Radiative return at NLOPS accuracy

This paper presents an updated version of the BabaYaga@NLO Monte Carlo generator that computes exact next-to-leading order (NLO) QED corrections matched to a Parton Shower for radiative return processes ($e^+e^-\to \pi^+\pi^-\gamma$ and $\mu^+\mu^-\gamma$) to enable precision measurements of the pion form factor and the muon anomalous magnetic moment at flavour factories.

The Gist

The Big Picture: The "Muon Mystery" and the "Flashlight"

Imagine the muon is a tiny, spinning top. Physicists have measured how much this top wobbles (its "anomalous magnetic moment") with incredible precision. However, to predict exactly how much it should wobble based on our current laws of physics (the Standard Model), we need to know how the muon interacts with a "cloud" of virtual particles popping in and out of existence.

The biggest piece of this puzzle is the pion form factor. Think of the pion as a fuzzy, soft ball rather than a hard marble. To understand how it interacts, we need to measure its "shape" (form factor) very carefully.

To measure this shape, scientists use particle colliders (flavour factories) that smash electrons and positrons together. They use a trick called "Radiative Return."

The Analogy: Imagine you are trying to hit a specific target on a wall, but you are standing too far away. You can't get close enough to see the details. So, you throw a heavy rock (a photon) at the wall before you throw your main ball. The rock hits the wall and bounces back, slowing you down just enough that your main ball now hits the target at the perfect speed.

• The Rock: A high-energy photon emitted by the electron or positron.
• The Slowdown: The collision happens at a lower energy, allowing scientists to scan a continuous range of energies without changing the machine's settings.

The Problem: The "Blurry Camera"

To get a perfect picture of the pion's shape, the scientists need to count exactly how many times this "slowdown" happens. But there's a catch: the universe is messy.

When the electron and positron collide, they don't just emit one "rock" (photon). They often emit a whole shower of tiny pebbles (soft photons) that are hard to see.

• Old Tools: Previous computer programs (like Phokhara) were like a camera with a slightly blurry lens. They could count the big rocks perfectly, but they missed the tiny pebbles or guessed their pattern. This introduced a "blur" (uncertainty) of about 0.5% in the results.
• The Goal: The authors wanted to build a camera with a super-sharp lens that could see every single pebble, no matter how small, to reduce that blur to almost zero.

The Solution: A "Smart Filter" and a "Traffic Cop"

The authors created a new, upgraded version of a computer program called BabaYaga@NLO. They didn't just add more data; they completely rewrote the logic of how the simulation handles the collision.

Here is how they did it, using two main concepts:

1. The "Exact Blueprint" (Fixed-Order Calculation)

First, they calculated the collision exactly for the most important scenarios:

• One big rock: The main event where one hard photon is emitted.
• Two big rocks: The event where two hard photons are emitted.
• The "Virtual" Ghosts: They also calculated the invisible, fleeting interactions (virtual corrections) that happen inside the collision.

They treated the pion not as a simple point, but as a complex object with an internal structure (the "form factor"), ensuring the math accounted for its "fuzziness."

2. The "Traffic Cop" (Parton Shower)

This is the novel part. In the real world, after the main collision, the particles might emit many more tiny photons. Calculating every single possibility for infinite photons is impossible.

So, they used a Parton Shower (PS) approach. Think of this as a Traffic Cop at a busy intersection.

• Instead of trying to predict every single car that might drive by, the Traffic Cop knows the rules of the road (the laws of physics).
• If a car (particle) is about to emit a photon, the Traffic Cop says, "Okay, based on the rules, there is a 90% chance you emit a tiny pebble, and a 10% chance you emit a medium one."
• The Traffic Cop then simulates this chain reaction, generating a realistic "shower" of photons.

The Magic Match: The authors' breakthrough was matching the "Exact Blueprint" (the hard, precise math for the big rocks) with the "Traffic Cop" (the simulation of the endless tiny pebbles).

