Higgs Decays at NLO in the SMEFT

This paper presents a comprehensive calculation of two-, three-, and four-body Higgs decay rates at next-to-leading order in both QCD and electroweak interactions within the dimension-6 SMEFT framework, implemented in the publicly available Monte Carlo program NEWiSH to assess the impact of these corrections on future collider projections.

The Gist

Imagine the Higgs boson as the universe's most famous "party crasher." It's a tiny, fleeting particle that gives mass to everything else, but it only exists for a split second before it bursts apart into other particles. Physicists at the Large Hadron Collider (LHC) and future colliders are obsessed with watching these parties to see if the crasher behaves exactly as predicted by the Standard Model (our current rulebook of physics) or if it's doing something weird that hints at New Physics.

This paper is like a massive, ultra-precise instruction manual for predicting exactly how the Higgs crashes a party, but with a twist: it accounts for the possibility that the rulebook itself might have hidden, tiny errors.

Here is the breakdown of what the authors did, using some everyday analogies:

1. The Goal: Measuring the "Fuzziness" of Reality

Think of the Standard Model as a high-definition map of a city. For years, it's been accurate enough to get us from point A to point B. But now, we have a new, super-powerful GPS (the HL-LHC and future colliders) that can see the map in 8K resolution.

At this new level of detail, we can't just use the old map. We need to account for the "fuzziness" of the road—tiny bumps, wind, and traffic that we ignored before. In physics terms, we need Next-to-Leading Order (NLO) calculations. This means calculating not just the main path, but also the tiny detours and side-streets (quantum loops) that particles take.

2. The Tool: The "SMEFT" (The Rulebook with Extra Pages)

The authors use a framework called SMEFT (Standard Model Effective Field Theory).

• The Analogy: Imagine the Standard Model is a cookbook with 100 recipes. We know these recipes work perfectly. But what if there are secret ingredients (New Physics) that are so expensive and rare that we can't buy them directly? We can only taste their effect in the final dish.
• SMEFT is like adding a "Notes" section to the cookbook. It says, "If you add a pinch of this invisible spice (a dimension-6 operator), the cake might rise 1% more or taste slightly sweeter."
• The paper calculates exactly how much the cake rises for every possible recipe (decay channel) when you add these invisible spices, including the tiny ripples in the batter (quantum corrections).

3. The Calculation: Counting Every Crumb

The authors calculated the decay rates for the Higgs boson breaking into:

• Two-body decays: Like a balloon popping into two pieces (e.g., Higgs $\to$ two photons).
• Three-body decays: Like a balloon popping into three pieces (e.g., Higgs $\to$ a Z boson and two electrons).
• Four-body decays: Like a balloon popping into four pieces.

The Challenge: Calculating these "pop" events is hard because particles interact with each other in complex ways.

• The "Narrow Width Approximation" (NWA): For the four-body decays, the authors used a clever shortcut. Imagine a domino effect: The Higgs breaks into a Z boson and two electrons. The Z boson is unstable and immediately breaks into two more electrons. Instead of calculating the whole chaotic explosion at once, they calculated the first break, then the second break, and multiplied them. It's like calculating the odds of a domino falling, then the next one falling, rather than simulating the whole chain reaction in one go. They proved this shortcut works well enough for most cases.

4. The Result: The "NEWiSH" Code

The authors didn't just write down numbers; they built a software tool called NEWiSH (New Standard Model Higgs).

• The Analogy: Think of this as a "Higgs Simulator" app. If you are a physicist, you can plug in your theory (how strong the invisible spices are), and the app tells you exactly what the Higgs decay rates should look like, including all the tiny quantum corrections.
• This tool is public, meaning anyone can download it to check their own theories against the data.

5. Why Does This Matter? (The "Detective Work")

The paper shows that when you include these ultra-precise corrections, things get correlated.

