Electroexcitation of nucleon resonances of the [70,1-] multiplet in a light-front relativistic quark model

TL;DR

This paper uses a light-front relativistic quark model to predict the core contributions to the electroexcitation of specific nucleon resonances, aiding interpretation of upcoming experimental data in the third resonance region.

Contribution

It provides theoretical predictions for nucleon resonance electroexcitation amplitudes, distinguishing quark core effects from meson-baryon contributions, especially relevant for upcoming CLAS experiments.

Findings

Quantified meson-baryon contributions to electroexcitation amplitudes.

Predicted core contributions for N(1520)3/2-, N(1535)1/2-, and N(1675)5/2-.

Provided Q2-dependent electroexcitation amplitude estimates.

Abstract

We utilize a light-front relativistic quark model to predict the 3q core contribution to the electroexcitation of nucleon resonances of the [70,1-] multiplet on the proton and neutron at Q2 < 5GeV2. The investigation is stimulated in large degree by expected progress in the studies of the electroexcitation of nucleon resonances in the third resonance region in the CLAS experiment. For the resonances N(1520)3/2-, N(1535)1/2-, and N(1675)5/2-, experimental data on electroexcitation amplitudes on the proton are available in a wide range of Q2. This allowed us to quantify the expected meson-baryon contributions to these amplitudes as a function of Q2.

Tables

Table 1. Table 1: Transverse transition helicity amplitudes at Q 2 = 0 superscript 𝑄 2 0 Q^{2}=0 for several states of the [ 70 , 1 − ] 70 superscript 1 [70,1^{-}] multiplet for proton and neutron (in units of 10 − 3 ​ G ​ e ​ V − 1 / 2 superscript 10 3 𝐺 𝑒 superscript 𝑉 1 2 10^{-3}GeV^{-1/2} ). The first two columns show the RPP estimates RPP . Columns 3 and 4 show the inferred meson-baruon contributions obtained by subtraction the values obtained in the LF RQM from experimental data. The quoted uncertainties are from the experimental estimates.

Resonance	proton	neutron
	$A_{1/2}{A}_{3/2}{A}_{1/2}{A}_{3/2}$	$A_{1/2}{A}_{3/2}{A}_{1/2}{A}_{3/2}$
	exp. RPP                  exp$-$LF RQM	exp. RPP                  exp$-$LF RQM
$N\left(1520\right){\frac{3}{2}}^{-}$	$-20\pm 5$        $140\pm 10$     $-17\pm 5$     $-174\pm 10$	$-50\pm 10$   $-115\pm 10$     $25\pm 10$    $131\pm 10$
$N\left(1535\right){\frac{1}{2}}^{-}$	$115\pm 15$                       $-54\pm 15$	$-75\pm 20$ $102\pm 20$
$\Delta \left(1620\right){\frac{1}{2}}^{-}$	$40\pm 15$                       $-152\pm 15$
$N\left(1675\right){\frac{5}{2}}^{-}$	$19\pm 8$            $20\pm 5$         $16\pm 8$         $15\pm 5$	$-60\pm 5$   $-85\pm 10$     $-13\pm 5$    $-23\pm 10$
$\Delta \left(1700\right){\frac{3}{2}}^{-}$	$140\pm 30$        $140\pm 30$     $-85\pm 30$     $-59\pm 30$

Equations

$$<N^{\ast },S_z^{\prime }|J_{em}^{\mu }|N,S_z>{|}_{\mathrm{P}\to \infty }$$

$$=3e{Q}_a\int {\Psi }^{\prime +}({\mathrm{p}}_a^{\prime },{\mathrm{p}}_b^{\prime },{\mathrm{p}}_c^{\prime }){\Gamma }_a^{\mu }\Psi ({\mathrm{p}}_a,{\mathrm{p}}_b,{\mathrm{p}}_c)d\Gamma ,$$

$$x_a{\Gamma }_a^x=2{\mathrm{p}}_{ax}+{\mathrm{Q}}_x+i{\mathrm{Q}}_y{\sigma }_z^{(a)},$$

$$x_a{\Gamma }_a^y=2{\mathrm{p}}_{ay}+{\mathrm{Q}}_y-i{\mathrm{Q}}_x{\sigma }_z^{(a)},$$

$${\Gamma }_a^0={\Gamma }_a^z=2\mathrm{P},$$

$${\mathbf{p}}_i=x_i\mathbf{P}+{\mathbf{q}}_{i\perp },\sum _i{\mathbf{q}}_{i\perp }=0,\sum _i{x}_i=1.$$

$$M_0^2={\left(\sum _i{\mathrm{p}}_i\right)}^2=\sum _i\frac{{\mathbf{q}}_{i\perp }^2+m_q^2}{x_i},$$

$${\mathrm{q}}_{iz}+{\omega }_i=M_0{x}_i,{\omega }_i=\sqrt{m_q^2+{\mathbf{q}}_i^2},$$

$${\mathrm{q}}_{iz}=\frac{1}{2}\left(x_i{M}_0-\frac{m_q^2+{\mathbf{q}}_{i\perp }^2}{x_i{M}_0}\right),$$

