Light-by-light scattering at three loops in massless QCD and QED: amplitudes and cross sections

This paper presents the first analytic calculation of three-loop massless QCD and QED helicity amplitudes for light-by-light scattering, which are used to derive NNLO differential cross-section predictions that show agreement with ATLAS experimental data from ultra-peripheral heavy-ion collisions.

The Gist

Imagine light not just as a beam that travels in a straight line, but as a bustling crowd of invisible dancers. In the classical world (think of old-school physics), these dancers never interact; they simply pass through each other like ghosts. If you shine two flashlights at each other, the beams cross, but they don't bounce off one another.

However, in the quantum world, things get weird. Thanks to the "vacuum" (empty space), which is actually teeming with virtual particles popping in and out of existence, light can interact with light. This phenomenon is called Light-by-Light (LbL) scattering. It's like two dancers briefly grabbing hands with invisible partners from the crowd before letting go and continuing on their way.

This paper is a massive achievement in calculating exactly how these dancers move when they interact, pushing the math to the highest level of precision ever attempted.

Here is the breakdown of what the scientists did, using simple analogies:

1. The Goal: Predicting the Dance Moves

The researchers wanted to predict the outcome of this light-by-light scattering with extreme accuracy. In physics, we calculate these outcomes using a "recipe" called perturbation theory, which involves adding layers of complexity:

• Level 1 (LO): The basic dance step.
• Level 2 (NLO): Adding a few extra spins and turns.
• Level 3 (NNLO): The full, complex choreography with every possible twist.

This paper calculates the Level 3 (NNLO) version. This is the "three-loop" calculation. Think of it as calculating the dance moves not just for one step, but for a sequence involving three layers of invisible, virtual interactions happening simultaneously. It's like trying to predict the exact path of a leaf falling in a hurricane, accounting for every tiny gust of wind.

2. The Challenge: A Mathematical Jungle

Calculating this is incredibly hard.

• The Diagrams: To do this, they had to draw and solve 1,296 different Feynman diagrams (mathematical pictures of particle interactions). That's like trying to solve 1,296 different puzzles at once.
• The Complexity: The math involves "tensors" (multi-dimensional numbers) that get huge and messy very quickly. If you tried to write out the full equation on paper, it would be longer than the entire text of War and Peace.
• The Solution: The team used a "Lorentz tensor decomposition." Imagine you have a giant, tangled ball of yarn. Instead of trying to pull the whole thing apart at once, they found a way to cut it into 8 specific, manageable strands. This made the problem solvable.

3. The Tools: The "Abelianisation" Trick

They worked in two different "universes" of physics: QCD (the physics of quarks and gluons, which is very complex and sticky) and QED (the physics of light and electrons, which is simpler).

• They first solved the complex QCD version.
• Then, they used a clever trick called "Abelianisation." Think of this as a translator. They took their complex QCD results and "translated" them into the simpler QED language. This allowed them to get the answer for light scattering without having to start from scratch.

4. The Result: A Compact Secret

Despite the massive amount of math (involving millions of terms), the final answer turned out to be surprisingly elegant.

• The authors describe the final formula as "remarkably compact."
• It's like finding that a chaotic, noisy room of 1,000 people actually follows a simple, rhythmic song.
• They expressed the answer using a special set of mathematical functions called Harmonic Polylogarithms. Think of these as the "alphabet" of this quantum dance. They found that the entire complex dance could be written using just 23 of these "letters."

5. The Real-World Test: The Heavy Ion Collision

Why does this matter?

• The Experiment: At the Large Hadron Collider (LHC), scientists smash heavy lead ions together. Because these ions are so heavy and charged, they create a massive cloud of light (photons) around them. When they pass close to each other (without hitting head-on), these clouds of light collide and scatter.
• The Check: The ATLAS experiment at the LHC has measured this scattering.
• The Match: The team took their new, ultra-precise "three-loop" prediction and compared it to the ATLAS data.
  • Old predictions (Level 1 and 2): Were okay, but had some gaps.
  • New prediction (Level 3): Fits the experimental data almost perfectly. It confirmed that our understanding of the Standard Model (the rulebook of the universe) is correct, even at this incredibly high level of detail.

