Higgs boson decay to massive bottom quarks at order $α_s^4$ induced by top-quark Yukawa couplings

This paper presents a calculation of the $\mathcal{O}(\alpha_s^4)$ corrections to the Higgs boson decay into massive bottom quarks induced by top-quark Yukawa couplings, finding that these contributions increase the decay width by 0.4% and significantly reduce scale dependence to 0.4%, thereby surpassing the precision expected at future lepton colliders.

The Gist

Imagine the Higgs boson as a very heavy, unstable celebrity at a party. It doesn't stay around for long; it immediately breaks apart into smaller particles. The most common way it does this is by splitting into a pair of bottom quarks (let's call them "bottoms"). In fact, about 58% of the time, this is exactly what happens.

Physicists want to predict exactly how often this happens (the "decay rate") with extreme precision. Why? Because if our prediction is slightly off, it might mean we are missing a piece of the puzzle about how the universe works.

Here is the story of this paper, explained without the heavy math:

1. The Problem: A Slow-Moving Puzzle

For decades, physicists have been trying to calculate this decay rate. They use a method called "perturbation theory," which is like building a tower of blocks.

• Level 1 (The Basics): You build the first block.
• Level 2 (Better): You add a second layer to get a more accurate shape.
• Level 3, 4, etc.: You keep adding layers to get it perfect.

Usually, each new layer makes the prediction slightly better, and the changes get smaller and smaller. But in this specific case (the Higgs decaying into bottoms), there's a weird glitch.

The "bottoms" are heavy, but there's an even heavier particle involved in the background: the top quark. Even though the top quark isn't in the final pile of debris, its "ghost" (via a force called the Yukawa coupling) influences the process.

When physicists tried to calculate the influence of this heavy top quark, they found that the math was behaving badly. Instead of the corrections getting smaller, they were getting bigger because of some tricky mathematical "logarithms" (think of these as runaway feedback loops). The tower of blocks was wobbling, and the prediction wasn't stable enough for the super-precise experiments planned for the future.

2. The Solution: The Fourth Layer

This paper is about calculating the fourth layer of the tower (technically called the $O(\alpha_s^4)$ correction).

Think of the calculation like tuning a radio.

• Previous attempts: You were close to the station, but there was static (uncertainty).
• This paper: The authors (Wang, Wang, and Wang) found a way to tune the radio perfectly, removing the static caused by the heavy top quark's influence.

They focused on a specific part of the calculation where the top quark's influence interacts with the bottom quarks in a complex way (the $C_1C_1$ channel). This was the hardest part to solve because it involved "massive" particles that made the math incredibly messy, like trying to solve a Rubik's cube while it's spinning.

3. The "Aha!" Moment

When they finally cracked the code and added this fourth layer, two amazing things happened:

1. The Prediction Shifted: The calculated rate of decay increased by 0.4%.

   • Why this matters: Future particle colliders (like a "Higgs Factory") are going to be so precise they can measure changes as small as 0.2%. If the theorists don't get this 0.4% right, the experimentalists will think they've discovered new physics when they've actually just made a math error. This paper ensures the math is right.

2. The Stability Improved: Before this calculation, the result depended heavily on an arbitrary choice of "scale" (like measuring a room with a ruler that stretches and shrinks). This uncertainty was about 0.7%. After adding this new layer, the uncertainty dropped to 0.4%.

   • Analogy: It's like going from guessing the weight of a suitcase by eye (wobbly) to weighing it on a high-tech scale (stable).

4. The Real-World Impact

Why do we care about a 0.4% change?

• Weighing the Invisible: The bottom quark is too heavy to weigh directly on a scale. Instead, we infer its mass by watching how the Higgs boson decays.
• The Result: Because this paper made the Higgs decay prediction so much more precise, we can now use future experiments to measure the mass of the bottom quark with a precision of 0.36%. That is incredibly precise!

Summary

Think of this paper as the final polish on a diamond. The diamond (the Higgs decay) was already beautiful, but it had some scratches (mathematical uncertainties) caused by the heavy top quark. The authors used advanced mathematical tools (like "Master Integrals" and "Elliptic functions"—imagine them as specialized polishing cloths) to buff out those scratches.

Now, when the next generation of particle accelerators turns on, they will have a crystal-clear theoretical map to compare their data against, ensuring that any new discoveries are real and not just a result of messy math.

Technical Summary

1. Problem Statement

The decay of the Higgs boson into a bottom-antibottom quark pair ($H \to b\bar{b}$) is the dominant decay channel in the Standard Model (SM). Precision measurements of the bottom-quark Yukawa coupling at future colliders (e.g., HL-LHC, $e^+e^-$ Higgs factories) require theoretical predictions with matching precision (sub-percent level).

While the dominant contribution arises from the bottom-quark Yukawa coupling ($C_2$), the top-quark Yukawa coupling ($C_1$) induces significant corrections. Specifically:

• The interference term ($C_1 C_2$) and the pure top-induced term ($C_1 C_1$) exhibit logarithmic and power enhancements (e.g., $\log^j(m_H^2/m_b^2)$) due to soft massive quark effects.
• Previous calculations up to Next-to-Next-to-Leading Order (NNLO) and Next-to-Next-to-Next-to-Leading Order (N3LO) showed that the $C_1 C_1$ contribution increases the decay width by $\sim 1\%$ at N3LO, exceeding naive perturbative expectations.
• The Gap: A complete calculation of the fourth-order QCD corrections ($O(\alpha_s^4)$, or N4LO) for the $C_1 C_1$ channel, including full dependence on the finite bottom-quark mass ($m_b$), was missing. This is crucial for reducing scale uncertainty and testing the convergence of the perturbative series.

