Nucleon Form Factors Understood by Vector Meson Exchange Earle Lomon(MIT)Jlab12/12/08 Summary: Description of model, history & early success Details of present version Comparison with all nucleon elastic electromagnetic form factor data - polarized versus diff. cross section Another VMD approach, without hadronic form factors Relation to dipole ffs, extrapolation, onset of pQCD

Various approaches to modeling hadron structure: -Lattice QCD; good progress, but issues with isoscalar form factors and not yet able to address details. -Chiral PT; well suited to low momentum scale. -Light cone; basic structure with few things to adjust. For a detailed, compete description: -VMD+; hadronic picture with dominance of vector mesons + dispersive ππ, πρ and KK_bar contributions + asymptotics(approach to pQCD behavior).

VMD + DR + transition to pQCD at high Q^2 Asymptotic Q convergence obtained by hff-form factors at vector-meson/nucleon (quark) vertices. F. Iachello, A.D. Jackson and A. Lande, (no width,ρωφ), common hff Phys. Lett. B43(1973) 191 M.F. Gari and W. Krumplemann, (no width,ρωφ) ρω vs φ hff ->pQCDZ. Phys.A322(1985) 689; Phys. Lett.B173(1986) 10; Phys. Lett.B274(1992) 159 E.L. Lomon, (ρ width,ρρ’ωω’φ) Phys. Rev.C64(2001) ; (1) Phys. Rev.C66(2002) ; (2) nucl-th/ v2 (2006); (3)

Alternatively Asymptotic Q convergence obtained by precise cancellation of sum of vector meson pole and other terms. Requires added phenomenological pole terms. G. Hoehler et al., Nucl. Phys.B114(1976) 505 M.A. Belushkin, H.-W. Hammer and U. -G. Meissner, Phys. Rev.C75(2007)

The extended Gari-Krümpelmann models GKex (Lomon) include coupling to the photons through vector meson exchange terms [VMD - ρ (width included), ω, φ, ρ’ and ω’ mesons] and a transition at high momentum transfers to pQCD, controlled by a hadron/quark form factor and Λ QCD. Fitted to the dcs (Rosenbluth) data for G Mp and G Mn and low Q G Ep and G En, and to polarization data for R p =μ p G Ep /G Mp and R n =μ n G En /G Mn. All the model parameters are fixed at reasonable values, determining the pQCD normalization.

SPECIFICS OF THE Gkex NUCLEON EMFF MODEL In fitting the nucleon emff data including the new R n and R p results [2,3] (and preliminary high Q R n ), we use the extended GK model DR-GK ′(1) of [1] with the addition of a pole term for the well established isoscalar vector meson ω′(1419), whose mass is lower than that of the already included isovector vector meson ρ′(1450). The choice of the particular hadronic form factor parameterization DR-GK ′(1) was made because of its low χ2 value and the fact that its predicted values of R p were a little closer to the data than those of the other extended models, in addition to it having the following good physical properties:

(1)It uses the cut-off Λ 2 for the helicity flip meson-nucleon form factors, rather than the cut-off Λ 1 used by other versions. (2) The normalization of the pQCD limit is controlled by the quark-nucleon cut-off Λ D instead of Λ 2, while the evolution of the logarithmic dependence on Q2 depends on Λ QCD. (3) Fitted to the data set of 2001 it finds Λ QCD =.1163, close to the expected value. The form factors are not very sensitive to this parameter which is fixed at.15 for the fits to the new data sets.

The relevant formulas follow: The emff of a nucleon are defined by the matrix elements of the electromagnetic current J μ = e u bar (p ′) {γ μ F 1 N (Q 2 ) +[ i/(2m N )]σ μν Q ν F 2 N (Q 2 )} u(p), where N is the neutron, n, or proton, p, and −Q 2 = (p’ − p) 2 is the square of the invariant momentum transfer. F 1 N (Q 2 ) and F 2 N (Q 2 ) are respectively the Dirac and Pauli form factors, normalized at Q 2 = 0 as F 1 p (0) = 1, F 1 n (0) = 0, F 2 p (0) = κ p, F 2 n (0) = κ n.

