Three loop QCD corrections to electroweak radiative parameters

This paper reevaluates three-loop QCD corrections to electroweak vacuum polarization functions to determine ${\mathcal{O}}(\alpha \alpha_s^2)$ contributions to radiative parameters, resulting in improved predictions for the $W$ boson mass and the $\overline{\mathrm{MS}}$ electric charge that are significant for future FCC precision targets.

The Gist

Imagine the Standard Model of particle physics as the ultimate "Rulebook of the Universe." For decades, scientists have been checking this rulebook against real-world experiments, looking for tiny cracks or inconsistencies that might hint at new, undiscovered physics.

Recently, the discovery of the Higgs boson completed the list of known particles in the rulebook. But now, the game has changed. Instead of looking for new particles, scientists are playing a game of "extreme precision." They are measuring things like the weight of the W boson (a particle that carries the weak nuclear force) with such incredible accuracy that even the tiniest theoretical error could ruin the comparison.

This paper is about fixing the math to match that extreme precision.

The Problem: The "Blurry Lens"

Think of the W boson's mass as a target on a dartboard.

• Old measurements: The dartboard was fuzzy. We could hit the general area, but we couldn't tell if we were off by a millimeter.
• New measurements (Future Colliders): Scientists are building new machines (like the Future Circular Collider) that will throw darts with such precision that they can hit the exact center of the bullseye.

However, to know if you hit the bullseye, your prediction of where the dart should land must be just as sharp. If your math is slightly "blurry," you might think you missed the target when you actually hit it, or vice versa.

The "blur" in this case comes from Quantum Chromodynamics (QCD). This is the theory of how quarks (the building blocks of protons and neutrons) interact. At the quantum level, particles are constantly popping in and out of existence, creating a "cloud" of virtual particles around the W boson. These clouds shift the W boson's mass slightly.

For years, scientists calculated these shifts using "one-loop" or "two-loop" math (like taking a photo with a 10-megapixel camera). But for the new, super-precise experiments, that's not enough. They need a "three-loop" calculation (like a 100-megapixel camera) to see the tiny details.

The Solution: The "Three-Loop" Upgrade

The authors of this paper, Tanmoy Pati, Narayan Rana, and Alessandro Vicini, went back and recalculated these quantum shifts using three-loop QCD corrections.

Here is what they did, using some analogies:

1. The Vacuum Polarization (The Quantum Foam):
   Imagine the vacuum of space isn't empty; it's like a bubbling foam of virtual particles. When a W boson moves through this foam, it drags some of the foam with it, making it effectively heavier or lighter. The authors recalculated exactly how much "foam" is dragged along, but this time, they included the most complex interactions (three loops) that were previously too hard to calculate.

2. The Missing Ingredients:
   In previous calculations, they mostly focused on the heavy "top" and "bottom" quarks (the big, heavy ingredients in the soup). This paper added the contributions from the lighter quarks (the lighter ingredients) at this high level of complexity. It's like realizing that while the heavy spices define the flavor, the subtle hints of the light herbs actually change the taste just enough to matter when you are a professional chef.

3. The Result: Sharper Predictions:
   By including these new, complex calculations, the authors sharpened the theoretical prediction for the W boson's mass.

   • The Shift: They found that the predicted mass shifts by a small but significant amount.
   • Why it matters: Future experiments aim to measure the W mass with an uncertainty of less than 1 MeV (a tiny fraction of a proton's weight). The shift caused by these new calculations is large enough that if we didn't include them, our theoretical prediction would be wrong by the time the new experiments arrive.

The "Electric Charge" Connection

The paper also updated the value of the electric charge at high energies. Think of electric charge not as a fixed number, but as a value that "runs" or changes depending on how close you get to a particle (like a zoom lens changing focus). The authors calculated how this "zoom" behaves when you include the three-loop QCD effects, giving a more accurate value for the fundamental constants of nature.

The Bottom Line

This paper is like upgrading the GPS software for a self-driving car.

• The Car: The new, super-precise particle colliders of the future.
• The Road: The laws of physics.
• The Old GPS: Previous calculations (1-loop and 2-loop). It worked fine for driving to the grocery store (past experiments).
• The New GPS: This paper (3-loop calculations). It is necessary to navigate the "highway" of future precision physics without crashing into a wall of theoretical error.

In short: The authors have performed a massive, high-precision mathematical cleanup of the Standard Model's predictions. They found that when you look at the universe with the highest possible magnification, the "quantum foam" shifts the weights of particles in a way that was previously invisible. Now that they've accounted for it, we are ready to test the laws of physics with unprecedented accuracy.

Technical Summary

1. Problem Statement

The Standard Model (SM) of particle physics is currently under intense scrutiny due to the absence of direct New Physics signals at the Large Hadron Collider (LHC). Consequently, the field has shifted toward precision phenomenology, where minute discrepancies between theoretical predictions and experimental data could reveal physics beyond the SM.

Future colliders, such as the High-Luminosity LHC and proposed electron-positron colliders (e.g., FCC-ee), aim to measure electroweak (EW) parameters with unprecedented precision:

• W boson mass ($m_W$): Target uncertainty $\sim 1$ MeV (current $\sim 13$ MeV).
• Effective weak mixing angle ($\sin^2 \theta_{eff}^f$): Target uncertainty $\sim 0.6 \times 10^{-5}$.

To match these experimental goals, theoretical predictions must reach comparable accuracy. A critical bottleneck is the calculation of mixed QCD-Electroweak corrections at the three-loop level, specifically the $O(\alpha \alpha_s^2)$ contributions to the vacuum polarization functions. Previous calculations lacked full three-loop light-quark contributions and sufficient series expansion terms, limiting the precision of derived parameters like $\Delta \rho$, $\Delta r$, and $\Delta \kappa$.

