Radiative correction to the charge asymmetry in $e^{+}e^{-}\to\mu^{+}\mu^{-}$ process

This paper presents the calculation of next-to-next-to-leading order (NNLO) QED radiative corrections to the $C$-odd part of the differential cross section for the $e^{+}e^{-}\to\mu^{+}\mu^{-}$ process, thereby completing the analytical determination of the full NNLO differential cross section when combined with the authors' previous work.

The Gist

The Big Picture: A Cosmic Dance of Particles

Imagine a grand ballroom where electrons and positrons (the antimatter twins of electrons) meet to dance. They spin around each other and then vanish, reappearing as a new pair of dancers: muons (heavier cousins of electrons). This process, called $e^+e^- \to \mu^+\mu^-$, is one of the most fundamental "moves" in the universe's rulebook, known as Quantum Electrodynamics (QED).

For decades, physicists have known the basic steps of this dance (the "Born approximation"). But modern experiments are so precise that they can see the tiny, subtle wobbles and extra spins that happen when the dancers interact with invisible "ghosts" of energy (virtual particles) or emit soft photons (light).

This paper is about calculating those tiny wobbles with extreme precision. Specifically, the authors are looking at a very specific kind of wobble: asymmetry.

The "Left-Right" Imbalance

In a perfect, simple world, if you watch this dance, the muons would be just as likely to spin off to the left as to the right. The dance would be perfectly symmetrical.

However, the universe is a bit quirky. When you account for the complex interactions of quantum mechanics, the dance becomes slightly lopsided. The muons might prefer to spin slightly more to the right than the left. This is called Charge Asymmetry (or Forward-Backward Asymmetry).

• The Analogy: Imagine a coin toss. In a simple world, it's 50/50. But in this quantum world, the coin is slightly weighted. The authors are trying to calculate exactly how much that coin is weighted, not just once, but with incredible detail.

The "Next-Next-Next" Level of Detail

Physics calculations are done in layers of complexity, like peeling an onion:

1. Level 1 (Leading Order): The basic coin toss.
2. Level 2 (NLO): Accounting for the wind blowing on the coin.
3. Level 3 (NNLO): Accounting for the wind, the humidity, the spin of the earth, and the tiny vibrations of the table.

This paper calculates the NNLO (Next-to-Next-to-Leading Order) corrections. This is the third layer of detail. It is the difference between a rough sketch and a high-definition photograph.

The Two Main Ingredients

To get this level of precision, the authors had to solve two massive puzzles:

1. The "Ghost" of the Electron Mass

In these calculations, the electron is treated as having almost no mass, but not zero mass. If you treat it as zero, the math explodes (infinite numbers). If you treat it as heavy, the math gets too messy.

• The Metaphor: Imagine trying to balance a pencil on its tip. If the tip is perfectly sharp (zero mass), it falls instantly. If the tip is a flat block (heavy mass), it's easy. The authors had to calculate the balance for a tip that is almost a point but has a tiny, finite width. They had to track how this tiny width creates "logarithmic" effects (huge numbers that grow slowly) in the calculation.

2. The "Hadronic" Soup

Sometimes, the energy in the collision briefly turns into a cloud of protons and neutrons (hadrons) before turning back into muons. This is called Hadronic Vacuum Polarization.

• The Metaphor: Imagine the dancers are spinning, and for a split second, the floor turns into a thick, sticky mud (the hadron cloud) that slows them down or changes their path, before snapping back to a smooth floor. The authors calculated exactly how this "mud" distorts the dance.

What Did They Actually Do?

The authors didn't just guess; they performed a massive analytical calculation.

• The Math: They used advanced tools (like "Master Integrals" and "Polylogarithms") to solve the equations governing these particle interactions.
• The Result: They produced a complete, exact formula for the "lopsidedness" of the muon dance.
• The "C-Odd" Part: They focused specifically on the part of the calculation that changes sign if you swap left and right (C-odd). This is the part responsible for the asymmetry.

Why Does This Matter?

The paper states that this work completes the analytical calculation of this process at the NNLO level.

• The "Hello World" Upgrade: The authors call this process the "Hello, World!" of physics textbooks—the simplest example. By solving the "Hello, World" problem with the highest possible precision, they are providing a benchmark.
• Checking the Rules: If future experiments measure this asymmetry and it doesn't match their calculation, it would mean the "rulebook" (Standard Model) is wrong, hinting at new, undiscovered physics.
• Background Noise: They also mention this process is a "background" for other experiments. Think of it like trying to hear a whisper (a rare new particle) in a noisy room. To hear the whisper, you need to know exactly how loud the room's normal noise (this muon dance) is.

Summary in a Nutshell

The authors have built the most precise map ever created for a specific particle dance ($e^+e^- \to \mu^+\mu^-$). They calculated exactly how the dance leans to one side due to complex quantum effects, including the tricky influence of the electron's tiny mass and the temporary appearance of hadronic "mud." This map allows scientists to distinguish between the known rules of physics and potential new discoveries in future experiments.

Technical Summary

Technical Summary: Radiative correction to the charge asymmetry in $e^+e^- \to \mu^+\mu^-$ process

Problem and Motivation
Muon pair production in electron-positron annihilation ($e^+e^- \to \mu^+\mu^-$) is a fundamental process in Quantum Electrodynamics (QED). While its Born approximation is simple, modern experimental precision at colliders requires theoretical predictions beyond the Leading Order (LO). Next-to-Leading Order (NLO) corrections have been established for decades. However, the pursuit of Next-to-Next-to-Leading Order (NNLO) precision is now driven by the requirements of ongoing and future experiments.

