aa r X i v : . [ nu c l - t h ] D ec A Field-Theoretic Parametrization ofLow-Energy Nucleon Form Factors
Brian D. Serot ∗ Department of Physics and Nuclear Theory CenterIndiana University, Bloomington, IN 47405 (Dated: November 3, 2018)
Abstract
A field-theoretic parametrization is proposed for nucleon electromagnetic form factors at mo-mentum transfer less than 600 MeV. The parametrization is part of a larger effective field theorylagrangian that is Lorentz covariant and chiral symmetric, and that has been used to successfullydescribe bulk and single-particle properties of medium to heavy mass nuclei. The parametrizationis based on vector meson dominance and a derivative expansion of nucleon couplings to the elec-tromagnetic fields. At lowest order in the expansion, it is possible to fit all four parameters tomodern data on the rms radii of the nucleon form factors. At next-to-leading order it is possibleto fit the form factors to within a few percent up to momentum transfers of 600 MeV. The vec-tor meson dominance contributions are crucial in this fit, since a simple expansion in powers ofmomentum transfer would require many, many terms to achieve comparable accuracy. The abilityto fit single-nucleon form factors up to 600 MeV momentum transfer makes possible the study oftwo-body electromagnetic exchange currents within this effective field theory framework.
PACS numbers: 14.20.Dh; 25.30.Bf; 12.40.Vv; 11.10.-z ∗ Electronic address: [email protected] . INTRODUCTION Lorentz-covariant meson–baryon effective field theories of the nuclear many-body prob-lem (often called quantum hadrodynamics or QHD) have been known for many years toprovide a realistic description of the bulk properties of nuclear matter and heavy nuclei.(For reviews, see Refs. [1, 2, 3, 4, 5, 6].) Recently, a QHD effective field theory (EFT) hasbeen proposed [7, 8, 9, 10, 11, 12] that includes all the relevant symmetries of the under-lying QCD. In particular, the spontaneously broken SU (2) L × SU (2) R chiral symmetry isrealized nonlinearly. The motivation for this EFT and some calculated results are discussedin Refs. [6, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21].This QHD EFT has three desirable features: (1) It uses the same degrees of freedomto describe the currents and the strong-interaction dynamics; (2) It respects the same in-ternal symmetries, both discrete and continuous, as the underlying QCD (before and afterelectromagnetic interactions are included); and (3) Its parameters can be calibrated usingstrong-interaction phenomena, like π N scattering and the empirical properties of finite nuclei(as opposed to electroweak interactions with nuclei). It thus provides a natural framework,based on a single lagrangian, for discussing the roles of one-body and two-body currents innuclear electromagnetic interactions.The nucleon electromagnetic (EM) structure (form factors) is described in this EFT usinga combination of vector meson dominance (VMD) [7, 22, 23, 24, 25, 26, 27] and a derivativeexpansion for nucleon interactions with the EM field. In the applications of this EFT tonuclear structure noted above, however, only the lowest-order derivative couplings wereincluded, so that the form factors provided an accurate description of the single-nucleonelectron scattering data only up to roughly 250 MeV momentum transfer. In contrast, ifone is to study two-body (exchange) currents, one must reproduce the single-nucleon formfactors accurately up to at least 600 MeV momentum transfer, where two-body contributionsare expected to be visible. This momentum scale should be accessible in this low-energyhadronic EFT [7, 8, 28].Our motivation for this study of nucleon form factors is twofold. First, we want to updatethe lowest-order fits of Ref. [7] to include the large amount of low-energy, high-precision datathat became available in the early 2000’s. Second, we extend the fits to the next order inmomentum transfer and show that the form factors will accurately reproduce the empiricalresults up to roughly 600 MeV momentum transfer. This will make them suitable for studiesof exchange currents within the QHD EFT.In the past ten or fifteen years, much new data on the nucleon EM form factors have beenobtained using both unpolarized electron scattering and polarization transfer. (For recentreviews, see Refs. [29, 30].) There have also been numerous attempts at fitting the improveddata set; for example, see Refs. [31, 32, 33]. For the