page_title: Bethe–Salpeter equation
section_title: Rainbow-ladder approximation
hierarchical_section_title: Bethe–Salpeter equation / Rainbow-ladder approximation
caption_attribution_description: English: The Bethe-Salpeter equation in Ladder approximation
context_page_description: The Bethe–Salpeter equation describes the bound states of a two-body quantum field theoretical system in a relativistically covariant formalism. The equation was actually first published in 1950 at the end of a paper by Yoichiro Nambu, but without derivation.
  Due to its generality and its application in many branches of theoretical physics, the Bethe–Salpeter equation appears in many different forms. One form, that is quite often used in high energy physics is
  where Γ is the Bethe–Salpeter amplitude, K the interaction and S the propagators of the two participating particles.
  In quantum theory, bound states are objects that live for an infinite time, thus the constituents interact infinitely many times. By summing up, infinitely many times, all possible interactions that can occur between the two constituents, the Bethe–Salpeter equation is a tool to calculate properties of bound states. Its solution, the Bethe–Salpeter amplitude, is a description of the bound state under consideration.
  As it can be derived via identifying bound-states with poles in the S-matrix, it can be connected to the quantum theoretical description of scattering processes and Green's functions.
context_section_description: In principle the interaction kernel K contains all possible two-particle-irreducible interactions that can occur between the two constituents. Thus, in practical calculations one has to model it and only choose a subset of the interactions. As in quantum field theories, interaction is described via the exchange of particles (e.g. photons in quantum electrodynamics, or gluons in quantum chromodynamics), the most simple interaction is the exchange of only one of these force-particles.
  As the Bethe–Salpeter equation sums up the interaction infinitely many times, the resulting Feynman graph has the form of a ladder (or rainbow).
  While in quantum electrodynamics the ladder approximation caused problems with crossing symmetry and gauge invariance and thus crossed ladder terms had to be included, in quantum chromodynamics this approximation is used phenomenologically quite a lot to calculate hadron masses, since it respects Chiral symmetry breaking and therefore is an important part of the generation of these masses.
caption: A graphical representation of the Bethe–Salpeter equation in Ladder-approximation