About the magnitude of the $\gamma^{\ast}N\to N(1520)$ transverse
amplitudes near $Q^{2}=0$

In the last two decades there was a significant progress in the experimental study of the electromagnetic structure of the nucleon ($N$) and the nucleon resonances ($N^{\ast}$). The helicity amplitudes associated with the $\gamma^{\ast}N\to N^{\ast}$ transitions have been measured in detail for the $\Delta(1232)$, $N(1440)$, $N(1520)$, and $N(1535)$ resonances in a range from $Q^{2}=0.25$ GeV² up to 4 or 6 GeV² [1, 3, 2, 4, 5, 6]. The measured helicity amplitudes are: the transverse amplitudes $A_{1/2}$ and $A_{3/2}$ (for spin $J\geq 3/2$) and the longitudinal amplitude $S_{1/2}$. Near the photon point, however, there are still some uncertainties associated with the shape associated with the helicity amplitudes [1, 7, 8]. The selection from the Particle Data Group (PDG) at $Q^{2}=0$ has a large band of variation [9], and for most resonances there are no data below $Q^{2}=0.25$ GeV² [1, 10].

Among the best known experimental resonances the $N(1520)\frac{3}{2}^{-}$ (spin $J=\frac{3}{2}$ and negative parity, $P=-$) has properties that differ from the other low-lying nucleon excitations. The transverse amplitudes $A_{1/2}$ and $A_{3/2}$ have completely different magnitudes near the photon point [9], and the helicity amplitudes are related by two conditions near the pseudothreshold point, where $Q^{2}=-(M_{R}-M)^{2}$ [1, 11] ($M$ is the mass of the nucleon and $M_{R}$ is the mass of the nucleon resonance). Most transitions are constrained by only one condition [1, 7, 11]. Although these constraints are valid in a region not directly accessed by electron scattering on nucleons ($Q^{2}<0$), that may not be probed directly in physical experiments, the relations may have a significant impact on the shape of the helicity amplitudes at low $Q^{2}$, when the masses of the nucleon and the nucleon resonance are close [1, 7, 8, 12, 13, 14]. Numerically, the pseudothreshold occurs when $Q^{2}\simeq-0.38$ GeV².

In the present work, we study the magnitude of the $\gamma^{\ast}N\to N(1520)$ transverse amplitudes near $Q^{2}=0$, based on the analytic structure of the transition current and on the correlations between the amplitudes in the low-$Q^{2}$ region. We start by reviewing what we know about the transverse amplitudes in three kinematic regions.

Near the pseudothreshold, in addition to the condition associated with Siegert’s theorem [15, 8, 16, 17], one has the relation $A_{3/2}=\sqrt{3}A_{1/2}$ [1, 11, 7]. In the large-$Q^{2}$ region, theoretical calculations based on constituent quark-counting rules and perturbative QCD arguments indicate that there is a strong dominance of the $A_{1/2}$ amplitude over the $A_{3/2}$ amplitude ($|A_{1/2}|\gg|A_{3/2}|$) [1, 2, 5, 18]. Finally, near $Q^{2}=0$, one can quote the information from the PDG [9]

	$\displaystyle A_{3/2}=+(140\pm 5)\times 10^{-3}\;\mbox{GeV}^{-1/2},$
	$\displaystyle A_{1/2}=-(22.5\pm 7.5)\times 10^{-3}\;\mbox{GeV}^{-1/2}.$		(1)

From these results, we can conclude that there is a considerable suppression of $A_{1/2}$ relative to $A_{3/2}$ at the photon point.

We can summarize our knowledge of the ratio $A_{1/2}/A_{3/2}$, in the three regimes, as

$\displaystyle{\cal R}\equiv\frac{A_{1/2}}{A_{3/2}}=\left\{\begin{array}[]{ccc}\frac{1}{\sqrt{3}}&\mbox{if}&Q^{2}=-(M_{R}-M)^{2}\cr\vspace{.1cm}-\epsilon&\mbox{if}&Q^{2}=0\cr\vspace{.1cm}\infty&\mbox{if}&Q^{2}=+\infty\cr\end{array},\right.$

(5)

where $\epsilon$ represents a small positive value, $\epsilon\simeq 0.18\simeq\frac{1}{5}$, according to the experimental data (1).