present study, we are most interested inthe work of Kelly [32], who achieved excellent fits with a small number of free parameters.In particular, the fits are good enough over the momentum transfer range of interest to usthat we will simply fit our EFT parameters to Kelly’s analytic results rather than to thedata itself. Since our best fits reproduce Kelly’s at the few percent level, this procedure isjustified.One of our interesting results is that a straightforward Q expansion of Kelly’s analyticresults is inadequate unless many, many terms are retained. (Here Q ≡ − q is the squareof the spacelike four-momentum transfer.) The presence of the VMD contributions in theEFT approach greatly improves the situation. Moreover, it is important to include the new2FT parameters in such a way that the error is minimized for the whole relevant range of Q , not just Q → II. RE-FIT OF LOWEST-ORDER PARAMETERS
In this section, we consider the form factors as described in Ref. [7]. We follow theconventions of Refs. [7, 12]. Rather than work with the Dirac ( F ) and Pauli ( F ) formfactors, defined in terms of the nucleon EM vertex asΓ µ = F ( Q ) γ µ + F ( Q ) iσ µν q ν M , (1)where M is the nucleon mass and F contains the anomalous magnetic moment, here wewill primarily concentrate on the Sachs form factors G E ( Q ) = F ( Q ) − τ F ( Q ) , G M ( Q ) = F ( Q ) + F ( Q ) , (2)where τ ≡ Q / M ≡ − q / M in terms of the four-momentum q µ , and we have notdistinguished the charge states. The charge states are written in terms of the isoscalar (0)and isovector (1) parts as, for example, F p = F (0) + F (1) , F n = F (0) − F (1) . (3)The simple parametrizations used by Kelly [32] take the form G ( Q ) = P nk = 0 a k τ k P n +2 k = 1 b k τ k , (4)which guarantees the correct asymptotic dependence at large Q : G ( Q ) ∝ Q − . This willnot concern us, as we are interested in parametrizing the form factors at small Q . With n = 1 and a = 1, this parametrization gives excellent fits to G Ep , G Mp /µ p , and G Mn /µ n (where µ i is the full magnetic moment) using four parameters each [32]. For G En , Kellyfollows the so-called Galster parametrization [34]: G En ( Q ) = Aτ Bτ G D ( Q ) , (5)where the dipole form factor is G D ( Q ) ≡ Q / Λ ) , Λ = 0 .
71 GeV , (6)and A and B are fitted parameters.For our parametrization, we use set Q2 of Ref. [7]. This provides an accurate fit to bulkand single-particle nuclear properties and leaves the anomalous coupling to the isoscalar3 ABLE I: Coupling parameters from set Q2 [7]. Note that the nucleon couplings to the omegaand rho mesons ( g v and g ρ ) are determined from the empirical properties of nuclei. β (0) β (1) g v g ρ f v f ρ . − . . . . vector meson (“omega”) undetermined; we will determine it here. We will need the massparameters M = 939 MeV = 4 . − ,m v = 782 MeV = 3 .
963 fm − ,m ρ = 770 MeV = 3 .
902 fm − , (7)the anomalous magnetic moments λ p = 1 . , λ n = − . , (8)the couplings in Table I, and the electromagnetic coupling g γ = 5 . ρ → e + e − = 6 . G Ep ( Q ) = F (0)1 + F (1)1 − Q M ( F (0)2 + F (1)2 )= 1 − ( β (0) + β (1) ) Q M − g v g γ (cid:18) − f v Q M (cid:19) Q Q + m v − g ρ g γ (cid:18) − f ρ Q M (cid:19) Q Q + m ρ − λ p Q M . (9)If we define the mean-square radius as r Ep ≡ − G Ep ( Q )d Q (cid:19) Q = 0 , (10)then r Ep = 12 (cid:20) (cid:18) β (0) M + 2 g v g γ m v (cid:19) + 6 (cid:18) β (1) M + g ρ g γ m ρ (cid:19)(cid:21) + 3 λ p M ≡ (cid:0) h r i (0)1 + h r i (1)1 (cid:1) + 3 λ p M , (11)where the mean-square radii on the right-hand side are the isoscalar and isovector values forthe Dirac form factor F . Inserting the Q2 parameters leads to( r Ep ) / = 0 .
862 fm , (12)4hich also agrees with the result in Ref. [32].Turning now to G En , we have G En ( Q ) = F (0)1 − F (1)1 − Q M ( F (0)2 − F (1)2 )= − ( β (0) − β (1) ) Q M − g v g γ (cid:18) − f v Q M (cid:19) Q Q + m v + g ρ g γ (cid:18) − f ρ Q M (cid:19) Q Q + m ρ − λ n Q M , (13)so that r En = 12 (cid:20) (cid:18) β (0) M + 2 g v g γ m v (cid:19) − (cid:18) β (1) M + g ρ g γ m ρ (cid:19)(cid:21) + 3 λ n M = 12 (cid:0) h r i (0)1 − h r i (1)1 (cid:1) + 3 λ n M . (14)If we set h r i (0)1 = h r i (1)1 , as in Ref. [7], we then find r En = − .
127 fm , in significantdisagreement with Kelly’s value of − . ± .
003 fm . We conclude that the new datashows that h r i (0)1 − h r i (1)1 = 0 . . (15)With this information, together with Eq. (11), we can determine two distinct radii for theDirac form factor: h r i / = 0 .
