The feasibility of single ${\Lambda}$ production via ${\ell}^{-}$ $+$ $p$ ${\to}$ ${\Lambda}$ $+$ ${\nu}_{\ell}$ at $e^{+}e^{-}$ colliders

Technical Summary: Feasibility of Single $\Lambda$ Production via $\ell^- + p \to \Lambda + \nu_\ell$ at $e^+e^-$ Colliders

Problem Statement
The paper investigates the feasibility of observing single $\Lambda$ hyperon production through the lepton-nucleon deep inelastic scattering (LNDIS) process, $\ell^- + p \to \nu_\ell + \Lambda$, within the experimental environment of electron-positron ($e^+e^-$) colliders. While the inverse process, hyperon semileptonic decay (HSD, $\Lambda \to p + \ell^- + \bar{\nu}_\ell$), is a standard tool for probing baryon structure and extracting the Cabibbo–Kobayashi–Maskawa (CKM) matrix element $V_{us}$, it suffers from experimental challenges including the short $\Lambda$ lifetime, undetectable neutrinos, and difficult background separation. The authors propose utilizing the LNDIS process at $e^+e^-$ colliders as a complementary probe. In this scenario, incident charged leptons ($\ell^- = e^-, \mu^-$) originate from the decays of resonances ($J/\psi, \psi(2S), \Upsilon(1S), \Upsilon(2S), Z^0$) produced in $e^+e^-$ collisions. These leptons then scatter off stationary protons embedded in the detector materials (e.g., beam pipes). A key motivation is to test the universality of lepton-gauge boson couplings and potentially resolve the existing $2.5\sigma$ discrepancy in the unitarity of the CKM matrix.

Methodology
The study employs a theoretical framework to calculate the differential and total cross sections for the LNDIS process.

• Kinematics: The incident leptons are modeled as originating from resonance decays ($R \to \ell^+\ell^-$) at various center-of-mass (CM) energies. The scattering occurs on stationary protons in the surrounding detector materials.
• Form Factors: The hadronic transition matrix elements are parameterized using baryonic transition form factors. The authors utilize the $z$-expansion scheme to extrapolate these form factors from the time-like region (relevant for HSD, where $q^2 \approx 0$) to the deep space-like region (relevant for LNDIS, where $Q^2 = -q^2$ can be large).
• Theoretical Frameworks: Two distinct frameworks are used to determine the form factors:
  1. Quantum Chromodynamics (QCD) Sum Rules (QCDSR).
  2. Lattice QCD (LQCD).
• Calculation: The differential cross section $d\sigma/d\cos\theta$ is derived using helicity amplitudes and the squared scattering amplitude. The total cross sections are integrated over the kinematic phase space for various resonance energies.
• Event Yield Estimation: The expected number of signal events ($N_{sig}$) is calculated by multiplying the computed cross sections by the integrated luminosity and the effective proton areal density ($n_t$) of the detector beam pipes (specifically modeling the BESIII beam pipe composition of Au, Be, and oil).

Key Results

• Cross Section Behavior: The calculated cross sections increase with the center-of-mass energy of the incident lepton. The maximum cross section is observed when the incident lepton originates from $Z^0$ boson decays.
• Sensitivity to Form Factors: The results exhibit extreme sensitivity to the choice of theoretical framework for the form factors.
  • Cross sections calculated using QCDSR form factors are approximately three to four orders of magnitude larger than those calculated using LQCD form factors.
  • For the $Z^0$ case, the QCDSR prediction is $\sim 342.8$ fb, while the LQCD prediction is $\sim 1284$ fb (with very large uncertainties, e.g., $+7269/-766$ fb).
• Kinematic Distribution: The differential cross section is dominated by the form factors $F_2$ and $G_2$ in the high $Q^2$ region. The distribution is forward-peaked (small $\theta$), resembling a $t$-channel process, though the $H_{0,0}$ contribution is suppressed by $\sin^2\theta$ at very small angles.
• Event Yields: Despite the high flux of resonances at current (BESIII, Belle II) and future (STCF, CEPC, FCC-ee) facilities, the expected signal yields are negligible.
  • For BESIII (with $>10^{10}$ $J/\psi$ events), the expected yield is approximately $2 \times 10^{-6}$ events.
  • For future high-luminosity facilities, the yields remain well below 1 event (e.g., $\sim 0.01$ to $0.1$ events) even under optimistic assumptions using QCDSR cross sections.

Significance and Claims
The paper concludes that the observation of the single $\Lambda$ production process $\ell^- + p \to \nu_\ell + \Lambda$ at $e^+e^-$ colliders is currently highly challenging and likely unfeasible given the extremely low expected event rates.

• Theoretical Uncertainty: The primary limitation is the lack of precise data on baryonic form factors in the deep space-like region ($Q^2 > 0.1$ GeV$^2$). The current reliance on extrapolating from the $q^2 \approx 0$ region introduces significant theoretical uncertainties, making the predicted cross sections unreliable.
• Implication of Observation: The authors state that because the Standard Model prediction yields effectively zero events, any future experimental observation of this process at $e^+e^-$ colliders would likely indicate a significant deviation from present theoretical predictions, potentially signaling new physics.
• Future Needs: The study underscores the critical need for improved, precise determinations of baryonic transition form factors in the space-like region to reduce theoretical uncertainties and enable meaningful constraints on $V_{us}$ and baryon structure.

In summary, while the proposed mechanism offers a unique kinematic environment for studying LNDIS, the paper serves as a feasibility study demonstrating that current experimental capabilities and theoretical knowledge are insufficient for a positive detection, highlighting the process as a potential, albeit currently inaccessible, probe for new physics.