$$M_0=\sum _i{\omega }_i,\sum _i{\mathbf{q}}_i=0.$$

$$\Psi =U^{+}({\mathrm{p}}_a)U^{+}({\mathrm{p}}_b)U^{+}({\mathrm{p}}_c){\Psi }_{fss}\Phi ({\mathbf{q}}_a,{\mathbf{q}}_b,{\mathbf{q}}_c).$$

$$U({\mathrm{p}}_i)=\frac{m_q+M_0{x}_i+i{ϵ}_{lm}{\sigma }_l{\mathrm{q}}_{im}}{\sqrt{(m_q+M_0{x}_i{)}^2+{\mathbf{q}}_{i\perp }^2}}.$$

$$d\Gamma =\frac{1}{(2\pi {)}^6}\frac{d{\mathbf{q}}_{b\perp }d{\mathbf{q}}_{c\perp }d{x}_{b}d{x}_c}{4x_a{x}_b{x}_c}.$$

$$<N^{\ast }({\frac{1}{2}}^{-})|J_{em}^{\mu }|N>\equiv e\bar{u}(P^{\prime }){\gamma }_5{\tilde{J}}^{\mu }u(P),$$

$$<N^{\ast }({\frac{3}{2}}^{-})|J_{em}^{\mu }|N>\equiv e{\bar{u}}_{\nu }(P^{\prime }){\Gamma }^{\nu \mu }u(P),$$

$$<N^{\ast }({\frac{5}{2}}^{-})|J_{em}^{\mu }|N>\equiv e{\bar{u}}_{\nu {\nu }_1}(P^{\prime })k^{{\nu }_1}{\gamma }_5{\Gamma }^{\nu \mu }u(P),$$

$${\tilde{J}}^{\mu }=\left(k/k^{\mu }-k^2{\gamma }^{\mu }\right)G_1+\left[k/{\mathcal{P}}^{\mu }-(\mathcal{P}k){\gamma }^{\mu }\right]G_2,$$

$${\Gamma }^{\nu \mu }(Q^2)=G_1{\mathcal{H}}_1^{\nu \mu }+G_2{\mathcal{H}}_2^{\nu \mu }+G_3{\mathcal{H}}_3^{\nu \mu },$$

$${\mathcal{H}}_1^{\nu \mu }=k/g^{\nu \mu }-k^{\nu }{\gamma }^{\mu },$$

$${\mathcal{H}}_2^{\nu \mu }=k^{\nu }{P}^{\prime \mu }-(k{P}^{\prime })g^{\nu \mu },$$

$${\mathcal{H}}_3^{\nu \mu }=k^{\nu }{k}^{\mu }-k^2{g}^{\nu \mu },$$

$$\frac{1}{2P}<N^{\ast },\frac{1}{2}|J_{em}^{0,z}|N,\frac{1}{2}>{|}_{P\to \infty }=Q^2{G}_1(Q^2),$$

$$\frac{1}{2P}<N^{\ast },\frac{1}{2}|J_{em}^{0,z}|N,-\frac{1}{2}>{|}_{P\to \infty }=$$

$$=-\frac{M+m}{2}Q{G}_2(Q^2).$$

$$\frac{1}{2P}<N^{\ast },\frac{3}{2}|J_{em}^{0,z}|N,\frac{1}{2}>{|}_{P\to \infty }=$$

$$-\frac{Q}{\sqrt{2}}\left[G_1(Q^2)-\frac{M+m}{2}{G}_2(Q^2)\right],$$

$$\frac{1}{2P}<N^{\ast },\frac{3}{2}|J_{em}^{0,z}|N,-\frac{1}{2}>{|}_{P\to \infty }=\frac{Q^2}{2\sqrt{2}}{G}_2(Q^2),$$

$$<N^{\ast },\frac{3}{2}|J_{em}^x+i{J}_{em}^y|N,-\frac{1}{2}>{|}_{P\to \infty }=\frac{Q^3}{\sqrt{2}}{G}_3(Q^2).$$

$$\frac{1}{2P}<N^{\ast },\frac{5}{2}|J_{em}^{0,z}|N,\frac{1}{2}>{|}_{P\to \infty }=$$

$$-Q^2\left[G_1(Q^2)+\frac{M-m}{2}{G}_2(Q^2)\right],$$

$$\frac{1}{2P}<N^{\ast },\frac{5}{2}|J_{em}^{0,z}|N,-\frac{1}{2}>{|}_{P\to \infty }=-\frac{Q^3}{2}{G}_2(Q^2),$$

$$<N^{\ast },\frac{5}{2}|J_{em}^x+i{J}_{em}^y|N,-\frac{1}{2}>{|}_{P\to \infty }=Q^4{G}_3(Q^2).$$

