Inclusive charm and bottom quark pair production cross sections at hadron colliders at next-to-next-to-leading-order accuracy

This paper presents a comprehensive study of inclusive charm and bottom quark pair production cross sections across a wide range of collision energies using next-to-next-to-leading-order (NNLO) calculations with the new MaunaKea code, demonstrating that these advanced predictions significantly enhance accuracy and reduce theoretical uncertainties compared to previous NLO results, thereby achieving agreement with experimental data and offering valuable constraints on gluon density and the bottom-quark mass.

The Gist

Imagine the universe as a giant, high-speed racetrack where tiny particles called protons zoom around and crash into each other. When they collide, they sometimes smash together hard enough to create heavy, short-lived particles called charm and bottom quarks. These are like the "heavyweights" of the particle world.

This paper is essentially a massive scorecard and rulebook check for these collisions. The authors, a team of physicists, did three main things:

1. Gathering the Evidence (The "Scorecard")

First, they went on a treasure hunt through decades of scientific experiments. They collected over 100 different measurements of how often these heavy quarks are created in collisions ranging from low-energy crashes (like a bicycle bumping into a wall) to the most powerful crashes ever made by humans (like two freight trains hitting each other at full speed).

They had to be very careful because scientists don't see the quarks directly; they see the "debris" (other particles) the quarks leave behind. To figure out the total number of quarks, they had to use a "conversion factor" (called a fragmentation fraction).

• The Analogy: Imagine you are trying to count how many apples were in a truck, but you can only see the apple cores in the trash. You have to guess how many whole apples were there based on the cores. The paper found that the "guessing rules" have changed over time. In the past, scientists assumed the cores came from a standard apple. But new data from the Large Hadron Collider (LHC) suggests that in these high-energy crashes, the "apples" might be slightly different—perhaps more "baryon" apples and fewer "meson" apples than expected. This changes the final count.

2. Building a Better Calculator (The "Rulebook")

Next, the authors built a new, super-advanced calculator called MaunaKea.

• The Old Way: For a long time, scientists used a calculator that was "pretty good" (Next-to-Leading Order, or NLO). It was like using a map with some missing roads.
• The New Way: This new calculator is "Next-to-Next-to-Leading Order" (NNLO). Think of this as upgrading from a paper map to a live, 3D GPS that accounts for traffic, road construction, and detours.
• The Result: When they ran the new calculator, it predicted that twice as many heavy quarks are created as the old calculator thought. However, the new calculator is also much more confident in its answer. The "margin of error" (the uncertainty) shrank by half. It's like going from saying, "We think there are between 10 and 100 apples," to saying, "We are sure there are between 45 and 55 apples."

3. Comparing the Scorecard to the Calculator

Finally, they compared their new, super-precise calculator predictions against the real-world data they collected in step 1.

• The Match: The new predictions fit the real-world data very well across the entire range of energy levels, from the smallest crashes to the biggest ones.
• The "Why": The paper explains that the old calculator was missing some important "traffic rules" (higher-order corrections). Once those were added, the numbers lined up perfectly.

Key Takeaways for the Future

The paper points out two specific things that could help scientists in the future:

1. The Gluon Mystery: Heavy quarks are mostly made by "gluons" (the glue holding particles together). At very high energies, the paper suggests that measuring these quarks could help us map out the "gluon density" in a part of the proton that is currently a "blind spot" (a place where we don't have enough data). It's like using a flashlight to see into a dark corner of a room we've never looked at before.
2. Weighing the Bottom Quark: For the heavier "bottom" quarks, the paper suggests that if we can measure collisions at specific low energies (between 10 and 100 GeV) more precisely, we could determine the exact "weight" (mass) of the bottom quark more accurately. It's like trying to weigh a feather; you need a very sensitive scale and a very specific setup to get the right number.

