Search for $CP$ violation in $Ξ_c^+\toΣ^+h^+h^-$ and $Λ_c^+\to ph^+h^-$ at Belle II

Technical Summary: Search for CP Violation in $\Xi^+_c \to \Sigma^+ h^+ h^-$ and $\Lambda^+_c \to p h^+ h^-$ at Belle II

Problem and Motivation
Charge-parity (CP) symmetry violation has been established in the decays of strange, charmed, and bottom mesons, as well as recently in bottom baryons. While the Standard Model (SM) successfully explains CP asymmetries in strange and bottom decays, the measured CP asymmetry in charm decays (specifically the difference between $D^0 \to K^+K^-$ and $D^0 \to \pi^+\pi^-$) lacks a clear SM interpretation. Theoretical models suggest this asymmetry could arise from physics beyond the SM or from nonperturbative QCD effects. Furthermore, the LHCb collaboration's separate determinations of these asymmetries indicate an unexpectedly large breaking of U-spin symmetry, which posits invariance under the exchange of down and strange quarks.

While U-spin symmetry predicts specific relations between CP asymmetries in three-body, singly-Cabibbo-suppressed hadronic charmed-baryon decays, these asymmetries had not been measured individually or in their predicted sums. This paper addresses the gap in experimental data by reporting the first measurements of CP asymmetries for the decays $\Xi^+_c \to \Sigma^+ h^+ h^-$ and $\Lambda^+_c \to p h^+ h^-$ (where $h = K, \pi$).

Methodology
The analysis utilizes 428 fb$^{-1}$ of $e^+e^-$ collision data collected by the Belle II experiment at the SuperKEKB collider, corresponding to an integrated luminosity near the $\Upsilon(4S)$ resonance. The study focuses on four signal decay modes:

1. $\Xi^+_c \to \Sigma^+ K^+ K^-$
2. $\Xi^+_c \to \Sigma^+ \pi^+ \pi^-$
3. $\Lambda^+_c \to p K^+ K^-$
4. $\Lambda^+_c \to p \pi^+ \pi^-$

The decay-rate CP asymmetry, $A_{CP}$, is defined as the difference in decay rates between the particle and antiparticle, normalized by their sum. To extract this, the collaboration measures the yield asymmetry ($A_N$) and corrects for production asymmetries ($A_p$) and detection asymmetries ($A_d$).

• Production Asymmetry Suppression: Since $A_p$ is an odd function of the cosine of the polar angle ($\cos \theta$) of the charmed baryon's momentum, the analysis averages the yield asymmetries measured in the forward ($\cos \theta > 0$) and backward ($\cos \theta < 0$) regions to suppress this effect.
• Detection Asymmetry Cancellation: The analysis employs control channels to cancel detection asymmetries.
  • For $\Xi^+_c \to \Sigma^+ h^+ h^-$, the asymmetry is determined by subtracting the yield asymmetry of the control channel $\Lambda^+_c \to \Sigma^+ h^+ h^-$. This cancels the detection asymmetry of the final-state baryon ($\Sigma^+$).
  • For $\Lambda^+_c \to p h^+ h^-$, the analysis subtracts the asymmetry of $\Lambda^+_c \to p \pi^+ K^-$ and adds the asymmetry of $D^0 \to \pi^+ K^- \pi^+ \pi^-$. This procedure cancels detection asymmetries for the proton, kaon, and pion.
  • Kinematic weighting is applied to control channel candidates to ensure their momentum and $\cos \theta$ distributions match those of the signal channels, ensuring the cancellation of detection effects is valid.
• Yield Extraction: Yields are determined by performing simultaneous extended unbinned maximum likelihood fits to the invariant mass distributions of signal and control candidates, separated by charge and forward/backward regions. Signal shapes are modeled with double-sided Crystal Ball functions, and backgrounds with straight lines.

Key Contributions and Results
This work presents the first measurements of CP asymmetries for individual hadronic three-body charmed-baryon decays. The measured asymmetries are:

• $A_{CP}(\Xi^+_c \to \Sigma^+ K^+ K^-) = (3.7 \pm 6.6 \pm 0.6)\%$
• $A_{CP}(\Xi^+_c \to \Sigma^+ \pi^+ \pi^-) = (9.5 \pm 6.8 \pm 0.5)\%$
• $A_{CP}(\Lambda^+_c \to p K^+ K^-) = (3.9 \pm 1.7 \pm 0.7)\%$
• $A_{CP}(\Lambda^+_c \to p \pi^+ \pi^-) = (0.3 \pm 1.0 \pm 0.2)\%$

In all cases, the first uncertainty is statistical and the second is systematic. The results are consistent with CP symmetry (zero asymmetry).

The paper also derives the sums of these asymmetries, which are predicted to be zero under U-spin symmetry:

• $A_{CP}(\Xi^+_c \to \Sigma^+ \pi^+ \pi^-) + A_{CP}(\Lambda^+_c \to p K^+ K^-) = (13.4 \pm 7.0 \pm 0.9)\%$
• $A_{CP}(\Xi^+_c \to \Sigma^+ K^+ K^-) + A_{CP}(\Lambda^+_c \to p \pi^+ \pi^-) = (4.0 \pm 6.6 \pm 0.7)\%$

These sums are consistent with the U-spin symmetry prediction of zero.

Significance and Claims
The authors state that these are the world's first measurements of CP asymmetries for individual hadronic three-body charmed-baryon decays. The results agree with CP symmetry and are consistent with U-spin symmetry predictions. The paper emphasizes that the uncertainties are predominantly statistical in origin. Consequently, the authors conclude that future measurements utilizing larger datasets from the Belle II experiment will be crucial for precisely searching for CP violation and testing U-spin sum rules in the charm sector. The paper does not claim evidence of new physics but establishes the baseline measurements necessary for such future tests.