# Improved empirical parametrizations of the and transition amplitudes and the Siegert’s theorem

###### Abstract

In the nucleon electroexcitation reactions, , where is a nucleon resonance (), the electric amplitude , and the longitudinal amplitude , are related by , in the pseudo-threshold limit (), where and are respectively the energy and the magnitude of three-momentum of the photon. The previous relation is usually refereed to as the Siegert’s theorem. The form of the electric amplitude, defined in terms of the transverse amplitudes and , and the explicit coefficients of the relation, depend on the angular momentum and parity () of the resonance . The Siegert’s theorem is the consequence of the structure of the electromagnetic transition current, which induces constraints between the electromagnetic form factors in the pseudo-threshold limit. In the present work, we study the implications of the Siegert’s theorem for the and transitions. For the transition, in addition to the relation between electric amplitude and longitudinal amplitude, we also obtain a relation between the two transverse amplitudes: , at the pseudo-threshold. The constraints at the pseudo-threshold are tested for the MAID2007 parametrizations of the reactions under discussion. New parametrizations for the amplitudes , and , for the and transitions, valid for small and large , are proposed. The new parametrizations are consistent with both: the pseudo-threshold constraints (Siegert’s theorem) and the empirical data.

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^{†}preprint:

## I Introduction

The information relative to the structure of the electromagnetic transition between the nucleon and a nucleon excitation (), can be parametrized in terms of helicity amplitudes dependent on the photon polarization states, and the transfer momentum squared , which is restricted to the region NSTAR ; Aznauryan12 . For the transitions

Although those amplitudes are in principle independent functions, there are relations between them in the limit where the photon momentum vanishes. This limit is called the pseudo-threshold limit, corresponding to the case where both the nucleon and the resonance are at rest. At the pseudo-threshold limit (), one has , where is the photon energy, and , are respectively the resonance and the nucleon masses. Since the pseudo-threshold limit is defined by , with , it belongs to the unphysical region, where the helicity amplitudes cannot be measured from the transition. The extension of models for the region is important, however, for studies of reactions such as the Dalitz decay () and Dalitz decays of other resonances DeltaTL ; WhitePaper .

At the pseudo-threshold the matrix element of the electric multipole , defined by the spatial current density , and the matrix element of the Coulomb multipole , defined by the charge density , can be related by Buchmann98 . This result, obtained in the limit of the long wavelengths (), is usually refereed as the Siegert’s theorem Buchmann98 ; Atti78 ; AmaldiBook ; Drechsel92 ; Tiator16 . Although defined below , the relations between amplitudes can be used to test analytic properties of theoretical models and to test the consistency of phenomenological parametrizations.

The exact proportionality between the electric amplitude and the scalar amplitude depends on the angular momentum-parity state () of the resonance. The constraints for the helicity amplitudes can in general be derived from the analysis of the transition currents, expressed in a covariant form in terms of the properties of the nucleon and the resonance, which define a minimal number of independent structure form factors Bjorken66 ; Devenish76 ; Jones73 .

In Ref. newPaper , the implications of the pseudo-threshold limit for the transition form factors and helicity amplitudes, and their implications in the parametrizations of the data are discussed in detail. In the present work we discuss the consequences of the pseudo-threshold limit for the and transitions.

For the transition, we will conclude that, in the pseudo-threshold limit, one has , and is the mass. Note that the previous relation differs from the usual form Tiator2006 ; Drechsel2007 , by a factor . This difference has implications in shape of the parametrizations of the data, as we will show. [Along the paper, we will interpret the factors like or , as functions defined also for , with the result given by the limit , when the limit exists.] , where

As for the transition, the pseudo-threshold limit induces two constraints in the helicity amplitudes. The trivial constraint is expressed as is now defined in terms of the mass (). In addition, one has also the relation , at the pseudo-threshold. , where

The explicit form of the electric and the scalar amplitudes will be defined later for cases and . Defining , we can express the correlation between the electric and scalar amplitudes (Siegert’s theorem) as transition, and transition. for the for the

In order to take into account the constraints from the pseudo-threshold limit, in this work we present new parametrizations of the and helicity amplitudes. We will conclude at the end, that, an overall description of the data for low and large , including the pseudo-threshold, is possible using smooth representations of the helicity amplitudes. The presented parametrizations are compared with the MAID2007 parametrizations Tiator2006 ; Drechsel2007 ; MAID2009 . Parametrizations for very large , that simulate the expected falloff from perturbative QCD (pQCD), will be proposed.

