Fully differential Higgs boson pair production at N$^3$LO with top quark mass effects

This paper presents the first fully differential predictions for Higgs-boson pair production via gluon-gluon fusion at N$^3$LO in the heavy-top limit, demonstrating a threefold reduction in scale uncertainties and incorporating top-quark mass effects to provide highly precise theoretical benchmarks for LHC searches.

The Gist

Imagine the universe as a giant, complex machine, and the Higgs boson as a crucial gear that gives other particles their weight. Physicists want to understand how this gear works by smashing particles together at incredible speeds in the Large Hadron Collider (LHC). Specifically, they are trying to see what happens when two Higgs bosons are created at the same time. This is like trying to catch two rare, elusive butterflies in a storm to see how they interact.

This paper is a massive leap forward in the "instruction manual" (theoretical prediction) for how to spot these butterfly pairs. Here is the breakdown in simple terms:

1. The Problem: A Very Heavy Loop

To create two Higgs bosons, the particles smash together and create a temporary "loop" involving a top quark (the heaviest known particle).

• The Analogy: Imagine trying to predict the path of a ball rolling through a maze. The maze is made of the top quark. Because the top quark is so heavy, the maze is incredibly complex.
• The Old Way: For years, scientists used a shortcut called the "Heavy Top Limit." They pretended the top quark was infinitely heavy, which smoothed out the maze into a simple, flat floor. This made the math easier, but it wasn't perfectly accurate, especially when the particles were moving very fast.
• The New Way: This paper calculates the path through the actual maze (with the real weight of the top quark) but only for the first step of the journey (Next-to-Leading Order). However, for the main, complex parts of the journey, they use the "smooth floor" shortcut but calculate it to an unprecedented level of detail.

2. The Breakthrough: Calculating to "N3LO"

The paper reports the first-ever fully differential predictions at N3LO (Next-to-Next-to-Next-to-Leading Order).

• The Analogy: Think of calculating the weather.
  • LO (Leading Order): "It might rain." (Very rough guess).
  • NLO: "It will rain in the afternoon." (Better).
  • NNLO: "It will rain at 3 PM with 50% humidity." (Very good).
  • N3LO: "It will rain at 3:04 PM, with 50% humidity, and the drops will hit the ground at a 45-degree angle." (Extremely precise).
• What they did: They calculated the "weather" of the Higgs boson collision with this extreme level of precision. They didn't just calculate the total amount of rain (total cross-section); they calculated exactly where and how it falls (differential distributions), such as the speed and angle of the Higgs bosons.

3. The Results: Sharper Focus

• Reducing Uncertainty: Before this paper, the "forecast" had a big margin of error (like saying "it might rain between 10 AM and 6 PM"). The new N3LO calculations shrink that window significantly, reducing the uncertainty by about three times. Now, the prediction is precise enough to be in the "percent level."
• The Shape of the Storm: They found that while the total amount of "rain" didn't change much, the shape of the storm did. The new calculations change how the Higgs bosons are distributed in terms of their speed and direction. This is crucial because if the "forecast" (theory) doesn't match the "actual weather" (experiment), it could mean there is new physics hiding in the data.

4. Fixing the "Heavy Top" Shortcut

Since the "Heavy Top" shortcut isn't perfect when particles move fast, the authors combined their ultra-precise "smooth floor" calculations with a more accurate "real maze" calculation for the first step.

• The Analogy: Imagine you have a super-detailed map of a city (N3LO) but you know the map is slightly wrong about the height of the buildings. You take a rough, low-resolution photo of the actual buildings (NLO with real mass) and use it to correct the heights on your super-detailed map.
• The Outcome: This hybrid approach gives the most accurate picture of the Higgs boson pair production to date. They found that the "real mass" of the top quark significantly changes the predictions, especially for Higgs bosons moving at high speeds or in specific directions.

5. Why This Matters (According to the Paper)

The paper states that this level of precision is essential for ongoing experiments at the LHC.

• The Goal: Scientists are looking for signs that the Higgs potential (the energy field that gives particles mass) behaves differently than the Standard Model predicts.
• The Need: To find these tiny differences, you need a "ruler" (theoretical prediction) that is incredibly precise. If your ruler is blurry, you can't tell if the object you are measuring is slightly different or if your ruler is just wrong. This paper provides a much sharper ruler.

