Search for the radiative decays $D^0\to \gamma \bar K_1(1270)^0$ and $D^+\to \gamma K_1(1270)^+$

Using 20.3 fb⁻¹ of data collected by the BESIII detector, a search for the radiative decays $D^0\to \gamma \bar K_1(1270)^0$ and $D^+\to \gamma K_1(1270)^+$ yielded no significant signals, resulting in the first experimental upper limits on their branching fractions and providing the first test of the Vector Meson Dominance mechanism in these charmed meson decays.

The Gist

The Big Picture: A Cosmic Detective Story

Imagine the universe is a giant, high-speed racetrack. In this race, tiny particles called D mesons (think of them as heavy, short-lived delivery trucks) are created and zoom around at nearly the speed of light.

Physicists at the BESIII experiment (a massive camera and sensor array in China) are the detectives watching this race. They are looking for a very specific, rare event: a delivery truck (the D meson) suddenly stopping, dropping off its cargo, and shooting out a single, bright flash of light (a photon, or $\gamma$) in the process.

Specifically, they wanted to see if these trucks could turn into a specific type of "axial-vector" particle called a $K_1(1270)$ while flashing that light.

The Analogy: The "Tag and Track" Game

Because these particles live for such a tiny fraction of a second, you can't just watch them decay in a vacuum. The scientists use a clever trick called the "Tag and Track" method, which is like a game of "Find the Missing Twin."

1. The Setup: The machine smashes electrons and positrons together. This creates a pair of D mesons: a "positive" one ($D^+$) and a "negative" one ($D^-$), or a neutral pair ($D^0$ and $\bar{D}^0$). They are born together, like identical twins.
2. The Tag (The Anchor): The scientists catch one of the twins (the "Tag") and identify exactly what it is by looking at how it breaks apart. Once they know, "Okay, that's definitely a $D^-$," they know with 100% certainty that the other twin (the "Signal") must be a $D^+$.
3. The Track (The Hunt): Now that they know the Signal twin exists, they look at what it did. Did it decay into the specific pattern they are looking for? Did it shoot out a photon and turn into a $K_1$ meson?

The Search: Looking for a Ghost

The scientists analyzed 20.3 billion (20.3 fb$^{-1}$) of these collision events. That is a massive amount of data—like watching every car pass through a busy intersection for a whole year.

They were looking for two specific "ghosts":

• Ghost 1: A neutral D meson turning into a photon and a neutral $K_1$.
• Ghost 2: A positive D meson turning into a photon and a positive $K_1$.

The Theory:
Before this experiment, physicists had two main ideas about how this could happen:

1. The "Short-Cut" (Short-distance): A direct, rare quantum jump.
2. The "Detour" (Long-distance/VMD): The particle takes a scenic route, temporarily turning into other particles (like a rho meson) before becoming the final product. This "Vector Meson Dominance" (VMD) theory predicts these decays should happen, but very rarely.

The Result: The "Silent" Search

After sifting through the mountains of data, the result was... silence.

• Did they find the ghosts? No.
• Did they see a signal? No. The data looked exactly like random background noise (like static on an old TV).

However, in science, "not finding it" is still a huge discovery. It's like searching a haystack for a specific needle and finding nothing. You can't say the needle isn't there, but you can say: "If the needle is there, it must be smaller than this specific size."

The Conclusion: Setting the Limits

Because they didn't find the decay, the team set Upper Limits. Think of this as setting a "Maximum Speed Limit" for how often this event could be happening without them seeing it.

• For the neutral D meson ($D^0$), they said: "This decay happens less than 7.7 times out of every 10,000 tries."
• For the positive D meson ($D^+$), they said: "This decay happens less than 3.9 times out of every 100,000 tries."

Why Does This Matter?

Even though they didn't find the particle, this is a first-ever test of the "Detour" theory (Vector Meson Dominance) for this specific type of particle.

• If the numbers had been higher: It would have meant our current understanding of the universe (the Standard Model) was missing a piece, or that "New Physics" (unknown forces) was at play.
• Since the numbers are low: It confirms that our current theories are holding up. The universe is behaving exactly as the "Detour" theory predicts: these decays are incredibly rare, and the "Short-Cut" is even rarer.

Summary in One Sentence

The BESIII team acted as cosmic detectives, scanning billions of particle collisions for a rare flash of light, and while they didn't find the specific "ghost" they were hunting, they successfully proved that if it exists, it's hiding much more effectively than we thought, confirming our current maps of the subatomic world are accurate.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the search for rare radiative decays of charmed mesons into axial-vector mesons:

• $D^0 \to \gamma \bar{K}_1(1270)^0$
• $D^+ \to \gamma K_1(1270)^+$

Context:

• In the Standard Model, short-distance contributions to weak radiative decays of charmed hadrons are expected to be negligible compared to long-range processes.
• Long-range processes, specifically the Vector Meson Dominance (VMD) mechanism, are predicted to dominate these decays.
• While radiative decays to vector mesons (e.g., $D \to \gamma \rho, \gamma K^*$) have been measured, decays to axial-vector mesons ($K_1$) have not been observed.
• Theoretical predictions for the branching fraction (BF) of $D^+ \to \gamma K_1^+$ are approximately $(1.3 \text{--} 1.5) \times 10^{-5}$. The $D^0$ channel is constrained to be between $O(10^{-6})$ and $O(10^{-4})$.
• Goal: This study represents the first experimental test of the VMD mechanism in radiative decays of charmed mesons to axial-vector mesons.

