Disperon QED

The paper introduces "disperon QED," a novel method leveraging dispersion relations, automated tools like OpenLoops, effective field theory, and threshold subtraction to efficiently handle data input for hadronic vacuum polarization insertions in two-loop Monte Carlo calculations, specifically demonstrated through the $ee\to\pi\pi$ process in McMule.

The Gist

The Big Picture: Solving a "Black Box" Problem in Physics

Imagine you are trying to predict the outcome of a complex game of billiards (particle physics). You have a super-computer simulation (Monte Carlo code) that can calculate exactly how the balls will bounce if they are perfect, solid spheres.

But in the real world, the "balls" (particles like pions) aren't perfect spheres. They are fuzzy, squishy clouds made of even smaller stuff (quarks and gluons). When these fuzzy balls interact, the computer gets confused because it doesn't know the exact internal rules of the cloud.

Physicists have experimental data (measurements) that tell us how these fuzzy balls behave, but they can't just plug that raw data into the computer's math engine. The engine expects clean, simple formulas, not messy experimental numbers.

The Problem: How do we feed messy, real-world data into a clean, mathematical simulation, especially when the particles are involved in complex loops (like a ball bouncing off a wall, hitting another ball, and coming back)?

The Solution: The authors created a new method called Disperon QED.

---

The Core Idea: The "Disperon" (The Magic Particle)

To solve this, the authors invented a fictional, temporary particle they call a Disperon.

Think of a Disperon as a "data carrier" or a "messenger."

• The Old Way: Trying to calculate the interaction directly with the messy data was like trying to drive a car through a swamp. The wheels (the math) would get stuck.
• The New Way (Disperon QED): Instead of driving through the swamp, they build a bridge. The Disperon is the bridge.

Here is how it works:

1. The Dispersion Relation: This is a mathematical rule that says, "The messy behavior of the fuzzy ball can be described as a sum of many different types of heavy, invisible particles."
2. The Transformation: The authors take the messy data and pretend it is actually a stream of these heavy "Disperons" passing through the system.
3. The Magic: Because the Disperon is just a standard particle (like a heavy photon), the computer's existing tools (like OpenLoops) can easily calculate how it behaves. The computer doesn't care that the Disperon is fake; it just does the math.
4. The Final Step: Once the computer finishes the calculation, the authors "sum up" all the different types of Disperons to reconstruct the original messy data.

Analogy: Imagine you want to know the exact flavor of a complex soup, but your blender can only handle single ingredients.

• Old way: You try to blend the whole soup at once, and the blender breaks.
• Disperon way: You pretend the soup is made of 1,000 different types of "Flavor Cubes." You tell the blender, "Calculate the taste of a carrot cube, then a potato cube, then a celery cube." The blender handles each cube easily. At the end, you add up the tastes of all the cubes to get the flavor of the soup.

Handling the "Heavy" Stuff: The EFT Trick

There is a catch. Some of these "Disperons" are incredibly heavy (very high energy). If you try to calculate them directly, the computer slows down to a crawl or crashes because the numbers get too big.

To fix this, the authors use a technique called Disperon Effective Theory (DET).

• The Metaphor: Imagine you are trying to calculate the wind resistance on a giant ship. If the ship is huge, you don't need to calculate the movement of every single water molecule. You can just use a simple rule: "Big ships move slowly."
• The Application: For very heavy Disperons, the authors switch from the complex "full calculation" to a simplified "rule of thumb" (Effective Theory). This keeps the simulation fast and stable without losing accuracy.

The "Threshold" Problem: Avoiding the Cliff

When calculating these loops, there is a specific point (called a "threshold") where the math goes wild, like a car driving off a cliff. This happens when the energy is just enough to create a new particle.

• The Fix: The authors developed a "safety net" called Threshold Subtraction.
• The Metaphor: Imagine you are walking on a tightrope. Right in the middle, there is a wobbly spot that makes you dizzy. Instead of trying to walk perfectly over it, you lay down a temporary plank (the subtraction) that smooths out the wobble. You walk across the plank, and then you remove the plank and add back the exact amount you "borrowed" to stay safe. This allows the calculation to cross the dangerous point without crashing.

