Hunting for $B\bar B$ molecular state $X_{b0}$ via radiative transition of $\Upsilon(10753)$

This paper investigates the radiative decay $\Upsilon(10753) \to \gamma X_{b0}$ within nonrelativistic effective field theory, predicting a branching fraction of $10^{-6}-10^{-5}$ dominated by $B_1^{(\prime)}$ meson loops and suggesting this channel as a promising avenue for discovering the $B\bar{B}$ molecular state $X_{b0}$.

The Gist

Imagine the subatomic world as a giant, chaotic dance floor where particles are constantly pairing up, breaking apart, and spinning in complex patterns. For decades, physicists have been trying to map out the "family tree" of these particles. Most fit neatly into the expected categories, but every now and then, a "rogue" particle shows up that doesn't fit the rules. These are called exotic states, and they are the mystery guests of the particle physics party.

This paper is a theoretical investigation into how we might find one specific, elusive guest at this party: a particle called $X_{b0}$.

Here is the breakdown of the paper's story, using everyday analogies:

1. The Setting: A Heavy-Weight Dance Floor

The story takes place in the "bottomonium" sector. Think of this as a heavy-duty dance floor where particles made of a "bottom" quark and its anti-particle spin around.

• The Host: The main character here is a particle called $\Upsilon(10753)$. Think of this particle as a DJ who is actually a mix of two different styles (a "4S" style and a "3D" style). It's energetic and sits right at the edge of the dance floor where new pairs can be formed.
• The Mystery Guest ($X_{b0}$): Physicists suspect there is a particle called $X_{b0}$ hiding near the edge of the floor. It's not a single dancer, but rather a molecular state—a very loose, weakly bound pair of two other dancers (a $B$ meson and an anti-$B$ meson) holding hands just barely. It's like two people dancing so close they are practically one unit, but if you pull them apart, they separate easily.

2. The Problem: How Do We Spot the Guest?

The $X_{b0}$ is very heavy and doesn't show up easily in standard experiments. It's like trying to find a specific, shy guest at a crowded concert who refuses to stand on the stage.

• The Strategy: The authors propose a specific way to "spot" this guest. They suggest looking for a radiative decay.
• The Analogy: Imagine the DJ ($\Upsilon(10753)$) is spinning a record. Suddenly, the DJ stops, throws a glowing spotlight (a photon, or light particle) into the air, and in that flash of light, the shy guest ($X_{b0}$) appears. The paper calculates how bright that spotlight needs to be and how often this flash happens.

3. The Mechanism: The "Triangle" Shortcut

How does the DJ turn into a spotlight and a guest? The paper suggests a process involving intermediate loops.

• The Analogy: Think of it as a relay race. The DJ doesn't turn directly into the guest. Instead, the DJ first passes a baton to a temporary runner (a pair of bottom mesons), who runs a quick lap around a track, passes the baton to another runner, and then the final transformation happens.
• The Two Paths: The authors looked at two different "tracks" (loops) the particles could take:
  1. The S-Wave Track: A path involving standard, slow-moving dancers.
  2. The P-Wave Track: A path involving faster, spinning dancers (specifically a type called $B'_1$).
• The Discovery: The math shows that the P-Wave track is the winner. It's like finding out that the relay race is much faster if the runners spin while they run. The paper concludes that the "spinning" path contributes almost entirely to the creation of the $X_{b0}$, while the standard path is negligible.

4. The Results: How Likely Is It?

The authors ran the numbers to predict how often this "flash of light" event happens.

• The Prediction: They estimate that for every million times the DJ spins, this specific event (creating the $X_{b0}$ and a photon) happens between 1 and 10 times.
• The "Width" Factor: They also checked if the "spinning" dancers ($B'_1$) were very unstable (having a large "width" or short lifespan). They found that even if these dancers are very jittery and short-lived, it doesn't change the result much. The signal remains stable.
• The Binding Energy: They tested different "tightness" levels for the $X_{b0}$ molecule (how close the two dancers are holding hands). They found that as long as the bond is weak (which is expected for a molecule), the signal is strong enough to be seen.

5. The Conclusion: A Promising Hunt

The paper ends with a clear message: Keep looking for this particle using this specific method.

• Because the predicted signal (the branching fraction) is between $10^{-6}$ and $10^{-5}$, it is small but definitely within the reach of current high-energy physics experiments (like those at the Super KEKB collider).
• Finding this particle would be a huge win. It would confirm that the "bottomonium" family has a "spin partner" to a famous particle called $X(3872)$ (which was found in the "charm" sector years ago). It would prove that heavy quarks follow a specific symmetry rule, much like how every family has a set of cousins that look and act similarly.

In short: The authors have drawn a map showing the most efficient route to find a hidden, weakly-bound particle ($X_{b0}$) by watching a heavy particle ($\Upsilon(10753)$) emit a flash of light. Their calculations suggest the path is clear, the signal is detectable, and the "spinning" intermediate particles are the key to making it happen.

