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Smeared spectral functions from lattice QCD: opportunities and challenges

This paper summarizes recent developments, opportunities, and persistent challenges in reconstructing smeared spectral functions from Euclidean correlators in lattice QCD, a crucial step for calculating various physical observables despite the intrinsic difficulties of the ill-posed inverse problem.

Original authors: Shoji Hashimoto

Published 2026-09-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shoji Hashimoto

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, particles do not exist in isolation; they are constantly interacting, transforming, and decaying into other forms of matter. To understand the fundamental laws of nature, physicists must measure the properties of these fleeting interactions. One of the most powerful tools for this is a theoretical framework called quantum chromodynamics, which describes how quarks and gluons—the building blocks of protons and neutrons—stick together. However, a major hurdle exists: the equations that govern these particles are often easiest to solve in a mathematical realm where time flows differently, known as Euclidean time. In this realm, the data looks like a smooth, fading signal, but the physical reality researchers seek is a sharp, detailed map of energy levels and resonances. Reconstructing that sharp map from the smooth signal is notoriously difficult, much like trying to hear a specific instrument in a symphony when all you have is a recording of the entire orchestra muffled by thick walls.

This challenge lies at the heart of a recent contribution by Shoji Hashimoto, a theoretical physicist at the High Energy Accelerator Research Organization in Japan, presented at the 43rd International Symposium on Lattice Field Theory. The paper addresses a long-standing problem in lattice quantum chromodynamics: how to extract meaningful physical information from computer simulations that operate in this simplified, Euclidean time. For decades, scientists have struggled to reverse-engineer the specific energy spectrum of particles from the broad, averaged data produced by these simulations. Hashimoto's work does not claim to have solved this impossible puzzle once and for all. Instead, it offers a pragmatic shift in perspective. Rather than demanding a perfect, point-by-point picture of every energy level, the paper argues that physicists should focus on "smeared" spectral functions. These are weighted averages of the energy spectrum, where the sharp details are intentionally blurred just enough to make them calculable, yet precise enough to answer specific physical questions.

The paper outlines a wide range of opportunities where this approach can succeed. Many important physical quantities, such as the contribution of hadrons to the magnetic moment of the muon or the rates of certain particle decays, are naturally defined as these weighted averages. In these cases, the "smearing" is not an artificial trick but a built-in feature of the observable itself. By accepting this blur, researchers can use lattice data to calculate these quantities with controlled uncertainties. The author highlights several specific successes and promising avenues, including the calculation of inclusive hadronic tau decays and the rates of semileptonic decays for heavy mesons like the B and D particles. These calculations are crucial for testing the Standard Model of particle physics and searching for new physics beyond it. For instance, understanding the decay of B mesons helps physicists resolve tensions in the Cabibbo-Kobayashi-Maskawa matrix, a set of parameters that describes how quarks change flavor.

However, the paper is equally clear about the challenges that remain. The core difficulty is that the mathematical problem of reversing the signal is "ill-posed," meaning that tiny errors in the input data can lead to massive errors in the output. The author explains that while smoothing the data makes the problem solvable, it also limits the resolution. With the current precision of computer simulations, the energy resolution is likely limited to a range of 20 to 40 percent. This means that while broad features of the particle spectrum can be seen, fine details are lost. The paper explicitly warns against the hope that simply increasing the amount of computer power will solve this; the limitation is fundamental to the nature of the data, not just a lack of statistics. Consequently, for processes that require extremely fine resolution, such as the rare decay of a B meson into a kaon and a pair of leptons, the current methods face a significant hurdle. The physical structures in these decays, such as charmonium resonances, can be separated by energies as small as tens of MeV, which is far below the current resolution limit of roughly 100 MeV or more.

Hashimoto's analysis suggests that the path forward is not to force a perfect reconstruction but to carefully match the method to the physics. For some questions, the current level of smearing is sufficient to provide reliable answers. For others, particularly those involving complex intermediate states or rare decays, the resolution is not yet fine enough. The paper argues that researchers must now combine their reconstruction techniques with additional physical insights about the spectrum to make progress. The goal is no longer to see every single detail of the subatomic landscape, but to construct specific, phenomenologically useful observables with known and controlled errors. By accepting the limits of what can be seen and focusing on what can be reliably measured, the field can continue to make progress in understanding the fundamental forces of nature, even if the full, sharp picture remains just out of reach.

Technical Summary: Smeared Spectral Functions from Lattice QCD

1. The Problem
The central challenge addressed in this contribution is the reconstruction of spectral functions, $\rho(E)$ , from Euclidean lattice correlation functions, $C(t)$ . The relationship between these quantities is governed by a Laplace transform:
$C(t) = \int_0^\infty dE \, \rho(E) e^{-Et}$
Reconstructing the local function $\rho(E)$ from discrete, noisy Euclidean data is an intrinsically ill-posed inverse problem. The singular values of the Laplace kernel decay exponentially, meaning that high-energy (fine-structure) modes are exponentially suppressed in the correlator. Consequently, attempting to recover a pointwise spectral function is unstable and generally impossible without model-dependent assumptions, as statistical noise dominates the high-frequency components required for resolution.

The paper reframes this problem by noting that many phenomenologically relevant observables do not require a pointwise spectral function. Instead, they are integrals of the spectral function weighted by a specific kernel $K(E)$ :
$S_K[\rho] = \int_0^\infty dE \, K(E) \rho(E)$
The author distinguishes two roles for the kernel (smearing):

Physical Definition: The smearing is inherent to the observable (e.g., the Hadronic Vacuum Polarization (HVP) contribution to the muon $g-2$ ).
Regulator: The smearing is introduced to regularize singular kernels (e.g., step functions or $i\epsilon$ prescriptions in scattering amplitudes) to make the reconstruction feasible, with the physical limit ( $\sigma \to 0$ or $\epsilon \to 0$ ) taken via extrapolation.

