Pion radiative decays of excited hidden-charm pentaquark molecules: from $\Sigma_c^{(*)}\bar{D}^{(*)}(2S)$ molecules to the reported $P_c$ states

This paper systematically investigates the pion-emission decays of excited hidden-charm pentaquark molecules composed of $\Sigma_c^{(*)}$ and radially excited $\bar{D}^{(*)}(2S)$ states into known ground-state $P_c$ molecules, revealing that decay widths are highly sensitive to spin structures and coupled-channel interferences to provide decisive experimental signatures for future verification at LHCb and PANDA.

The Gist

Imagine the universe is filled with a vast, invisible library of tiny building blocks called hadrons. Most of these blocks are like standard Lego sets: you have a few basic pieces (quarks) snapped together in predictable ways. But sometimes, nature builds something stranger—exotic structures that don't fit the standard rules. One of the most exciting recent discoveries in this library is a family of "hidden-charm pentaquarks," named $P_c$ states.

Think of these $P_c$ states not as single, solid bricks, but as loose couples dancing together. Specifically, they are pairs made of a heavy "charmed baryon" (a heavy dancer) and an "anti-charmed meson" (a lighter partner). Scientists believe these pairs are held together by a gentle force, much like two people holding hands while spinning, rather than being glued into a single rigid block.

The Big Question: Are There "Older" Versions?

In 2015 and 2019, the LHCb experiment at CERN discovered three specific versions of these dancing couples: $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$. These are the "ground state" couples—their partners are in their most relaxed, lowest-energy pose.

But just like a human can stretch their arms or jump up high, these particle couples can also get "excited." The authors of this paper ask a simple question: What if the lighter partner in the dance is jumping up a step?

In the world of atoms, electrons can jump to higher energy levels. In this particle world, the "anti-charmed meson" partner can jump to a higher energy orbit, called a 2S state. This creates a new, excited version of the $P_c$ molecule. The paper investigates these "excited dancers" and asks: How do they calm down and return to their normal, ground-state dance?

The Mechanism: The "Pion" Balloon

The paper proposes a specific way these excited molecules lose their extra energy. Imagine the excited dancer is holding a small, invisible balloon called a pion (a type of light particle).

1. The Jump: The excited molecule (the dancer with the balloon) is unstable.
2. The Release: To calm down, the molecule pops the balloon. The balloon flies away (this is the pion emission).
3. The Landing: The molecule settles back down into a stable, ground-state dance (one of the known $P_c$ states we already know).

The authors used a sophisticated mathematical toolkit (the Chiral Quark Model) to calculate exactly how fast this "balloon popping" happens and how much energy is released. They treated the molecules like complex waves, calculating how the different parts of the dance interfere with each other.

The Surprising Results: A Game of Interference

The most fascinating part of the paper is that the outcome depends entirely on the spin (the direction of the dance spin) and how the different parts of the molecule mix together.

• The "Constructive" Dance: For some specific excited molecules, the different ways they can pop the balloon work together perfectly, like a choir singing in harmony. This results in a loud, fast decay (a wide decay width of several MeV). For example, an excited molecule can quickly turn into the known $P_c(4440)$ state.
• The "Destructive" Dance: For other configurations, the different ways to pop the balloon cancel each other out, like noise-canceling headphones. This makes the decay extremely slow or almost impossible. The paper found that for certain excited states, the path to becoming $P_c(4457)$ is blocked by this "destructive interference," making the decay width tiny (less than 0.3 MeV).

Why This Matters

The authors argue that if we can build a machine sensitive enough to see these "balloon popping" events, we can solve a mystery.

Currently, we know the $P_c$ states exist, but we don't know for sure if they are truly "molecules" (loose couples) or something else. If future experiments (like the upgraded LHCb or the PANDA experiment) see these specific excited molecules decaying into the known $P_c$ states by releasing a pion, it would be the smoking gun. It would prove that these particles are indeed molecular structures with internal energy levels, just like atoms.

Summary in a Nutshell

• The Subject: The paper looks for "excited" versions of mysterious particle couples called hidden-charm pentaquarks.
• The Process: It calculates how these excited couples lose energy by shooting out a tiny particle called a pion, turning into the "ground state" couples we already know.
• The Discovery: The speed of this process depends heavily on the spin of the particles. Sometimes the process is fast and easy; other times, the physics of the situation cancels it out, making it very rare.
• The Goal: These calculations provide a "recipe" for future experiments. If they see these specific decays, it confirms that pentaquarks are molecular structures, opening a new chapter in understanding how the strong force holds the universe together.

Technical Summary

Technical Summary: Pion Radiative Decays of Excited Hidden-Charm Pentaquark Molecules

Problem Statement
Following the LHCb Collaboration's discovery of the hidden-charm pentaquark states $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$, which are widely interpreted as hadronic molecules composed of a charmed baryon and an anti-charmed meson ($\Sigma_c^{(*)}\bar{D}^{(*)}$), a critical theoretical question remains: Do excited molecular partners of these states exist? Specifically, if the ground states are composed of ground-state mesons, general hadron spectroscopy suggests the existence of states built from radially excited anti-charmed mesons, denoted as $\Sigma_c^{(*)}\bar{D}^{(*)}(2S)$. The central problem addressed in this work is to determine the properties of these excited molecular candidates and, crucially, to calculate their decay mechanisms into the known ground-state $P_c$ molecules. The authors posit that pion-emission decays ($P_c' \to P_c + \pi$) serve as the most direct probe for identifying these excited states and establishing their dynamical connections to the observed spectrum.

