Survey of nucleon electromagnetic form factors

# Survey of nucleon electromagnetic form factors

I. C. Cloët, G. Eichmann, B. El-Bennich, T. Klähn and C. D. Roberts
###### Abstract

A dressed-quark core contribution to nucleon electromagnetic form factors is calculated. It is defined by the solution of a Poincaré covariant Faddeev equation in which dressed-quarks provide the elementary degree of freedom and correlations between them are expressed via diquarks. The nucleon-photon vertex involves a single parameter; i.e., a diquark charge radius. It is argued to be commensurate with the pion’s charge radius. A comprehensive analysis and explanation of the form factors is built upon this foundation. A particular feature of the study is a separation of form factor contributions into those from different diagram types and correlation sectors, and subsequently a flavour separation for each of these. Amongst the extensive body of results that one could highlight are: , owing to the presence of axial-vector quark-quark correlations; and for both the neutron and proton the ratio of Sachs electric and magnetic form factors possesses a zero.

## 1 Introduction

Owing in part to the relatively simple nature of the virtual photon as a probe, a reliable explanation of electromagnetic form factors provides information on the distribution of a nucleon’s characterising properties; e.g., total- and angular-momentum, amongst its QCD constituents. Since contemporary experiments employ ; i.e., momentum transfers in excess of the nucleon’s mass, a veracious understanding of the body of extant data requires a Poincaré covariant description of the nucleon. Poincaré covariance and the vector exchange nature of QCD guarantee the existence of nonzero quark orbital angular momentum in a hadron’s rest-frame bound-state amplitude [1, 2].

In fact the challenge is compounded owing, e.g., to the running of the dressed-quark mass [3, 4, 5, 6, 7, 8]. This entails that a quantum field theoretic treatment of hadron structure and electromagnetic interactions is generally necessary in order to provide understanding in terms of QCD’s genuine elementary degrees of freedom. The dressed light-quark mass function at infrared momenta is roughly -times larger than the current-quark mass. This marked enhancement is a corollary of dynamical chiral symmetry breaking (DCSB) and owes primarily to a dense cloud of gluons that clothes a low-momentum quark [9]. (The dressing gluons also acquire mass dynamically [10].) It means that the Higgs mechanism is largely irrelevant to the bulk of normal matter in the universe. Instead the single most important mass generating mechanism for light-quark hadrons is the strong interaction effect of DCSB; e.g., one can identify DCSB as being responsible for 98% of a proton’s mass. It has long been argued that form factors are a sensitive probe of this effect [11].

Recent years have seen rapid experimental and theoretical progress in the study of nucleon electromagnetic form factors, which is reviewed, e.g., in Refs. [12, 13]. Despite this, questions remain unanswered, amongst them: can one formulate an impulse-like approximation for hadron form factors and, if so, in terms of which degrees of freedom; is there a valid mapping of form factors into statements about the distribution of charge and magnetisation within a nucleon; and what role is played by pseudoscalar mesons in hadron electromagnetic structure and can one describe this in a quantitative, model-independent fashion? Herein we contribute to the discussion of these issues.

In Sect. 2 we recapitulate briefly upon a Poincaré covariant Faddeev equation for the nucleon, in which the primary element is the dressed-quark with its strongly momentum dependent mass function. The Faddeev equation solution defines a nucleon’s dressed-quark core. The study of baryons in this way sits squarely within the ambit of the application of Dyson-Schwinger equations (DSEs) in QCD [14]. Since the DSEs admit a nonperturbative symmetry-preserving truncation scheme [15, 16, 17, 18], which e.g. has enabled the proof of numerous exact results for pseudoscalar mesons [19, 20, 21, 22], the approach holds particular promise as a means of unifying the treatment of meson and baryon observables that preserves all global and local corollaries of DCSB without fine-tuning [23]. The coupling of a photon to the nucleon’s dressed-quark core is detailed in Sect. 3.

In Sect. 4 we discuss the interpretation of form factors and present a perspective on the circumstances under which the three dimensional Fourier transform of a Breit-frame Sachs form factor can reasonably be understood in terms of a charge or magnetisation density.

