Nucleon resonances and transition form factors

The simplest process used in the study of resonance transition amplitudes is single pion or kaon production, e.g. $ep\to e\pi^{+}n$. Single $\pi^{+}$ and $\pi^{0}$ production are most suitable for the study of the lower-mass excited states as they couple dominantly to the excited states with masses up to $\approx 1.7$ GeV. It may then be useful to describe in more detail the analysis techniques and the formalism used. The unpolarized differential cross section for single pseudoscalar meson production can be written in the one-photon exchange approximation as:

$\displaystyle\frac{d\sigma}{dE_{f}d\Omega_{e}d\Omega_{\pi}}$

$\displaystyle=$

$\displaystyle\Gamma{\frac{d\sigma}{d\Omega_{\pi}}}\,,$

(1)

where $\Gamma$ is the virtual photon flux,

$\displaystyle\Gamma=\frac{\alpha_{em}}{2\pi^{2}Q^{2}}\frac{(W^{2}-M^{2})E_{f}}{2ME_{i}}\frac{1}{1-\epsilon},$

(2)

where $M$ is the proton mass, $W$ the mass of the hadronic final state, $\epsilon$ is the photon polarization parameter, $Q^{2}$ the photon virtuality, $E_{i}$ and $E_{f}$ represent the initial and the final electron energies, respectively. Moreover,

$\displaystyle\epsilon=\left[1+2\left(1+\frac{\nu^{2}}{Q^{2}}\right)\tan^{2}\frac{\theta_{e}}{2}\right]^{-1}$

(3)

and

	$\displaystyle\frac{d\sigma}{d\Omega_{\pi}}$	$\displaystyle=$	$\displaystyle\sigma_{T}+\epsilon\sigma_{L}+\epsilon\sigma_{TT}\cos 2\phi_{\pi}$
			$\displaystyle\qquad\qquad\qquad+\sqrt{2\epsilon(1+\epsilon)}\sigma_{LT}\cos{\phi}_{\pi}\,.$

The kinematics for single $\pi^{+}$ production is shown in Fig. 2.

The response functions in (1) are given by:

	$\displaystyle\sigma_{T}$	$\displaystyle=$	$\displaystyle\frac{\vec{\it p}_{\pi}W}{2KM}(|H_{1}|^{2}+|H_{2}|^{2}+|H_{3}|^{2}+|H_{4}|^{2}),$		(5)
	$\displaystyle\sigma_{L}$	$\displaystyle=$	$\displaystyle\frac{\vec{\it p}_{\pi}W}{2KM}(|H_{5}|^{2}+|H_{6}|^{2}),$		(6)
	$\displaystyle\sigma_{TT}$	$\displaystyle=$	$\displaystyle\frac{\vec{\it p}_{\pi}W}{2KM}Re(H_{2}H_{3}^{*}-H_{1}H_{4}^{*}),$		(7)
	$\displaystyle\sigma_{LT}$	$\displaystyle=$	$\displaystyle\frac{\vec{\it p}_{\pi}W}{2KM}Re[H_{5}^{*}(H_{1}-H_{4})+H_{6}^{*}(H_{2}+H_{3})]\,,$		(8)

where $\vec{\it p}_{\pi}$ is the pion 3-momentum in the hadronic center-of-mass system, and $K$ is the equivalent real photon lab energy needed to generate a state with mass $W$:

$\displaystyle K=\frac{W^{2}-M^{2}}{2M}\,.$

(9)

The helicity amplitudes $H_{i},i=1$–6, can be expanded into Legendre polynomials:

$\displaystyle H_{1}$

$\displaystyle=$

$\displaystyle\frac{1}{\sqrt{2}}\sin\theta\cos{\frac{\theta}{2}}\sum_{l=1}^{\infty}(B_{l+}-B_{(l+1)-})(P^{\prime\prime}_{l}-P^{\prime\prime}_{l+1})$

