We present new parameterizations of vector and axial nucleon form factors. We
maintain an excellent descriptions of the form factors at low momentum
transfers, where the spatial structure of the nucleon is important, and use the
Nachtman scaling variable xi to relate elastic and inelastic form factors and
impose quark-hadron duality constraints at high momentum transfers where the
quark structure dominates. We use the new vector form factors to re-extract
updated values of the axial form factor from neutrino experiments on deuterium.
  We obtain an updated world average value from neutrino-d and pion
electroproduction experiments of M_A = 1.014 +- 0.014 GeV/c2. Our
parameterizations are useful in modeling neutrino interactions at low energies
(e.g. for neutrino oscillations experiments). The predictions for high momentum
transfers can be tested in the next generation electron and neutrino scattering
experiments.Comment: 5 pages, 3 figures, to be published in EPJ

Avvakumov, S.

Bodek, A.

Bradford, R.

Budd, H.

English

arXiv

Springer - Publisher Connector

Eur. Phys. J. C 53, 349–354 (2008) THE EUROPEAN
PHYSICAL JOURNAL C
DOI 10.1140/epjc/s10052-007-0491-4
Regular Article – Experimental Physics
Vector and axial nucleon form factors:
A duality constrained parameterization
A. Bodeka, S. Avvakumov, R. Bradford, H. Budd
Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627-0171, USA
Received: 12 October 2007 / Revised version: 21 November 2007 /
Published online: 15 December 2007 − © Springer-Verlag / Societa` Italiana di Fisica 2007
Abstract. We present new parameterizations of vector and axial nucleon form factors. We maintain an excel-
lent descriptions of the form factors at low momentum transfers, where the spatial structure of the nucleon
is important, and use the Nachtman scaling variable ξ to relate elastic and inelastic form factors and impose
quark–hadron duality constraints at high momentum transfers where the quark structure dominates. We use
the new vector form factors to re-extract updated values of the axial form factor from neutrino experiments
on deuterium. We obtain an updated world average value from νµd and pion electroproduction experiments
of MA = 1.014± 0.014 GeV/c
2. Our parameterizations are useful in modeling neutrino interactions at low
energies (e.g. for neutrino oscillations experiments). The predictions for high momentum transfers can be
tested in the next generation electron and neutrino scattering experiments.
PACS. 13.40.Gp; 13.15.+g; 13.85.Dz; 14.20.Dh; 25.30.Bf; 25.30.Pt
1 Introduction
The nucleon vector and axial elastic form factors have been
measured for more than 50 years in e−N and νN scat-
tering. At low Q2, a reasonable description of the proton
and neutron elastic form factors is given by the dipole ap-
proximation. The dipole approximation is a lowest-order
attempt to incorporate the non-zero size of the proton into
the form factors. The approximation assumes that the pro-
ton has a simple exponential spatial charge distribution,
ρ(r) = ρ0e
−r/r0 , where r0 is the scale of the proton radius.
Since the form factors are related in the non-relativistic
limit to the Fourier transform of the charge and magnetic
moment distribution, the above ρ(r) yields the dipole form
deﬁned by:
GV,AD (Q
2) = CV,A
/(
1+
Q2
M2V,A
)2
.
Here CV,A = (1, gA), gA = −1.267, M2V = 0.71 (GeV/c)
2,
andMA is the axial mass.
Since MA is not equal to MV, the distribution of elec-
tric and axial charge are diﬀerent. However, the magnetic
moment distributions were assumed to have the same spa-
tial dependence as the charge distribution (i.e., form factor
scaling). Recent measurements from Jeﬀerson Lab show
that the ratio of
µpGEp
GMp
falls at high Q2 challenging the
validity of form factor scaling [1] and resulting in new up-
a e-mail: bodek@pas.rochester.edu
dated parameterizations of the form factors [2], [3]. In this
paper we present parameterizations that simultaneously
satisfy constraints at low Q2 where the spatial structure of
the nucleon is important, and at high Q2 where the quark
structure is important. A violation of form-factor scaling is
expected from quark–hadron duality.
We use our new vector form factors to re-extract up-
dated values of the axial form factor from a re-analysis
of previous neutrino scattering data on deuterium and
present a new parameterization for the axial form factor
within the framework of quark–hadron duality.
