Abstract
A simple model for QCD dynamics in which the DGLAP integro-differential equation may be solved analytically has been considered in our previous papers arXiv:1611.08787 [hep-ph] and arXiv:1906.07924 [hep-ph]. When such a model contains only one term in the splitting function of the dominant parton distribution, then Bessel function appears to be the solution to this simplified DGLAP equation. To our knowledge, this model with only one term in the splitting function for the first time has been proposed by Blümlein in arXiv:hep-ph/9506403. In arXiv:1906.07924 [hep-ph] we have shown that a dual integro-differential equation obtained from the DGLAP equation by a complex map in the plane of the Mellin moment in this model may be considered as the BFKL equation. Then, in arXiv:1906.07924 we have applied a complex diffeomorphism to obtain a standard integral from Gradshteyn and Ryzhik tables starting from the contour integral for parton distribution functions that is usually taken by calculus of residues. This standard integral from these tables appears to be the Laplace transformation of Jacobian for this complex diffeomorphism. Here we write up all the formulae behind this trick in detail and find out certain important points for further development of this strategy. We verify that the inverse Laplace transformation of the Laplace image of the Bessel function may be represented in a form of Barnes contour integral.
Highlights
It often happens that a solution to an integro-differential equation is obtained in a form of contour integrals in one or more complex planes
We have considered such a possibility in the previous paper [1] in which we transformed the contour integral representing the solution to the DGLAP integro-differential equation in a simple model of QCD dynamics from this obtained form of a contour integral in the complex plane of the Mellin moment to the Laplace transformation of the corresponding Jacobian
In the previous paper [1] we have communicated that the BFKL equation [45, 46, 47, 48, 49] may be obtained from the DGLAP equation via a complex map in the domain of the contour integral (7) that represents the solution to this DGLAP integro-differential equation
Summary
It often happens that a solution to an integro-differential equation is obtained in a form of contour integrals in one or more complex planes. We have considered such a possibility in the previous paper [1] in which we transformed the contour integral representing the solution to the DGLAP integro-differential equation in a simple model of QCD dynamics from this obtained form of a contour integral in the complex plane of the Mellin moment to the Laplace transformation of the corresponding Jacobian These Jacobians may appear to be multivalued functions of new complex variables and integration over cuts may be required. At the NNLO Mellin space solutions with the running coupling have been worked out in several numerical codes, for example [20], and various later numerical software packages may be found in the citations of [20] When these first order differential equations for the Mellin moments are solved, the usual way is to convert these moments back to the Bjorken x-space by making the inverse Mellin transformation which may be performed by evaluation of residues on the complex plane of the Mellin moment [4, 21, 22].
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