The complete three-loop unpolarized and polarized massive operator matrix elements and asymptotic Wilson coefficients

This paper presents the recently completed calculations of three-loop unpolarized and polarized massive operator matrix elements and asymptotic Wilson coefficients for deep-inelastic scattering, while also providing efficient numerical representations of massless coefficients, splitting functions, and target-mass corrections to support QCD fitting.

The Gist

Imagine the universe is a giant, incredibly complex Lego set. Physicists are the master builders trying to understand how the smallest bricks (quarks and gluons) snap together to form everything we see. To do this, they use a set of mathematical rules called Quantum Chromodynamics (QCD).

However, these rules are notoriously difficult to calculate. It's like trying to predict the exact path of a single Lego brick in a hurricane, but the hurricane is made of invisible energy, and the brick has a heavy weight attached to it.

This paper is a massive report from a team of theoretical physicists who have just finished a "three-loop" calculation. In the world of particle physics, a "loop" is like a round-trip journey a particle takes in a virtual sense.

• One-loop is a simple detour.
• Two-loops is a complex detour with a side trip.
• Three-loops is a journey so convoluted it involves multiple dimensions, time travel, and a map that keeps changing.

Here is what they achieved, broken down into simple concepts:

1. The Heavy Hitters: The "Heavy" Quarks

Most of the time, physicists study the lightest particles. But the universe also has "heavy" particles, specifically the Charm and Bottom quarks.

• The Analogy: Imagine you are trying to measure the speed of a race car (the light particles). But suddenly, a heavy truck (the heavy quark) joins the race. The truck changes the wind, the road friction, and the way the race car moves.
• The Problem: For a long time, the math to describe how this "heavy truck" affects the race was too messy to solve perfectly, especially when the race car is moving very fast (high energy).
• The Solution: This paper provides the complete, perfect map for how these heavy trucks behave at the highest level of precision (three-loops) when the race is fast enough. They calculated the "Operator Matrix Elements" (OMEs), which are essentially the "instruction manuals" for how these heavy particles interact with the rest of the universe.

2. The Two-Mass Puzzle

Usually, physicists calculate the effect of one heavy truck at a time. But in reality, you might have a Charm truck and a Bottom truck on the track at the same time.

• The Analogy: It's like trying to predict the weather when two different types of storms are colliding. The math gets incredibly messy because the storms interact with each other.
• The Solution: The team solved the math for both heavy quarks interacting simultaneously. They found that when you have both, the effect is huge—about 50% of the total correction comes from their interaction. Ignoring this would be like ignoring the wind when calculating a plane's fuel consumption.

3. The "Asymptotic" Shortcut

The paper focuses on a specific region where the energy is very high ($Q^2 \gg m^2$).

• The Analogy: Imagine you are driving a car. At low speeds, the engine's quirks matter a lot. But if you drive at 200 mph, the engine settles into a smooth, predictable rhythm. You don't need to know every tiny vibration; you just need the "asymptotic" (long-distance) behavior.
• The Solution: They calculated the "asymptotic" behavior. This is a shortcut that allows scientists to get incredibly accurate results without doing the impossible math of the full, messy low-speed calculation. It's like using a GPS that knows the highway traffic patterns perfectly, so you don't need to count every pothole.

4. The Toolbox for Everyone

Calculating this was like building a new kind of super-computer from scratch. They invented new mathematical languages (like "harmonic sums" and "elliptic integrals") to describe the shapes of these particle interactions.

• The Gift: They didn't just keep the math to themselves. They turned their complex formulas into computer code (Fortran libraries) and made them public.
• The Impact: Now, any scientist anywhere in the world can plug these numbers into their own simulations. It's like they built a new, ultra-precise ruler and handed it to the whole scientific community.

Why Does This Matter?

The ultimate goal of all this is to measure two fundamental things with extreme precision:

1. The Strong Force ($\alpha_s$): The glue that holds the universe together.
2. The Mass of the Charm Quark: How heavy that specific "brick" is.

Currently, different experiments give slightly different answers for these values, causing a debate in the physics community.

• The Analogy: It's like three different scales in a grocery store giving you slightly different weights for the same apple. You don't know which one is right.
• The Result: With these new, ultra-precise "instruction manuals" (the three-loop calculations), scientists can re-analyze data from the Electron-Ion Collider (EIC) and other experiments. They expect this new precision to finally settle the debate, telling us exactly how strong the universe's glue is and how heavy the charm quark really is.

In summary: This paper is the completion of a 20-year marathon of mathematical gymnastics. The team has built the most precise "heavy particle" map ever created, turned it into software for everyone to use, and paved the way for us to finally understand the fundamental building blocks of our universe with crystal-clear clarity.

Technical Summary

1. Problem Statement

The precision determination of fundamental QCD parameters, specifically the strong coupling constant $\alpha_s(M_Z^2)$ and the charm quark mass $m_c$, relies heavily on Deep-Inelastic Scattering (DIS) data. To achieve a theoretical precision of $O(1\%)$ or better, calculations must account for heavy-flavor corrections (charm and bottom quarks) at the three-loop level ($O(\alpha_s^3)$).

Previous theoretical frameworks faced several limitations:

• Kinematic Restrictions: Full phase-space calculations for massive corrections were only available up to two loops ($O(\alpha_s^2)$).
• Asymptotic Region: In the region where the virtuality $Q^2 \gg m_Q^2$ (where $m_Q$ is the heavy quark mass), the asymptotic heavy-flavor corrections were incomplete at three loops.
• Mathematical Complexity: The solution spaces for three-loop massive Operator Matrix Elements (OMEs) involve complex structures beyond standard harmonic sums, including generalized harmonic sums, cyclotomic sums, nested binomial sums, and iterated integrals over quadratic forms and elliptic letters.
• Two-Mass Complexity: The simultaneous presence of charm and bottom quark masses introduces a three-scale problem, leading to even more intricate mathematical structures and new special constants.

