Improved empirical parametrizations of the $\gamma^\ast N \to N(1535)$ transition amplitudes and the Siegert's theorem

Some empirical parametrizations of the $\gamma^\ast N \to N(1535)$ transition amplitudes violates the Siegert's theorem, that relates the longitudinal and the transverse amplitudes, in the pseudo-threshold limit (nucleon and resonance at rest). In the case of the electromagnetic transition from the nucleon (mass $M$) to the resonance $N(1525)$ (mass $M_R$), the Siegert's theorem is sometimes expressed by the relation $|{\bf q}| A_{1/2}= \lambda S_{1/2}$ in the pseudo-threshold limit, when the photon momentum $|{\bf q}|$ vanishes, and $\lambda = \sqrt{2} (M_R -M)$. In this article, we argue that the Siegert's theorem should be expressed by the relation $A_{1/2} = \lambda \frac{S_{1/2}}{ |{\bf q}|}$, in the limit $|{\bf q}| \to 0$. This result is a consequence of the relation $S_{1/2} \propto |{\bf q}|$, when $|{\bf q}| \to 0$, as suggested by the analysis of the transition form factors and by the orthogonality between the nucleon and $N(1535)$ states. We propose then new empirical parametrizations for the $\gamma^\ast N \to N(1535)$ helicity amplitudes, that are consistent with the data and the Siegert's theorem. The proposed parametrization follow closely the MAID2007 parametrization, except for a small deviation in the amplitudes $A_{1/2}$ and $S_{1/2}$ when $Q^2<1.5$ GeV$^2$.


Introduction
The information relative to the structure of the electromagnetic transitions between the nucleon and the nucleon excitations (γ * N → N * ) has been parametrized using different forms [1,2]. The representations in terms of helicity amplitudes, longitudinal and transverse, can be defined independently of the proprieties of the resonances. Alternatively, one can use a representation in terms of structure form factors, that emphasize precisely the symmetries associated with the nucleon resonances. The helicity amplitudes and the structure form factors are functions of the transition fourmomentum transfer (q) squared, q 2 , but are often represented in terms of Q 2 = −q 2 , particularly in nucleon electroexcitation reactions (Q 2 > 0). In general the different helicity amplitudes are independent functions, except in some specific limits. The same holds for the form factors.
Taking the case of the nucleon as example: the electric and the magnetic form factors, G E and G M , are independent functions, except in the threshold limit, Q 2 = −4M 2 , where G E = G M (threshold of the γ * → NN reaction). In the case of the γ * N → N * transitions, there are constraints between helicity amplitudes, or between form factors, at the pseudo-threshold limit. The pseudothreshold limit is the limit where the photon momentum |q| vanishes, and both particles, the nucleon (N) and the resonance, labeled here in general as R, are at rest. In the pseudo-threshold Q 2 = Q 2 PS = −(M R − M) 2 [3,4]. The condition that expresses the relation between different amplitudes (or form factors) at the pseudothreshold is usually referred as the Siegert's theorem. The Siegert's theorem was introduced first in studies related with nuclear physics [3,5] and was later used in pion electroproduction reactions [6,7,8,9].
In this work, we study in particular the constraints of the Siegert's theorem in the γ * N → N(1535) transition, where N(1535) is a spin 1 2 state with negative parity (J P = 1 2 − ). We will show in particular that some parametrizations of the γ * N → N(1535) transition amplitudes, like the MAID2007 parametrization [8,9,10], are not consistent with the Siegert's theorem. In order to grant that the Siegert's theorem is valid, one needs to ensure that S 1/2 ∝ |q|, near |q| = 0. In the present article, we propose then new parametrizations for the amplitudes A 1/2 and S 1/2 , that are consistent with both, the empirical data and the Siegert's theorem.
The consequences of the Siegert's theorem for the γ * N → ∆(1232) and γ * N → N(1520) helicity amplitudes are discussed in a separate article [11].