This article is organized as follows: In Sec. II, we discuss the formalism associated with the electromagnetic transition current, helicity amplitudes and transition form factors. In Secs. III and IV, we study the and transitions, respectively. Parametrizations of the data appropriate for very large are discussed In Sec. V. Finally in Sec. VI, we present our summary and conclusions.

## Ii Generalities

We introduce now the formalism associated with the transition, where is the nucleon () and is a resonance. The case corresponds to the resonance; the case corresponds to the resonance. The variable represents the mass of the resonance under discussion [ or ].

We start with the discussion of the relation between the electromagnetic transition current and the helicity amplitudes. Next, we look for the properties of the amplitude . Before discussing in detail the transitions and , we present some useful notation.

### ii.1 Electromagnetic current and helicity amplitudes

In general, the transitions can be characterized in terms of transition form factors, to be defined later, or by the helicity amplitudes defined at the resonance rest frame. At the rest frame, the initial () and final () momenta can be represented, choosing the photon momentum, , along the -axis, as

(1) |

In the previous equations, is the magnitude of the photon (and nucleon) three-momentum, given by

(2) |

with . The nucleon energy and the photon energy can be expressed covariantly as and respectively.

The transverse () and the longitudinal () amplitudes, are defined at the rest frame Aznauryan12 ; N1520 , as

(3) | |||

(4) | |||

where () is the final (initial) spin projection, , () are the photon polarization vectors, and is the electromagnetic transition current operator in units of the elementary charge . In addition, is the fine-structure constant and .

The properties associated with the structure of the resonance are then encoded in the electromagnetic transition current operator . In the case of a transition between a spin state () and a spin state () we can project the current into the asymptotic states using

(6) | |||||

where and are respectively the Rarita-Schwinger and the Dirac spinors, , and is an operator dependent of the parity, to be defined in the following sections for the case of the (positive parity) and the (negative parity).

### ii.2 Scalar amplitude

In the case of the current conservation, one can replace , in the definition of the scalar amplitude (LABEL:eqS12), and write

(7) |

where the brackets represent the projection into the spin states with , defined at the resonance rest frame. If the current is not conserved, or the current operator is truncated, we cannot use Eq. (7), as discussed in Refs. Drechsel84 ; Buchmann98 .

Using Eq. (7), we can conclude that the scalar amplitude, near the pseudo-threshold, can be expressed for cases, as

(8) |

where is a form factor (dependent on the parity ), to be defined later, and is a parity-dependent operator, given by and .

Using the properties of Dirac and Rarita-Schwinger spinors, we can conclude that , near the pseudo-threshold NDelta ; NDeltaD . Applying those results, one obtains for resonance and for resonance. and

Note that, the result , with for positive parity, and , for negative parity, leads to , in the pseudo-threshold limit, if the form factor has no singularities in this limit, as expected Devenish76 . The result is equivalent to the orthogonality between the nucleon and the resonance states. The same property can be observed in the transition, where is a state newPaper .

It is interesting to note that, the dependence of a function , near the pseudo-threshold, can be inferred directly from the graph of the function in terms of . Since the derivative in , can be determined by the derivative in , given by

(9) |

we conclude that, at the pseudo-threshold (limit ), the derivative will be infinite (vertical line), when , and finite, only when with (we are interested only in the natural powers ). To summarize: the graphs with an infinite derivative at the pseudo-threshold are the representation of functions ; the graphs with finite derivative at the pseudo-threshold represent functions with .

### ii.3 Notation

In the following sections, we will study separately the resonances and . To convert helicity amplitudes into form factors, we use the factor Aznauryan12 ; N1520 ; Delta1600

(10) |

The factor will be used for the case , and the factor will be used for the case . For convenience we also define .

In the next two sections, we will also define the magnetic, electric and scalar amplitudes: and , where is an integer and is the parity, for the case . For a more detailed discussion about the multipole amplitude notation see Refs. Devenish76 ; Drechsel2007 .

## Iii transition

The transition current can be determined using Eq. (6), with the operator Aznauryan12 ; Devenish76 ; Jones73 ; NDelta

where () are structure form factors dependent on . The four form factors are not all independent, only three of them are independent. Using the current conservation condition, , we can conclude that NDelta ; NDeltaD

(12) |

Instead of the elementary form factors , alternatively we can use the multipole form factors: magnetic dipole (), electric quadrupole () and Coulomb quadrupole (), defined as Jones73 ; NDelta ; NDeltaD

(13) | |||||

(14) | |||||

(15) | |||||

where and

(16) |

is a new auxiliary form factor.