The Sachs form factors, directly obtained from experiment, are G EN (Q 2 ) = F 1 N (Q 2 ) − τ F 2 N (Q 2 ) τ= Q 2 /(4M N 2 ) G MN (Q 2 ) = F 1 N (Q 2 ) + F 2 N (Q 2 ). Expressed in terms of the isoscalar and isovector electromagnetic currents 2F i p = F i is + F i iv, 2F i n = F i is − F i iv (i=1,2).

The GKex model has the following form for the four isotopic emff F 1 iv (Q 2 ) = N/2[{ (1 + Q 2 /0.3176) − 2 }/ (1 + Q 2 /0.5496)]F 1 α (Q 2 ) + (g ρ ′ /f ρ ′ ) [m ρ ′ 2 /(m ρ ′ 2 + Q 2 )]F 1 α (Q 2 ) + − g ρ ′ /f ρ ′ ]F 1 D (Q 2 ) F 2 iv (Q 2 ) = N/2[{ (1 + Q 2 /0.1422) − 1 }/ (1 + Q 2 /0.5362)] F 2 α (Q 2 ) + (κ ρ ′ g ρ ′ /f ρ ′ )[m ρ ′ 2 /(m ρ ′ 2 + Q 2 )] F 2 α (Q 2 ) + [κ ν − N/2- κ ρ ′ g ρ ′ /f ρ ′ ] F 2 D (Q 2 )

F 1 is (Q 2 ) = (g ω /f ω ) [m ω 2 /(m ω 2 + Q 2 )]F 1 α (Q 2 ) + (g ω ′ /f ω ′ )[m ω ′ 2 /(m ω ′ 2 + Q 2 )]F 1 α (Q 2 ) + (g φ /f φ )[m φ 2 /(m φ 2 + Q 2 )] F 1 φ (Q 2 ) + [1 − g ω /f ω − g ω ′ /f ω ′ ]F 1 D (Q 2 ) F 2 is (Q 2 ) = (κ ω g ω /f ω )[m ω 2 /(m ω 2 + Q 2 )]F 2 α (Q 2 ) +(κ ω ′ g ω ′ /f ω ′ )[m ω ′ 2 /(m ω ′ 2 + Q 2 )]F 2 α (Q 2 ) + (κ φ g φ /f φ )[m φ 2 /(m φ 2 + Q 2 )]F 2 φ (Q 2 ) +[κ s − κ ω g ω /f ω − κ ω ′ g ω ′ /f ω’ − κ φ g φ/ /f φ ]F 2 D (Q 2 ) Note: The pQCD terms include intermediate Q contributions which normalize the ff’s at Q=0.

For GKex the above hadronic form factors are parameterized in the following way: F 1 α,D (Q 2 ) = [Λ 1,D 2 /(Λ 1,D 2 + Q T 2 )][Λ 2 2 /(Λ Q T 2 )] F 2 α,D (Q 2 ) = [Λ 1,D 2 /(Λ 1,D 2 + Q T 2 )][Λ 2 2 /(Λ Q T 2 )] 2 where α = ρ, ω and Λ i,D is Λ i for the F i α, Λ D for the F i D, F 1 φ (Q 2 ) = F 1 α [Q 2 /(Λ Q 2 )] 3/2, F 1 φ (0) = 0 F 2 φ (Q 2 ) = F 2 α [(Λ 1 2 / μ φ 2 )(Q 2 + μ φ 2 )/(Λ Q 2 )] 3/2 with Q T 2 = Q 2 ln[(Λ D 2 + Q 2 )/Λ QCD 2 ]/ln(Λ D 2 /Λ QCD 2 ).

This parameterization guarantees that the normalization conditions of the current at Q 2 =0 are met and that asymptotically F 1 i ∼ [Q 2 ln (Q 2 /Λ QCD 2 ) ] − 2, F 2 i ∼ F 1 i /Q 2 (i = is, iv) as required by PQCD. The form factor F 1 φ (Q 2 ) vanishes at Q 2 = 0, and both it and F 2 φ (Q 2 ) decrease more rapidly at large Q 2 than the other meson form factors. This conforms to the Zweig rule imposed by the s bar s structure.

