Lattice determination of the neutrino background for $J/ψ\rightarrow γ+ \textrm{invisible}$

This paper presents the first lattice QCD calculation of the irreducible Standard Model background for the $J/\psi \to \gamma + \text{invisible}$ decay, determining the branching fraction for $J/\psi \to \gamma\nu\bar{\nu}$ to be $1.00(9)(7)\times 10^{-10}$ to provide a critical benchmark for dark matter searches.

The Gist

Imagine you are a detective trying to find a ghost in a very crowded, noisy room. The "ghost" in this story is Dark Matter, a mysterious substance that makes up most of the universe but refuses to interact with light or ordinary matter. Scientists want to catch a glimpse of it by watching heavy particles called J/ψ (pronounced "J-psi") decay. Specifically, they are looking for a J/ψ to turn into a single flash of light (a photon) and then vanish completely. If it vanishes, it might have turned into a dark matter particle.

However, there is a problem: Neutrinos.

Neutrinos are tiny, ghost-like particles that are part of the Standard Model of physics. They also make the J/ψ disappear into nothingness when it decays. To the detector, a neutrino looks exactly like dark matter. It's like trying to find a specific rare bird in a forest, but every time you look, you see a common pigeon that looks exactly the same. If you don't know exactly how many pigeons are there, you can't be sure if you've found the rare bird.

The Paper's Mission
This paper is the first time scientists have used a super-powerful mathematical simulation (called Lattice QCD) to count exactly how many "pigeons" (neutrinos) are hiding in the forest. They wanted to calculate the exact rate at which a J/ψ turns into a photon and a pair of neutrinos ($J/\psi \to \gamma \nu \bar{\nu}$).

How They Did It: The "Pixelated Universe"
To do this, the researchers didn't use a telescope; they used a computer to build a 3D grid (a lattice) that represents space and time.

• The Grid: Imagine a giant, invisible fishnet stretched across the universe. They placed the J/ψ particle on this net.
• The Simulation: They watched how the J/ψ interacted with the grid, emitting a photon and a neutrino pair. Because the strong force holding the J/ψ together is incredibly complex (like a tangled ball of yarn), they couldn't just use simple math. They had to simulate the "yarn" knotting and untying on the grid.
• Cleaning the Signal: They had to be very careful to make sure they were only seeing the J/ψ and not "echoes" of heavier, excited versions of the particle. They used a technique called a "multi-state fit," which is like tuning a radio to filter out static and hear only the clear station.
• The Scale: They ran this simulation on three different sizes of grids (fine, medium, and coarse) to ensure their results weren't just an artifact of the grid's size. They then mathematically smoothed these results together to predict what would happen in the real, continuous world.

The Result
The team calculated the "branching fraction," which is essentially the probability of this specific event happening.

• The Number: They found that for every 10 billion J/ψ particles that decay, about 1 of them will turn into a photon and neutrinos.
• The Precision: Their calculation is extremely precise: $1.00 \times 10^{-10}$. They even provided a "margin of error" to show how confident they are.

Why This Matters
The paper explains that future experiments, like the Super Tau Charm Facility (STCF), are being built to be so sensitive that they will be able to detect signals at this exact level ($10^{-10}$).

Before this paper, scientists didn't have a precise number for the "neutrino background." It was like trying to weigh a feather on a scale that was already vibrating with an unknown amount of wind. Now, they have a precise measurement of the "wind" (the neutrinos).

The Bottom Line
By providing this exact number, the paper gives experimentalists a baseline. When they run their experiments in the future, they can subtract this known neutrino background from their data. If there is any signal left over after they subtract the neutrinos, that leftover signal could be the elusive Dark Matter.

In short: This paper didn't find Dark Matter, but it built the perfect ruler to measure the noise so that, in the future, we might finally hear the whisper of the dark.

Technical Summary

Technical Summary: Lattice Determination of the Neutrino Background for $J/\psi \to \gamma + \text{invisible}$

Problem Statement
The search for light dark matter and sterile neutrinos via radiative decays of heavy quarkonia (e.g., $J/\psi \to \gamma + \text{invisible}$) is a primary goal in modern particle physics. While experimental upper limits on these branching fractions have improved significantly, upcoming facilities like the Super Tau Charm Facility (STCF) are projected to reach sensitivities down to the $10^{-10}$ level. At this sensitivity, the Standard Model (SM) irreducible background from the decay $J/\psi \to \gamma \nu \bar{\nu}$ becomes dominant. To distinguish potential new physics signals from this background, a precise, model-independent determination of the SM neutrino background is indispensable. Previous phenomenological estimates relied on approximations (e.g., non-relativistic color-singlet models) that introduced uncontrolled systematic uncertainties.