Siegert's theorem
The parametrization of the current associated with a transition between the nucleon (state J P = 1 2 + ) and a J P = 1 2 − resonance can be represented in terms of two form factors, h 1 and h 3 according with Ref. [4]. At the pseudo-threshold those form factors are related by the condition [4] The functions h 1 , h 3 can be related with the helicity amplitudes by h 1 = − √ 2S 1/2 /(|q|b) and h 3 = and e is the elementary electric charge. The helicity amplitudes A 1/2 (transverse) and S 1/2 (longitudinal) will be defined precisely later [see Eqs. (11)- (12)].
A direct consequence of the Eq. (1) is where we define Note, that, we chose to include the ratio S 1/2 /|q| in the previous relation. In the case |q| = 0, the factor S 1/2 /|q| is interpreted as the limit |q| → 0. This point is important, since it is assumed that A 1/2 and S 1/2 /|q| have the same order in |q|, for small values of |q|. The consequence of this observation is that if A 1/2 = O(1), meaning that A 1/2 converges to a constant in the pseudothreshold limit, one can write also S 1/2 = O(|q|), near |q| = 0. In this article, we will assume then, that, the amplitudes A 1/2 and S 1/2 behave, near the pseudo-threshold, as The structure given by Eqs. (4), near the pseudothreshold can be derived from the analysis of the multipole transition amplitudes [3,6,5,7,8].
In order to understand the meaning of the second relation in (4), we look for the charge density operator, J 0 (zero component of the transition current), in the pseudo-threshold limit. The charge operator can be defined in terms of the Dirac (F 1 ) and Pauli (F 2 ) form factors [see Eq. (10)]. When J 0 is projected into the spin states, which we represent by J 0 , at the resonance rest frame, one obtains where u R (u) is the Dirac spinor of the resonance (nucleon) and with In the case where the initial and final state have the same spin projection, we can conclude, that, in the pseudothreshold limit at the R rest frame: (ū R γ 5 u) ∝ |q|. Thus The previous condition defines the orthogonality between the nucleon and the resonance states when J 0 → 0, which implies thatF 1 = O(1), (F 1 → constant) or thatF 1 scales with some power of |q|, in the pseudo-threshold limit. The orthogonality between states at the pseudo-threshold generalizes the nonrelativistic definition of orthogonality between states with different masses when the recoil (and the mass difference) is neglected (Q 2 = −q 2 = 0).
Since the amplitude S 1/2 can also be defined by J 0 , assuming current conservation 1 , in the cases where the spin projections are conserved (photon with zero spin projection), we can also write J 0 ∝ S 1/2 . Combined this result with the result (8), we conclude, that the orthogonality between the states, defined at the pseudothreshold, implies In the following, we will also show that the first condition in (4), Therefore, the combination of the result (9) and A 1/2 ∝ F 1 , is compatible with the Siegert's theorem (2), apart from normalization factors. To prove the relation (2), we need to look for the explicit parametrization of the amplitudes A 1/2 and S 1/2 .
We introduce next the formalism associated with the electromagnetic transition current, the electromagnetic form factors and the helicity amplitudes in the γ * N → N(1535) transition. Later, we discuss the implications of the Siegert's theorem in the structure of the transition form factors.

γ * N → N(1535) transition
The γ * N → N(1535) transition can be represented, omitting the asymptotic states, in the units of the elementary electric charge e, as [2,12,13]  where F 1 and F 2 are respectively the Dirac and Pauli form factors, as mentioned before. Given the structure of Eq. (10), we can ensure, that, both components of the current, the Dirac and the Pauli terms, are conserved separately.

Helicity amplitudes (at the R rest frame)
Since the transition γ * N → N(1535) correspond to a transition between two states with spin 1 2 (transition 1 2 there are only two helicity amplitudes to be considered, the transverse (A 1/2 ) and the longitudinal (S 1/2 ) amplitudes. Those amplitudes are defined, at the resonance rest frame, as follows [2]: where Q = Q 2 (assuming that Q 2 > 0), as before |q| is the photon (and nucleon) momentum, and ε λ (λ = 0, +) is the photon polarization vector. The momentum |q| is determined by where Based on the current (10), we can write the amplitudes [2,12,13,14], as , as before, andF 1 is defined by Eq. (6). The factor e appears because the current J µ is defined in units of the elementary electric charge.
In Eq. (15), we decompose the amplitude S 1/2 into aF 1 term and a term in |q| 2 , in order to facilitate the following discussion.
Based on Eqs. (14)- (15), we can conclude that if the term |q| 2 F 2 can be dropped in comparison withF 1 , we obtain immediately the Siegert's theorem condition, since in the pseudo-threshold limit, We look now for the results of the MAID2007 parametrization. The results for the amplitude S 1/2 and A 1/2 |q|/λ are presented in the Fig. 1. One can note in the figure, that |q|A 1/2 λS 1/2 , since the functions differ at the pseudo-threshold, Q 2 = Q 2 PS ≃ −0.36 GeV 2 , when we start to draw the lines.