2. Methodology

The authors employed an Effective Field Theory (EFT) framework where the top quark is integrated out.

• Effective Lagrangian: The calculation utilizes an effective Lagrangian with two renormalized operators:

  • $O_1 = (G_{\mu\nu}^a)^2$ (Gluonic operator, induced by top quark).
  • $O_2 = m_b \bar{b}b$ (Bottom Yukawa operator).
    The decay width is decomposed into three channels based on Wilson coefficient combinations: $\Gamma_{C_2 C_2}$, $\Gamma_{C_1 C_2}$, and $\Gamma_{C_1 C_1}$. This paper focuses on the $C_1 C_1$ channel at $O(\alpha_s^4)$.

• Optical Theorem: The decay width is calculated via the imaginary part of the forward scattering amplitude $H \to b\bar{b} \to H$:
  $$\Gamma_{H \to b\bar{b}} = \frac{1}{m_H} \text{Im}_{b\bar{b}} (\Sigma)$$
  The calculation involves three-loop Feynman diagrams containing two $O_1$ vertices.

• Master Integrals (MIs):

  • The amplitudes were reduced to a set of 38 Master Integrals (MIs) using Integration-by-Parts (IBP) identities (via the package Kira).
  • The MIs were categorized into two topological sectors:
    1. $\tilde{M}_{i, b\bar{b}}$: Contributions from final states $b\bar{b}, b\bar{b}g, b\bar{b}gg, b\bar{b}q\bar{q}$. These were solved using canonical differential equations, rationalizing square roots via variable transformations ($z \to w, y$), yielding results in terms of Multiple Polylogarithms (MPLs).
    2. $M_{i, 4b}$: Contributions from the $b\bar{b}b\bar{b}$ final state. These involve elliptic integrals. The authors utilized a regular basis where only the finite ($O(\epsilon^0)$) parts are required. These were expressed as one-fold or two-fold integrals involving complete elliptic integrals of the first ($K$) and second ($E$) kinds.
  • New Integrals: Three new MIs ($M_{36}, M_{37}, M_{38}$) were computed analytically. $M_{36}$ was solved via MPLs, while $M_{37}$ and $M_{38}$ (the "top sector") were solved using differential equations leading to elliptic integral representations.

• Asymptotic Expansion: Analytic results were derived for the limit $z = m_H^2/m_b^2 \to \infty$. The authors verified that power corrections ($O(z^{-1})$) are negligible ($<0.1\%$), justifying the use of asymptotic forms for phenomenology.

3. Key Contributions

1. First $O(\alpha_s^4)$ Calculation for $C_1 C_1$: The paper provides the first analytic calculation of the N4LO QCD corrections to the $H \to b\bar{b}$ decay width specifically for the top-quark induced $C_1 C_1$ channel, retaining full $m_b$ dependence.
2. Elliptic Master Integrals: The successful analytic computation of the top-sector master integrals ($M_{37}, M_{38}$) which involve elliptic structures, demonstrating a practical application of $\epsilon$-factorized differential equations for observables with non-trivial geometric objects.
3. Power Enhancement Analysis: The authors explicitly characterize the power-enhanced logarithmic structure ($\log^4 z$) in the $C_1 C_1$ channel, distinguishing it from the $C_1 C_2$ and $C_2 C_2$ channels.

4. Results

Using input parameters ($m_H = 125.09$ GeV, $m_b(m_b) = 4.18$ GeV, $\alpha_s(m_Z) = 0.1180$), the numerical results are:

• Magnitude of Correction: The $O(\alpha_s^4)$ correction in the $C_1 C_1$ channel increases the decay width by 0.4%.
  • This is larger than the $O(\alpha_s^4)$ correction in the dominant $C_2 C_2$ channel (which decreases the width by 0.1%).
  • The magnitude (0.4%) exceeds the projected experimental precision of future lepton colliders (0.21%).
• Scale Uncertainty Reduction: Including these corrections significantly reduces the renormalization scale dependence.
  • Scale uncertainty drops from 0.7% at N3LO to 0.4% at N4LO.
• Final Prediction: The total decay width in the $\overline{\text{MS}}$ scheme is:
  $$\Gamma_{H \to b\bar{b}}^{\text{N4LO}} = 2.421^{+0.008}_{-0.010}(\text{scale}) \pm 0.005(\alpha_s) \text{ MeV}$$
• Bottom Quark Mass Extraction: If the decay width is measured with 0.21% precision, the bottom quark mass $m_b(m_H)$ can be extracted with a theoretical precision of approximately 0.36%.

5. Significance

• Theoretical Precision: This work bridges a critical gap in the perturbative expansion of the Higgs decay width, ensuring that theoretical uncertainties do not limit the interpretation of future high-precision data from the HL-LHC or future Higgs factories.
• Convergence of Perturbation Series: The result confirms that while the $C_1 C_1$ channel is power-enhanced, the inclusion of higher orders stabilizes the prediction and reduces scale dependence, validating the perturbative approach even in the presence of large logarithms.
• Phenomenological Impact: The 0.4% shift is non-negligible for precision Higgs physics. Ignoring this term would lead to a bias in the extraction of the bottom Yukawa coupling and the bottom quark mass.
• Future Work: The authors note that the remaining $O(\alpha_s^4)$ correction in the $C_1 C_2$ interference channel is the next logical step to achieve a complete N4LO prediction for the total width.