Quark scalar, axial, and pseudoscalar charges in the Schwinger-Dyson formalism

We calculate the scalar, axial, and pseudoscalar charges of the quark in the Schwinger-Dyson formalism of Landau gauge QCD. It is found that the dressed quark-scalar density of the valence quark is significantly enhanced against the bare-quark contribution, and the result explains qualitatively the phenomenologically known value of the pion-nucleon sigma term and also that given by lattice QCD. Moreover, we show that the Richardson’s interquark potential suppresses the quark-scalar density in the Higashjima-Miransky approximation. This fact suggests that the quark-scalar density is an observable that is sensitive to quark confinement. For the quark axial charge, we find that it is suppressed due to the gluon dynamics. The result of the quenched analysis agrees qualitatively with the experimental data of the isovector axial coupling constant $g_A$. We show that the suppression of the quenched axial charge is due to a mechanism similar to that of the quark tensor charge. In the Schwinger-Dyson equation with the leading unquenching quark-loop contribution the quark axial charge is more suppressed, due to the anomaly effect. The quark pseudoscalar density is found to be large, and is divergent as the bare quark becomes massless. This result is in agreement with the phenomenological current algebraic analysis, and explains well the dominance of the pion-pole contribution.

Figure 1

The running strong coupling used in this work. We use the running coupling with the simple infrared regularization with ${\mathrm{\Lambda }}_{\text{QCD}}=900\mathrm{MeV}$.

Figure 2

The quark self-energy $\mathrm{\Sigma }(p_E^2)$ solved with the Schwinger-Dyson equation in the chiral limit.

Figure 3

The quark wave-function renormalization $Z(p_E^2)$ solved with the Schwinger-Dyson equation in the chiral limit.

Figure 4

The Schwinger-Dyson equation for the quark-scalar density expressed diagrammatically. The last two-loop diagram is the isoscalar unquenching effect, which is absent for the isovector quark-scalar density.

Figure 5

The $S_1$ and $S_2$ functions (not renormalized) solved with the Schwinger-Dyson equation for the quark-scalar density with the integral cutoff $\mathrm{\Lambda }=10\mathrm{GeV}$.

Figure 6

The convergence of the isovector quark-scalar density $S_1(0)$ vs the number of iterations of the Schwinger-Dyson equation with the initial conditions $S_1(p_E^2)=1$ and $S_2(p_E^2)=0$. The scalar density was renormalized at $\mu =2\mathrm{GeV}$.

Figure 7

The Z graph of the quark propagation with time in the horizontal direction. This describes quark-pair creation and annihilation in the intermediate state. The propagation of the virtual quark pairs are restricted by the interquark potential due to the gluon exchange.

Figure 8

Comparison of the $S_1$ and $S_2$, functions (not renormalized) solved with the quark-scalar density SDE with the integral cutoff $\mathrm{\Lambda }=10\mathrm{GeV}$ in the Higashijima-Miransky approximation with and without the Richardson ansatz.

Figure 9

Schematic picture of the exchange-current contribution from the unquenching quark-loop. The gray blob represents the scalar operator insertion.

Figure 10

The $S_1$ function (not renormalized) obtained by solving the Schwinger-Dyson equation with the $S_2$ function set to zero. The $S_1$ function solved with the full contribution ($S_1$ and $S_2$) is also shown for comparison.

Figure 11

The Schwinger-Dyson equation for the quark axial charge expressed diagrammatically.

Figure 12

Unquenching quark-loop diagram of the quark axial charge. This diagram contributes to the chiral anomaly, so the loop momentum of the integral must be specified to be consistent with the vector Ward identity.

Figure 13

The $G_1$, $G_2$, and $G_3$ functions solved with the Schwinger-Dyson equation for the quark axial charge with the integral cutoff $\mathrm{\Lambda }=10\mathrm{GeV}$. The $G_3$ function is resized with $p^2$. The renormalization scale is taken as $\mu =2\mathrm{GeV}$. “Anomaly insertion” corresponds to the solution of the quark axial SDE with the effect of the unquenching quark loop replaced by the axial anomaly current.

Figure 14

The convergence of the renormalized $G_1$ function at the origin vs the number of iterations of the Schwinger-Dyson equation with the initial conditions $G_1(p_E^2)=1$, $G_2(p_E^2)=0$, and $G_3(p_E^2)=0$. The initial value $G_1(p_E^2)=1$ is the bare-quark axial charge.

Figure 15

The schematic picture of the quark spin flip with the gluon emission. The spins of the quark and the gluon are, respectively, $s_q=\frac{1}{2}$ and $s_g=1$

Figure 16

The $G_1$ function obtained by solving the Schwinger-Dyson equation with the $G_2$ and $G_3$ functions set to zero. The $G_1$ function solved with the full contribution ($G_1$, $G_2$, and $G_3$) is also shown for comparison.

Figure 17

Diagrammatic picture of the Schwinger-Dyson equation for the quark pseudoscalar density.

Figure 18

Quark-loop diagram coupled to three gluons. The grey blob represents the pseudoscalar operator insertion. This contribution is not vanishing.

Figure 19

The $P$ function (not renormalized) obtained by solving the Schwinger-Dyson equation with different current-quark masses.

Figure 20

The convergence of the renormalized $P$ function at $p_E^2=0$ vs the number of iterations of the Schwinger-Dyson equation with the initial conditions $P(p_E^2)=1$ for different current-quark masses. The initial value $P(p_E^2)=1$ is the bare-quark pseudoscalar density.

Figure 21

The pion-pole contribution is generated by the ladder diagram.

Figure 22

Inner quark-loop of the quark scalar charge diagram.