⚛️ phenomenology

history: A tool for fully-differential cross sections at next-to-next-to-leading order

The paper introduces $\texttt{history}$ , a process-independent software tool based on a fully-local nested soft-collinear subtraction scheme that calculates fully-differential cross sections for colour-singlet production in hadronic collisions up to next-to-next-to-leading order in QCD, with initial implementations for Higgs production via gluon fusion and Higgs-Strahlung.

Original authors: Sven Yannick Klein, Lukas Simon

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sven Yannick Klein, Lukas Simon

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

Imagine you are trying to predict exactly how a complex machine will behave when you smash two giant, fast-moving trains together. In the world of particle physics, these "trains" are protons (from the Large Hadron Collider), and the "smash" creates a shower of new particles.

Physicists want to know not just if a specific particle (like the Higgs boson) is created, but exactly how it flies off, how fast it goes, and what else is created with it. This is called calculating a "fully-differential cross section."

The paper introduces a new software tool called history (short for "Higgs theory") that does this calculation with extreme precision. Here is how it works, explained through simple analogies:

1. The Problem: The "Infinite Noise" of the Universe

When you smash protons together, the math gets messy. In Quantum Chromodynamics (QCD), the theory describing these collisions, the calculations involve "infinities."

Think of it like trying to listen to a whisper in a hurricane.

The Whisper: The actual signal you want to measure (the Higgs boson).
The Hurricane: The "infrared divergences." These are mathematical infinities that happen when particles are emitted with zero energy or travel in the exact same direction as other particles.

If you try to add up all the possibilities to get a final answer, the "noise" (infinities) drowns out the "signal," and the math breaks.

2. The Solution: The "Noise-Canceling Headphones"

To fix this, physicists use a technique called Subtraction. Imagine you have a recording of the hurricane. You create a perfect "anti-noise" recording that cancels out the wind exactly, leaving only the whisper.

The history software uses a specific type of noise-canceling called the Nested Soft-Collinear (NSC) Subtraction Scheme.

"Soft": When a particle has almost zero energy (like a gentle breeze).
"Collinear": When a particle flies in the exact same direction as another (like two cars merging onto a highway).
"Nested": The software handles these problems in layers. It peels away the noise step-by-step, ensuring that every single "infinite" possibility is mathematically cancelled out by a corresponding "anti-infinite" calculation.

The magic of history is that this noise-canceling system is process-independent. It's like having a universal noise-canceling headphone that works whether you are listening to a whisper in a library, a shout in a stadium, or a siren in a city. You don't need to build new headphones for every new sound; you just plug in the specific "sound" (the physics of the collision) you want to study.

3. The Map: Navigating the "Phase Space"

Once the noise is canceled, you still have to calculate the result. This involves integrating over "phase space," which is a fancy way of saying "all the possible ways the particles could fly out."

Imagine trying to paint a picture of every possible outcome of a car crash. There are infinite angles, speeds, and directions.

The history software creates a highly organized map of this territory.
It divides the map into specific "sectors" (like neighborhoods on a map) where the rules are simple.
It then uses a Monte Carlo method (essentially a super-advanced dice-rolling simulation) to sample millions of points on this map to find the average result with high precision.

4. What Did They Test It On?

To prove their new tool works, the authors tested it on two very famous "crashes" at the Large Hadron Collider:

Gluon Fusion (The Heavy Higgs): Two gluons (particles that hold the nucleus together) smash together to create a Higgs boson. This is like two heavy weights colliding to create a new, heavy object.
Higgs-Strahlung (The Associated Higgs): A quark and an antiquark collide to create a W or Z boson (a heavy force carrier) which then "shoots" off a Higgs boson. This is like a billiard ball hitting another ball, which then knocks a third ball into the pocket.

They compared their results with other famous, trusted software (like SusHi, NNLOJET, and vh@nnlo). The results matched perfectly, down to the tiny decimal places.

5. Why Does This Matter?

The Large Hadron Collider is now a "precision machine." It's no longer just about discovering new particles; it's about measuring them so precisely that we can see if the Standard Model of physics is slightly wrong.

The Goal: To predict experimental results with such accuracy that if the real world deviates even slightly, we know there is "New Physics" hiding there.
The Tool: history provides a flexible, open-source framework that allows scientists to plug in different collision scenarios and get these ultra-precise predictions without having to reinvent the math for every single new experiment.

In summary: The authors built a universal, high-precision calculator that filters out the mathematical "static" of particle collisions. It allows physicists to predict exactly what happens when protons smash together, helping us understand the fundamental laws of the universe with unprecedented clarity.

1. Problem Statement

The Large Hadron Collider (LHC) has transitioned from a discovery machine to a precision facility, requiring theoretical predictions with accuracy matching experimental data (percent-level or better). While Next-to-Leading Order (NLO) and Next-to-Next-to-Leading Order (NNLO) calculations are standard for many processes, achieving fully-differential NNLO predictions remains a significant challenge.

