First Measurement of the Decay Dynamics in the Semileptonic Transition of the $D^{+(0)}$ into the Axial-vector Meson $\bar K_1(1270)$

Using 20.3 fb$^{-1}$ of $e^+e^-$ collision data collected by the BESIII detector, this study presents the first measurement of the decay dynamics and branching fractions for the semileptonic transitions $D^{+(0)}\to \bar K_1(1270) e^+ \nu_e$, including the extraction of hadronic form factors and the setting of upper limits for the $K_1(1400)$ channel.

The Gist

Imagine the subatomic world as a bustling, chaotic dance floor where tiny particles are constantly spinning, colliding, and transforming into one another. For decades, physicists have been trying to understand the choreography of these dances, specifically how heavy particles (like the D meson) break apart into lighter ones.

This paper is like a high-definition recording of a very specific, rare dance move that no one has ever fully analyzed before. Here is the story of what the BESIII Collaboration discovered, explained in everyday terms.

The Main Event: A Heavy Breakup

Think of the D meson as a heavy, unstable suitcase. Inside this suitcase are three smaller items: a charm quark and a strange quark (the "contents"). Sometimes, this suitcase spontaneously opens up, spilling its contents and transforming into a new set of particles.

In this specific study, the D meson decays (breaks apart) into:

1. An electron (a tiny, fast-moving particle).
2. A neutrino (a ghost-like particle that barely interacts with anything).
3. A K1(1270) meson (a complex, spinning "axial-vector" particle made of a kaon and a pion).

The Analogy: Imagine a heavy, spinning top (the D meson) suddenly shattering. One piece flies off as a spark (the electron), another vanishes into thin air (the neutrino), and the third piece is a wobbling, spinning gyroscope (the K1 meson) that immediately starts to wobble and break apart further into a kaon and two pions.

The Mystery: The "Spin" of the Breakup

Physicists have been great at predicting how these suitcases break apart when they turn into simple, flat objects (like a pancake). But when they turn into complex, spinning objects (like the K1 meson), the rules get murky.

The team wanted to measure two specific things about this breakup:

• The "Shape" of the Force (Form Factors): Think of the force holding the particles together as a rubber band. When the D meson breaks, how much does that rubber band stretch? The paper measured two specific "stretchiness" values, named $r_A$ and $r_V$. These are like measuring the tension in the rubber band to understand the strength of the glue holding the particles together.
• The "Up-Down" Tilt (Asymmetry): When the K1 meson breaks apart, does it prefer to spin "up" or "down"? The team measured this tilt, called $A'_{ud}$. It's like checking if a spinning coin lands on heads or tails more often.

The Experiment: The "Double-Tag" Trick

How do you catch a ghost (the neutrino) and a fleeting breakup? You can't see the neutrino, so you have to infer its presence by what's missing.

The scientists used a clever trick called Double-Tagging:

1. The Setup: They smashed electrons and positrons together to create pairs of D mesons (a D and an anti-D).
2. The Tag: They caught one of the D mesons in a "net" (reconstructed its decay) to know exactly what it was. This is the "Tag."
3. The Hunt: Because they knew the total energy of the collision, they knew exactly what the other D meson (the "Signal") should look like. If they saw the electron and the K1 meson, but energy was missing, they knew the neutrino was there.

It's like watching a magic show where two identical boxes are opened. If you see exactly what's in Box A, you know exactly what should be in Box B. If Box B has a rabbit and a hat, but is missing a wand, you know the magician hid the wand somewhere else (the neutrino).

The Results: New Rules for the Dance Floor

This paper is a big deal because it's the first time anyone has successfully measured these specific "stretchiness" values for this type of heavy-to-light particle transition.

• The Measurements: They found specific values for the "stretchiness" ($r_A$ and $r_V$). These numbers are like a fingerprint for the strong force.
• The Theory Check: Before this, scientists had many different theories (guesses) about what these numbers should be, based on complex math. The new measurements act like a referee. They said, "Okay, Theory A and Theory B are wrong. Only Theory C (specifically a method called 3PSR with a specific mixing angle) matches our data." It's like finally solving a puzzle and realizing only one specific piece fits.
• The "Ghost" Search: They also looked for a heavier, rarer version of the K1 meson (the K1(1400)). They didn't find it. This is important because it sets a limit: "If this particle exists, it's so rare that we can't see it yet."

