Abstract
In this paper we proceed to study the high energy behavior of a scattering amplitudes in the Yang-Mills theory with the Higgs mechanism for the gauge boson mass. The spectrum of the $j$-plane singularities of the $t$-channel partial waves, and corresponding eigenfunctions of the BFKL equation in leading log approximation (LLA), were previously calculated by us numerically in arXiv:1401.4671 . Here we develop a semiclassical approach and investigate the influence of the impact parameter exponential decrease, existing in this model, on the high energy asymptotic behaviour of the scattering amplitude. This approach is much simpler than the numerical calculations, and reproduces our earlier numerical results. The analytical (semianalytical) solutions which have been found in this paper, can be used to incorporate correctly the large impact parameter behavior in the framework of CGC/saturation approach. This behaviour is interesting as provides the high energy amplitude for the electroweak theory, which can be measured experimentally.
Highlights
Another facet of this model is that it is a possible candidate for an effective theory equivalent to perturbative QCD, in the region of distances (r ) shorter than 1/m, where m denotes the gluon mass
A plausible scenario is that the Higgs gauge theory describes QCD in the kinematic region r ≤ 1/m, while for r ∼ 1/ QCD > 1/m the non-perturbative QCD approach takes over, and it leads to the confinement of quarks and gluons, which is missing in the theory with a massive gluon
In our previous paper we studied the BFKL equation with massive gluons in the lattice and proved that its spectrum coincides with the spectrum of the massless BFKL equation
Summary
Another facet of this model is that it is a possible candidate for an effective theory equivalent to perturbative QCD, in the region of distances (r ) shorter than 1/m, where m denotes the gluon mass. In this paper we propose using the solution to the BFKL equation with massive gluons, which has the advantage of being simple and semianalytic. Having this solution in hand, we are able to progress to more difficult problems, e.g. a generalization of the main equations of the CGC/saturation approach [9,29–42]. For q = 0 the kernel (1) simplifies considerably and yields a homogeneous BFKL equation for the Yang–Mills theory with the Higgs mechanism, ωf ( p). Changing the notation of the wave function f ( p) to φE (κ), we obtain the one-dimensional BFKL equation. +iν these eigenfunctions are normalized and form a complete set of functions
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