Renormalization of three-quark operators with up to two derivatives at three loops

This paper presents the analytic three-loop $\overline{\mathrm{MS}}$ renormalization constants and anomalous dimensions for three-quark operators with up to two covariant derivatives in QCD, while also providing two-loop RI'/MOM matching results for their amputated four-point Green's functions to facilitate lattice QCD calculations.

The Gist

Imagine the universe is built out of tiny, invisible LEGO bricks called quarks. Three of these bricks snap together to form protons and neutrons, which make up the nucleus of every atom in your body. But these aren't static blocks; they are a chaotic, buzzing swarm of energy and force.

Physicists want to understand exactly how these three quarks move and interact inside a proton. To do this, they use a mathematical "map" called a Distribution Amplitude (DA). Think of this map as a blueprint that tells you the probability of finding the quarks in specific arrangements or speeds.

However, there's a problem. The rules of the game (Quantum Chromodynamics, or QCD) are incredibly complex. When physicists try to calculate these maps using their standard tools, they run into a mathematical nightmare: their equations spit out "infinity."

The Problem: Infinite Noise

Imagine you are trying to listen to a single violin in a stadium full of screaming fans. The "screaming fans" are the mathematical infinities that appear when you try to calculate the interactions of quarks. To hear the violin (the real physics), you have to filter out the noise.

In physics, this filtering process is called Renormalization. It's like putting on noise-canceling headphones that are tuned specifically to cancel out the infinities, leaving you with a clean, finite signal that matches reality.

The Challenge: The "Three-Loop" Puzzle

For decades, physicists have been trying to calculate these maps with increasing precision.

• One Loop: A rough sketch.
• Two Loops: A detailed drawing.
• Three Loops: A photorealistic, 3D model.

The paper you asked about is a massive achievement in reaching that Three-Loop level of precision. But it's not just about drawing a better picture; it's about solving a puzzle that gets exponentially harder with every step.

Here are the specific hurdles the authors (Kniehl and Veretin) had to jump over:

1. The "Ghost" Bricks (Evanescent Operators)

When doing these calculations, physicists use a trick where they pretend the universe has a slightly different number of dimensions (like 3.999 dimensions instead of 4). In this weird math-world, some "bricks" (mathematical structures) exist that don't exist in our real 4D world. These are called Evanescent Operators.

Think of them as ghosts. They are invisible in the real world, but if you don't account for them while you are building your math, they will haunt your final result and make it wrong. The authors had to develop a special "ghost-hunting" method to ensure these invisible structures didn't mess up the calculation of the real quark interactions.

2. The "Spinning Top" Problem (Spin and Derivatives)

The paper looks at quarks that are not just sitting still, but moving and spinning.

• N=0: The quarks are just sitting there (the simplest case).
• N=1 & N=2: The quarks are moving and spinning faster.

Imagine trying to balance a spinning top on a needle. As you add more spin (derivatives), the top gets wobblier. The math gets messy because the different ways the quarks can spin and move start to mix together, like different colored dyes swirling in water. The authors had to untangle this mix to see the individual colors (the specific properties of the proton) clearly.

3. The Lattice Connection

Why do we care about this? Because there are two ways to study protons:

1. Theory: Using the complex math described above (Perturbative QCD).
2. Experiment: Using supercomputers to simulate the proton on a grid (Lattice QCD).

To make these two methods agree, they need to speak the same language. The authors calculated a "translation dictionary" (called Conversion Factors) that allows scientists to take results from the supercomputer simulations and translate them into the language of the theoretical formulas.

The Big Picture: Why This Matters

This paper is like upgrading the GPS in a self-driving car.

• Before, the car (our understanding of the proton) could get you to the right city, but the turn-by-turn directions were a bit fuzzy.
• With this new "Three-Loop" calculation, the directions are now precise down to the inch.

This precision is crucial for the future of particle physics. As we build bigger, more powerful particle colliders (like the future Electron-Ion Collider), we need to know the internal structure of protons with extreme accuracy. If our "map" of the proton is slightly off, we might miss new discoveries or misinterpret the data.

In summary:
These two physicists spent years solving a incredibly difficult math puzzle. They figured out how to remove the "infinite noise," banish the "mathematical ghosts," and untangle the "spinning tops" to create the most precise map yet of how three quarks dance together inside a proton. This map will help experimentalists and theorists work together to unlock the secrets of the universe's building blocks.

Technical Summary

1. Problem Statement

The paper addresses a critical gap in the theoretical understanding of Baryonic Distribution Amplitudes (DAs), which are non-perturbative functions describing the structure of nucleons in exclusive processes. While DAs are complementary to parton distribution functions, their determination is challenging because they cannot be accessed via perturbation theory alone.

The primary theoretical tool for determining DAs is the calculation of their Mellin moments ($N$) using Lattice QCD. To connect Lattice QCD results (calculated in schemes like RI′/SMOM) with continuum perturbative QCD (usually in the $\overline{\text{MS}}$ scheme), one requires precise conversion factors and anomalous dimensions.

