First-principle evaluation of inclusive hadronic $\tau$ decays in QCD+QED

This paper proposes a first-principles strategy to extend lattice QCD calculations of inclusive hadronic $\tau$ decays to include QED and isospin-breaking effects, presenting preliminary results within the RM123 framework to ultimately improve the determination of the CKM matrix element $|V_{us}|$.

The Gist

Imagine you are trying to understand the life story of a very short-lived celebrity: the Tau particle ($\tau$).

The Tau is like a famous actor who lives for only a split second before "retiring" (decaying) into a crowd of other particles. Sometimes, it retires into a specific group of particles called "hadrons" (like protons and pions). By studying exactly how often the Tau retires into these specific groups, physicists can solve a massive cosmic mystery: determining the value of a number called $|V_{us}|$, which is a key piece of the "Standard Model" puzzle that explains how the universe is built.

For a long time, scientists have been trying to calculate this number using two different methods:

1. The "Real World" Method: Watching actual Tau particles decay in particle accelerators.
2. The "Simulation" Method: Using supercomputers to simulate the laws of physics from scratch (First Principles).

The Problem: The Simulation Was Too Simple

Until now, the "Simulation Method" had a major flaw. The computer simulations assumed that the universe was perfectly symmetrical and ignored two tiny but important things:

• Electromagnetism: The force of electricity and magnetism (QED).
• Isospin Breaking: The tiny difference between the mass of an "up" quark and a "down" quark.

Think of it like trying to bake a perfect cake but ignoring the fact that sugar and salt taste different, or that the oven temperature fluctuates. The simulation was "close enough" for a rough draft, but when scientists compared the simulation results to the real-world data, they found a 3-sigma tension (a significant disagreement). It was like the simulation predicted the cake would be sweet, but the real cake tasted slightly sour.

The Solution: A New, More Detailed Recipe

This paper presents a new strategy to fix the simulation. The authors, led by Matteo Di Carlo, are upgrading the computer model to include QCD + QED.

Here is how they are doing it, using some everyday analogies:

1. The "Time-Reverse" Camera

In the real world, a Tau particle decays forward in time. But in the computer simulation, time flows differently (it's "Euclidean").

• The Analogy: Imagine you have a security camera recording a crime in reverse. You see the broken vase reassembling itself and the shards flying back into the table.
• The Trick: The scientists realized that even though the simulation runs in "reverse time," the mathematical pattern of the decay is hidden inside the data. They use a clever mathematical tool (called the HLT method) to act like a "time-reversal filter." It takes the blurry, reverse-time data and reconstructs the clear, forward-time picture of the decay rate.

2. Breaking the Problem into Three "Teams"

Calculating the effect of electricity and mass differences on the Tau decay is incredibly hard. To make it manageable, the authors split the problem into three teams, using the RM123 framework:

• Team Leptonic (The "Tau's Personal Bubble"):

  • What it is: This looks at how the Tau particle itself interacts with photons (light particles) before it decays.
  • Analogy: Imagine the Tau is a celebrity walking down the street. This team calculates how the paparazzi (photons) bother the celebrity before they even enter the building. It doesn't matter what happens inside the building yet; we just need to know how the crowd affected the celebrity's mood.
  • Status: The paper shows preliminary results for this team.

• Team Factorizable (The "Party Inside"):

  • What it is: This looks at how the particles inside the hadronic cloud interact with each other via photons.
  • Analogy: Now the celebrity is inside the building. This team calculates how the guests (quarks) interact with each other using flashlights (photons). Crucially, the celebrity (Tau) is just watching from the outside and doesn't interact directly with the guests' flashlights. The two groups are separate.
  • Status: The paper also shows preliminary results for this team.

• Team Non-Factorizable (The "Messy Interaction"):

  • What it is: This is the hardest part. It's when the Tau particle and the hadronic cloud both exchange photons with each other simultaneously.
  • Analogy: This is like the celebrity walking into the room and immediately high-fiving a guest while holding a flashlight, while the guest is also holding a flashlight. Everyone is tangled up. You can't separate the celebrity's actions from the guest's actions.
  • Status: This is the "Next Step." The authors haven't solved this yet, but they have a plan to tackle it.

The Current Status: A Promising Start

The authors have run their upgraded simulation on a specific computer setup (using "electro-quenched" approximation, which means they included the light effects but temporarily ignored the heavy "sea" of virtual particles).

• The Result: They successfully reconstructed the "Leptonic" and "Factorizable" parts of the decay.
• The Visuals: The paper includes graphs (Figures 1 and 2) that look like noisy static turning into a clear signal. The "red dot" on the graphs represents the sweet spot where their mathematical filter works best, showing that the simulation is stable and reliable.

Why Does This Matter?

If they can finish the job (solving the "Messy Interaction" team and perfecting the math), they will have a First-Principles calculation of the Tau decay rate.

This is a "Holy Grail" moment for physics because:

1. It will tell us if the "3-sigma tension" (the disagreement between simulation and reality) was just a mistake in the old, simple simulation, or if it points to New Physics (something we don't know about yet).
2. It will give us a precise value for $|V_{us}|$, helping us understand why the universe has the specific mix of matter it does.

In summary: This paper is the blueprint for upgrading a blurry, black-and-white simulation of the universe into a high-definition, color 3D movie. They have successfully rendered the background and the main characters; now they just need to render the complex interactions between them to see the full picture.

Technical Summary

1. Problem Statement

The paper addresses the need for a first-principles determination of inclusive hadronic $\tau$ lepton decay rates within the full Standard Model framework, specifically including QCD+QED (Quantum Chromodynamics plus Quantum Electrodynamics).

