Beyond QED: Electroweak and hadronic extensions of McMule

This paper presents an overview of the McMule Monte Carlo framework and its recent extensions to include low-energy electroweak effects and non-perturbative hadronic contributions via disperon QED, highlighting their applications to precision experiments like MOLLER and the technical challenges of consistent effective field theory matching.

The Gist

Imagine you are trying to measure the weight of a feather with a scale that is so sensitive it can detect the weight of a single atom. To get a perfect reading, you can't just look at the feather; you have to account for the wind, the humidity, the tiny vibrations of the table, and even the gravitational pull of a passing fly.

In the world of particle physics, McMule is that ultra-sensitive scale. It is a computer program (a "Monte Carlo framework") designed to predict how subatomic particles like electrons and muons behave when they bounce off each other or decay. For years, McMule has been the gold standard for calculating the effects of QED (Quantum Electrodynamics), which is the physics of light and electricity. It's incredibly precise, calculating effects up to the "third decimal place" of complexity (known as NNLO).

However, the universe is messy. Sometimes, the "wind" isn't just light; it's the Weak Force (electroweak) or the Strong Force (hadronic, involving protons and pions). The paper by Sophie Kollatzsch explains how McMule has been upgraded to handle these messier, more complex forces.

Here is a breakdown of the two major upgrades, explained with everyday analogies:

1. The "Ghostly Cloud" Upgrade: Hadronic Effects

The Problem:
In the past, McMule treated particles like billiard balls. But in reality, particles are often surrounded by a "cloud" of virtual particles popping in and out of existence. One specific cloud is made of pions (heavy cousins of electrons). When an electron moves, it drags this cloud with it.

The problem is that this cloud is "non-perturbative." In math-speak, this means you can't just add up the pieces one by one to get the answer; the pieces are too tangled. It's like trying to predict the path of a leaf in a hurricane by calculating the wind speed of every single air molecule.

The Solution: "Disperon QED"
The authors invented a new method called Disperon QED.

• The Analogy: Imagine you are trying to calculate the drag on a boat moving through water. Instead of trying to model every single water molecule (which is impossible), you use a "dispersion relation." This is like saying, "We know how the water behaves at the surface, and we know how it behaves deep down. Let's just measure the water at the surface and mathematically 'fill in the blanks' for the deep water."
• The "Disperon": They created a fake particle called a "disperon" (a portmanteau of dispersion and photon). This particle acts as a stand-in for the messy cloud.
• The Two-Step Dance:
  1. Low Energy: For the "easy" part of the calculation, they use a powerful tool called OpenLoops to do the heavy lifting.
  2. High Energy: For the "hard" part where the math gets unstable, they switch to a simplified "Effective Field Theory" (EFT). Think of this as switching from a high-definition camera to a sketch artist. You don't need every pixel; you just need the general shape to get the right answer.
• The Result: They can now calculate how electrons interact with pions ($e^- e^- \to \pi \pi$) with extreme precision, which is crucial for experiments like MUonE that are trying to measure the "size" of the proton's cloud.

2. The "Weak Force" Upgrade: Electroweak Effects

The Problem:
There is a force called the Weak Force (responsible for radioactive decay). At low energies, it's usually very weak and can be ignored. But for a specific experiment called MOLLER, which shoots electrons at other electrons, this force becomes the star of the show.

The MOLLER experiment wants to measure a tiny difference in how left-handed electrons scatter compared to right-handed electrons. This difference tells us about the Weak Mixing Angle, a fundamental number in the Standard Model. To measure this, they need to subtract the "noise" of the Weak Force from the "signal" of the Strong Force.

The Solution: LEFT (Low-Energy Effective Field Theory)

• The Analogy: Imagine you are listening to a symphony. If you want to hear the violin solo, you don't need to know the physics of the entire orchestra's sound waves. You just need to know how the violin interacts with the air in the room.
• The Method: Instead of calculating the full, complex Standard Model (which includes heavy particles like the Z boson that are too heavy to be created in these low-energy experiments), McMule now uses LEFT.
  • It "integrates out" the heavy particles, replacing them with simple "rules" (called Wilson coefficients) that tell the program how the light particles should behave.
  • This makes the math much simpler and faster, allowing them to calculate the "Weak Force noise" with the same precision as the "Light Force signal."
• The "Gamma-Z Mixing" Issue: There is a tricky part where the photon (light) and the Z boson (weak force) mix together. It's like two radio stations broadcasting on the same frequency. The paper shows that while different models of this "static" exist, the final result for the MOLLER experiment is robust. No matter which model you use, the answer stays within the margin of error.

