The ultrafine splitting of heavy quarkonium with next-to-next-to-next-to-next-to-leading-order accuracy

This paper presents a theoretical computation of the hyperfine splitting for P-wave heavy quarkonium states with next-to-next-to-next-to-next-to-leading-order accuracy and next-to-next-to-next-to-next-to-leading-logarithmic resummation, followed by a phenomenological analysis applied to bottomonium, charmonium, the $B_c$ system, and various leptonic and atomic systems.

The Gist

Imagine the universe is filled with tiny, invisible dance partners. Sometimes, they are heavy particles like quarks (the building blocks of protons and neutrons) dancing together to form "quarkonium." Other times, they are lighter particles like electrons and positrons dancing in "positronium."

For decades, physicists have been trying to predict exactly how these dance partners move and how much energy they have. But there's a tricky detail: sometimes, the two partners spin in the same direction, and sometimes they spin in opposite directions. This difference in spinning creates a tiny, tiny gap in their energy levels. Physicists call this the hyperfine splitting.

For a long time, we could only guess this gap with a rough map. This new paper is like upgrading from a sketch on a napkin to a satellite image with laser precision.

Here is the story of what they did, explained simply:

1. The Goal: Measuring the "Ultrafine" Gap

Think of the energy levels of these particles like rungs on a ladder. Most of the time, we care about the big gaps between the rungs. But this paper focuses on the ultrafine splitting.

Imagine you are looking at a ladder rung. To the naked eye, it looks like a single flat bar. But if you zoom in with a super-microscope, you see that the bar is actually two bars stacked incredibly close together, separated by a microscopic sliver of space. That sliver is the "ultrafine splitting."

The authors wanted to calculate the size of that sliver for heavy particles (like bottom quarks and charm quarks) with the highest possible accuracy ever attempted: N4LO. That's a fancy code for "Next-to-Next-to-Next-to-Next-to-Leading-Order." In plain English, they didn't just calculate the main effect; they calculated the main effect, plus the tiny corrections to the main effect, plus the tiny corrections to those corrections, and so on, four times deep.

2. The Toolkit: Building a Better Map

To do this, they had to build a new, incredibly detailed map of the forces between these particles. They used a tool called pNRQCD (a type of "Effective Field Theory").

Think of this like trying to describe a car race.

• The Old Way: You just say, "The cars go fast." (This is the basic physics).
• The New Way: You describe the engine, the aerodynamics, the friction of the tires, the wind resistance, and even how the driver's weight shifts when they turn.

The authors had to calculate the "forces" (potentials) between the particles at a level of detail no one had done before. They had to account for:

• Spin: How the particles rotate.
• Relativity: How the particles move near the speed of light.
• Quantum Jitters: The weird, fuzzy nature of the quantum world where particles pop in and out of existence.

They had to solve complex math problems involving "loops" (imagine a particle traveling in a circle and interacting with itself) and "dimensions" (math that works in 4, 5, or even more dimensions to make the numbers add up correctly).

3. The Surprise: The "Accidental Cancellation"

When they put all these pieces together, they found something surprising.

In previous calculations (which were already very good), the result for this energy gap was very small. It looked like the universe was being very quiet. But the authors realized that this smallness was likely a coincidence.

Imagine two people pushing a heavy box from opposite sides. If they push with exactly the same force, the box doesn't move. It looks like there is no force at all. But if you measure their strength more precisely, you realize they are actually pushing very hard; they just happened to cancel each other out perfectly.

The authors found that the "forces" pushing the energy gap up and the forces pushing it down were canceling each other out almost perfectly in previous calculations. When they added their new, ultra-precise corrections, the "box" started to move. The true size of the energy gap is actually larger than we thought, and the previous small result was just an "accidental cancellation."

4. The Results: From Quarks to Atoms

They tested their new, super-precise formula on three different systems:

1. Bottomonium: Heavy quarks dancing together.
2. Charmonium: Slightly lighter quarks dancing.
3. Bc: A mix of two different heavy quarks.

They also applied their math to atomic physics (like Positronium, which is an electron and a positron dancing).

