Electroweak Higgs boson pair production: Updated inclusive cross sections

This paper presents updated inclusive cross sections for electroweak Higgs boson pair production at LHC-relevant energies, calculated at N$^3$LO QCD+NLO EW for vector-boson fusion and NNLO QCD for associate production, covering both Standard Model predictions and scenarios with anomalous trilinear Higgs self-couplings.

The Gist

Imagine the Large Hadron Collider (LHC) as the world's most powerful particle "smasher." Its main job is to smash protons together to see what tiny pieces fall out. One of the most famous pieces found was the Higgs boson, a particle that gives mass to everything else.

Now, scientists are preparing for the "High-Luminosity" phase of the LHC. Think of this as upgrading the smasher from a standard hammer to a giant, high-speed pneumatic drill that can hit the target much harder and much more often. The goal? To study the Higgs boson not just as a single particle, but to see what happens when two of them are created at the same time.

This paper is like a updated instruction manual for the scientists running the experiments. It provides the most precise "theoretical maps" of how often two Higgs bosons should appear when they are created through specific, less common methods.

Here is a breakdown of the paper's key points using everyday analogies:

1. The Goal: Catching a Rare Double-Event

Usually, when you smash protons, you get one Higgs boson. Getting two at once is like trying to catch two specific, rare butterflies in a hurricane. It's incredibly hard.

• The Main Event: The most common way to get two Higgs bosons is through "gluon-gluon fusion" (smashing them together directly).
• The Side Acts: This paper focuses on two other, rarer ways they can appear:
  • Vector Boson Fusion (VBF): Imagine two protons passing each other. They don't smash head-on; instead, they exchange a "force carrier" (like a messenger particle), and that messenger splits into two Higgs bosons. It's like two people throwing a ball at each other, and the ball magically splits into two new balls mid-air.
  • Associated Production (Vhh): Imagine a Higgs boson pair being born while a "vector boson" (a W or Z particle, which are like heavy-duty force carriers) is also flying away. It's like a mother duck (the vector boson) leading two ducklings (the Higgs bosons) out of the pond.

2. Why This Paper Matters: The "Recipe" for Success

The scientists at the LHC (ATLAS and CMS) are trying to measure the trilinear Higgs self-coupling.

• The Analogy: Think of the Higgs boson as a spring. If you push two springs together, how hard do they push back? That "push back" strength is the self-coupling.
• The Problem: If we get this number wrong, our understanding of the universe's stability is wrong.
• The Solution: To measure this, the experimentalists need to know exactly what the "Standard Model" (the current best theory) predicts. If they see more or fewer double-Higgs events than the theory predicts, it means there is "new physics" hiding in the numbers.

This paper updates those predictions. It's like a chef updating a recipe book. "Hey, we used to think this cake would weigh 500 grams, but with our new, more precise measuring tools, it actually weighs 502 grams." This tiny difference is crucial for the bakers (experimentalists) to know if their cake is perfect or if they need a new ingredient.

3. The "High-Tech" Math (Simplified)

The paper is full of terms like N3LO QCD and NNLO EW. Let's translate those:

• QCD (Quantum Chromodynamics): This is the math of the "strong force" (the glue holding particles together).
• LO, NLO, NNLO, N3LO: These stand for "Leading Order," "Next-to-Leading Order," etc.
  • LO (Level 1): A rough sketch. "The cake is round."
  • NLO (Level 2): A better drawing. "The cake is round with a slight bump."
  • N3LO (Level 4): A hyper-realistic 3D model. "The cake is round, with a specific texture, a tiny crumb on the left, and a precise temperature gradient."
• The Paper's Achievement: The authors have calculated these "Level 4" predictions for the VBF and Associated Production methods. They are using the most up-to-date "ingredients" (mathematical constants and particle masses) to ensure the numbers are as accurate as humanly possible.

4. The "What If" Scenarios

The paper also calculates what would happen if the Higgs boson's "springiness" (self-coupling) was different from what we expect.

