Radiative decays of the 1$P$, 1$D$, 2$S$, and 2$P$ $\Lambda_c$ and 1$D$, 2$S$, and 2$P$ $\Xi_c$ charmed baryons

Using the constituent quark model, this paper analyzes the radiative decays of various excited $\Lambda_c$ and $\Xi_c$ charmed baryons in the flavor anti-triplet, providing branching ratios and decay widths that aid in identifying resonances and clarifying the nature of the $\Xi_c(3055)$ and $\Xi_c(3080)$ states.

The Gist

Imagine the subatomic world as a bustling city where particles are the citizens. Among these citizens are charmed baryons, which are like heavy, three-person families made of quarks. Specifically, this paper focuses on two types of families: the $\Lambda_c$ (a family with two light members and one "charmed" heavy member) and the $\Xi_c$ (a family with one light, one strange, and one charmed member).

For a long time, scientists have been finding excited versions of these families—like children growing up and jumping around. But sometimes, two different families look so similar (they have the same weight and behave similarly) that it's hard to tell them apart. It's like trying to distinguish between two identical twins just by looking at their height; you need a different test.

This paper is about a specific test: radiative decay.

The "Flashlight" Test

Think of these excited baryon families as people holding a flashlight. When they are excited, they eventually calm down to a resting state. To do this, they sometimes flash a beam of light (a photon) to release their extra energy. This is called a "radiative decay."

The authors of this paper acted like forensic accountants for these particles. They didn't just guess how much light these families would flash; they used a detailed mathematical model (the "Constituent Quark Model") to calculate exactly how bright that flash should be for different types of families.

What They Did

The researchers looked at several generations of these families:

1. The Ground Floor: The calm, resting families.
2. The First Floor (P-wave): Families that are slightly excited, spinning or moving in a specific way.
3. The Second Floor (2S, 2P, 1D, 2D): Families that are even more excited, jumping higher or spinning differently.

They calculated the "brightness" (decay width) of the flash for many different scenarios, including some very complex configurations that had never been calculated before, such as families where the members are moving in mixed patterns (like a dance where one partner spins while the other jumps).

Solving the Mystery of the "Twins"

The most exciting part of their work is solving a real-world mystery involving two specific particles discovered by the LHCb experiment: $\Xi_c(3055)$ and $\Xi_c(3080)$.

For a long time, scientists weren't sure what kind of "dance" these particles were doing (their quantum numbers).

• The $\Xi_c(3055)$: The LHCb experiment recently figured out this particle is a "D-wave" dancer with a specific spin. The authors of this paper ran their calculations using this new information. They found that the "brightness" of the light flash their model predicted matches the experimental data perfectly. It's like confirming that the twins are indeed the same person because their fingerprints match.
• The $\Xi_c(3080)$: This one is still a bit of a mystery. The authors proposed two possibilities:
  1. It could be the "twin" of the 3055, just with a slightly different spin (a 5/2+ dancer).
  2. Or, it could be a completely different type of dancer (a 1/2+ dancer, perhaps a "radial" jumper).

The paper provides a list of "branching ratios," which are like probabilities. They say: "If the $\Xi_c(3080)$ is Type A, it will flash light in this specific pattern. If it is Type B, it will flash in a totally different pattern." This gives experimentalists a clear checklist to look for in their data to finally identify what this particle really is.

The "Safety Net" of Uncertainty

One unique thing about this paper is that the authors didn't just give a single number. They acknowledged that their model and the experimental measurements have small errors (like a ruler that might be slightly off). They used a computer simulation (Monte Carlo method) to run the calculation thousands of times with slightly different inputs. This gave them a range of likely answers rather than a single guess, making their conclusions much more reliable.

Summary

In short, this paper is a theoretical guidebook for physicists. It calculates exactly how heavy, excited particle families should emit light when they calm down. By comparing these calculations to real-world observations, the authors:

1. Confirmed the identity of the $\Xi_c(3055)$.
2. Provided a roadmap of "what to look for" to finally identify the $\Xi_c(3080)$.
3. Filled in the blanks for many other excited particle states that haven't been fully understood yet.

They didn't invent new technology or cure diseases; they simply provided the precise "blueprints" needed to help experimentalists identify the true nature of these tiny, heavy particles in the universe.

Technical Summary

Technical Summary: Radiative Decays of Excited Charmed Baryons in the Flavor Antitriplet

Problem Statement
The spectroscopy of singly heavy baryons has advanced significantly with the recent observation of numerous excited states by collaborations such as LHCb, Belle, and BESIII. While mass measurements and decay widths have been established for many states, determining their quantum numbers ($J^P$) and internal configurations remains a critical challenge, particularly when multiple resonances share similar masses and strong decay widths. Specifically, the nature of the $\Xi_c(3055)^{+,0}$ and $\Xi_c(3080)^{+,0}$ states has been a subject of theoretical debate. Although recent LHCb data suggests $\Xi_c(3055)$ corresponds to a $1D$-wave excitation, the assignment of $\Xi_c(3080)$ remains open, with possibilities including the $1D$ state with $J^P=5/2^+$ or a radially excited $2S$ state with $J^P=1/2^+$. Electromagnetic decay widths (EMDs) serve as a crucial tool for distinguishing these assignments, yet comprehensive calculations for higher-shell states, particularly those involving specific orbital modes and mixed configurations in the flavor antitriplet ($\bar{3}_F$), have been limited.

