Renormalization and Factorization Scale-Invariant Predictions for the Higgs Rare Decay $H\to J/ψ+γ$ via the Principle of Maximum Conformality

This paper applies the Principle of Maximum Conformality to the next-to-next-to-leading-order perturbative QCD calculation of the rare Higgs decay $H\to J/\psi+\gamma$, successfully eliminating renormalization and factorization scale ambiguities to provide a robust, scale-invariant prediction for the decay width that offers a precise probe for the charm-quark Yukawa coupling.

The Gist

Imagine you are a detective trying to solve a very specific, rare crime: a Higgs boson (the "God Particle") decaying into a charm-quark pair (which forms a particle called $J/\psi$) and a photon (a particle of light).

This crime is fascinating because it could tell us exactly how heavy the charm quark is and how strongly it interacts with the Higgs. However, the math used to predict how often this happens is incredibly messy. It's like trying to calculate the exact trajectory of a rocket, but your calculator keeps giving you different answers depending on what time of day you press the "equals" button.

Here is a simple breakdown of what this paper does to fix that mess.

1. The Problem: The "Arbitrary Ruler"

In physics, when we calculate these rare events, we use a method called perturbation theory. Think of this as building a tower block by block.

• The Blocks: Each layer of the tower represents a more complex calculation (Leading Order, Next-to-Leading Order, etc.).
• The Wobble: The problem is that the size of the blocks depends on a "scale" (a variable physicists call $\mu$). In the old way of doing things, physicists had to guess what this scale should be.
  • Analogy: Imagine you are baking a cake and the recipe says, "Add sugar to taste." If you guess "a little," the cake is bland. If you guess "a lot," it's too sweet. The old method was like guessing the sugar amount. If you guessed a different "scale" (time of day), you got a different cake (prediction). This made the predictions unreliable.

2. The Solution: The "Maximum Conformality" (PMC) Rule

The authors of this paper used a new method called the Principle of Maximum Conformality (PMC).

• The Analogy: Instead of guessing the sugar amount, PMC is like having a smart kitchen scale that automatically adjusts the ingredients based on the exact chemistry of the batter. It removes the "guesswork" entirely.
• How it works: The math has certain "noise" terms (called $\beta$-terms) that cause the scale-dependence. PMC identifies these noise terms and absorbs them into the definition of the strong force itself.
• The Result: Once you do this, the "ruler" disappears. The prediction no longer changes based on your arbitrary choice. It becomes scale-invariant. It's like finding the true weight of the cake, regardless of who is weighing it.

3. The "Two-Step" Cleanup

The paper had to clean up two different types of mess:

1. The Renormalization Scale (The "Time" Problem): This is the main "guessing" problem mentioned above. PMC fixes this by setting the "effective" energy of the interaction to the exact right value (about 9.17 GeV in this case), rather than letting the physicist pick a random number.
2. The Factorization Scale (The "Distance" Problem): This is a trickier, newer application. In quantum mechanics, particles interact at different distances. Usually, the math changes depending on how you define "short distance" vs. "long distance."
   • The Innovation: This paper is the first to show that PMC can also fix this second problem. They treated the "long-distance" part of the math (the $J/\psi$ structure) and the "short-distance" part (the Higgs interaction) separately, using their own internal rules to cancel out the dependence.
   • Analogy: It's like realizing that the distance between two cities doesn't change just because you switch from measuring in miles to kilometers. PMC ensures the answer is the same no matter which "unit" you use.

4. The Verdict: A Precise Prediction

By using this "smart scale" (PMC) and cleaning up both types of mess, the authors got a very clear, stable result.

• Old Way: The prediction was wobbly. If you changed the scale, the result could swing wildly (like a pendulum). The uncertainty was huge.
• New Way (PMC): The prediction is a solid, flat line.
  • The Number: They predict the decay happens at a rate of roughly $6.46 \times 10^{-11}$ GeV.
  • The Confidence: The "error bars" (uncertainty) are tiny and symmetric. It's no longer a guess; it's a precise measurement of what should happen.

