Color gauge invariant theory of diquark interactions

In the present work, we construct a color gauge invariant theory of diquark interactions. With the transformation rule of the diquark fields and the definition of the covariant derivatives under color SU(3) symmetry, we construct the gauge invariant Lagrangians describing the vertices of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>S</a:mi> <a:mo>−</a:mo> <a:mi>S</a:mi> <a:mo>−</a:mo> <a:mi>G</a:mi> </a:math> , <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>A</c:mi> <c:mo>−</c:mo> <c:mi>A</c:mi> <c:mo>−</c:mo> <c:mi>G</c:mi> </c:math> , <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>S</e:mi> <e:mi>Q</e:mi> </e:msub> <e:mo>−</e:mo> <e:msub> <e:mi>S</e:mi> <e:mi>Q</e:mi> </e:msub> <e:mo>−</e:mo> <e:mi>G</e:mi> </e:math> , <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>A</g:mi> <g:mi>Q</g:mi> </g:msub> <g:mo>−</g:mo> <g:msub> <g:mi>A</g:mi> <g:mi>Q</g:mi> </g:msub> <g:mo>−</g:mo> <g:mi>G</g:mi> </g:math> , and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>S</i:mi> <i:mi>Q</i:mi> </i:msub> <i:mo>−</i:mo> <i:msub> <i:mi>A</i:mi> <i:mi>Q</i:mi> </i:msub> <i:mo>−</i:mo> <i:mi>G</i:mi> </i:math> , where <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mi>S</k:mi> <k:mrow> <k:mo stretchy="false">(</k:mo> <k:mi>Q</k:mi> <k:mo stretchy="false">)</k:mo> </k:mrow> </k:msub> </k:math> is the light (heavy) scalar diquark field, <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msub> <o:mi>A</o:mi> <o:mrow> <o:mo stretchy="false">(</o:mo> <o:mi>Q</o:mi> <o:mo stretchy="false">)</o:mo> </o:mrow> </o:msub> </o:math> is the light (heavy) axial vector diquark field, and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>G</s:mi> </s:math> is the gluon field. And then we derive the one-gluon-exchange effective potentials for diquark-antidiquark interactions using the obtained Lagrangians. By comparing these potentials with those in the Godfrey-Isgur quark model, the coupling constants of the Lagrangians are determined. We find that the potentials are mainly made of Coulomb, contact, and tensor terms. The potential for the process <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>S</u:mi> <u:mover accent="true"> <u:mi>A</u:mi> <u:mo stretchy="false">¯</u:mo> </u:mover> <u:mo stretchy="false">→</u:mo> <u:mi>A</u:mi> <u:mover accent="true"> <u:mi>S</u:mi> <u:mo stretchy="false">¯</u:mo> </u:mover> </u:math> is 0, since <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:msub> <bb:mi mathvariant="script">L</bb:mi> <bb:mrow> <bb:mi>S</bb:mi> <bb:mi>A</bb:mi> <bb:mi>G</bb:mi> </bb:mrow> </bb:msub> <bb:mo>=</bb:mo> <bb:mn>0</bb:mn> </bb:math> . And the tensor terms proportional to <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:msup> <eb:mi>g</eb:mi> <eb:mn>2</eb:mn> </eb:msup> </eb:math> are negligible.