Sections 57 are extensive. They detail our computed results and the understanding they provide. All electromagnetic form factors of the proton and neutron are described along with their decomposition into individual flavour, diagram and diquark contributions, the meaning of which will subsequently become apparent.

We consider form factor contributions arising from pseudoscalar meson loops in Sect. 8 and exemplify the manner in which they add to the dressed-quark core results. We wrap-up in Sect 9.

## 2 Nucleon Model

In quantum field theory a nucleon appears as a pole in a six-point quark Green function. The pole’s residue is proportional to the nucleon’s Faddeev amplitude, which is obtained from a Poincaré covariant Faddeev equation that adds-up all possible quantum field theoretical exchanges and interactions that can take place between three dressed-quarks. Canonical normalisation of the Faddeev amplitude guarantees unit residue for the -channel nucleon pole in the three-quark vacuum polarisation diagram and entails unit charge for the proton.

A tractable truncation of the Faddeev equation is based [24] on the observation that an interaction which describes mesons also generates diquark correlations in the colour- channel [25]. The dominant correlations for ground state octet and decuplet baryons are scalar () and axial-vector () diquarks because, for example, the associated mass-scales are smaller than the baryons’ masses [26, 27], namely (in GeV)

 m[ud]0+=0.7−0.8,m(uu)1+=m(ud)1+=m(dd)1+=0.9−1.0. (1)

The kernel of the Faddeev equation is completed by specifying that the quarks are dressed, with two of the three dressed-quarks correlated always as a colour- diquark. As illustrated in Fig 1, binding is then effected by the iterated exchange of roles between the bystander and diquark-participant quarks.

The Faddeev equation that we employ is explained in Appendix A: Faddeev Equation. With all its elements specified, as described therein, the equation can be solved to obtain the nucleon’s mass and amplitude. Owing to Eq. (A.34), in this calculation the masses of the scalar and axial-vector diquarks are the only variable parameters. The axial-vector mass is chosen so as to obtain a desired mass for the ,111This is natural because the spin- and isospin- contains only an axial-vector diquark. The relevant Faddeev equation is not different in principle to that for the nucleon. It is described in Ref. [30]. and the scalar mass is subsequently set by requiring a particular nucleon mass.

We have written here of desired rather than experimental mass values because it is known that the masses of the nucleon and are materially reduced by pseudoscalar meson loop effects. This is detailed in Refs. [28, 29]. Hence, a baryon represented by the Faddeev equation described above must possess a mass that is inflated with respect to experiment so as to allow for an additional attractive contribution from the pseudoscalar mesons. As in previous work [30, 31, 32, 33] and reported in Table 1, we require GeV and GeV. The results and conclusions of our study are essentially unchanged should even larger masses and a smaller splitting be more realistic, a possibility suggested by Refs. [23, 34]. The relationship between the mass splitting and that between the axial-vector and scalar diquark correlations is sketched in Ref. [35].

## 3 Nucleon Electromagnetic Current

The nucleon’s electromagnetic current is

 Jμ(P′,P) = ie¯u(P′)\mathchar259\relaxμ(q,P)u(P), (2) = ie¯u(P′)(γμF1(Q2)+12MσμνQνF2(Q2))u(P), (3)

where () is the momentum of the incoming (outgoing) nucleon, , and and are, respectively, the Dirac and Pauli form factors. They are the primary calculated quantities, from which one obtains the nucleon’s electric and magnetic (Sachs) form factors

 GE(Q2)=F1(Q2)−Q24M2F2(Q2),GM(Q2)=F1(Q2)+F2(Q2). (4)

Static electromagnetic properties are associated with the behaviour of these form factors in the neighbourhood of . The nucleons’ magnetic moments are defined through

 μn=κn=GnM(0),μp=1+κp=GpM(0), (5)

where , , are referred to as the anomalous magnetic moments; and the electric and magnetic rms radii via

 r2p := −6ddsGpE(s)∣∣∣s=0,r2n:=−6ddsGnE(s)∣∣∣s=0, (6) (rμN)2 := −6ddslnGNM(s)∣∣∣s=0. (7)

In order to calculate the electromagnetic form factors one must know the manner in which the nucleon described in Sect. 2 couples to a photon. That is derived in Ref. [36], illustrated in Fig. 2 and detailed in Appendix C: Nucleon-Photon Vertex. As apparent in that Appendix, the current depends on the electromagnetic properties of the diquark correlations.