	$\displaystyle H_{2}$	$\displaystyle=$	$\displaystyle\sqrt{2}\cos{\frac{\theta}{2}}\sum_{l=1}^{\infty}(A_{l+}-A_{(l+1)-})(P^{\prime}_{l}-P^{\prime}_{l+1})$
	$\displaystyle H_{3}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{2}}\sin\theta\sin{\frac{\theta}{2}}\sum_{l=1}^{\infty}(B_{l+}+B_{(l+1)-})(P^{\prime\prime}_{l}+P^{\prime\prime}_{l+1})$
	$\displaystyle H_{4}$	$\displaystyle=$	$\displaystyle\sqrt{2}\sin{\frac{\theta}{2}}\sum_{l=1}^{\infty}(A_{l+}+A_{(l+1)-})(P^{\prime}_{l}+P^{\prime}_{l+1})$
	$\displaystyle H_{5}$	$\displaystyle=$	$\displaystyle\sqrt{2}\cos{\frac{\theta}{2}}\sum_{l=1}^{\infty}(C_{l+}-C_{(l+1)-})(P^{\prime}_{l}-P^{\prime}_{l+1})$
	$\displaystyle H_{6}$	$\displaystyle=$	$\displaystyle\sqrt{2}\sin{\frac{\theta}{2}}\sum_{l=1}^{\infty}(C_{l+}+C_{(l+1)-})(P^{\prime}_{l}+P^{\prime}_{l+1})\,,$		(10)

where the $A_{l+}$ and $B_{l+}$ etc., are the transverse partial wave helicity elements for $\lambda_{\gamma p}=\frac{1}{2}$ and $\lambda_{\gamma p}=\frac{3}{2}$, and $C_{\pm}$ the longitudinal partial wave helicity elements. In the subscript, $l+$ and $(l+1)-$ define the $\pi$ orbital angular momenta, and the sign $\pm$ is related to the total angular momentum $J=l_{\pi}\pm\frac{1}{2}$. In the analysis of data on the $N\Delta(1232)$ transition, linear combinations of partial wave helicity elements are expressed in terms of electromagnetic multipoles:

	$\displaystyle M_{l+}$	$\displaystyle=$	$\displaystyle\frac{1}{2(l+1)}[2A_{l+}-(l+2)B_{l+}]$		(11)
	$\displaystyle E_{l+}$	$\displaystyle=$	$\displaystyle\frac{1}{2(l+1)}(2A_{l+}+lB_{l+})$		(12)
	$\displaystyle M_{l+1,-}$	$\displaystyle=$	$\displaystyle\frac{1}{2(l+1)}(2A_{l+1,-}-lB_{l+1,-})$		(13)
	$\displaystyle E_{l+1,-}$	$\displaystyle=$	$\displaystyle\frac{1}{2(l+1)}[-2A_{l+1,-}+(l+2)B_{l+1,-}]$		(14)

	$\displaystyle S_{l+}$	$\displaystyle=$	$\displaystyle\frac{1}{l+1}\sqrt{\frac{{\vec{\it Q^{*}}^{2}}}{Q^{2}}}C_{l+}$		(15)
	$\displaystyle S_{l+1,-}$	$\displaystyle=$	$\displaystyle\frac{1}{l+1}\sqrt{\frac{{\vec{\it Q^{*}}^{2}}}{Q^{2}}}C_{l+1,-}\,,$		(16)

where $\vec{\it Q}^{*}$ is the photon 3-momentum in the hadronic rest frame. The electromagnetic multipoles are often used to describe the transition from the nucleon ground state to the $\Delta(1232)$, which is dominantly described as a magnetic dipole transition $M_{1+}$. The electromagnetic multipoles as well as the partial wave helicity elements are complex quantities and contain both non-resonant and resonant contributions. In order to compare the results to model predictions and LQCD, an additional analysis must be performed to separate the resonant parts $\hat{\rm A}_{\pm}$, $\hat{\rm B}_{\pm}$, etc., from the non-resonant parts of the amplitudes. In a final step, the known hadronic properties of a given resonance can be used to determine photocoupling helicity amplitudes that characterize the electromagnetic vertex:

	$\displaystyle\hat{A}_{l\pm}$	$\displaystyle=$	$\displaystyle\mp FC^{I}_{\pi N}A_{1/2},$		(17)
	$\displaystyle\hat{B}_{l\pm}$	$\displaystyle=$	$\displaystyle\pm F\sqrt{\frac{16}{(2j-1)(2j+3)}}C^{I}_{\pi N}A_{3/2},$		(18)
	$\displaystyle\hat{S}_{l\pm}$	$\displaystyle=$	$\displaystyle-F\frac{2\sqrt{2}}{2J+1}C^{I}_{\pi N}S_{1/2},$		(19)
	$\displaystyle F$	$\displaystyle=$	$\displaystyle\sqrt{\frac{1}{(2j+1)}\pi}\frac{K}{p_{\pi}}\frac{\Gamma_{\pi}}{\Gamma^{2}}$

where the $C^{I}_{\pi N}$ are isospin coefficients. The total transverse absorption cross section for the transition into a specific resonance is given by:

$\displaystyle\sigma_{T}=\frac{2M}{W_{R}\Gamma}(A^{2}_{1/2}+A^{2}_{3/2}).$

(20)

It is worth noting that the electric and the scalar transition amplitudes are connected at the so-called pseudo-threshold $Q^{2}_{pt}=-(W-M)^{2}$ through the Siegert theorem Siegert (1937), whose impact on the extraction of nucleon resonance transition amplitudes from pion electro-production is discussed in Tiator (2016); Ramalho (2016).

A model-independent determination of the amplitudes contributing to the electro-excitation of resonances in single pseudoscalar pion production $ep\to e^{\prime}N\pi$ (see kinematics of single pion production in Fig. 2) requires a large number of independent measurements at each value of the electron kinematics $W$, $Q^{2}$, the hadronic cms angle $\cos{\theta^{\pi}}$, and the azimuthal angle $\phi^{\pi}$ describing the angle between the electron scattering plane and the hadronic decay plane. Such a measurement requires full exclusivity of the final state and employing both polarized electron beams and the measurements of the nucleon recoil polarization.

Such measurements would in general require full $4\pi$ coverage for the hadronic final state. The only measurement that could claim to be complete was carried out at Jefferson Lab in Hall A Kelly et al. (2007) employing a limited kinematics centered at resonance for $\vec{e}p\to e^{\prime}\vec{p}\pi^{0}$ at $W=1.232$ GeV, and $Q^{2}\approx 1$ GeV². Figure 5 shows the 16 response functions extracted from this measurement. The results of this measurement in terms of the magnetic $N\Delta$ transition form factor and the quadrupole ratios are included in Fig. 4 among other data. They coincide very well with results of other experiments Aznauryan et al. (2009); Ungaro et al. (2006); Joo et al. (2002); Frolov et al. (1999) using different analysis techniques that may be also applied to broader kinematic conditions, especially higher mass resonances. Details of the latter are discussed in  Aznauryan and Burkert (2012a); Tiator et al. (2011). We briefly summarize them here:

• •

  Dispersion Relations have been employed in two ways: One is based on fixed-t dispersion relations for the invariant amplitudes and was successfully used throughout the nucleon resonance region. Another way is based on DR for the mulipole amplitudes of the $\Delta(1232)$ resonance, and allows getting functional forms of these amplitudes with one free parameter for each of them. It was employed for the analysis of the more recent data.

• •

  The Unitary Isobar Model (UIM) was developed in  Drechsel et al. (1999) from the effective Lagrangian approach for pion photoproduction Peccei (1969). Background contributions from t-channel $\rho$ and $\omega$ exchanges are introduced and the overall amplitude is unitarized in a K-matrix approximation.

• •

  Dynamical Models have been developed, as SAID from pion photoproduction data Arndt et al. (2002), the Sato-Lee model was developed in Sato and Lee (1996). Its essential feature is the consistent description of $\pi N$ scattering and the pion electroproduction from nucleons. It was utilized in the study of $\Delta(1232)$ excitations in the $ep\to ep\pi^{0}$ channel Sato and Lee (2001). The Dubna-Mainz-Taipei model Kamalov and Yang (1999) builds unitarity via direct inclusion of the $\pi N$ final state in the T-matrix of photo- and electroproduction.

The $\Delta^{++}$ isobar was first observed 70 years ago in Enrico Fermi’s experiment that used a $\pi^{+}$ meson beam scattered off the protons in a hydrogen target Anderson et al. (1952). The cross section showed a sharp rise above threshold towards a mass near 1200 MeV. While the energy of the pion beam was not high enough to see the maximum and the fall-off following the peak, a strong indication of the first baryon resonance was observed. It took 12 more years and the development of the underlying symmetry in the quark model before a microscopic explanation of this observation could emerge. There was, however, a problem; while the existence of the $\Delta^{+,0,-}$ could be explained within the model, the existence of the $\Delta(1232)^{++}$, which within the quark model would correspond to a state $|u{\uparrow}u{\uparrow}u{\uparrow}\rangle$, was forbidden as it would have an overall symmetric wave function. It took the introduction of para Fermi statistics Greenberg (1964) what later became ”color”, to make the overall wave function anti-symmetric. In this way the $\Delta^{++}(1232)$ resonance may be considered a harbinger of the development of QCD.