5. Why Does This Matter?

The Higgs boson is the only fundamental particle that gives mass to other particles. We know it interacts heavily with heavy particles (like the top quark), but we are still fuzzy on how it interacts with lighter ones (like the charm quark).

• The Goal: If we can measure this rare decay ($H \to J/\psi + \gamma$) in a real lab (like the Large Hadron Collider), we can compare the real data to this new, ultra-precise prediction.
• The Payoff: If the real data matches the prediction, we confirm our understanding of the Standard Model. If it doesn't match, it's a smoking gun for New Physics—something beyond our current understanding of the universe.

Summary

This paper is like upgrading from a hand-drawn map (where the scale depends on the artist's mood) to a GPS (which calculates the exact route based on real-time data). By applying the Principle of Maximum Conformality, the authors removed the "guesswork" from the math, giving physicists a crystal-clear target to aim for when hunting for new secrets of the universe.

Technical Summary

Here is a detailed technical summary of the paper "Renormalization and Factorization Scale-Invariant Predictions for the Higgs Rare Decay $H \to J/\psi + \gamma$ via the Principle of Maximum Conformality."

1. Problem Statement

The paper addresses the theoretical challenges in predicting the rare exclusive Higgs decay $H \to J/\psi + \gamma$ via the direct production mechanism. This decay channel is a crucial probe for extracting the charm-quark Yukawa coupling ($Hc\bar{c}$), which remains poorly constrained experimentally compared to third-generation couplings.

Key theoretical issues identified in previous calculations include:

• Renormalization Scale ($\mu_r$) Ambiguity: Conventional perturbative QCD (pQCD) predictions at Next-to-Next-to-Leading Order (N2LO) exhibit strong dependence on the arbitrary choice of the renormalization scale, leading to large theoretical uncertainties.
• Factorization Scale ($\mu_\Lambda$) Dependence: The decay width depends on the factorization scale due to the separation of short-distance coefficients (SDCs) and long-distance matrix elements (LDMEs). Conventional treatments often rely on potential-model wave functions or unified LDMEs that do not properly account for scale evolution.
• Poor Convergence: The perturbative series suffers from large collinear logarithms ($\alpha_s^n \ln^n(m_H^2/m_c^2)$) and renormalon terms ($n! \beta_0^n \alpha_s^n$), which degrade the convergence of the series and can even lead to unphysical results (e.g., negative decay widths) at certain scales.

2. Methodology

The authors employ the Principle of Maximum Conformality (PMC) to systematically eliminate scale ambiguities and improve the convergence of the pQCD series. The methodology involves three distinct steps:

A. Resummation of Yukawa Coupling Logarithms

The large logarithms associated with the renormalization of the charm-quark Yukawa coupling are resummed. This is achieved by converting the charm-quark pole mass ($m_c$) in the Yukawa coupling to the $\overline{\text{MS}}$-scheme running mass evaluated at the Higgs mass scale, $m_c(m_H)$. This step ensures the correct running behavior of the coupling.

B. Extraction of Scale-Independent LDME

Instead of using model-dependent wave functions, the authors extract the factorization-scale-dependent color-singlet LDME, $\langle J/\psi(\epsilon)|\psi^\dagger \sigma \cdot \epsilon \chi(\mu_\Lambda)|0\rangle$, from the experimentally measured leptonic decay width of the $J/\psi$ ($\Gamma_{J/\psi \to e^+e^-}$).

• By utilizing the Renormalization Group Equation (RGE) of the LDME, they identify and absorb the $\{\beta_i\}$-terms associated with the LDME's scale evolution.
• This procedure isolates the physical, factorization-scale-independent component of the series, effectively eliminating the $\mu_\Lambda$ dependence in the final prediction.

