Quark tensor charge and electric dipole moment within the Schwinger-Dyson formalism

We calculate the tensor charge of the quark in the QCD-like theory in the Landau gauge using the Schwinger–Dyson formalism. It is found that the dressed tensor charge of the quark is significantly suppressed against the bare quark contribution, and the result agrees qualitatively with the analyses in the collinear factorization approach and lattice QCD. We also analyze the quark confinement effect with the phenomenological strong coupling given by Richardson and find that this contribution is small. We show that the suppression of the quark tensor charge is due to the superposition of the spin flip of the quark arising from the successive emission of gluons that dress the tensor vertex. We also consider the relation between the quark and the nucleon electric dipole moments by combining with the simple constituent quark model.

Figure 1

The running strong couplings of QCD-like theory. We use the running couplings with the simple infrared regularization, the smooth infrared regularization, and the Richardson ansatz.

Figure 2

The quark self-energy $\Sigma (p_E^2)$ solved with the Schwinger–Dyson equation.

Figure 3

The quark wave function renormalization $Z(p_E^2)$ solved with the Schwinger–Dyson equation.

Figure 4

The quark self-energy $\Sigma (p_E)$ solved with the Schwinger–Dyson equation with the Higashijima–Miransky approximation.

Figure 5

The Schwinger–Dyson equation for the quark tensor charge expressed diagrammatically.

Figure 6

The $S_1$, $S_2$, and $S_3$, functions (not renormalized) solved with the Schwinger–Dyson equation for the quark tensor charge with the integral cutoff $\Lambda =20\mathrm{GeV}$. The left column shows the results calculated without approximation and the right column with the Higashijima–Miransky approximation. The $S_3$ function was resized with $p^2$.

Figure 7

The $S_1$ function (not renormalized) obtained by solving the Schwinger–Dyson equation with $S_2$ and $S_3$ functions set to zero. The $S_1$ function solved with the full contribution ($S_1$, $S_2$, and $S_3$) is also shown for comparison.

Figure 8

The convergence of $S_1$ function (not renormalized) at the origin in the number of iterations of the Schwinger–Dyson equation with the initial conditions $S_1(p^2)=1$, $S_2(p^2)=0$, and $S_3(p^2)=0$.

Figure 9

Expansion of the Schwinger–Dyson equation for the quark tensor charge. Each iteration gives the perturbative truncated contribution to the corresponding order.

Figure 10

The schematic picture of the quark spin flip with the gluon emission.

Figure 11

The $S_1$ function (not renormalized) calculated with resized self-energy.