Next-to-leading order QCD and relativistic corrections to $Z \to J/\psi+\Upsilon(nS)$

This paper calculates the decay widths and branching fractions for $Z \to J/\psi+\Upsilon(nS)$ at future super $Z$ factories within the nonrelativistic QCD framework, revealing that large negative relativistic and QCD corrections significantly reduce the leading-order predictions and are essential for reliable estimations of these rare decay channels.

The Gist

Imagine the universe is a giant, high-energy dance floor. At the center of this floor spins the Z-boson, a fundamental particle that acts like a massive, energetic DJ. Occasionally, this DJ decides to "spin out" two new dancers: a J/ψ (a charm quark pair) and an Υ (a bottom quark pair).

This paper is about predicting exactly how often this rare dance happens and how fast the music plays, but with a twist: the authors realized that the old predictions were like using a blurry, low-resolution map. They decided to redraw the map with high-definition precision, accounting for two major sources of "noise" that were previously ignored.

Here is the breakdown of their work using simple analogies:

1. The "Blurry Map" vs. The "High-Def Map"

In physics, scientists use math to predict how particles behave.

• The Old Way (Leading Order): Imagine trying to predict the path of a car driving through a city by only looking at a straight line on a map. You ignore traffic lights, potholes, and detours. This gives you a rough idea, but it's not accurate. In physics, this is called the "Leading Order" (LO) calculation.
• The New Way (Next-to-Leading Order + Relativistic): The authors in this paper said, "Let's add the traffic lights and potholes!"
  • QCD Corrections (The Traffic): They added the effects of the "strong force" (gluons), which act like heavy traffic jams or sudden detours that slow the particles down.
  • Relativistic Corrections (The Speed): They also accounted for the fact that these particles are moving incredibly fast (close to the speed of light), which changes their mass and behavior, much like how a fast-moving object looks different than a slow one.

2. The Big Surprise: The "Double Negative"

When the authors added these high-definition corrections to their calculations, they found something shocking: The dance happens much less often than we thought.

• The Analogy: Imagine you estimated that 100 people would show up to a party. Then, you realized that half the people got stuck in traffic (QCD correction) and the other half decided to stay home because they were too tired from running too fast (Relativistic correction).
• The Result: Instead of 100 people, maybe only 15 show up.
• The Paper's Finding: Both the "traffic" (QCD) and the "speed" (Relativistic) corrections were large and negative. They canceled out most of the original prediction. If you only used the old "blurry map," you would have been wildly optimistic about how often this event happens.

3. Why Does This Matter? (The "Super Z Factory")

The authors are writing this for the future. They are looking ahead to massive new particle colliders (like the CEPC or FCC-ee) that will act as "Super Z Factories."

• The Analogy: Think of these factories as a machine that produces billions of Z-bosons (the DJ) every year.
• The Challenge: Because the "dance" (Z → J/ψ + Υ) is so rare, you need to produce a lot of Z-bosons to see it even once.
• The Payoff: Even with the "double negative" corrections making the event rarer, the sheer volume of these new factories means we might finally catch a glimpse of this dance. The authors predict that at the FCC-ee, we might see about 70 events of this specific decay.

4. The "Recipe" They Used (NRQCD)

To do this math, they used a framework called NRQCD (Non-Relativistic Quantum Chromodynamics).

• The Analogy: Think of the heavy quarks (charm and bottom) as two heavy dancers holding hands. Because they are heavy, they don't spin wildly; they move in a somewhat predictable, "non-relativistic" way relative to each other.
• The authors broke the problem into two parts:
  1. Short-distance: The moment the Z-boson splits them apart (calculated with high-precision math).
  2. Long-distance: How the heavy dancers settle into their final formation (calculated using experimental data from other experiments).

The Bottom Line

This paper is a reality check. It tells experimentalists at future particle colliders: "Don't expect to see this event as often as the old textbooks said. The math is more complex, and the event is rarer."

However, it's also good news. Because the authors provided a precise, high-definition prediction (including all the "traffic" and "speed" factors), the experimentalists know exactly what to look for. If they see the event at the rate predicted in this paper, it proves our understanding of the universe's fundamental forces is correct. If they don't see it, or see it at a different rate, it might mean there is new physics hiding in the shadows!

In short: They took a rough guess, added the messy details of reality, and found the event is rarer than expected—but still possible to find with the right tools.

Technical Summary

1. Problem Statement

The paper addresses the theoretical prediction of the rare exclusive decay process $Z \to J/\psi + \Upsilon(nS)$ (where $n=1, 2, 3$). While previous studies have investigated same-flavor channels (e.g., $Z \to J/\psi + J/\psi$ or $Z \to \Upsilon + \Upsilon$), a complete theoretical treatment for this mixed-flavor channel ($c\bar{c} + b\bar{b}$) incorporating both Next-to-Leading Order (NLO) QCD corrections and relativistic corrections was missing.

Given the high luminosity of future $e^+e^-$ colliders (such as the CEPC, FCC-ee, and Super Z factories) operating at the Z-pole, precise predictions are essential to determine if these rare events are observable. The authors note that leading-order (LO) calculations alone are insufficient because higher-order effects are known to be large and potentially destructive in similar processes.