The nucleon to $\Delta(1232)\frac{3}{2}^{+}$ transition is now well measured in a large range of $Q^{2}$ Ungaro et al. (2006); Joo et al. (2002); Frolov et al. (1999). At the real photon point, it is explained by a dominant magnetic transition from the nucleon ground state to the $\Delta(1232)$ excited state. Additional contributions are related to small D-wave components in both the nucleon and the $\Delta(1232)$ wave functions leading to electric quadrupole and scalar quadrupole transitions. These remain in the few % ranges at small $Q^{2}$. The magnetic transition is to $\approx 65\%$ given by a simple spin flip of one of the valence quarks as seen in Fig. 4. The remaining 35% of the magnetic dipole strength is attributed to meson-baryon contributions. There exist extensive theoretical reviews of the $N\Delta(1232)$ transition in the lower $Q^{2}$ range are available Pascalutsa and Vanderhaeghen (2006), and more recent reviews that cover the full $Q^{2}$ covered by data Aznauryan and Burkert (2012a); Tiator et al. (2011).

$\rm G_{Mn,Ash}$ is shown normalized to the dipole form factor, but indicates a much faster $Q^{2}$ fall-off compared to that. In comparison to the advanced LFRQM with momentum-dependent constituent quark mass, and with the Dyson-Schwinger Equation (DSE-QCD) results, there is good agreement at the high-$Q^{2}$ end of the data. The discrepancy at small $Q^{2}=0$ is likely due to the meson-baryon contributions at small $Q^{2}$, which are not modeled in either of the calculations.

The quadrupole ratio $R_{EM}$ shows no sign of departing significantly from its value at $Q^{2}=0$, even at the highest $Q^{2}\approx 6.5$ GeV². Both calculations barely depart from $R_{EM}=0$, and remain near zero at all $Q^{2}>2$ GeV². This indicates that the negative constant value shown by the data is likely due to meson-baryon contributions that are not included in the theoretical models. For the scalar quadrupole ratio $R_{SM}$ the asymptotic prediction in holographic QCD (hQCD)  Grigoryan et al. (2009) is:

$\displaystyle R_{SM}=\frac{ImS_{1+}}{ImM_{1+}}\to-1,~{}~{}{\rm at~{}Q^{2}\to\infty},$

(24)

while $R_{EM}$ in hQCD is predicted to approach +1 asymptotically. The $R_{SM}$ data show indeed a strong trend towards increasing negative values at larger $Q^{2}$, semi-quantitatively described by both calculations at $Q^{2}<4$ GeV². The Dyson-Schwinger equation approach predicts a flattening of $R_{SM}$ at $Q^{2}>4$ GeV², while the Light Front relativistic Quark Model predicts a near constant negative slope of $R_{SM}(Q^{2})$ also at higher $Q^{2}$.

The Roper resonance, discovered in 1964 Roper (1964) in a phase shift analysis of elastic $\pi N$ scattering data, has been differently interpreted for half a century. In the non-relativistic quark model (nrQM), the state is the first radial excitation of the nucleon ground state with a mass expected around 1750 MeV, much higher than the measured Breit-Wigner mass of $\approx 1440$ MeV. This discrepancy is now understood as the consequence of a dynamical coupled channel effect that shifts the mass below the mass of the $N(1535){1/2}^{-}$ state, the negative-parity partner of the nucleon Suzuki et al. (2010). Another problem with the quark model was the sign of the transition form factor $A_{1/2}(Q^{2}=0)$, predicted in the nrQM as large and positive, while experimental analyses showed a negative value.

These discrepancies resulted in different interpretations of the state that could only be resolved with electroproduction data from CLAS at Jefferson Lab, the development of continuous QCD approximations in the Dyson-Schwinger equation approach Segovia et al. (2015) and Light Front Relativistic QM with momentum-dependent quark masses Aznauryan and Burkert (2012b) shown in Fig. 6, and Lattice data Mathur et al. (2005); Lin and Cohen (2012). A recent review of the history and current status of the Roper resonance, is presented in a colloquium-style article published in Review of Modern Physics Burkert and Roberts (2019).

