⚛️ high-energy experiments

QCD sum-rule determination of the axial-vector mixing angle in the \texorpdfstring{ $B_c(1P)$ }{Bc(1P)} sector

Using QCD sum rules, this paper determines the mixing angle between the $1^1P_1$ and $1^3P_1$ axial-vector states in the $B_c(1P)$ sector to be $(43.3\pm0.2)^\circ$ , indicating significant mixing between these two states.

Original authors: T. M. Aliev, S. Bilmis, M. Savci

Published 2026-07-02

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: T. M. Aliev, S. Bilmis, M. Savci

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the subatomic world as a giant, chaotic dance floor where particles are constantly bumping into each other. In this paper, the authors are trying to figure out the specific dance moves of a very special pair of dancers: the $B_c$ meson.

The Special Dance Partner

Most particles in this "heavy" family are like couples dancing with partners of the same type (two heavy dancers, or two light dancers). But the $B_c$ meson is unique because it's made of two different heavy partners: a "bottom" quark and a "charm" quark. Because they are different, they don't follow the usual rules of symmetry, which makes their dance moves (or quantum states) a bit more complicated and interesting.

The Confusing Mix-Up

The scientists are focusing on a specific level of energy where these particles can spin in two slightly different ways, which we'll call Style A and Style B.

Style A is like a dancer spinning with their arms crossed in a specific pattern.
Style B is the same dancer spinning, but with their arms uncrossed in a different pattern.

In the real world, nature doesn't always keep these styles separate. Instead, the physical particle we actually see is a mixture of both styles, like a smoothie made of two different fruits. The question the paper asks is: How much of Style A is in the mix, and how much is Style B?

This "recipe" is determined by a single number called the mixing angle. Think of this angle as the dial on a radio. If you turn the dial to 0 degrees, you hear only Style A. If you turn it to 90 degrees, you hear only Style B. The authors wanted to find out exactly where the dial is set for the $B_c$ meson.

The Detective Work: QCD Sum Rules

To find this angle, the authors used a mathematical tool called QCD Sum Rules. You can think of this as a sophisticated "sound check" or a way of listening to the echoes of the dance floor to figure out what the dancers are doing without seeing them directly.

The Setup: They created two mathematical "microphones" (called currents) to listen to the particle. One microphone was tuned to hear Style A, and the other to hear Style B.
The Echo: They calculated how these microphones would pick up the signal if the particle were purely Style A, purely Style B, or a mix of both.
The Calculation: By comparing the signals from the "microphones" and using a complex mathematical filter (called the Operator Product Expansion) to account for the background noise of the universe (gluon condensates), they could isolate the exact ratio of the mix.

The Result

After doing all the heavy math and running thousands of computer simulations to check for errors, the authors found the answer:

The mixing angle is approximately 43.3 degrees.

What does this mean in plain English?

It's not 0 degrees (pure Style A) and it's not 90 degrees (pure Style B).
It's right in the middle, leaning slightly toward the middle.
This tells us that the physical $B_c$ particle is a strong mixture of both styles. It's not mostly one or the other; it's a genuine blend of both.

Why This Matters (According to the Paper)

The paper notes that other scientists have guessed this angle before, but their guesses varied wildly, ranging from about 17 degrees to 55 degrees. Some models said it was mostly Style A, others said mostly Style B.

The authors' result of 43.3 degrees sits comfortably in the middle of these predictions. This is important because recently, the LHCb experiment (a giant particle detector) saw some new, blurry structures in the data that look like these excited $B_c$ particles. To understand exactly what those blurry structures are, scientists need to know the "recipe" (the mixing angle) to interpret the data correctly.

A Note on Limitations

The authors are careful to mention that their calculation is done at a "leading order," which is like taking a high-quality photo but not a 4K video. They acknowledge that if they added even more detailed corrections (like next-to-leading order), the number might shift slightly. However, for now, this is the best independent estimate they can provide using their specific mathematical method.

In summary: The paper uses advanced math to determine that the $B_c$ meson is a roughly 50/50 (but slightly weighted) mix of two different quantum dance styles, providing a crucial piece of the puzzle for understanding recent experimental discoveries.

Technical Summary: QCD Sum-Rule Determination of the Axial-Vector Mixing Angle in the $B_c(1P)$ Sector

Problem Statement
The spectroscopy of the $B_c$ meson family, composed of two heavy quarks of different flavors ( $\bar{b}c$ ), offers a unique testing ground for nonperturbative Quantum Chromodynamics (QCD). Unlike charmonium or bottomonium, the $B_c$ system lacks charge conjugation as a good quantum number. Consequently, the two axial-vector configurations with quantum numbers $J^P=1^+$ , specifically the spin-singlet $1^1P_1$ and the spin-triplet $1^3P_1$ , are not eigenstates of the Hamiltonian but mix to form the physical $B_c(1P)$ states.