In Summary:
This paper is a masterclass in mathematical precision. It takes a notoriously difficult physics problem (two Higgs bosons created via a heavy top quark loop) and calculates it with the highest possible accuracy currently available. By refining the "map" of how these particles behave, it allows experimentalists at the LHC to look for new physics with much sharper eyes, reducing the "fog" of theoretical uncertainty that has obscured the view for years.

Technical Summary

Technical Summary: Fully Differential Higgs Boson Pair Production at N3LO with Top Quark Mass Effects

Problem and Motivation
Higgs-boson pair production ($gg \to hh$) via gluon-gluon fusion is the dominant channel for probing the Higgs potential and the trilinear self-coupling ($\lambda_{3h}$) at the Large Hadron Collider (LHC). While Next-to-Leading Order (NLO) QCD corrections are available with full top-quark mass dependence, they leave significant scale uncertainties (approx. $\pm 13\%$) and introduce large theoretical uncertainties related to the top-quark mass renormalization scheme (up to 30% for certain distributions). Higher-order calculations in the Heavy Top-Quark Limit (HTL), where $m_t \to \infty$, have been achieved up to Next-to-Next-to-Next-to-Leading Order (N3LO) for inclusive cross sections and the di-Higgs invariant mass distribution. However, fully differential predictions at N3LO in the HTL, particularly for other kinematic observables, were previously unavailable or only approximate. Furthermore, the HTL approximation breaks down when the partonic center-of-mass energy approaches $2m_t$, necessitating the inclusion of finite top-quark mass effects for reliable phenomenology.

Methodology and Computational Framework
The authors perform the first fully differential N3LO QCD calculation for $gg \to hh$ in the HTL, combined with NLO predictions including full top-quark mass dependence ($N_{LO}^{mt}$).

1. HTL Decomposition: The calculation in the HTL is organized using an effective Lagrangian where top-quark loops are integrated out into local contact interactions. The squared amplitude is decomposed into three classes based on the number of effective vertex insertions:

   • Class-a: Two effective vertices (dominant, $O(\alpha_s^2)$ at LO).
   • Class-b: Three effective vertices ($O(\alpha_s^3)$ at LO).
   • Class-c: Four effective vertices ($O(\alpha_s^4)$ at LO).
     To reach N3LO accuracy ($O(\alpha_s^5)$), the authors compute N3LO corrections for Class-a, NNLO corrections for Class-b, and NLO corrections for Class-c.

2. Computational Techniques:

   • Class-a: Calculated using the NNLOJET framework. The authors relate the di-Higgs cross section to the single-Higgs cross section (known at N3LO) via a kinematic mapping. They employ the $q_T$-slicing method to handle infrared divergences, separating the phase space into a low-$q_T$ region (computed via $q_T$-resummation in Soft-Collinear Effective Theory) and a high-$q_T$ region (computed as $hh + \text{jet}$ at NNLO).
   • Class-b: The NNLO corrections are computed using a combination of NNLOJET (for the low-$q_T$ resummation part) and MadGraph5_aMC@NLO (for the high-$q_T$ real radiation part), utilizing the FKS subtraction scheme.
   • Class-c: Computed entirely at NLO using MadGraph5_aMC@NLO with FKS subtraction.
   • Validation: The implementation is validated against known inclusive results from iHixs2 and differential results from MadGraph5_aMC@NLO at lower orders, showing perfect agreement. The independence of results from the slicing parameter $p_T^{\text{veto}}$ is verified, confirming the cancellation of power corrections.

3. Incorporating Top-Quark Mass Effects:
   To account for finite top-quark mass effects, the authors combine the HTL N3LO results with full $m_t$-dependent NLO results (computed using the ggxy generator). They evaluate three combination schemes:

   • Additive ($N_{kLO} \oplus N_{LO}^{mt}$): Adds HTL higher-order corrections to the full $m_t$ NLO result.
   • Born-improved ($N_{kLO}^{B-i} \oplus N_{LO}^{mt}$): Reweights HTL corrections by the ratio of LO full-$m_t$ to LO HTL cross sections.
   • Multiplicative ($N_{kLO} \otimes N_{LO}^{mt}$): Reweights HTL predictions by the ratio of NLO full-$m_t$ to NLO HTL cross sections.
     The paper identifies the multiplicative scheme as the most perturbatively stable for differential distributions.