Specific gaps identified:

• Previous calculations of $\overline{\text{MS}}$ anomalous dimensions for three-quark operators were limited to $N=0$ at three loops, or $N=1, 2$ only at one or two loops.
• The conversion factors from RI′/SMOM to $\overline{\text{MS}}$ were known for $N=0$ at two loops, but not for $N=1$.
• Calculations involving higher Mellin moments ($N > 0$) introduce covariant derivatives, leading to operator mixing and increased complexity.
• Standard dimensional regularization ($d = 4-2\epsilon$) introduces complications regarding evanescent operators (operators that vanish in $d=4$ but mix with physical ones in $d \neq 4$) and the definition of the $\gamma_5$ matrix.

2. Methodology

The authors employ a rigorous perturbative QCD approach to compute renormalization constants and anomalous dimensions up to three loops and conversion factors up to two loops.

• Operator Definition: They study non-local three-quark operators with open Dirac spinor indices, expanded via the Operator Product Expansion (OPE) into local operators with up to two covariant derivatives ($N=0, 1, 2$).
• Renormalization Scheme:
  • They utilize a specific variant of the $\overline{\text{MS}}$ scheme advocated in Ref. [13]. This scheme ensures that evanescent operators vanish strictly in $d=4$, allowing calculations to be performed using only physical four-dimensional operators. This approach systematically avoids ambiguities associated with the $\gamma_5$ matrix in dimensional regularization.
  • They work in linear covariant gauge and verify that the resulting anomalous dimensions are gauge-independent.
• Computational Techniques:
  • Feynman Diagrams: Generated using QGRAF.
  • Algebraic Manipulation: Color and spinor traces are evaluated using custom FORM routines.
  • Integral Reduction: Feynman integrals are reduced to a set of Master Integrals using the Laporta algorithm (integration by parts) implemented in FIRE.
  • Infrared Rearrangement: To separate ultraviolet (UV) divergences from infrared (IR) ones, they use the method of global infrared rearrangement, reducing expressions to massive tadpole diagrams.
  • RI′/SMOM Calculation: For the conversion factors, they compute amputated four-point Green's functions with specific kinematics (all external legs having non-zero invariant mass) using FIESTA and CUBA for numerical evaluation of two-loop master integrals.

3. Key Contributions

The paper significantly advances the state of the art in perturbative QCD for baryonic systems:

1. Three-Loop Anomalous Dimensions:

   • Calculated for the general three-quark operator with open spinor indices for $N=1$ and $N=2$ (previously only $N=0$ was known at three loops).
   • Calculated for specific spin/chirality states ($J=1/2$ and $J=3/2$) for $N=0, 1, 2$.
   • Confirmed previous two- and three-loop results for $N=0$.

2. Two-Loop Conversion Factors:

   • Evaluated the RI′/SMOM to $\overline{\text{MS}}$ conversion factors for $N=1$ at two loops. This is essential for matching recent lattice QCD simulations to continuum perturbation theory.

3. Handling of Evanescent Operators:

   • Successfully applied a scheme that avoids the mixing of physical and evanescent operators by working with open spinor indices, thereby bypassing the notorious $\gamma_5$ problem in dimensional regularization.

4. Data Availability:

   • Provided all results (including those not printed in the text, such as $N=2$ anomalous dimensions) in machine-readable format, including both analytic expressions in terms of master integrals and high-precision numerical values.

4. Key Results

• Anomalous Dimensions ($\gamma$):

  • For $N=0$ (open spinor), the authors derived explicit analytic forms for $\gamma^{(1)}$, $\gamma^{(2)}$, and $\gamma^{(3)}$ involving symmetric combinations of $\Gamma$-matrix structures ($C_0, C_2, \dots$) and evanescent terms ($C_6, C_{642}$).
  • For $N=1$ and $N=2$, the anomalous dimensions are presented as matrices acting on the operator multiplets. The mixing patterns grow rapidly with $N$ (scaling as $\sim 64 \times (N+1)(N+2)/2$ components for leading twist).
  • Explicit matrix elements for the $N=1$ and $N=2$ cases are provided in the text and Appendix, expressed in terms of the strong coupling $a = \alpha_s/4\pi$, the number of flavors $n_f$, and Riemann zeta values ($\zeta_3$).

• Conversion Factors:

  • The form factors $f_j$ for the RI′/SMOM scheme were computed at two loops for $N=1$.
  • The results are expressed in terms of 44 two-loop master integrals (evaluated numerically with $\sim 10^{-6}$ precision) and 2 one-loop integrals (evaluated analytically).

• Gauge Independence:

  • The calculated anomalous dimensions were confirmed to be independent of the gauge parameter, as expected for physical observables.

5. Significance

This work is a foundational step for the next generation of Lattice QCD studies of baryon structure:

• NNLO Precision: The new results allow lattice simulations to extract the first two Mellin moments ($N=0, 1$) of baryonic DAs with Next-to-Next-to-Leading Order (NNLO) accuracy, with $N^3$LO evolution.
• Bridging Schemes: By providing the two-loop RI′/SMOM to $\overline{\text{MS}}$ conversion factors for $N=1$, the paper enables a direct and precise comparison between state-of-the-art lattice data and continuum perturbative predictions.
• Theoretical Robustness: The successful application of the evanescent-operator-free scheme to higher Mellin moments ($N=1, 2$) demonstrates a robust method for handling complex operator mixing in dimensional regularization, setting a precedent for future high-order calculations in QCD.

In summary, this paper removes a major bottleneck in the precision determination of nucleon structure functions, facilitating more accurate predictions for exclusive processes at current and future high-energy facilities like the Electron-Ion Collider.