Possible mixing between elementary and bound state fields in the $t\bar{t}$ production excess at the LHC

This paper investigates a potential mixing between a toponium bound state and an elementary field to explain the CMS $t\bar{t}$ production excess, finding that the Multicritical Point Principle restricts the mixing angle to be less than $13^\circ$ in a minimal model and less than $1^\circ$ within specific Two-Higgs-Doublet Model scenarios.

The Gist

The Big Mystery: A Glitch in the Matrix

Imagine the Large Hadron Collider (LHC) as the world's most powerful particle collider, smashing protons together to see what's inside. Recently, the CMS experiment noticed something strange: when they smashed protons to create Top quarks (the heaviest particles in the Standard Model), they saw a little "bump" or excess in the data right at the edge of where these particles should form.

Think of it like this: You are baking cookies. You expect a certain number of cookies based on your recipe. But suddenly, you find a few extra cookies appearing right at the edge of the tray. Are they burnt? Are they a new flavor? Or is your oven acting up?

Physicists have two main theories about these extra cookies (the Top-Anti-Top pairs):

1. The "Ghost" Theory: They are a Toponium ($\eta_t$). This is a "bound state," meaning a Top quark and an Anti-Top quark are holding hands so tightly they act like a single, temporary molecule before falling apart. It's like a dance couple spinning so fast they look like a single blur.
2. The "New Particle" Theory: There is a completely new, fundamental particle (let's call it $\Psi$) that we've never seen before, which is also showing up in the data.

The Main Idea: The "Mixing" Cocktail

This paper asks a fascinating question: What if both are true?

Imagine you have two drinks:

• Drink A: A complex cocktail made of two ingredients mixed together (the Toponium).
• Drink B: A pure, single-ingredient juice (the new elementary particle $\Psi$).

The author, Yoshiki Matsuoka, suggests that these two drinks might be mixing together in the glass. The "excess" we see at the LHC isn't just one or the other; it's a new, hybrid drink ($\Psi'$) created by blending the bound state and the new particle.

The Rules of the Game: The "Multicritical Point"

To figure out how much of Drink A and Drink B are in the mix, the author uses a special rulebook called the Multicritical Point Principle (MPP).

Think of the universe's energy levels like a landscape with hills and valleys. The MPP is like a cosmic law that says: "The universe prefers to sit in two different valleys at the same time, and both valleys must be at the exact same height."

By applying this rule, the author calculates exactly how much the "Toponium" and the "New Particle" can mix without breaking the laws of physics. It's like a tightrope walker trying to balance two heavy weights; if they mix too much, the whole system crashes.

The Two Scenarios Tested

The author tested two different ways this mixing could happen:

Scenario 1: The Minimal Model (The "Simple Mixer")

In this scenario, the new particle ($\Psi$) is a bit of a loner. It only talks to the Top quark and ignores everyone else.

• The Result: The math says the mixing angle (how much they blend) can be up to 13 degrees.
• The Analogy: This is like mixing a little bit of orange juice into your apple juice. You can taste the orange, but it's still mostly apple. This scenario fits the data nicely and feels "natural."

Scenario 2: The 2HDM Model (The "Complex Mixer")

Here, the author tries to fit this new particle into a more complex theory called the Two-Higgs-Doublet Model (2HDM). This is like trying to mix your new drink into a full-blown cocktail party with many other guests (other Higgs particles).

• The Result: The math gets very strict. To make this work without breaking the rules of the universe, the mixing angle must be tiny—less than 1 degree.
• The Analogy: This is like trying to mix a drop of food coloring into a swimming pool. You can technically do it, but the color change is so faint it's almost invisible.
• The Problem: The author calls this "unnatural." It feels like the universe is forcing the particles to stay apart so perfectly that it requires a lot of "fine-tuning" (like balancing a pencil on its tip). Also, this scenario predicts other particles that should be very heavy, which contradicts what we are seeing in the data.