Technical Summary

Technical Summary: Hunting for $B\bar{B}$ molecular state $X_{b0}$ via radiative transition of $\Upsilon(10753)$

Problem Statement
The paper addresses the search for the bottomonium counterpart of the charmonium exotic state $X(3872)$, specifically the scalar partner $X_{b0}$ with quantum numbers $J^{PC} = 0^{++}$. While the $X(3872)$ is widely interpreted as a loosely bound $D\bar{D}^*$ molecular state, the existence and properties of its heavy-flavor partner $X_{b0}$ (a $B\bar{B}$ molecule near the $B\bar{B}$ threshold) remain largely unexplored experimentally. Previous theoretical studies have focused on the vector partner $X_b$ ($1^{++}$) or the production of $X_{b0}$ via $\Upsilon(4S)$ radiative decays. However, the $\Upsilon(10753)$ resonance, recently identified by the Belle collaboration with a mass of approximately 10753 MeV, offers a new production mechanism. The $\Upsilon(10753)$ lies above the open-bottom threshold and provides sufficient phase space for the radiative production of $X_{b0}$. The central problem is to calculate the partial decay width and branching fraction for the process $\Upsilon(10753) \to \gamma X_{b0}$ to determine if this channel is viable for experimental discovery.

Methodology
The authors employ Nonrelativistic Effective Field Theory (NREFT) to calculate the radiative decay amplitude. The theoretical framework involves several key components:

1. State Interpretations:

   • The $X_{b0}$ is modeled as a weakly bound $B\bar{B}$ molecular state ($J^{PC}=0^{++}$) with a binding energy $\epsilon_X$ ranging from 0 to 10 MeV.
   • The $\Upsilon(10753)$ is treated as an $S-D$ mixed state, a linear combination of the $\Upsilon(4S)$ and $\Upsilon_1(3^3D_1)$ states, with a mixing angle $\theta \approx 33^\circ$.

2. Mechanism:

   • The decay proceeds via intermediate meson loops (triangle diagrams).
   • Two types of loops are considered:
     • $B^{(*)}\bar{B}^{(*)}$ loops: Involving $S$-wave bottom mesons, where the initial bottomonium couples in a $P$-wave.
     • $B^{(')}_1\bar{B}^{(*)}$ loops: Involving $P$-wave bottom mesons ($B'_1$ and $B_1$), where the initial bottomonium couples in an $S$-wave.
   • The $B^{(')}_1$ mesons are treated with their finite widths using a Breit-Wigner parameterization for the spectral function.

3. Lagrangians and Couplings:

   • The interactions are described using heavy-quark effective theory Lagrangians. The couplings of the bottomonium states to bottom meson pairs are extracted from unquenched quark model predictions (Ref. [51]).
   • The coupling of the $X_{b0}$ to the $B\bar{B}$ pair is determined by the binding energy relation derived from Weinberg's compositeness condition.
   • Radiative transitions involving photons are handled via effective Lagrangians containing magnetic and electric dipole terms, with coupling constants ($\beta, \tilde{\beta}, C$) fixed by experimental partial widths of $B^* \to \gamma B$ and $B'_{1} \to \gamma B$.

4. Power Counting:

   • A power counting analysis is performed using the velocity $v$ of intermediate mesons as the expansion parameter to estimate the relative importance of the different loop contributions.

Key Contributions and Results

• Dominance of $P$-wave Loops: The power counting analysis and numerical results demonstrate that the decay is dominated by the loops involving $P$-wave $B^{(')}_1$ mesons (Fig. 2 in the paper). These contributions are approximately 20 times larger than those from the $S$-wave $B^{(*)}\bar{B}^{(*)}$ loops (Fig. 1).
• Predicted Decay Widths: For a binding energy range of $\epsilon_X = 0 - 10$ MeV, the calculated partial decay width for $\Upsilon(10753) \to \gamma X_{b0}$ is predicted to be 0.2 – 1.5 keV.
• Branching Fractions: This partial width corresponds to a branching fraction in the range of $10^{-6} - 10^{-5}$.
• Sensitivity to Parameters:
  • The decay width increases monotonically with the binding energy $\epsilon_X$.
  • The result is found to be insensitive to the full width of the $B'_1$ meson. Even when varying the $B'_1$ width from 0 to 300 MeV, the total decay width varies only slightly (within the range of 1.02–1.12 keV for a fixed binding energy of 5 MeV).
  • The uncertainty arising from the $4S-3D$ mixing angle $\theta$ is estimated to be small ($\sim O(0.05)$).
• Ratio to Vector Partner: The paper also calculates the ratio $R = \mathcal{B}[\Upsilon(10753) \to \gamma X_b] / \mathcal{B}[\Upsilon(10753) \to \gamma X_{b0}]$, finding that while the absolute widths depend on binding energy, this ratio is relatively stable.

Significance and Claims
The authors claim that their study identifies the radiative decay $\Upsilon(10753) \to \gamma X_{b0}$ as a promising channel for searching for the $X_{b0}$ state. The predicted branching fraction ($10^{-6} - 10^{-5}$) is considered "appreciable" and potentially accessible to current or near-future experiments, such as those at SuperKEKB (Belle II).

The observation of the $X_{b0}$ via this channel would be crucial for:

1. Testing Heavy-Quark Symmetries: Specifically, Heavy Quark Spin Symmetry (HQSS), which predicts the existence of the $X_{b0}$ as the scalar partner of the $X_b$.
2. Understanding the Exotic Hadron Spectrum: Confirming the molecular nature of bottomonium-like states and completing the understanding of the heavy-quark sector's exotic spectrum.

The paper concludes that while the $X_{b0}$ is heavy and difficult to produce directly in $e^+e^-$ collisions, the radiative transition from the $\Upsilon(10753)$ provides a viable production mechanism that warrants experimental investigation.