Methodology
The study employs a two-step theoretical framework combining the One-Boson-Exchange (OBE) model and the Chiral Quark Model:

1. Mass Spectrum and Wave Functions: The authors first utilize OBE effective potentials to predict the mass spectrum and bound state solutions for $\Sigma_c^{(*)}\bar{D}^{(*)}(2S)$ molecular candidates. They note that in the heavy quark limit, the potentials for ground-state baryons interacting with radially excited mesons share the same form as those with ground-state mesons, but with a larger reduced mass leading to potentially deeper binding energies. The spatial wave functions for the constituents are modeled using Simple Harmonic Oscillator (SHO) functions, while the inter-constituent wave functions are obtained by numerically solving the Schrödinger equation with the OBE potentials, explicitly including coupled-channel effects among different $\Sigma_c^{(*)}\bar{D}^{(*)}$ components.
2. Decay Width Calculation: The pion-emission decay widths are calculated within the Chiral Quark Model. The effective Lagrangian describing quark-pseudoscalar meson interactions is expanded in a non-relativistic form up to the order of $1/m^2$. The decay amplitude is constructed by evaluating the matrix element of the effective Hamiltonian between the initial excited molecular state and the final ground-state molecular state plus a pion. The calculation accounts for the interference of various coupled-channel contributions (e.g., transitions involving $\Sigma_c\bar{D}(2S)$, $\Sigma_c\bar{D}^*(2S)$, and $\Sigma_c^*\bar{D}^*(2S)$ components).

Key Contributions and Results
The paper systematically investigates the pion-emission transitions from various excited $\Sigma_c^{(*)}\bar{D}^{(*)}(2S)$ molecular configurations to the known $P_c$ states. The results are highly sensitive to the spin structures and coupled-channel interferences:

• $\Sigma_c\bar{D}(2S)/\Sigma_c\bar{D}^*(2S)/\Sigma_c^*\bar{D}^*(2S)$ with $I(J^P) = 1/2(1/2^-)$:

  • The decay to $P_c(4440)$ is significant, with widths reaching several MeV.
  • The decay to $P_c(4457)$ is strongly suppressed (below 0.3 MeV) due to destructive interference between coupled-channel amplitudes.
  • The decay to $P_c(4312)$ is also on the order of several MeV.
  • The decay to $P_c(4380)$ is suppressed (~0.25 MeV) because the dominant $\Sigma_c^*\bar{D}$ component of $P_c(4380)$ requires a heavy-quark spin flip, and the contributing channel is suppressed by the small $\Sigma_c^*\bar{D}^*(2S)$ component in the initial state.

• $\Sigma_c\bar{D}^*(2S)/\Sigma_c^*\bar{D}^*(2S)$ with $I(J^P) = 1/2(1/2^-)$:

  • Decays to $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$ all exhibit widths of approximately 1 MeV, dominated by the $\Sigma_c\bar{D}^*(2S) \to \Sigma_c\bar{D}^* + \pi$ mechanism.
  • The decay to $P_c(4380)$ is also around 1 MeV but arises from different dominant mechanisms involving $\Sigma_c^*\bar{D}^*(2S)$ components.

• States with $I(J^P) = 1/2(3/2^-)$:

  • Generally, decay widths are smaller compared to the $1/2^-$ cases due to weaker spin-spin interactions and destructive interference.
  • For the $\Sigma_c\bar{D}^*(2S)/\Sigma_c^*\bar{D}(2S)/\Sigma_c^*\bar{D}^*(2S)$ system, the width to $P_c(4312)$ is extremely small (tens of keV).
  • For the $\Sigma_c^*\bar{D}(2S)/\Sigma_c^*\bar{D}^*(2S)$ system, widths to $P_c(4380)$, $P_c(4440)$, and $P_c(4457)$ are on the order of a few MeV, with $P_c(4380)$ being the largest due to the large probability of the $\Sigma_c^*\bar{D}$ component in the final state.

• $\Sigma_c^*\bar{D}^*(2S)$ with $I(J^P) = 1/2(5/2^-)$:

  • The decay to $P_c(4380)$ drops sharply (~0.03 MeV).
  • The channel to $P_c(4457)$ becomes relatively prominent (up to ~0.75 MeV).
  • Decays to $P_c(4312)$ and $P_c(4440)$ are nearly zero.

• General Trends: The authors note that neutral pion emission widths are approximately half of the charged pion emission widths due to isospin relations. Overall, $P_c(4380)$ is identified as the dominant decay recipient for low-spin excited states, while $P_c(4312)$ is strongly suppressed across all spin configurations.

Significance and Claims
The paper claims that pion-emission decays provide a decisive experimental signature for unveiling the excited molecular spectrum of hidden-charm pentaquarks. By establishing a specific decay connection between radially excited anti-charmed meson-based molecules and the observed $P_c$ states, the work offers a method to distinguish different spin-parity assignments of excited molecular pentaquarks. The authors assert that the predicted decay widths (ranging from keV to several MeV) are within the reach of current and future experiments, specifically citing the upgraded LHCb and the PANDA experiment as facilities capable of verifying these predictions. The observation of such signals would provide strong evidence for the molecular interpretation of pentaquarks and open a new chapter in exotic hadron spectroscopy.