$$A_{\frac{1}{2}}=b\left[2Q^2{G}_1-(M^2-m^2)G_2\right],$$

$$S_{\frac{1}{2}}=-b\frac{K}{\sqrt{2}}{\tilde{S}}_{\frac{1}{2}},$$

$${\tilde{S}}_{\frac{1}{2}}=2(M-m)G_1+(M+m)G_2,$$

$$b\equiv e\sqrt{\frac{Q_{+}}{8m(M^2-m^2)}},$$

$$K\equiv \frac{\sqrt{Q_{+}{Q}_{-}}}{2M},$$

$$Q_{\pm }\equiv (M\pm m{)}^2+Q^2.$$

$$A_{1/2}=h_{3}X,A_{3/2}=\mp \sqrt{3}{h}_{2}X,$$

$$S_{1/2}=\mp h_1\frac{K}{\sqrt{2}M}X,$$

$$X\equiv K^{l-1}e\sqrt{\frac{Q_{\pm }}{32Jm(M^2-m^2)}},l=J-\frac{1}{2},$$

$$h_1(Q^2)=\mp 4M{G}_1(Q^2)+4M^2{G}_2(Q^2)+$$

$$2(M^2-m^2-Q^2)G_3(Q^2),$$

$$h_2(Q^2)=-2(\mp M+m)G_1(Q^2)-$$

Full text

I Introduction

Experiments on the new generation of electron beam facilities CEBAF(Jefferson Lab), MAMI(Mainz), and MIT-Bates led to dramatic progress in the investigation of the electroexcitation of nucleon resonances, and significant role in the interpretation of new data belongs to quark models, in particular, to light-front relativistic quark models (LF RQM). The CLAS measurements at Jefferson Lab made possible, for the first time, the determination of the electroexcitation amplitudes of the Roper resonance $N(1440)\frac{1}{2}^{+}$ on the proton in a wide range of photon virtuality up to $Q^{2}=4.5~{}$GeV2 CLAS2009 . The comparison of these results with the LF RQM predictions Capstick1995 ; Aznauryan2007 was crucial for identification of the $N(1440)\frac{1}{2}^{+}$ as a predominantly radial excitation of a three-quark (3$q$) ground state, with additional non-3-quark contributions needed to describe the low $Q^{2}$ behavior of the amplitudes. The $\gamma^{*}p\rightarrow\Delta(1232)\frac{3}{2}^{+}$ transition amplitudes have been measured in a more wide range of $Q^{2}$ ($0.06\div 8~{}$GeV2) CLAS2009 ; Stave ; Sparveris ; Mertz ; Kunz ; Frolov ; Vilano ; KELLY . The obtained data strongly confirm the meson-cloud contribution as a source of the long-standing descreapancy between the data and quark model predictions for the magnetic-dipole form factor of this transition, and the ’bare’ contribution to this form factor, obtained within dynamical reaction model Sato2001 ; Lee2004 ; EBAC is very close to the LF RQM predictions Riska2004 ; Aznauryan2015 ; Aznauryan2016 . Above $2~{}$GeV2, the LF RQM Aznauryan2015 reproduces observed in experiment smallness of the ratio $R_{EM}$, as well the negative sign and sharply growing absolute value of the ratio $R_{SM}$ for the $\gamma^{*}p\rightarrow\Delta(1232)\frac{3}{2}^{+}$ transition. A very interesting conclusion was made from the results on the $\gamma^{*}p\rightarrow N(1675)\frac{5}{2}^{-}$ amplitudes extracted from CLAS data CLAS2015 . A special feature of the resonance $N(1675)\frac{5}{2}^{-}$ is the strong suppression of the transverse helicity amplitudes for its excitation through quark transition from the proton. This feature allowed one to draw conclusion regarding the dominant strength of the meson-baryon contribution to the $\gamma^{*}p\rightarrow N(1675)\frac{5}{2}^{-}$ transverse helicity amplitudes Aznauryan2015_1 which is supported by the results of the dynamical coupled-channels approach EBAC .

Experiments on meson electroproduction on new electron beam facilities have been performed on the proton target and, in the whole, allowed extraction of the electroexcitation ampltudes for the resonances $\Delta(1232)\frac{2}{2}^{+}$ CLAS2009 ; Stave ; Sparveris ; Mertz ; Kunz ; Frolov ; Vilano ; KELLY and $N(1535)\frac{1}{2}^{-}$ CLAS2009 ; Thompson ; Denizli ; Armstrong ; Dalton in the range of $Q^{2}$ up to $8~{}$GeV2, for the $N(1440)\frac{1}{2}^{+}$, $N(1520)\frac{3}{2}^{-}$, $N(1675)\frac{5}{2}^{-}$, $N(1680)\frac{5}{2}^{+}$, and $N(1710)\frac{1}{2}^{+}$ at $Q^{2}<4.5~{}$GeV2 CLAS2009 ; CLAS2015 ; Mokeev1 ; Mokeev2 ; Mokeev3 , and for the $\Delta(1620)\frac{1}{2}^{-}$, $N(1650)\frac{1}{2}^{-}$, $\Delta(1700)\frac{3}{2}^{-}$, and $N(1720)\frac{3}{2}^{+}$ at $Q^{2}=0.65\div 1.3~{}$GeV2 Mokeev1 ; Mokeev2 ; Mokeev3 . Currently new data are in preparation by the CLAS collaboration for the $ep\to ep\pi^{0}$ process in the same kinematics region as the CLAS data in the $ep\to en\pi^{+}$ channel CLAS2015 ; CLAS2008 . The two-channel analysis will allow for the separation of all resonances in the third nucleon resonance region at $Q^{2}<4.5~{}$GeV2. Other processes, such as $en(p_{s})\to ep\pi^{-}(p_{s})$ on deuterium target and $ep\to ep\pi^{+}\pi^{-}$ are also in preparation.