2. Methodology

Data Sample:

• Source: $e^+e^-$ annihilation data collected by the BESIII detector at the BEPCII collider.
• Center-of-Mass Energy: $\sqrt{s} = 3.773$ GeV (the $\psi(3770)$ resonance).
• Integrated Luminosity: $20.3 \text{ fb}^{-1}$.
• Environment: At this energy, $D\bar{D}$ pairs are produced without accompanying hadrons, providing a clean environment for analysis using the tag technique.

Analysis Strategy:
The analysis employs a Double-Tag (DT) method to suppress background:

1. Single-Tag (ST): One $D$ meson (the tag side, either $D^-$ or $\bar{D}^0$) is fully reconstructed in hadronic modes (e.g., $K^+\pi^-\pi^-$, $K^0_S\pi^-$, etc.).
2. Double-Tag (DT): The signal side (the other $D$ meson) is searched for the specific radiative decay modes.
3. Signal Reconstruction:
   • $D^0 \to \gamma \bar{K}_1(1270)^0$ with $\bar{K}_1^0 \to K^-\pi^+\pi^0$.
   • $D^+ \to \gamma K_1(1270)^+$ with $K_1^+ \to K^+\pi^+\pi^-$.
   • The radiative photon is identified as the highest energy photon on the signal side.
4. Background Suppression:
   • Kinematic Constraints: Beam-constrained mass ($M_{BC}$) and energy difference ($\Delta E$) variables are used for both tag and signal sides.
   • Particle Identification (PID): Strict criteria for $K/\pi$ separation using $dE/dx$ and Time-of-Flight (TOF).
   • Veto Cuts: Events with extra $\pi^0$ candidates or extra charged tracks are rejected. Specific vetoes are applied to remove backgrounds from $D \to K_1 \pi^0$ and $D \to K \omega \pi$ decays.
   • Optimization: Mass windows for $K\pi\pi$ candidates are optimized using the Punzi figure-of-merit ($\epsilon / (1.5 + \sqrt{B})$).

Statistical Analysis:

• Signal yields are extracted using a 2D unbinned maximum-likelihood fit to the distribution of $\Delta E_{sig}$ (signal energy difference) versus $\cos \theta_H$ (helicity angle of the $K_1$ decay).
• Signal shapes are derived from Monte Carlo (MC) simulations; background shapes are modeled using Kernel Density Estimation (KDE) from inclusive MC samples.
• Since no significant signal is observed, upper limits on the branching fractions are calculated using the Frequentist approach at the 90% confidence level (CL).

3. Key Contributions

1. First Search: This is the first experimental attempt to observe $D \to \gamma K_1$ decays, filling a gap in the understanding of charmed meson radiative decays.
2. VMD Test: It provides the first empirical constraint on the Vector Meson Dominance mechanism specifically for transitions involving axial-vector mesons in the charm sector.
3. Comprehensive Systematics: The paper details a rigorous evaluation of systematic uncertainties, including:
   • Additive: Background shape modeling (varied via alternative MC fractions and smoothing parameters).
   • Multiplicative: Tracking efficiency, PID, photon/$\pi^0$ selection, MC modeling, and branching fractions of intermediate states.
   • Total systematic uncertainties are estimated at ~20.7% for $D^0$ and ~19.8% for $D^+$.

4. Results

• Observation: No significant signal was observed in either decay channel.
  • $D^0 \to \gamma \bar{K}_1(1270)^0$: Signal yield $199^{+69}_{-68}$.
  • $D^+ \to \gamma K_1(1270)^+$: Signal yield $2^{+8}_{-7}$.
• Upper Limits (90% CL):
  • $\mathcal{B}(D^0 \to \gamma \bar{K}_1(1270)^0) < \mathbf{7.7 \times 10^{-4}}$
  • $\mathcal{B}(D^+ \to \gamma K_1(1270)^+) < \mathbf{3.9 \times 10^{-5}}$

5. Significance

• Theoretical Consistency: The derived upper limits are consistent with current theoretical expectations (which predict the $D^+$ mode around $10^{-5}$ and constrain the $D^0$ mode to $10^{-6} \text{--} 10^{-4}$).
• Constraint on New Physics: While the results do not show an enhancement, they set strict bounds on potential contributions from New Physics or enhanced long-range interactions that could have boosted these branching fractions by orders of magnitude.
• Future Directions: These limits serve as a baseline for future high-statistics experiments (e.g., at the Super Tau-Charm Factory) to potentially observe these rare decays or further constrain the VMD parameters in the charm sector.

In summary, the BESIII Collaboration successfully performed the first search for these specific radiative decays, establishing stringent upper limits that align with Standard Model predictions and providing a crucial reference point for theoretical models of charmed meson decays.