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We investigate the flavor decomposition of the electromagnetic form factors of the nucleon, based on the chiral quark–soliton model (

Electromagnetic form factors (EMFFs) are the most fundamental observables that reveal the charge and magnetization structures of the nucleon. A series of recent measurements of the EMFFs has renewed the understanding of the internal structure of the nucleon and has posed fundamental questions about its nonperturbative nature. The results of the ratio of the proton EMFFs,

Assuming isospin and charge symmetries, neglecting the strangeness in the nucleon, and using both the experimental data for the proton and neutron EMFFs, Cates et al. [

In this context, we investigated the flavor structure of the nucleon EMFFs within the framework of the self-consistent SU(2) and SU(3) chiral quark–soliton models (

In this work, we present the results of the flavor-decomposed up- and down-quark EMFFs based on the SU(3)

The present work is sketched as follows. In

The matrix element of a flavor vector current between the two nucleon states is expressed in terms of the flavor Dirac and Pauli FFs:

In the Breit frame,

In SU(3) flavor the nucleon EMFFs are expressed in terms of the triplet and octet vector form factors:

The matrix elements given in Eq. (

The Dirac

The mass term

The single-quark Hamiltonian

The integration over the pion field

The

Here, the matrix

After minimizing the action in Eq. (

Treating

Within the collective quantization procedure the nucleon states given in Eq. (

In flavor SU(3) the eigenfunctions of the SU(3) symmetric part of the Hamiltonian turn out to be the SU(3) Wigner

In contrast to the SU(2) case, the nucleon state is no longer a pure octet state but is a mixed state with those in higher representations arising from flavor SU(3) symmetry breaking, i.e.,

A detailed formalism for the zero-mode quantization can be found in Refs. [

All the results presented in the following were computed completely within the model, in the same level of approximation, to maintain consistency. In particular, magnetization observables are presented not in terms of the physical nuclear magneton but, instead, in terms of the model nuclear magneton, i.e., defined as the model value for the nucleon mass, which, at the level of approximation used in this work, is

We want to mention that the ratio between the model nuclear magneton and the physical one is the same as that between the value of

To address the properties of the baryon octet immediately implies flavor structures of the SU(3) baryons. However, it simultaneously indicates the question of how accurate the

The Sachs EM form factors [

On the other hand, the Sachs form factors have a merit that in the Breit frame they may be apparently interpreted as the Fourier transform of the charge and magnetization distributions inside a nucleon. This comes from the fact that in the Breit frame the proton does not exchange energy with the virtual photon with momentum

In the left panel of

The ratio of the proton magnetic FF to the electric FF:

In order to decompose the proton EMFFs into flavor ones, we need to compute the singlet vector form factors of the proton. Then, we are able to express the flavor-decomposed EMFFs of the proton in terms of the singlet, triplet, and octet FFs of the proton:

The flavor-decomposed magnetic moments are defined as

SU(3) | |||

SU(2) | |||

Ref. [ |

The Sachs FFs for the different quark flavors are presented in

Ratios of the nucleon Sachs flavor FFs to the dipole parametrizations (Eq. (

The Dirac (

As mentioned in the introduction, pQCD with factorization schemes [

Dirac FFs

The flavor-decomposed Dirac (

Note, however, that

In

The flavor-decomposed Dirac and Pauli FFs weighted by

At

ThAnomalous magnetic moments

SU(3) | |||||

SU(2) | |||||

Exp. & Phen. |

We are now in a position to discuss the quark transverse charge densities inside both unpolarized and polarized nucleons. The traditional charge and magnetization densities in the Breit framework are defined ambiguously because of the Lorentz contraction of the nucleon in its moving direction [

By definition, Eqs. (

In the upper panel of

Transverse charge densities inside a proton (upper left panel) and a neutron (upper right panel), and the transverse magnetization densities inside a proton (lower left panel) and a neutron (lower right panel). Notations are the same as in

The lower panel of

In

Flavor-decomposed transverse charge and magnetization densities inside a proton. Those for the up quark are shown in the upper panel, the down ones in the middle panel, and the strange charge and magnetization densities in the lower panel. Notations are the same as in

As discussed, the SU(3) transverse charge density was very different from the SU(2) one. We can understand the reason for this from the results of the flavor-decomposed transverse charge densities. The transverse charge densities inside a proton and a neutron can be respectively expressed in terms of the flavor-decomposed ones:

Since

When the nucleon is transversely polarized along the

Transverse charge densities inside a transversely polarized nucleon. The upper-left and upper-right panels show the transverse charge densities inside a proton and a neutron, respectively, being polarized along the

It is very instructive to examine the transverse charge densities inside the transversely polarized nucleon for each flavor, since they reveal with more detail the inner structure of the nucleon.

Flavor-decomposed transverse charge densities inside a transversely polarized nucleon. The upper-left, upper-middle, and upper-right panels show the up, down, and strange transverse charge densities inside a proton and a neutron, respectively, being polarized along the

In the present work, we aimed to investigate the electromagnetic properties of the nucleon, based on the SU(2) and SU(3) chiral quark–soliton models with symmetry-preserving quantization employed. We considered the rotational

We first presented the results of the ratio of the magnetic form factor to the electric form factor of the proton. It was shown that the results from the SU(2) chiral quark–soliton model described the experimental data very well, whereas those of SU(3) seemed slightly underestimated in higher

The Dirac and Pauli form factors were predicted to be asymptotically proportional to

Having performed the 2D Fourier transform of the nucleon electromagnetic form factors, we were able to produce the charge densities in the transverse plane inside a proton. As expected, both the SU(2) and SU(3) transverse charge densities were positive in the proton. However, as for the neutron case, the result from SU(2) was opposite to that from SU(3): the negative charge was located in the center of the neutron while the positive one was distributed in the outer part within the SU(3) chiral quark–soliton model; it was the other way around in the SU(2) model. The explanation for this comes from the decomposed-flavor transverse charge densities in the SU(3) model. In particular, the component of the strange quark played an essential role in spite of the smallness of its magnitude. Since the up-quark component mainly contributed to the transverse charge density inside a proton, the strange transverse charge density was almost negligible. On the other hand, the up- and down-quark contributions were nearly canceled out in such a way that the negative charge remained in the center of the neutron with small magnitude. Then the contribution of the strange quark came into play, so that the transverse charge densities inside a neutron finally became negative in the center.

When the proton was polarized along the positive

Since the transverse charge densities inside unpolarized and polarized nucleons pave a novel way for understanding the internal structure of the nucleon, it is interesting to investigate them for other baryons such as the

H.-Ch.K. is grateful to R. Woloshyn and P. Navratil for their hospitality during his stay at TRIUMF, where part of the present work was done. The present work was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (Grant Number: NRF-2015R1D1A1A01060707). A.S. is grateful to S. Riordan for sharing information on experimental data on the nucleon electromagnetic form factors.

Open Access funding: SCOAP