2. Methodology and Reconstruction Framework
The paper surveys the theoretical landscape for solving this "smeared" inverse problem.

Linear Reconstruction: If the kernel $K(E)$ can be approximated by a linear combination of exponentials (e.g., $K(E) \approx \sum g_t e^{-Et}$ ), the smeared observable can be computed directly from the Euclidean correlator as a linear sum $\sum g_t C(t)$ .
SVD and Resolution Limits: Using Singular Value Decomposition (SVD) of the Laplace kernel, the author demonstrates that the energy resolution $\Delta E$ is logarithmically dependent on the precision of the correlator data. Specifically, $\Delta E/E \sim \pi^2 / (2 \ln(1/\Delta))$ , where $\Delta$ is the relative error in the correlator. With current lattice precision ( $\sim 10^{-5}$ to $10^{-8}$ ), the realistic energy resolution is estimated at 20–40%.
Basis and Criteria: Various methods (Maximum Entropy, Bayesian approaches, Machine Learning, Backus-Gilbert, Chebyshev approximations, etc.) are reviewed. The author argues that these methods represent different "lenses" applied to the same underlying data. While they differ in basis choice (powers of $e^{-E}$ , Chebyshev polynomials, SVD modes) and regularization criteria, none can create new information or overcome the fundamental ill-posedness of the inverse problem.
Approximation vs. Bounds: The paper contrasts approximation methods (estimating truncation errors in polynomial expansions) with rigorous bound methods (convex optimization to maximize/minimize the smeared quantity subject to positivity constraints).

3. Key Physics Applications (Opportunities)
The paper categorizes applications into three classes where the smeared spectral approach is viable:

Smeared $R$ -ratio and Inclusive Hadronic $\tau$ Decays:
- The ETMC collaboration has successfully used Gaussian smearing kernels to compute the $R$ -ratio, providing a benchmark where the lattice result is compared to a resolution-limited experimental observable rather than a pointwise spectrum.
- Inclusive $\tau$ decays involve a smooth polynomial kernel (phase space) integrated against the spectral function. The sharp kinematic cutoff is regularized using a sigmoid function, allowing for the determination of CKM matrix elements $|V_{ud}|$ and $|V_{us}|$ .
Inclusive Semileptonic Decays ( $B$ and $D$ mesons):
- These rates are expressed as integrals over hadronic spectral densities with kinematic kernels (Heaviside functions).
- Recent lattice calculations for inclusive $D_s$ decays have reached a stage of quantitative error estimates.
- For $B$ decays, challenges remain due to larger recoil momenta leading to noisier correlators. Strategies involve combining non-perturbative calculations at charm masses with heavy-quark scaling.
Hadronic Amplitudes and Energy Filters:
- Maiani-Testa Theorem: The paper discusses using energy filters (Heaviside or sigmoid functions of the Hamiltonian) to isolate specific energy regions in matrix elements, bypassing the limitation that Euclidean correlators only access the ground state at large times.
- Rare Processes: Amplitudes for processes like $D_s \to \ell \nu \gamma^*$ and neutral $D^0-\bar{D}^0$ mixing are formulated as integrals involving a kernel $1/(E' - E - i\epsilon)$ . The parameter $\epsilon$ acts as a regulator.
- $B \to K^{(*)} \ell^+ \ell^-$ : This is highlighted as a major challenge. The amplitude involves long-distance charm-loop contributions. A pilot study exists for the small-recoil region, but resolving the complex charmonium resonance structure (widths of tens to hundreds of MeV) requires an $\epsilon$ significantly smaller than currently accessible.

4. Results and Limitations

Resolution Constraint: The paper concludes that the achievable smearing width (or regulator $\epsilon$ ) is fundamentally limited by the signal-to-noise ratio and the time range of the Euclidean correlator. A realistic resolution is estimated to be $\epsilon \gtrsim (0.2\text{--}0.4)E$ .
Finite Volume Effects: For smeared spectra involving multi-particle states, finite-volume effects scale as power laws ( $1/L^3$ ) if the smearing width is too narrow to cover the discrete energy levels. A sufficiently broad smearing is required to suppress these artifacts exponentially, but this must be balanced against the need to resolve physical structures.
Extrapolation Reliability: For applications requiring the limit $\epsilon \to 0$ (e.g., rare decays), the reliability of the extrapolation depends on the physical spectrum. If the physical amplitude contains structures narrower than the achievable $\epsilon$ (e.g., narrow charmonium resonances), the extrapolation becomes non-trivial and requires additional physics input regarding the analytic structure of the amplitude.

5. Significance and Claims
The author's primary claim is modest but transformative: Smearing does not solve the inverse problem once and for all; rather, it changes the question.

Instead of asking whether a pointwise spectral function can be fully reconstructed, the field should ask which weighted integrals of the spectral function are accessible with controlled uncertainties given current lattice data.

For observables with naturally smooth kernels (like inclusive $\tau$ decays), this approach is already successful.
For observables requiring sharp resolution (like rare $B$ decays or $D^0-\bar{D}^0$ mixing), the current resolution limits the direct extraction of the physical limit. The paper argues that for these cases, the zero-smearing limit is not just a reconstruction problem but a physics problem requiring an understanding of the underlying spectrum and intermediate states.

The paper concludes that the most productive path forward is to construct phenomenologically useful smeared observables with controlled errors, potentially combining reconstruction methods with controlled physics inputs, rather than striving for a model-independent, pointwise reconstruction of the spectral function.

Technical Summary: Smeared Spectral Functions from Lattice QCD

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