Bethe–Salpeter equation
The Bethe–Salpeter equation (BSE, named after Hans Bethe and Edwin Salpeter)[1] is an integral equation, the solution of which describes the structure of a relativistic two-body (particles) bound state in a covariant formalism quantum field theory (QFT).
[2] Due to its common application in several branches of theoretical physics, the Bethe–Salpeter equation appears in many forms.
In quantum theory, bound states are composite physical systems with lifetime significantly longer than the time scale of the interaction breaking their structure (otherwise the physical systems under consideration are called resonances), thus allowing ample time for constituents to interact.
By accounting all possible interactions that can occur between the two constituents, the BSE is a tool to calculate properties of deep-bound states.
In the case where particle-pair production can be ignored, if one of the two fermion constituent is significantly more massive than the other, the system is simplified into the Dirac equation for the light particle under the external potential of the heavy one.
The crucial step is now, to assume that bound states appear as poles in the Green function.
, and then makes an Ansatz for the Green function in the vicinity of the pole as where P is the total momentum of the system.
If one plugs that Ansatz into the Dyson equation above, and sets the total momentum "P" such that the energy-momentum relation holds, on both sides of the term a pole appears.
To obtain the above form one introduces the Bethe–Salpeter amplitudes "Γ" and gets finally which is written down above, with the explicit momentum dependence.
As the Bethe–Salpeter equation sums up the interaction infinitely many times from a perturbative view point, the resulting Feynman graph resembles the form of a ladder (or rainbow), hence the name of this approximation.
While in QED the ladder approximation caused problems with crossing symmetry and gauge invariance, indicating the inclusion of crossed-ladder terms.
[4] Such an Ansatz for the dressed quark-gluon vertex within the rainbow-ladder truncation respects chiral symmetry and its dynamical breaking, which therefore is an important modeling of the strong nuclear interaction.
As an example the structure of pions can be solved applying the Maris—Tandy Ansatz from the Bethe—Salpeter equation in Euclidean space.
In the rainbow-ladder approximation this Interaction kernel does not depend on the total momentum of the Bethe–Salpeter amplitude, in which case the second term of the normalization condition vanishes.
An alternative normalization based on the eigenvalue of the corresponding linear operator was derived by Nakanishi.
Such singularities are usually located when the constituent momentum is timelike, which are not directly accessible from Euclidean-space solutions of this equation.
Instead one develop methods to solve these type of integral equations directly in the timelike region.
[7] In the case of scalar bound states through a scalar-particle exchange in the rainbow-ladder truncation, the Bethe—Salpeter equation in the Minkowski space can be solved with the assistance of Nakanishi integral representation.
[8] Many modern quantum field theory textbooks and a few articles provide pedagogical accounts for the Bethe–Salpeter equation's context and uses.
