Impact of $N^*$ and $Λ^*$ resonances on $CP$ violation in $Λ_b^0$ decays

Abstract: The four-body decay $Λ<em>b^0\to pK<sup>-π<sup>+π<sup>-$ has led to the first observation of baryonic $CP$ violation. However, the underlying subprocesses $Λ_b^0\to N^* M$ and $Λ_b^0\to Λ^* M$, as well as the roles of excited nucleon ($N<sup>*$) and hyperon ($Λ<sup>*$) resonances, remain largely unexplored. Within the constituent quark model, we identify the relevant resonant states contributing to these underlying two-body transitions, including $N(1535)$, $N(1520)$, $Λ(1670)$, $Λ(1690)$, together with the remaining $1P$-wave baryon states. We obtain the resonant branching fraction $${\cal B}(Λ_b^0\to pK^{-π^{+π^-)}} =(30.0^{+2.8+4.0}</em>{-1.3-3.4}\pm1.8)\times10<sup>{-6}$$, while the resulting $${\cal A}_{CP}(Λ_b^0\to pK<sup>-π<sup>+π<sup>-)=(3.18\pm0.11\pm0.13\pm0.11)\%$$ provides a natural interpretation of the first observed baryonic $CP$ asymmetry. Our analysis establishes the first comprehensive framework for quantifying the impact of excited baryon resonances in multi-body beauty-baryon decays, with the associated mechanism generally applicable to baryonic $CP$ asymmetries.

• The paper demonstrates that incorporating N* and Λ* resonances via a CQM framework explains observed CP asymmetries, notably predicting ~7% in dominant channels.
• It decomposes the four-body Λb⁰ decay into resonance-driven two-body subprocesses, yielding branching fraction and asymmetry results consistent with experiments.
• The study establishes testable constraints from quark model selection rules and provides predictions for related decay channels, guiding future experimental investigations.

Impact of $N^*$ and $\Lambda^*$ Resonances on $CP$ Violation in $\Lambda_b^0$ Decays

Introduction and Motivation

The recent experimental observation of $CP$ violation in the four-body decay $\Lambda_b^0 \to p K^- \pi^+ \pi^-$, with $CP$ asymmetry at $(2.45 \pm 0.46 \pm 0.10)\%$ and $5.2\sigma$ significance, marks a pivotal advance in the study of baryonic $CP$ phenomena. Unlike prior developments in mesonic systems, this baryonic $\Lambda^*$0-asymmetry provides new opportunities and challenges for understanding matter-antimatter asymmetry and testing the Standard Model (SM) in the baryon sector. However, the underlying hadronic dynamics—particularly the roles of excited baryonic resonances $\Lambda^*$1 and $\Lambda^*$2 in mediating these multi-body decays—remained theoretically unresolved. This work establishes a constituent quark model (CQM) framework for quantifying the contributions of $\Lambda^*$3 and $\Lambda^*$4 resonances to $\Lambda^*$5 violation in $\Lambda^*$6 decays (2606.08248).

Theoretical Framework: Constituent Quark Model Treatment

The analysis decomposes the four-body $\Lambda^*$7 decay into resonance-dominated two-body subprocesses:

• $\Lambda^*$8,
• $\Lambda^*$9,
• $CP$0,

where $CP$1 and $CP$2 are excited nucleons and hyperons, and $CP$3 ($CP$4, $CP$5, $CP$6) and $CP$7 ($CP$8, $CP$9) are intermediate mesonic states selected kinematically. The CQM is employed to construct the relevant baryonic and mesonic wave functions, including flavor, color, spin, and spatial components using Jacobi coordinates and harmonic oscillator bases.

The effective Hamiltonian is built from leading order contributions and QCD penguin operators, with Wilson coefficients and CKM factors identified for transitions $\Lambda_b^0$0 and $\Lambda_b^0$1. The generalized factorization approach is implemented, parameterizing nonfactorizable effects via an effective color number $\Lambda_b^0$2 variation.

Crucially, the work identifies all relevant $\Lambda_b^0$3-wave $\Lambda_b^0$4 and $\Lambda_b^0$5 resonances contributing in the relevant kinematic region, including $\Lambda_b^0$6, $\Lambda_b^0$7, $\Lambda_b^0$8, $\Lambda_b^0$9, $CP$0, $CP$1, $CP$2, and $CP$3. All possible channels (27 in total) are considered to exhaust the resonance contributions.