Therefore, in the near future CLAS experiment will provide us with rich information on the electroexcitation of the nucleon resonances from the multiplet $[70,1^{-}]$ at $Q^{2}<4.5~{}$GeV2, and our goal in the present investigation is to extend our previous results on the electroexcitation of the $N(1520)\frac{3}{2}^{-}$ and $N(1535)\frac{1}{2}^{-}$ within LF RQM Aznauryan2012 by comprehensive investigation of electroexcitation of all resonances assigned to the $[70,1^{-}]$-plet on the proton and neutron.

We use an approach based on the LF dynamics which presents the most suitable framework for describing the transitions between relativistic bound systems Drell ; Terentiev ; Brodsky . In early works by Berestetsky and Terent’ev Terentiev , the approach was based on the construction of the generators of the Poincaré group in the LF. It was later formulated in the infinite momentum frame (IMF) Terentiev1 ; Aznauryan1982 . This allowed one to demonstrate more clearly that diagrams which violate impulse approximation, i.e. the diagrams containing vertices like $\gamma^{*}\rightarrow q{\bar{q}}$, do not contribute. The interpretation of results for the $\gamma^{*}N\rightarrow N(N^{*})$ transitions in terms of the vertices $N(N^{*})\leftrightarrow 3q$ and corresponding wave functions became more evident. A similar approach was developed and used in the investigation of electroexcitation of nucleon resonances in Ref. Capstick1995 within LF Hamiltonian dynamics Keister . Both approaches use complete set of orthogonal wave functions that correspond to the classification of the nucleon and nucleon resonances within the group $SU(6)\times O(3)$ in the c.m.s. of constituent quarks. It was shown in Ref. Aznauryan1982 that the wave functions of the system of quarks in the IMF and in their c.m.s. are related through Melosh rotations of quark spin matrices Melosh . The same result was obtained in Ref. Capstick1995 within LF Hamiltonian dynamics.

The paper is organized as follows. In Sec. II we present the LF RQM formalism to compute the $\gamma^{*}N\rightarrow N^{*}$ transition amplitudes. We specify the IMF where the LF RQM is built and the relations between the $<N^{*}|J_{em}^{\mu}|N>$ matrix elements and the $N(N^{*})\leftrightarrow 3q$ wave functions in this frame. Further, the relations between these matrix elements and the $\gamma^{*}N\rightarrow N^{*}$ form factors and transition helicity amplitudes are presented. In Sec. III we discuss the mixings of the states $N(1535)\frac{1}{2}^{-}$ and $N(1650)\frac{1}{2}^{-}$, and $N(1520)\frac{3}{2}^{-}$ and $N(1700)\frac{3}{2}^{-}$. We discuss and present the available information on the corresponding mixing angles. The results are presented in Sec. IV and further summarized and discussed in Sec. V.

II The $\gamma^{*}N\rightarrow N^{*}$ transition amplitudes

in LF RQM

The $\gamma^{*}N\rightarrow N^{*}$ transition amplitudes have been evaluated within the approach of Ref. Aznauryan1982 where the LF RQM is formulated in the infinite momentum frame chosen in such a way, that the initial hadron moves along the $z$-axis with the momentum ${\rm P}\rightarrow\infty$, the virtual photon momentum is ${\rm k}^{\mu}=\left(\frac{M^{2}-m^{2}-\mathbf{Q}^{2}_{\perp}}{4{\rm P}},\mathbf{Q}_{\perp},-\frac{M^{2}-m^{2}-\mathbf{Q}^{2}_{\perp}}{4{\rm P}}\right)$, the final hadron momentum is ${\rm P^{\prime}=P+k}$, and ${\rm Q^{2}\equiv-k^{2}}=\mathbf{Q}_{\perp}^{2}$; $m$ and $M$ are masses of the nucleon and resonance, respectively. In this frame, the matrix elements of the electromagnetic current for the $\gamma^{*}N\rightarrow N^{*}$ transition have the form:

[TABLE]

where $S_{z}$ and $S^{\prime}_{z}$ are the projections of the hadron spins on the $z$-direction. In Eq. (1), it is supposed that the photon interacts with quark $a$ (the quarks in hadrons are denoted by $a,b,c$), $Q_{a}$ is the charge of this quark in units of $e$ ($e^{2}/4\pi=1/137$); $\Psi$ and $\Psi^{\prime}$ are wave functions in the vertices $N(N^{*})\leftrightarrow 3q$; ${\rm p}_{i}$ and ${\rm p}^{\prime}_{i}$ ($i=a,b,c$) are the quark momenta in IMF; $d\Gamma$ is the phase space volume; $\Gamma_{a}^{\mu}$ corresponds to the vertex of the quark interaction with the photon:

[TABLE]

where $x_{i}$ ($i=a,b,c$) is the fraction of the initial hadron momentum carried by the quark:

[TABLE]

The invariant mass of the system of initial quarks has the form:

[TABLE]

$m_{q}$ is the quark mass.