• Context: Recent lattice QCD calculations in isosymmetric QCD (isoQCD) have revealed a $\approx 3\sigma$ tension between the CKM matrix element $|V_{us}|$ extracted from inclusive $\tau$ decays and that from kaon decays. Since the isoQCD results are fully non-perturbative, the discrepancy cannot be attributed to Operator Product Expansion (OPE) approximations used in previous studies.
• Goal: To extend these calculations beyond the isosymmetric limit by incorporating isospin-breaking effects (due to $m_u \neq m_d$) and electromagnetic effects (QED) directly from first principles. This is crucial for resolving the $|V_{us}|$ tension and providing complementary data for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment ($g-2$).

2. Methodology

The authors propose a strategy combining Euclidean lattice correlation functions with the Hansen-Lupo-Tantalo (HLT) spectral reconstruction method, implemented within the RM123 framework.

A. Theoretical Framework

• Weak Effective Hamiltonian: The decay $\tau \to X \nu_\tau$ is described by the weak Hamiltonian $H_W$. The inclusive decay rate $\Gamma$ is related to the squared amplitude $|A(m_\tau)|^2$.
• Euclidean Correlator: The authors establish a relation between the physical decay rate and a specific Euclidean 4-point correlation function $C(t_+, t, t_-)$. By taking the ratio of this 4-point function to a standard $\tau$-lepton 2-point function, they define a ratio $R(t)$ that admits a spectral representation:
  $$R(t) = \int_0^\infty \frac{d\omega}{2\pi} e^{-\omega t} |A(\omega)|^2$$
• Spectral Reconstruction: To extract the physical rate at $\omega = m_\tau$, the inverse problem (recovering $|A(\omega)|^2$ from $R(t)$) is solved using the HLT strategy. This involves approximating a smeared Dirac delta function as a linear combination of decaying exponentials to reconstruct the spectral density from lattice data.

B. The RM123 Expansion

Instead of simulating full QCD+QED directly (which is computationally expensive), the authors use the RM123 approach, expanding the correlator in powers of the electromagnetic coupling $\alpha_{em}$ and the quark mass difference $(m_u - m_d)$ around isoQCD.
The radiative correction is decomposed into three distinct, infrared-finite contributions:

1. Leptonic Corrections: Photons attached only to the external $\tau$ lepton line.
2. Factorizable Corrections: Photons exchanged only among quark lines (hadronic sector), including quark-mass counterterms.
3. Non-Factorizable Corrections: Photons exchanged between the lepton and hadronic sectors.

C. Implementation Strategy

• Leptonic & Factorizable Terms: These can be reformulated using standard Euclidean hadronic two-point functions (vector and axial currents) computed in QCD+QED, modified by specific radiative kernels ($\delta K$ for leptonic, $K$ for factorizable).
• Non-Factorizable Terms: These require a more complex 4-point correlation function involving the neutrino, $\tau$, and hadronic electromagnetic currents, which is currently under investigation.
• Lattice Setup: Preliminary results were generated using $N_f=2+1+1$ Wilson-clover twisted-mass fermions on the D96 ensemble ($a \approx 0.0569$ fm, $L \approx 5.46$ fm) in the electro-quenched approximation (sea quarks do not feel QED effects).

3. Key Contributions

• Formalism Extension: Successfully extended the HLT spectral reconstruction strategy from isoQCD to the QCD+QED framework, explicitly handling the decomposition of radiative corrections.
• Decomposition Strategy: Demonstrated that leptonic and factorizable corrections can be isolated and computed using modified kernels applied to standard hadronic correlators, significantly simplifying the lattice implementation for these terms.
• Preliminary Results: Provided the first lattice-based reconstruction of the leptonic and factorizable contributions to the inclusive $\tau$ decay rate in the electro-quenched approximation.

4. Results

• Reconstruction Quality: The authors presented preliminary reconstructions of the spectral densities for both factorizable and leptonic corrections at a smearing parameter $\sigma/m_\tau = 0.040$.
• Stability: The reconstructed spectral densities (shown in Figures 1 and 2) exhibit stability under variations of the regularization parameter $\lambda$ in the minimization functional.
• Comparison: The quality of the reconstruction (quantified by the functional $A[g]$) is comparable to previous isoQCD studies, suggesting the methodology is robust even with the added complexity of QED kernels.
• Current Limitations: The results are currently limited to the electro-quenched approximation (neglecting QED effects on sea quarks) and do not yet include non-factorizable corrections or non-perturbative renormalization of the weak Hamiltonian.

5. Significance and Future Outlook

• Resolving Tensions: This work is a critical step toward a fully non-perturbative determination of $|V_{us}|$ that includes all Standard Model effects, potentially resolving the existing tension between $\tau$ and kaon decay extractions.
• Broader Impact: The methodology provides a pathway to calculate the isovector component of the hadronic vacuum polarization for the muon $g-2$, a major topic in current particle physics.
• Next Steps: The authors identify two main challenges for a complete calculation:
  1. Non-Factorizable Corrections: Developing efficient numerical strategies to compute the complex 4-point functions involving photon exchange between leptons and hadrons.
  2. Renormalization: Performing a non-perturbative renormalization of the weak effective Hamiltonian in QCD+QED to handle operator mixing induced by chiral symmetry breaking on the lattice.

In summary, the paper establishes a viable, first-principles framework for calculating inclusive $\tau$ decays with QED corrections, successfully demonstrating the reconstruction of the dominant leptonic and factorizable terms and paving the way for a complete Standard Model calculation.