Gluon bound state and asymptotic freedom derived from the Bethe–Salpeter equation

Abstract

In this paper we study the two-body bound states for gluons and ghosts in a massive Yang-Mills theory which is obtained by generalizing the ordinary massless Yang-Mills theory in a manifestly Lorentz covariant gauge. First, we give a systematic derivation of the coupled Bethe-Salpeter equations for gluons and ghosts by using the Cornwall-Jackiw-Tomboulis effective action of the composite operators within the framework of the path integral quantization. Then, we obtain the numerical solutions for the Bethe-Salpeter amplitude representing the simultaneous bound states of gluons and ghosts by solving the homogeneous Bethe-Salpeter equation in the ladder approximation. We study how the inclusion of ghosts affects the two-gluon bound states. Moreover, we show explicitly that the approximate solutions obtained for the gluon-gluon amplitude are consistent with the ultraviolet asymptotic freedom signaled by the negative β function.

Nonperturbative Ball-Chiu construction of the three-gluon vertex

“…18 is present in all such analyses 9 . Previous studies indicate that in order to obtain masses compatible with lattice simulations, the total integrated strength of the kernel must undergo a considerable suppression [42,45]. A need for an analogous attenuation has been also observed in the recent study of hybrid mesons [46]; the required suppression has been implemented by resorting to a simplified Ansatz for the form factors associated with the tensors comprising Γ (0) αµν (q, r, p).…”

“…In addition, in recent years, the paramount importance of IΓ αµν for a plethora of nonperturbative phenomena has become increasingly evident among practitioners, leading to a vigorous activity for unraveling its infrared properties . In particular, distinct but equally remarkable aspects of the three-gluon vertex are intimately associated with the emergence of a gluonic mass scale [29][30][31][32][33][34][35][36][37][38][39][40][41], the masses and properties of glueballs [42][43][44][45], and the potential formation of hybrids and exotics states [46].…”

“…In particular, while the gluon acquires dynamically an effective mass, the ghost remains massless even nonperturbatively; thus, loops containing gluons give rise to "protected" logarithms, whilst loops containing ghosts to divergent ones [18]. From the phenomenological perspective, the infrared suppression of IΓ αµν , and the overall attenuation of the interaction strength that this causes to the Bethe-Salpeter kernels [42,45], appears to be instrumental for the formation of glueball states with masses compatible with those obtained from lattice simulations [52]. Moreover, the necessity of a considerable suppression has become evident also in a recent study of the hybrid states, in the framework of the Faddeev equations [46].…”

Ghost dynamics in the soft gluon limit

“…In particular, the nonperturbative masslessness of the ghost is responsible for the vanishing of the gluon spectral density at the origin [66][67][68][69], and for the infrared suppression of the three-gluon vertex [52,53,[61][62][63]. In that sense, the ghost dynamics leave their imprint on a variety of fundamental phenomena, such as chiral symmetry breaking and the generation of quark constituent masses [1,[70][71][72][73][74], the emergence of a mass gap in the gauge sector of the theory [9,65,75,76], and the dynamical formation of hadronic bound states [3,20,[77][78][79] and glueballs [80][81][82][83].…”