Descriptions of the baryon resonance transitions form factors, including the Roper resonance $N(1440)\frac{1}{2}^{+}$, have also been carried out within holographic models de Teramond and Brodsky (2012); Ramalho and Melnikov (2018). In the range $Q^{2}<0.6$ GeV², calculations based on meson-baryon degrees of freedom and effective field theory Bauer et al. (2014) have been successfully performed, as may be seen in Fig. 6. Earlier model descriptions, such as the Isgur-Karl model that describe the nucleon as a system of 3 constituent quarks in a confining potential and a one-gluon exchange contribution leading to a magnetic hyperfine splitting of states Isgur and Karl (1978, 1977), and the relativized version of Capstick Capstick and Isgur (1986) have popularized the model that became the basis for many further developments and variations, e.g. the light front relativistic quark model, and the hypercentral quark model Giannini et al. (2003). Other models were developed in parallel. The cloudy bag model Thomas et al. (1981) describes the nucleon as a bag of 3 constituent quarks surrounded by a cloud of pions. It has been mostly applied to nucleon resonance excitations in real photoproduction, $Q^{2}=0$ Bermuth et al. (1988); Thomas et al. (1981), with some success in the description of the $\Delta(1232)\frac{3}{2}^{+}$ and the Roper resonance transitions.

There is agreement with the data at $Q^{2}>1.5$ GeV² for these two states, while the meson-baryon contributions for the $\Delta(1232)$ are more extended, and agreement with the quark based calculations is reached at $Q^{2}>4$ GeV². The calculations deviate significantly from the data at lower $Q^{2}$, which indicates the presence of non-quark core effects. For the Roper resonance such contributions have been described successfully in dynamical meson-baryon models Obukhovsky et al. (2011) and in effective field theory Bauer et al. (2014). Calculations on the Lattice for the N-Roper transition form factors $F_{1}^{pR}$ and $F_{2}^{pR}$, which are combinations of the transition amplitudes $A_{1/2}$ and $S_{1/2}$, have been carried out with dynamical quarks Lin and Cohen (2012). The results agree well with the data in the range $Q^{2}<1.0$ GeV², where data and calculations overlap Fig. 7.

New electroproduction data on the Roper Štajner et al. (2017) and on several higher mass states have been obtained in the 2-pion channel, specifically in $ep\to e^{\prime}p\pi^{+}\pi^{-}$ Mokeev et al. (2016).

The mass of the Roper state has been computed on the Lattice and extrapolated to the physical pion mass, showing good agreement with the physical value measured with a Breit-Wigner parametrization. It should be noted that the Roper mass measured at the pole in the complex plane is significantly different from the value obtained using the BW ansatz.

Supported by an extensive amount of single pion electroproduction data, covering the full phase space in the pion polar and azimuthal center-of-mass angles, and accompanied by several theoretical modeling, we can summarize our current understanding of the $N(1440)\frac{1}{2}^{+}$ state as follows:

• •

  The Roper resonance is, at heart, the first radial excitation of the nucleon.

• •

  It consists of a well-defined dressed-quark core, which plays a role in determining the system’s properties at all length scales, but exerts a dominant influence on probes with $Q^{2}>m_{N}^{2}$, where $m_{N}$ is the nucleon mass.

• •

  The core is augmented by a meson cloud, which both reduces the Roper’s core mass by $\approx 20\%$, thereby solving the mass problem that was such a puzzle in constituent quark model treatments, and, at low $Q^{2}$, contributes an amount to the electroproduction transition form factors that is comparable in magnitude with that of the dressed quark core, but vanishes rapidly as $Q^{2}$ is increased beyond $m_{N}^{2}$.

As stated in the conclusions of  Burkert and Roberts (2019): ”The fifty years of experience with the Roper resonance have delivered lessons that cannot be emphasized too strongly. Namely, in attempting to predict and explain the QCD spectrum, one must fully consider the impact of meson-baryon final state interactions and the coupling between channels and states that they generate, and look beyond merely locating the poles in the S-matrix, which themselves reveal little structural information, to also consider the $Q^{2}$ dependencies of the residues, which serve as a penetrating scale-dependent probe of resonance composition.”

