Observation and investigation of the $T_{c\bar{c}1}(4430)^{+}$ structure in $B^{+} \to \psi(2S) K_{\text{S}}^{0} \pi^{+}$ decays

Using a four-dimensional amplitude analysis of proton-proton collision data from the LHCb experiment, this study confirms the existence of the exotic $T_{c\bar{c}1}(4430)^{+}$ structure in $B^{+} \to \psi(2S) K_{\text{S}}^{0} \pi^{+}$ decays and investigates its properties through both Flatté parametrization and triangle singularity mechanisms.

The Gist

Imagine the universe as a giant, cosmic LEGO set. For decades, physicists have known the basic rules: you can build "normal" structures using specific bricks. In the world of subatomic particles, these normal structures are mesons (made of two bricks stuck together) and baryons (made of three).

But for a long time, physicists suspected there were "exotic" structures—complex, four-brick creations that didn't fit the standard rules. One of the most famous suspects is a particle called $T_{c\bar{c}}1(4430)^+$. It's like a mysterious, four-brick LEGO tower that was spotted once before, but nobody was 100% sure how it was built or if it was a real, stable object or just a weird optical illusion caused by the way the pieces were thrown together.

This new paper from the LHCb collaboration at CERN is like a high-definition, slow-motion replay of that event, confirming the tower exists and giving us a much better look at its blueprint.

Here is the breakdown of what they found, using simple analogies:

1. The Setup: A Cosmic Pinball Machine

The scientists used the Large Hadron Collider (LHC) to smash protons together at nearly the speed of light. Think of this as a massive, chaotic pinball machine. When the balls (protons) hit each other, they shatter and create a shower of new particles.

The team specifically looked for a specific "cascade" of events: A heavy particle called a $B^+$ meson decays (breaks apart) into three things:

1. A $\psi(2S)$ (a heavy, excited version of a particle).
2. A $K^0_S$ (a strange particle).
3. A $\pi^+$ (a pion).

2. The Mystery: The "Ghost" in the Machine

The scientists had a theory about how this decay should happen. They expected the $B^+$ to break into a $\psi(2S)$ and a temporary, short-lived particle called a $K^*$ (like a spinning top that immediately falls apart into the $K^0_S$ and $\pi^+$).

They ran the simulation (the "blueprint") based on this theory.

• The Result: The simulation matched the data perfectly for two of the three particles.
• The Problem: When they looked at the relationship between the $\psi(2S)$ and the $\pi^+$, the data didn't match the blueprint. There was a "bump" or a "hump" in the data around a specific energy level (4.45 GeV) that the standard theory couldn't explain.

It was like listening to a song where the drums and bass fit perfectly, but the melody had a weird, unexpected note that the composer didn't write.

3. The Discovery: The Exotic Tetraquark

To explain that weird "note," the scientists added a new ingredient to their model: the $T_{c\bar{c}}1(4430)^+$.

Think of this particle as a quartet.

• Standard particles are duos (quark + anti-quark) or trios (three quarks).
• This exotic particle is a tetraquark: it's made of four "bricks" stuck together (a charm quark, an anti-charm quark, an up quark, and a down quark).

When they added this "quartet" to their math, the weird bump in the data vanished, and the model fit the reality perfectly. It's like realizing the song wasn't missing a note; it was actually a harmony between four instruments, not just three.

The Evidence:

• The Shape: The "bump" wasn't random noise. When they plotted the data on a graph (called an Argand diagram), the points traced a perfect circle. In physics, a circle like this is the fingerprint of a real, resonant particle, proving it's not just a statistical fluke.
• The Confidence: They are so sure of this that the statistical significance is over 16 sigma. To put that in perspective: if you flipped a coin and got heads 16 times in a row, that's a coincidence. This result is like flipping a coin and getting heads 16 times every single time you try, for a billion years. It is a definitive discovery.

4. The "How": Two Possible Blueprints

Now that they know the tower exists, they asked: How is it built? They tested two main theories:

Theory A: The Molecular Bond (The Flatté Model)
Imagine the four bricks aren't glued tightly together, but are two pairs of bricks holding hands loosely, like a molecule.

