Tackling inverse problems for PDFs from lattice QCD

This presentation outlines the integration of recent advancements in extracting parton distribution functions (PDFs) from lattice QCD with established methods for solving inverse problems in spectral function reconstruction.

The Gist

The Big Picture: Seeing the Invisible Inside a Proton

Imagine a proton (a particle inside an atom) not as a solid ball, but as a high-speed train packed with tiny passengers called "partons" (quarks and gluons).

Physicists want to know the Parton Distribution Functions (PDFs). In our analogy, this is like wanting a precise map that tells us: "If you look at this train, what is the probability of finding a passenger sitting in seat 10, seat 50, or seat 100?"

Knowing this map is crucial. It helps us understand how the universe works, how to build better particle colliders, and even how to detect new physics beyond our current theories.

The Problem: The "Frozen" Camera

The problem is that we can't take a photo of this train while it's moving at the speed of light.

• The Real World: To get the map, we need to see the partons moving freely. This requires "Real-Time" physics.
• The Lattice QCD (Our Tool): The supercomputers we use to simulate these particles (Lattice QCD) are stuck in a different dimension called Euclidean time. Think of this as a camera that can only take frozen, blurry snapshots of the train. It can see the train from the side, but it cannot see the passengers moving forward.

Because of this, we can't just "read" the map directly. We have to take the blurry, frozen snapshots and try to reverse-engineer the moving train. This is called an Inverse Problem.

The Challenge: The "Fuzzy" Puzzle

The paper explains that trying to reconstruct the moving train from frozen snapshots is like trying to solve a puzzle where:

1. You are missing pieces: We can only see a small part of the "Ioffe time" (a specific coordinate system). It's like trying to guess the shape of a whole mountain by only looking at the very bottom.
2. The puzzle is broken: If you try to force the pieces together without help, the result is chaotic. A tiny error in your snapshot (like a smudge on the lens) gets blown up into a massive, wild error in your final map. This is called an ill-posed problem.

The Solution: Using "Prior Knowledge" as a Guide

Since the puzzle is broken, we can't just guess. We need to use Prior Information (what we already know about the train) to guide our reconstruction. This is called Regularization.

The paper compares three different "detectives" (methods) trying to solve this puzzle:

1. The Backus-Gilbert Method (The "Average" Detective)

• How it works: This method tries to find a smooth average that fits the data.
• The Flaw: It's too cautious. If the real map has a sharp spike (a sudden peak in passengers), this method smooths it out into a hill. It's like trying to draw a jagged mountain range with a thick marker; you lose all the sharp details.
• Verdict: It works okay if you have perfect data, but with our "frozen" data, it misses the important peaks.

2. The Maximum Entropy Method (MEM) (The "Smooth" Detective)

• How it works: This method assumes the simplest, smoothest possible map that still fits the data. It uses a rule called "Maximum Entropy" to avoid inventing fake details.
• The Strength: It is very good at ignoring noise. It won't let a tiny smudge on the lens turn into a fake mountain.
• The Result: In the paper's tests, this method did an excellent job of recovering the true map, even with limited data. It's the most reliable "safe" bet.

3. The Bayesian Reconstruction (BR) Method (The "Flexible" Detective)

• How it works: This method is similar to MEM but uses a different set of rules. It tries to be more flexible.
• The Flaw: Because it's more flexible, it sometimes gets "jittery." If the data is limited, it might start drawing ringing artifacts—fake wiggles and waves in the map that aren't actually there. It's like a musician playing a note that is slightly out of tune, creating an echo that wasn't in the original song.
• Verdict: It works well for simple shapes, but for complex shapes, it can get confused and create fake features.

4. Neural Networks (The "AI" Detective)

• How it works: This uses Artificial Intelligence (like the tech behind ChatGPT) to learn the shape of the map.
• The Potential: It's very powerful and can learn complex patterns. However, the paper notes that we need to be careful about how we "train" the AI so it doesn't just memorize the noise.

The "Aha!" Moment: Teamwork Across Fields

The most exciting part of the paper is the call for collaboration.