• They ran simulations where the coupling was zero (no springiness), twice as strong, or three times as strong.
• Why? This gives the experimentalists a "menu" of possibilities. If they see a result that matches the "3x strength" menu, they know they've discovered something new!

5. The Bottom Line

This document is a reference guide for the next decade of particle physics.

• For the Theorists: It confirms that their math is solid and up-to-date.
• For the Experimentalists (ATLAS/CMS): It tells them, "If you run the machine for 10 years, expect to see this many double-Higgs events. If you see more, look closer; you might have found a crack in the Standard Model."

In short, this paper is the calibration sheet for the most sensitive scale in the universe, ensuring that when we weigh the future of the universe, we get the right number.

Technical Summary

Here is a detailed technical summary of the paper "Electroweak Higgs boson pair production: Updated inclusive cross sections" (LHCHWG-2026-001).

1. Problem Statement

Following the discovery of the Higgs boson in 2012, the primary objective of the High-Luminosity LHC (HL-LHC) phase is the precise determination of the Higgs trilinear self-coupling ($\lambda_{hhh}$). While gluon-gluon fusion ($ggF$) dominates the Higgs pair ($hh$) production cross-section, it is insufficient for fully constraining anomalous couplings.

The paper addresses the need for updated, high-precision theoretical predictions for the two sub-dominant electroweak (EW) production channels:

1. Vector-Boson Fusion (VBF): $pp \to hhjj$
2. Associated Production with a Vector Boson (Vh): $pp \to Vhh$ (where $V = W^\pm, Z$)

These channels provide complementary information, particularly regarding the coupling of two Higgs bosons to two vector bosons ($\kappa_{2V}$), which cannot be isolated in VBF alone but can be constrained independently in $Vhh$ production. The authors aim to provide state-of-the-art cross-section predictions for LHC Run 3 (13.6 TeV) and the HL-LHC (14 TeV), incorporating the latest parton distribution functions (PDFs) and higher-order perturbative corrections.

2. Methodology and Calculational Setup

The calculations follow the guidelines of the LHC Higgs Cross Section Working Group (LHCHWG).

• Input Parameters:

  • PDFs: PDF4LHC21_40 set with $\alpha_s(M_Z) = 0.118$.
  • Masses/Widths: $m_t = 172.5$ GeV, $M_W = 80.379$ GeV, $M_Z = 91.1876$ GeV. The Higgs width is set to zero.
  • Coupling Scheme: The $G_F$ scheme is used for electroweak couplings.
  • Anomalous Couplings: Predictions are provided for the Standard Model (SM) and for anomalous trilinear couplings defined by $\kappa_\lambda = \lambda_{hhh}/\lambda_{hhh}^{SM} \in \{0, 1, 2, 3\}$.

• Vector-Boson Fusion (VBF) $hh$ Production:

  • Approximation: Calculations utilize the "factorised approximation," omitting $s$-channel contributions and treating $t$- and $u$-channel topologies separately (omitting interferences).
  • Perturbative Order: N$^3$LO QCD + NLO EW.
  • Tools:
    • PROVBFHH v2.1.0: Used for N$^3$LO QCD corrections (based on HOPPET and parametrised coefficient functions).
    • RECOLA + MoCaNLO: Used for NLO EW corrections.
  • Scale Definition: $\mu = \sqrt{m_h^2/4 + p_{T,hh}^2}$.
  • Combination: EW corrections are applied multiplicatively to the N$^3$LO QCD result. Photon-induced contributions are omitted due to PDF inconsistencies and their small magnitude (<1%).

• Associated Production ($Vhh$):

  • Perturbative Order: NNLO QCD.
  • Tools: Inclusive cross-sections based on [35], updated with the PDF4LHC21 set.
  • Scale Definition: Central scale $\mu_R = \mu_F = M_{Vhh}$ (invariant mass of the $Vhh$ system).
  • Specifics:
    • $W^\pm hh$: Treated as Drell-Yan-like production ($pp \to V^* \to Vhh$).
    • $Zhh$: Includes contributions from gluon-initiated processes (triangle, box, pentagon diagrams) involving heavy fermions. A bottom quark mass of $m_b = 4.9$ GeV is used for these diagrams.