The Conclusion: Which Story Wins?

The paper concludes that Scenario 1 (The Minimal Model) is the most likely explanation.

• Why? It explains the "extra cookies" (the LHC excess) without requiring the universe to be weirdly precise or unnatural. It allows for a healthy amount of mixing (up to 13 degrees) that fits the data perfectly.
• The Takeaway: If the LHC is seeing a new particle, it's probably a simple mix of a Top-quark molecule and a new fundamental particle. If we try to force this into a more complex theory (Scenario 2), the math gets messy and requires the particles to barely mix at all, which seems unlikely.

Summary in a Nutshell

The universe might be showing us a "glitch" where Top quarks are forming a temporary bond. This paper suggests that this bond is secretly mixing with a brand-new, invisible particle. By using a cosmic balancing rule, the author shows that a simple mix works best, while a complex mix requires the universe to be too perfect to be true. Future experiments will need to check if this "hybrid drink" is really what's causing the extra particles.

Technical Summary

1. Problem Statement

Recent data from the CMS Collaboration at the Large Hadron Collider (LHC) indicates an excess in the invariant-mass spectrum of top-antitop ($t\bar{t}$) pairs near the production threshold ($\sqrt{s} \approx 13$ TeV). While this excess has been interpreted as a pseudo-scalar toponium bound state ($\eta_t$, $J^{PC}=0^{-+}$), alternative hypotheses suggest the existence of an unknown elementary (pseudo-)scalar field coexisting with the bound state.

The paper addresses two specific theoretical challenges:

1. Vacuum Stability: Previous studies suggest that if only the toponium bound state exists within the Standard Model (SM), the Higgs vacuum stability may be compromised. Introducing an additional elementary field could stabilize the vacuum.
2. Nature of the Excess: Determining whether the observed excess arises purely from a bound state, a pure elementary particle, or a mixing between the two.

2. Methodology

The author employs a hybrid theoretical framework combining Effective Field Theory (EFT) with high-scale constraints:

• Bardeen-Hill-Lindner (BHL) Framework: To treat the composite toponium state ($\eta_t$) consistently with elementary fields, the paper uses the BHL prescription. This involves:
  • Introducing a four-fermion interaction for top quarks at a cutoff scale $\Lambda$.
  • Applying a Hubbard–Stratonovich transformation to linearize the interaction into an auxiliary scalar field.
  • Imposing a compositeness condition at the cutoff scale $\Lambda$: the wavefunction renormalization $Z_2(\Lambda) = 0$ and the Yukawa coupling $y_{\eta_t}(\Lambda) \to \infty$. This ensures $\eta_t$ behaves as a bound state at high energies but an effective elementary field at low energies ($\mu \le \Lambda$).
• Multicritical Point Principle (MPP): The MPP is used as a high-scale organizing principle to constrain the model's parameters. It requires the one-loop effective potential $V_{\text{eff}}$ to have degenerate stationary minima at the electroweak scale ($\mu_{EW}$) and a high scale ($\mu_c \gg \mu_{EW}$), such that $V_{\text{eff}}(\mu_{EW}) \approx V_{\text{eff}}(\mu_c) \approx 0$.
• Renormalization Group Evolution (RGE): The paper calculates one-loop RGEs to evolve couplings from the cutoff scale $\Lambda$ (approx. 345–400 GeV) up to the MPP scale ($\mu_c \approx 10^{12.3}$ GeV).
• Mixing Formalism: The model considers the mixing between the elementary field $\Psi$ and the bound state $\eta_t$. The physical mass eigenstates ($\Psi', \eta'_t$) are defined by a mixing angle $\theta$:
  $$\begin{pmatrix} \eta'_t \\ \Psi' \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \eta_t \\ \Psi \end{pmatrix}$$
  The mass matrix includes off-diagonal terms arising from top-quark loop effects.