• Group A: Physicists studying Parton Distribution Functions (PDFs) at normal temperatures (T=0).
• Group B: Physicists studying Spectral Functions (how particles behave in hot, dense matter, like inside a star or the early universe) at high temperatures (T>0).

The Connection: Both groups face the exact same mathematical nightmare: trying to reconstruct a moving picture from frozen, noisy snapshots.

Group B has been fighting this battle for decades. They have developed sophisticated tools (like MEM and BR) to handle the "fuzzy puzzle." The paper argues that Group A (PDFs) should borrow these tools from Group B. By working together, the PDF community can use these proven methods to get much better maps of the proton's interior.

Summary

• The Goal: Map the inside of a proton.
• The Obstacle: Our computers can only see "frozen" snapshots, making the math impossible to solve directly.
• The Fix: We must use smart mathematical tricks (Regularization) that combine the data with what we already know.
• The Winner: The Maximum Entropy Method (MEM) seems to be the most robust tool for this job right now, preventing fake errors while finding the true shape.
• The Future: By sharing techniques between different areas of physics, we can finally get a clear, high-definition picture of the building blocks of our universe.

Technical Summary

1. Problem Statement

The extraction of Parton Distribution Functions (PDFs) from Lattice QCD (LQCD) simulations faces a fundamental theoretical and numerical challenge.

• Euclidean vs. Light-Cone: LQCD simulations are performed in Euclidean time, whereas PDFs are defined via matrix elements on the light-cone (real-time physics). Direct evaluation of light-cone correlations is impossible in Euclidean space.
• The Inverse Problem: Current approaches (Quasi-PDFs and Pseudo-PDFs) attempt to access light-cone physics by evaluating space-like correlation functions at large momenta. This requires an Inverse Fourier Transform (IFT) to convert data in "Ioffe time" ($\nu$) to momentum fraction ($x$).
• Ill-Posed Nature: The IFT is a well-posed problem only if the full Brillouin zone (infinite Ioffe time) is accessible. In practice, LQCD data is limited to a finite range of Ioffe time ($\nu_{max}$) and contains statistical noise. This renders the inverse problem ill-posed (in the sense of Hadamard):
  1. Non-uniqueness: Infinitely many PDFs can reproduce the sparse, noisy input data.
  2. Instability: The solution is exponentially sensitive to small errors in the input data. The eigenvalues of the discretized kernel matrix decay exponentially as the accessible Ioffe time range shrinks, leading to noise amplification upon naive inversion.

2. Methodology

The paper surveys and benchmarks various regularization strategies required to solve this ill-posed inverse problem. The core methodology involves treating the extraction as a statistical inference problem where prior information is introduced to select a unique, physically meaningful solution.

A. Mathematical Formulation

The relationship between the matrix element $M(\nu)$ and the PDF $f(x)$ is modeled as a convolution:
$$Q(\nu) = \int dx K(x, \nu) f(x)$$
Discretizing this leads to a matrix equation $Q = K \cdot f$. Since $K$ is ill-conditioned for limited $\nu$, regularization is mandatory.

B. Reconstruction Approaches Surveyed

The author categorizes methods into three main classes:

1. Parametric Methods:

   • Assumes a specific functional form (e.g., $f(x) = c x^a (1-x)^b$) and fits parameters via $\chi^2$.
   • Limitation: Strongly biased by the assumed ansatz; cannot capture unexpected features.

2. Linear Non-Parametric Methods:

   • Backus-Gilbert (BG): Constructs a linear combination of data to maximize resolution (minimize the width of the resolution function).
   • Limitation: Cannot incorporate complex priors (like positivity) easily; tends to produce artificial smoothing and fails to resolve sharp peaks.

