Light-by-light scattering: asymptotic expansions, Coulomb resummation and NLO corrections

This paper refines theoretical predictions for light-by-light scattering by improving two-loop helicity amplitudes with asymptotic expansions and Coulomb resummation, providing state-of-the-art Standard Model cross-section calculations and a new Monte Carlo event generator named LbLatNLO.

The Gist

The Big Picture: When Light Bumps into Light

Imagine a world where two beams of light can pass through each other like ghosts, never interacting. That is how light behaves in our everyday world and in classical physics (think of two flashlights crossing paths in a dark room). They just keep going.

However, in the quantum world, light is not just a wave; it's also made of particles (photons) that can interact with each other. This phenomenon is called Light-by-Light (LbL) scattering. It's like two cars driving toward each other, but instead of crashing, they briefly turn into a cloud of "ghost particles" (virtual particles), swap some energy, and then turn back into cars, continuing on their way.

This paper is about calculating exactly how often this happens and how strong the interaction is, with extreme precision. The authors have built a new, super-accurate "calculator" (a computer program) to predict these events for scientists working at the Large Hadron Collider (LHC) and future particle accelerators.

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The Three Main Ingredients

The paper tackles three specific challenges to make these predictions perfect. Here is how they did it, using simple metaphors:

1. The "Zoom Lens" Problem (Asymptotic Expansions)

The Problem: Calculating how light scatters involves complex math with many variables. When the energy is very low (like a slow-moving car) or very high (like a bullet train), the standard math formulas become unstable. It's like trying to measure the thickness of a hair with a ruler meant for measuring a football field; the numbers get messy, and computers start making mistakes due to "rounding errors."

The Solution: The authors created two special "zoom lenses" for their math.

• Low-Energy Lens: For slow interactions, they simplified the complex formulas into a series of easy steps (like a Taylor series).
• High-Energy Lens: For fast interactions, they did the same thing but focused on the dominant effects.
  The Analogy: Imagine trying to navigate a city. If you are walking slowly, you need a detailed street map. If you are flying in a jet, you need a high-altitude satellite view. The authors built both maps so the computer never gets lost, no matter how fast the particles are moving. This makes the calculations stable and fast.

2. The "Traffic Jam" at the Threshold (Coulomb Resummation)

The Problem: When two particles are about to collide and form a temporary pair (like an electron and a positron), they feel a strong pull toward each other, similar to magnets snapping together. In physics, this is called the "Coulomb force." Near the exact moment they form, the math predicts a "singularity"—a point where the numbers blow up to infinity, which doesn't make physical sense. It's like a traffic jam where the cars are so close they stop moving, causing a mathematical crash.

The Solution: The authors used a technique called Coulomb Resummation. Instead of trying to calculate the pull of the magnets one by one (which leads to the infinite crash), they calculated the total effect of the magnets all at once.
The Analogy: Think of a crowd of people trying to squeeze through a narrow door. If you try to count how many people pass through every second, you might get a chaotic, infinite number right at the bottleneck. Instead, the authors calculated the "flow rate" of the entire crowd as a single, smooth stream. This fixes the "infinite" problem and gives a realistic number for how many particles pass through.

3. The "Next-Gen" Calculator (NLO Corrections & The Event Generator)

The Problem: The first attempt to calculate this (called "Leading Order") is like a sketch. It's good, but it misses the fine details. To get a photograph, you need to add more layers of detail (Next-to-Leading Order, or NLO). This involves accounting for extra "ghost particles" popping in and out of existence.
The Solution: The authors didn't just write down the math; they built a software tool called LbLatNLO.
The Analogy: If the math formulas are the recipe for a cake, LbLatNLO is the fully automated bakery. You tell it what ingredients (particle masses, energy levels) you want, and it bakes the cake (simulates the collision) and tells you exactly how it will taste (the probability of the event).