Why Does This Matter?

Think of McMule as the GPS for particle physicists.

• Before: The GPS could tell you how to get to the store using only the main roads (QED).
• Now: The GPS has been updated to include backroads, construction zones, and traffic jams caused by other types of vehicles (Hadronic and Electroweak effects).

This upgrade is critical because experiments like MOLLER and P2 are pushing the boundaries of human knowledge. They are looking for "New Physics"—signs that the Standard Model is wrong or incomplete. If the GPS (McMule) isn't perfectly calibrated, we might think we found a new planet when we were actually just looking at a cloud.

By mastering these "messy" forces, the McMule team ensures that when we finally see something new in the data, we know for a fact that it's a discovery, not just a calculation error.

Technical Summary

Here is a detailed technical summary of the paper "Beyond QED: Electroweak and hadronic extensions of McMule" by Sophie Kollatzsch.

1. Problem Statement

Low-energy scattering and decay processes are critical laboratories for testing the Standard Model (SM) and extracting fundamental parameters. However, achieving the precision required by modern experiments (e.g., MUonE, MOLLER, P2) necessitates theoretical predictions beyond pure Quantum Electrodynamics (QED).

• The Challenge: While QED corrections up to Next-to-Next-to-Leading Order (NNLO) are well-established, incorporating Electroweak (EW) and hadronic effects introduces significant conceptual and computational difficulties.
• Specific Hurdles:
  • Hadronic Effects: Non-perturbative inputs, such as the pion vector form factor ($F_\pi^V$) and Hadronic Vacuum Polarization (HVP), cannot be treated using standard perturbative techniques when they appear inside loop integrals. Their momentum dependence complicates analytical evaluation.
  • Electroweak Effects: At low energies ($\sqrt{s} \ll M_Z$), EW effects are suppressed but crucial for parity-violating observables (e.g., the left-right asymmetry $A_{LR}$). Calculating these requires handling large logarithms ($\log(\mu_s/\mu_h)$) arising from the hierarchy between low-energy scales ($\mu_s$) and heavy EW scales ($\mu_h \sim M_Z, M_W$).
  • $\gamma-Z$ Mixing: The non-perturbative contribution to $\gamma-Z$ mixing ($\Pi_{\gamma Z}$) is a major source of theoretical uncertainty for parity-violating experiments, with existing models showing discrepancies.

2. Methodology

The paper focuses on McMule, a Monte Carlo framework designed for fully differential calculations of leptonic processes. The authors extend McMule beyond pure QED using two primary strategies:

A. Hadronic Extensions: Disperon QED

To handle non-perturbative hadronic inputs in loop amplitudes, the authors employ a method called disperon QED, which combines OpenLoops (for amplitude generation) with Effective Field Theory (EFT) techniques.

• Dispersion Relations: Instead of standard loop integration, the momentum dependence of form factors (like $F_\pi^V$) or HVP is expressed via dispersion relations. For example, the form factor is written as an integral over a spectral parameter $s_1$:
  $$F_\pi^V(q^2) = \dots + \frac{1}{\pi} \int_{4m_\pi^2}^\infty ds_1 \frac{\text{Im} F_\pi^V(s_1)}{q^2 - s_1 + i\delta}$$
• The "Disperon" Concept: This relation replaces the standard photon propagator with a propagator of a massive particle (the "disperon") with mass $\sqrt{s_1}$.
• Hybrid Calculation Strategy: To manage numerical stability and efficiency, the integration over $s_1$ is split into two regions using a cutoff $s_{cut}$:
  1. Low $s_1$ ($4m_\pi^2 \to s_{cut}$): Calculated using OpenLoops with a dedicated disperon QED model.
  2. High $s_1$ ($s_{cut} \to \infty$): Calculated using a Disperon Effective Theory. This EFT integrates out the heavy disperon, retaining terms up to a fixed order in the inverse mass expansion.
• Threshold Subtraction: A universal counterterm is formulated to subtract singularities that arise when $s_1 = q^2$ in the dispersive integral, ensuring numerical stability.