Now we define the c.m.s. of initial quarks with the quark three-momenta $\mathbf{q}_{i}~{}(i=a,b,c)$, where quark transverse momenta are given by Eqs. (5), and the z-components are defined as:

[TABLE]

For the final state quarks, the quantities defined by Eqs. (5-9) are expressed through $\mathbf{P}^{\prime}$, ${\rm p}^{\prime}_{i}$, $\mathbf{q}^{\prime}_{i}$, and $M^{\prime}_{0}$.

According to results of Ref. Aznauryan1982 , the wave function $\Psi$ in Eq. (1) is related to the wave function in the c.m.s. of quarks defined according to Eqs. (5-9) through Melosh matrices Melosh :

[TABLE]

Here we have separated the flavor-spin-space ($\Psi_{fss}$) and spatial ($\Phi$) parts of the c.m.s. wave function. The Melosh matrices are

[TABLE]

We construct the flavor-spin-space parts of the wave functions in the c.m.s. of quarks by utilizing the rules Capstick1995 ; Isgur1 that correspond to the classification of the nucleon and nucleon resonances within the group $SU(6)\times O(3)$.

The phase space volume in Eq. (1) has the form:

[TABLE]

II.1 The relations between matrix elements (1)

and the $\gamma^{*}N\rightarrow N^{*}$ transition helicity amplitudes

Electroexcitation of the states with $J^{P}=\frac{1}{2}^{-}$ and $J^{P}=\frac{3}{2}^{-},\frac{5}{2}^{-}$, that enter the multiplet $[70,1^{-}]$, is described, respectively, by two and three form factors, which we define according to Refs. Aznauryan_review ; Devenish in the following way:

[TABLE]

where

[TABLE]

${\cal P}\equiv\frac{1}{2}(P^{\prime}+P)$, $u(P),u(P^{\prime})$ are the Dirac spinors, and $u_{\nu}(P^{\prime})$, $u_{\nu\nu_{1}}(P^{\prime})$ are the generalized Rarita-Schwinger spinors.

In the LF RQM under consideration, the form factors $G_{i}(Q^{2})$ are derived through the matrix elements (1). For the $J^{P}={\frac{1}{2}}^{-}$ resonances, the relations between form factors and the matrix elements (1) are following:

[TABLE]

For the $J^{P}={\frac{3}{2}}^{-}$ resonances, these relations are following:

[TABLE]

For the $J^{P}={\frac{5}{2}}^{-}$ resonances, we have:

[TABLE]

The relations between the $\gamma^{*}N\rightarrow N^{*}{\frac{1}{2}}^{-}$ helicity amplitudes and the form factors $G_{1}(Q^{2}),G_{2}(Q^{2})$ are following:

[TABLE]

For the resonances with $J^{P}={\frac{3}{2}}^{-}$ and ${\frac{5}{2}}^{-}$ we have:

[TABLE]

where

[TABLE]

and the upper and lower signs correspond, respectively, to $J^{P}={\frac{3}{2}}^{-}$ and ${\frac{5}{2}}^{-}$ resonances.

III Mixing of $N(1535)\frac{1}{2}^{-}$, $N(1650)\frac{1}{2}^{-}$, and

$N(1520)\frac{3}{2}^{-}$, $N(1700)\frac{3}{2}^{-}$

The multiplet $[70,1^{-}]$ consists of the following states: $N\frac{1}{2}^{-}(^{2}8_{1/2})$, $N\frac{3}{2}^{-}(^{2}8_{3/2})$, $N\frac{1}{2}^{-}(^{4}8_{1/2})$, $N\frac{3}{2}^{-}(^{4}8_{3/2})$, $N\frac{5}{2}^{-}(^{4}8_{5/2})$, $\Delta\frac{1}{2}^{-}(^{2}10_{1/2})$, and $\Delta\frac{3}{2}^{-}(^{2}10_{3/2})$, where we use the notation ${}^{2S+1}SU(3)_{J}$, which gives the assignment of the state according to the $SU(3)$ group, $S$ is the total spin of the quarks, and $J$ is the spin of the resonance. The resonances with $J^{P}=\frac{1}{2}^{-}$ and $\frac{3}{2}^{-}$ can be composed, respectively, from the states ${}^{2}8_{1/2},~{}^{4}8_{1/2}$ and ${}^{2}8_{3/2},~{}^{4}8_{3/2}$, and therefore can be mixings of these states:

[TABLE]

There is information on the mixing angles $\theta_{S}$ and $\theta_{D}$, obtained from the description of resonance masses within quark model with QCD-inspired interquark forces Isgur2 and from experimental data on the decay widths of the resonances in the $\pi N$ channel Hey . The results of Ref. Hey are based on the relations:

[TABLE]

that follow from the $SU(6)_{W}$-symmetry. The same relations have been obtained in Ref. Aznauryan1985 within the LF RQM by relating the $<\pi N|N^{*}>$ amplitudes to the matrix elements of the axial-vector current $<N^{*}|J^{\mu}_{ax}|N>$ using the hypothesis of partially conserved axial-vector current (PCAC) in the way suggested in Ref. PCAC . The results of Ref. Hey are based on early data. Using recent data RPP , we have revised the values of the mixing angles extracted from the $\pi N$ widths of the resonances. As a result, we have obtained

[TABLE]

instead of $\theta_{S}=-31.9^{\circ}$ and $\theta_{D}=10.4^{\circ}$ in Ref. Hey . Large difference in $\theta_{S}$ is caused mainly by the significant change of the $N(1535)\frac{1}{2}^{-}\rightarrow\pi N$ width, that resulted in increasing of the ratio of the mean values of the $N(1535)\frac{1}{2}^{-}$ and $N(1650)\frac{1}{2}^{-}$ $\pi N$ decay widths from 0.3 to 0.8.

The mixing angles obtained from the description of masses Isgur2 are following:

[TABLE]

IV Results

In this Section we present our results for the $3q$ core contribution to the helicity transition amplitudes for the electroexcitation of the resonances of the multiplet $[70,1^{-}]$ on the proton and neutron (Figs. 1-12). The spacial part of the wave functions and parameters of the model have been specified in Ref. Aznauryan2012 via description of the nucleon electromagnetic form factors by combining $3q$ and pion-cloud contributions in the LF dynamics. Good description of the nucleon electromagnetic form factors up to $Q^{2}=16~{}$GeV2 has been obtained with the nucleon wave function in the form:

[TABLE]

and by employing two forms of the spatial wave function:

[TABLE]

with the following oscillator parameters and running quark masses:

[TABLE]

For the resonances of the $[70,1^{-}]$-plet, the results for the transition amplitudes obtained with the wave functions (50,51) and corresponding parameters (52,53) are very close to each other. The role of running quark mass becomes visible above $3~{}$GeV2. At $Q^{2}=5~{}$GeV2, it increases the transition helicity amplitudes by $25-35\%$ and $10-15\%$ for the wave functions (50) and (51), respectively.

Meson electroproduction gives strong evidence, that baryon resonances are not excited from quark transition alone, but there can be significant contribution from meson-baryon interaction, including pion-loop contributions generated by nearly massles pions. The common feature of all approaches that account for meson-baryon contributions is the fact that they are more rapidly losing their strength when $Q^{2}$ increases in comparison to the $3q$ contributions. For the $N(1535)\frac{1}{2}^{-}$ and $N(1520)\frac{3}{2}^{-}$, it is expected, that meson-baryon contributions can be neglected at $Q^{2}>2~{}$GeV2 EBAC . There are accurate data for the electroexcitation of these resonances on the proton, respectively, at $Q^{2}<8$ and $4.5~{}$GeV2. Therefore, the weight of the $3q$ contributions to the $N(1535)\frac{1}{2}^{-}$ and $N(1520)\frac{3}{2}^{-}$:

[TABLE]

we find from experimental values of the transition helicity amplitudes, assuming that at $Q^{2}>2~{}$GeV2 they are dominated by the $3q$ contributions. The weight factors $c_{N^{*}}$ for the $N(1535)\frac{1}{2}^{-}$ and $N(1520)\frac{3}{2}^{-}$ are presented in the Captions to Figs. 1 and 6.

IV.1 Mixings and the results for the

$N(1535)\frac{1}{2}^{-}$, $N(1650)\frac{1}{2}^{-}$ and $N(1520)\frac{3}{2}^{-}$, $N(1700)\frac{3}{2}^{-}$

The results for the resonances $N(1535)\frac{1}{2}^{-}$, $N(1650)\frac{1}{2}^{-}$ and $N(1520)\frac{3}{2}^{-}$, $N(1700)\frac{3}{2}^{-}$ are shown in Figs. 1-4 and 6-9 taking into account mixings discussed in Section III. It can be seen, that the amplitudes for the resonances $N(1650)\frac{1}{2}^{-}$ and $N(1700)\frac{3}{2}^{-}$, taken as pure ${}^{4}8_{1/2}$ and ${}^{4}8_{3/2}$ states, are significantly smaller than the amplitudes for the $N(1535)\frac{1}{2}^{-}$ and $N(1520)\frac{3}{2}^{-}$. For this reason, the mixings play significant role in the electroexcitation of the $N(1650)\frac{1}{2}^{-}$ and $N(1700)\frac{3}{2}^{-}$, and in Figs. 3,4 and 8,9, we present three kind of curves: thin solid curves for the unmixed states ($\theta_{S}=\theta_{D}=0$) and thick solid and dashed curves, respectively, for mixing angles from Eqs. (47) and (48). For the resonances $N(1535)\frac{1}{2}^{-}$ and $N(1520)\frac{3}{2}^{-}$, the corresponding curves are very close to each other.

