Revealing the $D_0^*(2300)$ two-pole structure from lattice data and the SU(3) limit

This paper analyzes lattice QCD data using unitarized chiral perturbation theory to reveal that the experimental $D_0^*(2300)$ resonance corresponds to a two-pole structure, where a lower pole ($D_0^*(2100)$) behaves like the $\sigma$ meson and an upper pole relates to the $\mathbf{6}$ representation, with their distinct behaviors across chiral trajectories offering new insights into their underlying quark-gluon dynamics.

The Gist

Imagine the subatomic world as a bustling city where tiny particles called "mesons" are constantly bumping into each other, forming temporary partnerships, and sometimes splitting apart. For years, physicists have been trying to understand a specific, somewhat mysterious character in this city: a particle called $D^*_0(2300)$.

Think of this particle as a "ghost" that appears in experiments but is hard to pin down. The big question has been: Is it a single, solid object (like a brick), or is it a fleeting "dance" between two other particles coming together?

This paper is like a high-tech detective story where the authors use a powerful tool called Lattice Quantum Chromodynamics (LQCD)—which is essentially a super-accurate computer simulation of the universe's fundamental forces—to solve the mystery. They also use a mathematical framework called UChPT (Unitarized Chiral Perturbation Theory) to interpret the data.

Here is the breakdown of their findings using simple analogies:

1. The "Two-Pole" Mystery

For a long time, scientists thought the $D^*_0(2300)$ was just one particle. However, this paper reveals that it's actually two different "poles" (mathematical points that represent resonances or states) acting together.

• The Analogy: Imagine you hear a strange sound in a room. At first, you think it's one person humming. But after analyzing the sound waves carefully, you realize it's actually two people humming at slightly different pitches, creating a complex harmony.
• The Discovery: The authors found two distinct "voices" in the data:
  • The Lower Pole ($D^*_0(2100)$): This one is like a very tight hug between two particles (a $D$ meson and a pion). It is almost entirely made of these two dancing partners. The authors call this a "molecular state."
  • The Higher Pole ($D^*_0(2300)$): This one is the one we actually see in experiments. It's a bit more complex. It can be a resonance (a short-lived dance) or a "virtual state" (a ghostly presence that almost forms a bond but doesn't quite stick).

2. Changing the "Weather" (Pion Mass)

In the real world, the "weight" of the particles (specifically the pion) is fixed. But in computer simulations, scientists can change this weight to see how the particles behave under different conditions. The authors tested the particles as they changed the "pion mass" from light (real life) to very heavy (theoretical limits).

• The Analogy: Imagine watching a dance couple in different weather. In a light breeze (light pion mass), they dance freely. As the wind gets heavier and heavier (increasing pion mass), their dance changes.
• The Finding for the Lower Pole: As the "wind" got heavier, the lower pole split into two. One became a "bound state" (they stuck together permanently), and the other became a "virtual state" (they hovered near each other but didn't stick). This behavior is very similar to a famous particle called the $\sigma$ (sigma) resonance in a different part of physics.
• The Finding for the Higher Pole: This one was stubborn. No matter how heavy the "wind" got, its mass stayed roughly the same. Why? Because it has a "hidden secret": it is strongly connected to channels involving strange quarks (like $D\eta$ and $D_s\bar{K}$). It's like a dancer who is so focused on a specific partner that changing the weather doesn't affect their position.

3. The "SU(3) Limit" and the Hidden Component

The authors pushed their simulation all the way to a theoretical limit called the SU(3) limit, where the masses of different quarks become equal. This is like testing the dance in a perfectly symmetrical, frictionless room.