B. Electroweak Extensions: Low-Energy EFT (LEFT)

To address EW effects at low energies, the authors utilize Low-Energy Effective Field Theory (LEFT).

• Framework: The calculation combines NNLO QED with NLO EW effects within LEFT, including operators up to mass dimension 6.
• Resummation: LEFT naturally facilitates the resummation of large logarithms ($\log(\mu_s/\mu_h)$) via Renormalization Group Evolution (RGE) of Wilson coefficients.
• Implementation: All EW effects are encoded in Wilson coefficients, allowing flexible testing of different renormalization schemes and new physics scenarios.

3. Key Contributions

• Implementation of Disperon QED in McMule: The first systematic integration of non-perturbative hadronic form factors and HVP into NNLO Monte Carlo calculations for processes like $e^+e^- \to \pi^+\pi^-$.
• LEFT Framework for MOLLER: A new computational framework for the MOLLER experiment that consistently combines NNLO QED with NLO EW corrections and NLL resummation.
• Analysis of $\gamma-Z$ Mixing Models: A quantitative assessment of the uncertainty introduced by different models for the hadronic $\gamma-Z$ mixing term ($\Pi_{\gamma 3}$), comparing the standard alphaQED model against a lattice-informed model.

4. Results

• Hadronic Process ($e^+e^- \to \pi^+\pi^-$):
  • The authors successfully computed Next-to-Leading Order (NLO) mixed corrections involving pion form factors.
  • Stability Check: The results for the differential distribution of the average angle $\theta_{avg}$ were shown to be stable for cutoff values $s_{cut} \gtrsim 0.5 \text{ GeV}^2$. The final calculation used $s_{cut} = 4 \text{ GeV}^2$.
  • The method successfully reduced the complex two-loop problem involving HVP to a manageable one-loop integral over the dispersive parameter.
• Electroweak Process (MOLLER Experiment):
  • Resummation Impact: The study demonstrated that while NLO EW corrections reduce the scale dependence, the combination of NNLO EW with NLL running is essential for a fully satisfactory description of the left-right asymmetry $A_{LR}$.
  • Model Dependence: The impact of using different models for $\Pi_{\gamma 3}$ (hadronic $\gamma-Z$ mixing) on $A_{LR}$ was quantified.
    • At the $Z$-pole scale ($\mu_s = M_Z$), the model difference causes a $1-1.5%$ shift.
    • However, for the MOLLER experiment's preferred lower scale ($\mu_s \sim 5 \text{ GeV}$), the difference drops to the sub-percent level.
    • Conclusion: Given MOLLER's target experimental precision of $\sim 2\%$, the specific choice of the $\Pi_{\gamma 3}$ model (when used in an RGE-improved setting) does not significantly impact the final prediction.

5. Significance

• Precision Frontier: This work bridges the gap between pure QED precision and the inclusion of complex hadronic/EW effects, enabling McMule to support next-generation experiments like MUonE, MOLLER, and P2.
• Theoretical Robustness: By validating the "disperon" approach, the authors provide a robust method for handling non-perturbative inputs in loop calculations, which is essential for reducing theoretical uncertainties in HVP determinations (crucial for the muon $g-2$ anomaly).
• Experimental Support: The results confirm that current theoretical uncertainties regarding $\gamma-Z$ mixing models are under control for the MOLLER experiment, allowing the collaboration to focus on NNLO EW calculations as the next priority.
• Future Outlook: The framework is being extended to NNLO EW accuracy and applied to more complex processes (e.g., $e^+e^- \to \pi^+\pi^-\gamma$) and lepton-proton scattering (P2 experiment), solidifying McMule's role in the global effort to understand low-energy QCD and EW physics.