The Big Win:

• For Heavy Quarks: Their new calculations fit the experimental data (what we see in real life) much better than before, especially for the bottom quark system.
• For Atoms: They solved a recent argument in the physics community. Two different groups had calculated the energy gap for Positronium and got different answers. The authors of this paper recalculated it and showed that the older, established answer was correct, and the newer, conflicting one had a mistake.

5. Why Does This Matter?

You might ask, "Why do we care about a tiny sliver of energy in a particle we can't even see?"

1. Testing the Rules of the Universe: The Standard Model is our rulebook for how particles behave. If our calculations don't match the experiments, it means we are missing a piece of the puzzle (maybe new physics!). By calculating the "sliver" with extreme precision, we are stress-testing the rulebook.
2. Measuring the Strong Force: The size of this gap depends on the "strong force" (the glue that holds atoms together). By measuring the gap and comparing it to their math, we can determine the strength of this force more accurately than ever before.
3. Future Tech: While this seems abstract, understanding these fundamental forces is the foundation for everything from medical imaging to future quantum computers.

The Bottom Line

This paper is a masterpiece of "microscopic accounting." The authors didn't just count the dollars; they counted the pennies, the fractions of pennies, and the dust motes on the pennies. They showed us that the universe is even more intricate and precise than we thought, and they provided the most accurate map yet for how these tiny quantum dancers spin and interact.

Technical Summary

Here is a detailed technical summary of the paper "The ultrafine splitting of heavy quarkonium with next-to-next-to-next-to-next-to-leading-order accuracy".

1. Problem Statement

The paper addresses the calculation of the hyperfine splitting of P-wave heavy quarkonium states (specifically the "ultrafine splitting," $\Delta$) with unprecedented precision.

• Definition: The ultrafine splitting is defined as the energy difference between the singlet $^1P_1$ state and the center of gravity of the triplet $^3P_J$ states:
  $$\Delta \equiv E(1P_1) - E(3P)_{\text{c.o.g}} = E(1P_1) - \frac{1}{9} \left( 5E(3P_2) + 3E(3P_1) + E(3P_0) \right)$$
• Significance: This quantity is unique because its leading contribution starts at order $O(m\alpha_s^5)$, making it extremely small (hence "ultrafine"). It is sensitive to light-fermion effects and offers a potential avenue for precise determination of the strong coupling constant $\alpha_s$.
• Goal: To achieve N4LO (Next-to-Next-to-Next-to-Next-to-Leading Order) accuracy, corresponding to $O(m\alpha_s^6)$, and to partially address N4LL (Next-to-Next-to-Next-to-Next-to-Leading Logarithmic) accuracy by resumming hard logarithms. Previous calculations had only reached N3LO/N3LL.

2. Methodology

The authors utilize Non-Relativistic QCD (NRQCD) and potential NRQCD (pNRQCD) effective field theories. The methodology involves a systematic expansion in powers of $1/m$(heavy quark mass) and$\alpha_s$.

A. Theoretical Framework

• pNRQCD Lagrangian: The problem is reduced to solving a Schrödinger equation with a Hamiltonian containing a static potential and relativistic corrections ($\delta h$).
• Matching Schemes: The authors compute potentials in different matching schemes (Wilson-loop, off-shell, and on-shell) and demonstrate their equivalence after eliminating energy-dependent terms via field redefinitions.
• Dimensional Regularization: Calculations are performed in general $D = 4 - 2\epsilon$ dimensions to handle ultraviolet divergences, with a specific focus on the consistent treatment of Dirac and Pauli matrices in $D$ dimensions (using spin projectors to avoid ambiguities with the Levi-Civita tensor).

