Role of diquark correlations and the pion cloud in nucleon elastic form factors

Electromagnetic form factors of the nucleon in the spacelike region are investigated within the framework of a covariant and confining Nambu–Jona-Lasinio model. The bound-state amplitude of the nucleon is obtained as the solution of a relativistic Faddeev equation, where diquark correlations appear naturally as a consequence of the strong coupling in the color $\overline{3} qq$ channel. Pion degrees of freedom are included as a perturbation to the “quark-core” contribution obtained using the Poincaré covariant Faddeev amplitude. While no model parameters are fit to form-factor data, excellent agreement is obtained with the empirical nucleon form factors (including the magnetic moments and radii) where pion loop corrections play a critical role for $Q^2\lesssim 1{\mathrm{GeV}}^2$. Using charge symmetry, the nucleon form factors can be expressed as proton quark sector form factors. The latter are studied in detail, leading, for example, to the conclusion that the $d$-quark sector of the Dirac form factor is much softer than the $u$-quark sector, a consequence of the dominance of scalar diquark correlations in the proton wave function. On the other hand, for the proton quark sector Pauli form factors we find that the effect of the pion cloud and axial-vector diquark correlations overcomes the effect of scalar diquark dominance, leading to a larger $d$-quark anomalous magnetic moment and a form factor in the $u$-quark sector that is slightly softer than in the $d$-quark sector.

Figure 1

The NJL gap equation in the Hartree-Fock approximation, where the thin line represents the elementary quark propagator, $S_0^{-1}\left(k\right)=k-m+i\varepsilon$, and the shaded circle represents the $\overline{q}q$ interaction kernel given in Eq. (2). Higher-order terms, attributed to meson loops, for example, are not included in the gap equation kernel.

Figure 2

NJL Bethe-Salpeter equation for the quark-antiquark $t$ matrix, represented as the double line with the vertices. The single line corresponds to the dressed quark propagator and the BSE $\overline{q}q$ interaction kernel, consistent with the gap equation kernel used in Eq. (5), is given by Eq. (2).

Figure 3

Homogeneous Faddeev equation for the nucleon in the NJL model. The single lines represent the quark propagator and the double lines the diquark propagators. Both scalar and axial-vector diquarks are included in these calculations.

Figure 4

Feynman diagrams representing the nucleon electromagnetic current. The diagram on the left is called the quark diagram and the diagram on the right the diquark diagram. In the diquark diagram the photon interacts with each quark inside the nonpointlike diquark.

Figure 5

Inhomogeneous BSE whose solution gives the quark-photon vertex, represented as the large shaded oval. The small dot is the inhomogeneous driving term, while the shaded circle is the $\overline{q}q$ interaction kernel given in Eq. (2). Only the $\rho$ and $\omega$ interaction channels contribute. This integral equation can equivalently be represented using the elementary quark-photon interaction and the $\rho$ and $\omega t$ matrices, given in Eqs. (12) and (13). This case is depicted by the second equality.

Figure 6

(Top) Dressed-up quark form factors. The dashed line is the Dirac form factor obtained from the BSE of Eq. (59); the solid and dash-dotted lines are, respectively, the Dirac and Pauli form factors generated by also including the pion loop corrections illustrated in Fig. 8. (Bottom) Dressed-down quark form factors. Each curve represents an analogous form factor to those in the top panel.

Figure 7

Pion loop contribution to the dressed quark self-energy. The pion couples to the dressed quark via ${\gamma }_5{\tau }_i$ and the pion $t$ matrix is approximated by its pole form.

Figure 8

Pion loop contributions to the quark-photon vertex. The quark wave function renormalization factor $Z$ represents the probability of striking a dressed quark without a pion cloud. In the first two diagrams the photon couples to the dressed quark with a vertex of the general form given by Eq. (53) and defined by Eqs. (58) and (60). The shaded oval in the third diagram represents the quark-pion vertex, which we approximate by its pole form. It is therefore given by $({\ell }^{\prime }+\ell )F_{\pi }\left(Q^2\right)$, where $F_{\pi }\left(Q^2\right)$ is the usual pion form factor [see Eq. (86) and associated discussion].