3. Bayesian Inference (Non-Linear):

   • Utilizes Bayes' theorem: $P[f|Q, I] \propto P[Q|f, I] \times P[f|I]$.
   • Likelihood ($P[Q|f, I]$): Measures agreement with data (typically $\chi^2$).
   • Prior ($P[f|I]$): Encodes regularization via a functional $S[f, m, \alpha]$, where $m(x)$ is a default model and $\alpha$ controls uncertainty.
   • Maximum Entropy Method (MEM): Uses Shannon-Jaynes entropy as the regulator. Known for strong smoothing, preventing ringing but potentially hiding physical peaks.
   • Bayesian Reconstruction (BR): Uses axioms tailored for 1D problems, often resulting in a Gamma distribution prior. Less smoothing than MEM but prone to "ringing" artifacts if data is sparse.
   • Neural Networks (NN): Uses NNs as a flexible basis for the PDF, trained with a loss function containing data fidelity and regularization terms (effectively a Bayesian prior).

3. Key Contributions and Results

The paper provides a comparative benchmark of these methods using mock PDFs converted into Ioffe-time matrix elements, simulating realistic LQCD conditions (limited $\nu_{max} \approx 10$, $N_\nu = 12$ data points).

Benchmark Scenarios

Two mock PDFs were tested:

• Model A: Monotonically increasing as $x \to 0$.
• Model B: Increases then decreases, vanishing at $x=0$ (phenomenologically realistic).

Performance Findings

1. Backus-Gilbert (BG):

   • Raw Data: Failed to reproduce the true PDF, missing peaked structures.
   • Preconditioned: Improved significantly when the data was first fitted to a phenomenological model, with the BG method only reconstructing the deviations from that model. However, it still struggled with the small-$x$ region and failed to capture the true solution within error bands.

2. Maximum Entropy Method (MEM):

   • Performance: Excellent agreement with both mock PDFs.
   • Mechanism: The inherent artificial smoothing of MEM (Bryan's implementation) stabilized the reconstruction, effectively suppressing ringing artifacts caused by the limited data range.
   • Result: Successfully recovered the underlying physics despite the ill-posed nature of the data.

3. Bayesian Reconstruction (BR):

   • Performance: Accurate for Model A.
   • Failure Mode: Suffered from significant ringing artifacts for Model B (the vanishing intercept case).
   • Cause: The weaker regularization of BR compared to MEM allowed noise to manifest as oscillations, particularly when the Ioffe time range was short. Extending $\nu_{max}$ to 20 significantly improved BR results.

4. Uncertainty Quantification:

   • The paper emphasizes that a successful reconstruction must quantify two sources of uncertainty:
     1. Statistical: From Monte Carlo noise (assessed via Jackknife resampling).
     2. Systematic: From the choice of regularization (assessed by varying the default model $m(x)$ and hyperparameters).
   • Bayesian methods are highlighted as superior for this because they explicitly separate likelihood (data) and prior (regularization) uncertainties.

4. Significance

• Cross-Disciplinary Collaboration: The paper argues for a strong synergy between the PDF community ($T=0$) and the Spectral Function community (finite temperature, $T>0$). Both face identical ill-posed inverse problems (unfolding spectral functions from Euclidean correlators). The $T>0$ community has decades of experience with MEM, BR, and uncertainty quantification that is directly transferable to PDF extraction.
• Validation of Bayesian Methods: The study demonstrates that while linear methods (BG) have limitations in handling limited data without strong parametric assumptions, Bayesian methods (specifically MEM and BR) offer viable paths forward.
• Path to Precision: The work suggests that with improved matrix elements, better data quality, and rigorous application of Bayesian regularization (learning from spectral function studies), reliable extraction of challenging quantities (like the gluon PDF) from Lattice QCD is achievable.
• Addressing the "Ringing" vs. "Smoothing" Trade-off: The benchmark highlights the critical need to understand the specific artifacts of different regularization schemes (MEM smooths, BR rings) to distinguish physical features from numerical artifacts.

In conclusion, the paper establishes that extracting PDFs from Lattice QCD is an ill-posed inverse problem solvable only through rigorous regularization. It advocates for the adoption of Bayesian inference techniques, refined by the experience of the finite-temperature spectral function community, to achieve reliable, uncertainty-quantified results.