The longitudinal polarization fraction of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>D</a:mi><a:mo>*</a:mo></a:msup></a:math> meson is measured in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>B</c:mi><c:mn>0</c:mn></c:msup><c:mo stretchy="false">→</c:mo><c:mrow><c:msup><c:mi>D</c:mi><c:mrow><c:mo>*</c:mo><c:mo>−</c:mo></c:mrow></c:msup><c:msup><c:mi>τ</c:mi><c:mo>+</c:mo></c:msup><c:msub><c:mi>ν</c:mi><c:mi>τ</c:mi></c:msub></c:mrow></c:math> decays, where the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi>τ</f:mi></f:math> lepton decays to three charged pions and a neutrino, using proton-proton collision data collected by the LHCb experiment at center-of-mass energies of 7, 8 and 13 TeV and corresponding to an integrated luminosity of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mrow><h:mn>5</h:mn><h:mtext> </h:mtext><h:mtext> </h:mtext><h:msup><h:mrow><h:mi>fb</h:mi></h:mrow><h:mrow><h:mo>−</h:mo><h:mn>1</h:mn></h:mrow></h:msup></h:mrow></h:math>. The <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msup><j:mi>D</j:mi><j:mo>*</j:mo></j:msup></j:math> polarization fraction <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msubsup><l:mi>F</l:mi><l:mi>L</l:mi><l:msup><l:mi>D</l:mi><l:mo>*</l:mo></l:msup></l:msubsup></l:math> is measured in two <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msup><n:mi>q</n:mi><n:mn>2</n:mn></n:msup></n:math> regions, below and above <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mn>7</p:mn><p:mtext> </p:mtext><p:mtext> </p:mtext><p:msup><p:mi>GeV</p:mi><p:mn>2</p:mn></p:msup><p:mo>/</p:mo><p:msup><p:mi>c</p:mi><p:mn>4</p:mn></p:msup></p:math>, where <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msup><r:mi>q</r:mi><r:mn>2</r:mn></r:msup></r:math> is defined as the squared invariant mass of the <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi>τ</t:mi><t:msub><t:mi>ν</t:mi><t:mi>τ</t:mi></t:msub></t:math> system. The <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msubsup><v:mi>F</v:mi><v:mi>L</v:mi><v:msup><v:mi>D</v:mi><v:mo>*</v:mo></v:msup></v:msubsup></v:math> values are measured to be <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mn>0.52</x:mn><x:mo>±</x:mo><x:mn>0.07</x:mn><x:mo>±</x:mo><x:mn>0.04</x:mn></x:math> and <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mn>0.34</z:mn><z:mo>±</z:mo><z:mn>0.08</z:mn><z:mo>±</z:mo><z:mn>0.02</z:mn></z:math> for the lower and higher <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:msup><bb:mi>q</bb:mi><bb:mn>2</bb:mn></bb:msup></bb:math> regions, respectively. The first uncertainties are statistical and the second systematic. The average value over the whole <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:msup><db:mi>q</db:mi><db:mn>2</db:mn></db:msup></db:math> range is <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:mrow><fb:msubsup><fb:mrow><fb:mi>F</fb:mi></fb:mrow><fb:mrow><fb:mi>L</fb:mi></fb:mrow><fb:mrow><fb:msup><fb:mrow><fb:mi>D</fb:mi></fb:mrow><fb:mrow><fb:mo>*</fb:mo></fb:mrow></fb:msup></fb:mrow></fb:msubsup><fb:mo>=</fb:mo><fb:mn>0.41</fb:mn><fb:mo>±</fb:mo><fb:mn>0.06</fb:mn><fb:mo>±</fb:mo><fb:mn>0.03</fb:mn></fb:mrow></fb:math>. These results are compatible with the Standard Model predictions. © 2024 CERN, for the LHCb Collaboration 2024 CERN