This process serves dual purposes: as a key channel for luminosity determination and as a significant background for searches for rare events and New Physics. Furthermore, precise theoretical predictions are essential for measurements of observables like the pion form factor, which contributes to the hadronic vacuum polarization correction for the muon anomalous magnetic moment ($(g-2)_\mu$).

The differential cross section for this process naturally decomposes into C-even and C-odd parts. The C-even part, symmetric under $\theta \to \pi - \theta$, was calculated exactly in the muon mass in a previous work by the authors [1]. The C-odd part, which is responsible for angular and forward-backward asymmetries, vanishes at the Born level. Consequently, its leading contribution appears only at NLO, making the two-loop (NNLO) correction particularly significant relative to the C-even cross section. This paper focuses on completing the analytical calculation of the NNLO differential cross section by addressing the C-odd part.

Methodology
The authors calculate the NNLO QED corrections to the C-odd part of the differential cross section. The calculation involves evaluating two-loop virtual amplitudes and real radiative corrections.

1. Amplitude Decomposition: The full NNLO amplitude is decomposed into gauge-invariant sets of $n\gamma$-reducible diagrams (diagrams containing $n$ intermediate photon lines). The authors focus on the $A^{(2)}_{2\gamma}$ amplitude (two-loop, two-photon reducible), which introduces a new ingredient compared to previous calculations.
2. Mass Treatment: The muon mass $m_\mu$ is kept exact throughout the calculation to ensure applicability to processes involving taus and to handle specific background measurements. The electron mass $m_e$ is treated as a small parameter to regularize collinear divergences, resulting in large logarithms of the form $\ln(s/m_e^2)$. Power corrections in $m_e$ are neglected.
3. Renormalization and Factorization: Dimensional regularization ($d=4-2\epsilon$) is used for UV and IR divergences. An on-shell renormalization procedure eliminates UV divergences. Soft IR divergences are factorized using the standard formalism where the amplitude is expressed as $A = e^V H$, with $V$ containing the soft singularities and $H$ being the finite hard amplitude.
4. Master Integrals: The calculation of the hard amplitude $H^{(2)}_{2\gamma}$ relies on the evaluation of master integrals.
   • Diagrams involving electron mass logarithms (where $L_e=1$) utilize results from Ref. [6], which employ a Frobenius expansion in $m_e$.
   • Diagrams without electron mass logarithms are calculated separately, allowing the setting of $m_e=0$ to simplify the computation.
   • The authors derive 42 master integrals for the muonic contribution to the 2$\gamma$-reducible amplitude using IBP reduction (LiteRed) and solving differential equations in $\epsilon$-form (Libra). The results are expressed in terms of Goncharov's polylogarithms $G(\vec{w}|\beta)$.
5. Hadronic Vacuum Polarization (HVP): The contribution of hadronic vacuum polarization is included via a subtracted dispersion relation. The imaginary part of the polarization operator is extracted from the ratio $R(s)$ of hadronic to muon pair production cross sections. This contribution is integrated over the spectral function to obtain the HVP correction to the amplitude.
6. Asymptotic Analysis: The authors calculate both threshold and high-energy asymptotics. The high-energy limit involves expanding polylogarithms and using the PSLQ algorithm to express coefficients in terms of classical polylogarithms.

Key Contributions and Results

• Analytical Calculation: The paper provides the exact analytical expression for the C-odd part of the NNLO differential cross section for $e^+e^- \to \mu^+\mu^-$. This completes the NNLO calculation of the differential cross section, complementing the C-even part calculated in Ref. [1].
• Hard Amplitudes: The authors present the hard invariant amplitudes $H^{(2)}_{2\gamma}$ in terms of Goncharov's polylogarithms. These expressions are provided in computer-readable ancillary files.
• Asymmetry Corrections:
  • The relative corrections to the C-odd cross section, $\delta^{(1)}_{C-odd}$ (NLO) and $\delta^{(2)}_{C-odd}$ (NNLO), are derived.
  • The angular asymmetry $A(c)$ and the forward-backward asymmetry $A_{FB}$ are calculated up to NNLO.
  • The authors demonstrate that the terms depending on the soft photon cutoff $\omega_0$ cancel out in the definition of the asymmetry $A^{(2)}$, rendering the result independent of the soft cutoff parameter.
• Numerical Results:
  • For a beam energy of 1 GeV, the one-loop C-odd correction is of the order of 10%, while the two-loop correction is of the order of 1%. The two-loop correction has the opposite sign to the one-loop contribution.
  • The two-loop contribution to the forward-backward asymmetry ($A^{(2)}_{FB}$) is found to be at the level of $10^{-4}$.
  • The finite value of $A^{(2)}_{FB}$ at the muon pair production threshold is calculated to be $5/3 \alpha^2$.
  • The HVP contribution to the forward-backward asymmetry is evaluated, showing behavior dependent on experimental data for hadronic cross sections.
• Z-boson Contribution: The paper includes an estimate of the tree-level Z-boson exchange contribution to the forward-backward asymmetry for comparison, noting it becomes significant at higher energies (around 2 GeV).

Significance
The paper claims to complete the analytical calculation of the $e^+e^- \to \mu^+\mu^-$ differential cross section at NNLO in QED. By providing the C-odd part, the authors enable precise theoretical predictions for angular and forward-backward asymmetries. These results are intended to support high-precision experiments at $e^+e^-$ colliders, particularly for luminosity determination and background estimation for New Physics searches. The work also provides the necessary theoretical input for processes where this reaction serves as a background for measuring the pion form factor, relevant to the $(g-2)_\mu$ anomaly. The results are applicable to arbitrary polarization of the involved particles, though the paper focuses on the unpolarized case for numerical illustrations.