Window quantities for the hadronic vacuum polarization contributions to the muon anomalous magnetic moment in spacelike and timelike domains

This paper establishes the precise mathematical relations and necessary edge-effect corrections required to ensure the mutual equivalence of window quantities for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment across spacelike and timelike domains, thereby enabling hybrid assessments using diverse data sources like lattice QCD, MUonE, and $R$-ratio measurements.

The Gist

The Big Picture: The "Magnetic Spin" Mystery

Imagine a muon (a heavy cousin of the electron) as a tiny, spinning top. Because it has an electric charge, it acts like a tiny magnet. Physicists have a very precise prediction for how strong this magnet should be, based on the Standard Model of physics (our current best rulebook for how the universe works).

However, when scientists actually measure the muon's magnetism in experiments, it spins slightly differently than the rulebook predicts. This tiny difference is called the anomalous magnetic moment. It's like if a compass needle pointed slightly off from true North. This discrepancy might be the first clue that there is "new physics" out there—something we haven't discovered yet.

The Problem: The "Fuzzy" Middle Ground

To calculate the theoretical prediction, scientists have to account for the muon interacting with a cloud of virtual particles popping in and out of existence. The biggest source of uncertainty comes from Hadronic Vacuum Polarization (HVP).

Think of the HVP as a thick, foggy cloud of "stuff" (quarks and gluons) that the muon spins through. Calculating how the muon interacts with this fog is incredibly hard because the rules of quantum physics (specifically Quantum Chromodynamics, or QCD) get messy and unpredictable at low energies. It's like trying to predict the exact path of a leaf swirling in a turbulent storm.

Currently, scientists use two main ways to measure this "fog":

1. The "Spacelike" Method: Looking at the fog from a distance (using theoretical functions called $\bar{\Pi}$ and the Adler function $D$).
2. The "Timelike" Method: Watching the fog collide and break apart (using experimental data on electron-positron collisions, called the $R$-ratio).

Usually, these two methods should give the exact same answer. But in the real world, data is messy, and different experiments cover different parts of the energy spectrum.

The Solution: The "Window" Analogy

This is where the paper comes in. The author, A.V. Nesterenko, introduces the concept of "Window Quantities."

Imagine you are trying to measure the total noise in a long hallway. Instead of listening to the whole hallway at once, you put up a window (a specific range of time or energy) and only listen to the sound coming through that window.

• Abrupt Windows: Like a shutter that snaps open and shut instantly.
• Smooth Windows: Like a curtain that fades in and out gently.

By using these windows, scientists can focus on specific energy ranges where they have good data (e.g., using lattice QCD simulations for the middle range and experimental data for the edges). This allows them to mix and match the best data from different sources.

The Twist: The "Edge Effect"

Here is the main discovery of the paper.

If you look through a window, the view inside the frame is clear. But what happens right at the edges of the window?

• In the "Spacelike" view, the math works one way.
• In the "Timelike" view, the math works another way.

The author shows that if you just compare the numbers inside the window, they won't match. It's like looking at a painting through a square window versus a circular window; the edges distort the view differently.

The paper proves that the two methods are only equivalent if you add a "correction factor" for the window edges.

• The Analogy: Imagine you are counting apples in a basket. If you use a square box to scoop them out, you might miss a few apples stuck in the corners. If you use a round bowl, you miss different ones. To get the true total, you have to add a "correction" for the apples you missed in the corners of your specific scoop.

The author derives the exact mathematical formulas for these "corner corrections" (called edge effects) for four different types of windows (sharp vs. smooth, symmetric vs. asymmetric).

Why This Matters

1. Mixing Data: This allows scientists to combine data from different experiments (like the MUonE experiment at CERN and lattice simulations) without getting confused. They can say, "We use the lattice data for the middle chunk and the experimental data for the edges," and mathematically stitch them together perfectly.
2. Checking for New Physics: By making these calculations more precise and removing the "fuzziness" caused by how we look at the data, we can see the discrepancy between theory and experiment more clearly. If the gap remains after fixing the "window edge" errors, it is stronger evidence that new, undiscovered particles exist.

Summary in One Sentence

This paper provides the mathematical "glue" needed to stitch together different ways of measuring the muon's magnetic behavior, ensuring that when we look through different "windows" of energy, we aren't fooled by the distortions at the edges, allowing us to hunt for new physics with greater precision.

Technical Summary

1. Problem Statement

The muon anomalous magnetic moment ($a_\mu$) is a critical precision test of the Standard Model (SM). A persistent discrepancy exists between experimental measurements (BNL E821, Fermilab E989) and theoretical predictions, potentially indicating new physics. The dominant source of theoretical uncertainty is the Hadronic Vacuum Polarization (HVP) contribution ($a_\mu^{\text{HVP}}$).

Currently, $a_\mu^{\text{HVP}}$ is evaluated using two distinct approaches:

1. Spacelike (Euclidean) Domain: Using the subtracted hadronic vacuum polarization function $\bar{\Pi}(Q^2)$ or the Adler function $D(Q^2)$. This is the method used by lattice QCD simulations and the MUonE experiment.
2. Timelike (Minkowski) Domain: Using the $R$-ratio of electron-positron annihilation into hadrons ($e^+e^- \to \text{hadrons}$). This is the traditional data-driven method.

The Core Issue: Recent efforts to combine inputs from different energy ranges (e.g., lattice data at intermediate energies and $R$-ratio data at low/high energies) utilize "window quantities." These involve integrating over specific kinematic intervals modulated by a "window function." However, it was not rigorously established whether window quantities calculated in the spacelike domain (using $\bar{\Pi}$ or $D$) are mathematically equivalent to those in the timelike domain (using $R$) when the window function is non-trivial (i.e., not constant). Specifically, the impact of "window edge effects" on the equivalence of these representations was unclear.

