Nucleon Form Factors Understood by Vector Meson Exchange

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Nucleon Form Factors Understood by Vector Meson
Exchange Earle Lomon(MIT) Jlab 12/12/08 S
ummary 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
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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 pp, p? and KK_bar contributions
asymptotics(approach to pQCD behavior).
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VMD DR transition to pQCD at high Q2
Asymptotic Q convergence obtained by hff-form
factors at vector-meson/nucleon (quark)
vertices. F. Iachello, A.D. Jackson and A.
Lande, (no width,??f), common hff Phys. Lett.
B43(1973) 191 M.F. Gari and W. Krumplemann, (no
width,??f) ?? vs f hff -gtpQCD Z.
Phys.A322(1985) 689 Phys. Lett.B173(1986)
10 Phys. Lett.B274(1992) 159 E.L. Lomon, (?
width,????f) Phys. Rev.C64(2001) 035204
(1) Phys. Rev.C66(2002) 045501
(2) nucl-th/0609020v2 (2006) (3)
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• 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) 035202

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The extended Gari-Krümpelmann models GKex (Lomon)
include coupling to the photons through vector
meson exchange terms VMD - ? (width included),
?, f, ? and ? mesons and a transition at high
momentum transfers to pQCD, controlled by a
hadron/quark form factor and ?QCD. Fitted to the
dcs (Rosenbluth) data for GMp and GMn and low Q
GEp and GEn , and to polarization data for
RpµpGEp/GMp and RnµnGEn/GMn. All the model
parameters are fixed at reasonable values,
determining the pQCD normalization.
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SPECIFICS OF THE Gkex NUCLEON EMFF
MODEL In fitting the nucleon emff data
including the new Rn and Rp results 2,3 (and
preliminary high Q Rn ), 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 ?'(1419), whose mass is lower than
that of the already included isovector vector
meson ?'(1450). The choice of the particular
hadronic form factor parameterization DR-GK '(1)
was made because of its low ?2 value and the fact
that its predicted values of Rp were a little
closer to the data than those of the other
extended models, in addition to it having the
following good physical properties
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• It uses the cut-off ?2 for the helicity flip
  meson-nucleon form factors, rather than the
  cut-off ?1 used by other versions.
• (2) The normalization of the pQCD limit is
  controlled by the quark-nucleon cut-off ?D
  instead of ?2, while the evolution of the
  logarithmic dependence on Q2 depends on ?QCD.
• (3) Fitted to the data set of 2001 it finds ?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.

