Observation of a new excited charm-strange meson $D_{s1}(2933)^+$ in $B^0\to D^+ D^- K^+ \pi^-$ decays

Using 13 TeV proton-proton collision data from the LHCb experiment, researchers observed a new excited charm-strange meson, $D_{s1}(2933)^+$, with a statistical significance exceeding 10 standard deviations, measuring its mass at approximately 2933 MeV and identifying it as a candidate for a $D_s(2P^{(\prime)}_{1})^+$ state with $J^P = 1^+$.

The Gist

Imagine the universe as a giant, bustling construction site. For decades, physicists have been trying to understand the blueprints of this site, specifically how the "bricks" of matter (quarks) stick together to build larger structures called mesons.

One of the most interesting neighborhoods on this site is the "Charm-Strange" district. Here, a heavy "charm" quark and a lighter "strange" quark hold hands to form a meson. Scientists have mapped out the ground floor of this district (the basic, stable bricks) and a few rooms on the first floor (excited states). But there are still empty spaces on the upper floors where they expect to find new, heavier rooms.

The Big Discovery
In this paper, the LHCb collaboration (a team of scientists working at the Large Hadron Collider in Europe) announces they have finally found a new room on the upper floor. They call it $D_{s1}(2933)^+$.

Think of it like this:

• The Search: Imagine you are listening to a crowded party (the particle collisions). You know there are many people talking (background noise), but you are looking for a specific song being played by a specific band (the new particle).
• The Method: Instead of just listening for a single note, the scientists used a technique called "amplitude analysis." This is like taking a recording of the entire party and using a super-computer to separate every single voice, instrument, and echo. They looked at how the particles bounced off each other in the decay of a B-meson (a heavy particle that acts like a delivery truck dropping off packages).
• The Find: In the chaos of the data, they found a distinct "echo" or resonance at a mass of 2933 MeV (a unit of energy/mass). This echo was so loud and clear that there was less than a 1 in a trillion chance it was just random noise. It was a statistical certainty.

What is this new particle?
The scientists measured two main things about this new "room":

1. Its Weight (Mass): It weighs about 2933 units.
2. Its Lifespan (Width): It is very unstable, living for only a tiny fraction of a second before falling apart. The "width" of 72 units tells us how quickly it decays.
3. Its Shape (Spin): They determined its "spin-parity" is $1^+$. In simple terms, this describes how the particle spins and how it looks if you were to take a mirror image of it. This specific shape is the "fingerprint" that confirms it is a $1^+$ state.

Why does this matter?
For a long time, the "Charm-Strange" district has been a bit of a mystery. Some particles found there were lighter than scientists expected, leading to wild theories that they might be "molecules" (two particles loosely stuck together) or "tetraquarks" (four particles stuck together in a weird cluster).

This new discovery, $D_{s1}(2933)^+$, fits perfectly into the standard "quark model" predictions. It looks like a $2P$ state, which is a fancy way of saying it's the second excited version of a standard charm-strange meson.

The Analogy of the Guitar String
Think of a quark and an antiquark connected by a spring (the strong force).

• The ground state is the spring sitting still.
• The first excited state is the spring vibrating gently (like the first note on a guitar string).
• This new $D_{s1}(2933)^+$ is like the string vibrating in a more complex, higher-energy pattern (the second harmonic).

The Conclusion
By finding this new particle, the scientists have filled in a missing piece of the puzzle. It confirms that our "blueprint" of how quarks build matter is largely correct, at least for this part of the universe. It helps explain why some other particles in this neighborhood behave the way they do and rules out some of the more exotic theories that suggested these particles were made of four quarks instead of two.

In short: The LHCb team listened to the universe's noise, found a new, clear signal, and confirmed that the "Charm-Strange" neighborhood has a new, heavy, spinning resident that fits exactly where the textbooks said it should be.

Technical Summary

1. Problem Statement

The spectroscopy of charm-strange ($c\bar{s}$) mesons is a critical testing ground for Quantum Chromodynamics (QCD) in the non-perturbative regime. While ground states and several low-lying excitations are established, significant puzzles remain:

• Mass Discrepancies: The masses of $D_{s0}^*(2317)^+$ and $D_{s1}(2460)^+$ are $\sim 100$ MeV lower than quark-model predictions for $P$-wave excitations, leading to hypotheses involving molecular or tetraquark structures.
• Unobserved States: Higher excitations (e.g., $2P$, $1D$, $2S$) remain largely unobserved or poorly characterized.
• The $D_{s0}(2590)^+$ Puzzle: Recent observations of this state show a mass discrepancy with quark-model predictions, suggesting the need for refined models including coupled-channel effects or $1S$-$2S$ mixing.
• Spin-Parity Access: Traditional $DK$ spectra only allow access to natural spin-parity states ($0^+, 1^-, 2^+, \dots$). There is a need to explore decay modes that allow access to unnatural spin-parity states to complete the $D_s$ spectrum map.

2. Methodology

The analysis utilizes an amplitude analysis (Dalitz plot analysis) of the full five-dimensional phase space of the decay $B^0 \to D^+D^-K^+\pi^-$.

