MATRIX HAWAII: PineAPPL interpolation grids with MATRIX

This paper introduces a new interface between PineAPPL and MATRIX that enables the calculation of fully differential cross-section interpolation grids at NNLO QCD and NLO EW accuracy, allowing for efficient predictions with arbitrary parton distribution functions and facilitating global PDF analyses without relying on approximate K-factors.

The Gist

Imagine you are trying to predict the weather for a specific city, but you have to do it for thousands of different possible climate scenarios (some with more humidity, some with more wind, some with different ocean currents).

In the world of particle physics, the "weather" is how particles smash together in the Large Hadron Collider (LHC), and the "climate scenarios" are the Parton Distribution Functions (PDFs). These PDFs are essentially maps describing how the tiny building blocks inside a proton (quarks and gluons) are arranged.

For a long time, if scientists wanted to know what would happen if they changed the "climate" (the PDFs), they had to run the entire, incredibly complex weather simulation from scratch. This is like re-running a supercomputer simulation for every single possible wind speed. It takes days, weeks, or even months of computing time.

The Problem: The "Re-Cooking" Dilemma

The paper introduces a tool called Matrix Hawaii. To understand what it does, let's use a cooking analogy.

Imagine Matrix is a world-class chef who can cook a perfect, multi-course meal (a precise calculation of particle collisions) using a specific set of ingredients (a specific PDF). However, if you want to see how the meal tastes with slightly different ingredients, the chef has to start over, chopping, sautéing, and baking everything again. This is slow and expensive.

Previously, scientists tried to cheat by using a shortcut called a "K-factor." This is like taking the meal cooked with Ingredient A, and just multiplying the taste by a simple number to guess how it would taste with Ingredient B.

• The Catch: This shortcut assumes the change in ingredients affects every part of the dish (the soup, the steak, the dessert) in exactly the same way. In reality, changing the ingredients might make the soup taste great but ruin the dessert. The paper argues that this "multiplication shortcut" is often too rough and can lead to wrong conclusions about the recipe.

The Solution: The "Universal Recipe Card"

Matrix Hawaii is the new interface that solves this. It works like this:

1. The Master Grid: The chef (Matrix) cooks the meal once, but instead of just serving the plate, they create a Universal Recipe Card (an interpolation grid). This card doesn't list specific ingredients; it lists the structure of the dish in a way that is independent of the specific ingredients used.
2. Instant Adaptation: Now, if you want to know how the dish tastes with a new set of ingredients, you don't need the chef to cook again. You just take the Recipe Card and the new ingredients, and a simple, fast machine (PineAPPL) instantly tells you the result. It takes a fraction of a second instead of days.
3. The "Hawaii" Twist: This specific tool is special because it can handle the most complex "recipes" (calculations) known to physics, including those that require NNLO (Next-to-Next-to-Leading Order) accuracy. This is the highest level of precision currently available for many processes. It also combines two types of "flavors" (QCD and Electroweak corrections) that were previously hard to mix.

What Did They Prove?

The authors didn't just build the tool; they tested it to make sure it works.

• The Taste Test: They cooked the same "meals" (particle collision predictions) using the old slow method (Matrix directly) and the new fast method (Matrix Hawaii + Recipe Cards).
• The Result: The results were identical, down to a tiny fraction of a percent. The "Recipe Card" method is just as accurate as cooking from scratch, but infinitely faster.
• The K-factor Reality Check: They also tested the "multiplication shortcut" (K-factors) against the new "Recipe Card" method. They found that while the shortcut works okay in some simple cases, it can be significantly wrong (off by several percent) when the "ingredients" (PDFs) change drastically. This suggests that for the most precise science, we should stop using the shortcut and start using the Recipe Cards.

Why Does This Matter?

• Speed: Scientists can now test thousands of different "ingredient" combinations in the time it used to take to test one.
• Accuracy: It removes the need for the rough "K-factor" shortcuts, leading to more precise maps of the proton's structure.
• Future-Proofing: As the Large Hadron Collider gets more powerful and data gets more precise, these Recipe Cards allow the community to update their predictions instantly without needing to rebuild the entire supercomputer infrastructure.

In short, Matrix Hawaii is a tool that turns a slow, repetitive cooking process into a fast, flexible system, allowing physicists to explore the universe with unprecedented speed and precision.

Technical Summary

Technical Summary: MATRIX HAWAII – PineAPPL Interpolation Grids with MATRIX

Problem Statement
As the Large Hadron Collider (LHC) approaches its high-luminosity phase, experimental measurements of Standard Model processes have reached a level of precision where theoretical predictions must match this accuracy. A critical bottleneck in achieving this precision is the determination of Parton Distribution Functions (PDFs). PDF fits require repeated convolutions of theoretical cross-section predictions with varying PDF sets. At Next-to-Next-to-Leading Order (NNLO) in Quantum Chromodynamics (QCD), these calculations are computationally expensive, making the iterative process of PDF determination prohibitively costly.

