Conformal moments of the two-loop coefficient functions in DVCS

This paper introduces a new technique to calculate the conformal moments of two-loop coefficient functions in Deeply Virtual Compton Scattering, enabling the extraction of generalized parton distributions from experimental data with NNLO accuracy via the Mellin-Barnes approach.

The Gist

The Big Picture: Taking a 3D X-Ray of the Proton

Imagine the proton (the core of every atom) not as a solid marble, but as a bustling, chaotic city made of tiny particles called quarks and gluons. Physicists want to take a high-resolution 3D "X-ray" of this city to see exactly where these particles are, how they move, and how they interact.

To do this, they use a process called Deeply Virtual Compton Scattering (DVCS). Think of this as firing a high-speed electron (like a camera flash) at a proton. The electron hits a quark, the proton wobbles, and a photon (light) is emitted. By studying the light that comes out, scientists can reconstruct the 3D map of the proton's interior.

However, there is a problem: The math describing this interaction is incredibly messy. It's like trying to solve a puzzle where the pieces are constantly changing shape and the instructions are written in a language that gets harder with every step.

The Problem: The "Translation" Barrier

In physics, there are two main ways to describe these interactions:

1. Momentum Space: Describing particles by how fast they are moving. This is like describing a city by the speed of every car on every street. It's intuitive but mathematically messy when you try to combine different effects.
2. Conformal Moments (The "Harmonic" View): Describing the system using specific mathematical patterns (like musical notes or waves). This is like describing the city by its overall "vibe" or rhythm.

The paper's authors explain that to get the most precise 3D map of the proton (specifically to reach NNLO accuracy, which is the "4K Ultra HD" of particle physics), scientists need to translate the messy "Momentum Space" data into the clean "Conformal Moments" language.

The Challenge: The rules for this translation (called "coefficient functions") have been known for simple cases, but for the most precise, two-loop calculations, the math becomes a tangled knot of complex numbers and logarithms that is nearly impossible to untangle by hand.

The Solution: A New "Magic Wand" Technique

The authors (Braun, Gotzler, and Manashov) developed a new technique to untangle this knot. Here is how they did it, using an analogy:

The Analogy: The Echo Chamber

Imagine you are in a room with a very strange echo.

• The Goal: You want to know exactly what the echo sounds like for a specific song (the complex math function).
• The Old Way: You try to sing the song, record the echo, and analyze the sound wave by wave. This is slow and prone to errors.
• The New Way (The Authors' Method): They realized that the room has a special property. If you shout a specific "base note" (a mathematical pattern called a Gegenbauer polynomial), the room echoes it back perfectly, just louder or softer. These notes are "eigenfunctions"—they don't change shape, only volume.

The authors realized they could use a set of "magic wands" (mathematical operators) that act on the messy equations.

1. The Wands: These wands transform a complex function into a simpler one.
2. The Secret: Because the "base notes" (Gegenbauer polynomials) are special, applying a wand to them just multiplies them by a known number.
3. The Trick: Instead of calculating the messy integral directly, they applied these wands to a set of simple, known functions. By mixing and matching the results, they could build up the answer for the complex functions they actually needed.

It's like figuring out the recipe for a complex cake by knowing exactly how a mixer affects flour, sugar, and eggs separately, rather than trying to taste the batter while it's still mixing.

Why This Matters: The "GPD" Map

The ultimate goal of this math is to extract Generalized Parton Distributions (GPDs).

• Analogy: If a standard map of the proton shows you where the quarks are (like a street map), the GPD map shows you where they are and how fast they are moving and how they are spinning, all at once. It's a 3D hologram of the proton's internal structure.

To get this hologram from experimental data, scientists need to use a specific mathematical tool called the Mellin-Barnes approach. This tool works best when the data is in the "Conformal Moments" format.

Before this paper: Scientists had the raw data and the 3D map goal, but they were missing the precise "translation dictionary" (the two-loop coefficient functions) to convert the data into the right format. They were stuck with blurry, low-resolution maps.

After this paper: The authors have provided the complete, high-precision dictionary. They calculated the "moments" for all the necessary two-loop functions.

The Results and Impact

1. Precision: They provided explicit formulas for these moments. This allows future experiments (like those at the Electron-Ion Collider in the US or the EicC in China) to analyze data with NNLO accuracy. This means the 3D maps of the proton will be incredibly sharp.
2. Efficiency: Their method is much faster and less prone to error than trying to calculate these integrals directly. They used computer software (HyperInt) to handle the heavy lifting, but the mathematical framework they built is the real breakthrough.
3. Consistency: They checked their work and found that the results obey a beautiful mathematical symmetry (the "reciprocity relation"). This is like checking a puzzle and realizing all the pieces fit together perfectly, giving them high confidence that the answer is correct.

Summary

In short, this paper is about solving a massive mathematical translation problem.

The authors invented a clever "shortcut" using the natural symmetry of the universe to translate messy, high-speed particle data into a clean, organized format. This allows physicists to finally build the most detailed 3D maps of the proton ever created, helping us understand the fundamental glue that holds our universe together.

Technical Summary

1. Problem Statement

The extraction of Generalized Parton Distributions (GPDs) from experimental data in Deeply Virtual Compton Scattering (DVCS) requires high-precision theoretical inputs. While the two-loop (Next-to-Next-to-Leading Order, NNLO) coefficient functions (CFs) for DVCS have been calculated in momentum fraction space [10–14], utilizing these results for data analysis presents a significant challenge.