• The scientists checked if this particle was trying to break apart into a specific pair of other particles ($D^*_1$ and $D$).
• The Finding: They found that if this "molecular" bond exists, it's very weak. The particle prefers to stay together and decay into the $\psi(2S)$ and $\pi^+$ rather than breaking into those other pairs. They set a strict limit on how likely that break-up is.

Theory B: The Triangle Illusion (The Triangle Singularity)
Imagine a billiard ball hitting another ball, which hits a third, and the path of the third ball creates a "ghost" peak that looks like a new particle but is actually just a trick of geometry.

• The scientists tested a complex "triangle" scenario where particles bounce off each other in a specific sequence to create this bump.
• The Finding: This "geometric trick" also fits the data quite well.

The Conclusion: Both theories (a real particle or a geometric trick) fit the data. The paper doesn't definitively say which one it is yet, but it proves that something exotic is happening. It's like seeing a shadow on the wall; you know there's an object casting it, even if you aren't 100% sure if the object is a dog or a statue yet.

Why Does This Matter?

This is a big deal for the "Standard Model" of physics.

1. It confirms the existence of "Exotic" matter: It proves that nature can build complex structures out of four quarks, not just two or three.
2. It solves a 20-year mystery: This particle was first spotted in 2008. It took nearly two decades and a massive amount of data to confirm it wasn't a mistake.
3. It opens a new door: Understanding how these four-brick structures hold together helps us understand the "glue" (the strong nuclear force) that holds the entire universe together.

In a nutshell: The LHCb team took a massive amount of data, found a weird "ghost" in the numbers, and proved it was actually a real, four-piece particle family member. They've confirmed it exists, measured its weight and spin, and are now trying to figure out exactly how its internal parts are glued together.

Technical Summary

1. Problem and Motivation

The nature of exotic hadrons—particles that do not fit the conventional quark model of $q\bar{q}$ mesons or $qqq$ baryons—remains a central open question in Quantum Chromodynamics (QCD). The charged charmonium-like structure $T_{c\bar{c}1}(4430)^+$ (historically referred to as $Z(4430)^+$) was first observed in the decay $\bar{B}^0 \to \psi(2S)K^-\pi^+$. Its minimal quark content is $c\bar{c}u\bar{d}$, and its spin-parity was determined to be $J^P = 1^+$.

However, the physical nature of this state is debated:

• Dynamical Interpretations: It could be a genuine exotic state, either a hadronic molecule (e.g., $D^*_1 D$) or a compact tetraquark.
• Kinematic Interpretations: It could be a kinematic effect, such as a triangle singularity arising from rescattering processes in the decay chain, rather than a bound state.

Previous studies were limited to the isospin-related channel $\bar{B}^0 \to \psi(2S)K^-\pi^+$. This paper addresses the need to investigate the same structure in the charged $B^+$ decay channel ($B^+ \to \psi(2S)K^0_S\pi^+$) to confirm its existence via isospin symmetry and to probe its nature using different theoretical models (molecular vs. triangle singularity) within a full amplitude analysis.

2. Methodology

The analysis utilizes proton-proton collision data collected by the LHCb experiment at a center-of-mass energy of $\sqrt{s} = 13$ TeV, corresponding to an integrated luminosity of $5.4 \text{ fb}^{-1}$.

Data Selection and Reconstruction:

• Signal: $B^+ \to \psi(2S)K^0_S\pi^+$ candidates are reconstructed via $\psi(2S) \to \mu^+\mu^-$ and $K^0_S \to \pi^+\pi^-$.
• Background Suppression: A Boosted Decision Tree (BDT) classifier is used to suppress combinatorial background.
• Signal Yield: An unbinned extended maximum-likelihood fit to the $B^+$ invariant mass yields a signal count of $9600 \pm 100$.
• Background Subtraction: The sFit technique is employed to assign signal weights to candidates, allowing for background subtraction in the subsequent amplitude analysis without explicitly modeling the background shape in the fit.

Amplitude Analysis:
A four-dimensional amplitude analysis is performed using the helicity formalism. The kinematics are described by four independent variables:

1. $m_{K\pi}$: Invariant mass of the $K^0_S\pi^+$ system.
2. $\cos \theta_{K^*}$: Helicity angle of the $K^*$ decay.
3. $\cos \theta_{\psi}$: Helicity angle of the $\psi(2S)$ decay.
4. $\phi$: Angle between the $K^*$ and $\psi(2S)$ decay planes.

