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Register a new userBound state equation for the Nakanishi weight function
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The bound state Bethe-Salpeter amplitude was expressed by Nakanishi using a two-dimensional integral representation, in terms of a smooth weight function $g$, which carries the detailed dynamical information. A similar, but one-dimensional, integral representation can be obtained for the Light-Front wave function in terms of the same weight function $g$. By using the generalized Stieltjes transform, we first obtain $g$ in terms of the Light-Front wave function in the complex plane of its arguments. Next, a new integral equation for the Nakanishi weight function $g$ is derived for a bound state case. It has the standard form $g= N g$, where $N$ is a two-dimensional integral operator. We give the prescription for obtaining the kernel $ N$ starting with the kernel $K$ of the Bethe-Salpeter equation. The derivation is valid for any kernel given by an irreducible Feynman amplitude.
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The bound state Bethe-Salpeter amplitude was expressed by Nakanishi in terms of a smooth weight function g. By using the generalized Stieltjes transform, we derive an integral equation for the Nakanishi function g for a bound state case. It has the standard form g= Vg, where V is a two-dimensional integral operator. The prescription for obtaining the kernel V starting with the kernel K of the Bethe-Salpeter equation is given.
The Bethe-Salpeter amplitude $Phi(k,p)$ is expressed, by means of the Nakanishi integral representation, via a smooth function $g(gamma,z)$. This function satisfies a canonical equation $g=Ng$. However, calculations of the kernel $N$ in this equation, presented previously, were restricted to one-boson exchange and, depending on method, dealt with complex multivalued functions. Although these difficulties are surmountable, but in practice, they complicate finding the unambiguous result. In the present work, an unambiguous expression for the kernel $N$ in terms of real functions is derived. For the one-boson scalar exchange, the explicit formula for $N$ is found. With this equation and kernel, the binding energies, calculated previously, are reproduced. Their finding, as well as calculation of the Bethe-Salpeter amplitude in the Minkowski space, become not more difficult than in the Euclidean one. The method can be generalized to any kernel given by irreducible Feynman graph. This generalization is illustrated by example of the cross-ladder kernel.
We present a method to directly solving the Bethe-Salpeter equation in Minkowski space, both for bound and scattering states. It is based on a proper treatment of the singularities which appear in the kernel, propagators and Bethe-Salpeter amplitude itself. The off-mass shell scattering amplitude for spinless particles interacting by a one boson exchange is computed for the first time.
In this paper, we study the finite temperature-dependent Schr{o}dinger equation by using the Nikiforov-Uvarov method. We consider the sum of the Cornell, inverse quadratic, and harmonic-type potential as the potential part of the radial Schr{o}dinger equation. Analytical expressions for the energy eigenvalues and the radial wave function are presented. Application of the results for the heavy quarkonia and $B_c$ meson masses are good agreement with the current experimental data except for zero angular momentum quantum numbers. Numerical results for the temperature dependence indicates a different behaviour for different quantum numbers. Temperature-dependent results are in agreement with some QCD sum rule results from the ground states.
We present a derivation of the Gribov equation for the gluon/photon Greens function D(q). Our derivation is based on the second derivative of the gauge-invariant quantity Tr ln D(q), which we interpret as the gauge-boson `self-loop. By considering the higher-order corrections to this quantity, we are able to obtain a Gribov equation which sums the logarithmically enhanced corrections. By solving this equation, we obtain the non-perturbative running coupling in both QCD and QED. In the case of QCD, alpha_S has a singularity in the space-like region corresponding to super-criticality, which is argued to be resolved in Gribovs light-quark confinement scenario. For the QED coupling in the UV limit, we obtain a propto Q^2 behaviour for space-like Q^2=-q^2. This implies the decoupling of the photon and an NJLVL-type effective theory in the UV limit.
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