It is known, that the results for the $\gamma^{*}N\rightarrow N^{*}$ transition amplitudes extracted from experimental data contain an additional sign related to the vertex of the resonance coupling to the final state hadrons (see, for example, Ref. Aznauryan_review ). In the electroproduction of pions on nucleons this is the relative sign between the $\pi NN^{*}$ and $\pi NN$ vertices. For the resonances of $[70,1^{-}]$-plet, this sign has been found in Ref. Aznauryan1985 in the LF approach based on PCAC (see also Section III). In Ref. Aznauryan1985 , the electroexcitation of the resonances of $[70,1^{-}]$-plet on the proton and neutron has been investigated at $Q^{2}=0$, and the results for the transverse transition helicity amplitudes have been presented taking into account the relative sign between the $\pi NN^{*}$ and $\pi NN$ vertices. This sign is taken into account also in the results obtained in the present investigation and shown in Figs. 1-12. We mention, that from the relations (41,42,45) it follows that in all considered cases of mixings, the relative sign between the $\pi NN(1535)$ and $\pi NN(1650)$ vertices is negative. This is important for understanding of the results for the $N(1650)\frac{1}{2}^{-}$.

IV.2 SQTM and the results for the

$N(1675)\frac{1}{2}^{-}$

Now we comment on the results for the $N(1675)\frac{5}{2}^{-}$, Figs. 11,12. The approximation of the single quark transition model (SQTM) Hey_Weyers ; Babcock_Rosner ; Cottingham ; SQTM leads to selection rules, which for the resonances of the $[70,1^{-}]$-plet result in the suppression of the transition from the proton to the states with $S=\frac{3}{2}$ for the transverse helicity amplitudes. These are the states $N\frac{1}{2}^{-}(^{4}8_{1/2})$, $N\frac{3}{2}^{-}(^{4}8_{3/2})$, and $N\frac{5}{2}^{-}(^{4}8_{5/2})$. According to our results, relativistic effects violate this suppression weakly. For the $J=\frac{1}{2}$ and $\frac{3}{2}$ states, this can be seen from Figs. 3, 8, where the amplitudes for the electroexcitation of $N\frac{1}{2}^{-}(^{4}8_{1/2})$ and $N\frac{3}{2}^{-}(^{4}8_{3/2})$ are given by the thin solid lines. For the resonance $N(1675)\frac{5}{2}^{-}$, we also have small violation of the suppression of the transverse helicity amplitudes for the electroexcitation on the proton (see Fig. 11). In contrast with proton, electroexcitation amplitudes on the neutron are large. In both cases, for proton and neutron, close predictions have been obtained in the quark model of Ref. Giannini .

IV.3 Inferred meson-baryon contributions

For the resonances $N(1520)\frac{3}{2}^{-}$, $N(1535)\frac{1}{2}^{-}$, and $N(1675)\frac{5}{2}^{-}$, experimental data on electroexcitation amplitudes on the proton are available in wide range of $Q^{2}$. This allowed us to quantify the expected meson-baryon contributions to these amplitudes at $Q^{2}<2-3~{}$GeV2. The meson-baryon contributions inferred from the difference of the LF RQM predictions and the data are shown in Figs. 1, 6, 11 by thin dashed lines. They correspond approximately to the mean values of experimental data. The spread of these contributions can be deduced from the spread and errors of experimental data.

The constituent quark and inferred meson-baryon contributions can be associated, respectively, with the bare and meson-cloud contributions of the dynamical coupled-channels approaches that incorporate hadronic and electromagnetic channels. Much progress has been made recently within the EBAC/Argonne-Osaka coupled-channels analyses EBAC ; EBAC1 ; EBAC2 that include pion photo- and electroproduction data. However, only preliminary results are available from the analyses that are based on the complete set of the CLAS pion electroproduction data in the whole $Q^{2}$ range up to $4.5~{}$GeV2 and from two channels $ep\to ep\pi^{0}$ and $ep\to en\pi^{+}$ INT ; INT1 . The results of the coupled-channels analyses are related to the resonance pole positions; with this in Refs. EBAC ; EBAC1 the absolute values of the meson cloud contributions continued to the real axis and evaluated at $W=1.535,~{}1.52$, and $1.625$ GeV, respectively, for the resonances $N(1520)\frac{3}{2}^{-}$, $N(1535)\frac{1}{2}^{-}$, and $N(1675)\frac{5}{2}^{-}$ are presented.

All inferred meson-baryon contributions have clear peak at $Q^{2}=0$, except the contributions for the $N(1520)\frac{3}{2}^{-}$ $A_{1/2}(Q^{2})$ amplitude and for the $N(1535)\frac{1}{2}^{-}$ $S_{1/2}(Q^{2})$ amplitude. Such pronounced peak is specific for the corresponding meson cloud contributions in the coupled-channels analyses EBAC ; EBAC1 ; INT ; INT1 . Concerning the $A_{1/2}(Q^{2})$ amplitude for the $N(1520)\frac{3}{2}^{-}$, we mention that in all coupled-channels analyses EBAC ; EBAC1 ; INT ; INT1 the results for the meson cloud contribution are by order of magnitude and $Q^{2}$ dependence very close to our result.