Form factors
We turn now for the analysis of the transition form factors. The transition form factors F 1 and F 2 can be determined inverting Eqs. (14)- (15). The results are For the convenience of the discussion we multiply F 2 by η, given by Eq. (7). From Eqs. (17)-(18), we can conclude, that, in the sumF 1 = F 1 + ηF 2 , all terms cancel, except for the term A 1/2 /(2b), as expected from Eq. (14). From the equations, we can also conclude that if the factor R = A 1/2 − λS 1/2 /|q| does not vanish (R 0), or it does not vanish fast enough with |q| when |q| → 0, then the form factors F 1 and ηF 2 diverge in the limit |q| → 0.
We can show however, that, if we represent any of the functions A 1/2 and S 1/2 /|q|, by a non-singular function F of Q 2 , we can write R = O(|q| n ) with n ≥ 2, since in the expansion of a function F(Q 2 ) in powers of |q|, near |q| = 0, the first term vanishes. This result is the consequence of the relation dF d|q| =

Modified MAID parametrization
We consider now parametrizations of the γ * N → N(1535) helicity amplitudes, that differs from the MAID2007 parametrization.
Since the proposed parametrization is based in the form of the MAID2007 parametrization, but is also compatible with the Siegert's theorem, we label it as MAID-SG parametrization (SG holds for Siegert).
In the MAID-SG parametrization one uses where the a 0 , a 1 , a 4 , s 1 , s 2 and s 4 are adjustable parameters and s ′ 0 will be fixed by the Siegert's theorem condition (2). Comparatively to the original MAID2007 parametrization [8,10], we replaced s 0 → (2M R |q|)s ′ 0 /Q 2 + and add an extra term in Q 4 for S 1/2 . The extra term (s 2 Q 4 ) is important in order to obtain a parametrization based on small coefficients (between 10 −3 and 10 3 ), in the spirit of the previous MAID parametrizations. The factor (2M R |q|)/Q 2 + is included to give the correct behavior (proportional to |q|) near |q| = 0, and preserve the high Q 2 behavior of the parametrization, since 2M R |q|/Q 2 Note that, using Eqs. (19) and (20), one has A 1/2 = O(1) and S 1/2 = O(|q|), when |q| → 0. However, to ensure the Siegert's theorem, we still need to constrain the value of s ′ 0 by Eq. (2). We fit all the coefficients to the MAID data [10]. Since the MAID analysis gives negligible error bars for the amplitude S 1/2 when Q 2 > 1.5 GeV 2 , for the propose of the fit we use an error of 0.01×10 −3 GeV −1/2 . The coefficients determined by the best fit are presented in Table 1.
Although we could impose the Siegert's theorem refitting only the amplitude S 1/2 , for a question of consistence one chose to fit both amplitudes simultaneously. The coefficients associated with the new fit based on Eqs. (19)-(20) are presented in Table 1, in comparison with the MAID2007 parametrization, which violates the Siegert's theorem. To facilitate the comparison with MAID2007, we replace s 0 by (2M R |q|)s ′ 0 /Q 2 + . The results for the amplitudes A 1/2 and S 1/2 in the MAID-SG parametrization are presented in Fig. 2 (solid line), and are compared with the result from MAID2007 (dashed line). It is interesting to see that the two parametrizations are almost undistinguished for Q 2 > 1.5 GeV 2 . From the figure, we conclude, that, the constraints of the Siegert's theorem, can by included in the parametrization of the γ * N → N(1535) helicity amplitudes, without a significant loss of accuracy.
The results for the amplitudes are consistent with the Siegert's theorem expressed in the form of Eq. (16), combined withF 1 = O(1). Using the new parametrization for the amplitudes A 1/2 and S 1/2 , it is possible now to look the form factors (18). The results for the form factors are presented in the Fig. 3. In the figure, it is clear, that F 1 and F 2 are finite at the pseudo-threshold, as one expects from the dependence R = O(|q| 2 ), discussed previously. We can calculate the explicit dependence of R near the pseudo-threshold, using the functions A, S defined by A ≡ A 1/2 and S 1/2 ≡ (2M R |q|)s ′ 0 /Q 2 + S . One obtains then neglecting terms in O(|q| 4 ). In Eq.(21), A, S and A ′ , S ′ represent respectively the functions and the derivatives in the limit Q 2 = Q 2 PS . In Fig. 3, one can also see, that the functionF 1 is dominated by the form factors F 1 , for larger values of Q 2 . It is also possible to observe that the form factor F 2 has large values for Q 2 < 0.5 GeV 2 , but decreases significantly for larger values of Q 2 , and it is negligible for Q 2 > 1.5 GeV 2 . A consequence of the result F 2 ≃ 0, is that the amplitudes A 1/2 and S 1/2 are correlated by the relation (M R +M) 2 , for Q 2 > 1.5 GeV 2 [13]. As discussed in Refs. [13,15] the result F 2 ≃ 0, suggests that there is a cancellation between the valence quark contributions and the meson cloud contributions.

