Large Relativistic Corrections to Nonrelativistic $M1$ Transitions in Heavy Quarkonium

This paper demonstrates that contrary to the common assumption of small relativistic effects in heavy quarkonium, relativistic corrections to nonrelativistic $M1$ transitions are unexpectedly large, ranging from approximately 66% to 83% for both charmonium and bottomonium decays, as revealed by a comprehensive Bethe-Salpeter equation analysis including higher-order multipole contributions.

The Gist

Imagine the universe as a giant, bustling dance floor. On this floor, there are two main types of couples dancing: Charmonium (a pair of "Charm" quarks) and Bottomonium (a pair of "Bottom" quarks).

For decades, physicists have believed that because Bottom quarks are so heavy, they dance very slowly and carefully. Because they move so slowly, scientists thought they could ignore the complex, relativistic rules of Einstein's universe and just use simple, "non-relativistic" math (like Newtonian physics) to predict how they move. It was like assuming a slow-moving turtle doesn't need to worry about the wind.

The Big Surprise:
This paper says, "Wait a minute! Even the slow turtles are feeling the wind!"

The authors, a team of physicists, used a super-advanced mathematical tool called the Bethe-Salpeter (BS) equation to look at these dancing couples when they change their dance moves by emitting a flash of light (a photon). Specifically, they looked at transitions where a "Vector" dancer (spinning one way) turns into a "Pseudoscalar" dancer (spinning the other way).

The Old Way vs. The New Way

The Old Way (Non-Relativistic):
Imagine watching the dancers from far away. You only see the main move: a simple spin. In physics, this is called the M1 transition. It's the "main course" of the meal. For a long time, scientists thought this was the only thing that mattered, especially for the heavy Bottomonium couples. They thought the "relativistic corrections" (the fancy, high-speed details) were too small to notice.

The New Way (Relativistic):
The authors put on high-definition, 3D glasses and zoomed in. They realized that the dancers aren't just doing one move. They are actually doing a complex routine that includes:

1. M1: The main spin (Non-relativistic).
2. E2, M3, E4: These are the "relativistic corrections." Think of them as the dancers' footwork, arm waves, and subtle body leans that happen because they are moving fast enough to feel the effects of relativity.

The "Relativistic Shock"

The paper's biggest discovery is that these "extra moves" (E2, M3, E4) are huge.

• For the Charm couples (Charmonium): The extra moves account for 68% to 83% of the total energy! It's like saying that when you order a burger, the bun and the pickles (the extra moves) actually make up most of the meal, not the patty (the main move).
• For the Bottom couples (Bottomonium): Even though these dancers are heavier and slower, the extra moves still account for 66% to 75% of the total energy!

The Metaphor:
Imagine you are trying to predict how much a car will cost to run.

• The Old Theory: "It's a heavy truck, so it only uses gas for the engine. The wind resistance and tire friction are negligible."
• This Paper: "Actually, for this specific type of driving, the wind resistance and tire friction are costing you more than the engine itself!"

Why Does This Matter?

1. It breaks the rules of "Heavy = Slow": We used to think that because Bottom quarks are heavy, they are safe from relativistic chaos. This paper proves that for these specific "light-emitting" transitions, the chaos is dominant.
2. It explains the "Forbidden" moves: In the old model, some transitions were predicted to be almost impossible (very small numbers). But when you add the "extra moves" (the relativistic corrections), the numbers jump up to match what we actually see in experiments.
3. It's a mix of waves: The authors explain that in the old view, a particle is like a pure "S-wave" (a simple circle). But in reality, because of relativity, the particle is a messy mix of S-waves, P-waves, and D-waves (circles, figure-eights, and complex loops) all happening at once. This mix is what creates the huge "extra moves."

The Takeaway

This paper is a wake-up call. It tells us that even in the heavy, slow world of Bottomonium, we cannot ignore the complex, high-speed rules of the universe. If we want to understand how these particles decay and emit light, we have to stop looking at just the "main move" and start appreciating the entire, chaotic, relativistic dance routine.

In short: The "small corrections" everyone ignored are actually the main event.

Technical Summary

1. Problem Statement

Heavy quarkonia (charmonium $c\bar{c}$ and bottomonium $b\bar{b}$) are traditionally modeled using non-relativistic frameworks (such as NRQCD or Heavy Quark Effective Theory) because the heavy quark masses imply low velocities ($\langle v^2 \rangle \approx 0.3$ for charmonium and $\approx 0.1$ for bottomonium). Consequently, relativistic corrections are often assumed to be negligible.

However, the authors identify a critical gap in this assumption:

• Leading-Order M1 Transitions: Processes where the Magnetic Dipole (M1) transition is the leading-order contribution (e.g., vector to pseudoscalar transitions like $\psi \to \gamma \eta_c$) have been suspected to suffer from large relativistic corrections, but this has not been systematically quantified across various states.
• Excited States: While ground states may fit non-relativistic models, excited states often exhibit significant deviations in velocity expectations ($\langle v^2 \rangle$), suggesting that non-relativistic approximations fail for higher-order multipoles.
• The Core Question: Do processes dominated by M1 transitions universally exhibit large relativistic corrections, and how significant are these corrections even for the heavier bottomonium system?

2. Methodology

The authors employ the Relativistic Bethe-Salpeter (BS) equation method (specifically its instantaneous version, the Salpeter equation) to calculate electromagnetic radiative decays.

