Event generation for future DIS experiments

This paper presents state-of-the-art, next-to-leading-order hadron-level predictions for deep-inelastic scattering with multiple final-state particles, consistently merged into single samples to support physics studies at future colliders including the Electron-Ion Collider, LHeC, and FCC-eh.

The Gist

Imagine you are trying to understand the internal structure of a proton (a tiny building block of matter) by smashing an electron into it at incredibly high speeds. This is what "Deep Inelastic Scattering" (DIS) is all about. It's like firing a high-speed bullet at a complex machine to see how its gears fly apart.

This paper is about building the best possible simulation software to predict exactly what happens when these collisions occur at future, super-powerful particle accelerators. The authors are essentially creating a "flight simulator" for physicists so they know what to expect when they turn on these new machines.

Here is a breakdown of their work using simple analogies:

1. The Problem: The "Too Simple" Map

In the past, scientists used computer programs to predict these collisions. Think of these old programs as using a basic road map.

• The Old Way (LO + PS): They would calculate the main crash (the electron hitting the proton) very precisely, but when it came to the debris flying off (the "jets" of particles), they just guessed based on simple rules. It was like saying, "If you crash a car, maybe a few pieces will fly off," without calculating exactly how many or how fast.
• The Limitation: This worked okay for simple crashes, but at the new, higher energies planned for the future, the debris gets messy. You might get 1, 2, 3, or even 4 pieces flying off in different directions. The old maps couldn't handle the complexity of "multijet" chaos.

2. The Solution: The "High-Definition" Simulation

The authors used a sophisticated software called SHERPA to create a new, high-definition simulation.

• The "Merging" Trick: Imagine you are painting a picture. You have a high-detail brush for the main subject (the core collision) and a rougher brush for the background. The authors developed a technique to seamlessly merge these two brushes.
  • They calculate the most important parts of the crash with extreme precision (Next-to-Leading Order, or NLO).
  • They calculate the extra, messy bits (the extra jets) with a slightly less precise but faster method.
  • They then "stitch" these two calculations together so there are no gaps or double-counting. This is called MEPS@NLO.

3. The Test Drive: Three Different Tracks

The authors tested their new simulation on three different "race tracks" (future colliders):

• Track 1: The Electron-Ion Collider (EIC)

  • The Analogy: This is the current "test track" being built in the US. It's the most advanced project right now.
  • The Result: The authors confirmed that their new simulation matches what we already know from past experiments (like HERA). They found that if you ignore the "merging" (the extra debris), your prediction is wrong by a factor of 2 in certain areas. The new simulation fixes this.

• Track 2: The LHeC (Large Hadron-Electron Collider)

  • The Analogy: This is a proposed track in Europe that would use the existing giant LHC tunnel but shoot electrons at protons. It's much faster (higher energy) than the EIC.
  • The Result: As the speed increases, the "debris" gets more energetic. The authors found that the "merging" effect (accounting for extra jets) remains crucial even at higher energies. It's only when the energy gets extremely high (around 1000 GeV²) that the simple "main crash" calculation starts to catch up, but for most of the track, the detailed simulation is needed.

• Track 3: The FCC-eh (Future Circular Collider)

  • The Analogy: This is the "dream track," a hypothetical machine even bigger and faster than the LHeC.
  • The Result: Here, the energy is so high that the "debris" (jets) flies off with incredible force. The authors found that the "merging" corrections (the need to account for extra jets) stretch to even higher energy levels than before. The simple maps fail completely here; you absolutely need their high-definition simulation to get the right answer.

4. The Key Takeaway

The paper argues that for these future experiments to succeed, physicists cannot rely on old, simplified models.

• The Metaphor: If you are trying to predict the weather, a simple "it's sunny" forecast works for a picnic. But if you are launching a rocket, you need a model that accounts for wind shear, humidity, and pressure changes at every altitude.
• The Claim: The authors show that for the new, high-energy colliders, the "wind shear" (the extra jets) is a dominant force. Their new method (MEPS@NLO) is the only way to accurately predict the "weather" of these particle collisions, especially in the lower-energy zones where the debris is most chaotic.

Summary

The authors have upgraded the "flight simulator" for particle physics. They proved that to understand the future of particle collisions at the EIC, LHeC, and FCC-eh, you must use a simulation that perfectly combines the precise calculation of the main crash with a realistic prediction of all the messy debris flying off. Without this upgrade, our predictions for these future machines would be significantly off.

Technical Summary

Technical Summary: Event Generation for Future DIS Experiments

Problem Statement
The planning and operation of next-generation lepton-hadron colliders, specifically the Electron-Ion Collider (EIC), the Large Hadron-Electron Collider (LHeC), and the Future Circular Collider in electron-hadron mode (FCC-eh), require precise theoretical predictions for hadronic final states. While rigorous factorization theorems allow cross-sections to be expressed as convolutions of perturbative matrix elements and non-perturbative parton distribution functions (PDFs), current phenomenological studies often rely on Leading Order (LO) matrix elements matched to parton showers (LO+PS). These approaches may be insufficient for describing the dominant dynamics at low virtualities ($Q^2 \simeq \mathcal{O}(1\text{--}10 \text{ GeV}^2)$) and for capturing the effects of additional hard scales introduced by multi-jet configurations. There is a need for state-of-the-art predictions that consistently merge matrix elements of several partonic multiplicities at Next-to-Leading Order (NLO) precision to ensure reliable descriptions of the hadronic final state across the full kinematic range of future facilities.