In summary: This paper updated the math used to predict heavy particle creation, fixed the way we count them based on new experimental clues, and confirmed that our new, more complex math matches reality much better than the old, simpler math. It also highlights where we need to look next to understand the invisible "glue" inside protons and the exact weight of heavy particles.

Technical Summary

Technical Summary: Inclusive Charm and Bottom Quark Pair Production Cross Sections at Hadron Colliders at Next-to-Next-to-Leading-Order Accuracy

Problem Statement
While substantial progress has been made in perturbative Quantum Chromodynamics (pQCD) regarding heavy-quark production, a systematic comparison of the full set of available inclusive charm ($c\bar{c}$) and bottom ($b\bar{b}$) production cross sections with Next-to-Next-to-Leading-Order (NNLO) predictions has been lacking. Previous theoretical state-of-the-art for these processes was largely limited to Next-to-Leading-Order (NLO) accuracy, often supplemented by resummation in General-Mass Variable-Flavour-Number Schemes (GM-VFNS). Theoretical uncertainties in these calculations remain sizable, driven by missing higher-order corrections, Parton Distribution Function (PDF) uncertainties, heavy-quark mass definitions, and the strong coupling constant. Furthermore, experimental extractions of inclusive cross sections from hadronic final states are complicated by non-universal fragmentation fractions observed at the LHC, where heavy-quark baryon-to-meson ratios differ significantly from those measured in $e^+e^-$ collisions. There is a need to confront fixed-order NNLO predictions with experimental data across a wide range of center-of-mass energies ($\sqrt{s} \approx 10$ GeV to 400 TeV) to assess perturbative convergence and constrain gluon densities at very small momentum fractions ($x$).

Methodology
The authors compile and analyze over 50 experimental measurements of inclusive $c\bar{c}$ and $b\bar{b}$ cross sections from proton-proton ($p$-$p$), proton-antiproton ($p$-$\bar{p}$), and proton-nucleus ($p$-$A$) collisions. Experimental data are derived from measurements of heavy-flavor hadrons (mesons and baryons) or their decay products (leptons, $J/\psi$), extrapolated to the full phase space using fragmentation fractions and Monte Carlo (MC) simulation factors.

Theoretical predictions are generated using a new open-source fixed-order code, MaunaKea, which implements NNLO matrix elements for heavy-quark pair production within the Fixed-Flavour-Number (FFN) scheme. The code utilizes the coefficient functions from the top++ framework, interfaced with PineAPPL for fast interpolation of PDFs. Key methodological features include:

• Schemes: Calculations are performed in the FFN scheme (where heavy quarks are produced explicitly in the hard scattering) and compared against NLO predictions in the SACOT-$m_T$ GM-VFNS scheme (which resums collinear logarithms).
• PDF Sets: Predictions are computed using three modern NNLO PDF sets: NNPDF4.0, CT18, and MSHT20.
• Uncertainty Quantification: Theoretical uncertainties are assessed via 7-point scale variations (factorization $\mu_F$ and renormalization $\mu_R$), variations in heavy-quark pole masses ($m_c, m_b$), and variations in the strong coupling constant $\alpha_s(m_Z)$.
• Scale Choices: Two scale prescriptions are explored: a static scale $\mu = 2m_Q$ and a dynamic scale $\mu = \sqrt{(2m_Q)^2 + \langle p_T \rangle^2}$ to better capture high-energy kinematics.
• Corrections: Experimental data from $p$-$\bar{p}$ and $p$-$A$ collisions are corrected to $p$-$p$-equivalent values using nuclear PDFs (EPPS21) and antiproton PDFs where necessary.