• Before: You had to choose: either use the precise math (but miss the tiny pebbles) OR use the Traffic Cop (but miss the precise details of the big rocks).
• Now: They combined them. The Traffic Cop handles the tiny pebbles, but it is constantly corrected by the Exact Blueprint to ensure the big rocks are counted perfectly.

Why This Matters (The Results)

The paper presents a "validation test" to prove their new camera works.

1. No More "Blind Spots": They showed that their results don't change based on arbitrary settings (like how they define a "hard" vs. "soft" photon). This proves the math is solid.
2. The "Three-Rock" Test: They tested a scenario where three hard photons are emitted. Their simulation matched the results of other, independent, super-complex calculations almost perfectly.
3. The "Percent" Difference: They found that the "tiny pebbles" (higher-order corrections) actually change the results by about 1% to 3% in certain situations.
   • Why is this important? Because the experiments are trying to measure things with 0.1% precision. If you ignore the 1% effect of the tiny pebbles, your measurement is wrong. The old tools missed this; the new tool catches it.

The Bottom Line

The authors have built a super-accurate simulator for particle collisions.

• What it does: It predicts exactly what happens when electrons and positrons collide and emit photons, including the messy, invisible showers of tiny particles.
• Why it's better: It combines the best of two worlds: the precision of exact math for the main event and the realism of a simulation for the background noise.
• The Impact: This tool allows scientists to measure the "shape" of the pion with much higher confidence. This, in turn, helps solve the mystery of the muon's wobble, potentially revealing if there is new physics beyond our current understanding of the universe.

The code is now available for other scientists to use, acting as a new, sharper lens for the entire field of particle physics.

Technical Summary

Technical Summary: Radiative Return at NLOPS Accuracy

Problem Statement
The measurement of the pion form factor, $F_\pi(Q^2)$, via the radiative return method at flavor factories is a critical input for the data-driven dispersive calculation of the leading-order hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment, $a_\mu$. While the dispersive approach remains a key tool for determining the HVP contribution, tensions exist between different experimental measurements and between data-driven and lattice QCD results. Precise theoretical predictions are required to resolve these discrepancies.

Current Monte Carlo (MC) tools, such as Phokhara, incorporate complete Next-to-Leading Order (NLO) radiative corrections for processes like $e^+e^- \to X^+X^-\gamma$ ($X=\{\pi, \mu\}$). However, these generators lack the exclusive exponentiation of multiple photon emission. Consequently, their theoretical accuracy is limited to approximately 0.5%, a value often quoted as a systematic uncertainty by experiments. There is a specific gap in the literature for an event generator that combines exact NLO corrections with a Parton Shower (PS) to describe exclusive multiple photon emission, particularly for processes with a hard photon in the final state ($2 \to 3$ processes).

Methodology
The authors present a calculation of the exact NLO corrections matched to a Parton Shower (NLOPS) for the processes $e^+e^- \to \mu^+\mu^-\gamma$ and e^e^- \to \pi^+\pi^-\gamma. The calculation is implemented in an updated version of the MC event generator BabaYaga@NLO.

Key methodological components include:

