Holographic spin alignment for vector mesons

We develop a general framework for studying the spin alignment <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>ρ</a:mi><a:mn>00</a:mn></a:msub></a:math> for flavorless vector mesons by using the gauge/gravity duality. Focusing on the dilepton production through vector meson decay, we derive the relation between production rates at each spin channel and meson’s spectral function in a thermal medium, which can be evaluated by holographic models for a strongly coupled system. As examples, we study <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>ρ</c:mi><c:mn>00</c:mn></c:msub></c:math> for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>J</e:mi><e:mo>/</e:mo><e:mi>ψ</e:mi></e:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>ϕ</g:mi></g:math> mesons, induced by the relative motion to a thermal background, within the soft-wall model. Spectral functions for <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>J</i:mi><i:mo>/</i:mo><i:mi>ψ</i:mi></i:math> at <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>T</k:mi><k:mo>=</k:mo><k:mn>150</k:mn><k:mtext> </k:mtext><k:mtext> </k:mtext><k:mi>MeV</k:mi></k:math> have significant resonance peaks, indicating that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>c</m:mi><m:mo>−</m:mo><m:mover accent="true"><m:mi>c</m:mi><m:mo stretchy="false">¯</m:mo></m:mover></m:math> can exist as quasistable <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>J</q:mi><q:mo>/</q:mo><q:mi>ψ</q:mi></q:math> mesons. However, no significant peaks for the <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>ϕ</s:mi></s:math> meson are observed at the same temperature, implying that <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>ϕ</u:mi></u:math> mesons are melting down at this temperature and spectral functions are interpreted as probabilities of unstable <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>s</w:mi><w:mo>−</w:mo><w:mover accent="true"><w:mi>s</w:mi><w:mo stretchy="false">¯</w:mo></w:mover></w:math> pairs. These pairs are related to free-streaming <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>ϕ</ab:mi></ab:math> mesons by an instantaneous freeze-out assumption, which is performed by sandwiching the in-medium spectral function between the meson’s free-streaming propagators. Then only states with invariant masses closing to the meson’s vacuum mass survive during the freeze-out stage and contribute to final results. We show that <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi>ρ</cb:mi><cb:mn>00</cb:mn></cb:msub></cb:math> in the helicity frame for <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>J</eb:mi><eb:mo>/</eb:mo><eb:mi>ψ</eb:mi></eb:math> and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>ϕ</gb:mi></gb:math> mesons have positive and negative deviations from <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mrow><ib:mn>1</ib:mn><ib:mo>/</ib:mo><ib:mn>3</ib:mn></ib:mrow></ib:math> at <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mi>T</kb:mi><kb:mo>=</kb:mo><kb:mn>150</kb:mn><kb:mtext> </kb:mtext><kb:mtext> </kb:mtext><kb:mi>MeV</kb:mi></kb:math>, respectively, which consequently leads to different properties for their global spin alignments.