Two-loop quarkonium Hamiltonian in annihilation channel

This paper completes the full two-loop quarkonium Hamiltonian within the pNRQCD framework by calculating the annihilation channel component and providing a generalized color structure applicable to various gauge groups.

The Gist

The Cosmic Dance of the Tiny: A Simple Guide to the "Quarkonium Hamiltonian"

Imagine you are trying to understand the most intricate dance in the universe. This dance isn't performed by humans, but by quarks—the tiny, fundamental building blocks that make up almost everything we see.

Specifically, this paper is about a very special type of dance called Quarkonium.

1. The Setting: The "Quarkonium" Ballroom

In the world of particle physics, some quarks are "heavyweights" (like Charm quarks or Bottom quarks). Because they are so heavy, they tend to pair up—a quark and an anti-quark—and orbit each other like a tiny, high-speed solar system. This pair is what we call Quarkonium.

To understand how these pairs behave, scientists need a "rulebook" that describes all the forces acting on them. In physics, this rulebook is called a Hamiltonian.

2. The Problem: The "Annihilation" Twist

Most studies of these quark pairs focus on how they orbit each other (the "non-annihilation" channel). It’s like studying two dancers spinning around a center point.

However, there is a dramatic twist in this dance: sometimes, the two dancers collide so intensely that they both vanish in a flash of light, turning into pure energy. This is called Annihilation.

Until now, scientists had a great rulebook for the "spinning" part of the dance, but the rulebook for the "vanishing" part (the annihilation channel) was incomplete. This paper is the missing chapter that finally explains exactly what happens during that explosive finale.

3. The Method: The "Mathematical Microscope"

How do you calculate something so small and violent? The authors use a technique called Effective Field Theory (pNRQCD).

Think of this like a mathematical microscope. If you want to study a forest, you don't need to track every single atom in every leaf; you look at trees and branches. Similarly, instead of trying to solve the impossibly complex equations of the entire universe, these scientists use a "zoomed-in" version of the math that focuses specifically on the heavy quarks, making the math manageable but incredibly precise.

They performed a "two-loop" calculation. In physics, a "loop" is like a level of detail.

• One loop is like seeing the dancers' movements.
• Two loops is like seeing the sweat on their brows and the tension in their muscles.
  By going to "two loops," they are providing a level of precision that allows us to test our fundamental understanding of the universe (the Standard Model) to an extreme degree.

4. The Result: Completing the Map

The authors didn't just solve the problem for our specific universe (where the "color" of particles follows a specific rule called SU(3)). They created a universal formula.

It’s like they didn't just write a manual for one specific car; they wrote a master manual that works for any car, regardless of the engine type. This means their math can be applied to other theoretical "universes" or different types of particles that scientists might discover in the future.

Summary in a Nutshell

The Goal: To write the perfect "rulebook" for how heavy quark pairs behave.
The Achievement: They added the final, most difficult chapter: what happens when those pairs collide and disappear.
Why it matters: It allows scientists to use these tiny particles as ultra-precise tools to check if our current understanding of physics is correct or if there is something even deeper waiting to be discovered.

Technical Summary

Technical Summary: Two-loop Quarkonium Hamiltonian in the Annihilation Channel

Authors: Yukinari Sumino (Tohoku University) and Takahiro Ueda (Juntendo University)

1. Problem Statement

The precision study of heavy quarkonium systems (charmonium, bottomonium, and potentially toponium) is essential for testing the Standard Model and understanding Quantum Chromodynamics (QCD). To achieve high-precision theoretical predictions, calculations must reach the next-to-next-to-next-to-next-to-leading order ($\text{N}^4\text{LO}$).

While the two-loop heavy quarkonium Hamiltonian in the non-annihilation channel was recently computed, the calculation for the annihilation channel (where the quark and antiquark annihilate into gluons) remained a missing piece. This paper aims to complete the full two-loop quarkonium Hamiltonian by calculating the annihilation channel within the framework of potential Non-Relativistic QCD (pNRQCD).

2. Methodology

The authors employ a direct matching procedure between full QCD and the pNRQCD effective field theory (EFT). The technical workflow is as follows:

• Matching Procedure: They match the quark-antiquark on-shell elastic scattering amplitudes from full QCD to pNRQCD at order $\alpha_s^3$ and at each order of the $1/m$ expansion.
• Computational Steps in Full QCD:
  1. Spinor Projection: Amplitudes are projected onto a spinor basis to convert loop integrals into Lorentz scalar integrals.
  2. Reduction: Scalar integrals are reduced to a set of "master integrals" using Integration-By-Parts (IBP) identities (utilizing tools like Kira and FLINT).
  3. Differential Equations & EBR: The master integrals are solved using differential equations in the $1/m$ expansion, with boundary conditions determined via the Expansion-By-Regions (EBR) method.
• Simplification via Symmetry: The authors note that most diagrams in the annihilation channel are related to those in the non-annihilation channel via $s$-channel and $t$-channel exchange. They prove that a specific diagram that vanishes in the non-annihilation channel (due to color-singlet projection identities) does not vanish in the annihilation channel, requiring explicit calculation.
• Scale Analysis: Using EBR, they demonstrate that "soft" contributions vanish in the annihilation channel because the soft-scale momentum transfer becomes scaleless. Consequently, the calculation is restricted to the Hard-Hard (HH) region, which directly matches the EFT Hamiltonian.

3. Key Contributions

• Completion of the Hamiltonian: By providing the annihilation channel results, the authors complete the full two-loop quarkonium Hamiltonian.
• General Color Structure: Unlike previous works that focused on the $SU(N)$ color gauge group, this paper provides an expression with a more general color structure. This includes the independent color factor $\frac{d_{abc}^F d_{abc}^F}{N}$, making the results applicable to gauge groups beyond $SU(3)$.
• Spinor Basis Formalism: The paper defines a specific spinor basis and performs Fierz transformations to convert the results into the standard basis used in non-relativistic quantum mechanics.

4. Results

• Wilson Coefficients: The authors provide the one-loop and two-loop Wilson coefficients ($C_{An}^{\{i,j,n,2\ell\}}$) up to $\text{N}^4\text{LO}$.
  • The one-loop coefficients are finite and well-known.
  • The two-loop coefficients are also finite and include complex dependencies on the strong coupling constant $\alpha_s(k)$, the mass $m$, and the momentum transfer $k$.
• Verification: The results for the $SU(N)$ case were cross-checked and found to be in perfect agreement with previous NRQCD four-quark operator calculations.
• Master Integrals: The paper provides the explicit expansions of the two-loop HH master integrals in terms of the dimensional regularization parameter $\epsilon$.

5. Significance

This work is a significant step toward $\text{N}^4\text{LO}$ precision in heavy quarkonium physics. The completed Hamiltonian allows for highly accurate calculations of:

• Hyperfine and Fine Splittings: Specifically at order $\alpha_s^6 m$.
• Standard Model Precision: Improved theoretical precision in quarkonium observables allows for more stringent tests of QCD and the search for new physics in heavy quark systems.