The S-wave topped meson

Motivated by recent near-threshold enhancements in top-quark pair production, this paper utilizes the instantaneous Bethe-Salpeter formalism to calculate the S-wave spectral structures of heavy-light systems containing a single top quark ($t\bar{q}$, $t\bar{c}$, and $t\bar{b}$), identifying them as quasi-bound configurations with masses slightly above the top-quark mass and offering qualitative insights into their potential production and decay patterns.

The Gist

Imagine the subatomic world as a bustling construction site where tiny particles are constantly being built and torn apart. Usually, when a heavy particle like a "top quark" is created, it tries to grab a partner (like a lighter quark) to form a stable structure called a "meson"—think of it like a dancer grabbing a partner to start a waltz.

However, the top quark is a very special dancer. It is so heavy and unstable that it falls apart (decays) almost instantly—faster than the speed of light can travel across the size of an atom. In fact, it falls apart so quickly that it never gets the chance to finish the "waltz" of forming a stable dance partnership. It dies before the music even starts.

The Big Question
Scientists at the Large Hadron Collider (LHC) recently noticed something strange: a slight bump in the data where top quarks are created. It looked like the top quarks were briefly sticking together before falling apart, almost like a "ghost" of a dance partner. This got the authors of this paper thinking: What if we tried to calculate what these "ghost dances" would look like if they could exist for a split second?

The Experiment (The "What If" Scenario)
The authors used a sophisticated mathematical tool (the Bethe-Salpeter equation) to simulate these fleeting moments. They imagined three types of "topped" pairs:

1. A top quark paired with a generic light quark.
2. A top quark paired with a "charm" quark.
3. A top quark paired with a "bottom" quark.

They calculated the "weight" (mass) of these imaginary pairs. Think of it like calculating the weight of a shadow. Even though the shadow isn't a solid object, it has a shape and a size.

The Findings
The math gave them specific numbers. They found that if these pairs could exist for a tiny fraction of a second, they would weigh just a little bit more than a single top quark.

• For the pair with a bottom quark, the "ghost" would weigh about 5 to 6 GeV more than the top quark alone.
• For the pair with a charm quark, it would be about 2 GeV heavier.

The Catch (The Reality Check)
The authors are very careful to state: These are not real, stable particles. You cannot go to a lab and catch one in a jar. Because the top quark dies so fast, these "topped mesons" are more like quasi-bound configurations—a fancy way of saying "temporary arrangements that exist only in our models."

They are essentially saying: "Here is a theoretical map of where these fleeting structures might hang out in the energy spectrum. If you look for them in the data, these are the coordinates to check."

How They Might Appear and Disappear
The paper also sketches a rough idea of how these might show up and vanish:

• Creation: Imagine two protons smashing together, creating a top quark and an anti-quark. Before the top quark can die, it might briefly "hug" a nearby anti-quark from the vacuum.
• Destruction: The top quark immediately decays into a "W boson" (a force carrier) and a bottom quark. The result is a messy explosion of particles: a W boson, a bottom quark, and the leftover light quark.

The Bottom Line
This paper is a theoretical benchmark. It's like an architect drawing up blueprints for a building that might be too unstable to ever be constructed. The authors aren't claiming these buildings exist; they are providing a reference point. If future experiments at the LHC see strange signals that look like these "ghost dances," scientists can use these blueprints to understand what they are seeing.

In short: The top quark is too fast to form a real family, but this paper calculates what that family would look like if it could pause time for a nanosecond.

Technical Summary

Technical Summary: The S-wave Spectral Structure of Heavy-Light Systems Containing a Top Quark

Problem Statement
Motivated by recent near-threshold enhancements in top-quark pair ($t\bar{t}$) production reported by the CMS and ATLAS collaborations, this work investigates the existence and spectral properties of "topped" mesons—specifically heavy-light systems containing a single top quark ($t\bar{q}$, $t\bar{c}$, and $t\bar{b}$). The central physical challenge addressed is the hierarchy of timescales: the top quark decays via the weak interaction ($\tau_t \approx 5 \times 10^{-25}$ s) significantly faster than the characteristic QCD hadronization time ($\tau_{\text{had}} \sim 10^{-23}$ s). Consequently, the formation of conventional, long-lived hadrons containing a top quark is theoretically prohibited. The paper seeks to determine whether meaningful quasi-bound spectral structures can be defined for these systems within a theoretical framework, serving as a benchmark for understanding the interplay between perturbative and non-perturbative dynamics in the presence of an unstable heavy quark.

