Investigation on the photoproduction of bottom-charmed baryon within NRQCD

Technical Summary: Investigation on the photoproduction of bottom-charmed baryon within NRQCD

Problem Statement
Following the 2017 discovery of the doubly charmed baryon $\Xi_{cc}^{++}$, theoretical interest has shifted toward bottom-charmed baryons ($\Xi_{bc}$), which consist of two different heavy quarks ($b$ and $c$) and a light quark. Despite extensive theoretical predictions regarding their mass, lifetime, and production mechanisms, experimental confirmation of $\Xi_{bc}$ states remains elusive. While previous studies have focused primarily on S-wave production, the contributions of orbital excited P-wave states have been identified as non-negligible (reaching several percent) but require further in-depth analysis. This paper addresses the need for precise theoretical calculations to guide experimental exploration at future high-energy linear colliders, specifically the International Linear Collider (ILC) and the Compact Linear Collider (CLIC).

Methodology
The authors employ the Non-Relativistic Quantum Chromodynamics (NRQCD) factorization framework to analyze the photoproduction of ground and excited $\Xi_{bc}$ baryons. The study considers two primary production channels at $e^+e^-$ colliders:

1. Direct photoproduction: $\gamma + \gamma \to \Xi_{bc} + \bar{c} + \bar{b}$
2. Resolved photoproduction: $\gamma + g \to \Xi_{bc} + \bar{c} + \bar{b}$

The initial photons are modeled as originating from Laser Back-Scattering (LBS) of electron/positron beams, with parton gluons emitted from the resolved photon. The formation of the $\Xi_{bc}$ baryon is treated as a two-step process:

• Perturbative Stage: The production of a compact diquark state $\langle bc \rangle[n]$ along with anti-quarks $\bar{c}$ and $\bar{b}$. This is calculated using perturbative QCD (pQCD) at leading order ($O(\alpha_s^3)$), involving 20 Feynman diagrams for the $\gamma\gamma$ channel and 24 for the $\gamma g$ channel.
• Non-Perturbative Stage: The transition of the intermediate diquark state $\langle bc \rangle[n]$ into the physical baryon $\Xi_{bc}$ via hadronization, described by Long-Distance Matrix Elements (LDMEs).

The study investigates 12 intermediate diquark configurations characterized by their spin, orbital angular momentum, and color states:

• S-wave: $[1S_0]_{\bar{3}/6}$ and $[3S_1]_{\bar{3}/6}$
• P-wave: $[1P_1]_{\bar{3}/6}$, $[3P_0]_{\bar{3}/6}$, $[3P_1]_{\bar{3}/6}$, and $[3P_2]_{\bar{3}/6}$

Calculations are performed for center-of-mass energies $\sqrt{s} = 250, 500,$ and $1000$ GeV. The authors utilize specific input parameters for wave functions at the origin ($|R_{bc}(0)|$ and $|R'_{bc}(0)|$) derived from the K2O potential. A significant aspect of the methodology involves addressing theoretical uncertainties associated with the color-sextuplet ($h_6$) LDME, which lacks a clear correlation with potential models. The authors analyze scenarios where $h_6 = 0$, $h_6 = q^2 h_{\bar{3}}$, and $h_6 = h_{\bar{3}}$.

Key Results

• Cross Sections and Energy Dependence: The total cross-section increases with collision energy. At $\sqrt{s} = 250$ GeV, the direct $\gamma\gamma$ fusion channel dominates. However, at $\sqrt{s} \ge 500$ GeV, the resolved $\gamma g$ fusion channel becomes overwhelmingly dominant.
• State Contributions: Among all diquark states, the $[3S_1]_{\bar{3}}$ configuration contributes the most to the total production (approx. 42% in $\gamma\gamma$ and 39% in $\gamma g$). Within the P-wave sector, the $[3P_2]_{\bar{3}}$ state is the dominant contributor.
• P-wave vs. S-wave: The contribution of orbital excited P-wave $\Xi_{bc}$ baryons is found to be approximately 7% to 9% of the S-wave contribution across the studied energy range.
• Kinematic Distributions:
  • Transverse Momentum ($p_T$): In the low $p_T$ region, the resolved $\gamma g$ channel yields more events than the direct $\gamma\gamma$ channel. For $p_T > 40$ GeV, the trend reverses. Both channels show a downward trend in event rates as $p_T$ increases.
  • Rapidity ($y$): Events are concentrated within $y \in [-4.2, 4.2]$. The $\gamma g$ distribution exhibits a steeper slope compared to the more gradual $\gamma\gamma$ distribution.
  • Angular and Invariant Mass: The angular distributions ($\cos\theta_{35}, \cos\theta_{45}$) and invariant mass distributions ($s_{35}, s_{45}$) indicate that the $\Xi_{bc}$ baryon is most likely to move parallel to the two heavy antiquarks, with significant contributions when the heavy antiquarks move back-to-back.
• Event Rates: Assuming an integrated luminosity of $O(10^4) \text{ fb}^{-1}$, the estimated number of produced P-wave excited baryons ranges from $5.21 \times 10^4$ to $8.05 \times 10^4$ depending on energy. If these excited states decay to the ground state with $\sim 100\%$ probability, the total ground-state $\Xi_{bc}$ production could reach $\sim 1.10 \times 10^6$ events at 1000 GeV.
• Theoretical Uncertainty: The uncertainty regarding the color-sextuplet contribution ($h_6$) is significant. If $h_6$ is suppressed ($h_6=0$), the total event count decreases by up to 43.69% at 1000 GeV. However, the order of magnitude remains significant even under conservative assumptions.

Significance and Claims
The paper claims that the orbital excited P-wave contribution to bottom-charmed baryon photoproduction is non-negligible, providing a 7–9% enhancement to the S-wave production rate. This finding suggests that experimental searches at future linear colliders (ILC and CLIC) must account for excited states, as their decay could significantly enhance the observable yield of ground-state $\Xi_{bc}$ baryons. The study provides essential differential distributions (transverse momentum, rapidity, angular, and invariant mass) to guide experimental detection strategies. Furthermore, by quantifying the impact of the color-sextuplet uncertainty, the authors demonstrate that while theoretical precision is limited by non-perturbative effects, the overall feasibility of observing $\Xi_{bc}$ baryons remains robust across different collision energies.