Methodology
This work utilizes the SHERPA 3 event generator to produce state-of-the-art hadron-level predictions for Deep Inelastic Scattering (DIS). The methodology employs the CKKW merging approach to consistently combine matrix elements of different final-state multiplicities with parton-shower evolution.

• Framework and Algorithms: The study uses the MEPS@NLO (Multi-Jet Merging at NLO) approach. Matrix elements for single- and dijet production are calculated at NLO accuracy, while three- and four-jet final states are included at LO. Matching between NLO matrix elements and the parton shower is performed using the MC@NLO prescription.
• Process Coverage: Predictions are generated for both Neutral-Current (NC) and Charged-Current (CC) DIS.
  • For NC DIS: $e^-p \to e^- + 1, 2, j$ (NLO) $+ 3, 4, j$ (LO).
  • For CC DIS: $e^-p \to \nu + 1, 2, j$ (NLO) $+ 3, 4, j$ (LO).
  • Massive charm and bottom quarks are treated via separate LO matrix elements merged into the sample, utilizing massive splitting kernels in the parton shower.
• Scales and Parameters:
  • Parton densities are taken from the NNPDF30_nlo_as_0118 set.
  • The strong coupling is set to $\alpha_s(M_Z) = 0.118$.
  • A dynamic merging scale $Q_{cut}$ is employed, defined as $Q_{cut} = \bar{Q}_{cut} / \sqrt{1 + \bar{Q}_{cut}^2 / (S_{DIS} Q^2)}$, with $\bar{Q}_{cut} = 25$ GeV and $S_{DIS} = 0.4$. This allows the merging scale to lower at small virtualities to improve the description of that kinematic regime.
  • Factorization, renormalization, and shower starting scales are set equal to a core scale ($\mu_{core}$) determined by the underlying $2 \to 2$ topology (DIS-like, photon-parton, or purely QCD-induced scattering).
• Validation: The framework was previously validated against HERA data (H1 and ZEUS collaborations) and analytic resummation calculations, particularly for event-shape observables.

Key Contributions

1. Extension of MEPS@NLO to Future Colliders: While previous work (Ref. [50]) established the first MEPS@NLO predictions for the EIC, this contribution extends these high-precision simulations to the higher-energy LHeC ($\sqrt{s} \approx 1.2$ TeV) and FCC-eh ($\sqrt{s} \approx 3.5$ TeV) scenarios.
2. Systematic Comparison of Accuracy Levels: The paper provides a systematic comparison of four simulation setups:
   • LO+PS (Leading Order matched to Parton Shower)
   • MC@NLO (NLO matched to Parton Shower, single multiplicity)
   • MEPS@LO (Merged at Leading Order)
   • MEPS@NLO (Merged at Next-to-Leading Order)
3. Kinematic Analysis: The study analyzes distributions of momentum transfer ($Q^2$), Bjorken-$x$, jet multiplicity, and 1-jettiness ($\tau$) across the different collider energies.

Results

• Impact of Merging at Low $Q^2$: Across all collider scenarios (EIC, LHeC, FCC-eh), multijet merging induces significant corrections to the cross-section at low virtualities. The LO merged samples show characteristic corrections at small $Q^2$ and small $x$ due to the appearance of additional scales (e.g., jet transverse momenta).
• NLO Corrections vs. Merging: In the LHeC setup, MC@NLO corrections to the LO distribution are relatively small. However, the NLO merged sample (MEPS@NLO) shows that merging effects dominate over NLO virtual corrections up to $Q^2 \approx 100\text{--}200$ GeV$^2$. Only at very high $Q^2$ ($\gtrsim 1000$ GeV$^2$) do virtual corrections become significant enough that MEPS@NLO agrees better with MC@NLO than with MEPS@LO.
• Energy Dependence: As the center-of-mass energy increases from EIC to LHeC to FCC-eh, the region where merging corrections are dominant shifts to progressively higher $Q^2$ values. For FCC-eh, merging effects remain significant up to $Q^2 \approx 10^4$ GeV$^2$.
• Jet Multiplicity and 1-Jettiness: Merged samples predict large corrections for high jet multiplicities, where LO+PS or MC@NLO cross-sections are negligible. For 1-jettiness, the most significant corrections occur in the small $\tau$ region. At higher $\tau$, merged samples show slightly stronger enhancements due to the larger radiation phase space available at higher energies.

Significance
The paper asserts that merging corrections are a crucial ingredient for the phenomenology of future DIS experiments. Specifically, the consistent merging of matrix elements is essential for:

• Accurately describing the low $Q^2$ region.
• Providing a reliable interpolation to the photoproduction regime.
• Ensuring that theoretical predictions for future facilities (EIC, LHeC, FCC-eh) are robust against the introduction of additional hard scales in multi-jet final states.

The authors conclude that while the current work represents a significant step forward, future developments should aim for a fully consistent combination of photoproduction and DIS regimes, advanced treatments of QED corrections, and non-perturbative tunes utilizing the full scope of available DIS data.