Figure 9

Dressed quark body form factors associated with the pion loop corrections, where $f_1^{\left(q\right)}$ and $f_2^{\left(q\right)}$ originate from the second diagram in Fig. 8 and $f_1^{\left(\pi \right)}$ and $f_2^{\left(\pi \right)}$ from the third diagram.

Figure 10

(Top) Dressed-up quark Dirac form factors, which also include pion-cloud effects, separated into quark sectors. The solid line is the $u$-quark sector of the dressed-up quark and the dashed line represents $d$-quark sector. (Bottom) Dressed-up quark Pauli form factors separated into quark sectors. The solid line is the $u$-quark sector and the dashed line the $d$-quark sector.

Figure 11

Feynman diagrams that represent the diquark electromagnetic current. The shaded circles are the diquark Bethe-Salpeter vertices and the shaded ovals are the quark-photon vertex. The Feynman diagrams for the meson form factors are analogous. However, the flow of baryon number on one of the quark lines must be reversed.

Figure 12

Results for the scalar diquark and pion form factors. For the pion we just show the full results; however, for the scalar diquark form factors we show the cases when the dressed quarks are pointlike, the quark-photon vertex is given by the BSE and finally when pion loop corrections are also included.

Figure 13

Scalar diquark and pion form factors multiplied by $Q^2$. The pion form-factor data is from Refs. [59, 60, 61, 62, 63].

Figure 14

Axial-vector diquark body form factors. These body form factors must still be multiplied by the appropriate dressed quark form factors to obtain the axial-vector diquark form factors.

Figure 15

Form factors for an axial-vector diquark with quark content $\left\{ud\right\}$.

Figure 16

(Top) results for the charge, magnetic, and quadruple form factors of a $ud$ axial-vector diquark. (Bottom) results for the charge, magnetic and quadruple form factors of a ${\rho }^{+}$ meson.

Figure 17

Results for the $\mathrm{scalar}\leftrightarrow \mathrm{axial}-\mathrm{vector}$ diquark and $\pi \leftrightarrow \rho$ electromagnetic transition form factors.

Figure 18

Nucleon Dirac (top) and Pauli (bottom) body form factors which result from a vector coupling to the quarks in the Feynman diagrams of Fig. 4. To obtain their contribution to the nucleon form factors these results must be multiplied by the appropriate isospin factors, as in Eqs. (102) and (103), and the dressed quark Dirac form factors.

Figure 19

Nucleon Dirac (top) and Pauli (bottom) body form factors which result from a tensor coupling to the quarks in the Feynman diagrams of Fig. 4. To obtain their contribution to the nucleon form factors these results must be multiplied by the appropriate isospin factors, as in Eqs. (102) and (103), and the dressed quark Pauli form factors.

Figure 20

Results for the proton Dirac (top) and Pauli (bottom) form factors. In each case the dot-dashed curve [superscript (bare)] gives the result when the constituent quark form factors are those of an elementary Dirac particle, the dashed curve [superscript (bse)] includes the quark-photon vertex dressing effects from the BSE, and the solid curve is the full result which also includes pion loop effects. The dotted curve is the empirical result from Ref. [71].

Figure 21

Results for the neutron Dirac (top) and Pauli (bottom) form factors. In each case the dot-dashed curve [superscript (bare)] gives the results when the constituent quark form factors are those of an elementary Dirac particle, the dashed curve [superscript (bse)] includes the quark-photon vertex dressing effects from the BSE, and the solid curve is the full result which also includes pion loop effects. The dotted curve is the empirical result from Ref. [71].