At the pseudothreshold, the amplitudes have similar magnitudes (${\cal R}\simeq 0.6$). The suppression of $A_{3/2}$ at large-$Q^{2}$ is extensively discussed in the literature [1, 2, 18, 19]. The theoretical challenge is then to understand why is the ratio between the two amplitudes so small in absolute value near $Q^{2}=0$.

From the theoretical point of view, there is some debate about the nature of the $N(1520)$ resonance: if it is dominated by valence quark degrees of freedom, or alternatively, if it is dominated by baryon-meson molecular-like states [1, 19, 6, 20]. The magnitude of $A_{3/2}(0)$ is difficult to explain based solely on the quark core structure of the baryon states. Quark model calculations explain in general only about one-third or one-half of the measured value of the amplitude [18, 21, 22, 24, 23]. Those estimates are improved when explicit meson cloud dressing or quark-antiquarks excitations are taken into account in quark model calculations [24, 25]. Calculations based on dynamical coupled-channel reaction models, where the baryon resonances are described in terms of baryon-meson states [6, 26, 27], predict large contributions to the amplitude $A_{3/2}$ at low $Q^{2}$, on the order of 50% of the experimental values [20]. In the present work, we look for the origin of the difference of magnitudes between $A_{1/2}$ and $A_{3/2}$, based on the numerical contributions for each amplitude, without an explicit reference to the internal degrees of freedom.

The transverse amplitudes can be expressed in terms of the multipole form factors: the magnetic dipole ($G_{M}$) and the electric quadrupole ($G_{E}$) form factors, as defined by Devenish et al. [1, 2, 11],

$\displaystyle A_{1/2}=-\frac{1}{4F}T_{1},\hskip 25.6073ptA_{3/2}=-\frac{\sqrt{3}}{4F}T_{2},$

(6)

where

$\displaystyle T_{1}\equiv G_{E}-3G_{M},\hskip 25.6073ptT_{2}\equiv G_{E}+G_{M},$

(7)

and the factor $F$ takes the form

$\displaystyle F=\frac{1}{e}\frac{2M}{M_{R}-M}\sqrt{\frac{MM_{R}K}{(M_{R}+M)^{2}+Q^{2}}},$

(8)

with $K=\frac{M_{R}^{2}-M^{2}}{2M_{R}}$, $e=\sqrt{4\pi\alpha}$, and $\alpha\simeq 1/137$ is the hyperfine structure constant.

From the previous relations, we can conclude that $A_{1/2}\simeq 0$, near $Q^{2}=0$ is equivalent to the result $G_{E}\simeq 3G_{M}$. Notice, however, that this analysis only transfers the discussion from helicity amplitudes to the multipole form factors $G_{E}$ and $G_{M}$, and tells us nothing about the correlation between $G_{E}$ and $G_{M}$.

The results $A_{1/2}\simeq 0$ or $G_{E}\simeq 3G_{M}$ can be understood when we write the relations between the helicity amplitudes and the multipole form factors in terms the elementary form factors, defined by the gauge-invariant representation of the transition current for a $J^{P}=\frac{3}{2}^{-}$ nucleon resonance. The transition current can be expressed in terms of three independent gauge-invariant structures which define three independent forms factors that can be labeled as $G_{1}$, $G_{2}$, and $G_{3}$, and are free of kinematic singularities [1, 11]. For convenience, we call these functions elementary form factors.

Using the elementary form factors, we can re-write the transverse amplitudes (6) in the limit $Q^{2}=0$, as

	$\displaystyle A_{1/2}$	$\displaystyle=$	$\displaystyle-\frac{1}{4F_{0}}\,T_{1},$		(9)
	$\displaystyle A_{3/2}$	$\displaystyle=$	$\displaystyle-\frac{\sqrt{3}}{4F_{0}}\,\left[T_{1}-4\frac{M}{\sqrt{6}}\frac{M_{R}-M}{M_{R}}G_{1}\right],$		(10)

where $F_{0}={\cal B}\sqrt{M}$ and ${\cal B}=\frac{1}{e}\frac{M}{M_{R}}\sqrt{\frac{M_{R}}{K}}\simeq 3.67$ is dimensionless. The factor $T_{1}$, defined by Eqs. (7), takes the form

$\displaystyle T_{1}=-4\frac{M}{\sqrt{6}}\left[\frac{M}{M_{R}}G_{1}+\frac{1}{2}(M_{R}+M)G_{2}\right].$

(11)