Models Tested:

1. Baseline Model: Includes only known $K^*$ resonances decaying to $K^0_S\pi^+$ (e.g., $K^*(892)^+$, $K^*_0(700)^+$, etc.).
2. Model-Independent Approach: A cubic spline interpolation of amplitudes at six fixed $m_{\psi\pi}$ values to characterize any unmodeled structure.
3. Resonance Model: Adds a $T_{c\bar{c}}^+$ component decaying to $\psi(2S)\pi^+$, modeled with a relativistic Breit-Wigner function.
4. Flatté Parametrization: Tests the hypothesis that $T_{c\bar{c}1}(4430)^+$ is a molecular state coupling to the open-charm threshold $D^*_1(2600)^0 D^+$.
5. Triangle Singularity (TS): Tests a kinematic mechanism where the structure arises from the rescattering $\psi(4230)\pi^+ \to \psi(2S)\pi^+$ in a $B^+ \to \psi(4230)K^*(892)^+$ cascade.

3. Key Contributions and Results

Observation of the Structure:

• The baseline model (containing only $K^*$ resonances) fails to describe the data, specifically showing a significant discrepancy in the $\psi(2S)\pi^+$ invariant mass ($m_{\psi\pi}$) distribution in the range $4.2 < m_{\psi\pi} < 4.7 \text{ GeV}/c^2$.
• Adding a $T_{c\bar{c}}^+$ component significantly improves the fit quality.
• Significance: The statistical significance of the $T_{c\bar{c}}^+$ state with $J^P=1^+$ is found to be >16$\sigma$ (statistical) and remains >9$\sigma$ after including systematic uncertainties.

Measured Properties:
The properties of the observed structure in $B^+ \to \psi(2S)K^0_S\pi^+$ are consistent with the $T_{c\bar{c}1}(4430)^+$ observed in the neutral $B$ decay:

• Mass: $M = 4.452 \pm 0.016 \pm 0.055 \text{ GeV}/c^2$
• Width: $\Gamma = 0.174 \pm 0.019 \pm 0.083 \text{ GeV}$
• Spin-Parity: $J^P = 1^+$ (Other hypotheses like $0^-, 1^-, 2^-, 2^+$ are rejected with $>6\sigma$ significance).
• Fit Fraction: $f_{T_{c\bar{c}}} = (3.7 \pm 0.6 \pm 4.0)\%$.

Investigation of Nature (Molecular vs. Kinematic):

• Molecular Hypothesis (Flatté): The analysis investigates the coupling to the $D^*_1(2600)^0 D^+$ threshold. The fit yields a coupling ratio upper limit of $R = |g_2/g_1| < 6.8$ at 95% confidence level, constraining the strength of the decay into the open-charm channel relative to $\psi(2S)\pi^+$.
• Triangle Singularity Hypothesis: A model based on the $K^*(892)^+\psi(4230)\pi^+$ triangle diagram provides a reasonable description of the data, with a fit fraction of $(3.9 \pm 0.7 \pm 3.3)\%$, consistent with the Breit-Wigner result. The TS model exhibits a phase shift behavior similar to the resonance model.
• Conclusion on Nature: The data is consistent with both the resonance (Breit-Wigner) and the triangle singularity interpretations. Larger datasets are required to distinguish between these two physical origins.

4. Significance

• Confirmation of Isospin Symmetry: This is the first observation of the $T_{c\bar{c}1}(4430)^+$ structure in the $B^+$ decay channel, confirming its existence as an isospin partner to the state seen in $\bar{B}^0$ decays.
• Methodological Advance: This represents the first four-dimensional amplitude analysis of this specific decay mode, utilizing a model-independent approach to reveal the phase motion of the structure, which strongly suggests a resonant or singular behavior rather than a statistical fluctuation.
• Theoretical Constraints: The study provides crucial constraints on the nature of the state. While it cannot definitively rule out the triangle singularity mechanism, it places strict limits on the coupling of the state to the $D^*_1 D$ channel, challenging pure molecular interpretations that predict strong coupling to open charm.
• Future Outlook: The results highlight the need for higher statistics to distinguish between the dynamical (resonance) and kinematic (triangle singularity) origins of the $T_{c\bar{c}1}(4430)^+$, a key step in understanding the non-perturbative regime of QCD.