2 New parametrization
The new parameterizations presented in this paper are re-
ferred to as the duality based “BBBA07” parameteriza-
tion. Our updated parameterizations feature the following:
1. Improved functional form that adds an additional Q2
dependence using the Nachtman scaling variable ξ to
relate elastic and inelastic form factors. For elastic scat-
tering (x= 1)
ξp,n,N =
2
(1+
√
1+1/τp,n,N)
,
where τp,n,N =Q
2/4M2p,n,N . Here Mp,n,N are the pro-
ton (0.9383GeV/c2), neutron (0.9396GeV/c2), and
average nucleon mass (for proton, neutron, and axial
form factors, respectively).
350 A. Bodek et al.: Vector and axial nucleon form factors: A duality constrained parameterization
2. Yield the same values as Arrington and Sick [4] for
Q2 < 0.64 (GeV/c)2, while satisfying quark–hadron du-
ality constraints at high-Q2.
For vector form factors our ﬁt functions are AN (ξ) (i.e.
AEp(ξ
p),AMp(ξ
p), AEn(ξ
n),AMn(ξ
n)) multiplying an up-
dated Kelly [3] type parameterization of one of the proton
form factors. The Kelly parameterization is:
GKelly(Q2) =
∑m
k=0 akτ
k
p
1+
∑m+2
k=1 bkτ
k
p
,
where a0 = 1 andm= 1.
In our analysis, we use all the datasets used by Kelly [3],
updated to include the recent BLAST [7] results, to ﬁt
GEp, GEn, GMp/µp, and GMn/µn (µp = 2.7928, µn =
−1.9130). Our parameterization employs the published
Kelly functional form to GKellyEp , and an updated set of
parameters for GKelly-updMP (Q
2). The parameters used for
GKellyEp and G
Kelly-upd
Mp are listed in Table 1, and AN (ξ) is
given by
AN (ξ) =
n∑
j=1
Pj(ξ)
Pj(ξ) = pj
n∏
k=1,k =j
ξ− ξk
ξj− ξk
.
Each Pj is a LaGrange polynomial in ξ. The ξj are equidis-
tant “nodes” on an interval [0, 1], and pj are the ﬁt pa-
rameters that have an additional property AN (ξj) = pj .
The functional form AN (ξ) (for GEp, GMp, GEn, and
GMn) is used with seven pj parameters at ξj = 0, 1/6,
1/3, 1/2, 2/3, 5/6, and 1.0. In the ﬁtting procedure de-
scribed below, the parameters of AN (ξ) are constrained
to give the same vector form factors as the recent low
Q2 ﬁt of Arrington and Sick [4] for Q2 < 0.64 (GeV/c)2
(as that analysis includes coulombs corrections which
modify GEp, and two photon exchange corrections which
modify GMp and GMn). Since the published form fac-
tor data do not have these corrections, this constraint is
implemented by including additional “fake” data points
for Q2 < 0.64 (GeV/c)2.
Our ﬁts to the form factors are:
GMp(Q
2)/µp =AMp(ξ
p)GKelly-updMp (Q
2)
GEp(Q
2) =AEp(ξ
p)GKellyEp (Q
2)
GMn(Q
2)/µn =A
25,43
Mn (ξ
n)GMp(Q
2)/µp
GEn(Q
2) =A25,43En (ξ
n)GEp(Q
2)
(
aτn
1+ bτn
)
,
where we use our updated parameters in the Kelly param-
eterizations. For GEn the parameters a= 1.7 and b = 3.3
are the same as in the Galster [9–11] parametrization and
ensure that dGEn/dQ
2 at for Q2 = 0 is in agreement with
measurements.
For convenience, we also provide ﬁts for the form factors
GEp and GMp/µp that give very close to the same values,
Table 1. Parameters for GKellyEp and G
Kelly-upd
Mp . Our parame-
terization employs the as-published Kelly parameterization to
G
Kelly
Ep and an updated set of parameters for G
Kelly-upd
Mp (Q
2)
that includes the recent BLAST [7] results
a1 b1 b2 b3 χ
2/ndf
G
Kelly
Ep –0.24 10.98 12.82 21.97 0.78
G
Kelly-upd
Mp 0.1717 11.26 19.32 8.33 1.03
but use the dipole form instead:
GEp(Q
2) =AEp-dipole(ξ
p)GVD(Q
2)
GMp(Q
2)/µp =AMp-dipole(ξ
p)GVD(Q
2) .