2. Methodology

The author employs dispersive improved perturbation theory (DPT) and complex analysis techniques to derive explicit relations between window quantities in different domains.

• Mathematical Framework: The study utilizes contour integration in the complex $q^2$ plane. The function $\Pi(q^2)$ has a cut along the positive real axis (timelike), while the kernel function $K_R(q^2)$ has a cut along the negative real axis (spacelike).
• Window Functions: The paper analyzes four specific types of window functions $W_n(q^2)$:
  1. Constant: $W_1(q^2) = 1$ (No windowing).
  2. Abrupt Symmetric: Step functions covering equal intervals in spacelike ($Q^2_1 \le Q^2 \le Q^2_2$) and timelike ($s_1 \le s \le s_2$) domains where $|Q^2| = s$.
  3. Abrupt Asymmetric: Step functions covering different intervals in spacelike and timelike domains.
  4. Smooth Symmetric/Asymmetric: Functions using hyperbolic tangents ($\tanh$) to create smooth transitions, introducing poles in the complex plane.
• Derivation Strategy:
  • The author constructs closed integration contours in the complex $q^2$ plane.
  • By applying Cauchy's residue theorem and analyzing the discontinuities across the cuts, the author derives the relationship between the integrals of $\bar{\Pi}(Q^2)$, $D(Q^2)$, and $R(s)$.
  • Crucially, the derivation accounts for additional contributions arising from the boundaries of the window functions (edge effects) and the poles introduced by smooth windowing.

3. Key Contributions

The paper provides the first rigorous derivation of the explicit correction terms required to equate window quantities across different representations.

1. Proof of Non-Equivalence without Corrections: The study demonstrates that for non-constant window functions, the representations of $a_\mu^{\text{HVP}}$ in terms of $\bar{\Pi}$, $D$, and $R$ are not automatically equivalent.
2. Explicit Correction Formulas: The author derives explicit expressions for the "edge effect" contributions ($\Delta a_\mu^{\text{HVP}}$) that must be added to the Adler function ($D$) and $R$-ratio representations to match the $\bar{\Pi}$ representation.
   • For Abrupt Windows: Corrections arise from the discontinuities at the window edges (semicircles in the complex plane).
   • For Smooth Windows: Corrections arise from the infinite series of poles introduced by the smooth transition functions (hyperbolic tangents) in the complex plane.
3. Generalization: The results generalize previous work (Ref. [82, 83]) to include asymmetric kinematic intervals and smooth window functions, which are relevant for modern lattice QCD and MUonE analyses.
4. Hybrid Evaluation Framework: The paper establishes a formalism to calculate $a_\mu^{\text{HVP}}$ by splitting the kinematic range and using different inputs (e.g., $\bar{\Pi}$ for low energy, $D$ for high energy, $R$ for intermediate) while maintaining mathematical consistency via the derived correction terms.

4. Results

The author validates the derived relations using DPT with $n_f=3$ active flavors and PDG24 parameters.

• Numerical Verification:
  • Abrupt Symmetric Case: Calculated $a_\mu^{\text{HVP}}$ using $\bar{\Pi}$, $D$, and $R$ over the interval $[0.25, 0.50] \text{ GeV}^2$.
    • $\bar{\Pi}$ result: $(25.75 \pm 0.10) \times 10^{-10}$.
    • $D$ result (uncorrected): $(7.85 \pm 0.02) \times 10^{-10}$.
    • Correction $\Delta a_D$: $(17.90 \pm 0.07) \times 10^{-10}$.
    • $R$ result (uncorrected): $(183.55 \pm 0.60) \times 10^{-10}$.
    • Correction $\Delta a_R$: $(-157.80 \pm 0.51) \times 10^{-10}$.
    • Conclusion: $a_{\Pi} = a_D + \Delta a_D = a_R + \Delta a_R$, confirming the derived equivalence.
  • Similar successful verification was performed for asymmetric and smooth window functions.
• Hybrid Assessment: The paper demonstrates a "hybrid" calculation where the total $a_\mu^{\text{HVP}}$ is split into three regions:
  1. Low energy: Calculated via $\bar{\Pi}$.
  2. Intermediate: Calculated via $R$-ratio (with edge corrections).
  3. High energy: Calculated via Adler function $D$ (with edge corrections).
     The sum of these hybrid components matches the canonical full-range calculation, validating the method for combining disparate data sources.
• Updated SM Prediction: Using DPT, the author updates the SM prediction for the muon anomalous magnetic moment:
  • $a_\mu^{\text{SM}} = (11659185.25 \pm 3.00) \times 10^{-10}$.
  • This differs from the experimental average ($11659207.15 \pm 1.45 \times 10^{-10}$) by 6.6 standard deviations.

5. Significance

• Bridging Lattice and Data-Driven Methods: The derived relations are essential for the MUonE experiment and lattice QCD collaborations. They allow these groups to compare their spacelike results directly with traditional timelike $R$-ratio data, even when using different window functions or kinematic ranges.
• Resolving Discrepancies: By providing the exact mathematical corrections for "window edge effects," the paper removes a potential source of systematic error in hybrid calculations. This ensures that any remaining discrepancy between theory and experiment is due to physics (new particles/forces) rather than mathematical inconsistencies in the windowing procedure.
• Theoretical Rigor: The work clarifies the conditions under which different representations of HVP are equivalent, emphasizing that for piecewise continuous functions or non-trivial windowing, naive equivalence fails without specific boundary term corrections.

In summary, this paper provides the necessary mathematical toolkit to unify spacelike and timelike evaluations of the muon anomalous magnetic moment, enabling more precise and reliable hybrid assessments that are crucial for testing the Standard Model.