Probing the electromagnetic dipole moment of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>τ</mml:mi> </mml:math> lepton in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:m

High-precision measurements of the lepton’s electromagnetic dipole moments provide a powerful probe for testing the Standard Model. A nonzero value of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>τ</a:mi> </a:math> lepton’s electric(weak) dipole moment ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msubsup> <c:mi>d</c:mi> <c:mi>τ</c:mi> <c:mi>γ</c:mi> </c:msubsup> </c:math> , <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msubsup> <e:mi>d</e:mi> <e:mi>τ</e:mi> <e:mi>Z</e:mi> </e:msubsup> </e:math> ) would serve as a smoking gun of the new physics. On the one hand, current and future high energy colliders offer an ideal environment for such measurements. On the other hand, it is essential to investigate the optimal measurement method for extracting not only <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msubsup> <g:mi>d</g:mi> <g:mi>τ</g:mi> <g:mi>γ</g:mi> </g:msubsup> </g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msubsup> <i:mi>d</i:mi> <i:mi>τ</i:mi> <i:mi>Z</i:mi> </i:msubsup> </i:math> , but also the anomalous magnetic dipole moment ( <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mi>a</k:mi> <k:mi>τ</k:mi> </k:msub> </k:math> ). In this work, we analyze the precision of observables, particularly optimal observables, for determining these physics quantities through the process <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msup> <m:mi>e</m:mi> <m:mo>+</m:mo> </m:msup> <m:msup> <m:mi>e</m:mi> <m:mo>−</m:mo> </m:msup> <m:mo stretchy="false">→</m:mo> <m:msup> <m:mi>γ</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo>/</m:mo> <m:mi>Z</m:mi> <m:mo stretchy="false">→</m:mo> <m:msup> <m:mi>τ</m:mi> <m:mo>+</m:mo> </m:msup> <m:msup> <m:mi>τ</m:mi> <m:mo>−</m:mo> </m:msup> </m:math> with <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>τ</q:mi> </q:math> leptons undergoing (semi)leptonic and hadronic decays at the center-of-mass energies at the Z pole mass and significantly below <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msub> <s:mi>m</s:mi> <s:mi>Z</s:mi> </s:msub> </s:math> . By considering the full kinematic information of the decay products, we find that the sensitivities to the weak dipole moments, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>Im</u:mi> <u:mo stretchy="false">[</u:mo> <u:msubsup> <u:mi>d</u:mi> <u:mi>τ</u:mi> <u:mi>Z</u:mi> </u:msubsup> <u:mo stretchy="false">]</u:mo> </u:math> and <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi>Re</y:mi> <y:mo stretchy="false">[</y:mo> <y:msubsup> <y:mi>d</y:mi> <y:mi>τ</y:mi> <y:mi>Z</y:mi> </y:msubsup> <y:mo stretchy="false">]</y:mo> </y:math> , can reach <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mrow> <cb:msup> <cb:mrow> <cb:mn>10</cb:mn> </cb:mrow> <cb:mrow> <cb:mo>−</cb:mo> <cb:mn>21</cb:mn> </cb:mrow> </cb:msup> <cb:mtext> </cb:mtext> <cb:mtext> </cb:mtext> <cb:mi>e</cb:mi> <cb:mtext> </cb:mtext> <cb:mi>cm</cb:mi> </cb:mrow> </cb:math> at CEPC with <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:msqrt> <eb:mi>s</eb:mi> </eb:msqrt> <eb:mo>=</eb:mo> <eb:msub> <eb:mi>m</eb:mi> <eb:mi>Z</eb:mi> </eb:msub> </eb:math> . This compares to sensitivities for the electric dipole moments, <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mi>Im</gb:mi> <gb:mo stretchy="false">[</gb:mo> <gb:msubsup> <gb:mi>d</gb:mi> <gb:mi>τ</gb:mi> <gb:mi>γ</gb:mi> </gb:msubsup> <gb:mo stretchy="false">]</gb:mo> </gb:math> and <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"> <kb:mi>Re</kb:mi> <kb:mo stretchy="false">[</kb:mo> <kb:msubsup> <kb:mi>d</kb:mi> <kb:mi>τ</kb:mi> <kb:mi>γ</kb:mi> </kb:msubsup> <kb:mo stretchy="false">]</kb:mo> </kb:math> , of <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"> <ob:msup> <ob:mn>10</ob:mn> <ob:mrow> <ob:mo>−</ob:mo> <ob:mn>20</ob:mn> </ob:mrow> </ob:msup> <ob:mtext> </ob:mtext> <ob:mtext> </ob:mtext> <ob:mi>e</ob:mi> <ob:mtext> </ob:mtext> <ob:mi>cm</ob:mi> </ob:math> at Belle-II <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"> <qb:mi mathvariant="normal">ϒ</qb:mi> <qb:mo stretchy="false">(</qb:mo> <qb:mn>4</qb:mn> <qb:mi>S</qb:mi> <qb:mo stretchy="false">)</qb:mo> </qb:math> resonance and <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" display="inline"> <vb:msup> <vb:mn>10</vb:mn> <vb:mrow> <vb:mo>−</vb:mo> <vb:mn>17</vb:mn> </vb:mrow> </vb:msup> <vb:mtext> </vb:mtext> <vb:mtext> </vb:mtext> <vb:mi>e</vb:mi> <vb:mtext> </vb:mtext> <vb:mi>cm</vb:mi> </vb:math> at BEPCII <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"> <xb:mi>ψ</xb:mi> <xb:mo stretchy="false">(</xb:mo> <xb:mn>2</xb:mn> <xb:mi>S</xb:mi> <xb:mo stretchy="false">)</xb:mo> </xb:math> resonance. For <bc:math xmlns:bc="http://www.w3.org/1998/Math/MathML" display="inline"> <bc:msub> <bc:mi>a</bc:mi> <bc:mi>τ</bc:mi> </bc:msub> </bc:math> , we find that precisions of <dc:math xmlns:dc="http://www.w3.org/1998/Math/MathML" display="inline"> <dc:msup> <d