Instantaneous Bethe–Salpeter Equation and Its Analog: Breit-like Equation

Abstract

We take (µ ± e ∓ ) systems and consider the states with quantum number J P = 0 − as examples, to explore the different contents of the instantaneous Bethe-Salpeter (BS) equation and its analog, Breit equation, by solving them exactly. The results show that the two equations are not equivalent, although they are analogous. Furthermore, we point out that the Breit equation contains extra unphysical solutions, so it should be abandoned if one wishes to have an accurate description of the bound states for the instantaneous interacting binding systems. [5,6] un-properly. To explore the different contents i.e. the un-equivalence of the two equations, in the paper we restrict ourselves to the bound states of the systems (µ ± e ∓ ) systems with quantum number J P = 0 − (S-wave) to solve the two equations exactly and to examine the obtained solutions accordingly.BS equation for a fermion-antifermion system has the general formulation [4]:(1) where χ P (q) is the BS wave function, P is the total momentum, q is relative momentum, and V (P, q, k) is the kernel of the equation, p 1 , p 2 are the momenta of the constituents 1 and 2 respectively. The total momentum P and the relative momentum q are related to the momenta p 1 , p 2 as follows:If the kernel V (P, k, q) of the BS equation has the behavior: V (P, q, k)| P =0 = V ( q, k) ( P = 0 in center mass frame of the concerned bound state), the BS equation is called as an 'instantaneous BS equation'.For instance, in Coulomb gauge, the terms corresponding to the possible transverse-photon exchange between the two components in BS kernel for (µ ± e ∓ ) systems are considered as higher order, so the 'lowest order' BS equation kernel for the systems has the form:Namely, there are some physical systems whose bound states are described by an instantaneous BS equation indeed. * Not post-mail address.In Refs. [1,7], we showed how to solve an instantaneous BS equation exactly, and also showed the authors of Ref. [2,3] how to mislead the problem to its analog: Breit equation. For present convenience, let us repeat the main procedure in Refs. [1,7].Firstly, as done by the authors of Refs. [2,3], the 'instantaneous BS wave function' ϕ P ( q) asis introduced, then the BS equation Eq.(1) can be rewritten asHere S(1) f (p 1 ) and Sf (−p 2 ) are the propagators of the fermion and anti-fermion respectively andThe propagators (in C.M.S. i.e. P = 0) can be decomposed as:, where J = 1 for the fermion (i = 1) and J = −1 for the anti-fermion (i = 2). It is easy to check± can be considered as 'energy' projection operators, and 'complete' for the projection. For below discussions let us introduce the notations ϕ ±± P ( q) as:Because of the completeness of the projection for Λ ± , we have:

Spectrum for heavy quarkonia and mixture of the relevant wave functions within the framework of Bethe-Salpeter equation

“…Details about examining the equivalence may be found in ref. [10]. Alternately here we exactly start with the four equations to solve the BS-In equation.…”

Bethe-Salpeter equation for doubly heavy baryons in the covariant instantaneous approximation

“…III. V 1 and V 2 are the scalar confinement and one-gluon-exchange terms given in the following, respectively, (after imposing covariant instantaneous approximation [41][42][43][44]):…”

“…In the numerical calculations, we let 0 vary between 0:01 GeV 3 and 0:06 GeV 3 [8,39,49]. The light quark propagator can be written as the following form [9,44]:…”

“…In another formalism to solve the BS equation, the covariant instantaneous approximation is adopted in the kernel. In recent decades, this formalism has been successfully used to investigate heavy mesons, heavy baryons, and exotic states [7][8][9]12,[37][38][39][40][41][42][43][44].…”

Separation of different wave components in the Bethe–Salpeter wave function

“…In Cartesian coordinate system this mixing cannot be separated clearly, especially for high spin bound states, for example 2 + , 2 ++ , etc. We found that it can be done in spherical coordinate system [13,16,17]. …”