799 fm , h r i / = 0 .
780 fm , (16)in contrast to the assumption made in Ref. [7].We now consider the magnetic form factors, beginning with G Mp ( Q ) = F (0)1 + F (1)1 + F (0)2 + F (1)2 = 1 − ( β (0) + β (1) ) Q M − g v (1 + f v )3 g γ Q Q + m v − g ρ (1 + f ρ )2 g γ Q Q + m ρ + λ p . (17)As expected, for Q → G Mp ( Q ) → λ p = µ p . Thus we normalize G Mp by dividing by µ p , leading to the mean-square radius r Mp = 12(1 + λ p ) (cid:20) (cid:18) β (0) M + 2 g v g γ m v (cid:19) + 6 (cid:18) β (1) M + g ρ g γ m ρ (cid:19)(cid:21) + 12(1 + λ p ) (cid:18) f v g v g γ m v + 6 f ρ g ρ g γ m ρ (cid:19) = 12(1 + λ p ) (cid:0) h r i (0)1 + h r i (1)1 + ( λ p + λ n ) h r i (0)2 + ( λ p − λ n ) h r i (1)2 (cid:1) = 0 . , (18)5here the numerical value is taken from Kelly. Note that the magnetic radii depend on boththe Dirac and Pauli mean-square radii.For the neutron, G Mn ( Q ) = F (0)1 − F (1)1 + F (0)2 − F (1)2 = − ( β (0) − β (1) ) Q M − g v (1 + f v )3 g γ Q Q + m v + g ρ (1 + f ρ )2 g γ Q Q + m ρ + λ n . (19)The mean square radius is r Mn = 12 λ n (cid:20) (cid:18) β (0) M + 2 g v g γ m v (cid:19) − (cid:18) β (1) M + g ρ g γ m ρ (cid:19)(cid:21) + 12 λ n (cid:18) f v g v g γ m v − f ρ g ρ g γ m ρ (cid:19) = 12 λ n (cid:0) h r i (0)1 − h r i (1)1 + ( λ p + λ n ) h r i (0)2 − ( λ p − λ n ) h r i (1)2 (cid:1) = 0 . . (20)With the results in Eqs. (16), (18), and (20), it is now simple algebra to determine therms radii for the Pauli form factor, with the results h r i / = 1 .
30 fm , h r i / = 0 .
896 fm . (21)The isovector radius is consistent with Refs. [7, 35]. At this time, we have no way to estimateerrors for any of these derived radii.Equation (20) also shows the relationships between the Dirac and Pauli rms radii and the O ( Q ) parameters in the EM coupling expansion (i.e., β (0) , β (1) , f v , f ρ ). Simple inversion ofthese results allows us to present parameters determined from the new single-nucleon datawith the nuclear couplings of set Q2: β (0) = 0 . , β (1) = − . , f v = − . , f ρ = 4 . . (22) III. INCLUSION OF HIGHER-ORDER COUPLINGS
So far, we have adjusted the parameters that enter at O ( τ ) to achieve the modern valuesfor the four rms radii. Now we want to go to higher order in τ and see what adjustments arenecessary to reproduce the Sachs form factors up to Q ≈
600 MeV or τ ≈ .
1. We can relatethese adjustments to higher-order terms in the derivative expansion of the EM lagrangian.For the purposes of this work, we take the nucleon part of the EM lagrangian as L EM = − eA µ N γ µ
12 (1 + τ ) N − e M F µν N λσ µν N − e M ( ∂ ν F µν ) N βγ µ N − e M ( ∂ ν ∂ η F µη ) N λ ′ σ µν N − eM ( ∂ ∂ ν F µν ) N β ′ γ µ N − e M ( ∂ ∂ ν ∂ η F µη ) N λ ′′ σ µν N + · · · , (23)6here A µ and F µν are the electromagnetic fields. All of the constants λ, β, λ ′ , β ′ , λ ′′ havethe isospin structure λ = λ p
12 (1 + τ ) + λ n
12 (1 − τ ) = 12 ( λ p + λ n ) + 12 τ ( λ p − λ n ) ≡ λ (0) + λ (1) τ , etc . (24)As shown in Ref. [12], the isovector parts of these constants should be modified to includepion interactions to maintain the residual chiral symmetry in the lagrangian. When appliedto the nucleon form factors, however, these pions appear only in loops, which we are notconsidering, so we have omitted these terms.We have already included the λ and β parameters in the preceding section. It is easy tosee that the λ ′ constants enter the magnetic rms radii at the same order in Q as the vectormeson couplings f v and f ρ . Thus the λ ′ parameters are redundant in our approach and willnot be considered in the sequel. Thus the four new adjustable constants at our disposal arecontained in β ′ and λ ′′ .It is straightforward to work out the Feynman rules for the new vertices and to constructthe tree-level contributions to the form factors. One finds that these constants enter theDirac and Pauli form factors at O ( Q ). Thus they will enter the Sachs form factors at O ( τ )and O ( τ ).Our strategy is the following: We begin with the magnetic form factors, where both newterms enter at O ( τ ). We numerically adjust the coefficient of this term until we get a goodfit to the form factor up to τ = 0 .
