Multiplicity dependence of quarkonia production at ultrahigh energies

In this paper, we propose an approach in high-energy QCD, which allows us to calculate the inclusive quarkonia production at ultrahigh energies. This approach is based on <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>t</a:mi> </a:mrow> </a:math> -channel unitarity and on the expressions for dipole densities from the procedure of summing of large Pomeron loops we have developed in our previous papers. In the framework of this approach, we discuss the dependence of quarkonia production on the multiplicity of the accompanying hadrons. In this paper, we used the three gluon- fusion mechanism for quarkonia production, without assuming the multiplicity dependence of the saturation scale. We found the multiplicity distribution of the produced gluon with ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:msubsup> <c:mrow> <c:mi mathvariant="script">P</c:mi> </c:mrow> <c:mrow> <c:mi>n</c:mi> </c:mrow> <c:mrow> <c:mi>J</c:mi> <c:mo>/</c:mo> <c:mi mathvariant="normal">Ψ</c:mi> </c:mrow> </c:msubsup> </c:mrow> </c:math> ) and without ( <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msubsup> <g:mi mathvariant="script">P</g:mi> <g:mi>n</g:mi> <g:mtext>gluon</g:mtext> </g:msubsup> </g:math> ) the quarkonia production. It turns out that <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mrow> <j:mfrac> <j:mrow> <j:msubsup> <j:mrow> <j:mi mathvariant="script">P</j:mi> </j:mrow> <j:mrow> <j:mi>n</j:mi> </j:mrow> <j:mrow> <j:mi>J</j:mi> <j:mo stretchy="false">/</j:mo> <j:mi mathvariant="normal">Ψ</j:mi> </j:mrow> </j:msubsup> </j:mrow> <j:mrow> <j:msubsup> <j:mrow> <j:mi mathvariant="script">P</j:mi> </j:mrow> <j:mrow> <j:mi>n</j:mi> </j:mrow> <j:mrow> <j:mtext>gluon</j:mtext> </j:mrow> </j:msubsup> </j:mrow> </j:mfrac> <j:mo>∝</j:mo> <j:msup> <j:mrow> <j:mi>n</j:mi> </j:mrow> <j:mrow> <j:mn>2</j:mn> </j:mrow> </j:msup> </j:mrow> </j:math> , as has been expected and discussed in corresponding publications.