Why Should You Care?

You might ask, "Who cares about a spinning K1 meson?"

1. Understanding the Universe's Glue: This helps us understand the Strong Force, one of the four fundamental forces of nature. It's the glue that holds atoms together.
2. Searching for New Physics: The way these particles spin and decay is a sensitive test for the Standard Model (our current rulebook of physics). If the numbers were slightly off, it would suggest "New Physics"—like hidden particles or forces we haven't discovered yet. In this case, the results matched the Standard Model perfectly, which is good news for our current understanding, but it also means we need even more precise experiments to find the cracks in the theory.
3. Connecting to the Big Picture: These measurements help us understand how heavier particles (like B-mesons, which are related to the Higgs boson and the early universe) behave. It's like learning how a small gear turns to understand how the whole clock works.

The Bottom Line

The BESIII team took a massive dataset (20.3 inverse femtobarns—imagine a library of particle collisions), used a clever "double-tag" trick to isolate a rare event, and successfully measured the "dance moves" of a heavy particle turning into a spinning axial-vector meson. They didn't just find a new particle; they measured the rules of the dance with unprecedented precision, confirming some theories and ruling out others. It's a victory for precision physics, proving that even in the chaotic subatomic world, there is a rhythm we can finally hear.

Technical Summary

1. Problem and Motivation

The semileptonic (SL) decays of heavy mesons are crucial for testing the Standard Model (SM) and probing New Physics. While the hadronic form factors (FFs) for SL $D$ decays into $S$-wave states (pseudoscalar or vector mesons) are well-studied, those for $P$-wave states (axial-vector mesons) remain largely unexplored experimentally.

• Theoretical Gap: The physical mass eigenstates of strange axial-vector mesons, $K_1(1270)$ and $K_1(1400)$, are mixtures of the $^1P_1$ and $^3P_1$ states, determined by a mixing angle $\theta_{K_1}$. Theoretical predictions for the FFs of $D \to \bar{K}_1(1270)$ vary widely depending on the calculation method (e.g., LCSR, 3PSR, LFQM) and the assumed value of $\theta_{K_1}$.
• Physics Impact: Precise measurements of these FFs are essential to:
  1. Constrain the mixing angle $\theta_{K_1}$.
  2. Guide theoretical calculations for decays of $\tau$, $B$, and $D$ mesons.
  3. Extract photon polarization in $b \to s\gamma$ transitions (via the up-down asymmetry), which is a sensitive probe for right-handed couplings in New Physics.

2. Methodology

The analysis utilizes $e^+e^-$ collision data collected by the BESIII detector at a center-of-mass energy of $\sqrt{s} = 3.773$ GeV, corresponding to an integrated luminosity of 20.3 fb$^{-1}$.

• Double-Tag (DT) Technique: Since $D\bar{D}$ pairs are produced at rest without accompanying particles, the experiment employs the double-tag method.
  • Single Tag (ST): The $\bar{D}$ meson is reconstructed in specific hadronic decay modes (e.g., $\bar{D}^0 \to K^+\pi^-$, $D^- \to K^+\pi^-\pi^-$).
  • Signal Selection: The remaining tracks and showers in the event are used to reconstruct the signal decay $D^{+(0)} \to \bar{K}_1(1270)^{(0)} e^+ \nu_e \to K^- \pi^+ \pi^{0(-)} e^+ \nu_e$.
• Kinematic Variables:
  • $U_{miss}$: Defined as $E_{miss} - |\vec{p}_{miss}|$, used to extract signal yields by distinguishing signal from background.
  • Missing Momentum/Energy: Calculated based on the beam energy and the reconstructed signal candidates.
• Amplitude Analysis: An unbinned maximum likelihood fit is performed on the 5-body phase space.
  • Model: The decay amplitude is constructed using covariant tensors involving vector ($V$) and axial-vector ($A$) currents.
  • Form Factors: Parameterized using single-pole models: $V_{1,2}(q^2)$ and $A(q^2)$. The analysis extracts the ratios $r_A = A(0)/V_1(0)$ and $r_V = V_2(0)/V_1(0)$.
  • Sub-structures: The $\bar{K}_1(1270) \to K^- \pi^+ \pi^0$ decay is modeled via intermediate states: $\rho K$ (S-wave) and $\pi \bar{K}^*(892)$ (S- and D-waves). The contribution of $\bar{K}_1(1400)$ is also tested.
• Angular Analysis: A simultaneous fit to the distributions of $\cos \theta_L$ and $\cos \theta_K$ is performed to extract the up-down asymmetry ($A'_{ud}$).