3. Key Contributions

1. Updated Cross Sections: Provided inclusive cross-section values for VBF and $Vhh$ production at center-of-mass energies of 13 TeV, 13.6 TeV, and 14 TeV.
2. Higher-Order Precision:
   • VBF results are the first to combine N$^3$LO QCD with NLO EW corrections in the VBF approximation.
   • $Vhh$ results are updated to NNLO QCD with modern PDF sets.
3. Anomalous Coupling Scenarios: The paper provides a comprehensive table of cross-sections for various $\kappa_\lambda$ values, essential for interpreting experimental limits on the Higgs self-coupling.
4. Energy Dependence: Explicitly addresses the transition from current LHC energies (13.6 TeV) to the future HL-LHC design energy (14 TeV).

4. Key Results

The paper presents detailed tables (Tables 1–5) summarizing the cross-sections ($\sigma$) in femtobarns (fb) with associated uncertainties (scale and PDF).

• VBF $hh$ Production (Table 1 & 2):

  • At 13 TeV (SM, $m_h=125$ GeV): $\sigma \approx 1.687$ fb.
  • At 14 TeV (SM): $\sigma \approx 2.005$ fb.
  • Uncertainties: Scale uncertainties are negligible ($\sim 0.05\%$), while PDF uncertainties dominate ($\sim 2.7\%$).
  • Anomalous Couplings: For $\kappa_\lambda = 0$, the cross-section increases significantly ($\sim 4.5$ fb at 13 TeV), while for $\kappa_\lambda = 2$ and $3$, the values vary non-monotonically due to interference effects.

• $W^\pm hh$ Production (Tables 3 & 4):

  • $W^+ hh$: At 13 TeV (SM): $\sigma \approx 0.334$ fb. Scale uncertainty is small ($\sim 0.4\%$).
  • $W^- hh$: At 13 TeV (SM): $\sigma \approx 0.173$ fb. Scale uncertainty is larger ($\sim 1.3\%$) due to different parton luminosity weights ($q\bar{q}$ vs $qg$) compared to $W^+$.
  • Anomalous Couplings: Cross-sections for $\kappa_\lambda = 0$ are roughly half the SM value, while $\kappa_\lambda = 2$ and $3$ show significant enhancements.

• $Zhh$ Production (Table 5):

  • At 13 TeV (SM): $\sigma \approx 0.366$ fb.
  • Uncertainties: Scale uncertainties are notably higher ($\sim 3\%$) compared to $W^\pm hh$ due to the large scale dependence of the gluon-fusion initiated contributions ($gg \to Zhh$).
  • Anomalous Couplings: Similar trends to $W^\pm hh$, with $\kappa_\lambda = 0$ reducing the cross-section and higher $\kappa_\lambda$ values increasing it.

5. Significance

• Experimental Reference: These numbers serve as the definitive theoretical reference for the ATLAS and CMS collaborations for Run 3 and the HL-LHC. They are crucial for setting limits on the trilinear Higgs coupling ($\kappa_\lambda$) and the quartic coupling ($\kappa_{2V}$).
• Precision Benchmark: By pushing VBF predictions to N$^3$LO QCD + NLO EW, the theoretical uncertainty is reduced to the percent level, matching the anticipated experimental precision of the HL-LHC.
• Complementarity: The results highlight that while $ggF$ has the largest cross-section, the EW channels ($VBF$ and $Vhh$) are indispensable for disentangling different anomalous coupling structures, particularly the independent measurement of $\kappa_{2V}$ for $W$ and $Z$ bosons via $Vhh$ production.
• Future Readiness: The inclusion of 13.6 TeV and 14 TeV predictions ensures immediate applicability to current and upcoming data analyses.

In conclusion, this report provides the necessary high-precision theoretical inputs to maximize the physics potential of the HL-LHC in the search for new physics within the Higgs sector.