Estimates exist of the size of diquark correlations. For example, a first Faddeev equation study of nucleon form factors [37] found a scalar diquark radius of , where is the pion charge radius within the same model. One obtains a similar result in a DSE calculation [38] that provides a good description of pseudoscalar and vector meson properties; i.e.,

 r[ud]0+≈0.7fm,r(ud)1+∼0.8fm, (8)

where the last result is an estimate based on the -meson/-meson radius-ratio [39, 40]. From another perspective, numerical simulations of quenched lattice-regularised QCD suggest a scalar-diquark matter-radius [41]

 rρ[ud]0+=1.1±0.2fm. (9)

It is thus evident that diquark correlations within a baryon are not pointlike. Hence, with increasing , interaction diagrams in which the photon resolves a diquark’s substructure must be suppressed with respect to contributions from diagrams that describe a photon interacting with a bystander or exchanged quark. These latter are the only hard interactions with dressed-quarks allowed in a nucleon. One can therefore improve on Refs. [31, 32] by introducing a diquark form factor. This is expressed in Eqs. (C.13), (C.14) and (C.24).

We use a one-parameter dipole because the system involves two quarks. The parameter is a length-scale that characterises the diquark radius. In the absence of an explicit calculation of the axial-vector diquark’s radius, we employ the same value for scalar and axial-vector diquarks. Owing to differences between the formulation of our nucleon model and the DSE truncation employed in Ref. [38], the values quoted in Eq. (3) provide only a loose constraint on this parameter. It’s value does not have a large effect on form factors for GeV but does influence their evolution thereafter. For example, it influences the position of the zero in : a larger diquark radius shifting the zero further from the origin. Computations have been analysed with four values: , , and fm. Unless otherwise stated, the results reported herein were obtained with

 rqq=0.8fm. (10)

## 4 On Interpreting Form Factors

Now that the Faddeev equation and a consistent Ward-Takahashi-identity conserving current are completely defined, the calculation of a nucleons’ electromagnetic form factors is a straightforward numerical exercise. However, in light of Refs. [42, 43] we judge it worthwhile to comment on their putative interpretation in terms of charge and magnetisation densities before presenting our results.

Such an interpretation rests on the existence of a quantitatively reliable expression for the form factors in terms of a current in which the interacting constituents are well-defined and distinct, for then the charge and current carrying quanta are unambiguous. This is achieved through a current of impulse approximation type, which may include small non-single-particle contributions that arise owing to the Ward-Takahashi identity.

In QCD the relevant degrees of freedom change as the wavelength of the probe evolves. This feature is encoded, e.g., in the dressed-quark mass function, which is discussed in connection with Eq. (A.18). The nature of the mass function is model-independent and one consequence is that to a long wavelength probe a light-quark appears to have a large inertial mass MeV.

Figure 2 expresses a nucleon current in which the primary degrees of freedom are dressed-quarks. Along with the Faddeev equation described in App. A, it is an extension to baryons of the systematic and symmetry preserving rainbow-ladder truncation of QCD’s DSEs, which provides a sound description of pseudoscalar and vector mesons and, in particular, a veracious description of the pion as both a Goldstone mode and a bound state of dressed-quarks [14]. It is a valid impulse approximation, which provides a systematically improvable continuum prediction for nucleon form factors.

Subject to this understanding the question of whether a connection exists between the spatial distribution of charge or magnetisation and the three-dimensional Fourier transform of a Sachs form factor involves a consideration of recoil-corrections experienced by dressed-quarks. The interpretation is appropriate if recoil corrections are small and can be calculated perturbatively. In that case the relevant expectation values in quantum mechanics are validly approximated by the Fourier transform of the Sachs form factor.

