Study $γγ\to τ^+τ^-$ process including $τ^+ τ^-$ spin information in Pb-Pb ultraperipheral collision and at Lepton collider

1. Problem Statement

The paper addresses the need for high-precision theoretical predictions for the production of tau-lepton pairs ($\tau^+\tau^-$) via photon-photon ($\gamma\gamma$) fusion. While the $\tau$ lepton is a crucial probe for electroweak couplings and Beyond Standard Model (BSM) physics, its rapid decay ($\sim 2.9 \times 10^{-13}$ s) necessitates the reconstruction of its polarization and spin correlations through decay products.

Previous studies have focused on unpolarized cross-sections or leading-order (LO) spin effects. However, with recent observations of photon-induced $\tau$-pair production in Ultra-Peripheral Collisions (UPC) at the LHC and plans for future Lepton Colliders (LC), there is a critical need for:

1. Next-to-Leading Order (NLO) Electroweak (EW) precision calculations including full spin information.
2. A comprehensive analysis of spin correlations and quantum entanglement in the $\tau^+\tau^-$ system across different collider environments (Pb-Pb UPC, $e^+e^-$, and $\mu^+\mu^-$).

2. Methodology

The authors employ a rigorous theoretical framework combining the Equivalent Photon Approximation (EPA) with NLO EW corrections and spin density matrix formalism.

• Production Mechanisms:
  • Pb-Pb UPC: Utilizes the Equivalent Photon Approximation (EPA) based on the Electric Dipole Form Factor (EDFF). The photon flux is derived from coherently emitted photons of heavy ions, scaling as $Z^4$.
  • Lepton Colliders ($e^+e^-$ and $\mu^+\mu^-$): Uses the Weizsäcker-Williams Approximation (WWA) to describe initial-state radiation (bremsstrahlung) from incoming leptons.
• Theoretical Calculation:
  • The process is modeled as $\gamma(p_1, \lambda_1) + \gamma(p_2, \lambda_2) \to \tau^-(k_1, s_1) + \tau^+(k_2, s_2)$.
  • NLO EW Corrections: The authors calculate NLO electroweak corrections to the subprocess, extending their previous work on unpolarized production to include full spin dependence.
  • Spin Density Matrix: The production density matrix $R$ is decomposed using the spin density matrix formalism. It is characterized by a scalar coefficient (unpolarized rate) and a tensor $\tilde{C}_{ij}$ representing spin correlations.
  • Basis Definition: The spin correlations are expanded in an orthonormal basis $\{\hat{r}, \hat{k}, \hat{n}\}$ defined in the $\tau^+\tau^-$ zero-momentum frame (ZMF), where $\hat{k}$ is the flight direction, $\hat{p}$ is the photon direction, and $\hat{n} = \hat{p} \times \hat{k}$.
• Observables:
  • Spin Correlation Coefficients ($C_{ab}$): Defined as double spin asymmetries, e.g., $C_{rr}, C_{nn}, C_{kk}$.
  • Quantum Entanglement: Quantified using the concurrence and the rotation-invariant entanglement criterion $D = \frac{1}{3}\text{tr}[C]$. A state is entangled if $-1 \le D < -1/3$.

3. Key Contributions

1. First NLO EW Spin Analysis: This work provides the first NLO electroweak precision predictions for $\gamma\gamma \to \tau^+\tau^-$ that include full $\tau^+\tau^-$ spin information for both UPC and Lepton Collider environments.
2. Comprehensive Collider Comparison: It systematically compares the cross-sections and spin correlations across three distinct environments: LHC Pb-Pb UPC, future $e^+e^-$ colliders (e.g., CEPC, CLIC), and Muon colliders.
3. Entanglement Threshold Analysis: The paper moves beyond inclusive averages to perform a differential analysis of quantum entanglement as a function of the $\tau^+\tau^-$ invariant mass ($m_{\tau\tau}$), identifying specific kinematic regions where genuine entanglement occurs.

4. Key Results

A. Cross Sections and NLO Corrections

• Magnitude: The NLO EW corrections are numerically small (on the order of permille, i.e., $< 1\%$) for all considered observables.
• Pb-Pb UPC: The total cross-section is $\sim 1$ mb. The relative NLO correction ($\delta$) increases slightly with collision energy ($\sqrt{s_{NN}}$), reaching $\sim 0.96\%$ at 5.52 TeV.
• Lepton Colliders:
  • For $e^+e^-$, the cross-section ranges from 0.513 nb (at 250 GeV) to 1.344 nb (at 3 TeV). The relative NLO correction decreases with energy (from $\sim 0.8\%$ to $0.26\%$).
  • For $\mu^+\mu^-$, the cross-section is slightly lower than in $e^+e^-$ due to mass-dependent PDF effects. At 10 TeV, the NLO correction becomes negative, dominated by weak corrections.
• Conclusion: The NLO EW effects provide a stable, precise Standard Model (SM) baseline but do not drastically alter the LO predictions.

B. Spin Correlation Coefficients ($C_{ab}$)

• Threshold Behavior: Near the $\tau^+\tau^-$ production threshold ($m_{\tau\tau} \approx 2m_\tau$), the system is dominated by the $^1S_0$ state (spin singlet). This results in antiparallel spins, leading to negative values for $C_{rr}$ and $C_{kk}$ (approx. $-0.3$).
• High-Energy Behavior: As energy increases, helicity conservation drives the system toward a $^3S_1$-like state (spin triplet), causing $C_{rr}$ and $C_{kk}$ to become less negative (transitioning toward positive values).
• Angular Dependence:
  • $C_{nn}$ (normal to the scattering plane) remains large and negative due to total angular momentum conservation.
  • $C_{pp}$ (along the beam direction) is consistently small, indicating suppressed spin correlations along the photon beam axis.
• NLO Impact: NLO corrections induce only minor rescaling of the coefficients without changing their signs or energy trends.

C. Quantum Entanglement

• Inclusive Result: When integrating over the full phase space, the average entanglement criterion $D$ is greater than $-1/3$, indicating no net entanglement for the inclusive sample.
• Differential Result (Crucial Finding): When analyzing $D$ as a function of the invariant mass $m_{\tau\tau}$:
  • Near Threshold: $D$ drops significantly below $-1/3$ (reaching values near $-1$), confirming the presence of genuine quantum entanglement in the spin-singlet configuration.
  • High Mass: $D$ rises above the threshold, indicating a transition to separable or less entangled states.
• NLO Effect: NLO EW corrections cause a slight upward shift in $D$, slightly reducing the degree of entanglement near the threshold but preserving the qualitative conclusion.

5. Significance

• Precision Baseline: The results establish a high-precision SM baseline for future experiments at the LHC (Run 3 and HL-LHC) and future colliders (CEPC, CLIC, Muon Colliders). This is essential for distinguishing SM predictions from potential BSM signals (e.g., anomalous couplings or CP violation).
• Quantum Information at Colliders: The paper provides concrete evidence that high-energy colliders can serve as laboratories for studying quantum entanglement in massive particle systems. It demonstrates that while inclusive measurements might wash out entanglement, differential measurements near the kinematic threshold are required to observe it.
• Experimental Guidance: By identifying the specific kinematic region (near $m_{\tau\tau} \approx 2m_\tau$) where entanglement is maximal, the study guides experimentalists on how to optimize event selection to test Bell inequalities and study quantum correlations in the $\tau$ sector.

In summary, this work bridges high-energy particle physics and quantum information science, offering a precise theoretical toolkit for analyzing spin dynamics and quantum entanglement in $\tau$-pair production across the spectrum of modern and future colliders.