6. The "K-Factor": The Surprise

In physics, we often look at a "K-factor," which tells us how much the new, complex math changes the old, simple math.

• Usually, as you add more layers of complexity, the changes get smaller and smaller (like adding a pinch of salt to a soup).
• The Surprise: In this case, the "Level 3" correction was actually quite large—sometimes even bigger than the "Level 2" correction! It turned out that a specific, new type of interaction (a new "dance move" that only appears at this high level) was significantly boosting the result. This was a happy surprise that showed how much there is still to learn about these interactions.

Summary

This paper is a triumph of theoretical physics. The authors took a problem that was previously too messy to solve (light hitting light with extreme precision), built a new mathematical "ladder" to climb it, and found that the view from the top matches perfectly with what we see in the real world. They proved that even in the chaotic quantum vacuum, the universe follows a precise, calculable, and surprisingly elegant rhythm.

Technical Summary

1. Problem Statement

The paper addresses the theoretical calculation of Light-by-Light (LbL) scattering ($\gamma\gamma \to \gamma\gamma$), a fundamental process in Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD) where photons interact via virtual fermion loops.

• Context: While classically forbidden by the superposition principle, LbL scattering occurs in Quantum Field Theory due to vacuum polarization. It is a crucial probe for the Standard Model (SM) and potential New Physics (BSM).
• Experimental Relevance: The process is observed at the Large Hadron Collider (LHC) in Ultra-Peripheral Collisions (UPCs) of heavy ions (e.g., Pb-Pb), where the electromagnetic interaction dominates. Recent data from ATLAS and CMS require high-precision theoretical predictions to test the SM and constrain BSM models.
• The Gap: Prior to this work, theoretical predictions were limited to Next-to-Leading Order (NLO) (two-loop) accuracy. The authors aim to provide the first Next-to-Next-to-Leading Order (NNLO) prediction, requiring the computation of three-loop helicity amplitudes in both massless QCD and QED.

2. Methodology

The authors employed a sophisticated computational framework combining modern amplitude techniques, algebraic reduction, and numerical validation.

A. Theoretical Framework

• Process: $\gamma(p_1) + \gamma(p_2) \to \gamma(-p_3) + \gamma(-p_4)$ with massless external photons.
• Scheme: Calculations were performed in the 't Hooft-Veltman (tHV) dimensional regularization scheme ($d = 4-2\epsilon$), where loop momenta are $d$-dimensional but external momenta are strictly 4-dimensional.
• Decomposition:
  • Lorentz Structure: The amplitude was decomposed into 8 independent tensors ($T_i$) and corresponding form factors ($F_i$) using a specific tensor basis derived from spinor-helicity formalism.
  • Helicity Amplitudes: The 8 helicity configurations were reduced to 3 independent ones ($++++$, $-+++$, $--++$) via Bose symmetry and parity relations.
  • Gauge Invariance: The amplitude was split into gauge-invariant subsets based on color and flavor structures ($N^{(i,j)}$), distinguishing between pure QCD, pure QED, and mixed contributions.