For the states that are not affected by mixings, we present also in Table 1 the inferred meson-baryon contributions to the transverse transition helicity amplitudes at the photon point $Q^{2}=0$. According to our results, these contributions for the $N(1520)\frac{3}{2}^{-}$, $N(1535)\frac{1}{2}^{-}$, and $N(1675)\frac{5}{2}^{-}$ are dominated by the isovector component.

V Summary and discussion

In this paper we present the results of a comprehensive investigation of electroexcitation of nucleon resonances of the multiplet $[70,1^{-}]$ on the proton and neutron within LF RQM. The investigation was stimulated by the expected progress in the extraction of the electroexcitation amplitudes for these resonances from the CLAS data, and also by the experiments on deuterium target.

It is known, that the three-quark structure of baryons resulted in predictions of a wealth of excited states with underlying spin-flavor and orbital symmetry of $SU(6)\times O(3)$. In spite of the essentially non-relativistic nature of this symmetry, it describes well the observed quantum numbers and in many cases masses of the resonances in the first, second, and third nucleon resonance regions. The LF dynamics is known as most suitable framework for describing transitions of baryons composed of relativistic constituent quarks. The important feature of the LF approach of Ref. Aznauryan1982 , employed in the present investigation, as well of the LF approach of Ref. Capstick1995 , is the fact that these approaches could solve in uniform way the problem of construction of orthogonal set of wave functions for the relativistic quarks by preserving the $SU(6)\times O(3)$ symmetry. This has been done by setting the $SU(6)\times O(3)$ symmetry in the c.m.s. of constituent quarks defined by Eqs. (5-9). Then it was shown, that in the IMF or LF framework, which are used for calculation of the transition amplitudes, the flavour-spin-space part of wave functions are related to the wave functions in c.m.s. of quarks by quark spin rotations given by the Melosh matrices. Therefore, in our calculations we employ the flavor-spin-space parts of the wave functions that in the c.m.s. of quarks correspond to the classification of states within the group $SU(6)\times O(3)$.

The pairs of resonances $N(1535)\frac{1}{2}^{-}$, $N(1650)\frac{1}{2}^{-}$ and $N(1520)\frac{3}{2}^{-}$, $N(1700)\frac{3}{2}^{-}$ with the same spin-parity can be composed, respectively, from the states ${}^{2}8_{1/2},~{}^{4}8_{1/2}$ and ${}^{2}8_{3/2},~{}^{4}8_{3/2}$. Therefore, they can be mixings of these states. There is information on the mixing angles, obtained from the description of resonance masses within quark model with QCD-inspired interquark forces Isgur2 and from experimental data on the decay widths of the resonances in the $\pi N$ channel Hey . The results of Ref. Hey are based on the early data. Using recent data RPP , we have revised the values of the mixing angles extracted from the $\pi N$ widths of the resonances. In our calculations of the electroexcitation amplitudes for the $N(1535)\frac{1}{2}^{-}$, $N(1650)\frac{1}{2}^{-}$, $N(1520)\frac{3}{2}^{-}$, and $N(1700)\frac{3}{2}^{-}$ we have used two sets of mixing angles: obtained from the description of mass in Ref. Isgur2 and found in the present work from the $\pi N$ widths of the resonances. The calculated amplitudes for the electroexitation of the states ${}^{4}8_{1/2}$ and ${}^{4}8_{3/2}$ turned out significantly smaller than the amplitudes for the states ${}^{2}8_{1/2}$ and ${}^{2}8_{3/2}$. As a result, the mixings do not affect practically the electroexcitation amplitudes for the $N(1535)\frac{1}{2}^{-}$ and $N(1520)\frac{3}{2}^{-}$, but play a significant role for the $N(1650)\frac{1}{2}^{-}$ and $N(1700)\frac{3}{2}^{-}$.

The approximation of the single quark transition model Hey_Weyers ; Babcock_Rosner ; Cottingham ; SQTM leads to selection rules, which for the resonance $N(1675)\frac{5}{2}^{-}$ result in the suppression of the amplitudes $A_{1/2}(Q^{2})$ and $A_{3/2}(Q^{2})$ on the proton. According to our results, relativistic effects violate this suppression weakly, and we expect that experimental values of these amplitudes will be determined mostly by the meson-baryon contributions. In contrast with proton, the predicted electroexcitation amplitudes on the neutron for the $N(1675)\frac{5}{2}^{-}$ are large.

For the resonances $N(1520)\frac{3}{2}^{-}$, $N(1535)\frac{1}{2}^{-}$, and $N(1675)\frac{5}{2}^{-}$, experimental data on electroexcitation amplitudes on the proton are available in wide range of $Q^{2}$. This allowed us to present the expected meson-baryon contributions to these amplitudes at $Q^{2}<2-3~{}$GeV2 inferred from the difference of the LF RQM predictions and the data. The correspondence between these contributions and the meson cloud contributions obtained within the EBAC/Argonne-Osaka coupled-channels analyses EBAC ; EBAC1 ; EBAC2 ; INT ; INT1 is discussed in Sec. IV.3.

Acknowledgments. This work was supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under Contract No. DE-AC05-06OR23177, and the National Science Foundation, State Committee of Science of the Republic of Armenia, Grant No. 15T-1C223.