For convenience Eqs. (13)-(15) are expanded in powers of . For the sake of the discussion, we consider and , as our base for the form factors, following Jones and Scadron Jones73 , but we use also and , when necessary. For the multipole form factors we choose the Jones and Scadron representation. To convert to the alternative Ash representation, the functions and should be divided by the factor Drechsel2007 .

The helicity amplitudes (3)-(LABEL:eqS12) can be obtained from the form factors Aznauryan12 ; Drechsel2007 , using

(17) | |||

(18) | |||

(19) |

where is defined by Eq. (10).

The multipole amplitudes , can be defined directly in terms of the multipole form factors, or as a combination of the amplitudes Devenish76 ; Drechsel2007 ,

(20) | |||||

(21) | |||||

(22) |

The multipole amplitudes have the same dimensions as the helicity amplitudes. In this work we define the multipole amplitudes with the sign of the form factors. Other authors use different conventions of sign for the multipole amplitudes Devenish76 ; Drechsel2007 .

### iii.1 Pseudo-threshold limit

Now we consider the pseudo-threshold limit. Since the form factors (), are defined with no kinematic singularity, we can conclude from Eqs. (14)-(15) that Jones73

(23) |

when . A simple consequence of this result Jones73 , is

(24) |

To express the relation (24) in terms of helicity amplitudes, we use the relations (21)-(22) and

(25) |

In the previous relation we recall that and . Note in Eq. (25), that the common factor, , cannot be eliminated, unless we can prove that and , with , near the pseudo-threshold.

The relation (25) is consistent with

(26) |

near the pseudo-threshold. The previous forms were adopted by the MAID2007 parametrization MAID2009 . As for the amplitude , the MAID2007 parametrization gives , near the pseudo-threshold [which is equivalent to ]. The behavior of the multipole amplitudes near the pseudo-threshold is consistent with the results expected when the form factors are free of kinematic singularities at the pseudo-threshold Bjorken66 ; Devenish76 .

To satisfy the condition (25), it is necessary that both sides of the equation give the same numerical value. It is at that point that the MAID2007 parametrization fails, as we will show next.

To summarize: we conclude that the correlation between the form factors at the pseudo-threshold given by Eq. (24), usually refereed as the Siegert’s theorem, is not equivalent to the condition . The equivalent condition is the one expressed by Eq. (25).

The results of the MAID2007 parametrization for the form factors and , where , are presented in the top panel of Fig. 1, in comparison with the data from Ref. MokeevDatabase . The database from Ref. MokeevDatabase includes data for finite from Refs. Stave08 ; Data ; Aznauryan09 , and the world data average of at , extracted from the particle data group (PDG) result for at PDG .

From the top panel of Fig. 1, we can conclude, that, although the MAID2007 describes well the data for and , it fails to describe the relation (24). In the graph it is clear that . We discuss now alternative parametrizations of the form factors and , that are consistent with the Siegert’s theorem expressed in the form (24).

### iii.2 Improved parametrizations of and

A parametrization consistent with Eq. (24), inspired in the MAID2007 form is

(27) | |||

where is a dipole form factor, and and are adjustable. There are two main differences between the MAID2007 expressions and our expressions, apart from the constraint at pseudo-threshold discussed previously. The first difference is that we omitted the factor , in Eqs. (27)-(LABEL:eqGCsg). This factor appears in the MAID2007 parametrization because the form factors were defined originally in the Ash representation, and not in the Jones and Scadron representation (conversion factor) Jones73 . The second difference is the suppression of a factor , where is a new parameter, used in the MAID2007 parametrization of the function . This factor was added to the MAID2007 parametrization in an attempt to improve the quality of the fit near Tiator16 ; Drechsel2007 . We choose not to include that factor to avoid possible singularities in the timelike region, and also because the inclusion of higher powers in , as the terms associated with the coefficients and , may be sufficient to simulate the effect of an extra monopole factor in (in MAID2007: ). . The parameters

Apart from the two differences discussed above, the relevant difference between the present forms and the MAID2007 parametrization is that the coefficient is fixed by Eq. (24), once defined the remaining coefficients. We label the improved parametrization given by Eqs. (27)-(LABEL:eqGCsg), as the MAID-SG parametrization, since the new parametrization is consistent with the Siegert’s theorem (SG holds for Siegert).