Recent experimental observations by the LHCb Collaboration of orbitally excited $B_c^+$ structures in the $B_c^+\gamma$ mass spectrum have renewed the need for precise theoretical inputs to interpret these states. However, existing theoretical predictions for the $1^1P_1 - 1^3P_1$ mixing angle ( $\theta$ ) exhibit significant model dependence, ranging from approximately $20^\circ$ to over $50^\circ$ . This spread complicates the identification of physical states and the calculation of radiative transition patterns. The authors aim to provide an independent determination of this mixing angle using the QCD sum-rule method to complement existing lattice and quark-model studies.

Methodology
The authors employ the QCD sum-rule framework to derive a relation for the mixing angle. The theoretical approach involves the following steps:

Interpolating Currents: Two independent interpolating currents are constructed for the axial-vector $B_c(1P)$ system:
- $J^A_\mu = \bar{b}\gamma_\mu\gamma_5 c$
- $J^B_\mu = i\bar{b}\sigma_{\mu\alpha} \frac{p^\alpha}{m_b+m_c}\gamma_5 c$
  The factor $1/(m_b+m_c)$ in the second current ensures both currents share the same mass dimension.
Correlation Functions: The authors analyze the two-point correlation functions $\Pi_{\mu\nu}^{ij}(p)$ (where $i,j = A,B$ ) in the deep Euclidean region. These functions are decomposed into transverse ( $\Pi_1$ ) and longitudinal ( $\Pi_0$ ) amplitudes, with the spin-one contribution isolated via projection.
Operator Product Expansion (OPE): The correlation functions are calculated on the QCD side using the OPE in the fixed-point gauge. The expansion includes:
- Perturbative contributions (leading order).
- Non-perturbative contributions from the two-gluon condensate ( $\langle G^2 \rangle$ ).
- Non-perturbative contributions from the three-gluon condensate ( $\langle g_s^3 G^3 \rangle$ ).
  The massive quark propagators include background field contributions up to the three-gluon term.
Mixing Angle Derivation: The physical currents are defined as a rotation of the basis currents by an angle $\theta$ . The mixing angle is determined by the condition that the off-diagonal correlation function vanishes in the physical basis. This leads to the sum rule:
$\tan 2\theta = -\frac{2\Pi_{AB}^1(M^2, s_0)}{\Pi_{AA}^1(M^2, s_0) - \Pi_{BB}^1(M^2, s_0)}$
where $M^2$ is the Borel parameter and $s_0$ is the continuum threshold.
Numerical Analysis: The sum rule is evaluated using standard QCD input parameters (heavy quark masses in the $\overline{MS}$ scheme, gluon condensates). The analysis is performed within a stability window defined by:
- $7 \text{ GeV}^2 \le M^2 \le 9 \text{ GeV}^2$
- $53 \text{ GeV}^2 \le s_0 \le 55 \text{ GeV}^2$
  Uncertainties are estimated via Monte Carlo sampling of input parameters and auxiliary sum-rule parameters.

Key Results
The numerical analysis yields a mixing angle of:
$\theta_{B_c(1P)} = (43.3 \pm 0.2)^\circ$

Stability: The result is stable with respect to variations in the Borel parameter and continuum threshold within the chosen windows.
Condensate Contributions: The perturbative term provides the dominant contribution. The inclusion of two-gluon and three-gluon condensate terms results in only a negligible shift in the extracted angle.
Comparison: The derived value of $\approx 43.3^\circ$ falls within the broad spectrum of existing theoretical predictions (which range from $\approx 17^\circ$ to $55^\circ$ ) but is distinct from many potential model results that cluster around $20^\circ-30^\circ$ . It is notably higher than several quark model predictions but consistent with some coupled-channel and lattice QCD approaches.

Significance and Claims
The paper claims that the determined mixing angle indicates a "sizable mixing" between the $1^1P_1$ and $1^3P_1$ components. This implies that the physical $B_c(1P)$ axial-vector states are strong mixtures of these configurations rather than nearly pure states.

The authors position this result as a complementary, nonperturbative input for the interpretation of the recently observed $B_c(1P)$ structures in the $B_c^+\gamma$ spectrum. Specifically, the mixing angle is a critical parameter for:

Assigning quantum numbers to the experimentally observed structures.
Calculating radiative transition amplitudes.
Constraining phenomenological models used in theory-constrained experimental analyses.

The authors modestly note that their calculation is performed at leading order in perturbation theory. While common radiative corrections to the correlation functions largely cancel in the ratio used to extract the angle, channel-dependent corrections could introduce shifts. Therefore, they suggest that a dedicated Next-to-Leading-Order (NLO) calculation would be necessary for a fully systematic uncertainty estimation in future precision studies.