Center vortices and the SU(3) conformal window

A novel approach for estimating the lower end of the SU(3) conformal window is presented through the study of center vortex geometry and its dependence on the number of fermion flavors <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi>N</a:mi></a:mrow><a:mrow><a:mi>f</a:mi></a:mrow></a:msub></a:mrow></a:math>. Values ranging from <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:msub><c:mrow><c:mi>N</c:mi></c:mrow><c:mrow><c:mi>f</c:mi></c:mrow></c:msub><c:mo>=</c:mo><c:mn>2</c:mn><c:mi>–</c:mi><c:mn>8</c:mn></c:mrow></c:math> are utilized to infer an upper limit for vortex behavior in the low <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>N</e:mi><e:mi>f</e:mi></e:msub></e:math> phase, which may inform the transition to the conformal window. The simulations are performed at a single lattice spacing and pion mass, both fixed for all <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>N</g:mi><g:mi>f</g:mi></g:msub></g:math>. Visualizations of the center vortex structure in three-dimensional slices of the lattice reveal a growing roughness in the vortex matter as a function of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>N</i:mi><i:mi>f</i:mi></i:msub></i:math>, embodied by an increase in the density of vortex matter in the percolating cluster and a simultaneous reduction in secondary clusters disconnected from the percolating cluster in 3D slices. This is quantified by various bulk properties, including the vortex and branching point densities. A correlation of the vortex structure reveals a turning point near <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>N</k:mi><k:mi>f</k:mi></k:msub><k:mo>≃</k:mo><k:mn>5</k:mn></k:math> past which a randomness in the vortex field becomes the dominant aspect of its evolution with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>N</m:mi><m:mi>f</m:mi></m:msub></m:math>. As a byproduct, extrapolations to the vortex content of a uniform-random gauge field provide a critical point at which there must be a drastic shift in vacuum field structure. A precise estimate for the critical value is extracted as <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mrow><o:msubsup><o:mrow><o:mi>N</o:mi></o:mrow><o:mrow><o:mi>f</o:mi></o:mrow><o:mrow><o:mo>*</o:mo></o:mrow></o:msubsup><o:mo>=</o:mo><o:mn>11.43</o:mn><o:mo stretchy="false">(</o:mo><o:mn>16</o:mn><o:mo stretchy="false">)</o:mo><o:mo stretchy="false">(</o:mo><o:mn>17</o:mn><o:mo stretchy="false">)</o:mo></o:mrow></o:math>, close to various other estimates.