Consider the Breit frame and a photon probe with momentum . In the scattering process this momentum is absorbed by the dressed-quarks within the proton. It is elastic scattering so all the dressed-quarks must recoil together, which means they can each be considered as absorbing a momentum fraction222Faddeev and Bethe-Salpeter amplitudes are peaked at zero relative momentum. Hence, the domain of greatest support in the impulse approximation calculation is that with each quark absorbing . This is demonstrated explicitly, e.g., in Ref. [11]. . The magnitude of a recoil correction is then measured by the mass-squared scale

 sr:=q29. (11)

We will consider that recoil corrections are small so long as

 sr<19M2(sr), (12)

where is the dressed-quark mass function. This constraint means

 qto0.0pt∼

a value determined from Eqs. (A.18) – (A.22). This momentum bound corresponds to a length-scale

 λ0.1=0.49fm=0.57rp, (14)

where is the proton’s charge radius. Hence in the three-dimensional Fourier transform of a Sachs form factor, recoil corrections are on the order of 10% or less throughout the domain ; namely, over 81% of the nucleons’ volume.

In measuring the total charge one must evaluate

It is interesting to reckon the amount of charge that is contained within the domain on which recoil corrections are not negligible. It is

 Q0.1N=4π∫0.57rp0drr2ρN(r). (16)

A Gaußian charge form factor can be used to obtain an algebraic and hence easily understood estimate; viz.,

 Gp(q2)=e−16q2r2p, (17)

yields

 ρp(r)=3√6π4π2r3pe−3r2/[2r2p], (18)

from which follows

 Q0.1p=0.19. (19)

It is apparent that this region contains only 19% of the proton’s charge. Expressed another way, the domain on which recoil corrections can be neglected contains 81% of the proton’s charge. (For the neutron’s charge form factor the illustration can be made using a difference of two Gaußians, each of which may be said to represent either the - or -quark contribution to the form factor.) If instead of Eq. (4) one were to consider recoil corrections as small for , then the upper bound in Eq. (4) is and the region contains only 12% of the proton’s charge.

On the other hand, recoil corrections are certainly large and essentially nonperturbative for

 sr≳M2(sr)⇒q≳1GeV, (20)

a momentum boundary which corresponds to lengths

 λ1.0≲0.2fm=0.23rp. (21)

On this domain no quantum mechanical connection can be made between three-dimensional Fourier transforms of Sachs form factors and the density distribution of distinct charge and current carriers. It corresponds to 1.2% of the nucleon’s volume and contains just 1.6% of the proton’s charge.

This analysis elucidates the circumstances under which the three-dimensional Fourier transform of a Breit-frame Sachs form factor can be viewed as providing a useful, qualitatively and semi-quantitatively reliable description of the configuration space distribution of a nucleon’s charge or magnetisation over dressed-quarks. Dressed-quarks are an emergent feature of QCD. The requisite conditions pertain within 81% – 99% of a nucleon’s volume. Moreover, notwithstanding any caveats, Poincaré invariant form factors are always a gauge of a hadron’s structure because they are a measurable and physical manifestation of the nature of the hadron’s constituents and the dynamics that binds them together.

## 5 Calculated Form Factors

In the following two sections we present and discuss the results that our model of the dressed-quark core produces for nucleon form factors. Notably, we made significant modifications to the computer codes used to obtain the results in Ref. [31]. In addition to that described in App. D, which defines a convergent continuation of the Faddeev amplitude into the Breit frame, we succeeded in reducing execution times by an order of magnitude. These two improvements enabled us to use a desk-top computer and obtain, within hours, numerically accurate results for the form factors on the domain GeV.

In order to explain our results we must introduce our notation. The Pauli, Dirac and Sachs form factors are all represented by their usual symbols. Hence, the notation can be introduced by a single example. We choose the proton’s Dirac form factor, , and list the definitions in Appendix E: Form Factor Notation.

