VALO1.0: New real-photon parton distributions with Monte Carlo uncertainties

This paper presents VALO1.0, a new set of leading-order and next-to-leading-order real-photon parton distribution functions determined via a global QCD analysis of photon structure function data, which utilizes Monte Carlo replicas to quantify experimental uncertainties and provides open-source tools for their application.

The Gist

Imagine the universe is built out of tiny, invisible Lego bricks called quarks and gluons. Usually, we think of these bricks as being stuck inside heavy, solid blocks like protons (which make up the atoms in your body). But sometimes, these bricks can float freely inside a beam of light itself.

This paper is about creating a new, high-precision "map" of how these bricks are arranged inside a beam of light (a real photon). The authors call this new map VALO1.0 (which is Finnish for "light").

Here is the story of how they made this map, explained simply:

1. The Mystery of the "Ghost" Brick

Usually, when we shine a light, it just bounces off things. But in the world of high-energy physics, a photon (a particle of light) can act like a ghost. It can briefly turn into a swarm of quarks and gluons before turning back into light.

• The Direct Way: The photon hits something directly.
• The "Resolved" Way: The photon acts like a bag of quarks and gluons, and those particles hit the target.

To understand the "Resolved" way, physicists need to know exactly how many quarks and gluons are in that bag at any given moment. This is what a Parton Distribution Function (PDF) is: a recipe that tells you the probability of finding a specific type of brick inside the photon.

2. The Old Maps vs. The New Map

Before this paper, scientists had old maps (called GRV, CJK, etc.). These maps were drawn using math and some data, but they had a few problems:

• They didn't tell you how "fuzzy" or uncertain the map was.
• They were sometimes inconsistent with new, more precise data.

The authors of this paper decided to redraw the map from scratch using a massive amount of data collected over decades from giant particle colliders (like LEP, PETRA, and TRISTAN).

3. The "Monte Carlo" Cooking Method

Instead of trying to find just one perfect recipe, the authors used a clever statistical trick called Monte Carlo replicas.

• The Analogy: Imagine you are trying to bake the perfect cake, but you don't know the exact amount of sugar or flour. Instead of guessing once, you bake 100 different cakes.
• For each cake, you slightly tweak the ingredients based on the "noise" or small errors in your measuring tools.
• After baking 100 cakes, you taste them all.
  • The average taste of all 100 cakes becomes your "Central Recipe" (the best guess).
  • The difference between the cakes tells you how uncertain you are. If all 100 cakes taste almost the same, your recipe is very precise. If they taste wildly different, your recipe is shaky.

This is what the authors did. They generated 100 different versions of the photon map to see which ones fit the experimental data best. This allowed them to draw "uncertainty bands" (like a safety margin) around their map.

4. What They Found

After running their 100 "cakes" through the math, they discovered:

• The Quarks (The Main Ingredients): They found a very clear, stable picture of how quarks are arranged inside the photon. Whether they looked at the data with simple math (Leading Order) or complex math (Next-to-Leading Order), the quark map looked the same and was very reliable.
• The Gluons (The Glue):
  • At the complex level (NLO): They managed to pin down the gluon distribution reasonably well. It's like they finally figured out how much glue is in the bag.
  • At the simple level (LO): The gluon map was still a bit of a mystery. The 100 different "cakes" had very different amounts of glue, meaning the data wasn't strong enough yet to tell them exactly how the glue is distributed.

5. The Tools They Left Behind

The authors didn't just give you the map; they gave you the tools to use it and make better maps in the future:

• The Map (VALO1.0): Available for anyone to download in a standard format used by physicists.
• The Evolution Engine (γEKO): A piece of software that acts like a time machine. It takes the map at one energy level and "evolves" it to a higher energy level, showing how the quarks and gluons rearrange themselves as the photon gets more energetic.
• The Fitting Kit (VALOfitter): The actual software they used to bake the 100 cakes, now open for others to use.

Summary

In short, this paper is about taking a blurry, old photograph of the inside of a photon and turning it into a sharp, high-definition image with a clear "confidence rating." They used a massive dataset and a "100-cake" statistical method to create the most reliable map of light's internal structure to date, while admitting exactly where the map is still a little fuzzy (specifically regarding the "glue" or gluons at simple energy levels).

Technical Summary

Technical Summary: VALO1.0: New real-photon parton distributions with Monte Carlo uncertainties

Problem Statement
The paper addresses the need for modern, robust Parton Distribution Functions (PDFs) for the real photon to support phenomenological studies of Ultra-Peripheral Collisions (UPCs) at the Large Hadron Collider (LHC) and future photoproduction measurements at the Electron-Ion Collider (EIC). While previous global QCD analyses of photon PDFs exist, they often lack rigorous uncertainty quantification or rely on older datasets. The authors aim to determine new Leading Order (LO) and Next-to-Leading Order (NLO) photon PDFs by performing a global QCD analysis of world data on the photon structure function $F_2^\gamma$ measured in $e^+e^-$ scattering. A primary goal is to quantify the uncertainties of these PDFs using a Monte Carlo (MC) replica framework, a standard in modern proton PDF fits but less common in photon PDF determinations.

