Search for a command to run...
We study the two-point field-strength correlation <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>g</a:mi><a:mn>2</a:mn></a:msup><a:mo stretchy="false">⟨</a:mo><a:msubsup><a:mi>G</a:mi><a:mrow><a:mi>μ</a:mi><a:mi>ν</a:mi></a:mrow><a:mi>a</a:mi></a:msubsup><a:mo stretchy="false">(</a:mo><a:mi>s</a:mi><a:mo stretchy="false">)</a:mo><a:msubsup><a:mi>G</a:mi><a:mrow><a:mi>α</a:mi><a:mi>β</a:mi></a:mrow><a:mi>b</a:mi></a:msubsup><a:mo stretchy="false">(</a:mo><a:msup><a:mi>s</a:mi><a:mo>′</a:mo></a:msup><a:mo stretchy="false">)</a:mo><a:mo stretchy="false">⟩</a:mo></a:math> in the Landau gauge in SU(2) and SU(3) quenched lattice QCD, as well as the gluon propagator <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msup><i:mi>g</i:mi><i:mn>2</i:mn></i:msup><i:mo stretchy="false">⟨</i:mo><i:msubsup><i:mi>A</i:mi><i:mi>μ</i:mi><i:mi>a</i:mi></i:msubsup><i:mo stretchy="false">(</i:mo><i:mi>s</i:mi><i:mo stretchy="false">)</i:mo><i:msubsup><i:mi>A</i:mi><i:mi>ν</i:mi><i:mi>b</i:mi></i:msubsup><i:mo stretchy="false">(</i:mo><i:msup><i:mi>s</i:mi><i:mo>′</i:mo></i:msup><i:mo stretchy="false">)</i:mo><i:mo stretchy="false">⟩</i:mo></i:math>. The Landau-gauge gluon propagator <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msup><q:mi>g</q:mi><q:mn>2</q:mn></q:msup><q:mo stretchy="false">⟨</q:mo><q:msubsup><q:mi>A</q:mi><q:mi>μ</q:mi><q:mi>a</q:mi></q:msubsup><q:mo stretchy="false">(</q:mo><q:mi>s</q:mi><q:mo stretchy="false">)</q:mo><q:msubsup><q:mi>A</q:mi><q:mi>μ</q:mi><q:mi>a</q:mi></q:msubsup><q:mo stretchy="false">(</q:mo><q:msup><q:mi>s</q:mi><q:mo>′</q:mo></q:msup><q:mo stretchy="false">)</q:mo><q:mo stretchy="false">⟩</q:mo></q:math> is well described by the Yukawa-type function <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msup><y:mi>e</y:mi><y:mrow><y:mo>−</y:mo><y:mi>m</y:mi><y:mi>r</y:mi></y:mrow></y:msup><y:mo>/</y:mo><y:mi>r</y:mi></y:math> with <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>r</ab:mi><ab:mo>≡</ab:mo><ab:mo stretchy="false">|</ab:mo><ab:mi>s</ab:mi><ab:mo>−</ab:mo><ab:msup><ab:mi>s</ab:mi><ab:mo>′</ab:mo></ab:msup><ab:mo stretchy="false">|</ab:mo></ab:math> for <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>r</eb:mi><eb:mo>=</eb:mo><eb:mn>0.1</eb:mn><eb:mi>–</eb:mi><eb:mn>1.0</eb:mn><eb:mtext> </eb:mtext><eb:mtext> </eb:mtext><eb:mi>fm</eb:mi></eb:math> in both SU(2) and SU(3) QCD. Next, motivated by color-magnetic instabilities in the QCD vacuum, we investigate the perpendicular-type color-magnetic correlation, <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mrow><gb:msub><gb:mrow><gb:mi>C</gb:mi></gb:mrow><gb:mrow><gb:mo>⊥</gb:mo></gb:mrow></gb:msub><gb:mo stretchy="false">(</gb:mo><gb:mi>r</gb:mi><gb:mo stretchy="false">)</gb:mo><gb:mo>≡</gb:mo><gb:msup><gb:mrow><gb:mi>g</gb:mi></gb:mrow><gb:mrow><gb:mn>2</gb:mn></gb:mrow></gb:msup><gb:mo stretchy="false">⟨</gb:mo><gb:msubsup><gb:mrow><gb:mi>H</gb:mi></gb:mrow><gb:mrow><gb:mi>z</gb:mi></gb:mrow><gb:mrow><gb:mi>a</gb:mi></gb:mrow></gb:msubsup><gb:mo stretchy="false">(</gb:mo><gb:mi>s</gb:mi><gb:mo stretchy="false">)</gb:mo><gb:msubsup><gb:mrow><gb:mi>H</gb:mi></gb:mrow><gb:mrow><gb:mi>z</gb:mi></gb:mrow><gb:mrow><gb:mi>a</gb:mi></gb:mrow></gb:msubsup><gb:mo