Figure 22

Results for the proton Sachs electric (top) and magnetic (bottom) form factors. In each case the dot-dashed curve [superscript (bare)] is the result when the constituent quark form factors are those of an elementary Dirac particle, the dashed curve [superscript (bse)] includes the quark-photon vertex dressing effects from the BSE, and the solid curve is the full result, which also includes pion loop effects. The dotted curve is the empirical result from Ref. [71].

Figure 23

Results for the neutron Sachs electric (top) and magnetic (bottom) form factors. In each case the dot-dashed curve [superscript (bare)] is the results when the constituent quark form factors are those of an elementary Dirac particle, the dashed curve [superscript (bse)] includes the quark-photon vertex dressing effects from the BSE, and the solid curve is the full result which also includes pion loop effects. The dotted curve is the empirical result from Ref. [71].

Figure 24

Proton up-quark-sector Dirac and Pauli form factors. The empirical results are obtained using Ref. [71] and Eq. (47).

Figure 25

Proton down-quark-sector Dirac and Pauli form factors. The empirical results are obtained using Ref. [71] and Eq. (47).

Figure 26

Total contributions to the proton $u$-sector form factors from each Feynman diagram in Fig. 4. These results include both the vector and the tensor coupling contributions and the sum gives the total $u$-sector Dirac and Pauli proton form factors (solid lines in Fig. 24).

Figure 27

Total contributions to the proton $d$-sector form factors from each Feynman diagram in Fig. 4. These results include both the vector and the tensor coupling contributions and the sum gives the total $d$-sector Dirac and Pauli proton form factors (solid lines in Fig. 25).

Figure 28

Quark-sector contributions to the proton Dirac form factor multiplied by $Q^4$. Experimental data are taken from Ref. [56].

Figure 29

Quark-sector contributions to the proton Pauli form factor multiplied by $Q^4$. Experimental data are taken from Ref. [56].

Figure 30

Exchange-type diagrams that do not appear in our present form-factor calculation because the static approximation is used for the quark exchange kernel.

Figure 31

Proton Sachs form-factor ratio. Data are from Refs. [1, 2, 3, 4, 5, 6].

Figure 32

Neutron Sachs form-factor ratio. Data are from Ref. [78].

Figure 33

Proton up-quark-sector Sachs form factors. The empirical results are obtained using Ref. [71] and Eq. (47).

Figure 34

Neutron up-quark-sector Sachs form factors. The empirical results are obtained using Ref. [71] and Eq. (47).

Figure 35

Total contributions to the proton $u$-sector Sachs form factors from each Feynman diagram in Fig. 4. These results include both the vector and the tensor coupling contributions and the sum gives the total $u$-sector Sachs proton form factors.

Figure 36

Total contributions to the neutron $u$-sector Sachs form factors from each Feynman diagram in Fig. 4. These results include both the vector and the tensor coupling contributions and the sum gives the total $u$-sector Sachs neutron form factors.

Figure 37

Total contributions to the proton Dirac form factor from each Feynman diagram in Fig. 4 for a vector coupling (top) and tensor coupling (bottom) to the dressed quarks. The sum of these ten contributions gives the total proton Dirac form factor illustrated in Fig. 20 (solid curve).

Figure 38

Total contributions to the proton Pauli form factor from each Feynman diagram in Fig. 4 for a vector coupling (top) and a tensor coupling (bottom) to the dressed quarks. The sum of these ten contributions gives the total proton Pauli form factor illustrated in Fig. 20 (solid curve).

Figure 39

Total contributions to the neutron Dirac form factor from each Feynman diagram in Fig. 4 for a vector quark coupling (top) and tensor quark coupling (bottom). The sum of these ten contributions gives the total neutron Dirac form factor illustrated in Fig. 21 (solid curve).

Figure 40

Total contributions to the neutron Pauli form factor from each Feynman diagram in Fig. 4 for a vector quark coupling (top) and tensor quark coupling (bottom). The sum of these ten contributions gives the total neutron Dirac form factor illustrated in Fig. 21 (solid curve).