Methodology
The analysis employs a global QCD fit to 157 experimental data points on $F_2^\gamma$ spanning kinematic ranges of $10^{-3} \le x \le 0.98$ and $1.86 \le Q^2 \le 390 \text{ GeV}^2$. The methodology is built on three pillars:

1. Theoretical Framework: The analysis utilizes the QCD collinear factorization theorem. The photon structure function is calculated to NLO accuracy in the $\overline{\text{MS}}$ and $\text{DIS}_\gamma$ factorization schemes. The authors implement the Zero-Mass Variable Flavor Number Scheme (ZM-VFNS) and solve the inhomogeneous DGLAP evolution equations, which include the point-like $\gamma \to q\bar{q}$ contribution. To handle the scale evolution, they developed a new open-source code, $\gamma\text{EKO}$, which solves the evolution equations in Mellin moment space, separating the homogeneous (hadronic) and inhomogeneous (point-like) components.
2. Statistical Framework: The authors utilize a Monte Carlo replica method. They generate 100 pseudo-data replicas based on the experimental measurements and their uncertainties. For each replica, the PDF parameters are determined by minimizing a loss function (using the iminuit package with a soft $L_1$ approximation to suppress outliers). The central PDF is defined as the average over all replicas, while uncertainties are quantified using both the standard deviation and the 68% confidence interval (CI).
3. Parametrization and Initial Conditions: At the input scale $Q_0^2 = 1 \text{ GeV}^2$, the photon PDFs are parametrized using a five-parameter "hadron-like" ansatz inspired by the Vector Meson Dominance (VMD) model. The ansatz assumes isospin symmetry for up and down quarks, a VMD-motivated suppression for strange quarks ($K_s \approx 0.3$), and a vanishing charm distribution at the input scale. The gluon distribution is constrained by quark counting rules at large $x$ ($b_g = 3$). The fit treats the normalization and shape parameters for up-quarks ($N_u, a_u, b_u$) and gluons ($N_g, a_g$) as free parameters.

Key Contributions

• VALO1.0 PDF Sets: The paper presents new LO and NLO photon PDF sets (VALO1.0) in both $\text{DIS}_\gamma$ and $\overline{\text{MS}}$ schemes. These are provided as LHAPDF6 grids containing 100 MC replicas and a central fit, facilitating their use in general-purpose event generators.
• Uncertainty Quantification: Unlike many previous parameterizations, VALO1.0 provides a rigorous statistical assessment of uncertainties derived from experimental data propagation via MC replicas.
• $\gamma\text{EKO}$ Code: The authors release an open-source code, $\gamma\text{EKO}$, which implements the scale evolution of photon PDFs in Mellin space, handling the inhomogeneous terms specific to photon evolution.
• VALOfitter Framework: The numerical fitting framework used to generate the results is made publicly available.

Results

• Fit Quality: The global fits converge well, with $\chi^2/\text{DOF} = 0.81$ at LO and $0.94$ at NLO. The results reproduce the experimental $F_2^\gamma$ data well across most kinematic regions, though some older datasets (JADE 1984, TOPAZ 1994) show larger deviations.
• Quark Distributions: The quark PDFs are found to be well-constrained and robust at both LO and NLO. The shapes and magnitudes are similar between orders, with small uncertainties across the $x$ range.
• Gluon Distributions:
  • NLO: The gluon distribution is reasonably well-constrained with modest uncertainties, though the parametrization imposes some rigidity at large $x$.
  • LO: The LO gluon distribution remains largely unconstrained by the $F_2^\gamma$ data. The MC replicas show strong deviations, clustering into two distinct solution groups, leading to non-Gaussian uncertainty estimates where the standard deviation and 68% CI yield different shapes.
• Scale Evolution: The evolved PDFs show the expected behavior: quark distributions evolve weakly, while the gluon distribution exhibits a rapid rise at small $x$. The authors note that at NLO, quark distributions become negative for very large $x$ ($x > 0.95$), a feature often neglected in other parameterizations but present in their unconstrained evolution.
• Comparison: The VALO1.0 PDFs broadly agree with existing parameterizations (GRV, CJKL, SaSG, SAL) in shape, with the most significant numerical differences appearing in the small-$x$ region where data is sparse.

Significance
The paper claims that VALO1.0 provides a necessary update for phenomenological applications in high-energy physics, particularly for processes involving real photons at the LHC and EIC. By providing PDFs with quantified uncertainties in a standard format (LHAPDF6), the work facilitates more reliable predictions for photoproduction processes. The authors emphasize that while the current fit is constrained by $F_2^\gamma$ data, the framework is designed to incorporate future data from HERA and the EIC to further constrain the gluon distribution and flavor separation. The release of the $\gamma\text{EKO}$ and VALOfitter tools is highlighted as a significant contribution to the community's ability to perform similar analyses.