Chiral susceptibility and the scalar Ward identity

The chiral susceptibility is given by the scalar vacuum polarization at zero total momentum. This follows directly from the expression for the vacuum quark condensate so long as a nonperturbative symmetry preserving truncation scheme is employed. For QCD in-vacuum the susceptibility can rigorously be defined via a Pauli-Villars regularization procedure. Owing to the scalar Ward identity, irrespective of the form or Ansatz for the kernel of the gap equation, the consistent scalar vertex at zero total momentum can automatically be obtained and hence the consistent susceptibility. This enables calculation of the chiral susceptibility for markedly different vertex Ansätze. For the two cases considered, the results were consistent and the minor quantitative differences easily understood. The susceptibility can be used to demarcate the domain of coupling strength within a theory upon which chiral symmetry is dynamically broken. Degenerate massless scalar and pseudoscalar bound-states appear at the critical coupling for dynamical chiral symmetry breaking.

Figure 1

Dressed quark propagator. Upper panel: $A(p^2)=1/Z(p^2)$, where $Z$ is the wave function renormalization function. Lower panel: $B(p^2)$, the scalar piece of the dressed-quark self-energy. In both panels, the dashed curve is calculated in rainbow-ladder truncation, Eq. (27), with $\mathcal{I}=D/{\omega }^2=4$; and the solid curve is calculated with the BC vertex Ansatz, Eq. (28), and $\mathcal{I}=2$.

Figure 2

$P=0$ scalar vertex, Eq. (18). Upper panel: $C(p^2)$, lower panel: $D(p^2)$. In both panels, the dashed curve is calculated in rainbow-ladder truncation, Eq. (27), with $\mathcal{I}=D/{\omega }^2=4$; and the solid curve is calculated with the BC vertex Ansatz, Eq. (28), and $\mathcal{I}=2$.

Figure 3

Integrand in Eq. (43). Dashed curve: calculated in rainbow-ladder truncation, Eq. (27), with $\mathcal{I}=D/{\omega }^2=4$; solid curve: calculated with BC vertex Ansatz, Eq. (28), and $\mathcal{I}=2$.

Figure 4

Dependence of the chiral susceptibility on the interaction strength in Eq. (31); viz., $\mathcal{I}:=D/{\omega }^2$: dashed curve: RL vertex, Eq. (27); solid curve: BC vertex, Eq. (28).

Figure 5

Influence of $\mathcal{I}:=D/{\omega }^2$ on the pointwise behavior of the dimensionless integrand in Eq. (43), evaluated with the BC vertex, Eq. (28).