Predictions for the scalar partner of the LHC tetraquark $X(6600)$

This paper predicts the existence of a lighter scalar partner, $X(6400)$, for the recently observed $X(6600)$ tetraquark based on CMS data, urging further experimental investigation to confirm an S-wave multiplet of $cc\bar{c}\bar{c}$ states and distinguish between competing theoretical models.

The Gist

Imagine the universe is filled with a vast zoo of tiny particles. For a long time, we knew about the "standard" animals: single particles like electrons and protons, and simple pairs like atoms. But recently, physicists have started spotting strange, exotic creatures made of four heavy particles stuck together. These are called tetraquarks.

This paper is like a detective story about a specific family of these exotic creatures made entirely of "charm" particles (a type of heavy quark). The authors are trying to figure out exactly what these creatures look like on the inside and predict what other members of their family might be hiding in the data.

Here is the breakdown of their findings in simple terms:

1. The Mystery of the "Four-Charmed" Family

Scientists at the Large Hadron Collider (LHC) have recently found three heavy particles that look like they are made of four charm quarks. They named them X(6600), X(6900), and X(7100) based on their weight (mass).

• The New Clue: A recent experiment by the CMS team at the LHC measured the "spin" and symmetry of the heaviest of these, X(6600). They found it acts like a 2++ particle.
• The Authors' Prediction: The authors of this paper had previously guessed that X(6600) would have these exact properties. They were right!

2. The Missing "Sibling" (The Scalar Partner)

Here is the main point of the paper: In the world of particle physics, particles usually come in families or "multiplets," much like a set of siblings. If you have one heavy sibling with a specific shape (spin), you usually expect to find a lighter sibling with a slightly different shape right next to it.

• The Theory: If X(6600) is a "tensor" particle (spin 2++), physics models say there must be a lighter "scalar" partner (spin 0++) sitting just below it in weight.
• The Prediction: The authors predict this missing sibling, which they call X(6400), should weigh about 6400 MeV (roughly 200 units lighter than X(6600)).
• The Evidence: The authors looked at the raw data from the CMS experiment again. They noticed a small, strange bump in the data right around 6400. The CMS team originally thought this bump was just "noise" or background static. However, the authors argue that this bump is actually the X(6400) particle they predicted. It's just quieter and harder to see than its heavier brother.

3. Two Ways to Build a House: Quarks vs. Diquarks

To understand what these particles are made of, physicists use two different "blueprints" or models:

1. The Quark Model: Imagine the particle is a house built with four individual bricks (quarks) arranged in a specific way.
2. The Diquark Model: Imagine the bricks come pre-glued in pairs. So, the house is built with two "double-bricks" (diquarks).

Why does this matter?
The authors say that if we can measure the exact weight of the missing sibling (X(6400)), we can tell which blueprint is correct.

• If the particle weighs ~6443, it supports the Quark Model (four individual bricks).
• If it weighs ~6513, it supports the Diquark Model (two glued pairs).

Currently, the data is a bit fuzzy, but the authors are urging scientists to look closer at that 6400 bump to settle the score.

4. How to Find the Hidden Sibling

The authors also explain how to find this hidden particle, because it's tricky:

• The "Silent" Decay: The heavier brother (X(6600)) is loud and easy to spot because it decays (breaks apart) into two specific particles called J/ψ. The lighter sister (X(6400)) is much quieter; it doesn't break apart into J/ψ as often. That's why it's been hiding in the data.
• The Better Search: The authors suggest looking for a different decay pattern. They predict the lighter sister is much more likely to break apart into a different pair of particles called $\eta_c \eta_c$. If scientists search for this specific combination, they might finally catch a clear glimpse of the X(6400).

5. The Big Picture

The paper concludes that if we can confirm the existence of this X(6400) particle and measure its weight, we will have found the first complete "S-wave multiplet" of these four-charm particles.

Think of it like finally finding the last missing piece of a puzzle. Once we have the whole family (the heavy one, the light one, and the middle ones), we will finally understand the fundamental rules of how heavy quarks stick together. This would be a huge breakthrough in understanding the "exotic" side of the universe's building blocks.

In short: The paper says, "We predicted a lighter partner for the particle X(6600). We think we see a faint hint of it in the data at 6400. If you look closer and search for a specific decay pattern, you will find it, and that will tell us exactly how these exotic particles are built."

Technical Summary

Problem
The paper addresses the theoretical interpretation of recently observed all-charm tetraquark states ($cc\bar{c}\bar{c}$), specifically the resonances $X(6600)$, $X(6900)$, and $X(7100)$ observed by the CMS, ATLAS, and LHCb collaborations. While the CMS Collaboration has provided refined measurements suggesting these states share the same $J^{PC} = 2^{++}$ quantum numbers and may represent radial excitations, the internal structure of these exotic hadrons remains debated. A critical unresolved issue is the nature of a lower-mass enhancement observed around 6400 MeV in experimental data (notably by ATLAS and recently in combined CMS Run 2+3 data). Theoretical models, specifically constituent quark models versus diquark models, predict different spectra and mass splittings for $S$-wave tetraquark multiplets. The authors aim to determine if the 6400 MeV structure corresponds to a predicted scalar partner ($0^{++}$) of the $X(6600)$ tensor state ($2^{++}$) and to provide precise mass and decay predictions that can discriminate between competing theoretical frameworks.