The paper addresses the completion of these calculations for both unpolarized and polarized cases, providing the necessary theoretical tools for next-generation QCD analyses.

2. Methodology

The authors employed a sophisticated combination of symbolic computation, algebraic algorithms, and numerical analysis to derive the results.

• Diagram Generation and Reduction: Feynman diagrams were generated using QGRAF and reduced to master integrals using FORM, Color, and Reduze 2.
• Calculation Strategies:
  • Hypergeometric Functions: Simpler topologies were solved using generalized hypergeometric functions and direct summation methods based on difference ring theory.
  • Mellin Moments & Guessing: For complex cases where the spanning functions were unknown, the method of arbitrary high Mellin moments was used. Recurrence relations were derived from a finite number of moments using guessing algorithms (e.g., Sigma package).
  • Differential Equations: For cases where recurrences were not first-order factorizable (specifically for $A_{gg,Q}^{(3)}$ and $\Delta A_{gg,Q}^{(3)}$), the authors utilized differential equations in $x$-space. They solved linear systems of coupled differential equations for master integrals using a generating function approach with a continuous variable $t \in [0, \infty[$.
  • Analytic Continuation: Solutions were obtained in the region $t \in [0, 1]$ and analytically continued to $t \in [1, \infty[$. The final results in $x$-space ($x = 1/t$) were constructed via overlapping logarithmically modulated Taylor expansions to ensure high numerical precision.
• Two-Mass Handling: For the two-mass case (charm and bottom), the authors expanded in the mass ratio around $m_c = m_b$ and utilized Aitken extrapolation to accelerate convergence. They also handled the decoupling of contributions in a two-mass Variable Flavor Number Scheme (VFNS).
• Numerical Implementation: To make the results usable for QCD fitting codes, the authors developed fast and precise numerical representations (Fortran libraries) for the massless Wilson coefficients, splitting functions, and target-mass corrections.

3. Key Contributions

• Complete Three-Loop OMEs: The paper presents the first complete calculation of the massive Operator Matrix Elements ($A_{ij}$) at three loops for both single-mass (charm or bottom) and two-mass (charm + bottom) cases, covering both unpolarized and polarized scenarios.
• Asymptotic Wilson Coefficients: Derivation of the associated asymptotic massive Wilson coefficients for structure functions $F_2(x, Q^2)$, $F_L(x, Q^2)$, $xF_3(x, Q^2)$, and polarized $g_1(x, Q^2)$ in the region $Q^2 \gg m_Q^2$.
• Mathematical Structures: The work extends the mathematical toolkit of QCD by successfully handling iterated integrals over root-valued letters, elliptic letters, and 2F1-structures, which are necessary for the two-mass case.
• Public Software: The authors released public Fortran codes (WILS3, SPLIT_U, SPLIT_P, TARGM) providing numerical implementations of:
  • Massless three-loop Wilson coefficients.
  • Unpolarized and polarized splitting functions (up to three loops).
  • Target-mass corrections.
  • These are designed for direct use in $x$-space QCD fitting codes.

4. Results

• Single-Mass Corrections: The constant part of the massive OME $A_{Qg}^{(3)}$ was calculated exactly, replacing previous approximate estimates. The results show that sub-leading terms in the small-$x$ region are of the same size or larger than the leading $\ln(x)/x$ term, significantly impacting the theoretical error budget.
• Two-Mass Corrections: The genuine two-mass corrections (arising at three loops) were found to contribute approximately 50% on average to the total $O(T_F^2 C_F A)$ contributions across all processes.
• Structure Function Predictions:
  • The new results allow for the first time a full NNLO QCD analysis of DIS data including heavy-flavor corrections at large $Q^2$.
  • In the large-$x$ region, non-singlet contributions dominate and are negative, reducing the massless contributions to $F_2$.
  • In the small-$x$ region, positive gluonic terms dominate, followed by negative pure-singlet terms.
• Impact on $\alpha_s$ and $m_c$: The inclusion of these corrections significantly reduces the theoretical uncertainty in the extraction of $m_c$ (previously $\delta_T m_c \approx 70$ MeV) and enables a cleaner extraction of $\alpha_s(M_Z^2)$, particularly through flavor non-singlet structure functions measured at future facilities like the Electron-Ion Collider (EIC).

5. Significance

• Precision Physics: This work provides the necessary theoretical foundation to match the precision of upcoming high-luminosity experiments (EIC, LHeC, FCC-he). It resolves discrepancies in previous $\alpha_s$ extractions by removing missing heavy-flavor theoretical uncertainties.
• Standard Model Tests: By enabling precise determination of $\alpha_s$ and quark masses, the results facilitate rigorous tests of the Standard Model and the running of the strong coupling constant.
• Methodological Advancement: The development of algorithms to handle non-factorizable differential equations and elliptic integrals in multi-loop QCD calculations sets a new standard for perturbative calculations in particle physics.
• Practical Utility: The release of numerical codes allows the global community to immediately incorporate these three-loop massive corrections into their QCD fitting frameworks, accelerating the analysis of current and future DIS data.

In conclusion, this paper represents a milestone in perturbative QCD, completing the three-loop heavy-flavor sector and providing the tools required to interpret high-precision deep-inelastic scattering data in the era of the Electron-Ion Collider.