The values A(ξ) = p1 at ξ1 = 0 (Q
2 = 0) for GMp, GEp,
GEn, GMn are set to to 1.0. The value A(ξ) = p7 at ξj = 1
(Q2→∞) for GMp andGEp is set to 1.0.
The value A(ξ)=pj at ξj=1 for GMn and GEn are ﬁxed
by constraints from quark–hadron duality. Quark–hadron
duality implies that the ratio of neutron and proton mag-
netic form factors should be the same as the ratio of the
corresponding inelastic structure functions F2n
F2p
in the ξ = 1
limit. (Here F2 = ξ
∑
i e
2
i qi(ξ))
G2Mn
G2Mp
=
F2n
F2p
=
1+4 d
u
4+ d
u
=
(
µ2n
µ2p
)
A2Mn(ξ = 1) .
We ran ﬁts with two diﬀerent values of du at the ξ=1
limit: d
u
= 0 and 0.2 (corresponding to F2n
F2p
= 0.25 and
0.4286). The ﬁt utilizing d
u
= 0 is A25Mn, and the ﬁt utiliz-
ing d
u
= 0.2 is A43Mn. The ﬁnal parameters for both cases of
d
u
, are given in Table 2 (or download computer code [31]).
The diﬀerence between these two sets is indicative of the
theoretical error of our parameterization.
Our parameterizations are within the error band of re-
cent theoretical ﬁts based in dispersion relations [12]. Since
our ﬁts are constrained to give the same vector form fac-
tors as the recent low Q2 ﬁt of Arrington and Sick [4] for
Q2 < 0.64 (GeV/c)2, they are in agreementwith the experi-
mental measurements of the proton and neutron rms radii.
(Note that as discussed in reference [13], the nucleon rms
radius should be determined from ﬁtting a polynomial of
second order to the low Q2 form factors. The commonly
used polynomial of ﬁrst order yields radius values which
are too small).
The value A(ξ) = pj at ξj = 1 for GEn is set by another
duality-motivated constraint. R is deﬁned as the ratio of
deep-inelastic longitudinal and transverse structure func-
tions. For inelastic scattering, as Q2→∞, Rn =Rp. If we
assume quark–hadron duality, the same should be true for
the elastic form factors at ξ=1 ( Q2→∞) limit:
Rn(x= 1;Q
2) =
4M2n
Q2
(
G2En
G2Mn
)
G2En/G
2
Mn =G
2
Ep/G
2
Mp .
A. Bodek et al.: Vector and axial nucleon form factors: A duality constrained parameterization 351
Table 2. Fit parameters for AN (ξ), the LaGrange portion of the new parameterization. Note A
25
Mn, A
25
En, and A
25
FA are con-
strained to have du = 0 at ξ = 1, and A
43
Mn, A
43
En, are constrained to have
d
u = 0.2
ξ,Q2 p1 p2 p3 p4 p5 p6 p7
0,0 0.167, 0.029 0.333, 0.147 0.500, 0.440 0.667, 1.174, 0.833, 3.668 1.0,∞
AEp 1. 0.9927 0.9898 0.9975 0.9812 0.9340 1.
AMp 1. 1.0011 0.9992 0.9974 1.0010 1.0003 1.