It is also worth noting here that our analysis assumes . Hence the only difference between the - and -quarks is their electric charge. Our equations, computer codes and results therefore exhibit the following charge symmetry relations:

 edFp,ui=euFn,di,euFp,di=edFn,ui,i=1,2. (22)

## 6 Proton Form Factors

### 6.1 Dirac proton

In Fig. 3 we depict the proton’s Dirac form factor and a breakdown into contributions from various subclasses of diagrams. The figures deserve careful study.

The upper left panel shows the -evolution of the quark, diquark and exchange (or two-body) contributions to the form factor. Their values measure, respectively, the probability that the photon interacts with a bystander quark or a diquark correlation, or acts in association with diquark breakup:

 quark−Pp,q1=0.47:diquark−Pp,c1=0.35:exchange−Pp,e1=0.18. (23)

These and analogous probabilities are collected in Table 2. For the diquark and exchange contributions switch in importance at GeV. Moreover, while the net result is always positive, the diquark contribution becomes negative at GeV. This panel here, and in kindred figures to follow, also displays a parametrisation of experimental results [44] for illustrative comparison with our computation. The manner by which that comparison should be understood is canvassed in Sect. 8.

A radius can be associated with each of the form factors. We exemplify its definition via ; viz.,

 (rp,q1)2:=−6Fp,q1(0)ddQ2Fp,q1(Q2)∣∣∣Q2=0, (24)

and remark that

 (rp1)2=Pp,q1(rp,q1)2+Pp,c1(rp,c1)2+Pp,e1(rp,e1)2. (25)

The calculated Dirac radii are reported in Table 3. Their values emphasise that so far as the Dirac form factor is concerned, the diquark component of the nucleon is softest.

The lower left panel provides a flavour decomposition of the quark, diquark and exchange contributions to the form factor. While the other two -quark components are positive definite, changes sign at GeV. Up quarks are doubly represented in the proton and from Table 2 it is evident that they are almost equally likely to be struck by a photon whether a bystander or a diquark participant. This explains the near equality of the radii associated with each term in the subclass of these form factor contributions in which a -quark is struck. The same is not true for the -quark, for which the probabilities show that it is more likely to be struck while a diquark participant. This signals that the -quark is less free to move throughout the proton’s volume and hence explains the small value of .

The upper right panel of Fig. 3 shows the -evolution of the contributions to that involve a scalar diquark, an axial-vector diquark, or one of each. It is clear from Table 2 that the scalar diquark component of the proton is dominant. All contributions are positive definite, and the relative strength of the axial-vector and mixed contributions switches at GeV. From Table 3 one reads that the softest contribution to the proton’s Dirac form factor is provided by diagrams involving an axial-vector diquark. One can picture this as stemming from the axial-vector correlation being more massive than the scalar and hence a bystander quark of any flavour ranges further from a collective centre-of-mass.

The lower right panel provides a flavour decomposition of the diquark contributions just discussed. All -quark components are positive definite. For the singly-represented -quark, however, each of the form factors changes sign: becomes positive at GeV; at GeV; and at GeV. Axial-vector contributions to the Dirac form factor are the softest in each flavour sector.

Evident in Table 3 is a notable feature of our calculation; viz.,

 rp,d1>rp,u1. (26)

Owing to charge symmetry this entails

 rn,u1>rn,d1, (27)

a result also obtained and explained in Ref. [23]. Equation (6.1) follows from the presence of axial-vector diquark correlations in the nucleon. One reads from Table 2 that the proton’s singly represented -quark is more likely to be struck in association with an axial-vector diquark correlation than with a scalar, and form factor contributions involving an axial-vector diquark are soft. On the other hand, the doubly-represented -quark is predominantly linked with harder scalar-diquark contributions.

### 6.2 Pauli proton

In Figs. 4 and 5 we depict the proton’s Pauli form factor and a breakdown into contributions from various subclasses of diagrams.