Methodology
The authors employ a phenomenological approach based on mass formulae derived in their previous work for $S$-wave tetraquarks in both constituent quark and diquark models.

1. Mass Relations: They utilize mass formulae expressed in terms of the multiplet center-of-mass mass ($M$) and coupling constants ($C_{cc}$ and $C_{c\bar{c}}$). They analyze the ratio of couplings $R = C_{c\bar{c}}/C_{cc}$, considering the symmetry limit ($R=1$) and varying $R$ to test model dependence.
2. Input Constraints: The tensor state mass ($M_2$) is fixed to the recent CMS measurement of $X(6600)$ ($6593 \pm 15$ MeV). Using this anchor, they calculate the masses of partner states (axial $1^{+-}$ and scalars $0^{++}$ and $0^{++\prime}$) for both models.
3. Decay Analysis: The authors calculate partial decay widths for transitions into charmonium pairs ($J/\psi J/\psi$, $\eta_c \eta_c$, $J/\psi \eta_c$). These calculations assume identical spatial wavefunctions for the initial and final states, relying on spin and color wavefunction overlaps to determine relative amplitudes. They compare the predicted branching ratios between quark and diquark models.
4. Data Interpretation: They re-examine the CMS and ATLAS data fits, specifically the inclusion of a Breit-Wigner amplitude (BW0) around 6400 MeV. They argue that the incoherent addition of this amplitude in CMS fits (unlike the coherent interference required for the $2^{++}$ states) suggests distinct quantum numbers, consistent with a scalar assignment.

Key Contributions and Results

• Confirmation of $X(6600)$: The authors note that the CMS confirmation of $J^{PC} = 2^{++}$ for $X(6600)$ validates their previous prediction that this state belongs to an $S$-wave multiplet.
• Prediction of Scalar Partner $X(6400)$: The paper argues that the enhancement near 6400 MeV is the scalar partner ($0^{++}$) of the $X(6600)$.
  • Mass Predictions: In the symmetry limit ($R=1$), the quark model predicts a scalar mass of $6443 \pm 19$ MeV, while the diquark model predicts $6513 \pm 16$ MeV. The ATLAS measurement of $6410 \pm 80$ MeV is consistent with the quark model prediction but less so with the diquark model.
  • Mass Ordering: The study confirms that in diquark models, the mass ordering is $M(0^{++}) < M(1^{+-}) < M(2^{++})$. In quark models, a heavier scalar $0^{++\prime}$ is also predicted, with the ordering $M(0^{++}) < M(1^{+-}) < M(2^{++}) < M(0^{++\prime})$.
• Decay Patterns:
  • The decay $0^{++} \to J/\psi J/\psi$ is predicted to be suppressed relative to $2^{++} \to J/\psi J/\psi$ (ratio $\sim 0.07$ in quark models, $\sim 0.19$ in diquark models), explaining the lower prominence of the 6400 MeV peak in $J/\psi J/\psi$ data.
  • Conversely, the decay $0^{++} \to \eta_c \eta_c$ is predicted to be prominent, with a ratio $\Gamma(0^{++} \to \eta_c \eta_c)/\Gamma(0^{++} \to J/\psi J/\psi)$ of 19 in quark models and 4.3 in diquark models.
  • The axial state $1^{+-}$ is predicted to decay prominently to $\eta_c J/\psi$.
• Model Discrimination: The paper establishes that measuring the mass of the scalar partner ($M_0$) alongside the tensor mass ($M_2$) allows for the prediction of the axial mass ($M_1$) via linear mass relations. These relations yield significantly different predictions for $M_1$ in quark versus diquark models, offering a robust experimental test to distinguish the degrees of freedom (quarks vs. diquarks) in these systems.

Significance
The paper claims that establishing an $S$-wave multiplet of $cc\bar{c}\bar{c}$ states, specifically by confirming the existence and properties of the scalar partner $X(6400)$, would constitute a breakthrough in exotic hadron research. The primary significance lies in the ability to discriminate between constituent quark and diquark models. The authors emphasize that the specific mass splitting between the $0^{++}$ and $2^{++}$ states, and the relative decay rates into $\eta_c \eta_c$ versus $J/\psi J/\psi$, provide "key experimental tests" to determine the internal structure of exclusively heavy quark exotics. They urge closer experimental scrutiny of the 6400 MeV region, suggesting that angular analyses including the 6400 MeV structure (treated incoherently with the higher states) and searches in the $\eta_c \eta_c$ channel are necessary to validate the proposed multiplet and resolve the theoretical debate.