AEp-dipole 1. 0.9839 0.9632 0.9748 0.9136 0.5447 –0.2682
AMp-dipole 1. 0.9916 0.9771 0.9801 1.0321 1.0429 0.5084
A25Mn 1. 0.9958 0.9877 1.0193 1.0350 0.9164 0.7300
A43Mn 1. 0.9958 0.9851 1.0187 1.0307 0.9080 0.9557
A25En 1. 1.1011 1.1392 1.0203 1.1093 1.5429 0.9706
A43En 1. 1.1019 1.1387 1.0234 1.1046 1.5395 1.2708
A
25-dipole
FA 1.0000 0.9207 0.9795 1.0480 1.0516 1.2874 0.7707
Fig. 1. Ratios of GEp (a), GMp/µp (b), GEn (c) and GMn/µn (d) to GD. The short-dashed line in each plot is the old Kelly
parameterizations (old Galster for GEn). The solid line is our new BBBA0725 parameterization for
d
u = 0.0, and the long-dashed
line is BBBA0743 for
d
u = 0.2. The values of ξ and the corresponding values of Q
2 are shown on the bottom and top axis
In order to constrain the ﬁts toGEn at highQ
2 we have
assumed that the values of
G2En
G2
Mn
are the same as the meas-
ured
G2Ep
G2
Mp
for the three highestQ2 data points forGEp, and
included these three “fake” data points in the GEn ﬁts. In
addition, the Rn = Rp condition yields the following con-
straint at ξ = 1:
A25,43En (ξ = 1) = P7 =
(
b
a
)(
1+4 du
4+ du
)1/2
,
352 A. Bodek et al.: Vector and axial nucleon form factors: A duality constrained parameterization
Fig. 2. The constraint used in ﬁtting GEn stipulates that
G2En/G
2
Mn =G
2
Ep/G
2
Mp at high ξ. The solid line is
GEp
|GMp|
and
|GEp|
|GMp|
, and the short-dashed line is GEn|GMn| and
|GEn|
|GMn|
where b/a = 1.7/3.3. As there are two parameter sets
A25,43Mn (ξ), we have produced two parameter sets A
25,43
En as
shown in Table 2.
The new form factors GEp, GMp/µp, GMn/µn, and
GEn are plotted in Fig. 1 as ratios to the dipole formG
V
D.
As seen in Table 2, AN (ξ) is not needed for GMp as it
is very close to 1.0. For GEp it yields a correction of 1%
at low Q2 (because it is required to agree with the ﬁts of
Arrington and Sick [4] which include two photon exchange
and Coulomb corrections). For GEn and GMn it is used to
impose quark–hadron duality asymptotic constraints. Fig-
ure 2 shows plots of the data and ﬁts to GEn|GMn|
and
GEp
|GMp|
(for the d
u
= 0 at ξ = 1 case).
Table 3.MA (GeV/c
2) values published by νµ-deuterium experiments [16–22] and updated corrections ∆MA when re-extracted
with updated BBBA200725 form factors, and ga =−1.267. Also shown is updatedMA from νµhydrogen→ µ
−n [23]
Experiment [16–22] QE Q2 range Eν Vector FF −ga,M
2
V MA ∆MA M
updated
A
νµd→ µ
−pps events GeV/c
2 GeV used [8–11] used (published) FF,RC GeV/c2
Mann73 166 .05–1.6 0.7 Bartl, Gen = 0 1.23 , .84
2 0.95± .12
Barish77 500 .05–1.6 0.7 Ollsn, Gen = 0 1.23 , .84
2 0.95± .09 –.026, .002
Miller82,77,73 1737 .05–2.5 0.7 Ollsn, Gen = 0 1.23 , .84
2 1.00± .05 –.030, .002 0.972± .05
Baker81 1138 .06–3.0 1.6 Ollsn, Gen = 0 1.23 , .84
2 1.07± .06 –.028, .002 1.044± .06
Kitagaki83 362 .11–3.0 20 Ollsn, Gen = 0 1.23 , .84
2 1.05+.12−.16 –.025, .001 1.026
+.12
−.16
Kitagaki90 2544 .10–3.0 1.6 Ollsn, Gen = 0 1.254 , .84
2 1.070+.040−0.045 –.036, .002 1.036
+.040
−0.045
Allasia90 552 .1–3.75 20 dipole, Gen = 0 1.2546, .84
2 1.080± .08 –.080, .002 1.002± .08
Av. νµd [16–22] 5780 above BBBA200725 1.267 , .71 1.051± .026 θ
−
µ , Eµ, θ, Pp 1.016± .026
π electrprd. [15] 1.014± .016
νµH→ µ−n [23] 13 0–1.0 1.1 dipole, Gen = 0 1.23 , .842 0.9±0.35 –.070, 0.01 .831±0.35
νµH→ µ−n [23] 13 0–1.0 1.1 BBBA200725 1.267 , .71 σQE θ
+
µ , Eµ 1.04±0.40
Average all 1.014± .014
3 Re-extraction of axial form factor
Using our updated BBBA200725 form factors and an up-
dated value gA =−1.267, we perform a complete reanalysis
of published ν quasielastic [16–22] (QE) data on deuterium
(νµn→ µ−p) using the procedure described in detail in [2,
14]. We extract new values of MA with updated form fac-
tors (FF) and also include radiative corrections [5, 6] (RC).