Why Does This Matter?

This paper isn't just about one specific experiment; it's about building a universal toolkit.

1. Solving Tensions: There are currently big disagreements in physics between different experiments (like measuring the size of a proton or the magnetic strength of a muon). This new method allows scientists to run much more precise simulations to see who is right.
2. Future Proofing: The authors tested this on a simple process ($e^+e^- \to \pi\pi$), but the method is designed to handle much more complex scenarios, like adding extra photons or dealing with protons instead of pions.
3. Automation: By turning messy data into "standard particles" (Disperons), they can use automated software that was already built for simple physics. This saves years of manual math work.

Summary

Disperon QED is a clever engineering trick. It takes messy, real-world experimental data, disguises it as a stream of fake "heavy particles" (Disperons) that computers love to calculate, uses a "rule of thumb" for the heaviest ones to keep things fast, and uses a "safety plank" to cross dangerous mathematical cliffs.

The result is a powerful new way to simulate the subatomic world with unprecedented precision, helping physicists solve the mysteries of why the universe behaves the way it does.

Technical Summary

1. Problem Statement

Low-energy scattering processes (e.g., $e^+e^- \to \pi^+\pi^-$ and lepton-proton scattering) are critical for addressing fundamental questions in particle physics, such as the hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment ($g-2$) and the proton radius puzzle. However, there are significant tensions between experimental data, lattice QCD calculations, and different experimental methods.

To resolve these discrepancies, precise Monte Carlo (MC) tools are required. While pure QED corrections for point-like particles have reached Next-to-Next-to-Leading Order (NNLO) and beyond, incorporating non-perturbative hadronic effects (such as HVP insertions and hadronic form factors) into loop calculations remains a major challenge.

• The Core Issue: Standard loop integration tools (like OpenLoops) cannot handle numerical input functions (e.g., HVP or Vector Form Factors, VFF) that depend on the loop momentum $k$.
• Current Limitations: Existing methods often rely on specific analytic parametrizations (e.g., Generalized Vector Meson Dominance) or "Form Factor $\times$ Scalar QED" ($F \times sQED$) approaches where the form factor is applied after loop integration. The latter is known to yield poor agreement with data (e.g., forward-backward asymmetry) because it fails to capture the correct momentum dependence within the loop.

2. Methodology: Disperon QED

The authors propose a novel framework called Disperon QED to integrate generic numerical hadronic data into automated loop calculations. The method relies on three main pillars:

A. The Disperon Concept

Instead of treating the hadronic vacuum polarization or form factor as a complex numerical function inside a loop, the method uses a dispersion relation to rewrite the propagator.

• A function $F(k^2)$ (like HVP $\Pi(k^2)$ or VFF $F_\pi^V(k^2)$) is expressed via a once-subtracted dispersion relation:
  $$\frac{F(k^2)}{k^2} = \frac{F(0)}{k^2} - \frac{1}{\pi} \int_{s_{thr}}^\infty ds_1 \frac{\text{Im} F(s_1)}{s_1(k^2 - s_1)}$$
• This transforms the problem into a standard loop integral where the photon propagator $1/k^2$ is replaced by a massive propagator $1/(k^2 - s_1)$.
• The authors introduce a fictitious massive vector boson called the "disperon" with mass $\sqrt{s_1}$. The loop calculation is performed in a theory containing photons and disperons.
• After the loop amplitude is calculated, the result is integrated numerically over the dispersion parameter $s_1$, weighted by the imaginary part of the hadronic input ($\text{Im} F(s_1)$).

B. Automation with OpenLoops

Because the disperon behaves like a massive particle with standard couplings, the authors implemented a custom model in OpenLoops (an automated tool for one-loop amplitudes).

• OpenLoops generates and evaluates the amplitudes for diagrams containing 0, 1, or 2 disperons.
• This allows the use of standard, high-performance loop technology without needing to derive analytic expressions for every specific hadronic input.

C. Handling Large Masses: Disperon Effective Theory (DET)

The dispersion integral extends to infinity. While the integrand is suppressed at large $s_1$, direct numerical integration with OpenLoops becomes unstable and slow for very large disperon masses.

