Multiplicity distribution of produced gluons in deep inelastic scattering: Main equations and their homotopy solutions for heavy nuclei

In this paper we discuss the multiplicity distribution in the deep inelastic processes in the frame work of high energy QCD. We obtained three results. First, we get the new derivation of the equations for the cross sections of productions of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>n</a:mi> </a:math> -cut Pomerons in the final states ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>σ</c:mi> <c:mi>n</c:mi> </c:msub> </c:math> ). These equations coincide with the equations that have been derived using the Abramovsky-Gribov-Kancheli cutting rules but based on the dipole approach to QCD. Second, we developed the homotopy approach for finding the solutions to these equations. It consists with the analytic solution for the first iteration and the converge procedure of calculating the next iterations using computing. Third, we found the analytical solution for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>σ</e:mi> <e:mi>n</e:mi> </e:msub> </e:math> at large <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>n</g:mi> <g:mo>≳</g:mo> <g:mi>N</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>z</g:mi> <g:mo stretchy="false">)</g:mo> <g:mo>=</g:mo> <g:mn>2</g:mn> <g:msub> <g:mi>N</g:mi> <g:mn>0</g:mn> </g:msub> <g:mi>z</g:mi> <g:mi>exp</g:mi> <g:mo stretchy="false">(</g:mo> <g:msup> <g:mi>z</g:mi> <g:mn>2</g:mn> </g:msup> <g:mo>/</g:mo> <g:mo stretchy="false">(</g:mo> <g:mn>2</g:mn> <g:mi>κ</g:mi> <g:mo stretchy="false">)</g:mo> <g:mo stretchy="false">)</g:mo> </g:math> with <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>z</o:mi> <o:mo>=</o:mo> <o:mi>ln</o:mi> <o:mo stretchy="false">(</o:mo> <o:msup> <o:mi>r</o:mi> <o:mn>2</o:mn> </o:msup> <o:msubsup> <o:mi>Q</o:mi> <o:mi>s</o:mi> <o:mn>2</o:mn> </o:msubsup> <o:mo stretchy="false">)</o:mo> </o:math> . Using this solution we calculate the entropy of the produced gluons at large <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>z</s:mi> </s:math> : <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msub> <u:mi>S</u:mi> <u:mi>E</u:mi> </u:msub> <u:mo>=</u:mo> <u:mi>ln</u:mi> <u:mrow> <u:mo stretchy="false">(</u:mo> <u:mi>N</u:mi> <u:mo stretchy="false">(</u:mo> <u:mi>z</u:mi> <u:mo stretchy="false">)</u:mo> <u:mo stretchy="false">)</u:mo> </u:mrow> </u:math> , where the saturation momentum <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msub> <ab:mi>Q</ab:mi> <ab:mi>s</ab:mi> </ab:msub> </ab:math> and all constants are discussed in the text.