Numerical Implementation and Results

Quark model parameters (constituent masses, oscillator parameters) are calibrated to existing strong decay and weak decay data and cross-checked for internal consistency. Theoretical uncertainties are propagated from CKM elements, $CP$4, and hadron wavefunction parameters.

The calculated resonance-driven branching fraction for $CP$5 is found to be

$CP$6

and the corresponding $CP$7 asymmetry is

$CP$8

These results are consistent within errors with the experimental measurement of the $CP$9 asymmetry and branching ratios.

A key numerical result is that the $\Lambda_b^0 \to p K^- \pi^+ \pi^-$0 resonance sector—specifically $\Lambda_b^0 \to p K^- \pi^+ \pi^-$1 and $\Lambda_b^0 \to p K^- \pi^+ \pi^-$2—dominates the $\Lambda_b^0 \to p K^- \pi^+ \pi^-$3 channel with a predicted $\Lambda_b^0 \to p K^- \pi^+ \pi^-$4 asymmetry of $\Lambda_b^0 \to p K^- \pi^+ \pi^-$5, in agreement with the experimental value for this channel. This is notably larger than the $\Lambda_b^0 \to p K^- \pi^+ \pi^-$6 level predicted for the penguin-dominated $\Lambda_b^0 \to p K^- \pi^+ \pi^-$7 and $\Lambda_b^0 \to p K^- \pi^+ \pi^-$8 subchannels, reflecting their lack of interference with tree-level contributions.

An explicit suppression is found for the $\Lambda_b^0 \to p K^- \pi^+ \pi^-$9 contribution, which is kinematically forbidden ($CP$0 below threshold), and for $CP$1 due to zero overlap of the spin structure in the CQM framework. This represents a nontrivial constraint for resonance modeling in baryonic weak decays.

The study also predicts, for the related tree-dominated channel $CP$2,

$CP$3

which is proposed as a testable prediction.

Theoretical and Phenomenological Implications

The analysis demonstrates that excited baryon resonances ($CP$4) are critical for the observed $CP$5 violation in multi-body $CP$6 decays. The hierarchy of $CP$7 asymmetry in different resonance subchannels is shown to correspond directly to the interplay of tree and penguin operator contributions at the amplitude level and their associated strong and weak phases. Moreover, the quark model-based selection rules (vanishing overlap for certain transitions) provide explicit, testable constraints for future measurements.

The results reinforce that the baryonic sector can yield strong $CP$8-asymmetry signals depending on resonance structure and the specific admixture of SM effective operators. This has significant implications for flavor physics experiments, as it shows that observable $CP$9 violation in baryons is dominated by hadronic mechanisms analogous to—but distinct from—mesonic cases.

Furthermore, the approach can be generalized to other beauty-baryon decay topologies and to the study of other resonance structures (e.g., $(2.45 \pm 0.46 \pm 0.10)\%$0, $(2.45 \pm 0.46 \pm 0.10)\%$1), supporting a unified understanding of $(2.45 \pm 0.46 \pm 0.10)\%$2 violation across baryonic systems.

Future Directions

This framework opens several research avenues:

• Improved treatment of nonfactorizable QCD effects and resonance-continuum interference using lattice QCD or amplitude analysis techniques.
• Systematic extension to charm baryon decays and multi-strange baryons.
• Investigation of possible new physics scenarios in baryonic $(2.45 \pm 0.46 \pm 0.10)\%$3-violating observables, exploiting differences between tree and penguin dominance in specific resonance channels.
• Cross-checks and refinements using accumulating higher-statistics data from LHCb and future hadron collider experiments.

Conclusion

This work provides the first comprehensive constituent quark model treatment of $(2.45 \pm 0.46 \pm 0.10)\%$4 and $(2.45 \pm 0.46 \pm 0.10)\%$5 resonance contributions to $(2.45 \pm 0.46 \pm 0.10)\%$6 violation in $(2.45 \pm 0.46 \pm 0.10)\%$7 decays. The predicted branching fractions and $(2.45 \pm 0.46 \pm 0.10)\%$8 asymmetries align with current experimental observations, and the formalism enables systematic identification of resonance contributions to multi-body baryonic decay amplitudes. The conclusions underscore the central role of excited baryon dynamics in baryonic $(2.45 \pm 0.46 \pm 0.10)\%$9 violation and highlight the predictive power of CQM-based approaches for future experimental verification.