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:

Heavy-light mesons in a relativistic model

“…( 16)-( 19), as pointed out in Refs. [32,33]. We find that our goal can be achieved by employing the constraint of Eqs.…”

Heavy-light mesons in a relativistic model

“…( 16)-( 19), as pointed out in Refs. [32,33]. We find that our goal can be achieved by employing the constraint of Eqs.…”

A review of the open charm and open bottom systems

“…This relativistic quark model is a type of the covariant exten sion of the GI model. One may consult [77][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92][93][94][95][96][97][98][99][100][101][102] for the other potential approaches.…”

A Proposal to the ‘12 vs. 32 Puzzle’

“…The detailed numerical calculations in this work are studied within the framework of the instantaneous Bethe-Salpeter (BS) methods [35,36], which have already been successfully used to cope with the doubly heavy baryons [37], the recently observed exotic pentaquarks [38] and the fully heavy tetraquarks [39], as well as have also been generally applied to the meson mass spectra [40,41] and the hadronic transitions and decays [42][43][44]. The theoretical calculations from BS methods achieved satisfactory consistences with the experimental results.…”

Heavy-light mesons in a relativistic model

“…( 16)-( 19), as pointed out in Refs. [32,33]. We find that our goal can be achieved by employing the constraint of Eqs.…”

A review of the open charm and open bottom systems

“…This relativistic quark model is a type of the covariant exten sion of the GI model. One may consult [77][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92][93][94][95][96][97][98][99][100][101][102] for the other potential approaches.…”

A Proposal to the ‘12 vs. 32 Puzzle’

“…The detailed numerical calculations in this work are studied within the framework of the instantaneous Bethe-Salpeter (BS) methods [35,36], which have already been successfully used to cope with the doubly heavy baryons [37], the recently observed exotic pentaquarks [38] and the fully heavy tetraquarks [39], as well as have also been generally applied to the meson mass spectra [40,41] and the hadronic transitions and decays [42][43][44]. The theoretical calculations from BS methods achieved satisfactory consistences with the experimental results.…”

Heavy-light mesons in a relativistic model

“…( 16)-( 19), as pointed out in Refs. [32,33]. We find that our goal can be achieved by employing the constraint of Eqs.…”

A review of the open charm and open bottom systems

“…This relativistic quark model is a type of the covariant exten sion of the GI model. One may consult [77][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92][93][94][95][96][97][98][99][100][101][102] for the other potential approaches.…”

A Proposal to the ‘12 vs. 32 Puzzle’

“…The detailed numerical calculations in this work are studied within the framework of the instantaneous Bethe-Salpeter (BS) methods [35,36], which have already been successfully used to cope with the doubly heavy baryons [37], the recently observed exotic pentaquarks [38] and the fully heavy tetraquarks [39], as well as have also been generally applied to the meson mass spectra [40,41] and the hadronic transitions and decays [42][43][44]. The theoretical calculations from BS methods achieved satisfactory consistences with the experimental results.…”