• It can simulate collisions at the LHC (using heavy lead ions).
• It can simulate collisions at future electron-positron colliders.
• It generates "unweighted events," which means physicists can use this software to generate fake data that looks exactly like real data, helping them design experiments and spot new physics.

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Why Does This Matter?

1. Testing the Standard Model: The Standard Model is our best theory of how the universe works. Light-by-light scattering is a very rare, delicate process. If our new, super-precise calculator predicts a result that doesn't match what the LHC sees, it could mean we've discovered a new particle or a new force (Physics Beyond the Standard Model).
2. New Physics Probes: This process is sensitive to "exotic" particles like axions (dark matter candidates) or extra dimensions. By knowing exactly what the "normal" background looks like, scientists can spot the tiny deviations caused by these mysterious new things.
3. Practical Tool: By releasing LbLatNLO to the public, the authors have given the whole physics community a powerful new tool. It's like giving everyone a high-precision GPS for navigating the quantum world.

Summary

This paper is a masterclass in precision. The authors took a notoriously difficult quantum calculation, fixed the math so it doesn't break at high or low speeds, smoothed out the "traffic jams" where particles attract each other, and packaged it all into a user-friendly software tool. This allows scientists to look for new physics with a sharper, clearer lens than ever before.

Technical Summary

1. Problem Statement

Light-by-light (LbL) scattering ($\gamma\gamma \to \gamma\gamma$) is a fundamental prediction of Quantum Electrodynamics (QED) arising from vacuum fluctuations. While historically difficult to observe, recent measurements at the Large Hadron Collider (LHC) in ultra-peripheral heavy-ion collisions (UPCs) and future prospects at free-electron lasers have renewed interest in this process.

The paper addresses several theoretical challenges in predicting LbL scattering cross sections with high precision:

• Numerical Instability: Full two-loop helicity amplitudes with complete fermion-mass dependence suffer from severe numerical cancellations, particularly in the low-energy (LE) and high-energy (HE) limits, requiring extreme arithmetic precision (quadruple or higher) to remain stable.
• Threshold Singularities: In the threshold region ($\sqrt{s} \to 2m_f$), fixed-order Next-to-Leading Order (NLO) calculations exhibit logarithmic Coulomb singularities ($\log(E_f)$), rendering perturbative predictions unreliable near the production threshold of fermion pairs.
• Lack of Unified Tools: There was a need for a comprehensive framework combining state-of-the-art NLO QCD and QED corrections, threshold resummation, and a user-friendly event generator for phenomenological studies across different collider environments (hadron and lepton colliders).

2. Methodology

The authors employ a multi-faceted approach combining analytic expansions, resummation techniques, and numerical implementation:

• Asymptotic Expansions:

  • Low-Energy (LE) Expansion: For $s, |t|, |u| \ll m_f^2$, the authors derive analytic expansions of one- and two-loop helicity amplitudes in powers of $m_f^{-2}$. They utilize the "expansion-by-region" method to reduce complex two-loop master integrals (originally expressed as Goncharov polylogarithms) to vacuum-type integrals, which are solved analytically. This provides terms up to $\mathcal{O}(m_f^{-10})$.
  • High-Energy (HE) Expansion: For $s, |t|, |u| \gg m_f^2$, they perform a series expansion in $m_f^2$ using the differential equation method for master integrals. This captures subleading-power Sudakov logarithms ($\log(m_f^2)$) which are crucial for numerical stability in the high-energy regime.

• Coulomb Resummation:

  • To cure the logarithmic singularities in the threshold region, the authors implement a Green's function approach. They resum the leading-power (LP) Coulomb interactions between the virtual fermion-antifermion pair.
  • A damping function is introduced to ensure the resummation is applied only in the non-relativistic region, preventing unphysical overshooting in the relativistic regime where Coulomb and non-Coulomb contributions must cancel delicately.

• Event Generator (LbLatNLO):

  • A new Fortran90 event generator, LbLatNLO, is developed. It incorporates the full two-loop massive amplitudes, their asymptotic expansions (for stability), and Coulomb-resummed results.
  • It supports photon collinear factorization for various initial states: monochromatic photons, hadron-hadron UPCs (using Charge Form Factor flux), and electron-positron colliders (using Improved Weizsäcker-Williams flux).