The left panel of Fig. 4 shows the -evolution of the quark, diquark and exchange contributions to the form factor. Listed in Table 4, their values measure, respectively, the contribution to the proton’s anomalous magnetic moment from the photon interacting with a bystander quark, a diquark or in association with diquark breakup. The net contribution from Diagrams 2 and 4 in Fig. 2 is negative. This remains the case until GeV, at which point the net diquark contribution changes sign, as was also the case in . The Pauli radii are listed in Table 5, from which it is evident that Diagrams 3, 5 and 6 in Fig. 2 provide the softest contribution.

The left panels in Fig. 5 provide a flavour decomposition of the quark, diquark and exchange contributions to the proton’s Pauli form factor. We remark that is positive definite whereas changes sign at GeV and at GeV. (The latter should be interpreted qualitatively because our calculations are not truly reliable beyond GeV.) It is evident upon comparison between Tables 3 and 5 that the pattern exhibited by the Pauli radii is kindred to that of the Dirac radii, with the origin alike.

The right panel of Fig. 4 shows the -evolution of the contributions to that involve a scalar diquark, an axial-vector diquark, or one of each. It is apparent from the figure and Table 4 that diagrams involving the scalar correlation are dominant on a material domain. These contributions to a nucleon’s Faddeev amplitude have the simplest rest-frame spin–angular-momentum structure [2, 45]. We find that the scalar and axial-vector contributions are positive definite whereas the mixed contribution changes sign at GeV. The latter provides a larger contribution to the proton’s magnetic moment than the axial-vector diagram. One reads from Table 5 that the softest contribution to the proton’s Pauli form factor is provided by the axial-vector diquark diagrams. This was also the case for the Dirac form factor. However, in contrast to , the mixed contribution to is hardest, a result which owes primarily to Diagram 4 and the simple Ansatz we have made for the interaction therein; viz., Eq. (C.35).

The right panels of Fig. 5 provide a flavour decomposition of the diquark contributions just discussed. It is curious that , a feature which highlights the presence and role of correlations in the nucleon’s Faddeev amplitude. The associated form factor becomes positive at GeV. The contribution with the simplest structure, , is positive definite whereas becomes negative at GeV. In association with the proton’s -quark, the axial-vector diagrams make a positive definite contribution, the scalar diquark form factor becomes negative at GeV and the mixed contribution is negative definite but small.

It is apparent from Table 5 that

 rp,d2>rp,u2, (28)

which entails . These orderings are the same as those exhibited by the Dirac radii, Eq. (6.1), but the separation in magnitudes is larger. The presence of axial-vector diquark correlations again plays a large role in producing these results. We note in addition that and , with the greater reduction in . Indeed, it is almost uniformly true that the quark-core Pauli form factors are harder than their Dirac counterparts. The reduction is marked for and the only exception to the rule is .

### 6.3 Pauli–Dirac proton ratio

In Fig. 6 we plot a weighted ratio of Pauli to Dirac form factors; viz.,

 Rp21(^Q2):=^Q2(ln[^Q2/^\mathchar259\relax2])2Fp2(^Q2)Fp1(^Q2),^Q2=Q2M2N,^\mathchar259\relax2=\mathchar259\relax2M2N. (29)

A perturbative analysis that considers effects arising from both the proton’s leading- and subleading-twist light-cone wave functions, the latter of which represents quarks with one unit of orbital angular momentum, suggests that this ratio should be constant for , where is a mass-scale that is said to correspond to an upper-bound on the domain of soft momenta [46].

We analysed our calculated result in this context and found that with this weighted ratio is a constant on ; by which we mean that the rms relative error with respect to the straight-line fit is % with an associated standard deviation of %. These numbers increase as the minimum value of included in the fit is decreased and, moreover, the value of comes to depend on this minimum value.

In the figure we also plot the ratio in Eq. (6.3) as evaluated from extant experimental data, available on the domain . Using , the result is described by a constant with rms relative error % and an associated standard deviation of %. It is evident in the figure that on the domain for which the ratio is well described by a constant, our model produces a result that lies above the experimental data. This is because thereupon our calculation underestimates experimental results for by % and overestimates those for by a similar amount. (See Sect. 8 for details.)