From the relations (9)–(11), we can then conclude that in the limit $Q^{2}=0$, the transverse amplitudes depend only on the values of the functions $G_{1}$ and $G_{2}$. We can also conclude that $A_{1/2}\simeq 0$, when $T_{1}$ is negligible, and as a consequence $A_{3/2}\propto\frac{M_{R}-M}{M_{R}}G_{1}$ is large when $G_{1}$ is large. The numerical result for $A_{1/2}$ is then explained when $G_{1}$ and $G_{2}$ are large and have opposite signs. In this case, there is a significant cancellation between the terms in $G_{1}$ and in $G_{2}$. We will conclude, however, that $T_{1}\simeq 0$ ($\epsilon\simeq 0$) provide only a rough explanation of the data. The values of $G_{1}$ and $G_{2}$ at $Q^{2}=0$ have corrections of the order of 30% and 20%, respectively, when we use the experimental ratio $A_{1/2}/A_{3/2}$ ($\epsilon\simeq 0.2$), instead of $A_{1/2}=0$ ($\epsilon=0$).

At this point, one can ask if the values of $G_{1}(0)$ and $G_{2}(0)$ can help to explain the $Q^{2}$ dependence of $A_{1/2}$ and $A_{3/2}$ in the range $Q^{2}=0$…1 GeV². A simple numerical calculation demonstrates, however, that the shape of the amplitude $A_{3/2}$ cannot be explained without an estimate of the derivative of the elementary form factors $G_{i}$. We conclude at the end that the $A_{1/2}$ and $A_{3/2}$ data can be well described when we consider simple multipole parametrizations of the form factors $G_{i}$, where the scale of variation is determined by the scale of the nucleon dipole form factor, used in parametrizations of the nucleon electromagnetic form factors and some $\gamma^{\ast}N\to N^{\ast}$ transition form factors.

We propose parametrizations of the $A_{1/2}$ and $A_{3/2}$ amplitudes based on our analysis of the amplitudes at $Q^{2}=0$. The parametrizations are consistent with the $Q^{2}=0$…1 GeV² data, within the uncertainties of the available data, and may be tested by future experiments in facilities like MAMI or JLab-12 GeV in the low-$Q^{2}$ region [10]. The precision of the present estimates can be improved once the uncertainties of the $A_{1/2}(0)$ and $A_{3/2}(0)$ data are reduced.

This article is organized as follows: in the next section we present the general formalism for the $\gamma^{\ast}N\to N^{\ast}$ transition form factors and helicity amplitudes for $J^{P}=\frac{3}{2}^{-}$ nucleon resonances, and discuss the relevant limits (pseudothreshold, photon point and large $Q^{2}$). Our numerical analysis of the elementary form factors at the photon point is presented in Sec. III. In Sec. IV, we derive parametrizations of the data based on the our analysis and discuss the limits of the parametrizations. We finalize in Sec. IV with the outlook and conclusions.

We discuss now the formalism associated with the $\gamma^{\ast}N\to N(1520)$ transition, and the definition of helicity amplitudes and multipole form factors.

Considering an initial nucleon with the momentum $p$ and a final nucleon resonance with momentum $p^{\prime}$, we can define

$\displaystyle q=p^{\prime}-p,\hskip 17.07182ptP={\textstyle\frac{1}{2}}(p^{\prime}+p),$

(12)

as the transfer momentum and the average of the baryons momentum, respectively.

The transition current between a nucleon and an $N^{\ast}$ $J^{P}=\frac{3}{2}^{-}$ state can be written as

$\displaystyle J^{\mu}=\bar{u}_{\alpha}(p^{\prime})\Gamma^{\alpha\mu}(P,q)\gamma_{5}u(p),$

(13)

where $u_{\alpha}$, $u$ are the resonance, and the nucleon spinors, respectively, and $\Gamma^{\alpha\mu}$ takes the form [1, 2, 11, 19, 28]

	$\displaystyle\Gamma^{\alpha\mu}(P,q)$	$\displaystyle=$	$\displaystyle\left(q^{\alpha}\gamma^{\mu}-{\not\!q}g^{\alpha\mu}\right)G_{1}+$		(14)
			$\displaystyle\left[q^{\alpha}P^{\mu}-(P\cdot q)g^{\alpha\mu}\right]G_{2}+$
			$\displaystyle\left(q^{\alpha}q^{\mu}-q^{2}g^{\alpha\mu}\right)G_{3}.$

In the previous relation, $G_{i}$ ($i=1,2,3$) are independent functions, free of kinematic singularities, refereed to hereafter as elementary form factors. Comparatively with other authors that use the Devenish convention for the operators, and define the second term of Eq. (14) in terms of $p^{\prime}=P+{\textstyle\frac{1}{2}}q$ [1, 2, 11], we follow the Jones and Scadron convention [29] and use $P$ to define the operator associated with $G_{2}$ [29, 19, 28, 12]. The conversion is trivial¹¹1To obtain the Devenish form factors [11] in terms of the Jones and Scadron form factors [29] we replace $G_{1}\to G_{1}$, $G_{2}\to G_{2}$, and $G_{3}\to G_{3}+{\textstyle\frac{1}{2}}G_{2}$..

