TauSpinner algorithms for including spin and New Physics effects in $\bar q q \rightarrow Z/γ^* \to ττ$ process

Technical Summary: TauSpinner Algorithms for Including Spin and New Physics Effects in $\bar{q}q \to Z/\gamma^* \to \tau\tau$

Problem Statement
The paper addresses the need for precise theoretical tools to investigate anomalous New Physics (NP) contributions to the dipole and weak dipole moments of the $\tau$ lepton. While the $\tau$-pair production process in $Z$-boson decay ($\bar{q}q \to Z/\gamma^* \to \tau\tau$) has been studied extensively at the LHC, precise measurements of transverse spin correlations remain underexplored. Existing general-purpose Monte Carlo (MC) generators (e.g., Pythia, Sherpa) often lack specific spin correlations or NP effects in generated event samples. The challenge lies in efficiently incorporating these complex spin-dependent effects and NP modifications into existing datasets without regenerating events, particularly given the intricate nature of transverse spin correlations and the necessity of including Electroweak (EW) corrections.

Methodology
The authors extend the TauSpinner program, a tool designed for event reweighting, to handle the $\bar{q}q \to \tau^+\tau^-$ process with full control over spin correlations and NP effects. The methodology involves:

1. Formalism Extension:

   • Improved Born Approximation (IBA): The matrix elements are calculated within the IBA framework, incorporating EW radiative corrections via form-factors (using the Dizet library) rather than simple effective couplings.
   • Dipole Moments: The formalism includes anomalous magnetic ($A$), electric ($B$), weak magnetic ($X$), and weak electric ($Y$) dipole moments for the $\tau$-lepton couplings to $\gamma$ and $Z$ bosons.
   • Phase-Shifts: A novel feature allows for the introduction of a phase-shift ($\Phi$) between the vector and axial-vector couplings of the $Z$ boson to $\tau$ leptons. This is implemented by rescaling EW form-factors to mimic NP effects, distinct from direct coupling redefinition.
   • Spin-Correlation Matrix ($R_{ij}$): Explicit analytical formulas for the spin-correlation matrix elements ($R_{ij}$) are derived, including contributions from $\gamma$, $Z$, and $Z\gamma$ interference. The authors provide explicit expressions for transverse-transverse (TT), transverse-normal (TN), and longitudinal elements in the IBA.

2. Algorithm Implementation:

   • Reweighting: The core algorithm calculates a weight ($wt$) for each event generated with a standard model (SM) matrix element. This weight is the product of a production weight ($wt_{prod}$) and a spin weight ($wt_{spin}$).
   • Convention Matching: A significant technical component involves reconciling the frame conventions and sign conventions of the theoretical matrix elements with those used in the TauSpinner and Tauola decay libraries. This includes specific rotations and sign flips for the spin-correlation matrix elements to ensure consistency when contracted with $\tau$ decay polarimetric vectors.
   • Frame Selection: The implementation supports different reference frames (Collins-Soper and Mustraal) for defining the transverse spin axes, with options to handle events with high-$p_T$ jets by adjusting reference frames to absorb hadronic activity.

3. Numerical Validation:

   • Simulations were performed using events generated by Pythia 8.3 for $pp$ collisions at 13 TeV, focusing on the invariant mass range $m_{\tau\tau} = 65\text{--}150$ GeV.
   • $\tau$ decays were modeled via $\tau^\pm \to \rho^\pm \nu_\tau \to \pi^\pm \pi^0 \nu_\tau$ using the Tauola library.
   • The impact of NP parameters ($\Phi = \pm 0.1$, $X=0.1$, $Y=0.1$) was evaluated against the SM baseline.

Key Contributions

• Algorithmic Extension: The paper presents the implementation of IBA with EW corrections and arbitrary dipole moments into the TauSpinner reweighting algorithm for the $\bar{q}q \to \tau\tau$ channel.
• Phase-Shift Formalism: It introduces a method to simulate NP effects via a phase-shift between vector and axial-vector $Z\tau\tau$ couplings, providing a flexible way to test CP-violating signatures.
• Explicit Formulas: The authors provide explicit analytical expressions for the spin-correlation matrix elements ($R_{ij}$) in the IBA, including the dependence on complex EW form-factors and the introduced phase-shifts.
• Technical Integration: The paper details the necessary adjustments for frame orientation and sign conventions required to interface the theoretical matrix elements with the TauSpinner/Tauola software stack.

Results

• Spin-Correlation Sensitivity:
  • The transverse-transverse correlation element ($r_{xx}$) is found to be close to 0.5 at the $Z$ peak and largely insensitive to the phase-shift $\Phi$.
  • The transverse-normal correlation element ($r_{xy}$) shows strong sensitivity to the phase-shift $\Phi$ and the weak electric form-factor $Y$.
  • The longitudinal-transverse element ($r_{tz}$) is sensitive to the weak anomalous magnetic moment $X$.
  • The element $r_{yy}$ is significantly affected by the weak magnetic moment $X$.
• Experimental Observables:
  • Kinematic distributions sensitive to longitudinal polarization ($E_\rho/E_\tau$, $E_{\pi^\pm}/E_\rho$, $m_{\rho\rho}/m_{\tau\tau}$) show minimal changes for the tested NP scenarios, except for effects induced by $X$ on $r_{tz}$.
  • Acoplanarity angles ($\Psi$ and $\phi^*$), which are sensitive to transverse spin correlations, exhibit phase shifts or modulation cancellations depending on the specific NP parameters (e.g., $\Phi$ shifts the phase of $\Psi$ and $\phi^*$; $Y$ alters the cosine-like modulation).
• Magnitude of Effects: The impact of the tested NP parameters on the integrated cross-section is negligible (per mille level) at the $Z$ pole, confirming that the primary sensitivity lies in spin-correlation observables.

Significance and Claims
The paper claims that the extended TauSpinner algorithm provides a robust tool for studying spin correlations and NP effects in $\tau$-pair production without the computational cost of full event regeneration. The authors emphasize that:

1. Sensitivity: Exploring spin correlations, particularly transverse ones, can significantly improve experimental sensitivity to New Physics effects that might otherwise be missed.
2. Bias Mitigation: Neglecting spin correlations in experimental analyses can bias cross-section measurements and potentially mimic the presence of New Physics.
3. Flexibility: The implementation allows for the simultaneous study of EW corrections and NP effects, which is crucial as EW corrections can introduce phases that mimic NP signatures.
4. Practicality: The tool is ready for use with existing event samples (e.g., from Pythia) and experimental data, provided the necessary configuration (PDFs, form-factors, and NP parameters) is set up as described in the appendices.

The authors conclude that while the current implementation uses simplified kinematics (2$\to$2 effective Born frames), it offers a necessary foundation for future analyses involving detector effects and more complex final states.