Key Results
Calculations are presented for proton-proton collisions at $\sqrt{s} = 14$ TeV under realistic fiducial cuts ($p_{T,h1} > 30$ GeV, $p_{T,h2} > 20$ GeV, $|y_h| < 2.4$).

1. Fiducial Cross Sections (HTL):

   • The N3LO QCD corrections increase the NNLO fiducial cross section by approximately 3.8%.
   • Scale uncertainties are significantly reduced: from $\pm 18\%$ at NLO to $\pm 5.2\%$ at NNLO, and down to $+1.4\%/-2.9\%$ at N3LO. This represents a reduction by a factor of roughly three compared to NNLO.
   • Class-a contributions dominate ($\sim 103\%$ of the total), while Class-b and Class-c are negligible ($\sim -3.6\%$ and $\sim 0.03\%$, respectively).

2. Differential Distributions (HTL):

   • Invariant Mass ($m_{hh}$): The N3LO corrections are relatively flat ($K \approx 1.04$) except near the threshold ($m_{hh} \to 2m_h$), where scale uncertainties remain large.
   • Transverse Momentum ($p_T$): The $p_T$ spectra of the leading and subleading Higgs bosons show significant shape modifications at N3LO. In the low-$p_T$ region, fixed-order predictions are unstable (negative corrections, large uncertainties), indicating the necessity of resummation. In the high-$p_T$ region ($>100$ GeV), N3LO corrections modify the shape of the NNLO distributions.
   • Azimuthal Angle: The distribution $|\phi_{h1} - \phi_{h2}|/\pi$ shows excellent convergence between NNLO and N3LO, with corrections reaching $\sim 10\%$ in specific bins.
   • Rapidity: Rapidity distributions show flat $K$ factors and N3LO bands lying within NNLO uncertainty bands.

3. Finite Top-Mass Effects:

   • Combining HTL N3LO with full $m_t$ NLO (via the multiplicative scheme) reveals that finite top-mass effects are essential even at low invariant masses ($m_{hh} \approx 250$ GeV).
   • Compared to the HTL, the full $m_t$ results show a factor-of-two enhancement in the threshold region and a suppression in the high-mass tail.
   • The $p_T$ spectra are softened in the tails when full mass dependence is included.
   • The azimuthal angle distribution is suppressed by $>10\%$ in the back-to-back region and by a factor of $\sim 1.7$ in the collinear region.
   • Scale uncertainties for the combined predictions are generally small, though the $N_{LO}^{mt}$ scale dependence differs slightly from the HTL case.

Significance and Claims
The paper claims to provide the first fully differential N3LO predictions for Higgs-boson pair production in the HTL. By combining these with NLO full top-mass effects, the authors present "one of the most precise parton-level differential predictions to date" for LHC searches.

Key claims regarding the impact of this work include:

• Uncertainty Reduction: The N3LO QCD corrections reduce the scale uncertainties of NNLO fiducial and differential predictions by approximately a factor of three, bringing theoretical uncertainty to the percent level in the HTL.
• Shape Modifications: The results demonstrate that N3LO corrections are not merely a normalization factor; they significantly alter the shapes of transverse momentum and azimuthal angle distributions, highlighting the necessity of including N3LO corrections for precision phenomenology.
• Top-Mass Necessity: The study confirms that finite top-quark mass effects cannot be neglected even at the lowest invariant masses and must be included at the highest available order (NLO in this context) to avoid significant distortions in differential distributions.

The authors modestly note that while scale uncertainties are under control, other theoretical uncertainties remain, including missing finite-$m_t$ corrections at NNLO and beyond, top-quark mass scheme uncertainties (OS vs. $\overline{MS}$), missing higher-order electroweak corrections, and bottom-quark loop contributions. They suggest that future work extending full mass dependence to NNLO could further reduce these uncertainties.