The $N(1520)\frac{3}{2}^{-}$ state corresponds to the lowest excited nucleon resonance with $J^{P}=\frac{3}{2}^{-}$. Its helicity structure is defined by the $Q^{2}$ dependence of the two transverse transition amplitudes $A_{1/2}$ and $A_{3/2}$. They are both shown in Fig. 8. A particularly interesting feature of this state is that at the real photon point, $A_{3/2}$ is strongly dominant, while $A_{1/2}$ is very small. However, at high $Q^{2}$, $A_{1/2}$ is becoming dominant, while $A_{3/2}$ drops rapidly. This behavior is qualitatively consistent with the expectation of asymptotic QCD, which predicts the transition helicity amplitudes to behave like:

$\displaystyle\hskip 71.13188ptA_{1/2}\propto\frac{a}{Q^{3}},A_{3/2}\propto\frac{b}{Q^{5}}\,.$

(25)

The helicity asymmetry

$\displaystyle\hskip 71.13188ptA_{hel}=\frac{A_{1/2}^{2}-A_{3/2}^{2}}{A_{1/2}^{2}+A_{3/2}^{2}},$

(26)

shown in Fig. 8, illustrates this rapid change in the helicity structure of the $\gamma_{v}pN(1520){3/2}^{-}$ transition. At $Q^{2}>2~{}$GeV², $A_{1/2}$ fully dominates the process.

This state is the parity partner state to the ground state nucleon, with the same spin 1/2 but with opposite parity, its quark content requires an orbital L=1 excitation in the transition from the proton. In the $SU(6)\otimes O(3)$ symmetry scheme, the state is a member of the $[70,1^{-}]$ super multiplet. This state couples equally to $N\pi$ and to $N\eta$ final state. It has therefore be probed using both decay channels $ep\to ep\eta$ and $ep\to eN\pi^{+,0}$. Because of isospin $I=1/2$ for nucleon states, the coupling to the charged $\pi^{+}n$ channel is preferred over $\pi^{0}p$ owing to the Clebsch-Gordon coefficients.

The $A_{1/2}$ helicity amplitude for the $\gamma pN(1535)\frac{1}{2}^{-}$ resonance excitation shown in Fig. 9 represents the largest range in $Q^{2}$ of all nucleon states for which resonance transition form factors have been measured as part of the broad experimental program at JLab.

For this state, as well as for the $N(1440)\frac{1}{2}^{+}$ state, advanced relativistic quark model calculations Aznauryan and Burkert (2015a), DSE-QCD calculations Segovia et al. (2015) and Light Cone sum rule results Anikin et al. (2015) are available, employing QCD-based modeling of the excitation of the quark core for the first time.

The transverse transition amplitude $A_{1/2}$ of $N(1535)\frac{1}{2}^{-}$ is a prime example of the power of meson electroproduction to unravel the internal structure of the resonance transition. In another section of the Volume ”50 Years of Quantum Chromodynamics”  Burkert et al. (2022), the nature of this state is discussed as a possible example of a dynamically generated resonance. The electroproduction data shown here reveal structural aspects of the state and its nature that require a different interpretation. The transition form factor $A_{1/2}$ of the state, shown in Fig. 9, is quantitatively reproduced over a large range in $Q^{2}$ by two alternative approaches, the LFRQM and the LCSR. Both calculations are based on the assumptions of the presence of a 3-quark core. Notice that there is deviation from the quark calculations at $Q^{2}<1-2$ GeV², highlighted as the shaded area in Fig. 9, which may be assigned to the presence of non-quark contributions. Attempts to compute the transition form factors within strictly dynamical models have not succeeded in explaining the available data Jido et al. (2008). The discrepancy could be resolved if the character of the probe, meson (pion) in the case of hadron interaction and short range photon interaction in the case of electroproduction, probe different parts of the resonance’s spatial structure: peripheral in case of meson scattering and short distance behavior in electro-production. The peripheral meson scattering and low $Q^{2}$ meson photo-production reveal the dynamical features of the state, whereas high $Q^{2}$ electroproduction reveals the structure of the quark core.

In previous discussions we have concluded that meson-baryon degrees of freedom provide significant strength to the resonance excitation in the low-$Q^{2}$ domain where quark based approaches LF RQM, DSE/QCD, and LC SR calculations fail to reproduce the transition amplitudes quantitatively. Our conclusion rests, in part, with this assumption. But, how can we be certain of the validity of this conclusion?