For the representation of the helicity amplitudes, defined at the resonance rest frame, it is convenient to introduce the magnitude of the transfer three-momentum $|{\bf q}|$. This variable can be written in a covariant form as

$\displaystyle|{\bf q}|=\frac{\sqrt{Q_{+}^{2}Q_{-}^{2}}}{2M_{R}},$

(15)

using the notation

$\displaystyle Q_{\pm}^{2}=(M_{R}\pm M)^{2}+Q^{2}.$

(16)

The magnetic dipole ($G_{M}$) and the electric quadrupole ($G_{E}$) form factors can be calculated inverting the relations (6) and (7)

	$\displaystyle G_{M}$	$\displaystyle=$	$\displaystyle-F\left[{\textstyle\frac{1}{\sqrt{3}}}A_{3/2}-A_{1/2}\right],$		(17)
	$\displaystyle G_{E}$	$\displaystyle=$	$\displaystyle-F\left[\sqrt{3}A_{3/2}+A_{1/2}\right].$		(18)

One can also relate the longitudinal (scalar) amplitude $S_{1/2}$ with the Coulomb quadrupole form factor $G_{C}$,

$\displaystyle S_{1/2}=-\frac{1}{\sqrt{2}F}\frac{|{\bf q}|}{2M_{R}}G_{C}.$

(19)

Using the expressions (13) and (14), we can write the magnetic dipole and the electric quadrupole form factors in terms of $G_{i}$, as [1, 2]

	$\displaystyle G_{M}$	$\displaystyle=$	$\displaystyle-Z_{R}\,Q_{-}^{2}\frac{G_{1}}{M_{R}},$		(20)
	$\displaystyle G_{E}$	$\displaystyle=$	$\displaystyle-Z_{R}\left\{[(3M_{R}+M)(M_{R}-M)-Q^{2}]\frac{G_{1}}{M_{R}}\right.$		(21)
			$\displaystyle\left.\frac{}{}+2(M_{R}^{2}-M^{2})G_{2}-4Q^{2}G_{3}\right\},$

where $Z_{R}=\frac{1}{\sqrt{6}}\frac{M}{M_{R}-M}$.

Using the previous equations, we conclude that

	$\displaystyle T_{1}$	$\displaystyle=$	$\displaystyle-Z_{R}\left\{4[M(M_{R}-M)-Q^{2}]\frac{G_{1}}{M_{R}}\right.$		(22)
			$\displaystyle\left.\frac{}{}+2(M_{R}^{2}-M^{2})G_{2}-4Q^{2}G_{3}\right\},$
	$\displaystyle T_{2}$	$\displaystyle=$	$\displaystyle-Z_{R}\left\{4(M_{R}-M)G_{1}\right.$		(23)
			$\displaystyle\left.+2(M_{R}^{2}-M^{2})G_{2}-4Q^{2}G_{3}\right\}.$

We can also write

	$\displaystyle T_{2}$	$\displaystyle=$	$\displaystyle T_{1}-4Z_{R}Q_{-}^{2}\frac{G_{1}}{M_{R}}$		(24)
		$\displaystyle=$	$\displaystyle T_{1}+4G_{M}.$

For future discussion, we write also the relation between the Coulomb quadrupole form factor and the elementary form factors,

	$\displaystyle G_{C}$	$\displaystyle=$	$\displaystyle Z_{R}\left[4M_{R}G_{1}+(3M_{R}^{2}+M^{2}+Q^{2})G_{2}\right.$		(25)
			$\displaystyle\left.+2(M_{R}^{2}-M^{2}-Q^{2})G_{3}\right].$

The previous relation can be used to calculate the amplitude $S_{1/2}$, according to Eq. (19). Notice that $S_{1/2}$ and $G_{C}$ cannot be measured at the photon point (because there are no real photons with zero polarization). The relation (25) can be used, however, to estimate $G_{C}$ and $G_{3}$ for values of $Q^{2}$ arbitrarily close to $Q^{2}=0$.

