Study of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>γ</mml:mi> <mml:mi>γ</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:msup> <mml:mi>τ</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>τ</mml:m

We study the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>γ</a:mi> <a:mi>γ</a:mi> <a:mo stretchy="false">→</a:mo> <a:msup> <a:mi>τ</a:mi> <a:mo>+</a:mo> </a:msup> <a:msup> <a:mi>τ</a:mi> <a:mo>−</a:mo> </a:msup> </a:math> process including full <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:msup> <d:mi>τ</d:mi> <d:mo>+</d:mo> </d:msup> <d:msup> <d:mi>τ</d:mi> <d:mo>−</d:mo> </d:msup> </d:math> spin information in Pb-Pb ultraperipheral collisions and at lepton colliders. We present the predictions for the corresponding cross sections and spin correlations at next-to-leading order (NLO) electroweak precision, and find that the NLO electroweak contributions are numerically small for the observables considered. Additionally, we use the spin correlations obtained in this paper to analyze the quantum entanglement in the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:msup> <f:mi>τ</f:mi> <f:mo>+</f:mo> </f:msup> <f:msup> <f:mi>τ</f:mi> <f:mo>−</f:mo> </f:msup> </f:math> system of the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>γ</h:mi> <h:mi>γ</h:mi> <h:mo stretchy="false">→</h:mo> <h:msup> <h:mi>τ</h:mi> <h:mo>+</h:mo> </h:msup> <h:msup> <h:mi>τ</h:mi> <h:mo>−</h:mo> </h:msup> </h:math> process. Our results show that there is a genuine entangled configuration near the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msup> <k:mi>τ</k:mi> <k:mo>+</k:mo> </k:msup> <k:msup> <k:mi>τ</k:mi> <k:mo>−</k:mo> </k:msup> </k:math> invariant mass threshold. This work is helpful for studying the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>τ</m:mi> </m:math> -pair production induced by photon-photon collision at high-energy colliders.