Methodology
This work presents the first ab initio lattice QCD calculation of the $J/\psi \to \gamma \nu \bar{\nu}$ decay width. The methodology employs the following strategies to ensure precision and control over systematic effects:

1. Factorization and Form Factors: The decay amplitude is factorized into a perturbative leptonic part and a non-perturbative hadronic part. The hadronic interaction is encoded in a function $H_{\mu\nu\alpha}$, which is parameterized by a single form factor $F_{\gamma\nu\bar{\nu}}(q^2)$.
2. Scalar Function Method: Instead of calculating form factors at discrete lattice momenta and interpolating (which introduces model dependence), the authors utilize the "scalar function method." This involves constructing a scalar function $I(E_\gamma, \Delta t)$ by integrating the Euclidean three-point correlation function over spatial coordinates with a Bessel function kernel ($j_0$). This allows for the extraction of the form factor with full $q^2$ distribution coverage in the phase space.
3. Excited-State Suppression: To mitigate contamination from excited states, the authors employ a multi-state fit using a two-state ansatz. They analyze the time dependence of the correlation functions and extract the ground-state contribution in the limit of large time separation ($\Delta t \to \infty$).
4. Finite-Volume and Discretization Control:
   • Finite-Volume: A spatial volume integral with a truncation range $R$ is used to monitor finite-volume effects. The authors demonstrate that these effects are exponentially suppressed and negligible for the charmonium system.
   • Continuum Extrapolation: Calculations are performed on three gauge ensembles with different lattice spacings ($a \approx 0.067, 0.085, 0.098$ fm) generated by the Extended Twisted Mass Collaboration (ETMC). The results are extrapolated to the continuum limit ($a \to 0$) assuming a linear $a^2$ behavior, consistent with the automatic $O(a)$ improvement of twisted-mass fermions.
5. Renormalization: The electromagnetic and weak currents are renormalized using constants $Z_V$ and $Z_A$ determined in previous works and via the RI-MOM scheme.

Key Results
The authors compute the dimensionless ratio $R_f = \Gamma_{\gamma\nu\bar{\nu}} / f_{J/\psi}$ and perform the continuum extrapolation to obtain the physical branching fraction. The final result is:
$$\text{Br}(J/\psi \to \gamma \nu \bar{\nu}) = 1.00(9)(7) \times 10^{-10}$$
where the first uncertainty is statistical (including uncertainties from the continuum extrapolation) and the second is the estimated systematic uncertainty.

• Comparison with Models: This result is compared to a previous phenomenological prediction based on the non-relativistic color-singlet model, which estimated $0.7 \times 10^{-10}$. The lattice result provides a more precise, parameter-free benchmark, differing slightly but remaining within the same order of magnitude.
• Systematic Checks: The authors verified the robustness of their result by excluding the coarsest lattice ensemble ($a_{98}$) from the extrapolation, yielding $1.07(13) \times 10^{-10}$. The consistency between the full and reduced datasets supports the absence of significant residual $O(a)$ discretization errors.
• Neglected Contributions: Disconnected diagrams are noted to be negligible due to Okubo-Zweig-Iizuka (OZI) suppression in the charmonium system.

Significance and Claims
The paper claims to deliver the first ab initio lattice QCD benchmark for the neutrino background in $J/\psi \to \gamma + \text{invisible}$ decays. Its significance lies in:

1. Enabling Future Searches: By providing a precise prediction ($10^{-10}$ level), the work enables experiments like the STCF to accurately subtract the SM background, thereby allowing them to probe new physics in the invisible sector without being limited by theoretical uncertainty.
2. Generalizability: The methodology is presented as generalizable to other quarkonium channels, such as $\Upsilon \to \gamma + \text{invisible}$ and $\phi \to \gamma + \text{invisible}$, providing a theoretical roadmap for dark matter searches across different energy scales.
3. Theoretical Rigor: The calculation avoids the approximations inherent in phenomenological models, offering a rigorous, non-perturbative determination of the decay width that is free from uncontrolled systematic uncertainties associated with model assumptions.