Key Contributions

1. MaunaKea Code: Introduction of a new NNLO calculation tool for inclusive heavy-quark production, distinct from top++ by its dynamic handling of light quark flavors ($N_f$) and integration with PineAPPL for efficient PDF evaluation.
2. Comprehensive Data Compilation: A systematic collection of $\sim$50 measurements for both charm and bottom cross sections, including re-evaluations of fiducial cross sections to inclusive values using updated fragmentation fractions (accounting for LHC-specific baryon enhancements).
3. NNLO vs. Data Comparison: The first systematic confrontation of inclusive $c\bar{c}$ and $b\bar{b}$ data with NNLO FFN predictions across $\sqrt{s} \approx 10$ GeV to 400 TeV, including a detailed breakdown of theoretical uncertainties.
4. PDF and Mass Sensitivity: An analysis of how different PDF sets and heavy-quark mass definitions impact cross-section predictions, particularly in the small-$x$ regime relevant for future colliders (FCC) and ultra-high-energy cosmic rays.

Results

• Perturbative Convergence: NNLO cross sections are enhanced by up to a factor of two relative to NLO predictions for both charm and bottom production. The associated theoretical scale uncertainties are reduced by a similar factor, though they remain substantial (approx. $\pm 60\%$ for charm and $\pm 20\%$ for bottom at NNLO).
• Agreement with Data: When accounting for all theoretical uncertainties (scales, PDFs, masses), the NNLO predictions agree with experimental data over the full energy range.
  • Charm: Agreement depends on the PDF set and the charm mass value. CT18 (with lower $m_c$) tends to overshoot data slightly, while NNPDF4.0 (with higher $m_c$) undershoots, though both are consistent within uncertainties. The data lie near the upper edge of the NNPDF4.0 scale uncertainty band at LHC energies.
  • Bottom: Predictions from all three PDF sets are more consistent with each other due to similar mass values. The data are well-described within uncertainties, with bottom-quark mass variations being the dominant source of theoretical uncertainty, especially near threshold energies ($\sqrt{s} \approx 10-100$ GeV).
• Small-$x$ Behavior: At LHC and FCC energies, charm production probes gluon densities at $x \lesssim 10^{-6}$. The study highlights that MSHT20 grids become unreliable below $x=10^{-6}$ due to extrapolation issues, while NNPDF4.0 and CT18 extend to $x=10^{-9}$. However, the NNPDF4.0 gluon density shows a rapid decrease at very small $x$ in the absence of direct constraints, leading to potential non-physical cross-section behaviors in extrapolation regions.
• Scale Choice: The dynamic scale choice reduces scale uncertainties at high energies compared to the static scale, though the central values remain largely unchanged.

Significance and Claims
The paper claims that the inclusion of NNLO corrections is essential for a reliable description of heavy-quark production, significantly reducing theoretical uncertainties compared to NLO. The authors assert that:

• PDF Constraints: Existing LHC measurements of inclusive charm cross sections can serve as valuable input for future global PDF fits, providing constraints on the gluon density at very small $x$ ($x \approx 10^{-6}$) where current data is sparse. This could help anchor the small-$x$ behavior of PDFs and reduce reliance on theoretical extrapolation assumptions.
• Mass Constraints: Precise measurements of bottom cross sections at low energies ($\sqrt{s} \approx 10-100$ GeV) offer a pathway to constrain the bottom-quark pole mass, as the cross section is highly sensitive to $m_b$ near the production threshold.
• Future Projections: The study provides a state-of-the-art baseline for predictions at the Future Circular Collider (FCC) and for cosmic-ray interactions up to the GZK cutoff ($\sqrt{s} \approx 400$ TeV), emphasizing the need for improved control over low-$x$ parton dynamics and extended PDF grid coverage (down to $x \sim 10^{-10}$).
• Fragmentation: The work underscores the necessity of using in-situ fragmentation fractions or explicitly stating assumptions when extrapolating experimental hadronic data to inclusive quark-level cross sections, particularly given the observed violation of fragmentation universality at the LHC.

The authors conclude that while current NNLO calculations agree with data, the slow perturbative convergence and remaining uncertainties highlight the need for further theoretical progress (e.g., N$^3$LO or resummation) and improved experimental knowledge of heavy-flavor hadron production fractions.