# TRANSVERSE SPIN CORRELATIONS IN ${{\boldsymbol Z}}$ $\rightarrow$ ${{\boldsymbol \tau}^{+}}{{\boldsymbol \tau}^{-}}$

The correlations between the transverse spin components of ${{\mathit \tau}^{+}}{{\mathit \tau}^{-}}$ produced in ${{\mathit Z}}$ decays may be expressed in terms of the vector and axial-vector couplings:
$\mathit C_{\mathit TT}$ = ${\vert \mathit g{}^{{{\mathit \tau}}}_{A}\vert ^2−\vert \mathit g{}^{{{\mathit \tau}}}_{V}\vert ^2\over \vert \mathit g{}^{{{\mathit \tau}}}_{A}\vert ^2+\vert \mathit g{}^{{{\mathit \tau}}}_{V}\vert ^2}$
$\mathit C_{\mathit TN}$ = $-2{\vert \mathit g{}^{{{\mathit \tau}}}_{A}\vert \vert \mathit g{}^{{{\mathit \tau}}}_{V}\vert \over \vert \mathit g{}^{{{\mathit \tau}}}_{A}\vert ^2+\vert \mathit g{}^{{{\mathit \tau}}}_{V}\vert ^2}$ sin$(\Phi _{\mathit g{}^{{{\mathit \tau}}}_{\mathit V}}−\Phi _{\mathit g{}^{{{\mathit \tau}}}_{\mathit A}})$
$\mathit C_{\mathit TT}$ refers to the transverse-transverse (within the collision plane) spin correlation and $\mathit C_{\mathit TN}$ refers to the transverse-normal (to the collision plane) spin correlation.
The longitudinal ${{\mathit \tau}}$ polarization $\mathit P_{{{\mathit \tau}}}$ (= $−\mathit A_{{{\mathit \tau}}}$) is given by:
$\mathit P_{{{\mathit \tau}}}$ = $-2{\vert \mathit g{}^{{{\mathit \tau}}}_{A}\vert \vert \mathit g{}^{{{\mathit \tau}}}_{V}\vert \over \vert \mathit g{}^{{{\mathit \tau}}}_{A}\vert ^2+\vert \mathit g{}^{{{\mathit \tau}}}_{V}\vert ^2}$ cos $(\Phi _{\mathit g{}^{{{\mathit \tau}}}_{\mathit V}}−\Phi _{\mathit g{}^{{{\mathit \tau}}}_{\mathit A}})$
Here $\Phi$ is the phase and the phase difference $\Phi _{\mathit g{}^{{{\mathit \tau}}}_{\mathit V}}−\Phi _{\mathit g{}^{{{\mathit \tau}}}_{\mathit A}}$ can be obtained using both the measurements of $\mathit C_{\mathit TN}$ and $\mathit P_{{{\mathit \tau}}}$.

# $\boldsymbol C_{\boldsymbol TN}$ INSPIRE search

VALUE EVTS DOCUMENT ID TECN  COMMENT
$0.08$ $\pm0.13$ $\pm0.04$ 120k 1
 1997 D
ALEP ${\it{}E}^{\it{}ee}_{\rm{}cm}$= $91.2$ GeV
1  BARATE 1997D combine their value of $\mathit C_{\mathit TN}$ with the world average $\mathit P_{{{\mathit \tau}}}$ = $-0.140$ $\pm0.007$ to obtain tan $\Phi _{\mathit g{}^{{{\mathit \tau}}}_{\mathit V}}−\Phi _{\mathit g{}^{{{\mathit \tau}}}_{\mathit A}}$ = $-0.57$ $\pm0.97$.
References:
 BARATE 1997D
PL B405 191 Measurement of the Transverse Spin Correlations in the Decay ${{\mathit Z}}$ $\rightarrow$ ${{\mathit \tau}^{+}}{{\mathit \tau}^{-}}$