Proton tensor charges from a Poincaré-covariant Faddeev equation

The proton’s tensor charges are calculated at leading order in a symmetry-preserving truncation of all matter-sector equations relevant to the associated bound-state and scattering problems. In particular, the nucleon three-body bound-state equation is solved without using a diquark approximation of the two-body scattering kernel. The computed charges are similar to those obtained in contemporary simulations of lattice-regularized quantum chromodynamics, an outcome which increases the tension between theory and phenomenology. Curiously, the theoretical calculations produce a value of the scale-invariant ratio ($-{\delta }_{T}d/{\delta }_{T}u$) which matches that obtained in simple quark models, even though the individual charges are themselves different. The proton’s tensor charges can be used to constrain extensions of the Standard Model using empirical limits on nucleon electric dipole moments.

Figure 1

The tensor charge, Eq. (1), measures the net light-front distribution of transversely polarized quarks inside a transversely polarized proton.

Figure 2

Three-body equation in Eq. (19), solved herein for the proton’s mass and bound-state amplitude. Amplitude: vertex on the left-hand-side; spring with shaded circle: quark-quark interaction kernel in Eq. (20); and solid line with shaded circle: dressed-propagators for scattering quarks, obtained by solving a gap equation with the same interaction (Sec. 4b).

Figure 3

Evaluated on shell, i.e., at $P^2=-M_N^2$, the bound-state amplitude which satisfies this identity is canonically normalized. Amplitude: vertices on either side of the square brackets; and solid line with shaded circle: dressed-quark propagators (Sec. 4b). The explicit appearance of the bracketed term with inverse propagators emphasizes that the normalization condition overlaps the amplitude with the unamputated bound-state wave function, either on the left or right.

Figure 4

$a=3$ spinor component of the tensor current in Eq. (26): $\delta$, ${\delta }^{\prime }$ are spinor indices and $n$, $n^{\prime }$ are isospin indices: first term on the right-hand-side, impulse contribution; and second term, interaction correction. The new element is the dressed-quark-tensor vertex, ${\mathrm{\Gamma }}_{\mu \nu }$, described in Sec. 4c.

Figure 5

Inhomogeneous Bethe-Salpeter equation for the dressed-quark-tensor vertex, ${\mathrm{\Gamma }}_{\mu \nu }$, in rainbow-ladder truncation, which uses $S(k)$ from Eq. (35).

Figure 6

Scalar functions determining the $Q=0$ dressed-quark-tensor vertex in Eq. (38) obtained using the interaction in Eq. (21) and $S(p)$ computed from Eq. (35). Shown, too, is the sensitivity to a $\pm 10\%$ variation in the interaction’s range parameter, subject to Eqs. (22), (23).

Figure 7

Comparison of our prediction for the proton’s tensor charges, position 1—Eq. (41a), with those obtained using: lQCD (2—[26], 3—[27], 4—[28]); and a contact-interaction Faddeev equation (5—[33]). The renormalization scale is ${\zeta }^2=4{\mathrm{GeV}}^2$ in all these cases; and the gray bands depict the averages in Eq. (43). Position 6—projected errors achievable at JLab 12 with the Solenoidal Large Intensity Device (SoLID) [4], using Eq. (1) and anticipated transversity distribution data. The central values are chosen to match those estimated elsewhere [89] (7, ${\zeta }^2=2.4{\mathrm{GeV}}^2$) following an analysis of extant transversity distribution data. Earlier estimates from transversity distribution data are also depicted (8—[90], ${\zeta }^2=2.4{\mathrm{GeV}}^2$, and 9—[91], ${\zeta }^2=1{\mathrm{GeV}}^2$.)

Figure 8

Comparison of our prediction for the proton’s isovector tensor charge [position 1—Eq. (41b)] with those obtained using: lQCD (2—[26], 3—[27], 4—[28], 5—[29]); and a contact-interaction Faddeev equation (6—[33]). The renormalization scale is ${\zeta }^2=4{\mathrm{GeV}}^2$ in all these cases; and the gray band depicts the weighted average in Eq. (44). Position 7—estimate obtained using Eq. (1) and extant transversity distribution data [89] (${\zeta }^2=2.4{\mathrm{GeV}}^2$).

Figure 9

Ratio ($-{\delta }_{T}d/{\delta }_{T}u$). Position 1: our result; lQCD: 2—[26], 3—[27] and 4—[28]; contact-interaction Faddeev equation: 5—[33]. The gray band depicts the weighted average in Eq. (45); and the dashed horizontal line is the quark model result $(-{\delta }_{T}d/{\delta }_{T}u)=1/4$ [30]. Position 6—estimate obtained using Eq. (1) and extant transversity distribution data [89].