• The Twist: When they looked at the lower pole ($D^*_0(2100)$) in this perfect room, they found something surprising. In the real world, it's 99% a "molecule" (two particles dancing). But in this perfect SU(3) room, it became only about 63% a molecule.
• The Explanation: This means that in this specific theoretical limit, the particle needs a "third ingredient" to exist. The authors suggest this ingredient is a genuine quark-antiquark core (a $c\bar{q}$ state).
• The Analogy: Think of a cake. In our kitchen (real world), the cake is 99% flour and sugar (the two dancing particles). But in a magical kitchen (the SU(3) limit), the recipe changes, and you realize you actually need a secret egg (the quark core) to make the cake rise properly. Without that egg, the cake falls flat.

4. Why This Matters

The paper concludes that the $D^*_0(2300)$ is not just a simple brick; it is a complex system with two poles.

• One pole is a pure "molecular" dance between two particles.
• The other pole is a resonance that stays stable because of its connection to "strange" particles.
• Crucially, the study shows that depending on the conditions (the pion mass), the nature of these particles changes. Sometimes they are pure dances; sometimes they need a hidden core to exist.

In Summary:
The authors used computer simulations to show that the mysterious $D^*_0(2300)$ particle is actually a double act. One part is a pure partnership of two particles, while the other is a more complex entity that relies on "hidden strange" connections. They also discovered that if you change the fundamental rules of the universe (by changing particle masses), the "molecular" nature of these particles can fade, revealing a hidden core underneath. This helps explain why these particles are so hard to classify and supports the idea that they are dynamic, changing entities rather than static objects.

Technical Summary

Technical Summary: Revealing the $D^*_0(2300)$ Two-Pole Structure from Lattice Data and the SU(3) Limit

Problem Statement
The nature of the scalar charmed meson $D^*_0(2300)$ and its axial partner $D^*_1(2430)$ remains a subject of debate. While the quark model predicts these states to be broad $1P$ excitations, experimental observations and Lattice QCD (LQCD) data suggest they may be predominantly molecular states composed of two mesons. Furthermore, Unitarized Chiral Perturbation Theory (UChPT) predicts a two-pole structure for the $D^*_0(2300)$: a lower pole associated with the $\bar{3}$ representation and a higher pole linked to the $6$ representation. However, previous LQCD simulations in the $m_\pi \approx 200-400$ MeV range have struggled to confirm the existence of the second (higher energy) pole, and the connection between poles found near the physical point and those in the SU(3) limit remains unclear. Additionally, the quark mass dependence of these poles across different chiral trajectories has not been systematically investigated using high-precision LQCD data up to the SU(3) limit.

Methodology
The authors perform a comprehensive analysis of LQCD scattering data for light-pseudoscalar mesons interacting with charmed mesons using Next-to-Leading Order (NLO) UChPT. The study covers pion masses ranging from $m_\pi \simeq 230$ MeV to the SU(3) limit at $m_\pi \simeq 700$ MeV.

Key methodological components include:

• Data Integration: The analysis incorporates energy level data from multiple LQCD ensembles ($m_\pi = 239, 391, 688$ MeV) from the Hadron Spectrum Collaboration, including data for the $D\pi$, $D\eta$, $D_s\bar{K}$, and $DK$ coupled channels.
• Chiral Trajectories: The authors investigate the pion mass dependence of the poles along three distinct chiral trajectories:
  1. The symmetric line where $m_u = m_d = m_s$ (reached at $m_\pi \approx 420$ MeV).
  2. The trajectory where the strange quark mass is fixed to its physical value ($m_s = m_{s,phy}$), extending up to $m_\pi \approx 780$ MeV.
  3. The trajectory where the average light quark mass equals the strange quark mass ($m_s = m_{u(d)}$).
• Formalism: The interaction is derived from NLO UChPT, utilizing Low-Energy Constants (LECs) constrained by heavy meson masses and splittings. The scattering amplitude is implemented in a finite volume to match LQCD energy levels.
• Handling the SU(3) Limit: To address discrepancies in the SU(3) limit data (specifically from Ref. [28]), the authors introduce a bare $c\bar{q}$ component (a CDD pole) into the interaction potential. This allows for a combined fit of NLO UChPT and a bare quark-model-like state to describe the data near the SU(3) limit.
• Compositeness: The molecular nature (compositeness) of the states is evaluated using the Weinberg compositeness criterion.