B. Computational Workflow

1. Diagram Generation: Generated 1,296 Feynman diagrams at the three-loop level using Qgraf.
2. Algebraic Manipulation: Performed color and Lorentz tensor algebra using FORM.
3. Integral Reduction:
   • Mapped scalar integrals to known topologies.
   • Removed scaleless integrals (zero in dimReg).
   • Reduced ~1.05 million integrals to a basis of 486 Master Integrals (MIs) using Integration-By-Parts (IBP) identities.
   • Tools used: Reduze 2, Kira 3, and an in-house framework Finred.
   • Optimization: Employed syzygy relations and finite-field reconstruction methods to manage expression growth. A staged IBP reduction strategy with targeted multivariate partial fraction decomposition (using MultivariateApart) prevented intermediate expressions from becoming computationally intractable (avoiding gigabytes of data).
4. Master Integrals: The MIs were expressed in terms of Harmonic Polylogarithms (HPLs) up to transcendental weight 6. Analytic continuation was handled using shuffle algebra relations to cross branch cuts when moving between kinematic regions.
5. Abelianization: Pure QCD results were converted to QED and mixed QCD-QED amplitudes by replacing gluons with photons and adjusting color/flavor factors ($N_c \to 1$, $C_F \to Q_f^2$, etc.).
6. Renormalization:
   • UV: Renormalized in the $\overline{\text{MS}}$ scheme for couplings and the On-Shell (OS) scheme for external photon fields.
   • IR: The process is free of infrared divergences (the IR anomalous dimension vanishes for LbL), so no additional IR subtraction was required beyond UV renormalization.

3. Key Contributions

• First NNLO Prediction: The paper presents the first complete calculation of three-loop massless QCD and QED helicity amplitudes for LbL scattering.
• Compact Analytic Results: Despite the complexity of three-loop calculations, the final finite remainders are remarkably compact. They are expressed as linear combinations of 126 HPLs (reducible to a basis of 23 functions) with rational coefficients.
• Abelianization Strategy: The authors provide a systematic method to extract QED and mixed QCD-QED corrections from the pure QCD results by analyzing the gauge-invariant subsets.
• Numerical Stability: The analytic results are designed for fast and stable numerical evaluation, facilitating phenomenological applications.

4. Results

A. Amplitudes

• The finite remainders for the helicity amplitudes ($f_{ren}$) were derived for all independent configurations.
• All-Plus Configuration ($++++$): The result is particularly simple, showing a "weight drop" compared to the double-minus configuration. Explicit formulas are provided in the text.
• Imaginary Parts: While the one-loop amplitude is purely real (loop-induced), higher-order loops introduce significant imaginary parts, which grow with the loop order.

B. Differential Cross Sections

The authors computed the differential cross sections ($d\sigma/dm_{\gamma\gamma}$ and $d\sigma/dy_{\gamma\gamma}$) for Pb-Pb UPCs at $\sqrt{s_{NN}} = 5.02$ TeV.

• K-Factors:
  • NLO: Corrections are small ($\sim 0.5\% - 2\%$).
  • NNLO: Corrections are significant ($\sim 1\% - 3.5\%$), comparable to or larger than NLO corrections. This is largely driven by a new gauge-invariant topology ($N^{(3,4)}$) appearing at three loops.
• Comparison with Data:
  • The NNLO predictions show excellent agreement with ATLAS experimental data across all invariant mass bins (except the very last bin where LO was slightly low, but NNLO fits well).
  • The total integrated cross section is predicted as $\sigma_{NNLO} = 102.10^{+0.44}_{-0.25}$ nb, consistent with the ATLAS measurement of $120 \pm 22$ nb.
• Uncertainties: The relative uncertainty of the NNLO result is slightly larger than NLO, but the results remain within experimental error bars.

5. Significance

• Precision Physics: This work establishes the state-of-the-art theoretical precision for LbL scattering, essential for interpreting LHC heavy-ion data and searching for BSM physics (e.g., axion-like particles, anomalous gauge couplings).
• Validation of SM: The agreement between the NNLO prediction and ATLAS data reinforces the validity of the Standard Model in the non-linear photon sector.
• Methodological Benchmark: The successful computation of these amplitudes demonstrates the viability of modern IBP reduction and finite-field techniques for complex three-loop processes with multiple external legs.
• Future Directions: The authors note that while massless approximations work well at high invariant masses, future work must incorporate massive quark (charm, bottom, top) and lepton (tau) loops, particularly near production thresholds, to fully match experimental precision at lower energies.

In summary, this paper delivers a landmark theoretical achievement in high-energy physics, providing the necessary NNLO tools to rigorously test the Standard Model against experimental LbL scattering data.