1. Here we define a good fit as one that minimizes themaximum deviation between the fit and the data (i.e., Kelly’s results [32]) throughout thewhole interval 0 ≤ τ ≤ .
1. The result provides two constraints between the β ′ and λ ′′ . Wethen turn to the electric form factors, where the new terms enter at both O ( τ ) and O ( τ ).The coefficients of both of these terms are then adjusted, consistent with the constraint,until a good fit is obtained for 0 ≤ τ ≤ .
1. This procedure provides us with four numbersthat can be used to determine the β ′ and λ ′′ .Denoting the new contributions to the Sachs form factors as δG , we find δG Mp ( τ ) = 16( λ ′′ (0) + λ ′′ (1) − β ′ (0) − β ′ (1) ) τ , (25) δG Mn ( τ ) = 16( λ ′′ (0) − λ ′′ (1) − β ′ (0) + β ′ (1) ) τ . (26)Taking the best fit to the data leads to Figs. 1 and 2, and the constraints λ ′′ p − β ′ p = 2 . , λ ′′ n − β ′ n = − . . (27)The fits are accurate to a few percent.For the electric form factors, the new terms are δG Ep ( τ ) = − β ′ (0) + β ′ (1) ) τ − λ ′′ (0) + λ ′′ (1) ) τ , (28) δG En ( τ ) = − β ′ (0) − β ′ (1) ) τ − λ ′′ (0) − λ ′′ (1) ) τ (29)Taking the best fit subject to the constraints above produces Figs. 3 and 4, and the values β ′ p = − . , β ′ n = 0 . . (30)Combining these results with the constraints determines all four new parameters.Although the relative error in G En is as large as 30%, G En is small in the momentum-transfer range of interest. A more relevant comparison is given in Fig. 5, where all four ofthe empirical form factors are shown along with fits to G En and G Mn .7 τ G M p / µ p FIG. 1: Proton magnetic form factor as a function of τ . The curves represent the data (solid), thelowest-order fit (dashed), and the new fit (dot-dashed). IV. SUMMARY
In this paper we studied a field-theoretic parametrization of single-nucleon EM formfactors at low energy. This parametrization is part of a Lorentz-covariant, chiral invariant,hadronic effective field theory that was proposed to study the nuclear many-body problem.We thus have a single lagrangian that describes the nuclear structure, nuclear currents, andinteraction vertices at low energies. This is a natural framework for discussing the roles ofone-body and two-body currents in nuclear electromagnetic interactions.The parametrization of the form factors is based on a combination of vector meson dom-inance and a derivative expansion for nucleon interactions with the EM field. At leadingorder in derivatives, the form factors accurately reproduce the single-nucleon electron scat-tering data only up to roughly 250 MeV momentum transfer. To study two-body exchangecurrents, however, one must reproduce the form factors accurately up to at least 600 MeVmomentum transfer. This paper is proof of principle that by including the next-to-leadingorder (nonredundant) derivatives, one can adequately describe the form factors up to 600MeV with our form of parametrization. 8 τ G M n / µ n FIG. 2: Neutron magnetic form factor as a function of τ . The curves are identified as in Fig. 1. We also revisited the leading-order parametrization and used a modern data set to eval-uate the expansion coefficients. We found that it is now possible to determine all four rmsradii for the Dirac and Pauli, isoscalar and isovector form factors, and thus determine allfour leading-order coefficients from the single-nucleon data, unlike in Ref. [7].It is interesting that a simple power-series expansion in powers of the momentum transfersquared would require many, many terms to reproduce the form factors up to the desired 600MeV. (This is easy to see using ten minutes of simple calculations on mathematica withKelly’s [32] fits to the data.) Thus the vector meson dominance contributions are critical inallowing us to parametrize the desired data using only the next-to-leading order derivativeterms.While there is much work still to be done before two-body currents can be studied numer-ically within this effective field theory framework, we have shown that the one-body currentcan be adequately parametrized to make a study of two-body currents possible.9 τ G E p FIG. 3: Proton electric form factor as a function of τ . The curves are identified as in Fig. 1. Acknowledgments
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