internal structure of these exotic mesons, suggesting that while the basic building blocks are well understood, the forces between them create a rich and varied landscape that requires careful, detailed modeling to fully comprehend.

Technical Summary: Identification of $D_s1(2933)$ as the $2P_1$ State in the Quark Model

Problem Statement
The paper addresses the internal structure of the newly observed charm-strange meson $D_s1(2933)^+$ , reported by the LHCb Collaboration in the $B^0 \to D^+ D^- K^+ \pi^-$ decay. While the state has a mass of approximately 2933 MeV and quantum numbers $J^P = 1^+$ , its mass is significantly lower (by 80–100 MeV) than the prediction for the lowest $2P_1$ state ( $\sim 3.02$ GeV) provided by the conventional Godfrey-Isgur (GI) quark model. This discrepancy mirrors issues seen in other charmed-strange mesons like $D^*_{s0}(2317)$ and $D_s1(2460)$ , suggesting that the conventional "bare" quark model picture is insufficient. The central problem is to determine whether $D_s1(2933)$ is a standard radially excited $2P_1$ state of the $c\bar{s}$ system significantly modified by coupled-channel effects, or if it represents a different structure, such as a hadronic molecule.

Methodology
The authors investigate the $D_s1(2933)$ as the $2P_1$ excited state of the $c\bar{s}$ system, incorporating unquenched corrections via a coupled-channel approach. The methodology proceeds in three main stages:

Mass Matrix Construction and Coupled-Channel Corrections:
- The study begins with the bare mass matrix for the $2^1P_1$ and $2^3P_1$ states derived from the GI quark model.
- Coupled-channel corrections are evaluated using two-point self-energy loop diagrams involving intermediate meson pairs: $DK^*$ , $D^*K$ , $D^*K^*$ , $D_s\phi$ , $D^*_s\phi$ , and $D^*_s\eta$ .
- Effective Lagrangians are constructed for these interactions, and the self-energy corrections ( $\Sigma$ ) are calculated. A form factor $F(q^2)$ with a cutoff scale $\Lambda = \alpha \Lambda_{QCD}$ is introduced to regularize ultraviolet divergences.
- The physical mass matrix is obtained by adding the real part of the self-energy corrections to the bare mass matrix. The cutoff parameter $\alpha$ is determined by imposing the secular condition that the lower eigenvalue of this corrected matrix equals the experimental mass of 2933 MeV.
Decay Width Calculations:
- The $^3P_0$ model is employed to calculate the strong coupling constants for meson decays and the partial decay widths.
- The total width of the physical state is computed by summing the partial widths of dominant channels ( $DK^*$ , $D^*K$ , $D^*K^*$ , and $D^*_s\eta$ ).
- A sensitivity analysis is performed by adjusting the $^3P_0$ pair-creation coupling constant $\gamma$ to match the experimental total width.
Branching Ratio Analysis:
- The paper calculates the theoretical relative three-body branching ratio $R_{th}$ for the decay $D_s1(2933) \to D^+ K^+ \pi^-$ via two distinct intermediate pathways: $DK^*$ and $D^*_2 K$ .
- This theoretical ratio is compared against the experimental value $R_{exp}$ extracted from the LHCb amplitude analysis.

Key Results

Mass and Mixing Angle: By requiring the lower eigenvalue of the corrected mass matrix to match 2933 MeV, the authors determine a cutoff scale of $\alpha \approx 3.7$ ( $\Lambda \approx 0.82$ GeV). This yields a physical mass for the partner state ( $2P'_1$ ) at 3012 MeV and a $1P_1$ – $3P_1$ mixing angle of $-12.9^\circ$ . This mixing angle deviates significantly from the heavy-quark ideal mixing angle of $-35.3^\circ$ .
Alternative Assignment Rejected: An alternative hypothesis, assigning $D_s1(2933)$ as the higher-mass $2P'_1$ state, requires a cutoff of $\alpha \approx 13.9$ ( $\Lambda \approx 3.1$ GeV). The authors deem this physically unreasonable and note that such a high mass would overlap with the known $1P$ multiplet ( $D_s1(2460)$ and $D_s1(2536)$ ), making this assignment disfavored.
Decay Widths: Using a standard $^3P_0$ coupling $\gamma = 0.4$ , the predicted total width is 115. However, by constraining $\gamma$ to 0.32 to reproduce the experimental width of $72^{+18}_{-12}$ MeV, the model yields a total width of 71.8 MeV. This adjusted $\gamma$ remains within the range of uncertainties for quark model parameters.
Branching Ratios: The calculated theoretical ratio for the three-body decay is $R_{th} = 1.98$ . This is consistent with the experimental value $R_{exp} = 1.27^{+1.29}_{-0.95}$ within the large experimental uncertainties.
Dominant Channels: The analysis indicates that the $D^*K^*$ , $D_s\phi$ , and $D^*_s\phi$ intermediate states provide the primary contributions to the mass shift due to their proximity to the bare-state mass and the S-wave selection rules in the $^3P_0$ model.

Significance and Claims
The paper claims that the assignment of $D_s1(2933)$ as the $2P_1$ excited state of the $c\bar{s}$ system is supported by the data, provided that significant coupled-channel effects are included. The study demonstrates that:

The mass discrepancy between the GI quark model prediction and the observed $D_s1(2933)$ can be resolved through coupled-channel corrections with a physically reasonable cutoff scale.
The resulting mixing angle ( $-12.9^\circ$ ) is distinct from the ideal mixing limit, highlighting the importance of nearby S-wave channels in modifying the internal structure of axial-vector mesons.
The consistency between the predicted and experimental total widths and the three-body branching ratio fraction provides complementary evidence for this assignment.

The authors conclude that while $D_s1(2933)$ is likely the lower $2P_1$ state, its properties are strongly influenced by the nearby S-wave coupling channels. They suggest that more precise measurements of partial branching fractions, particularly the ratio $\Gamma(D_s1(2933) \to D^*K)/\Gamma(D_s1(2933) \to DK^*)$ , would provide a more stringent test of this assignment and aid in the search for the $2P'_1$ partner state.

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