• Fixed-Order Calculation: The authors compute the exact NLO corrections, including virtual and real contributions of order $\mathcal{O}(\alpha^4)$. This involves calculating soft+virtual (SV) corrections and real emission of an additional hard photon. The calculation accounts for Initial-State Radiation (ISR), Final-State Radiation (FSR), and Initial-Final Interference (IFI).
  • For the muon channel, standard QED is used.
  • For the pion channel, the calculation employs the QED $\oplus$ F$\times$sQED (Factorised scalar QED) approach, where scalar QED point-like amplitudes are multiplied by the pion form factor $F_\pi(Q^2)$ at appropriate virtualities.
• Parton Shower Matching: The authors develop a novel matching scheme to combine the exact NLO calculation with a PS algorithm that exponentiates leading logarithmic (LL) corrections for multi-photon emission.
  • Unlike previous formulations for $2 \to 2$ processes, the LL approximation for real photon emission is constructed on top of the exact matrix element for the two-photon emission process ($e^+e^- \to X^+X^-\gamma\gamma$).
  • A CKKW-like clustering algorithm is adapted from QCD to QED to select the "hardest" photons for the exact matrix element evaluation, ensuring infrared and collinear safety.
  • A correction factor, $F_{SV}$, is introduced to reproduce the exact SV correction in the one-photon sample while preserving the LL exponentiation for samples with $n \ge 1$ additional photons. This ensures the cancellation of the unphysical soft-hard separator parameter $\varepsilon$.
• Phase-Space Treatment: The implementation utilizes a multi-channel importance sampling technique to efficiently handle the phase-space integration, particularly addressing enhancements from soft/collinear singularities and non-perturbative effects like narrow resonances and the running of the electromagnetic coupling.

Key Contributions

1. First NLOPS Calculation for Radiative Return: This work provides the first computation of exact NLO corrections matched to a PS for $e^+e^- \to X^+X^-\gamma$ processes, filling a gap in the literature where such tools were previously unavailable for exclusive hard photon events.
2. Novel Matching Formulation: The paper introduces a specific formulation for matching NLO corrections to PS in $2 \to 3$ processes. By basing the LL approximation on the exact $2 \to 4$ kinematics (two hard photons) rather than $2 \to 2$, the authors avoid spurious enhancements near resonances that can arise in other matching schemes.
3. Implementation in BabaYaga@NLO: The calculation is fully integrated into BabaYaga@NLO, making it available for fully exclusive simulations and data analysis.
4. Validation: The authors perform extensive validation tests, comparing their NLO results with independent codes (McMule, Recola, Alpha, and Phokhara). They demonstrate agreement at the per-mille level for the muon channel and identify specific discrepancies with Phokhara in the pion channel angular distributions, attributed to the inclusion of previously omitted gauge-invariant subsets.

Results
Numerical results are presented for four experimental scenarios (KLOE I Large-Angle, KLOE II Small-Angle, BES3, and B-factory) using realistic event selection criteria.

• Cross Sections: The integrated cross sections show NLO corrections of approximately 1% for the KLOE Large-Angle scenario and per-mille levels elsewhere.
• Differential Distributions: The impact of NLO and higher-order (multi-photon) corrections is quantified.
  • NLO corrections are particularly large in the high invariant mass region ($M_{XX}$), reaching tens of percent for the muon channel due to soft photon emission.
  • Higher-order corrections (beyond NLO) are found to be non-negligible, reaching the percent level in high invariant mass regions for KLOE scenarios and up to a few percent for BES3 and B-factory scenarios.
  • The ratio of the pion to muon differential cross sections, used for normalizing hadronic data, shows higher-order corrections varying up to 2% in the KLOE Large-Angle setup.
• Gauge-Invariant Subsets: The analysis of ISR, FSR, and IFI contributions reveals that while FSR is suppressed in small-angle setups, IFI contributions (both even and odd under particle exchange) are significant in large-angle setups and affect forward-backward asymmetries.

Significance and Claims
The paper claims that the observed higher-order effects (beyond NLO) are comparable to or larger than the systematic uncertainties in current radiative return measurements at meson factories. Consequently, the authors assert that the new NLOPS formulation in BabaYaga@NLO is essential for achieving sub-percent precision in the extraction of the pion form factor.

The work highlights that accurate simulations for radiative return experiments require not only exact NLO corrections but also the inclusion of multiple photon contributions. The authors state that their code will likely impact future data analyses, potentially reducing theoretical uncertainties in the determination of the HVP contribution to $a_\mu$. They explicitly note that while their current treatment of the pion channel uses the F$\times$sQED approximation, future work will address structure-dependent corrections beyond this approximation.