C. PMC Scale Setting for Strong Coupling

The remaining perturbative series for the decay width is treated using the PMC.

• The $\{\beta_i\}$-terms (non-conformal contributions arising from renormalization) in the N2LO coefficients are identified and absorbed into the running of the strong coupling constant $\alpha_s$.
• This determines an effective, process-dependent PMC scale ($Q^*$), fixing the value of $\alpha_s$ at each order.
• The resulting series becomes conformal (independent of the initial renormalization scale choice) and renormalization-scheme invariant.

D. Uncertainty Estimation via Bayesian Analysis

To estimate the impact of unknown higher-order (UHO) contributions (specifically N3LO), the authors apply a Bayesian Analysis (BA).

• Using the converged PMC series, they assign a prior probability distribution to the unknown coefficients.
• This allows for a robust quantification of the theoretical uncertainty associated with truncating the series at N2LO.

3. Key Contributions

1. First Application of PMC to Factorization Scales: This work demonstrates, for the first time, how the PMC strategy can be extended to eliminate factorization-scale dependence by treating the RGEs of the LDME separately from the strong coupling.
2. Resolution of Scale Ambiguities: The method successfully removes both renormalization and factorization scale ambiguities, yielding a prediction that is invariant under scale variations.
3. Improved Convergence: By absorbing renormalon terms into the coupling, the PMC series exhibits significantly better convergence properties compared to the conventional scale-setting method.
4. Robust Uncertainty Quantification: The combination of PMC and Bayesian analysis provides a more reliable estimate of UHO uncertainties than traditional scale-variation methods.

4. Results

The authors calculate the total decay width for the direct production mechanism, $\Gamma_{\text{dir}}(H \to J/\psi + \gamma)$, at N2LO accuracy.

• Conventional Method:

  • Shows significant scale dependence. For $\mu_r \in [m_{J/\psi}, m_H]$, the scale uncertainty is approximately 16.31%.
  • Result: $\Gamma_{\text{dir}}^{\text{Conv}} = (6.7500^{+0.2893}_{-2.6522}) \times 10^{-11} \text{ GeV}$.
  • The large negative uncertainty arises from the divergence of the series at low scales.

• PMC Method:

  • The prediction is scale-invariant. The effective PMC scale is determined to be $Q^*_{\text{LL}} = 9.1706 \text{ GeV}$.
  • The convergence is characterized by $\kappa$-factors: $\kappa_{\text{PMC}} = \{1, 0.1613, 0.1231\}$, indicating a stable series compared to the conventional method.
  • Final Prediction:
    $$\Gamma_{\text{dir}}^{\text{PMC}} = (6.4574 \pm 0.3995) \times 10^{-11} \text{ GeV}$$
  • The total uncertainty ($\pm 0.3995 \times 10^{-11} \text{ GeV}$) is the quadratic sum of contributions from:
    • $\Delta \alpha_s(m_Z) = \pm 0.0009$
    • $\Delta \Gamma_{J/\psi \to e^+e^-} = \pm 0.10 \text{ keV}$
    • $\Delta m_c(m_c) = \pm 0.0046 \text{ GeV}$
    • Estimated N3LO contributions via Bayesian analysis.

5. Significance

• Precision Physics: The result provides a highly precise, scale-invariant theoretical baseline for the $H \to J/\psi + \gamma$ decay. This is essential for future high-luminosity experiments (LHC, Higgs factories) to accurately extract the $Hc\bar{c}$ Yukawa coupling.
• Validation of PMC: The study validates the PMC as a powerful tool for handling complex processes involving multiple scales (masses of Higgs, $J/\psi$, and charm quark) and multiple types of running parameters (coupling, mass, LDME).
• New Standard for Theory: By eliminating artificial scale dependencies, this approach sets a new standard for theoretical predictions in perturbative QCD, reducing the "theoretical noise" that often obscures signals of new physics or precise Standard Model tests.