Symmetry preserving truncations of the gap and Bethe-Salpeter equations

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, ${\mathrm{\Gamma }}_{\mu }^a$, as fundamental. We use a novel representation of ${\mathrm{\Gamma }}_{\mu }^a$, in terms of the gluon-quark scattering matrix, to develop a method capable of elucidating the unique quark-antiquark Bethe-Salpeter kernel, $\mathcal{K}$, that is symmetry consistent with a given quark gap equation. A strength of the scheme is its ability to expose and capitalize on graphic symmetries within the kernels. This is displayed in an analysis that reveals the origin of $H$-diagrams in $\mathcal{K}$, which are two-particle-irreducible contributions, generated as two-loop diagrams involving the three-gluon vertex, that cannot be absorbed as a dressing of ${\mathrm{\Gamma }}_{\mu }^a$ 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 ladderlike truncation; and, moreover, adding any number of similarly dressed crossed-box diagrams cannot improve the situation.

Figure 1

Quark self-energy, $\mathrm{\Sigma }(k)=i\gamma ·k[A(k^2)-1]+B(k^2)$, Eq. (1): solid line with open-circle, dressed-quark propagator $S(q)=1/[i\gamma ·q+\mathrm{\Sigma }(q)]$; open-circle “spring,” dressed-gluon propagator, $D_{\mu \nu }(k-q)$; and (red) shaded circle, dressed gluon-quark vertex, ${\mathrm{\Gamma }}_{\mu }^a(k,q)$. A coupling $g$ appears at each vertex.

Figure 2

Upper panel. BSE for a color-singlet vertex, ${\mathcal{G}}_M(k;P)$. The channel is defined by the inhomogeneity, e.g. with $g_M=\frac{1}{2}{\tau }^i{\gamma }_5{\gamma }_{\mu }$ one gains access to all states that communicate with an isovector axial-vector probe, such as the pion and $a_1$ meson. The interaction between the dressed valence constituents is completely described by the scattering kernel, $\mathcal{K}$. Lower panel. For $I\ne 0$ mesons, the Bethe-Salpeter kernel is a sum of two terms, Eq. (4). The interaction content of both is completely determined by that of the dressed gluon-quark vertex, explicitly for ${\mathcal{K}}_S$ and implicitly for ${\mathcal{K}}_{\mathrm{\Gamma }}$, Eq. (5).

Figure 3

Pion mass-squared vs current-quark mass, obtained with the BC Ansatz, Eq. (6), for ${\mathrm{\Gamma }}_{\mu }^a$ in the gap equation, Fig. 1: dashed (blue) curve, Bethe-Salpeter kernel in Fig. 2 is $\mathcal{K}={\mathcal{K}}_S$; dot-dashed (red) curve, kernel is ${\mathcal{K}}_{L-\mathrm{\Gamma }\mathrm{\Gamma }}$, the dressed-vertex ladderlike kernel; and solid (black) curve, complete BC-vertex-consistent kernel constructed following Refs. [33, 55]. Evidently, only the complete kernel is sufficient to ensure the existence of a Nambu-Goldstone pion. (Results obtained with the dressed-gluon line represented by the interaction in Ref. [28]: $D=0.5{\mathrm{GeV}}^2$, $\omega =0.5\mathrm{GeV}$, $\tau \to \infty$.)

Figure 4

Upper panel. DSE for the dressed gluon-quark vertex, ${\mathrm{\Gamma }}_{\mu }^a$. Only selected contributions are shown. The complete equation is depicted, e.g. in Fig. 2.6 of Ref. [1]. The shaded (blue) circle at the junction of three-gluon lines is the dressed three-gluon vertex. Lower panel. Some of the contributions to the quark-antiquark scattering kernel, $\mathcal{K}$, generated by the gap equation expressed in terms of the dressed gluon-quark vertex depicted in the upper panel. The last diagram drawn explicitly is an example of an $H$ diagram. It is 2PI; but cannot be expressed as a correction to either vertex in a ladder kernel nor as a member of the class of crossed-box diagrams.

Figure 5

Upper panel. DSE for the dressed gluon-quark vertex, ${\mathrm{\Gamma }}_{\mu }^a$, expressed with the antiquark providing the reference line and involving the $s$-channel 1PI gluon-quark scattering amplitude $\mathcal{C}$. Lower panel. In terms of the gluon-quark vertex, the equation for the quark self-energy is manifestly symmetric when expressed using $\mathcal{C}$: (a) rainbow term; and (b) all corrections.

Figure 6

In $I\ne 0$ channels, Eq. (3) produces this series of diagrams for the 2PI kernel of the symmetry-consistent BSE. The dashed (red) lines represent the act of functional differentiation and the arrows direct attention to the resulting kernel contribution, when that can be depicted simply.

Figure 7

Upper panel. In $I\ne 0$ channels, ${\mathcal{K}}_{\cancel{\mathcal{C}}}$ has the expansion depicted here, where the dashed (red) lines indicate the quark line that reacts to the functional differentiation in Eq. (3). The expansion necessarily involves $g+q\to (m+1)g+q$, $m\ge 1$, transition matrices, ${\mathcal{M}}_1$, etc. Lower panel. A focus on the first diagram on the rhs in the upper panel reveals the simple nature of its contribution to $\mathcal{K}$.

Figure 8

Explicating the origin of $H$-diagrams and crossed-box terms in the quark-antiquark scattering kernel, ${\mathcal{K}}^{I\ne 1}$. Left: elements in the gluon-quark scattering matrix, $\mathcal{C}$; and right: contributions they generate in ${\mathcal{K}}_{\cancel{\mathcal{C}}}$.