3. Key Contributions

This paper presents the first experimental determination of the following quantities in semileptonic $D$ decays to axial-vector mesons:

1. Hadronic Form Factors ($r_A$ and $r_V$): Direct measurement of the FF ratios for $D \to \bar{K}_1(1270)$.
2. Up-Down Asymmetry ($A'_{ud}$): The first measurement of this asymmetry in $D \to \bar{K}_1(1270) e^+ \nu_e$, which is directly related to photon polarization in $b \to s\gamma$ transitions.
3. Branching Fractions (BFs): Improved precision measurements for $D^+ \to \bar{K}_1(1270)^0 e^+ \nu_e$ and $D^0 \to K_1(1270)^- e^+ \nu_e$.
4. Search for $K_1(1400)$: The first search for the semileptonic decay into the higher mass axial-vector state $K_1(1400)$.

4. Key Results

A. Hadronic Form Factors
The amplitude analysis yielded the following form factor ratios:

• $r_A = (-11.2 \pm 1.0_{\text{stat}} \pm 0.9_{\text{syst}}) \times 10^{-2}$
• $r_V = (-4.3 \pm 1.0_{\text{stat}} \pm 2.5_{\text{syst}}) \times 10^{-2}$

B. Branching Fractions
The branching fractions were determined with improved precision:

• $\mathcal{B}(D^+ \to \bar{K}_1(1270)^0 e^+ \nu_e) = (2.27 \pm 0.11_{\text{stat}} \pm 0.07_{\text{syst}} \pm 0.07_{\text{input}}) \times 10^{-3}$
• $\mathcal{B}(D^0 \to K_1(1270)^- e^+ \nu_e) = (1.02 \pm 0.06_{\text{stat}} \pm 0.06_{\text{syst}} \pm 0.03_{\text{input}}) \times 10^{-3}$

C. Asymmetry and Polarization

• Up-Down Asymmetry: $A'_{ud} = 0.01 \pm 0.11$. This result is consistent with the Standard Model prediction ($0.092 \pm 0.022$).
• Longitudinal Polarization: The fraction of longitudinally polarized $\bar{K}_1(1270)$ is $F_L = 0.50 \pm 0.04$ (stat only), consistent with previous BESIII results but with four times higher precision.

D. $K_1(1400)$ Search
No significant signal was observed for $D \to \bar{K}_1(1400) e^+ \nu_e$. Upper limits at 90% confidence level were set:

• $\mathcal{B}(D^+ \to \bar{K}_1(1400)^0 e^+ \nu_e) < 1.4 \times 10^{-4}$
• $\mathcal{B}(D^0 \to K_1(1400)^- e^+ \nu_e) < 0.7 \times 10^{-4}$

E. Fit Fractions and Sub-structures
The dominant decay mode of $\bar{K}_1(1270)$ in this channel is found to be $\rho K$ (S-wave), with a fit fraction of $\sim 79\%$ for $D^+$ and $\sim 72\%$ for $D^0$. The contribution from $\bar{K}_1(1400)$ was found to be insignificant ($< 5\%$ for $D^+$ and $< 9\%$ for $D^0$ at 90% CL).

5. Significance

• Theoretical Constraint: The measured values of $r_A$, $r_V$, and the branching fractions are consistent with Three-Point QCD Sum Rule (3PSR) predictions only if the mixing angle $\theta_{K_1}$ is in the range $(61^\circ, 67^\circ)$. These results disfavor all other theoretical calculations (LCSR, LFQM, etc.) by more than $5\sigma$.
• New Physics Probes: By confirming the SM prediction for the up-down asymmetry, the result validates the current understanding of hadronic effects in $D$ decays. This provides a crucial baseline for future studies of $B \to K_1 \gamma$ decays, where deviations in photon polarization could signal New Physics (specifically right-handed currents).
• Methodological Advance: This work demonstrates the feasibility of performing full amplitude analyses on semileptonic decays involving axial-vector mesons, opening a new window for studying $P$-wave dynamics in heavy flavor physics.