Study of N(1520) and N(1535) structures via $\gamma ^*p\rightarrow N^*$ transitions

Abstract

The helicity amplitudes of the N(1520) and N(1535) resonances in the $\gamma ^*p\rightarrow N^*$ electromagnetic transition are studied in the constituent quark model using the impulse approximation, with the proton and resonances assumed to be in three-quark configurations. The comparison of theoretical results and experimental data on the helicity amplitudes $A_{1/2}$, $A_{3/2}$, and $S_{1/2}$ indicates that the N(1520) and N(1535) resonances are primarily composed of three-quark $L=1$ states but may contain additional components. However, it is improbable that contributions from meson clouds will be dominant at low $Q^2$.

Appendices

Appendix A: Matrix elements of $\gamma q\rightarrow q'$ for helicity amplitudes

The matrix elements of the single quark transition $\gamma q\rightarrow q'$ for the helicity $\lambda =0,1$ are derived in detail,

$$\begin{aligned}&T^0_{\uparrow \uparrow }=\ \left[ \dfrac{(E'+m)(E+m)}{4E'E}\right] ^\frac{1}{2}\left[ 1+ \dfrac{p_z'p_z+2p_-'p_+}{(E'+m)(E+m)}\right] , \nonumber \\&T^0_{\uparrow \downarrow }=\ \left[ \dfrac{(E'+m)(E+m)}{4E'E}\right] ^\frac{1}{2}\left[ \dfrac{\sqrt{2}(p_z'p_--p_-'p_z)}{(E'+m)(E+m)}\right] , \nonumber \\&T^0_{\downarrow \uparrow }=\ \left[ \dfrac{(E'+m)(E+m)}{4E'E}\right] ^\frac{1}{2}\left[ \dfrac{\sqrt{2}(-p_z'p_++p_+'p_z)}{(E'+m)(E+m)}\right] , \nonumber \\&T^0_{\downarrow \downarrow }=\ \left[ \dfrac{(E'+m)(E+m)}{4E'E}\right] ^\frac{1}{2}\left[ 1+\dfrac{p_z'p_z+2p_+'p_-}{(E'+m)(E+m)}\right] , \nonumber \\&T^+_{\uparrow \uparrow }=\ \left[ \dfrac{(E'+m)(E+m)}{4E'E}\right] ^\frac{1}{2}\left[ \dfrac{2p_+}{E+m}\right] , \nonumber \\&T^+_{\uparrow \downarrow }=\ \left[ \dfrac{(E'+m)(E+m)}{4E'E}\right] ^\frac{1}{2}\left[ -\dfrac{\sqrt{2}p_z}{E+m}+\dfrac{\sqrt{2}p_z'}{E'+m}\right] , \nonumber \\&T^+_{\downarrow \uparrow }=\ 0,\nonumber \\&T^+_{\downarrow \downarrow }=\ \left[ \dfrac{(E'+m)(E+m)}{4E'E}\right] ^\frac{1}{2}\left[ \dfrac{2p_+'}{E'+m}\right] , \end{aligned}$$

(A.1)

where E and $E'$ are respectively the energies of the initial and final interacting quarks with the dynamical quark mass of u and d quarks as m, and $p_{\pm }=\frac{1}{\sqrt{2}}(p_{x}\pm ip_{y})$.

Appendix B: Basis functions of symmetric, $\rho$ and $\lambda$ types

The basis functions $\Psi _n^{[3]_S}$ of symmetric type in Table 1 as well as the basis functions of $\rho$ and $\lambda$ types in Tables 2 and 3 are constructed from the general harmonic oscillator wave functions $\phi _{nlm}({\varvec{\rho }})$ and $\phi _{nlm}({\varvec{\lambda }})$,

$$\begin{aligned} \phi _{nlm}({\varvec{r}})&= R_{nl}(r)Y_{lm}({\varvec{{\hat{r}}}}),\nonumber \\ R_{nl}(r)&= \left[ \frac{2\alpha ^3n!}{(\frac{1}{2}+n+l)!}\right] ^\frac{1}{2} (\alpha r)^le^{-\frac{1}{2}\alpha ^2r^2}L^{l+\frac{1}{2}}_{n}(\alpha ^2r^2), \end{aligned}$$

(B.2)

with

$$\begin{aligned} {\varvec{\rho }}&= \frac{1}{\sqrt{2}}({\varvec{r}}_1-{\varvec{r}}_2),\nonumber \\ {\varvec{\lambda }}&= \frac{1}{\sqrt{6}}({\varvec{r}}_1+{\varvec{r}}_2-2{\varvec{r}}_3). \end{aligned}$$

(B.3)

where $\alpha$ is the length parameter of the harmonic oscillator, $Y_{lm}({\varvec{{\hat{r}}}})$ are the spherical harmonic, $L^{l+\frac{1}{2}}_{n}$ are the associated Laguerre polynomials.