It is curious that what might appear to be a low mass-scale, , should serve to produce a constant value for the ratio in Eq. (6.3) [31]. In seeking to understand the origin of this scale we analysed the pointwise behaviour of our calculated Faddeev amplitude. The dominant functions are , , , which was to be expected given the associated Dirac structures (see Eqs. (A.8) – (A.10).) A Gaußian can be fitted to the leading Chebyshev moment of each of these functions. That procedure yields the following widths (in units of ):

 ωs11=0.48,ωA13=0.47,ωA15=0.46. (30)

The similarity between these widths and is notable. It highlights the point that while per se is not an elemental input to our calculation, such a mass-scale can arise dynamically as a derivative quantity which may be expressed in the relative-momentum support of the Faddeev amplitude. A challenge now is to determine whether an algebraic relationship exists between in Eq. (6.3) and the widths characterising the Faddeev amplitude.

### 6.4 Sachs proton electric

In Fig. 7 we present the proton’s Sachs electric form factor and a separation into contributions from various subclasses of diagrams. While in principle, given Eqs. (3), these panels contain no information that cannot be constructed from material already presented, they are nevertheless practically useful and informative.

The upper left panel shows the -evolution of the quark, diquark and exchange (or two-body) contributions to . Their values have precisely the same value and interpretation as those associated with the Dirac form factor, which are presented in Table 2. It is notable that the quark contribution; namely, Diagram 1 in Fig. 2, possesses a zero at GeV. It is only because the diquark contribution remains positive until GeV and the exchange contribution is positive definite that the complete result for does not exhibit a zero until GeV.333A zero in was seen in the light-front constituent-quark model of Ref. [50]. In Ref. [51] it was shown to be a property of the scalar-diquark Faddeev model of Ref. [37] but its appearance and location were argued to be dependent on dynamics, consistent with Refs. [52, 53] and the present study. We list the Sachs radii in Table 6. In comparison with the Dirac radii in Table 3, they are relatively uniform owing to Foldy-term contributions.

The lower left panel provides a flavour decomposition of the quark, diquark and exchange contributions to the form factor. Once more their values have precisely the same value and interpretation as those associated with the Dirac form factor. has a zero at GeV and no contribution to is positive definite: possesses a zero at GeV; at GeV; and is negative on the domain GeV. On the other hand, has a zero at GeV but is negative definite. possesses a zero at GeV and at GeV. We list the - and -quark Sachs radii in Table 6. Their values are readily computed and understood from Tables 3 and 4.

The upper right panel of Fig. 7 shows the -evolution of the contributions to that involve a scalar diquark, an axial-vector diquark, or one of each. has a zero at GeV and at GeV, whereas is positive definite. The lower right panel provides a flavour decomposition of the diquark contributions. exhibits a zero at GeV, at GeV and is negative on the domain GeV. On the other hand, passes through zero at GeV and at GeV but is negative definite. The associated Sachs radii are listed in Table 6.

### 6.5 Sachs proton magnetic

In Figs. 8 and 9 we depict the proton’s Sachs magnetic form factor and a separation into contributions from various subclasses of diagrams. Again, while in principle these panels only contain information that can be constructed from material already presented, they are nonetheless practically useful and informative.

The left panel of Fig. 8 shows the -evolution of the quark, diquark and exchange contributions to the form factor. , and are positive definite and monotonically decreasing. On the other hand, the net contribution from Diagrams 2 and 4 in Fig. 2; namely, , is uniformly small, negative in the vicinity of GeV and again for GeV. The pattern is qualitatively similar in the flavour breakdown of these form factors, depicted in the left panels of Fig. 9.

The right panel of Fig. 8 exhibits the -evolution of the contributions to that involve a scalar diquark, an axial-vector diquark, or one of each. All contributions are positive definite, diagrams involving only a scalar diquark are dominant and contributions involving at least one axial-vector diquark are uniformly of comparable magnitude. The flavour breakdown is contained in the right panels of Fig. 9: all contributions are positive definite except , which is uniformly small but becomes positive at GeV and remains so until GeV. (NB. The latter should be interpreted qualitatively because the feature appears at a larger value of than we consider our computation reliable.)