B. Key Computations

To reach N4LO, the authors computed several new ingredients:

1. **Two-loop $1/m^2$Potential:** Computed the spin-dependent, velocity-independent potential ($V_{S^2}$) to two loops in the Wilson-loop matching scheme in$D$ dimensions.
2. One-loop $1/m^3$ Potential: Computed the spin-dependent potential to one loop in the off-shell matching scheme, correcting a missing diagram in previous literature regarding non-Abelian terms.
3. **Tree-level $1/m^4$Potentials:** Derived the full set of$O(\alpha_s/m^4)$tree-level potentials relevant for spin-dependent observables, including terms involving Wilson coefficients$c_W$and$c_D$.
4. Renormalization Group (RG) Running: Incorporated the running of Wilson coefficients (specifically the chromomagnetic moment $c_F$) to NLL accuracy and other coefficients to LL accuracy.
5. Quantum Mechanical Perturbation Theory: Applied first- and second-order perturbation theory to the derived potentials. This included calculating expectation values of single and double insertions of the potentials, utilizing the reduced Green's function for second-order terms.

3. Key Contributions

• N4LO Accuracy: This is the first calculation of the P-wave hyperfine splitting at N4LO accuracy ($O(m\alpha_s^6)$).
• N4LL Partial Resummation: The authors include the resummation of "hard" logarithms arising from the Wilson coefficients of NRQCD bilinear terms, providing a partial N4LL result.
• Scheme Independence: They explicitly prove that the final physical result for $\Delta$ is independent of the matching scheme (off-shell vs. Wilson-loop) used to derive the potentials, provided energy-dependent terms are handled correctly.
• Correction of Literature: The paper identifies and corrects a missing diagram in previous calculations of the $1/m^3$ potential (Ref. [35]) and resolves a discrepancy in QED calculations for positronium (Ref. [31]), confirming earlier results.
• Generalization to QED: The formalism is applied to QED systems (positronium, muonium, hydrogen, muonic hydrogen), providing N4LO results for these atomic systems.

4. Results

The paper provides analytical expressions for $\Delta$ for:

• Heavy Quarkonium: Bottomonium ($b\bar{b}$), Charmonium ($c\bar{c}$), and $B_c$ systems.
• QED Systems: Positronium, dimuonium, muonium, and muonic hydrogen.

Phenomenological Findings:

• Magnitude of Corrections: The N4LO correction is found to be larger than the leading non-vanishing order (N3LO). This suggests that the smallness of the N3LO result was due to an accidental cancellation between Abelian and non-Abelian contributions, and the N4LO result represents the "natural" size of the effect.
• Scale Dependence: The fixed-order N4LO result exhibits a strong dependence on the renormalization scale ($\mu$).
• Resummation Effect: The partial resummation of logarithms (N4LL) significantly reduces the scale dependence for bottomonium, stabilizing the prediction. However, the effect is less pronounced for charmonium and $B_c$, where the weak-coupling expansion is less reliable.
• Comparison with Experiment:
  • For bottomonium ($n=2$), the theoretical prediction with scale variation is in reasonable agreement with experimental data ($\Delta_{exp} \approx -0.57$ MeV).
  • For charmonium, the results worsen when moving to smaller, more natural renormalization scales, indicating that higher-order non-perturbative effects or full N4LL resummation (including ultrasoft logs) are needed.
• QED Results: The calculated value for positronium ($\Delta_{pos} \approx 0.46$ MHz) agrees with older literature (Refs. [28–30]) but contradicts a recent claim in Ref. [31]. The authors' communication with Ref. [31] led to the identification of an error in their calculation, resolving the conflict.

5. Significance

• Precision Frontier: This work establishes the theoretical foundation for N4LO/N4LL precision in heavy quarkonium spectroscopy, a necessary step for using these systems as precision probes of QCD and for extracting $\alpha_s$.
• Relativistic Effects: It demonstrates that relativistic corrections at this order are sizable and cannot be neglected for precision phenomenology.
• Atomic Physics: The results provide a rigorous test of QED in bound states, resolving discrepancies in the positronium spectrum and offering new predictions for muonic systems.
• Future Outlook: The authors note that while the "hard" logarithms are now resummed, the full N4LL accuracy requires the inclusion of "ultrasoft" effects (which contribute at N4LL for P-waves), a task reserved for future work.

In summary, this paper represents a major advancement in the precision calculation of heavy quarkonium energy levels, correcting previous theoretical inconsistencies and providing a robust framework for future high-precision tests of the Standard Model.