Pion transitions in the Born-Oppenheimer Effective Field Theory: a long distance approach

This paper proposes a Born-Oppenheimer effective field theory framework for pion transitions involving heavy quarkonium and exotic states with large sizes, deriving universal low-energy functions via a pion-string interaction Lagrangian to calculate and phenomenologically analyze long-distance dominated transition amplitudes where the standard QCD multipole expansion fails.

The Gist

Imagine the universe is filled with tiny, invisible strings made of pure energy. These strings connect heavy particles called "quarks," holding them together to form larger particles like protons, neutrons, and the exotic "heavy quarkonium" particles that scientists are trying to understand.

This paper is like a detective story about how these heavy particles change their energy by letting go of tiny bursts of energy called "pions" (which are like the smallest possible ripples in the fabric of the universe).

Here is the story in simple terms:

The Problem: The "Too Big" Puzzle

For a long time, scientists used a method called the "Multipole Expansion" to predict how these heavy particles behave. Think of this method like trying to describe a massive, fluffy cloud by looking at it through a tiny keyhole. It works great if the cloud is small and tight.

However, the scientists realized that many of these heavy particles (especially the "exotic" ones and the very excited ones) are actually huge and fluffy—much bigger than the "keyhole" of the old method. When they tried to use the old rules, the math broke down. It was like trying to use a ruler meant for a grain of sand to measure a mountain; the tool just wasn't designed for that scale.

The New Approach: The "String Theory" Map

To fix this, the authors (Joan Soto and Sandra Tomàs Valls) decided to look at the problem from the opposite direction. Instead of zooming in on the tiny details, they zoomed out to look at the long-distance behavior.

They imagined the heavy particles as being connected by a QCD string (a taut rubber band of energy). They asked: "If we have a giant rubber band, how does it wiggle when it interacts with the tiny pion ripples?"

They built a new set of rules (a mathematical "Lagrangian") that describes how these giant rubber bands talk to the pion ripples. This new map respects the symmetries of the universe, ensuring the physics makes sense whether you are looking at the string or the ripples.

The Discovery: Three Magic Numbers

By matching their new "string map" with the existing "heavy particle map," they discovered something beautiful: all the complicated, unknown parts of the interaction could be boiled down to just three universal constants (magic numbers).

Think of it like this: Instead of needing a different instruction manual for every single type of heavy particle, they found that there are only three "knobs" that control how these particles interact with pions at long distances. Once you know the settings of these three knobs, you can predict how almost any of these heavy particles will behave.

The Experiment: Testing the Theory

The authors didn't just stop at the math. They tried to figure out what those three "magic numbers" actually are by looking at real-world data from particle accelerators.

1. The Calibration: They used known transitions (where one heavy particle turns into another by releasing pions) to "tune" their three knobs. They found two possible sets of settings that fit the data.
2. The Predictions: Once tuned, they used these settings to predict what happens in other, more mysterious transitions.
   • They looked at Charmonium (heavy charm particles) and Bottomonium (heavy bottom particles).
   • They specifically looked at "Hybrids"—exotic particles where the rubber band itself is vibrating, not just the ends.

The Results: A New Identity for a Mystery Particle

Their predictions matched the experimental data quite well for most cases. However, the most exciting finding was about a specific particle called Υ(10860).

For a long time, scientists weren't sure if this particle was a standard "heavy quark pair" or something more exotic. The authors' calculations suggested that this particle acts very much like a Hybrid—a particle where the rubber band itself is excited. Their data strongly supports the idea that Υ(10860) is mostly a hybrid with just a tiny bit of the standard particle mixed in.

The Bottom Line

This paper provides a new, long-distance "rulebook" for understanding how heavy, exotic particles interact with the universe's smallest ripples. By realizing that some particles are too big for the old "close-up" rules, they developed a "wide-angle" lens that successfully predicts how these particles behave and helps identify the true nature of some of the universe's most mysterious building blocks.

In short: They replaced a broken, close-up microscope with a wide-angle telescope, found that everything is controlled by just three numbers, and used those numbers to solve a mystery about what a specific heavy particle actually is.

Technical Summary

Technical Summary: Pion transitions in the Born-Oppenheimer Effective Field Theory: a long distance approach

Problem Statement
The paper addresses the description of pion transitions involving heavy quarkonium and heavy exotic states (specifically hybrids) within the Born-Oppenheimer Effective Field Theory (BOEFT). A significant limitation of traditional approaches is the reliance on the QCD multipole expansion, which assumes the bound state size $r$ is small compared to the hadronic scale ($r\Lambda_{QCD} \ll 1$) and the transition time scale ($r\Delta E \ll 1$). However, many exotic states, such as heavy quarkonium hybrids and highly excited quarkonium states, are expected to be dominated by long-distance physics where $r\Lambda_{QCD} \gtrsim 1$. In this regime, the multipole expansion breaks down, and the standard short-distance behavior of the low-energy functions (LEF) in BOEFT (scaling as $r^2$) is insufficient. The authors aim to formulate pion transitions for these states and, crucially, to determine the long-distance behavior of the unknown LEF that arise in the BOEFT framework.