2. Methodology

The authors employed state-of-the-art perturbative techniques to re-evaluate the vacuum polarization functions for electroweak gauge bosons ($W$ and $Z$) at three loops.

• Diagram Generation & Algebra:
  • Generated Feynman diagrams using QGRAF. For the third-generation quarks ($t, b$), this yielded 70 diagrams for $\Pi_{ZZ}$ and 31 for $\Pi_{WW}$.
  • Processed raw outputs using in-house FORM routines to handle Dirac, Lorentz, and color algebra, resulting in $\sim 10^4$ scalar Feynman integrals.
• Integral Reduction:
  • Applied Integration-By-Parts (IBP) reduction using public codes Kira and LiteRed to reduce the scalar integrals to 94 Master Integrals (MIs).
• Evaluation of Master Integrals:
  • Instead of seeking full closed-form analytic solutions (which involve complex elliptic multiple polylogarithms), the authors solved the system of differential equations for the MIs using a series expansion around $q^2 = 0$.
  • This approach is valid because the expansion parameter $z_t = m_Z^2/m_t^2 \approx 0.28$ lies well within the radius of convergence.
  • Solutions were determined using the generalized Frobenius method via a custom Mathematica routine.
  • Boundary conditions at $q^2 \to 0$ were fixed using OS-reduction, regularity at pseudo-thresholds, and high-precision PSLQ fits based on numerical evaluations from AMFlow.
• Renormalization Scheme:
  • Adopted a mixed scheme: Heavy quark masses were renormalized in the On-Shell (OS) scheme, while the strong coupling $\alpha_s$ was treated in the $\overline{\text{MS}}$ scheme.
  • The electric charge was defined via Thomson scattering (OS) but converted to the $\overline{\text{MS}}$ scheme for running coupling calculations.

3. Key Contributions

The paper makes several novel theoretical advancements:

1. Complete $O(\alpha \alpha_s^2)$ Corrections: First-time calculation of the full three-loop QCD corrections to the vacuum polarization functions, including massless light-quark contributions which were previously missing in the literature.
2. Improved Series Expansion: The authors included significantly more terms in the series expansion (up to $O(z_t^{15})$) compared to previous seminal works (which often stopped at $O(z_t^2)$), ensuring numerical stability and higher accuracy.
3. New Analytic Expressions:
   • Provided a compact analytic expression for the three-loop correction to the $\rho$ parameter ($\Delta \rho^{(2)}$).
   • Derived new analytic results for the UV-finite renormalization combinations $\delta c_Z$ and $\delta c_W$, which are crucial for vector boson production cross-sections.
4. $\overline{\text{MS}}$ Electric Charge: Calculated the $O(\alpha \alpha_s^2)$ contribution to the running of the electric charge, $\Delta \alpha_{\overline{\text{MS}}}(m_Z^2)$.

4. Results

The authors present numerical results for key electroweak parameters using standard input values ($m_Z=91.1876$ GeV, $m_t=173.2$ GeV, etc.).

• Electric Charge ($\alpha_{\overline{\text{MS}}}$):
  • The new three-loop QCD correction to the shift $\Delta \alpha(m_Z^2)$ is $-1.88 \times 10^{-5}$.
  • This leads to a refined prediction: $\alpha^{-1}(m_Z^2) = 128.079 \pm 0.015$.
• $\rho$ Parameter ($\Delta \rho$):
  • Confirmed agreement with existing literature for heavy quark contributions but provided the first complete analytic form for the transcendental constants involved (including Clausen functions and log-sine integrals).
• Radiative Parameters ($\Delta r$ and $\Delta \kappa$):
  • Calculated coefficients $\Delta r^{(2)}$ and $\Delta \kappa^{(2)}$ up to $O(z_t^{15})$.
  • Scale Dependence: The inclusion of $O(\alpha \alpha_s^2)$ corrections significantly reduces the dependence on the renormalization scale $\mu_R$, stabilizing the predictions.
  • Light Quark Impact: The three-loop light-quark contribution to $\Delta r$ and $\Delta \kappa$ is non-negligible.
• Phenomenological Impact:
  • W Boson Mass ($m_W$): The new corrections induce a shift in the predicted $m_W$. The light-quark three-loop contribution alone shifts $m_W$ by $-0.23$ MeV, a magnitude comparable to the ultimate precision targets of future colliders.
  • Weak Mixing Angle: The shift in $\sin^2 \theta_{eff}^f$ due to light quarks is $-0.5 \times 10^{-5}$, which is significant given the projected experimental error of $0.6 \times 10^{-5}$.

5. Significance

This work represents a critical step toward the sub-permille theoretical accuracy required for the next generation of particle physics experiments.

• Future Colliders: The improved precision is essential for interpreting data from the FCC-ee and ILC, where statistical and systematic errors will be extremely low. Without these three-loop QCD corrections, theoretical uncertainties would dominate the error budget.
• New Physics Sensitivity: By reducing theoretical uncertainties in SM predictions for $m_W$ and $\sin^2 \theta_{eff}^f$, the community can more rigorously test the SM. Any remaining discrepancy between the refined theory and future data would provide a stronger, more robust signature of New Physics.
• Methodological Benchmark: The successful application of series expansions combined with modern IBP reduction and numerical boundary techniques sets a precedent for calculating complex multi-loop integrals in mixed QCD-EW sectors.

In summary, Patia, Rana, and Vicini have closed a critical gap in the theoretical prediction of electroweak observables, ensuring that the theoretical framework is ready to match the precision of upcoming experimental facilities.