We discuss now briefly the three relevant limits: the pseudothreshold, the photon point and the large-$Q^{2}$ limit.

In the analysis of the transverse amplitudes near $Q^{2}=0$, we consider different approximations. For the discussion, we convert the experimental data (1), into the dimensionless variables

	$\displaystyle\tilde{A}_{3/2}=F_{0}A_{3/2}(0)=+0.498\pm 0.018,$		(38)
	$\displaystyle\tilde{A}_{1/2}=F_{0}A_{1/2}(0)=-0.080\pm 0.027,$		(39)

based on the numerical result $F_{0}=3.67\sqrt{M}$.

Notice that, since the comparison between amplitudes is made in units $10^{-3}$ GeV^-1/2, the first quantity ($\simeq 500\times 10^{-3}$) can be regarded as a large number, and the second quantity ($\simeq-80\times 10^{-3}$) can be regarded as a small number.

We can use the results (38) and (39), to calculate the corresponding form factors $G_{1}$ and $G_{2}$ for $Q^{2}=0$. Inverting the relations (9)–(11), one obtains

	$\displaystyle MG_{1}={\cal R}\left(\frac{1}{\sqrt{3}}\tilde{A}_{3/2}-\tilde{A}_{1/2}\right),$		(40)
	$\displaystyle M^{2}G_{2}=-\frac{2M}{M_{R}+M}\frac{M}{M_{R}}{\cal R}\left(\frac{1}{\sqrt{3}}\tilde{A}_{3/2}-\frac{M_{R}}{M}\tilde{A}_{1/2}\right),$
			(41)

where ${\cal R}=\frac{\sqrt{6}M_{R}}{M_{R}-M}$. The factors $M$ and $M^{2}$ are included to generate dimensionless expressions.

In the introduction, we discussed the approximation $\tilde{A}_{1/2}=0$ ($T_{1}=0$), based on Eqs. (9) and (10). In that case, we obtain $G_{2}=-\frac{2}{M_{R}+M}\frac{M}{M_{R}}G_{1}$. Now, we can notice, using the relations (40) and (41), that the condition $T_{1}=0$ provides only a rough approximation, since $\tilde{A}_{1/2}$ is combined in fact with $\tilde{A}_{3/2}/\sqrt{3}$. Neglecting $\tilde{A}_{1/2}$ in the estimates of $G_{1}$ and $G_{2}$ has an impact of 28% for $G_{1}$ and of 17% for $G_{2}$.

The relations (40) and (41) can also be used to explain the significant cancellation in $T_{1}$. The effect can be observed when we write $T_{1}$ on the form $T_{1}=-4\frac{M}{\sqrt{6}}t_{1}$, where

$\displaystyle t_{1}=\frac{M}{M_{R}}G_{1}+\frac{1}{2}(M_{R}+M)G_{2}.$

(42)

For that purpose, we write the two terms as

	$\displaystyle\frac{M}{M_{R}}G_{1}$	$\displaystyle=$	$\displaystyle\frac{{\cal R}}{M_{R}}\left(\frac{1}{\sqrt{3}}\tilde{A}_{3/2}-\tilde{A}_{1/2}\right),$
	$\displaystyle\frac{1}{2}(M_{R}+M)G_{2}$	$\displaystyle=$	$\displaystyle-\frac{{\cal R}}{M_{R}}\left(\frac{1}{\sqrt{3}}\tilde{A}_{3/2}-\tilde{A}_{1/2}\right)$
			$\displaystyle+\frac{{\cal R}}{M_{R}}\frac{M_{R}-M}{M}\tilde{A}_{1/2}.$

In this form, one concludes that the first term of $\frac{1}{2}(M_{R}+M)G_{2}$ cancels the term in $G_{1}$, and only the term proportional to $\tilde{A}_{1/2}$ survives the sum. The correction term is 13% of the term in $G_{1}$. In units $10^{-3}$ the term in $G_{1}$ and the term in $G_{2}$ are large numbers with opposite signs.

The dominance of the amplitude $A_{3/2}$ is still explained by the small magnitude of $T_{1}$. When we can neglect $T_{1}$ in Eq. (32), we conclude that $T_{2}\propto G_{1}$. Thus, the amplitude $A_{3/2}$ is large when $T_{2}$ is large, and $T_{1}$ is small in comparison with $T_{2}$. However, when we look for (31): $T_{2}=-4\frac{M}{\sqrt{6}}t_{2}$, with $t_{2}=G_{1}+\frac{1}{2}(M_{R}+M)G_{2}$, we conclude that $T_{2}$ is large because there is only a partial cancellation between the two large terms.