Key difficulties include:

Infrared (IR) Divergences: Calculations at NNLO involve double-virtual, real-virtual, and double-real emission contributions, all containing IR singularities that must cancel out.
Phase-Space Integration: Integrating over the multi-particle phase space while handling these singularities numerically is computationally intensive.
Lack of Public Tools: While slicing methods (e.g., $q_T$ -slicing) are well-established, fully-local subtraction schemes (which are generally more efficient and precise) have historically lacked robust, public, and process-independent implementations. Existing tools like NNLOCAL and NNLOJET address this but are either restricted in scope or use specific subtraction schemes (e.g., antenna subtraction).

2. Methodology

The authors present history (short for Higgs theory), a Monte Carlo program designed to compute fully-differential cross sections for colour-singlet production processes in hadronic collisions up to NNLO in QCD.

Core Algorithm: Nested Soft-Collinear (NSC) Subtraction

The software implements the Nested Soft-Collinear (NSC) subtraction scheme, a fully-local extension of the Frixione-Kunszt-Signer (FKS) scheme to NNLO.

Process Independence: The subtraction algorithm is entirely process-independent. It isolates IR singularities using local operators ( $S$ for soft, $C$ for collinear, $CC$ for triple-collinear) without needing to know the specific matrix elements of the process being calculated.
Singularities Handling:
- Real Emission: Singularities are extracted using local subtraction terms that render the integrand finite in 4 dimensions.
- Integrated Counterterms: The singular parts are integrated analytically over the unresolved phase space, resulting in explicit poles in the dimensional regulator $\epsilon$ .
- Cancellation: The poles from the integrated counterterms cancel exactly against the poles from virtual corrections (double-virtual and real-virtual) and collinear PDF renormalization terms.
Phase-Space Parametrization: The code uses a specialized parametrization of the phase space (based on Refs. [67, 68]) that aligns with the subtraction scheme. It divides the phase space into sectors (double-collinear and triple-collinear partitions) to handle overlapping singularities efficiently, using variables like energy fractions and angular ordering to map the integration domain to $[0,1]$ .

Implementation Details

Matrix Elements: The framework requires the user to supply process-dependent matrix elements. The current release includes:
- Gluon Fusion: $pp \to H + X$ (in the heavy-top limit).
- Associated Production: $pp \to VH + X$ (Higgs-Strahlung, $V=W, Z$ ).
Computational Tools:
- Uses MadGraph5_aMC@NLO (standalone mode) for generating double-real emission amplitudes.
- Interfaces with OpenLoops2 for validation of one-loop amplitudes.
- Relies on external libraries: LHAPDF (PDFs), Cuba (VEGAS integration), Polylogarithm, and LoopTools.

3. Key Contributions

Public Release of history: The first public release of a fully-local NNLO subtraction tool that is process-independent in its subtraction logic.
Validation of NSC Scheme: A comprehensive demonstration that the NSC scheme can be implemented robustly for both gluon-induced (gluon fusion) and quark-induced (Drell-Yan-like) colour-singlet processes.
Modular Architecture: The code separates the subtraction logic from the matrix element generation, allowing for the extension to arbitrary colour-singlet processes (e.g., Drell-Yan, vector boson fusion) simply by providing the relevant matrix elements.
Comprehensive Benchmarks: The paper provides extensive validation against established codes, proving the reliability of the implementation.

4. Results

The authors validated the code by comparing results against independent, state-of-the-art calculations:

Gluon Fusion ( $pp \to H + X$ ):
- Total Cross Section: Compared against SusHi. Agreement was found within Monte Carlo uncertainties for the total inclusive cross section and individual partonic channels ($gg$, $qg$, $qq$).
- Differential Observables: The Higgs transverse momentum ( $p_T$ ) spectrum was compared against NNLOJET. The results matched perfectly across the entire kinematic range.
Associated Production ( $pp \to VH + X$ ):
- Total Cross Section: Compared against vh@nnlo. Excellent agreement was observed for both $WH$ and $ZH$ production across all partonic channels.
- Differential Observables: The invariant mass spectrum ( $M_{VH}$ ) was compared against vh@nnlo. The history prediction reproduced the spectrum at the permille level (0.1% precision) across the full kinematic range.

5. Significance

Advancing Precision Phenomenology: history provides a critical tool for the LHC high-luminosity phase, enabling precise theoretical predictions for Higgs and vector boson observables.
Efficiency and Flexibility: By using a fully-local subtraction scheme, the tool avoids the limitations of slicing methods (which introduce power corrections) and offers a flexible framework for future extensions to more complex processes.
Community Resource: The public availability of the code, combined with its modular design, lowers the barrier to entry for performing NNLO calculations, fostering further development in automated QCD computations.
Future Outlook: The authors aim to establish history as a general semi-automated framework for NNLO computations, with plans to incorporate more sophisticated amplitude providers and extend support to additional processes.

In summary, this paper introduces a robust, validated, and publicly available software tool that successfully implements a complex, fully-local NNLO subtraction scheme, bridging a critical gap in the toolkit available for precision high-energy physics.