Methodology
The authors employ the Constituent Quark Model (CQM) to analyze radiative decays of $\Lambda_c$ and $\Xi_c$ baryons belonging to the flavor antitriplet ($\bar{3}_F$). The study utilizes a Hamiltonian that includes constituent masses, a harmonic oscillator potential, and spin-orbit, isospin, and flavor-dependent terms. The baryon states are described using Jacobi coordinates ($\rho$ and $\lambda$) to separate the relative motion of light quarks from the light-quark pair and the charm quark.

The electromagnetic transition amplitudes are calculated analytically for transitions $A \to A'\gamma$, where $A$ and $A'$ denote initial and final baryon states. The formalism treats the emission of left-handed photons using an interaction Hamiltonian involving magnetic moments, spin-ladder, and momentum-ladder operators. Unlike previous studies that relied on the Close–Copley replacement to evaluate transition amplitudes, this work computes them analytically without such approximations.

The analysis covers:

1. $\Lambda_c$ baryons: Transitions from ground ($N=0$) and P-wave ($N=1$) states to ground states, as well as transitions from all second-shell ($N=2$) states to ground and P-wave final states.
2. $\Xi_c$ baryons: Transitions from second-shell ($N=2$) states to ground and P-wave final states (ground and P-wave transitions were addressed in prior work).

The study explicitly calculates EMDs for:

• $D_\rho$-wave states.
• $\rho-\lambda$ mixed configurations.
• Radially excited $\rho$-mode states.

Uncertainties are rigorously accounted for by propagating errors from both experimental mass measurements (PDG) and theoretical mass predictions (Ref. [98]) using a Monte Carlo bootstrap procedure (1000 iterations).

Key Contributions and Results
The paper provides the first calculation of electromagnetic decays for $D_\rho$-wave, $\rho-\lambda$ mixed, and radially excited $\rho$-mode states in singly charmed baryons of the flavor antitriplet. The results are presented in Tables I–IV, detailing predicted decay widths (in keV) and branching ratios for various channels.

• $\Xi_c(3055)^{+,0}$ Assignment: The authors interpret $\Xi_c(3055)$ as a $1D$-wave excitation with $J^P=3/2^+$ and $S=1/2$ in the flavor antitriplet. Calculations using both theoretical and experimental masses yield consistent results. The derived branching ratios, such as $\Gamma[\Xi_c(3055) \to \Xi_c \gamma] / \Gamma[\Xi_c(3055) \to \Xi'_c \gamma] \approx 8.6$ and $\Gamma[\Xi_c(3055) \to \Xi_c \gamma] / \Gamma[\Xi_c(3055) \to \Xi^*_c \gamma] \approx 39$, are presented as values that can confirm the LHCb assignment.
• $\Xi_c(3080)^{+,0}$ Assignment: Two scenarios are explored:
  1. $J^P = 5/2^+$ ($1D$ state): The predicted branching ratios (e.g., $\Gamma[\Xi_c(3080) \to \Xi_c \gamma] / \Gamma[\Xi_c(3080) \to \Xi'_c \gamma] \approx 8.5$) support the hypothesis that $\Xi_c(3080)$ is the partner of $\Xi_c(3055)$ in the $1D$ doublet.
  2. $J^P = 1/2^+$ ($2S$ state): Alternative calculations for a $2S$ configuration yield significantly different branching ratios (e.g., $\Gamma[\Xi_c(3080) \to \Xi_c \gamma] / \Gamma[\Xi_c(3080) \to \Xi'_c \gamma] \approx 0.013$).
     The authors emphasize that these distinct branching ratios serve as discriminators for future experimental studies to determine the correct quantum numbers of $\Xi_c(3080)$.

Significance and Claims
The paper claims that its primary significance lies in providing a comprehensive set of electromagnetic decay widths for second-shell charmed baryons, including previously uncalculated $D_\rho$, mixed, and radially excited modes. The authors assert that these results are particularly valuable for resonance identification when states have overlapping masses and strong widths.

The work explicitly states that it offers the first calculation of EMDs for $D_\rho$-wave, $\rho-\lambda$ mixed, and radially excited $\rho$-mode states in the flavor antitriplet. Furthermore, the inclusion of both experimental and model-dependent uncertainties via a Monte Carlo method is highlighted as a necessary step for statistically significant comparisons between theory and experiment, addressing a gap in previous literature. The authors conclude that their interpretation of $\Xi_c(3055)$ is consistent with recent LHCb determinations and that their branching ratio predictions provide a concrete framework for testing the nature of the $\Xi_c(3080)$ state.