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Proton and neutron form factors with quark orbital excitations

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Abstract

Nucleon form factors play an especially important role in studying the dynamics of nucleons and explicit structure of the wave functions at arbitrary nucleon velocity. The purpose of the paper is to explain theoretically all four nucleon form factors measured experimentally in the cross section measurements (by the Rosenbluth method), yielding almost equal normalized form factors \(G^p_E,G^p_M,G^n_M\)), as well as in the polarization transfer experiments, where a strongly decreasing proton electric form factor has been discovered. It is shown, using relativistic hyperspherical formalism, that the nucleon wave functions in the lowest (hypercentral) approximation provide almost equal normalized form factors as seen in the Rosenbluth cross sections, but in the higher components they contain a large admixture of the quark orbital momenta, which strongly decreases \(G^p_E\) and this effect is possibly detected in the polarization transfer method (not seen in the classical cross section experiments). Moreover, the same admixture of the higher components explains the small positive form factor \(G^n_E\), which is zero in the hypercentral approximation. The resulting form factors, \(G^p_M(Q),G^p_E(Q),G^n_M(Q)\) are calculated up to \(Q^2\approx 10\) \(\hbox {GeV}^2\), using the the Lorentz contracted nucleon wave functions and shown to be in reasonable agreement with experimental data.

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Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment:The main emphasis of the paper is on the possibly very important new physics in the well discussed topic of form factors-Lorentz contraction and angular momenta excitation in the baryon wave function.To check this approach the necessary data are assembled in Tables 1,2,3 and in the text. More complete data vs theory comparison is planned for the next papers.]

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Acknowledgements

The author is grateful for help and useful discussions to A. M. Badalian. This work was supported by the Russian Science Foundation in the framework of the scientific project, Grant 16-12-10414.

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Correspondence to Yu. A. Simonov.

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Communicated by Reinhard Alkofer.

Appendices

Appendix A1

The spin-flavor wave functions for proton and neutron

Below we denote the ud quarks with spins up or down as \(u_+,u_-,d_+,d_-\). Neutron.

$$\begin{aligned} \Psi ^{\mathrm{(sym)}}_n (\sigma ,f)= & {} \frac{1}{2\sqrt{3}} ( 2u_-d_+d_+ - d_-u_+d_+ - u_+d_-d_+\nonumber \\&+ 2 d_+u_-d_+ -d_+d_-u_+ -d_-d_+u_+ \nonumber \\&-d_+u_+d_- -u_+d_+d_- +2 d_+d_+u_-), \end{aligned}$$
(A1.1)
$$\begin{aligned} \Psi '_n(\sigma ,f)= & {} \frac{1}{2\sqrt{3}} ( d_-d_+u_+ -d_+d_-u_+ -2 d_+d_-u_+ \nonumber \\&+2 u_-d_+d_+ +u_+d_+d_- -d_+u_+d_-) \end{aligned}$$
(A1.2)
$$\begin{aligned} \Psi ^{''}_n(\sigma ,f)= & {} \frac{1}{6} (-d_+d_-u_+ -d_-d_+u_+ -4 d_+d_+u_- \nonumber \\&+2u_+d_-d_+ \nonumber \\&+2d_-u_+d_+ -d_+u_+d_- +2 U_-d_+d_+ \nonumber \\&-u_+d_+d_- +2 u_+d_-d_+ ) \end{aligned}$$
(A1.3)

Proton

$$\begin{aligned} \Psi ^{\mathrm{(sym)}}_p (\sigma ,f)= & {} \frac{1}{3\sqrt{2}} ( 2 u_+d_-u_+ + 2 d_-u_+u_+ -d_+u_-u_+ \nonumber \\&-u_-d_+u_+ -u_+u_-d_+ -u_-u_+d_+ \nonumber \\&-u_+d_+u_- -d_+u_+u_- +2 u_+u_+d_-), \end{aligned}$$
(A1.4)
$$\begin{aligned} \Psi '_p(\sigma ,f)= & {} \frac{1}{2\sqrt{3}} ( u_-u_+d_+ -u_+u_-d_+ -2 u_+d_-u_+ \nonumber \\&+2 d_-u_+u_+ + d_+u_+u_- - u_+d_+u_- ),\end{aligned}$$
(A1.5)
$$\begin{aligned} \Psi ^{''}_p (\sigma ,f)= & {} \frac{1}{6} ( - u_+u_-d_+ -u_-u_+d_+ \nonumber \\&-4 u_+u_+d_- +2 U_+d_-u_+ \nonumber \\&+2 u_-d_+u_+ -u_+d_+u_- \nonumber \\&+2d_-u_+u_+ -d_+u_+u_- +2 d_+u_-u_+ ).\nonumber \\ \end{aligned}$$
(A1.6)

The matrix elements,

$$\begin{aligned} (a'_i,a^{''}_i,b'_i,b^{''}_i,c'_i,c^{''}_i,d'_i,d^{''}_i) \end{aligned}$$
(A1.7)

general structure

$$\begin{aligned} a'_i,b'_i,c'_i,d'_i= & {} x_(a,b,c,d) ( 1, -1, 0 ), a^{''}_i,b^{''}_i,c^{''}_i,d^{''}_i\nonumber \\= & {} y_(a,b,c,d) (1,1,-2), \end{aligned}$$
(A1.8)

where

$$\begin{aligned} x_(a,b,c,d)= \frac{1}{2\sqrt{6}}, -\frac{1}{2\sqrt{6}}, - \frac{1}{2\sqrt{6}}, \frac{1}{6\sqrt{6}}, \end{aligned}$$
(A1.9)

and

$$\begin{aligned} y_(a,b,c,d)= \frac{1}{6\sqrt{2}}, -\frac{1}{6\sqrt{2}}, - \frac{\sqrt{2}}{12}, \frac{1}{18\sqrt{2}}. \end{aligned}$$
(A1.10)

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Simonov, Y.A. Proton and neutron form factors with quark orbital excitations. Eur. Phys. J. A 57, 228 (2021). https://doi.org/10.1140/epja/s10050-021-00546-0

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