• The Convolution Problem: The relationship between GPDs and DVCS amplitudes involves convolutions over momentum fractions. Direct numerical evaluation of these convolutions with NNLO CFs (which contain generalized polylogarithms of high transcendentality) is computationally expensive and unstable.
• The Mellin-Barnes Advantage: It is established that transforming the problem into "conformal moments" space (using Gegenbauer polynomials) diagonalizes the evolution equations and simplifies the convolution integrals via Mellin-Barnes techniques.
• The Gap: While one-loop conformal moments were known, the explicit expressions for the two-loop conformal moments of all DVCS coefficient functions (vector, axial-vector, flavor-singlet, flavor-nonsinglet, and gluon transversity) were missing. Calculating these directly via integration is extremely difficult due to the complexity of the integrands.

2. Methodology

The authors developed a novel, systematic technique to calculate these conformal moments without performing direct integration of the complex momentum-space functions. The core of their method relies on the properties of $SL(2, \mathbb{R})$ invariant operators.

• Eigenfunction Property: The Gegenbauer polynomials (which define the conformal moments) are eigenfunctions of $SL(2, \mathbb{R})$ invariant operators, denoted as $H_\alpha$.
  $$H_\alpha |j\rangle = E_\alpha(j) |j\rangle$$
  where $|j\rangle$ represents the Gegenbauer polynomial state and $E_\alpha(j)$ is the eigenvalue.
• Operator Mapping: Instead of integrating the complicated coefficient functions directly, the authors utilize the property that if a function $f(x)$ has a known moment, the moment of the function $H^\dagger_\alpha f$ is simply the product of the original moment and the eigenvalue $E_\alpha(j)$.
• Position Space Simplification: A crucial technical innovation is the use of position space representations for these invariant operators. In momentum fraction space, these operators are complex integral kernels. In position space (variables $w_1, w_2$), they take a simple form involving integrals over a conformal ratio $\tau$.
  $$[H(h)\phi](w_1, w_2) = \int d\alpha \int d\beta \, h(\tau) \phi(w^\alpha_{12}, w^\beta_{21})$$
• Basis Construction:
  1. The authors identify a basis of functions (harmonic polylogarithms) that constitute the two-loop CFs.
  2. They construct a set of 16 independent invariant operators (using kernels $h(\tau)$ such as $1, \bar{\tau}, \ln(1-\tau)$, etc.) acting on a simple seed function (e.g., $1/z$).
  3. By applying these operators, they generate a system of linear equations relating the unknown moments of the CFs to known eigenvalues and simpler moments.
  4. The integrals required to generate the new functions in the basis are computed using the HyperInt package, which handles multiple polylogarithms efficiently.
  5. The final expressions are simplified using PolyLogTools.

3. Key Contributions

• New Calculation Technique: The paper introduces a robust algebraic method to compute conformal moments by leveraging $SL(2, \mathbb{R})$ symmetry and position-space kernels, bypassing the need for direct, difficult integration of momentum-space CFs.
• Complete Set of NNLO Moments: The authors provide explicit analytical expressions for the conformal moments of all two-loop DVCS coefficient functions:
  • Quark (Flavor-nonsinglet and Flavor-singlet).
  • Gluon (Flavor-nonsinglet and Flavor-singlet).
  • Vector and Axial-vector channels.
  • Gluon transversity.
• Reciprocity Relation Verification: The authors verify that all derived moments satisfy the reciprocity relation (symmetry under $N \to -1-N$ in their asymptotic expansion). This serves as a powerful non-trivial check of the results' correctness.
• Analytic Continuation: The results are presented in terms of harmonic sums, allowing for immediate analytic continuation to the complex $N$-plane, which is essential for the Mellin-Barnes inversion required in phenomenological analyses.

4. Results

The paper presents the explicit formulas for the conformal moments $C^{(2)}(N)$ up to weight four harmonic sums. Key features of the results include:

• Structure: The moments are expressed as linear combinations of harmonic sums ($S_1, S_2, S_{-2}, S_{1,3}$, etc.), Riemann zeta values ($\zeta_2, \zeta_3$), and logarithms of the scale ratio $L = \ln(Q^2/\mu^2)$.
• Decomposition: The results are decomposed into color factors ($C_F, C_A, N_c$) and electric charge dependencies ($e_q^2, \Sigma_Q$).
• Validation:
  • The authors cross-checked their analytical results against direct numerical integration using HyperInt for integer values $N \le 20$, finding perfect agreement.
  • They confirmed that the results match machine precision when compared with numerical evaluations of the provided Mathematica expressions.

5. Significance

• Enabling NNLO Phenomenology: These results are the missing link required to perform NNLO global fits of GPDs. Without these conformal moments, the theoretical precision of DVCS analyses would be limited to NLO, creating a mismatch with the high-precision data expected from future facilities.
• Future Experiments: The work is critical for upcoming experiments at the Jefferson Lab (JLab), the Electron-Ion Collider (EIC), and the EIcC in China. Specifically, the NNLO corrections are expected to be substantial for gluon GPD contributions in the EIC kinematic range.
• Consistency with PDFs: Since GPDs must reduce to standard Parton Distribution Functions (PDFs) in the forward limit, NNLO accuracy in DVCS is necessary to maintain consistency with modern PDF determinations, which have already reached NNLO accuracy.
• Computational Efficiency: The derived expressions allow for the implementation of NNLO corrections in existing analysis codes (like the Gepard code) using the Mellin-Barnes approach, which is significantly faster and more stable than traditional convolution methods.

In summary, this paper provides the theoretical infrastructure necessary to push the extraction of the 3D nucleon structure (GPDs) to the NNLO level, utilizing a sophisticated symmetry-based technique to overcome the computational barriers of multi-loop QCD calculations.