The coefficients defined by the best fit of the functions (27)-(LABEL:eqGCsg) to the GeV data, are presented in the Table 1. The coefficients associated with the MAID2007 parametrization are also included in the table. Note, however, that only the coefficients can be directly compared, since different combinations of polynomials and exponentials may lead to similar functions.

The results for the MAID-SG parametrization for the form factors and are presented in the lower panel of Fig. 1. At this point we restrict the calculations to the region GeV, since the main goal at the moment is the study of parametrizations consistent with the Siegert’s theorem, near GeV. For larger there are discrepancies between the data from different groups Aznauryan09 ; NDeltaD , which are not relevant for the discussion near the pseudo-threshold. In Fig. 1, one can see that the MAID-SG parametrization is consistent with the data from Ref. MokeevDatabase and with the Siegert’s theorem. Note that, compared to the MAID2007 parametrization, the MAID-SG parametrization, gives smaller values for near the pseudo-threshold.

MAID-SG | |||||

MAID2007 | |||||

– | – | ||||

– | – |

In order to check in more detail the implications of Eq. (24), instead of looking for the form factors, we compare the MAID-SG parametrization with the data for and we use the MAID2007 parametrization, since it gives a very good description of the data and it is unconstrained at the pseudo-threshold. The results for the ratios and are presented in Fig. 2. In addition to the previous data, we present also the MAID data Drechsel2007 . In the figure, one note the different behavior between the MAID2007 parametrization and the MAID-SG parametrization. Part of this difference is a consequence of the discrepancy between the MAID data and the data used in our fit MokeevDatabase . We note, however, that, even the MAID2007 parametrization has problems in describing the MAID data for below 1 GeV. . To calculate

From Figs. 1 and 2, we can conclude, that, the MAID-SG parametrization gives a very good description of the low data ( GeV) for and . Both functions are smooth near the pseudo-threshold. Looking in more detail in Fig. 2, we see that the MAID-SG parametrization starts to fail when GeV for . As for , one can see that the function starts to decrease in magnitude for GeV. One can also note, that the data for is well approximated by a constant (similar to MAID2007). The failure of the parametrization for larger values of , is in part a consequence of the inclusion of exponential functions in the parametrization of the form factors.

The disadvantage of the use of parametrizations based on exponential factors is that those parametrizations are not valid, in general, for large interval in , or fail when increases. Latter on, in Sec. V, we discuss the possibility of extending the parametrization of the data for large .

We checked, however, if we can improve the description of the large region ( GeV), by enlarging the range of the data used in the fit to GeV, or GeV. Overall, we can improve the description of the data for GeV, but the description for the low region losses quality. In particular the result for is overestimated ( is underestimated), compared to the PDG result [ and ] PDG . Note that, when we extend the range of the fit for larger values of , up to 3 GeV or 4.1 GeV, we reduce the impact of the low region in our fit, which leads to a poor estimate of the form factors near the pseudo-threshold. We may then conclude, that, with the parametrizations (27)-(LABEL:eqGCsg), we can not describe well the low and the large regions simultaneously. For this reason we restrict, for now, our analysis to the low region.

It is worth to mention, that, the fit based on Eqs. (27)-(LABEL:eqGCsg), is very sensitive to the low data, in particular to the result at the photon point from PDG PDG . If the datapoint from PDG is replaced by another datapoint, or the errorbar is reduced in the fit, the results for the form factors may change significantly. We note in particular that, in some experiments like in Ref. LEGS01 , the value of is larger in absolute value, .

The Siegert’s theorem was already investigated in the context of the transition, within the quark model framework Buchmann98 ; Drechsel84 ; Weyrauch86 ; Bourdeau87 ; Capstick90 . It was found that the Siegert’s theorem can be violated when the operators associated with the current density or the charge density are truncated, or expanded in different orders, inducing a violation of the current conservation condition Buchmann98 ; Drechsel84 ; Weyrauch86 . From those studies, one can conclude that a consistent calculation, where the current is conserved, requires the inclusion of processes beyond the impulse approximation at the quark level (one-body currents), and that, the inclusion of higher order processes involving two-body currents, such as processes with quark-antiquark states and/or meson cloud contributions, is necessary to ensure the conservation of the transition current and the Siegert’s theorem Buchmann98 . Since the Siegert’s theorem is defined at the pseudo-threshold, when GeV, one may then conclude, that, processes beyond the impulse approximation are fundamental to describe the helicity amplitudes and the transition form factors at low .