Methodology
The authors employ a matching procedure between the BOEFT and the Effective String Theory (EST) of QCD:

1. BOEFT Framework: They establish the interaction Lagrangians for pion transitions involving ordinary quarkonium (spin-0 and spin-1) and exotic hybrids with light degrees of freedom (LDF) quantum numbers $1^{+-}$. These Lagrangians contain unknown radial functions $g_s(r)$ (for quarkonium) and $g_4(r)$ (for hybrids).
2. Effective String Theory (EST): To determine the long-distance behavior of these functions, the authors propose an interaction Lagrangian describing the coupling between pions and the QCD string. This Lagrangian respects both the symmetries of the EST (reparameterization and Poincaré invariance) and chiral symmetry.
3. Matching: They calculate specific amplitudes within the EST framework:
   • Elastic pion scattering off the string.
   • The decay of string excitations (specifically the $N=1$ $\Pi_u$ states) into pion pairs.
   • The quark mass dependence of the string tension.
     These amplitudes are then matched to the corresponding amplitudes derived from the BOEFT Lagrangians in the static limit.
4. Phenomenological Analysis: Using the matched results, the LEF are expressed in terms of three universal constants ($c_{\pi\pi}, c_{\pi}, c_E$, derived from string parameters $\lambda, \eta, \lambda'$). These constants are estimated by fitting to experimental decay widths of specific bottomonium transitions ($\Upsilon(3S) \to \Upsilon(2S)\pi^+\pi^-$, $\Upsilon(4S) \to \Upsilon(2S)\pi^+\pi^-$, and the non-resonant component of $\Upsilon(10860) \to \Upsilon(3S)\pi^+\pi^-$).

Key Contributions

• Long-Distance LEF Behavior: The paper derives the long-distance behavior of the BOEFT low-energy functions. It finds that the functions $g_s(r)$, which scale as $r^2$ at short distances, transition to a linear $r$ dependence at long distances. Similarly, the hybrid function $g_4(r)$, which scales as $r$ at short distances, becomes a constant at long distances.
• Universal Parameters: The long-distance dynamics are reduced to three universal constants. The authors provide explicit expressions for these constants in terms of the string tension and chiral parameters.
• String Tension Dependence: As a byproduct, the authors derive the light quark mass dependence of the string tension, including one-loop corrections, which may help explain differences in heavy quarkonium spectra observed in lattice QCD at different pion masses.
• Transition Amplitudes: Explicit formulas for the dipion invariant mass spectra and decay widths are provided for:
  • Transitions between highly excited quarkonium states.
  • Transitions between hybrids (with $1^{+-}$ LDF) and highly excited quarkonium states.

Results

• Parameter Sets: Solving the system of equations from the input data yields two distinct sets of solutions for the universal constants, denoted as Set 1 and Set 2.
• Predictions: Using these sets, the authors calculate dipion decay widths and invariant mass spectra for various transitions, including:
  • Charmonium: $\chi_{c1}(2P) \to \chi_{c1}(1P)\pi^+\pi^-$, $\psi(2S) \to J/\psi \pi^+\pi^-$.
  • Bottomonium: $\Upsilon(nS) \to \Upsilon(mS)\pi^+\pi^-$, $\Upsilon_1(3D) \to \Upsilon(2S)\pi^+\pi^-$, and transitions involving the $\Upsilon(10860)$ and $\Upsilon(11020)$.
  • Hybrid transitions: $\psi[1(s/d)_1] \to h_c(1P)\pi^+\pi^-$ and $\Upsilon[2(s/d)_1] \to h_b(1P)\pi^+\pi^-$.
• Comparison with Data:
  • For ground-state transitions (e.g., $\Upsilon(2S) \to \Upsilon(1S)$), the model underestimates the decay width by an order of magnitude, consistent with the expectation that compact states are not dominated by long-distance physics.
  • For the $\chi_{c1}(3872) \to \chi_{c1}(1P)\pi^+\pi^-$ transition, the results satisfy experimental constraints.
  • Regarding $\Upsilon(10860)$, the pure quarkonium interpretation ($5S$) yields predictions significantly smaller than the non-resonant experimental data. However, interpreting $\Upsilon(10860)$ as a hybrid state ($2(s/d)_1$) yields results consistent with experimental data for transitions to $h_b$ states, supporting the hypothesis that $\Upsilon(10860)$ is primarily a hybrid meson with a small quarkonium component.
  • Discrepancies remain for transitions involving $D$-waves (e.g., $\Upsilon(10753) \to \Upsilon(3S)\pi^+\pi^-$), where the predicted ratios differ significantly from experimental values.

Significance and Claims
The authors claim to have successfully formulated a long-distance approach to pion transitions in BOEFT, bridging the gap where the multipole expansion fails. By matching BOEFT to EST, they provide a systematic way to determine the unknown long-distance behavior of LEF using only three universal parameters.

The paper modestly notes that while the results show qualitative agreement with experimental data for many transitions, large theoretical uncertainties exist. These uncertainties arise because the transitions considered may not be purely long-distance dominated, suggesting that an interpolation between short- and long-distance regimes would be necessary for a more precise description. The authors emphasize that their results support the interpretation of $\Upsilon(10860)$ as a hybrid state and provide a framework for analyzing future data on highly excited quarkonium and exotic states, provided more precise experimental data on dipion spectra becomes available. They do not claim definitive proof of the hybrid nature of these states but rather that their long-distance formalism is consistent with such an interpretation.