stretchy="false">(</gb:mo><gb:mi>s</gb:mi><gb:mo>+</gb:mo><gb:mi>r</gb:mi><gb:mover accent="true"><gb:mrow><gb:mo>⊥</gb:mo></gb:mrow><gb:mrow><gb:mo stretchy="false">^</gb:mo></gb:mrow></gb:mover><gb:mo stretchy="false">)</gb:mo><gb:mo stretchy="false">)</gb:mo><gb:mo stretchy="false">⟩</gb:mo></gb:mrow></gb:math> (<tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:mrow><tb:mover accent="true"><tb:mrow><tb:mo>⊥</tb:mo></tb:mrow><tb:mrow><tb:mo stretchy="false">^</tb:mo></tb:mrow></tb:mover></tb:mrow></tb:math>: unit vector on the <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"><xb:mi>x</xb:mi><xb:mi>y</xb:mi></xb:math> plane), and the parallel-type one, <zb:math xmlns:zb="http://www.w3.org/1998/Math/MathML" display="inline"><zb:mrow><zb:msub><zb:mrow><zb:mi>C</zb:mi></zb:mrow><zb:mrow><zb:mo stretchy="false">∥</zb:mo></zb:mrow></zb:msub><zb:mo stretchy="false">(</zb:mo><zb:mi>r</zb:mi><zb:mo stretchy="false">)</zb:mo><zb:mo>≡</zb:mo><zb:msup><zb:mrow><zb:mi>g</zb:mi></zb:mrow><zb:mrow><zb:mn>2</zb:mn></zb:mrow></zb:msup><zb:mo stretchy="false">⟨</zb:mo><zb:msubsup><zb:mrow><zb:mi>H</zb:mi></zb:mrow><zb:mrow><zb:mi>z</zb:mi></zb:mrow><zb:mrow><zb:mi>a</zb:mi></zb:mrow></zb:msubsup><zb:mo stretchy="false">(</zb:mo><zb:mi>s</zb:mi><zb:mo stretchy="false">)</zb:mo><zb:msubsup><zb:mrow><zb:mi>H</zb:mi></zb:mrow><zb:mrow><zb:mi>z</zb:mi></zb:mrow><zb:mrow><zb:mi>a</zb:mi></zb:mrow></zb:msubsup><zb:mo stretchy="false">(</zb:mo><zb:mi>s</zb:mi><zb:mo>+</zb:mo><zb:mi>r</zb:mi><zb:mover accent="true"><zb:mrow><zb:mo stretchy="false">∥</zb:mo></zb:mrow><zb:mrow><zb:mo>^</zb:mo></zb:mrow></zb:mover><zb:mo stretchy="false">)</zb:mo><zb:mo stretchy="false">⟩</zb:mo></zb:mrow></zb:math> (<mc:math xmlns:mc="http://www.w3.org/1998/Math/MathML" display="inline"><mc:mrow><mc:mover accent="true"><mc:mrow><mc:mo stretchy="false">∥</mc:mo></mc:mrow><mc:mrow><mc:mo>^</mc:mo></mc:mrow></mc:mover></mc:mrow></mc:math>: unit vector on the <qc:math xmlns:qc="http://www.w3.org/1998/Math/MathML" display="inline"><qc:mi>t</qc:mi><qc:mi>z</qc:mi></qc:math> plane). These two quantities reproduce all the correlation of <sc:math xmlns:sc="http://www.w3.org/1998/Math/MathML" display="inline"><sc:msup><sc:mi>g</sc:mi><sc:mn>2</sc:mn></sc:msup><sc:mo stretchy="false">⟨</sc:mo><sc:msubsup><sc:mi>G</sc:mi><sc:mrow><sc:mi>μ</sc:mi><sc:mi>ν</sc:mi></sc:mrow><sc:mi>a</sc:mi></sc:msubsup><sc:mo stretchy="false">(</sc:mo><sc:mi>s</sc:mi><sc:mo stretchy="false">)</sc:mo><sc:msubsup><sc:mi>G</sc:mi><sc:mrow><sc:mi>α</sc:mi><sc:mi>β</sc:mi></sc:mrow><sc:mi>b</sc:mi></sc:msubsup><sc:mo stretchy="false">(</sc:mo><sc:msup><sc:mi>s</sc:mi><sc:mo>′</sc:mo></sc:msup><sc:mo stretchy="false">)</sc:mo><sc:mo stretchy="false">⟩</sc:mo></sc:math>, due to the Lorentz and global SU(<ad:math xmlns:ad="http://www.w3.org/1998/Math/MathML" display="inline"><ad:msub><ad:mi>N</ad:mi><ad:mi>c</ad:mi></ad:msub></ad:math>) color symmetries in the Landau gauge. Curiously, the perpendicular-type color-magnetic correlation <cd:math xmlns:cd="http://www.w3.org/1998/Math/MathML" display="inline"><cd:msub><cd:mi>C</cd:mi><cd:mo>⊥</cd:mo></cd:msub><cd:mo stretchy="false">(</cd:mo><cd:mi>r</cd:mi><cd:mo stretchy="false">)</cd:mo></cd:math> is found to be always negative for arbitrary <gd:math