• Theoretical Framework:
  • Unlike expansion methods (NRQCD) that truncate relativistic corrections at specific orders of $1/m_Q$ or $v$, the BS equation is an integral equation. Solving it iteratively effectively sums the relativistic expansion to infinite orders, capturing the complete relativistic corrections.
  • The method utilizes positive-energy wave functions derived from the Salpeter equation.
• Wave Function Structure:
  • In non-relativistic physics, a meson with total angular momentum $J$ corresponds to a single orbital angular momentum $L$ (e.g., $J=1$ is purely S-wave).
  • In the relativistic BS approach, the wave function for a state with definite $J^P$ is a mixture of multiple partial waves.
    • Vector Mesons ($1^-$): Contain S-wave, P-wave, and D-wave components.
    • Pseudoscalar Mesons ($0^-$): Contain S-wave and P-wave components.
• Multipole Decomposition:
  • The transition amplitude is calculated as an overlap integral of initial and final state wave functions.
  • Due to the mixed partial wave nature of the relativistic wave functions, the amplitude naturally decomposes into multiple multipole contributions:
    • M1 (Magnetic Dipole): Non-relativistic leading order ($S \times S'$).
    • E2 (Electric Quadrupole): First-order relativistic correction ($S \times P' + P \times S'$).
    • M3 (Magnetic Octupole): Second-order relativistic correction ($P \times P' + D \times S'$).
    • E4 (Electric Hexadecapole): Third-order relativistic correction ($D \times P'$).
  • The total decay width is calculated as $\Gamma_{total} = \Gamma_{M1} + \Gamma_{E2} + \Gamma_{M3} + \Gamma_{E4}$.

3. Key Contributions

• Systematic Investigation: The paper provides a comprehensive study of M1-leading transitions for both charmonium and bottomonium, covering ground and excited states ($nS, 1D \to mS$).
• Quantification of Relativistic Effects: It explicitly separates the non-relativistic M1 contribution from higher-order relativistic multipoles (E2, M3, E4) to quantify the magnitude of relativistic corrections.
• Resolution of Helicity vs. Multipole Confusion: The authors clarify a theoretical confusion regarding the number of independent amplitudes. They explain that while helicity conservation suggests a single amplitude, the lack of $L$ as a good quantum number in relativistic systems leads to multiple multipole amplitudes (M1, E2, M3, E4) contributing to a single helicity amplitude.
• Kinematic vs. Dynamic Analysis: The study distinguishes between kinematic effects (mass differences between $c$ and $b$ quarks) and dynamic effects (wave function overlaps), showing that relativistic dominance is a dynamic phenomenon.

4. Key Results

A. Charmonium ($\psi \to \gamma \eta_c$ and $\eta_c \to \gamma \psi$)

• Dominance of Relativistic Corrections: For most transitions, the non-relativistic M1 term is not the dominant contribution.
  • In $\psi(nS) \to \gamma \eta_c(mS)$ (for $n \ge 2$), the E2 contribution dominates over M1.
  • In $\psi(3770) \to \gamma \eta_c$, the M3 contribution is the largest.
• Magnitude of Corrections:
  • Relativistic corrections range from 68.1% to 83.2% for $\psi(nS) \to \gamma \eta_c(mS)$.
  • For $\eta_c(nS) \to \gamma \psi(mS)$, corrections are also significant (e.g., 38.6% for $2S \to 1S$, up to 90.9% for $3S \to 1S$).
• Node Suppression: Transitions involving excited states with nodes (e.g., $\psi(2S) \to \gamma \eta_c(1S)$) show suppression of the M1 term due to orthogonality, making the relativistic E2 term even more critical.

B. Bottomonium ($\Upsilon \to \gamma \eta_b$ and $\eta_b \to \gamma \Upsilon$)

• Unexpected Large Corrections: Despite the heavier mass of the bottom quark ($\langle v^2 \rangle \approx 0.1$), relativistic corrections remain dominant in specific channels.
  • For $\Upsilon(nS) \to \gamma \eta_b(mS)$, corrections range from 65.9% to 75.2%.
  • In $\Upsilon(2S) \to \gamma \eta_b(1S)$ and $\Upsilon(3S) \to \gamma \eta_b(1S)$, the E2 term dominates.
• Contrast with $\eta_b \to \Upsilon$: In the reverse process ($\eta_b \to \gamma \Upsilon$), the non-relativistic M1 term is dominant (relativistic corrections are smaller, ~38-50%), highlighting a kinematic difference between vector-to-pseudoscalar and pseudoscalar-to-vector transitions.
• D-wave States: For $\Upsilon(1D) \to \gamma \eta_b$, the M3 term dominates, similar to the charmonium D-wave case.

C. Comparison with Experiment

• The calculated decay widths for $\psi(1S) \to \gamma \eta_c(1S)$ and $\psi(2S) \to \gamma \eta_c(1S)$ agree well with PDG experimental data and CLEO branching ratios.
• The results for $\Upsilon(3S) \to \gamma \eta_b(1S)$ (10.8 eV) align well with experimental data (10.4 ± 2.4 eV).
• The study provides theoretical predictions for currently unobserved or poorly constrained states (e.g., $\eta_b(2S)$, $\Upsilon(1D)$).

5. Significance

• Challenge to Non-Relativistic Paradigm: The paper demonstrates that even for heavy bottomonium, the assumption of "small relativistic corrections" is invalid for M1-leading transitions. The corrections are not just perturbative; they are often the dominant mechanism driving the decay.
• Theoretical Precision: It establishes that accurate predictions for heavy quarkonium radiative decays require a fully relativistic treatment (BS equation) rather than truncated expansions, as the higher-order multipoles (E2, M3, E4) are essential.
• Experimental Guidance: The results provide precise theoretical benchmarks for ongoing and future experiments (BESIII, Belle II, LHCb) searching for bottomonium and charmonium states, particularly those involving hindered M1 transitions where non-relativistic models fail.
• Fundamental Insight: It clarifies the relationship between total angular momentum and orbital angular momentum in relativistic bound states, showing that $J$ does not map one-to-one to $L$, thereby necessitating a multi-multipole description for single-helicity transitions.