The last conclusion is particularly important for the transition, since there are strong evidences of importance of the meson cloud effects for all form factors at small NSTAR ; Aznauryan12 ; NDelta ; NDeltaD ; LatticeD ; Delta1600 . For the magnetic dipole form factor, , it is known that the meson cloud effects are small for GeV NSTAR ; Aznauryan12 ; NDelta ; NDeltaD . As for the quadrupole form factors and , there are indications that the meson cloud contributions may be important up to 4 GeV NDeltaD ; Buchmann04 . The meson cloud contributions for the quadrupole form factors will be discussed in detail in Sec. V.3.

It is important to mention that, the distinction between the valence quark degrees of freedom and the non-valence quark degrees of freedom is only well defined in a given framework. Therefore, valence quark contributions in a given model may appear as non-valence quark contributions in another model. One can nevertheless conclude that, independent of the model, the meson cloud contributions help in general to approach the estimates from quark models to the experimental data.

## Iv transition

The current associated with the transition is determined using Eq. (6), with the operator Aznauryan12 ; Devenish76 ; N1520

where () are structure form factors dependent on . As in the case of the , the four form factors are not all independent. In this case the current conservation implies that N1520

(30) |

The multipole form factors can be represented Aznauryan12 ; N1520 , as

(31) | |||||

(32) | |||||

(33) | |||||

where and

(34) |

Once again, the form factors are decomposed in powers of .

The helicity amplitudes can be determined Aznauryan12 ; N1520 by

(35) | |||

(36) | |||

(37) |

Similar to the case we also can define the amplitudes Devenish76

(38) | |||||

(39) | |||||

(40) |

### iv.1 Pseudo-threshold limit

From Eqs. (31)-(33), we can conclude that, at the pseudo-threshold Devenish76 ; Note1

(41) | |||

(42) |

In particular, Eq. (42) is a consequence of the result

(43) |

when , since the form factors () have no kinematic singularities at the pseudo-threshold Devenish76 .

A consequence of Eq. (42) is, that, at the pseudo-threshold

(44) |

where, as before . As for the result from Eq. (41), it implies that

(45) |

at the pseudo-threshold [see Eq (38)].

The relations (44) and (45) are consistent with the following behavior of the functions near the pseudo-threshold Devenish76

(46) |

The relation for can be derived directly from Eq. (31) when . The dependences of and are consistent with the expected result for the transverse amplitudes Bjorken66 .

Since the available data for the transition at finite is restricted to the reactions with proton target we will restrict our analysis to that case.

The results of the MAID2007 parametrization for the amplitudes and are presented in Fig. 3. At the top, we test the relation (44), multiplied by the factor . If the relation is satisfied, the solid line () and the long-dashed line () should converge to zero, both, at the pseudo-threshold, when GeV. In the same graph we also represent , that should also vanish at the pseudo-threshold, according to Eq. (41). At the bottom, we compare with , in order to test how broken is the relation (45), as a consequence of the violation of the condition , at the pseudo-threshold.

The results from Fig. 3 show that the relations (44) and (45), which are consequence of the pseudo-threshold limit (Siegert’s theorem), are broken by the MAID2007 parametrization. The failure of the MAID2007 parametrization is then the consequence of the dependences of , near the pseudo-threshold. Those dependences are in conflict with the expected dependence of the multipole amplitudes, expressed in Eqs. (46). Now we consider alternative parametrizations that are consistent with Eqs. (41) and (42) [or alternatively by Eqs. (44) and (45)].

### iv.2 Improved parametrizations of and

To obtain parametrizations of the and data, that include the constraints at the pseudo-threshold, we consider the following representations of the helicity amplitudes

(47) | |||

(48) | |||

(49) |

where , which can also be written in the form . In the expressions (47)-(49), the coefficients , and are adjustable parameters and are determined from the constraints at the pseudo-threshold, once the values of the remaining coefficients are fixed. Since, at the pseudo-threshold, we can write , using Eq. (45), we can conclude, that both and can be expressed in terms of [because and ].

Jlab-SG |
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