Currently, the lack of publicly available tools to generate NNLO QCD interpolation grids for a wide range of LHC processes forces global analyses to rely on "K-factor" approximations. In this approach, NNLO predictions are approximated by multiplying Next-to-Leading Order (NLO) results by a ratio (K-factor) of NNLO to NLO cross-sections. This method assumes K-factors are independent of the PDF set and partonic channels, an approximation whose reliability and potential biases on PDF extractions have not been fully quantified. Furthermore, existing tools often lack the capability to consistently combine QCD and Electroweak (EW) corrections in a grid format suitable for modern PDF fits.

Methodology
The authors present Matrix Hawaii, a new interface connecting the MATRIX computational framework with the PineAPPL library.

• MATRIX: A public framework capable of calculating single- and double-differential distributions at NNLO QCD accuracy (and NLO EW accuracy) for processes such as Drell-Yan (DY) production and top-quark pair ($t\bar{t}$) production. It utilizes the $q_T$-subtraction formalism to handle infrared divergences, which introduces a dependence on a technical slicing parameter, $r_{cut}$.
• PineAPPL: A library designed to generate and manipulate interpolation grids. These grids store partonic cross-sections in a PDF-independent manner, allowing for rapid convolution with arbitrary PDF sets a posteriori.
• The Interface (Matrix Hawaii): The core innovation is the integration of these two codes. The interface enables MATRIX to output its results directly into PineAPPL interpolation grids.
  • $r_{cut}$ Extrapolation: A critical technical challenge addressed is the removal of the $r_{cut}$ dependence inherent in MATRIX's $q_T$-subtraction method. The authors generalize the extrapolation procedure (fitting results to a polynomial in $r_{cut}$ and extrapolating to zero) to the level of interpolation grids. By leveraging the linearity of the convolution operation and the vector space properties of grids, they construct a final grid that represents the $r_{cut} \to 0$ limit without requiring the user to perform the extrapolation manually.
  • QCD + EW Combination: The interface supports the simultaneous expansion in strong and electroweak couplings, allowing for the generation of grids that include both NNLO QCD and NLO EW corrections.
  • CKM Matrix Handling: For charged-current processes, the interface implements a reweighting approach to restore Born-level Cabibbo-Kobayashi-Maskawa (CKM) matrix dependencies, which are otherwise approximated as trivial in the loop calculations due to renormalization limitations in external amplitude providers.

Key Contributions

1. First Public NNLO QCD Grid Tool: Matrix Hawaii is presented as the first publicly available tool capable of generating interpolation grids at NNLO QCD accuracy for a broad set of processes, including off-shell $Z/W$ boson production and $t\bar{t}$ production.
2. Validation of Interpolation: The authors validate the interface by comparing Matrix Hawaii grid predictions against direct MATRIX calculations. They demonstrate that interpolation errors are negligible (below the per-mille level) compared to the combined Monte Carlo integration and $r_{cut}$ extrapolation uncertainties.
3. K-Factor vs. Exact NNLO Study: The paper provides a proxy study comparing PDF fits using exact NNLO calculations (via grids) versus K-factor approximations. Using the CJ22 PDF set as a testbed, the authors vary input parameters to assess the impact of the approximation on the $\chi^2$ of data description.
4. Inclusion of EW Corrections: The tool successfully generates grids containing NLO EW corrections combined with NNLO QCD, demonstrating that EW effects can exceed NNLO QCD scale uncertainties in certain kinematic regions (e.g., central rapidity).

Results

• Validation: For Drell-Yan and top-pair production at 7 TeV and 8 TeV (ATLAS, LHCb, CMS datasets), the interpolation errors between Matrix Hawaii and direct MATRIX results are found to be well below the per-mille level, confirming the reliability of the grid generation.
• K-Factor Impact: The study of the K-factor approximation reveals that while the method performs reasonably well near the best-fit point of a specific PDF set, deviations can become significant (up to several percent) when moving away from the evaluation point. The authors find that the PDF dependence of K-factors is not negligible compared to the K-factors themselves in certain parameter variations, suggesting that relying on K-factors can introduce biases in PDF determinations, particularly for the gluon and sea-quark distributions.
• EW Effects: The inclusion of NLO EW corrections shows negative corrections of a few percent, which are comparable to or larger than the residual perturbative uncertainties at NNLO QCD in specific kinematic regions.

Significance and Claims
The paper claims that Matrix Hawaii is a crucial step toward making precise, fully differential NNLO QCD predictions available to the broader community in a format that facilitates efficient PDF determinations. By providing grids that encode exact NNLO information (rather than K-factors), the tool addresses the computational bottleneck of PDF fitting and removes the systematic uncertainties associated with the K-factor approximation.

The authors emphasize that while the K-factor approach may be acceptable in some contexts, the availability of exact NNLO grids makes the K-factor approach "obsolete" for the process classes supported by MATRIX (Drell-Yan, Higgs, diboson, triphoton, and top-pair production). The grids generated for this publication, along with the Matrix Hawaii code, are made publicly available to enable the community to perform PDF fits and uncertainty analyses without the need for repeated, expensive NNLO calculations. The tool also lays the groundwork for future PDF determinations at next-to-next-to-next-to-leading order (N3LO) by establishing a workflow for handling high-order corrections efficiently.