Bridging hadronic and vacuum structure by heavy quarkonia

We discuss the central and, mostly, spin-dependent potentials in heavy quarkonia <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mover accent="true"><a:mrow><a:mi>b</a:mi></a:mrow><a:mrow><a:mo stretchy="false">¯</a:mo></a:mrow></a:mover><a:mi>b</a:mi><a:mo>,</a:mo><a:mover accent="true"><a:mrow><a:mi>c</a:mi></a:mrow><a:mrow><a:mo stretchy="false">¯</a:mo></a:mrow></a:mover><a:mi>c</a:mi></a:mrow></a:math>, with two goals in mind. The first is phenomenological: using the splitting between the 1S and 2S pairs, as well as the 1P and 2P quartet masses, we obtain pertinent matrix elements of all these potentials. The second is theoretical: using standard wave functions, we compute these matrix elements, quantifying the perturbative, nonperturbative and “string” contributions. The model for nonperturbative effects is a “dense instanton liquid model,” in which the QCD vacuum is made mostly of (strongly color correlated) instanton-anti-instanton pairs or “molecules.” Their contributions are calculated via standard Wilson lines, with or without extra powers of gauge fields. We find a reasonable description of all central, spin-spin, and spin-orbit forces at distances <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:mi>r</g:mi><g:mo>=</g:mo><g:mn>0</g:mn><g:mi>–</g:mi><g:mn>0.7</g:mn><g:mtext> </g:mtext><g:mtext> </g:mtext><g:mi>fm</g:mi></g:mrow></g:math>, relevant for <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mover accent="true"><i:mi>b</i:mi><i:mo stretchy="false">¯</i:mo></i:mover><i:mi>b</i:mi></i:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mover accent="true"><m:mi>c</m:mi><m:mo stretchy="false">¯</m:mo></m:mover><m:mi>c</m:mi></m:math> quarkonia.