• The Analogy: Imagine you are trying to figure out if a suspect is guilty. If you only look at one piece of evidence (like a fingerprint), you might be wrong. But if you look at the fingerprint, the shoe print, the DNA, and the alibi all together, the picture becomes clear.
• The authors found that many different "spices" (operators) affect the Higgs in similar ways. If you only look at one decay channel, you might think a specific spice is present. But when you look at all the decays together (including the new NLO corrections), you realize that the "flavor" might actually come from a different spice.
• Future Colliders: They compared the current LHC (HL-LHC) with future "Z-factories" (Tera-Z). They found that for some mysteries, the LHC is great. But for others, the future Z-factories (which produce billions of Z bosons) will be the ultimate detectives, especially when combined with the new, precise Higgs data.

Summary

This paper is a masterclass in precision. The authors have updated the "rulebook" for how the Higgs boson decays, accounting for the tiniest quantum effects. They provided a free software tool (NEWiSH) so that physicists worldwide can use these new, ultra-precise predictions to hunt for New Physics.

The Takeaway: We are no longer just guessing if the Higgs is behaving normally; we are now measuring its behavior with such precision that even the tiniest deviation from the rulebook will be impossible to hide. It's the difference between looking at a blurry photo of a crime scene and having a high-definition, 3D reconstruction of every single detail.

Technical Summary

1. Problem Statement

Precise measurements of Higgs boson properties are central to the physics programs of the High-Luminosity LHC (HL-LHC) and future lepton colliders (e.g., FCC-ee, ILC). As experimental uncertainties decrease, theoretical predictions must match this precision to effectively distinguish Standard Model (SM) physics from Beyond the Standard Model (BSM) effects.

While Quantum Chromodynamics (QCD) corrections to dimension-6 Standard Model Effective Field Theory (SMEFT) processes are largely automated, Next-to-Leading Order (NLO) electroweak (EW) corrections in the SMEFT have historically been calculated on a case-by-case basis. There was a lack of a comprehensive, consistent calculation covering all two- and three-body Higgs decay channels at NLO in both QCD and EW interactions within the dimension-6 SMEFT framework. This gap limits the ability to perform global fits of SMEFT coefficients and to accurately interpret loop-induced operators (such as those affecting the Higgs trilinear coupling) that do not appear at tree level.

2. Methodology

The authors performed a complete NLO calculation for Higgs decays using the Warsaw basis of the dimension-6 SMEFT.

• Framework: The calculation truncates the SMEFT expansion at dimension-6 ($\mathcal{O}(1/\Lambda^2)$), neglecting dimension-5 operators (lepton number violation) and dimension-8 terms.
• Renormalization:
  • Two schemes were employed: an On-Shell (OS) scheme for SM parameters and an $\overline{\text{MS}}$ scheme for SMEFT Wilson coefficients.
  • A second approach used the $\overline{\text{MS}}$ scheme for light quark masses to assess scheme dependence.
  • Tadpole contributions were set to zero to equate the renormalized VEV with the potential minimum.
  • The Complex Mass Scheme was utilized for unstable particles (W, Z) to ensure gauge invariance, particularly crucial for processes where vector bosons go on-shell in real emission corrections.
• Calculation Tools:
  • Amplitudes were generated using the chain: FeynRules $\to$ FeynArts $\to$ FeynCalc.
  • Loop integrals were evaluated using Package-X and Collier, expressed in terms of Passarino-Veltman functions.
  • Dipole Subtraction: Real emission contributions ($h \to X + \gamma/g$) were handled using massive and massless dipole subtraction techniques to cancel infrared (IR) divergences. Special care was taken for non-inclusive observables (e.g., differential distributions in $h \to Z\ell^+\ell^-$) where lepton masses act as regulators.
• Four-Body Decays: For $h \to 4f$ processes, the authors combined their NLO three-body results ($h \to f\bar{f}V$) with known NLO results for $V \to f\bar{f}$ using the Narrow Width Approximation (NWA). They also provided full LO four-body results using the complex mass scheme to validate the NWA.
• Implementation: All results are encapsulated in a public Monte Carlo code named NEWiSH.