Key Contributions and Results

1. Confirmation of Two-Pole Structure: The analysis confirms the existence of two poles in the non-strange isospin $I=1/2$ sector.

   • Lower Pole ($D^*_0(2100)$): Located at $\sqrt{s} \approx 2094 - i111$ MeV at the physical pion mass. This pole is consistently a resonance in the $D\pi$ channel within the $1\sigma$ region. As the pion mass increases, this pole splits into a virtual state and a bound state, eventually connecting to the $\bar{3}$ representation in the SU(3) limit.
   • Higher Pole ($D^*_0(2300)$): Located at $\sqrt{s} \approx 2463 - i108$ MeV at the physical pion mass. In the $1\sigma$ region, this pole can manifest as either a resonance or a virtual state close to the $D\eta$ and $D_s\bar{K}$ thresholds. This pole is linked to the $6$ representation.

2. Chiral Trajectory Dependence:

   • The lower pole ($D^*_0(2100)$) exhibits behavior similar to the $\sigma$ resonance in $\pi\pi$ scattering, splitting as the pion mass increases. Its compositeness is found to be $X \approx 0.99$ at the SU(3) limit in the pure UChPT description, indicating a predominantly molecular nature.
   • The higher pole shows a distinct behavior: its mass remains fairly constant in the $Tr[M]=C$ trajectory due to its coupling to channels with hidden strangeness ($D\eta, D_s\bar{K}$). This pole is identified with the $6$ representation, suggesting it is less affected by the $Q\bar{q}$ (quark-antiquark) state in the SU(3) limit.

3. Resolution of SU(3) Limit Discrepancies:

   • When analyzing the SU(3) limit data from Ref. [28], the pure NLO UChPT description fails to reproduce the pole positions accurately.
   • By including a bare $c\bar{q}$ component (CDD pole), the fit quality improves significantly. In this scenario, the compositeness of the lower pole ($D^*_0(2100)$) in the SU(3) limit is reduced to $X \approx 0.63$, indicating a significant mixing with a genuine quark-antiquark component at high pion masses ($m_\pi \approx 700$ MeV).
   • The higher pole remains associated with the $6$ representation and is consistent with the findings of Ref. [28] when the bare component is included.

4. Comparison with Experiment and Lattice:

   • The extrapolated pole positions at the physical point are consistent with other UChPT analyses (Refs. [17, 25]) but differ from the PDG average and recent LQCD simulations near the physical point that only included $D\pi$ interpolators.
   • The results support the hypothesis that the experimental $D^*_0(2300)$ is related to the higher pole, while the lower pole ($D^*_0(2100)$) is a distinct state that may not have been fully resolved in previous experimental analyses.

Significance and Claims
The paper claims to provide the first investigation of the pion mass dependence of the two-pole structure of the $D^*_0(2300)$ across multiple chiral trajectories, including the SU(3) limit. The authors highlight that:

• The two-pole structure is robust across the studied pion mass range.
• The distinction between the $\bar{3}$ (lower pole) and $6$ (higher pole) representations is maintained throughout the chiral trajectories.
• The higher pole's stability in the $Tr[M]=C$ trajectory serves as a testable prediction for future LQCD simulations to confirm the nature of the $D^*_0(2300)$.
• The necessity of including a bare $c\bar{q}$ component to describe the SU(3) limit data suggests a transition in the nature of the $D^*_0(2100)$ from a predominantly molecular state at physical pion masses to a state with significant quark-model character at the SU(3) limit.

The work reinforces the two-pole hypothesis for the $D^*_0(2300)$ and offers a framework to reconcile UChPT predictions with LQCD data across the full range of quark masses.