2. Methodology

The authors employ the Non-Relativistic QCD (NRQCD) factorization framework to calculate the decay widths. The methodology involves the following key steps:

• Factorization Formalism: The decay amplitude is factorized into perturbatively calculable Short-Distance Coefficients (SDCs) and non-perturbative Long-Distance Matrix Elements (LDMEs). The amplitude is expanded up to $\mathcal{O}(\alpha_s)$ (NLO QCD) and $\mathcal{O}(v^2)$ (relativistic corrections), where $v$ is the relative velocity of the heavy quarks.
• Perturbative Calculation:
  • The process involves the production of intermediate heavy quark pairs $Q_1\bar{Q}_1$ ($c\bar{c}$) and $Q_2\bar{Q}_2$ ($b\bar{b}$) in the color-singlet, spin-triplet state ($^3S_1^{[1]}$).
  • Feynman Diagrams: The calculation includes 4 tree-level diagrams and 21 one-loop diagrams (virtual corrections).
  • Renormalization: Ultraviolet (UV) divergences are handled via on-shell renormalization of the heavy quark mass and field. Infrared (IR) divergences are regularized using dimensional regularization ($D=4-2\epsilon$). The $\gamma_5$ matrix is treated using the Larin prescription to handle axial currents, requiring a finite renormalization constant $Z_5$.
  • Expansion: The amplitude is expanded in powers of the relative momentum $q$ around $q=0$ using a Taylor expansion. The $\mathcal{O}(v^2)$ corrections are extracted by averaging the $q$-dependent tensors over solid angles.
• Numerical Inputs:
  • Quark masses: $m_c = 1.4 \pm 0.2$ GeV, $m_b = 4.6 \pm 0.1$ GeV.
  • Coupling constants: $\alpha_s(m_Z) = 0.118$, evolved via the two-loop RGE.
  • LDMEs: Color-singlet matrix elements $\langle \mathcal{O}_1 \rangle$ and squared relative momentum matrix elements $\langle q^2 \rangle$ are taken from lattice QCD and phenomenological fits.
  • Software: Calculations utilize FeynArts (diagram generation), FeynCalc (trace evaluation), Apart (partial fractioning), FIRE (IBP reduction), and LoopTools (numerical integration).

3. Key Contributions

• First Complete Calculation: This is the first study to provide a complete NLO QCD and relativistic correction analysis for the mixed-flavor $Z \to J/\psi + \Upsilon(nS)$ channel.
• Quantification of Corrections: The authors explicitly demonstrate that both NLO QCD and relativistic corrections are large and negative, leading to a significant cancellation with the Leading Order (LO) result.
• Uncertainty Analysis: A comprehensive error budget is provided, accounting for variations in the renormalization scale ($\mu_R$), heavy quark masses, and NRQCD matrix elements.

4. Key Results

The numerical results are presented for the decay widths ($\Gamma$) and branching fractions ($\mathcal{B}$) at the central renormalization scale $\mu_R = m_Z/2$.

Decay Widths (in units of $10^{-3}$ eV):
The corrections drastically reduce the LO predictions:

• $Z \to J/\psi + \Upsilon(1S)$:
  • LO: $283.75$
  • NLO QCD ($\mathcal{O}(\alpha_s)$): $-140.66$($\approx -50%$)
  • Relativistic ($\mathcal{O}(v^2)$): $-107.33$($\approx -38%$)
  • Total: $35.77^{+38.57}_{-35.77} \times 10^{-3}$ eV
• $Z \to J/\psi + \Upsilon(2S)$:
  • Total: $14.52^{+20.05}_{-14.52} \times 10^{-3}$ eV
• $Z \to J/\psi + \Upsilon(3S)$:
  • Total: $9.14^{+15.94}_{-9.14} \times 10^{-3}$ eV

Branching Fractions:
Using the total Z width $\Gamma_Z = 2.4955$ GeV, the predicted branching fractions are:

• $\mathcal{B}(Z \to J/\psi + \Upsilon(1S)) = 14.33^{+15.46}_{-14.33} \times 10^{-12}$
• $\mathcal{B}(Z \to J/\psi + \Upsilon(2S)) = 5.82^{+8.03}_{-5.82} \times 10^{-12}$
• $\mathcal{B}(Z \to J/\psi + \Upsilon(3S)) = 3.66^{+6.38}_{-3.66} \times 10^{-12}$

Observation at Future Colliders:
The authors estimate that at the FCC-ee, which is expected to produce $\sim 5 \times 10^{12}$ Z bosons, approximately 70 events of $Z \to J/\psi + \Upsilon(1S)$ could be observed. This suggests the channel is potentially accessible at future high-luminosity Z-pole factories.

5. Significance

• Theoretical Benchmark: The study establishes a rigorous benchmark for future experimental searches. It highlights that relying solely on LO calculations would lead to overestimations by an order of magnitude, potentially misleading experimentalists regarding the feasibility of detection.
• Validation of NRQCD: The results reinforce the necessity of including both $\mathcal{O}(\alpha_s)$ and $\mathcal{O}(v^2)$ terms for precision predictions in heavy quarkonium production, particularly in exclusive mixed-flavor channels.
• Experimental Guidance: By providing updated branching fractions with realistic uncertainty ranges, the paper guides the design of search strategies for the CEPC, FCC-ee, and Super Z factories, indicating that while rare, these decays are within reach of next-generation detectors.