In Table 7 we list the values of all the form factors, which measure, respectively, the contributions to the proton’s magnetic moment. These values can be obtained from , where, e.g., , etc. The magnetic radii are listed in Table 8. The general pattern has electric radii exceeding magnetic radii. The few exceptions are easily explained. For example, : this is primarily because and of significant magnitude in the neighbourhood of . As already noted, it is curious that this contribution to the proton’s anomalous magnetic moment is negative.

### 6.6 Sachs electric–magnetic proton ratio

We plot in Fig. 10 in comparison with contemporary data. A sensitivity to the proton’s electromagnetic current is evident, here expressed via the diquarks’ radius. Irrespective of that radius, however, the proton’s electric form factor possesses a zero and the magnetic form factor is positive definite. On GeV our result lies below experiment. As discussed in Sect. 8, this can likely be attributed to our omission of so-called pseudoscalar-meson-cloud contributions.

## 7 Neutron Form Factors

### 7.1 Dirac neutron

In Fig. 11 we depict the neutron’s Dirac form factor and a decomposition into contributions from various subclasses of diagrams. Owing to charge symmetry, Eqs. (5), it is unnecessary to present a flavour breakdown. For example, with the normalisation used in our figures, the curve that would be denoted by is simply negative- drawn from Fig. 3.

In addition to that of itself, the left panel panel depicts the -evolution of the quark, diquark and exchange contributions to this form factor. and are negative definite, and is only positive for GeV. On the other hand, the diquark contribution; viz., , is positive until GeV. The right panel renders the -dependence of contributions from diagrams containing a scalar diquark, an axial-vector diquark or one of each. is negative definite and is negative for GeV. is small at (only 3% of the other two form factors) and negative for GeV. These features reflect: the dominant role played in the Faddeev amplitude by the positively-charged scalar diquark; the fact that the -quark is singly-represented and only a bystander in combination with an axial-vector diquark; and the softness of the diquark correlations, which ensures that only a bystander quark can participate in the scattering process at large-.

We list computed Dirac radii connected with the neutron in Table 9. Two entries are imaginary because the associated form factors have an inflexion point away from . We do not currently attribute any real significance to this local feature, which for the neutron is particularly sensitive to details of the Ansatz employed for Diagrams 5 and 6 in Fig. 2; namely, the two-body piece of the current which is not yet well constrained.

### 7.2 Pauli neutron

In Fig. 12 we depict the neutron’s Pauli form factor and a decomposition into contributions from various subclasses of diagrams. Once more, owing to charge symmetry, Eqs. (5), it is unnecessary to present a flavour breakdown. For example, with the normalisation used in our figures, the curve that would be denoted by is simply negative- drawn from Fig. 4.

The left panel panel depicts the -evolution of itself, and that of the quark, diquark and exchange contributions to this form factor. , and are negative definite on the domain within which we consider our calculations accurate, and is negative until GeV. The right panel portrays the -dependence of contributions from diagrams containing a scalar diquark, an axial-vector diquark or one of each. and are negative definite, and is negative for GeV and always small in magnitude. These features are consistent with those of the Dirac form factor.

We list computed anomalous magnetic moments and Pauli radii connected with the neutron in Table 10. The small value of may be understood via a cancellation between and contributions. Along with the small value of , Eq. (C.36), this explains the size of . With the exception of the uniformly small , the Pauli radii follow the same pattern as those of the proton.

### 7.3 Neutron Pauli–Dirac neutron ratio

In Fig. 13 we plot the weighted ratio of Pauli to Dirac form factors in Eq. (6.3) for the neutron. This ratio is constant for the proton, Fig. 6, however, that is not the case for the neutron. Moreover, with our calculated neutron form factors there is no value of for which this ratio assumes a constant value.

The apparent cause of this behaviour is a zero in at GeV. This point lies beyond the upper bound of the domain within which we consider our computation reliable. On the other hand, its presence does influence the evolution of the ratio. This can be seen by analysing the ratio using Padé approximants on subdomains of GeV