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The relevant formulas follow The emff of a
nucleon are defined by the matrix elements of the
electromagnetic current Jµ ltN(p') Jµ N (p)gt
e ubar(p ') ?µF 1 N(Q2 ) i/(2mN)sµ? Q?F 2N
(Q2) u(p), where N is the neutron, n, or
proton, p, and -Q2 (p - p)2 is the square of
the invariant momentum transfer. F 1N (Q2 ) and
F 2N (Q2 ) are respectively the Dirac and Pauli
form factors, normalized at Q2 0 as F 1p (0)
1, F 1n (0) 0, F2p(0) ?p , F2n(0) ?n .
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The Sachs form factors, directly obtained from
experiment, are GEN(Q2 ) F1N (Q2 ) - t F2N (Q2
) t Q2/(4MN2) GMN(Q2 ) F1N (Q2 )
F2N(Q2 ) . Expressed in terms of the isoscalar
and isovector electromagnetic currents 2F ip F
iis F iiv , 2F in Fiis - Fiiv (i1,2).
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The GKex model has the following form for the
four isotopic emff F1iv(Q2 ) N/21.0317
0.0875(1 Q 2 /0.3176) - 2 / (1 Q2
/0.5496)F1a (Q 2 ) (g? '/f? ') m? '2/(m? '2
Q2 )F1a (Q 2) - g? '/f? 'F1D (Q 2)
F2iv(Q2 ) N/25.7824 0.3907(1 Q 2 /0.1422)
- 1/ (1 Q2 /0.5362) F2a (Q 2 ) (? ? 'g?
'/f? ')m? '2/(m? '2 Q2 ) F2a(Q2 ) ? ? -
6.1731 N/2- ? ? 'g? '/f? ' F2D (Q2 )
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F1is(Q2 ) (g ? /f?) m?2 /(m?2 Q2)F 1a (Q 2
) (g ? '/f ? ')m? '2 /(m? '2 Q2 )F 1a (Q 2
) (gf/ff)mf2/(mf2 Q2) F 1f (Q 2 ) 1 -
g ? /f? - g ? '/f ? 'F1D (Q 2) F2is(Q 2 ) (??
g ? /f?)m?2/(m?2 Q2)F 2a (Q 2 ) (? ? ' g ?
'/f ? ')m? '2 /(m? '2 Q 2)F 2a (Q 2 ) (?f g
f/ff)mf2/(mf2 Q2)F2f (Q 2 ) ?s - ?? g ?
/f? - ? ? 'g ? '/f ?- ?f g f//ff F2D (Q2
) Note The pQCD terms include intermediate Q
contributions which normalize the ffs at Q0.
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For GKex the above hadronic form factors are
parameterized in the following way F1a,D (Q2 )
?1,D2 /(?1,D2 QT2)?22 /(?22 QT2) F2a,D
(Q2 ) ?1,D2 /(?1,D2 QT2)?22 /(?22
QT2)2 where a ?, ? and ?i,D is ?i for the F
ia , ?D for the FiD , F1f(Q2) F1a Q2/(?12
Q2 )3/2, F1f(0) 0 F2f(Q2) F2a (?12/ µf
2)(Q2 µf 2)/(?12 Q2 )3/2 with QT2 Q2
ln(?D2 Q2)/?QCD2/ln(?D2 /?QCD2) .
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This parameterization guarantees that the
normalization conditions of the current at Q20
are met and that asymptotically F 1i Q2 ln
(Q2 /?QCD2) - 2, F2i F 1i /Q2 (i is,
iv) as required by PQCD. The form factor F1f
(Q2 ) vanishes at Q2 0, and both it and F2f (Q2
) decrease more rapidly at large Q2 than the
other meson form factors. This conforms to the
Zweig rule imposed by the sbars structure.
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TABLE I Model parameters. (08 model adds BLAST
and 3 high Q Rn) Common to all models are ?v
3.706, ?s -0.12, m? 0.776 GeV, m? 0.784
GeV, mf 1.019 GeV, m? ' 1.45 GeV and m?'
1.419 GeV. Parameters of Model GK(05/08) g ?
'/f ? ' 0.0072089/0.021039 ? ? ' 12.0/8.3743
g? /f? 0.7021/0.6887 ?? 0.4027/0.4097 gf
/ff -0.1711/-.1654 ?f 0.01/0.0025 µf
.2/.1162 g ? ' /f ? ' 0.164/0.2098 ? ? '
-2.973/-2.4847 ?1 0.93088/0.93933 ?2
2.6115/2.6710 ?D 1.181/1.1971 ?QCD0.150
(all ? in GeV) N 1.0 not varied
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TABLE I I Contributions to the standard
deviation, ?2 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 Rn,p values and
differential cross section data is used only for
the GMn.p values. Data type Data size
?2 05/08 05/08 GMp
68/68 51.5/51.9 GMn 39/39
124.9/125. 2 Rp 22/28 10.3/14.5
Rn 5/17 1.1/6.9 Total 134/147
187.8/198.5
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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- 2005 fit without the above
data. Thick Solid- 2001 fit with dcs
(Rosenbluth) data for G_En and G_Ep, and no
polarization data. Thin Solid- Galster Rn curve.

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Recently M.A. Belushkin, H.-W. Hammer and U. -G.
Meissner, Phys. Rev.C75(2007) 035202 BHM
have extended the Hoehler type model by
considering the KK_bar and ?p continua in
addition to the pp continuum, which they conclude
are adequately represented by simple poles and
adding a broad phenomenological contribution to
each isovector form factor at higher masses.
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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.
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The BHM superconvergent model requires the
following extra vector mesons to fit the data and
enforce superposition (masses in GeV/c2) ws1
1.124860 ws2 2.019536 wv1 1.062128 wv2 1.3
00946 wv3 1.493630 wv4 1.668522 wv5 2.9154
51 The BHM pQCD asymptotic behavior model
requires fewer extra vector mesons ws1
1.799639wv1 1.0 wv2 1.627379 wv3 1.779245
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In the following slides Gkex (thick solid),
BHM-SC (dashed), BHM-pQCD (dotted), Galster (thin
solid)
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In the next 4 figures the thick curve is the
model prediction, the thin curve is the pQCD limit
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RpµpGEp/GMp Curves
GKext05 (solid), QCD limit (dashed)
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• RnµnGEn/GMn
• Curves GKext05 (thick blue), QCD limit (dashed),
  Galster parameterization (solid).
• Data Madey et al. PRL 91, 122002-1 (2003) (blue
  boxes), Warren et al., PRL 92, 04230 (2004) (red
  triangles).

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Effect of rho width width included (dashed), no
width (red solid). This modest effect would be
reduced by refitting parameters.
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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 Q2 4 - 30 (GeV/c)2 , depending on
the component. The model, which was not fitted
to data at Q2 lt 0.4 (GeV/c)2, predicts smooth
structure in the form factors near Q2 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 ? and ? contributions of
opposite sign.
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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 ? and ? vector mesons and the lower
momentum contributions of the pQCD term (the
latter actually represents the non-resonant
contributions from intermediate Q).