To summarize, the combination of the results for the transverse amplitudes is a consequence of the large magnitude of the form factors $G_{1}$ and $-G_{2}$. In $A_{1/2}$, one has a significant cancellation between the term in $G_{1}$ and the term in $G_{2}$. In $A_{3/2}$, the term in $G_{1}$ is enhanced and the suppression between the terms is attenuated. We conclude also that in first approximation (leading order in $\tilde{A}_{1/2}$), one has $A_{3/2}\propto G_{1}$.

The values of $G_{1}$ and $G_{2}$ for $Q^{2}=0$ are presented in Table 1 for the cases $T_{1}=0$ and $T_{1}\neq 0$. The first row ($T_{1}=0$) gives the results when we use $A_{1/2}(0)=0$, and the second row gives the result when $T_{1}\neq 0$ is fixed by the experimental value of $A_{1/2}(0)$. In the second row, we include also $G_{3}(0)=0$. The last four rows and the effect of $G_{3}(0)$ are discussed in the next sections.

The comparison between the first two rows demonstrates how important the inclusion of the experimental value of $A_{1/2}(0)$, is instead of $A_{1/2}(0)=0$, in the determination of the first two elementary form factors. The effect can also be seen in the results for $G_{M}(0)$ and $G_{E}(0)$. The differences are about 20% for the magnetic form factor and 10% for the electric form factor.

In the next sections, we use the estimated values for $G_{1}$ and $G_{2}$ at $Q^{2}=0$ to test if we can derive parametrizations that may explain the experimental data for the amplitudes $A_{1/2}$ and $A_{3/2}$, up to a certain range of $Q^{2}$. Due to the approximated character of the parametrizations, we restrict the analysis to the region $Q^{2}<1$ GeV². To estimate the uncertainties of the parametrizations, we calculate also the uncertainties of $G_{1}$ and $G_{2}$ at $Q^{2}=0$, based on the relations (40) and (41) and the data (1), with the errors combined in quadrature. The numerical values for the uncertainties of $G_{1}(0)$ and $G_{2}(0)$ are included in the last row of Table 1 (between brackets). The relative errors are 7.8% for $G_{1}$ and 10.6% for $G_{2}$.

In this section, we discuss possible parametrizations of the amplitudes $A_{1/2}$ and $A_{3/2}$ for $Q^{2}\leq 1$ GeV² based on the values of $G_{1}(0)$ and $G_{2}(0)$ calculated in the previous section.

In the following, we consider the $\gamma^{\ast}N\to N(1520)$ helicity amplitude data from experiments at JLab/CLAS on single-pion electroproduction [35] and on charged double-pion electroproduction [36, 20], and the PDG selection for $Q^{2}=0$ [9]. These JLab/CLAS experiments determine the whole set of helicity amplitudes ($A_{1/2}$, $A_{3/2}$, and $S_{1/2}$). The $\pi N$ ($\sim 60\%$) and the $\pi\pi N$ ($\sim 30\%$) channels are the dominant $N(1520)$ decay channels [9]. There are additional data associated with different experiments for the transverse amplitudes [37], but the data analysis is based on the assumption that $S_{1/2}\equiv 0$, an approximation that is not valid at low $Q^{2}$ [35].

In a first stage, we ignore the role of the form factor $G_{3}$, setting $G_{3}\equiv 0$, since no information about $G_{3}$ can be obtained from the transverse amplitudes at $Q^{2}=0$. We notice, however, that $G_{3}$ contributes to the amplitudes $A_{1/2}$ and $A_{3/2}$ for $Q^{2}\neq 0$, since $T_{1}$ and $T_{2}$ include the term $4Z_{R}Q^{2}G_{3}$ [see Eqs.(22) and (23)]. Later on, we estimate the impact of nonzero values for $G_{3}(0)$.

From the previous section, we concluded already that $A_{1/2}(0)\simeq 0$ is not a very good approximation. In the following, we consider then parametrizations based on Eqs. (40) and (41) consistent with the experimental value of $A_{1/2}(0)$. The numerical values are included in the lower part of Table 1 (with $T_{1}=0.319$).

We divided our analysis into several steps.