3. Key Contributions and Scenarios

The paper analyzes two distinct scenarios for the elementary field $\Psi$:

Scenario A: Minimal Model

• Setup: $\Psi$ is an inert Higgs-like doublet that couples to the top quark via a new Yukawa interaction ($y_\Psi$) but has negligible couplings to other fermions.
• Constraints: The MPP conditions and experimental limits on the $t\bar{t}$ excess are used to constrain the parameter space.
• Results:
  • The MPP scale is determined to be $\mu_c \lesssim 10^{12.3}$ GeV.
  • The Yukawa coupling is constrained to $|y_\Psi(M_t)| \le 0.3$.
  • The mixing angle is constrained to $|\theta| \le 13^\circ$.
  • The model predicts a new scalar/pseudo-scalar with mass $\approx 365$ GeV and a relative decay width $\Gamma/M \approx 2\%$.

Scenario B: Two-Higgs-Doublet Model (2HDM) Embedding

• Setup: The framework is embedded into a 2HDM (specifically Types II and Y), where $\Psi$ corresponds to one of the Higgs scalars after basis transformation.
• Constraints: In addition to MPP, the model must satisfy vacuum stability conditions and experimental bounds on charged Higgs bosons ($H^\pm$) and oblique parameters ($S, T, U$).
• Results:
  • MPP and vacuum stability imply constraints on quartic couplings: $|\lambda_4(M_t)|, |\lambda_5(M_t)| \le 0.5$.
  • Experimental bounds on Type II and Y 2HDMs require the charged Higgs mass $M_{H^\pm} \gtrsim 800$ GeV.
  • This forces the pseudo-scalar mass $M_A \gtrsim 800$ GeV.
  • Consequently, the mixing angle is severely restricted to $|\theta| \le 1^\circ$.
  • The author notes that such a small mixing angle is technically unnatural in the absence of a protective symmetry, as it corresponds to the decoupling limit.

4. Results and Comparison

• Mixing Angle Constraints:
  • Minimal Scenario: $|\theta| \le 13^\circ$. This allows for observable mixing effects while remaining consistent with data.
  • 2HDM Scenario (Types II & Y): $|\theta| \le 1^\circ$. This effectively suppresses the mixing, making the elementary and bound state components nearly distinct.
• Naturalness: The minimal scenario is argued to be more "natural" and predictive because it does not require fine-tuning to achieve a small mixing angle. In contrast, the 2HDM embedding requires the mixing to be unnaturally small to satisfy the heavy charged Higgs mass constraints.
• Comparison with Other 2HDM Types: The minimal scenario is less constrained by flavor physics and lepton-related bounds compared to 2HDM Types I and X, making it a more attractive target for near-term experimental probes.

5. Significance

• Theoretical Unification: The paper successfully demonstrates a mechanism where a composite bound state (toponium) and an elementary field can coexist and mix, providing a consistent EFT description via the BHL prescription.
• Vacuum Stability: It offers a potential solution to the Higgs vacuum instability problem by introducing the elementary field $\Psi$, which is constrained by the MPP.
• Phenomenological Guidance: The study provides specific, testable predictions for the LHC:
  • If the excess is due to mixing, the minimal scenario is favored over the 2HDM embedding due to the naturalness of the mixing angle.
  • Future collider studies should focus on the line shapes of the threshold region and associated production channels ($gg \to R \to t\bar{t}$, $gg \to R \to \gamma\gamma$) to distinguish between the two mass eigenstates.
• Limitations: The analysis relies on one-loop RGEs and assumes specific decay widths and mass degeneracies. Higher-order corrections and a systematic survey of electroweak precision constraints are identified as necessary future steps.

In conclusion, the paper argues that while both scenarios can technically explain the $t\bar{t}$ excess, the minimal mixing scenario is phenomenologically superior due to its relaxed constraints and lack of unnatural fine-tuning, whereas the 2HDM embedding forces the system into a decoupling regime that suppresses the very mixing effects required to explain the data naturally.