Implication of the Siegert's theorem in other resonances
The constraints of the Siegert's theorem have implications also in the helicity amplitudes associated with other γ * N → N * transitions. In particular, the parametrization proposed here, can be used in the study of the γ * N → N(1650) transition, since it is also a 1 2 In the case of the γ * N → N(1520) transition the Siegert's theorem implies that 1 2 , is the electric amplitude in the transition, and λ R = √ 2(M R − M) (M R is the resonance mass) [11]. One can see then, that apart the factor 1/2 at the l.h.s., and the replacement A 1/2 → E, the condition is the same as for the γ * N → N(1535) transition.

MAID-SG
at pseudo-threshold (the r.h.s. and the l.h.s. vanish both), this is not sufficient to ensure that G E = κ G C . In the Fig. 4, we compare at the top the form factors G E and κ G C , given by the MAID2007 parametrization. It is clear in the graph, that, the Siegert's theorem is violated. At the bottom, we consider an improved parametrization where the Siegert's theorem is imposed and fitted to the G E and G C data (defining a new MAID-SG parametrization). In this case, one can see the convergence of G E to κ G C at the pseudo-threshold. The γ * N → ∆(1232) transition form factors and their relation with the Siegert's theorem are discussed in detail in Ref. [11].

Summary and conclusions
In the present article we discuss the implications of the constraints in the γ * N → N(1535) helicity amplitudes, when the nucleon and the resonance N(1535) are both at rest (pseudo-threshold limit). In this limit the transverse (A 1/2 ) and the longitudinal (S 1/2 ) amplitudes are related by the Siegert's theorem (2). We concluded, that the Siegert's theorem is the consequence of the orthogonality between the nucleon and resonance states.
From the analysis of the structure of the current and the transition form factors, we conclude also, that, the amplitudes A 1/2 and S 1/2 /|q| are both finite and nonzero in the pseudo-threshold limit [recall Eq. (16) with F 1 = O(1)]. Based on this result, we explain why the MAID2007 parametrization for the amplitudes A 1/2 and S 1/2 violates the Siegert's theorem, and propose an alternative parametrization, consistent with both the Siegert's theorem and the data. The new parametrization is similar to the MAID2007 parametrization for both amplitudes when Q 2 > 1.5 GeV 2 , but deviates from MAID2007 for smaller values of Q 2 . In the new parametrization, the amplitude S 1/2 differs more significantly from the MAID2007 parametrization for Q 2 < 0, and vanishes at the pseudo-threshold as expected (S 1/2 ∝ |q|).
We concluded also, that, the Dirac and Pauli form factors are free of singularities at the pseudo-threshold as expected from the Siegert's theorem, expressed under the condition A 1/2 − λS 1/2 /|q| = O(|q| 2 ), near the pseudo-threshold.
The methods proposed in this article to study the structure of the helicity amplitudes and the structure of the transition form factors in the γ * N → N(1535) transition, can be extended for the transitions γ * N → ∆(1232), γ * N → N(1520) [11] and others.