Symmetry preserving truncations of the gap and Bethe-Salpeter equations

Abstract

Ward-Green-Takahashi (WGT) identities play a crucial role in hadron physics, e.g. imposing stringent relationships between the kernels of the one-and two-body problems, which must be preserved in any veracious treatment of mesons as bound-states. In this connection, one may view the dressed gluon-quark vertex, Γ a µ , as fundamental. We use a novel representation of Γ a µ , in terms of the gluon-quark scattering matrix, to develop a method capable of elucidating the unique quarkantiquark Bethe-Salpeter kernel, K , that is symmetry-consistent with a given quark gap equation.A strength of the scheme is its ability to expose and capitalise on graphic symmetries within the kernels. This is displayed in an analysis that reveals the origin of H-diagrams in K , which are twoparticle-irreducible contributions, generated as two-loop diagrams involving the three-gluon vertex, that cannot be absorbed as a dressing of Γ a µ in a Bethe-Salpeter kernel nor expressed as a member of the class of crossed-box diagrams. Thus, there are no general circumstances under which the WGT identities essential for a valid description of mesons can be preserved by a Bethe-Salpeter kernel obtained simply by dressing both gluon-quark vertices in a ladder-like truncation; and, moreover, adding any number of similarly-dressed crossed-box diagrams cannot improve the situation.

Baryons as relativistic three-quark bound states

“…Now, for convenience we may define a quark-gluon vertex as an auxiliary function that represents the resummation of these higher order corrections [248][249][250][374][375][376][377][378]. Then, being a functional of the quark propagator Γ µ = Γ µ [S], it must be cut implicitly, see [379,380] for a discussion, where an integral representation of the vertex is necessary so as to avoid an ambiguity in defining the momenta. The downside of this is that functionally differentiating a quark-gluon vertex leads to a five-point function that must satisfy an auxiliary Bethe-Salpeter equation, derivable from the auxiliary equation for the vertex.…”

Scale-setting, flavor dependence, and chiral symmetry restoration

“…On the other hand, the foundation for Ref. [18] is a model interaction [70][71][72] tuned to achieve a good description of in-vacuum light-quark observables when used with the leading-order (rainbow-ladder) truncation [73] of the strong-interaction's matter-sector DSEs; and the study explored the effect of two vastly different assumptions about the flavourdependence of that interaction, thus determining 7 n cr f 9.…”

Aspects of open-flavour mesons in a comprehensive DSBSE study

“…From the theory point of view, one needs a thorough understanding of nonperturbative QCD, which is sought via different approaches such as the traditional quark model in relativistic forms [3][4][5][6][7][8][9][10][11], effective field theories [12][13][14][15], QCD sum rules (QSRs) [16][17][18][19], latticeregularized QCD [20][21][22][23][24], and covariant approaches to QCD bound states at various levels of sophistication and aspects pertaining thereto [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].…”