3. Key Contributions

• Analytic Asymptotic Expansions: The paper provides the first complete analytic expressions for the two-loop LE and HE expansions of LbL helicity amplitudes, including higher-order terms ($\mathcal{O}(m_f^{-10})$ in LE and $\mathcal{O}(m_f^{14})$ in HE) that were previously unavailable.
• Threshold Resummation: A rigorous derivation and implementation of Coulomb resummation for loop-induced processes, specifically tailored to handle the unique cancellation structure of LbL scattering.
• Public Software Release: The release of LbLatNLO, a tool capable of computing differential fiducial cross sections and generating unweighted Monte Carlo events with full NLO QCD+QED accuracy and Coulomb resummation.
• Comprehensive Phenomenology: A unified analysis of LbL scattering across a vast energy range ($10^{-6}$ GeV to $10^7$ GeV), covering contributions from all Standard Model fermions and the $W^\pm$ boson.

4. Results

• Numerical Stability: The asymptotic expansions significantly improve numerical stability. In the LE region, the expansions allow for double-precision calculations where full amplitudes would fail. In the HE region, the expansions remain reliable up to $s/m_f^2 \sim 10^5$, whereas exact double-precision evaluations become unstable much earlier.
• Cross Section Predictions:
  • Standard Model (SM) Cross Section: The total SM cross section spans 13 orders of magnitude. The electron loop dominates up to $\sqrt{s} \approx 156$ GeV, after which the $W^\pm$ loop dominates.
  • Hadron Colliders (LHC Pb-Pb): Comparisons with ATLAS and CMS data at $\sqrt{s_{NN}} = 5.02$ TeV show good agreement.
    • NLO QCD corrections enhance the LO cross section by $\sim 5-6\%$.
    • NLO QED corrections add $\sim 0.3-0.4\%$.
    • Partial NNLO effects (from squared two-loop amplitudes) contribute $\sim 0.2\%$.
    • Coulomb resummation effects are negligible ($<0.1\%$) after integration over fiducial bins due to phase-space suppression near threshold.
    • Theoretical predictions lie within experimental uncertainties, though a slight tension ($\sim 1.8\sigma$) remains with the ATLAS central value in the low invariant mass bin.
  • Lepton Colliders: Predictions for $e^+e^-$ colliders (Belle II, FCC-ee, CEPC) show that NLO corrections increase the LO cross section by $5-8\%$, depending on the energy scale.
• Helicity Dependence: The paper analyzes helicity-dependent cross sections ($\sigma_{\pm\pm}$ vs $\sigma_{\pm\mp}$). Near the $e^+e^-$ threshold, the same-helicity configuration dominates due to the formation of a pseudoscalar state ($J^{PC}=0^{-+}$).

5. Significance

• Precision Test of QED: The work provides the most accurate theoretical predictions for LbL scattering to date, essential for testing the non-linearities of the QED vacuum and the Standard Model.
• BSM Physics Probe: By refining the SM background, the results enable more sensitive searches for Beyond Standard Model (BSM) physics, such as axion-like particles, anomalous quartic gauge couplings, and new charged particles, which could manifest as deviations in the LbL cross section.
• Methodological Benchmark: The techniques developed for asymptotic expansions and Coulomb resummation in loop-induced processes serve as a template for other scattering amplitudes (e.g., $gg \to \gamma\gamma$).
• Experimental Utility: The LbLatNLO generator bridges the gap between complex theoretical calculations and experimental analysis, allowing physicists to simulate realistic event distributions with state-of-the-art corrections for current and future collider experiments.

In conclusion, this paper establishes a new standard for the theoretical description of light-by-light scattering, resolving long-standing numerical and analytical issues and providing the necessary tools for precision phenomenology in the post-LHC era.