• Solution: The authors developed Disperon Effective Theory (DET).
• For large $s_1$, the disperon is "integrated out," and the amplitude is expanded in powers of $1/s_1$ (analogous to Soft-Collinear Effective Theory).
• They derive dimension-6 and dimension-8 operators (matching tree-level diagrams) to approximate the amplitude for large $s_1$.
• Hybrid Strategy: The calculation uses OpenLoops for $s_1 < s_{cut}$ and the analytic DET expansion for $s_1 > s_{cut}$, ensuring both speed and numerical stability.

D. Threshold Subtraction

A specific singularity arises in $s$-channel processes when the dispersion parameter $s_1$ equals the center-of-mass energy squared $s$ (the threshold for producing the intermediate state on-shell).

• The authors perform a Landau analysis to identify the singularity structure.
• They construct a Counterterm (CT) that analytically subtracts the singular part of the integrand.
• The CT is universal and can be applied to various topologies (boxes, pentagons), ensuring the numerical integration remains finite and well-behaved.

3. Key Contributions

1. General Framework: Introduced "Disperon QED," a universal method to include generic numerical hadronic inputs (HVP, VFF) into automated loop calculations without relying on specific analytic parametrizations.
2. FsQED Implementation: Applied the method to Form-factor Scalar QED (FsQED), where the pion Vector Form Factor is inserted inside the loop. This corrects the deficiencies of the $F \times sQED$ approach.
3. Technical Innovations:
   • Development of DET to handle the high-mass region of dispersion integrals efficiently.
   • Derivation of a universal threshold subtraction scheme to handle $s_1 = s$ singularities.
   • Extension to resummed HVP, allowing for the inclusion of Dyson-resummed propagators within loops (crucial near resonances like $J/\psi$).
4. Implementation: Integrated these methods into the McMule Monte Carlo framework, a state-of-the-art NNLO tool for QED processes.

4. Results

The method was applied to the process $e^+e^- \to \pi^+\pi^-$ at $\sqrt{s} = 0.7$ GeV (relevant for CMD-3 experiment data).

• Validation: The DET results were validated against OpenLoops (in double and quadruple precision) and arbitrary-precision Mathematica calculations, showing excellent agreement.
• Phenomenology:
  • Mixed NLO Corrections: The inclusion of VFF inside the loop (FsQED) significantly impacts C-odd observables, such as the forward-backward asymmetry ($A_{FB}$). The results show that FsQED provides a substantial improvement over $F \times sQED$ and aligns well with experimental data.
  • NNLO HVP: The NNLO initial-state radiation (ISR) corrections with HVP insertions were computed. The effect of HVP resummation was found to be small for ISR (order $10^{-6}$) but noted to be potentially significant near resonances in other kinematic regions.
  • Asymmetry: The calculated forward-backward asymmetry $A_{FB}$ matches previous high-precision calculations and CMD-3 data, confirming the validity of the dispersion-based approach.

5. Significance and Outlook

• Bridging the Gap: Disperon QED bridges the gap between non-perturbative hadronic physics and high-order perturbative QED calculations, enabling the use of experimental data directly in loop integrals.
• Scalability: The method is designed to be general. While demonstrated on $e^+e^- \to \pi^+\pi^-$, the authors plan to extend it to more complex processes like $e^+e^- \to \pi^+\pi^-\gamma$ and lepton-proton scattering.
• Precision Physics: By providing a robust way to include hadronic structure in NNLO (and potentially higher) calculations, this tool is essential for reducing theoretical uncertainties in the determination of the muon $g-2$ and the proton radius.
• Future Applications: The framework is particularly promising for analyzing regions near resonances (e.g., $J/\psi$) where resummation effects are large, and for tackling the "proton radius puzzle" by improving the treatment of two-photon exchange (TPE) contributions in lepton-proton scattering.

In summary, Disperon QED transforms a difficult numerical integration problem into a standard particle physics calculation by introducing a "disperon" particle, supported by effective field theory techniques and automated tools, thereby enabling a new level of precision in low-energy QED phenomenology.