3. Key Contributions

• Comprehensive NLO Coverage: The paper provides the first complete set of NLO (QCD + EW) results for all two- and three-body Higgs decays in the CP-conserving dimension-6 SMEFT.
• New Results: It presents new NLO calculations for:
  • $h \to W f \bar{f}'$ (including both leptonic and hadronic modes).
  • $h \to Z q \bar{q}$ (quark modes).
  • $h \to Z \nu \bar{\nu}$.
• Flavor Treatment: The calculation includes all possible flavor interactions but drops contributions proportional to off-diagonal CKM matrix elements. It accounts for non-zero masses for top, bottom, charm, strange, tau, and muon where numerically significant.
• Differential Distributions: The code provides differential distributions (e.g., $d\Gamma/dm_{\ell\ell}$) for $h \to Z\ell^+\ell^-$ and $h \to 4\ell$, allowing for the application of experimental kinematic cuts.
• Public Code (NEWiSH): The release of the NEWiSH code allows the community to compute total widths and distributions for any combination of dimension-6 operators at one-loop order.

4. Key Results

• Numerical Predictions: The authors provide explicit numerical formulas for the partial widths $\Gamma_{\text{NLO}} = \Gamma_{\text{LO}} + \delta\Gamma_{\text{NLO}}$ for all relevant channels ($h \to Z\ell\ell, Z\nu\nu, Zq\bar{q}, W\ell\nu, Wq\bar{q}, f\bar{f}, VV', ggZ$).
• NWA Validity: The study confirms that the NWA works well (within $\sim 10\%$) for most operators in total $h \to 4f$ widths. However, for specific fermionic operators and differential distributions (e.g., $C_{\phi WB}$ in $4\mu$ channels), the NWA can fail significantly due to off-shell and interference effects.
• HL-LHC Projections:
  • NLO corrections introduce strong correlations between operators.
  • Including NLO EW corrections is essential for constraining the Higgs trilinear coupling ($C_\phi$), which only appears at one-loop in single Higgs production.
  • The combination of HL-LHC data with Electroweak Precision Observables (EWPOs) significantly tightens constraints on operator pairs like $(C_\phi, C_{\phi D})$ and $(C_\phi, C_{bW})$.
• Future Colliders (Tera-Z vs. Higgstrahlung):
  • Tree-Level Operators: For operators contributing to $Z$-pole observables at tree level (e.g., $C_{\phi D}, C_{\phi WB}$), Tera-Z data provides the strongest constraints. Higgs decay data offers marginal improvement.
  • Loop-Level Operators: For operators that only appear at loop level in $Z$ production (e.g., $C_{\phi B}, C_{\phi W}$), the combination of Higgstrahlung ($e^+e^- \to Zh$) production with NLO Higgs decay measurements improves sensitivity by one to two orders of magnitude compared to Tera-Z alone.
• Case Study (Scalar Singlet Model):
  • The authors matched a BSM scalar singlet model to the SMEFT.
  • They demonstrated that projected mass reach limits are highly sensitive to assumptions about the model's dimensional parameters (e.g., $m_\xi, m_\zeta$) and the symmetry breaking pattern ($Z_2$ symmetry).
  • NLO matching significantly alters the projected mass reach compared to LO matching, highlighting the necessity of higher-order calculations for reliable BSM exclusion limits.

5. Significance

This work represents a critical step toward NLO-accurate global fits of the SMEFT.

1. Precision Physics: It enables the interpretation of future collider data (HL-LHC, FCC-ee) with theoretical uncertainties reduced to match experimental precision.
2. Loop-Induced Sensitivity: It unlocks the sensitivity to operators that are invisible at tree level in certain processes, particularly those related to the Higgs self-coupling and loop-induced decays ($h \to \gamma\gamma, Z\gamma, gg$).
3. Theoretical Consistency: By providing a consistent framework where production and decay are treated at the same perturbative order, it reduces ambiguities in extracting Wilson coefficients.
4. Community Resource: The public availability of NEWiSH facilitates global analyses, allowing researchers to incorporate NLO EW/QCD effects into their global SMEFT fits without needing to re-calculate complex loop amplitudes.

The paper concludes that while Tera-Z data is powerful for tree-level operators, the full potential of future colliders to probe high-scale new physics (via loop-induced operators) relies heavily on the inclusion of NLO Higgs decay calculations, as demonstrated by the dramatic improvement in sensitivity when combining Higgstrahlung production with NLO decay channels.