xmlns:gd="http://www.w3.org/1998/Math/MathML" display="inline"><gd:mi>r</gd:mi></gd:math>, except for the same-point correlation at <id:math xmlns:id="http://www.w3.org/1998/Math/MathML" display="inline"><id:mi>r</id:mi><id:mo>=</id:mo><id:mn>0</id:mn></id:math>. In contrast, the parallel-type color-magnetic correlation <kd:math xmlns:kd="http://www.w3.org/1998/Math/MathML" display="inline"><kd:msub><kd:mi>C</kd:mi><kd:mo stretchy="false">∥</kd:mo></kd:msub><kd:mo stretchy="false">(</kd:mo><kd:mi>r</kd:mi><kd:mo stretchy="false">)</kd:mo></kd:math> is always positive. In the infrared region of <pd:math xmlns:pd="http://www.w3.org/1998/Math/MathML" display="inline"><pd:mi>r</pd:mi><pd:mo>≳</pd:mo><pd:mn>0.4</pd:mn><pd:mtext> </pd:mtext><pd:mtext> </pd:mtext><pd:mi>fm</pd:mi></pd:math>, <rd:math xmlns:rd="http://www.w3.org/1998/Math/MathML" display="inline"><rd:msub><rd:mi>C</rd:mi><rd:mo>⊥</rd:mo></rd:msub><rd:mo stretchy="false">(</rd:mo><rd:mi>r</rd:mi><rd:mo stretchy="false">)</rd:mo></rd:math> and <vd:math xmlns:vd="http://www.w3.org/1998/Math/MathML" display="inline"><vd:msub><vd:mi>C</vd:mi><vd:mo stretchy="false">∥</vd:mo></vd:msub><vd:mo stretchy="false">(</vd:mo><vd:mi>r</vd:mi><vd:mo stretchy="false">)</vd:mo></vd:math> strongly cancel each other, which leads to a significant cancelation in the sum of the field-strength correlations as <ae:math xmlns:ae="http://www.w3.org/1998/Math/MathML" display="inline"><ae:msub><ae:mo>∑</ae:mo><ae:mrow><ae:mi>μ</ae:mi><ae:mo>,</ae:mo><ae:mi>ν</ae:mi></ae:mrow></ae:msub><ae:msup><ae:mi>g</ae:mi><ae:mn>2</ae:mn></ae:msup><ae:mo stretchy="false">⟨</ae:mo><ae:msubsup><ae:mi>G</ae:mi><ae:mrow><ae:mi>μ</ae:mi><ae:mi>ν</ae:mi></ae:mrow><ae:mi>a</ae:mi></ae:msubsup><ae:mo stretchy="false">(</ae:mo><ae:mi>s</ae:mi><ae:mo stretchy="false">)</ae:mo><ae:msubsup><ae:mi>G</ae:mi><ae:mrow><ae:mi>μ</ae:mi><ae:mi>ν</ae:mi></ae:mrow><ae:mi>a</ae:mi></ae:msubsup><ae:mo stretchy="false">(</ae:mo><ae:msup><ae:mi>s</ae:mi><ae:mo>′</ae:mo></ae:msup><ae:mo stretchy="false">)</ae:mo><ae:mo stretchy="false">⟩</ae:mo><ae:mo>∝</ae:mo><ae:mspace linebreak="goodbreak"/><ae:msub><ae:mi>C</ae:mi><ae:mo>⊥</ae:mo></ae:msub><ae:mo stretchy="false">(</ae:mo><ae:mo stretchy="false">|</ae:mo><ae:mi>s</ae:mi><ae:mo>−</ae:mo><ae:msup><ae:mi>s</ae:mi><ae:mo>′</ae:mo></ae:msup><ae:mo stretchy="false">|</ae:mo><ae:mo stretchy="false">)</ae:mo><ae:mo>+</ae:mo><ae:msub><ae:mi>C</ae:mi><ae:mo stretchy="false">∥</ae:mo></ae:msub><ae:mo stretchy="false">(</ae:mo><ae:mo stretchy="false">|</ae:mo><ae:mi>s</ae:mi><ae:mo>−</ae:mo><ae:msup><ae:mi>s</ae:mi><ae:mo>′</ae:mo></ae:msup><ae:mo stretchy="false">|</ae:mo><ae:mo stretchy="false">)</ae:mo><ae:mo>≃</ae:mo><ae:mn>0</ae:mn></ae:math>. Finally, we decompose the field-strength correlation into quadratic, cubic, and quartic terms of the gluon field <se:math xmlns:se="http://www.w3.org/1998/Math/MathML" display="inline"><se:msub><se:mi>A</se:mi><se:mi>μ</se:mi></se:msub></se:math> in the Landau gauge. For the perpendicular-type color-magnetic correlation <ue:math xmlns:ue="http://www.w3.org/1998/Math/MathML" display="inline"><ue:msub><ue:mi>C</ue:mi><ue:mo>⊥</ue:mo></ue:msub><ue:mo stretchy="false">(</ue:mo><ue:mi>r</ue:mi><ue:mo stretchy="false">)</ue:mo></ue:math>, the quadra