• Data Sample: Proton-proton collision data collected by the LHCb experiment at a center-of-mass energy $\sqrt{s} = 13$ TeV (2016–2018), corresponding to an integrated luminosity of 5.4 fb$^{-1}$.
• Event Selection:
  • Reconstructed via $D^\pm \to K^\mp\pi^\pm\pi^\pm$.
  • Strict particle identification (PID) and vertex quality requirements.
  • A Gradient Boosted Decision Tree (GBDT) classifier was used to suppress combinatorial background, trained on simulated signal and sideband data.
  • A kinematic fit constraining the $B^0$ mass and $D^\pm$ masses improved resolution.
• Amplitude Model Construction:
  • The total decay amplitude $\mathcal{M}$ is a coherent sum of intermediate resonant and non-resonant components within the isobar model.
  • Two topologies were considered: Cascade ($B^0 \to P_1 T_1 \to P_1 P_2 T_2 \to P_1 P_2 P_3 P_4$) and Quasi-two-body ($B^0 \to R_1 R_2$).
  • Angular Formalism: Helicity formalism with Wigner $d$-functions and exponential azimuthal terms.
  • Lineshapes: Relativistic Breit-Wigner (RBW) functions were used for most resonances. For the overlapping $D_{s1}^*(2700)^+$ and $D_{s1}^*(2860)^+$ states, a quasi-model-independent (MI) spline parametrization was employed to handle interference.
  • Background Modeling: An extended unbinned maximum-likelihood fit was performed. The background fraction ($\beta \approx 6.8\%$) was determined from the $B^0$ mass fit and modeled using Kernel Density Estimation (KDE) on sideband data.
• Fit Strategy:
  1. An initial fit included all established conventional resonances ($D_{s1}(2536)^+$, $D_{s0}(2590)^+$, $D_{s1}^*(2700)^+$, $D_{s1}^*(2860)^+$, $D_{s3}^*(2860)^+$, and charmonia).
  2. A significant discrepancy was observed in the $m(D^+K^+\pi^-)$ distribution around 2950 MeV.
  3. A new resonance component was added to the model. Various $J^P$ hypotheses ($0^-, 1^+, 2^\pm$) were tested.
  4. The model with $J^P = 1^+$ yielded the lowest negative log-likelihood (NLL) value.
  5. Systematic uncertainties were evaluated by varying lineshape parametrizations, background treatments, and including potential additional resonances.

3. Key Contributions

• Discovery of a New State: The observation of a new excited charm-strange meson, denoted $D_{s1}(2933)^+$.
• Full Phase Space Analysis: Unlike previous restricted analyses (e.g., $m(K^+\pi^-) < 750$ MeV), this study utilizes the full phase space, enabling the reconstruction of states with unnatural spin-parity ($1^+$) that are inaccessible in simpler $DK$ spectra.
• Precision Amplitude Modeling: The implementation of a coherent sum of cascade and quasi-two-body topologies, including complex interference effects between overlapping states (specifically the $D_{s1}^*$ states) using spline parametrizations.
• Comprehensive Systematic Study: A rigorous breakdown of systematic uncertainties covering lineshape modeling, efficiency corrections, and background treatment.

4. Results

• Statistical Significance: The new state is observed with a statistical significance exceeding 10 standard deviations ($>10\sigma$).
• Measured Parameters:
  • Mass ($m_0$): $2933^{+6}_{-5}(\text{stat})^{+4}_{-3}(\text{syst})$ MeV.
  • Width ($\Gamma_0$): $72^{+18}_{-12}(\text{stat})^{+7}_{-10}(\text{syst})$ MeV.
  • Spin-Parity ($J^P$): Determined to be $1^+$.
• Fit Fractions: The contribution of the new state to the total decay rate is significant. The fit fractions for the $D_{s1}(2933)^+$ components are:
  • $D_{s1}(2933)^+ \to D^+(K^+\pi^-)_S$: $2.4^{+0.5}_{-0.7} \pm 0.4^{+0.4}_{-1.3}\%$
  • $D_{s1}(2933)^+ \to D^+ K_1^*(892)^0$: $8.9^{+1.8}_{-2.1} \pm 2.4^{+2.4}_{-1.8}\%$
  • $D_{s1}(2933)^+ \to K^+ D_2^*(2460)^0$: $7.0^{+6.0}_{-3.0} \pm 3.0^{+3.0}_{-3.7}\%$
• Consistency Checks: The measured mass and width of the previously observed $D_{s0}(2590)^+$ state are consistent with previous LHCb results ($m \approx 2606$ MeV, $\Gamma \approx 87$ MeV).

5. Significance

• Candidate for $2P$ Excitation: The properties of $D_{s1}(2933)^+$ are compatible with a conventional $D_s(2P^{(\prime)})_1^+$ state. This corresponds to the first radial excitation of the $P$-wave $D_s$ system, which is distinct from the lower-lying $D_{s1}(2460)^+$ and $D_{s1}(2536)^+$ states.
• Completing the Spectrum: This discovery fills a gap in the high-mass $D_s$ excitation map, providing crucial data to test quark models and lattice QCD calculations.
• QCD Dynamics: The observation helps clarify the nature of the charm-strange spectrum, potentially resolving discrepancies between quark-model predictions and experimental masses by providing a benchmark for higher excitations where coupled-channel effects or mixing might be better understood.
• Methodological Impact: The success of the full phase-space amplitude analysis demonstrates the power of multibody $B$-meson decays in uncovering states that are invisible in simpler two-body invariant mass spectra.

In conclusion, this paper reports the first observation of the $D_{s1}(2933)^+$ meson, a high-mass $1^+$ charm-strange state, significantly advancing the understanding of heavy-light meson spectroscopy and QCD dynamics.