Although of lower statistical signiﬁcance, for completeness
we also include all available antineutrino data on hydrogen
targets [23].
The average of the corrected measurements of MA
from Table 3 is MdeuteriumA = 1.016± 0.026GeV/c
2. This
is in agreement the average value of MpionA = 1.014±
0.016GeV/c2 extracted from pion electroproduction ex-
periments after corrections for hadronic eﬀects [15]. The
average of the νµ and electroproduction values is
Mworld-averageA = 1.014±0.014GeV/c
2 .
This precise MA is smaller than the recent results (for
Q2 > 0.25 (GeV/c)2) reported by MiniBoone [28] on a car-
bon target (M carbonA = 1.25± 0.12GeV/c
2) and by the
K2K [29] collaboration on oxygen (MoxygenA = 1.20±
0.12GeV/c2). Both experiments use updated vector form
factors. Although the collaborations attribute the larger
MA to nuclear eﬀects, there are theoretical arguments that
MA in nuclear targets should be smaller [30] than (or the
same [26]) as in deuterium. This MA discrepancy is im-
portant for ν oscillations experiments since it aﬀects the
normalization (at high energies the QE cross section is ap-
proximately proportional to MA) and non-linearity of the
QE cross section, which is relevant to the extraction of ν
mass diﬀerence and mixing angle.
For deep-inelastic scattering, the vector and axial parts
ofF2 are equal. Local quark–hadron duality at largeQ
2 im-
A. Bodek et al.: Vector and axial nucleon form factors: A duality constrained parameterization 353
Fig. 3. (a) FA(Q
2) re-extracted from neutrino-deuterium data divided by GAD(Q
2) [32]. (b) FA(Q
2) from pion electroproduction
divided by GAD(Q
2) [32], corrected for for hadronic eﬀects [15]. Solid line – duality based ﬁt; Short-dashed line – FA(Q
2)A2=V 2.
Dashed-dot line – constituent quark model [27]
plies that the axial and vector components of F elastic2 are
also equal, which yields:
[
FA(Q
2)A2=V 2
]2
=
(
GVE
)2
(Q2)+ τN
(
GVM (Q
2)
)2
/(1+ τN) ,
where
GVE(Q
2) =GEp(Q
2)−GEn(Q
2) , and
GVM (Q
2) =GMp(Q
2)−GMn(Q
2) .
We extract values of FA(Q
2) from the diﬀerential cross
sections using the procedure of [14]. The overall normaliza-
tion is set by the theoretical QE cross section [32]. We then
do a duality based ﬁt to FA(Q
2) (including pion electropro-
duction data) of the form:
FA(Q
2) =A25FA(ξ
N )GAD(Q
2) .
We impose the constraint A25FA(ξ1 = 0) = p1 = 1.0. We
also constrain the ﬁt by requiring that A25FA(ξ
N ) yield
FA(Q
2) = FA(Q
2)A2=V 2 by including additional “fake”
data points) for ξ > 0.9 (Q2 > 7.2 (GeV/c)2).
Figure 3a shows FA(Q
2) extracted from neutrino-
deuterium experiments divided by GAD(Q
2) [32]. Figure 3b
shows FA(Q
2) extracted from pion electroproduction ex-
periments divided by GAD(Q
2) [32]. These pion electropro-
duction values can be directly compared to the neutrino re-
sults because they are multiplied by a factor FA(Q
2,MA =
1.014GeV/c2)/FA(Q
2,MA = 1.069GeV/c
2) to correct for
∆MA= 0.055GeV/c
2 originating from hadronic eﬀects [15].
The solid line is our duality based ﬁt. The short-dashed
line is FA(Q
2)A2=V 2. The dashed-dot line is a constituent-
quark model [27] prediction.
4 Conclusion
In conclusion, our new parameterizations of vector and
axial nucleon form factors use quark–hadron duality con-
straints at high momentum transfers, and maintain a very
good descriptions of the form factors at low momentum
transfers. Our new parameterizations are useful in mod-
eling ν interactions for oscillations experiments. Our pre-
dictions [33] for GEn(Q
2) and FA(Q
2) at high (Q2) can
be tested in future e-N and ν-N experiments. at Jeﬀerson
Laboratory and at Fermilab (MINERvA) [34].
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