Methodology
The authors employ the instantaneous Bethe-Salpeter (Salpeter) approximation using constituent-type quark propagators. This phenomenological model is chosen for its utility in meson spectroscopy and its ability to provide a relativistic framework for comparing different flavor combinations.

• Formalism: The calculation utilizes the time-ordered vacuum-to-bound-state matrix element to define the Bethe-Salpeter (BS) wave function. The equation is reduced to a three-dimensional form in the center-of-mass frame under the instantaneous approximation, where the interaction kernel depends only on the transverse momenta.
• Potential Model: The interquark interaction is modeled as a sum of a long-range linear confining potential (Lorentz scalar) and a short-range one-gluon-exchange potential (Lorentz vector). A screening factor ($e^{-\alpha r}$) is introduced to parametrize potential screening effects and avoid infrared divergence.
• Parameters: The model parameters (quark masses, strong coupling constant $\alpha_s$, and potential parameters $\alpha$ and $\lambda$) are fixed phenomenologically by fitting to experimental data for lighter mesons, consistent with Particle Data Group (PDG) values. The top quark mass is set to the PDG central value.
• Scope: The study calculates S-wave ($J^P = 0^-$ and $1^-$) states up to the fourth radial excitation ($n=4$) for the $t\bar{b}$, $t\bar{c}$, and $t\bar{q}$ ($q=u,d,s$) systems.

Key Results
The numerical evaluation yields discrete eigenvalues representing the mass spectra of these configurations.

• Mass Degeneracy: Consistent with the large top quark mass, the mass splitting between the $0^-$ and $1^-$ S-wave states is found to be negligible within numerical accuracy ($M_{0^-} \approx M_{1^-}$).
• Spectral Positions: The masses of the calculated configurations lie close to the top-quark mass ($m_t \approx 173$ GeV, though the table in the text lists values in units of 100 GeV, implying a total mass scale around 177-178 GeV for the ground state). The paper explicitly quantifies the excitation energies relative to the top quark mass:
  • $t\bar{b}$ System: The first four $0^-$ radial states are approximately 5.1, 5.4, 5.6, and 5.7 GeV above the top-quark mass.
  • $t\bar{c}$ System: The corresponding values are approximately 1.9, 2.2, 2.5, and 2.6 GeV above the top-quark mass.
  • $t\bar{q}$ ($u,d,s$) Systems: These states are even closer to the top mass, with the ground state roughly 3.4–3.6 GeV above $m_t$ (inferred from the table values of ~1.734–1.738 $\times$ 100 GeV vs $m_t$).
• Wave Functions: The radial components of the Salpeter wave functions for the $t\bar{b}$ system are presented, illustrating the nodal structures for the first four excitations.

Significance and Claims
The paper explicitly frames its findings with significant caveats regarding physical interpretation:

1. Quasi-bound Nature: The authors emphasize that the discrete eigenvalues obtained are not predictions for fully formed, conventional hadrons. Due to the top quark's rapid decay, these results should be interpreted as "model-dependent reference positions" for possible quasi-bound heavy-light configurations.
2. Theoretical Benchmark: The study serves as a theoretical benchmark for future phenomenological work. It provides a systematic systematization of mass scales and spectral structures for top-light correlations, assuming such correlations could be probed.
3. Qualitative Phenomenology: The paper offers a qualitative discussion of production and decay. It suggests that at hadron colliders, such configurations might arise if a produced top quark correlates with a nearby antiquark before decaying. The dominant decay mode is identified as the weak decay of the top quark within the system ($t \to W^+ b$), leading to final states like $W^+ + b + \bar{q}$.
4. Limitations: The authors state that the work does not include the finite width of the top quark in the propagator or interaction kernel, nor does it provide quantitative estimates for production cross-sections, branching fractions, or background suppression. The discussion of production mechanisms (e.g., gluon-gluon fusion, quark-antiquark annihilation) is schematic, intended only to guide future dedicated studies.

In conclusion, the paper establishes a model-based spectral map for top-flavored heavy-light mesons, clarifying that while these states are not conventional hadrons, their calculated mass scales and wave function structures offer a necessary reference for interpreting potential future experimental anomalies or for developing more complete theories that incorporate the top quark's finite width.