Spin effects in the tau-lepton pair induced by anomalous magnetic and electric dipole moments

1. Problem Statement

The paper addresses the search for New Physics (NP) through the study of anomalous magnetic and electric dipole moments of the $\tau$ lepton. While the Standard Model (SM) predicts specific values for these moments, deviations could indicate NP contributions (e.g., from heavy particles in loops).

• Context: Recent measurements at the Belle experiment and observations of $\tau$-pair production at the LHC (in $pp$ and $PbPb$ collisions) have renewed interest in this area.
• Challenge: The $\tau$ lepton decays rapidly, making direct spin measurement impossible. Instead, spin information must be reconstructed from the kinematics of decay products.
• Core Issue: To accurately isolate NP signals (dipole moments) from SM backgrounds, one must rigorously account for spin correlations between the produced $\tau^-$ and $\tau^+$ pairs. Neglecting these correlations or treating them incorrectly can lead to significant biases in cross-section calculations and obscure NP signatures.

2. Methodology

The authors employ a theoretical framework combined with Monte Carlo (MC) simulation techniques to model $\tau$-pair production and decay.

Theoretical Framework

• Processes Studied:
  1. Photon-Photon Fusion: $\gamma\gamma \to \tau^-\tau^+$ (relevant for peripheral $PbPb$ collisions and ultra-peripheral $pp$ collisions).
  2. Quark-Antiquark Annihilation: $q\bar{q} \to \tau^-\tau^+$ (dominant in $pp$ collisions).
• Formalism:
  • The interaction vertices ($\gamma\tau\tau$ and $Z\tau\tau$) are extended to include dipole form-factors.
  • Electromagnetic Vertex: Includes Dirac ($F_1$), Pauli ($F_2 \equiv A$, anomalous magnetic), and electric dipole ($F_3 \equiv B$) form-factors.
  • Weak Vertex: Includes vector/axial-vector couplings and weak anomalous magnetic ($X$) and electric ($Y$) form-factors.
  • Effective Field Theory (SMEFT): The dipole moments are linked to dimension-6 Wilson coefficients ($C^{(6)}_{\tau B}, C^{(6)}_{\tau W}$), allowing constraints on high-energy theories.
  • Electroweak Corrections: For $q\bar{q}$ processes, the Improved Born Approximation (IBA) is used to include electroweak corrections.
• Spin Correlation Matrix:
  • The squared matrix element is expressed in terms of a spin-correlation matrix $R_{ij}$, where $i,j$ represent spin components of the $\tau^-$ and $\tau^+$.
  • The matrix elements depend on the invariant mass ($m_{\tau\tau}$) and scattering angle ($\theta$).
  • Symmetry properties of $R_{ij}$ are analyzed to identify which elements vanish in the SM but become non-zero with NP (e.g., off-diagonal elements related to CP violation).

Simulation and Reweighting

• Event Generation: Events are generated using Pythia 8.3 for $pp$ collisions at $\sqrt{s} = 13$ TeV.
• Decay Modeling: $\tau$ decays (specifically $\tau \to \rho \nu \to \pi \pi^0 \nu$) are modeled using the Tauola library.
• Reweighting Algorithm: The TauSpinner program is used to reweight the generated events. This allows the inclusion of spin correlations and NP dipole moments without regenerating events from scratch.
  • The authors define a specific transformation of the spin-correlation matrix elements ($R_{ij}$) to match the coordinate systems used in TauSpinner and Tauola.

3. Key Contributions

1. Implementation in TauSpinner: The authors successfully implemented extensions to the SM amplitudes for $\gamma\gamma \to \tau\tau$ and $q\bar{q} \to \tau\tau$ within the TauSpinner framework, specifically accounting for electromagnetic and weak dipole moments.
2. Spin Correlation Analysis: A detailed breakdown of the spin-correlation matrix elements ($R_{ij}$) was performed, identifying which components are sensitive to specific dipole moments ($A, B, X, Y$).
3. Observable Identification: The study identifies specific kinematic observables constructed from decay products (pions) that are sensitive to spin correlations and dipole moments, distinguishing them from those that are insensitive.
4. Coordinate System Unification: The paper clarifies the transformation between the theoretical spin-correlation frame and the experimental decay frame used in Monte Carlo tools, ensuring accurate reweighting.

4. Results

The authors present numerical results for various kinematic distributions, comparing the SM ($A=B=X=Y=0$) against scenarios with non-zero dipole moments.

For $\gamma\gamma \to \tau\tau$ (Low $m_{\tau\tau}$: 5–50 GeV)

• SM Effects: Spin correlations significantly affect the shape of the invariant mass ratio distribution ($m_{\rho\rho}/m_{\tau\tau}$), causing a 10–15% deviation around $m_{\rho\rho}/m_{\tau\tau} \approx 0.5$.
• NP Effects:
  • Transverse momentum ($p_T^\pi$) and energy ratios ($E_\rho/E_\tau$) are largely insensitive to spin correlations in the SM but show shape changes when NP is introduced.
  • The ratio $(SM+NP)/SM$ increases with $p_T^\pi$.
  • However, the authors note that without detailed detector studies, the utility of these shape changes for extracting NP parameters is unclear.

For $q\bar{q} \to \tau\tau$ (Near $Z$-peak: 65–150 GeV)

• Sensitivity: In the $Z$-peak region, the electromagnetic form-factors ($A, B$) have negligible impact, but the weak form-factors ($X, Y$) are the primary targets.
• Acoplanarity Angles ($\Psi$ and $\phi^*$):
  • These angles are highly sensitive to transverse-transverse and transverse-normal spin correlations.
  • $\Psi$ (Beam-Pion Angle): Sensitive to the $r_{yy}$ component. Non-zero $X$ flattens the cosine-like distribution.
  • $\phi^*$ (Angle between decay planes): Sensitive to the $r_{xy}$ component. Non-zero $Y$ shifts the phase of the distribution.
  • The distributions exhibit periodic characteristics dependent on the sign of the product of energy asymmetries ($y_{\tau^+} \cdot y_{\tau^-}$).
• Other Observables: The ratio $m_{\rho\rho}/m_{\tau\tau}$ and energy ratios show sensitivity to the longitudinal spin correlation component ($r_{tz}$) when $X \neq 0$.

5. Significance

• Enhanced Sensitivity: The paper demonstrates that exploiting spin correlations in the final $\tau$ pair significantly enhances the sensitivity of LHC experiments to New Physics signatures, particularly anomalous dipole moments.
• Bias Mitigation: It highlights that ignoring spin correlations can act as a bias in cross-section measurements, potentially masking or mimicking NP effects.
• Experimental Strategy: The identification of specific observables (like acoplanarity angles $\Psi$ and $\phi^*$) provides a concrete roadmap for experimentalists at the LHC to search for CP-violating and CP-conserving dipole moments in $\tau$-pair production.
• Tool Development: The integration of these effects into TauSpinner makes these advanced calculations accessible for reweighting existing LHC datasets, facilitating immediate phenomenological studies without the computational cost of full event generation.

In conclusion, the work bridges the gap between theoretical SMEFT predictions and experimental observables by providing a robust, reweighting-based method to analyze $\tau$-pair spin correlations, thereby offering a powerful tool to constrain the anomalous magnetic and electric dipole moments of the $\tau$ lepton.