The $N(1675)\frac{5}{2}^{-}$ resonance allows us to test this assumption, quantitatively. Figure 10 shows our current knowledge of the transverse helicity amplitudes $A_{1/2}(Q^{2})$ and $A_{3/2}(Q^{2})$, for proton target compared to RQM Aznauryan and Burkert (2017) and hypercentral CQM Santopinto and Giannini (2012) calculations. The specific quark transition for a $J^{P}=\frac{5}{2}^{-}$ state belonging to the $[SU(6)\otimes O(3)]=[70,1^{-}]$ supermultiplet configuration, in non-relativistic approximation prohibits the transition from the proton in a single quark transition. This suppression, known as the Moorhouse selection rule Moorhouse (1966), is valid for the transverse transition amplitudes $A_{1/2}$ and $A_{3/2}$ at all $Q^{2}$. It should be noted that this selection rule does apply to the transition from a proton target, it does not apply to the transition from the neutron, which is consistent with the data. Modern quark models that go beyond single quark transitions, confirm quantitatively the suppression resulting in very small amplitudes from protons but large ones from neutrons. Furthermore, a direct computation of the hadronic contribution to the transition from protons confirms this (Fig. 10). The measured helicity amplitudes off the protons are almost exclusively due to meson-baryon contributions as the dynamical coupled channel (DCC) calculation indicates (dashed line). The close correlation of the DCC calculation and the measured data for the case when quark contributions are nearly absent, supports the phenomenological description of the helicity amplitudes in terms of a 3-quark core that dominate at high $Q^{2}$ and meson-baryon contributions that can make important contributions at lower $Q^{2}$.

Knowledge of the helicity amplitudes in a large $Q^{2}$ allows for the determination of the transition charge densities on the light cone in transverse impact parameter space ($b_{x},b_{y}$) Tiator and Vanderhaeghen (2009). The relations between the helicity transition amplitudes and the Dirac and Pauli resonance transition form factors are given by:

	$\displaystyle A_{1/2}=e\frac{Q_{-}}{\sqrt{K}(4M_{N}M^{*})^{1/2}}\{F_{1}^{NN^{*}}+F_{2}^{NN^{*}}\}~{}~{}~{}~{}~{}$		(27)
	$\displaystyle S_{1/2}=e\frac{Q_{-}}{\sqrt{K}(4M_{N}M^{*})^{1/2}}\left(\frac{Q_{+}Q_{-}}{2M^{*}}\right)\frac{(M^{*}+M_{N})}{Q^{2}}$
	$\displaystyle\times\{F_{1}^{NN^{*}}-\frac{Q^{2}}{(M^{*}+M_{N})^{2}}F_{2}^{NN^{*}}\}\,,~{}~{}~{}~{}~{}$		(28)

Similar transverse charge transition densities can be defined for $J^{P}=\frac{3}{2}^{+}$ states such as the $\Delta(1232)\frac{3}{2}^{+}$. This has been studied in Carlson and Vanderhaeghen (2008) and is shown in Fig. 12.

A comparison of $N(1440)\frac{1}{2}^{+}$ and $N(1535)\frac{1}{2}^{-}$ is shown in Figure 11. There are clear differences in the charge transition densities between the two states. The Roper state has a softer positive core and a wider negative outer cloud than $N(1535)\frac{1}{2}^{-}$ and develops a larger shift in $b_{y}$ when the proton is polarized along the $b_{x}$ axis.

Many of the exited states for which there is information about the transition form factors available have been assigned as members of the $[SU(6),L^{P}]$ = $[70,1^{-}]$ super multiplet of the $[SU(6)\otimes O(3)]$ symmetry group. In a model, where only single quark transitions to the excited states are considered Hey and Weyers (1974); Cottingham and Dunbar (1979); Burkert et al. (2003), only 3 of the amplitudes need to be known to determine the remaining 16 transverse helicity amplitudes for all states in $[70,1^{-}]$ including on neutrons. However, the picture is now more complicated due to the strong admixture of meson-baryon components to the single quark transition especially in the lower $Q^{2}$ range. This requires a model to separate the single quark contributions from the hadronic part before projections for other states can be made Ramalho (2014).