TABLE I: Model parameters. (08 model adds BLAST and 3 high Q Rn) Common to all models are κ v = 3.706, κ s = −0.12, m ρ = GeV, m ω = GeV, m φ = GeV, m ρ ′ = 1.45 GeV and m ω′ = GeV. Parameters of Model GK(05/08) g ρ ′ /f ρ ′ = / κ ρ ′ =12.0/ g ω /f ω =0.7021/0.6887κ ω =0.4027/ g φ /f φ = −0.1711/ κ φ =0.01/ μ φ =.2/.1162 g ω ′ /f ω ′ =0.164/ κ ω ′ = −2.973/ Λ 1 = / Λ 2 =2.6115/ Λ D =1.181/ Λ QCD =0.150 * (all Λ in GeV) N =1.0 * * not varied

TABLE I I: Contributions to the standard deviation, χ2 for each form factor, from the present data set. The number of data points contributing is in the second column. Only polarization data is used for the R n,p values and differential cross section data is used only for the G Mn.p values. Data type Data size χ2 05/08 05/08 G Mp 68/ /51.9 G Mn 39/ / R p 22/ /14.5 R n 5/17 1.1/6.9 Total 134/ /198.5

In the following figures the curves are: Red-2008 fit that includes low Q Rn and Rp BLAST data and preliminary high Q Rn Jlab data. Dashed fit without the above data. Thick Solid fit with dcs (Rosenbluth) data for G_En and G_Ep, and no polarization data. Thin Solid- Galster Rn curve.

Recently M.A. Belushkin, H.-W. Hammer and U. -G. Meissner, Phys. Rev.C75(2007) [BHM] have extended the Hoehler type model by considering the KK_bar and ρπ continua in addition to the ππ continuum, which they conclude are adequately represented by simple poles and adding a broad phenomenological contribution to each isovector form factor at higher masses.

The asymptotic momentum transfer behavior is restricted by a superconvergent requirement in one fit, but by an explicit pQCD behavior in another version. As there are no hadronic form factors, the required asymptotic behavior is obtained by a restriction on the sum of the coupling strengths and masses. This results in requiring vector mesons with unobserved and some low masses.

The BHM superconvergent model requires the following extra vector mesons to fit the data and enforce superposition: (masses in GeV/c^2) ws1 = ; ws2 = wv1 = ; wv2 = ; wv3 = wv4 = ; wv5 = The BHM pQCD asymptotic behavior model requires fewer extra vector mesons: ws1 = wv1 = 1.0; wv2 = ; wv3 =

In the following slides: Gkex (thick solid), BHM-SC (dashed), BHM-pQCD (dotted), Galster (thin solid)

In the next 4 figures the thick curve is the model prediction, the thin curve is the pQCD limit

R p =μ p G Ep /G Mp Curves; GKext05 (solid), QCD limit (dashed)

R n =μ n G En /G Mn Curves; GKext05 (thick blue), QCD limit (dashed), Galster parameterization (solid). Data; Madey et al. PRL 91, (2003) (blue boxes), Warren et al., PRL 92, (2004) (red triangles).

Effect of rho width: width included (dashed), no width (red solid). This modest effect would be reduced by refitting parameters.

CONCLUSIONS A model that transitions from VMD to pQCD, and fits all the nucleon elastic emff well, does not approach pQCD for the form factors until Q 2 = (GeV/c) 2, depending on the component. The model, which was not fitted to data at Q 2 < 0.4 (GeV/c) 2, predicts smooth structure in the form factors near Q 2 = 0.2 (GeV/c) 2, similar to that seen in recent BLAST results. However the structure of Rp in the Blast Data may have a larger amplitude than the model. The model structure is due to the large Pauli scalar and vector ω and ω’ contributions of opposite sign.

The Gkex model is a slightly better overall fit to the data than the BHM-pQCD, and substantially better than the BHM-SC model in the present data range. The BHM models more rapidly change slope after that range. New, higher Q, Rn measurements now being analyzed should easily differentiate between the models. The width of the rho meson has only a minor effect. The behavior of the Breit Frame configuration space charge distributions of the neutron and proton are as expected. They are mainly determined, as are the momentum space distributions of the form factors, by the ρ and ω vector mesons and the lower momentum contributions of the pQCD term (the latter actually represents the non-resonant contributions from intermediate Q).