Nucleon form factors and moments of generalized parton distributions using
N
f
¼ 2 þ 1 þ 1 twisted mass fermions
C. Alexandrou,
1,2
M. Constantinou,
1
S. Dinter,
3
V. Drach,
3
K. Jansen,
1,3
C. Kallidonis,
1
and G. Koutsou
2
1
Department of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
2
Computation-based Science and Technology Research Center, Cyprus Institute, 20 Kavafi Street, 2121 Nicosia, Cyprus
3
NIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany
(Received 27 March 2013; published 16 July 2013)
We present results on the axial and the electromagnetic form factors of the nucleon, as well as on the
first moments of the nucleon generalized parton distributions using maximally twisted mass fermions. We
analyze two N
f
¼ 2 þ1 þ 1 ensembles having pion masses of 213 MeV and 373 MeV each at a different
value of the lattice spacing. The lattice scale is determined using the nucleon mass computed on a total of
17 N
f
¼ 2 þ1 þ 1 ensembles generated at three values of the lattice spacing, a. The renormalization
constants are evaluated nonperturbatively with a perturbative subtraction of Oða
2
Þ terms. The moments of
the generalized parton distributions are given in the
MS scheme at a scale of ¼ 2 GeV. We compare
with recent results obtained using different discretization schemes. The implications on the spin content of
the nucleon are also discussed.
DOI: 10.1103/PhysRevD.88.014509 PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.Aw
I. INTRODUCTION
Recent progress in the numerical simulation of lattice
quantum chromodynamics (LQCD) has been remarkable.
The improvements in the algorithms used and the increase
in computational power have enabled simulations to be
carried out at near physical parameters of the theory. This
opens up exciting possibilities for ab initio calculation of
experimentally measured quantities, as well as for predict-
ing quantities that are not easily accessible to experiment.
Understanding nucleon structure from first principles is
considered a milestone of hadronic physics, and a rich
experimental program has been devoted to its study, start-
ing with the measurements of the electromagnetic form
factors initiated more than 50 years ago. Reproducing these
key observables within the LQCD formulation is a prereq-
uisite to obtaining reliable predictions on observables that
explore physics beyond the standard model.
A number of major collaborations have been studying
nucleon structure within LQCD for many years. However,
it is only recently that these quantities can be obtained with
near physical parameters both in terms of the value of the
pion mass and with respect to the continuum limit [1–11].
The nucleon electromagnetic form factors are a well-suited
experimental probe for studying nucleon structure and thus
provide a valuable benchmark for LQCD. The nucleon
form factors connected to the axial-vector current are
more difficult to measure and therefore less accurately
known than its electromagnetic form factors. A notable
exception is the nucleon axial charge g
A
which is accu-
rately measured in decays. The fact that g
A
can be
extracted at zero momentum transfer and that it is techni-
cally straightforward to compute in LQCD, due to its
isovector nature, makes it an ideal benchmark quantity
for LQCD. The generalized parton distributions (GPDs)
encode information related to nucleon structure that com-
plements the information extracted from form factors
[12–14]. They enter in several physical processes such as
deeply virtual Compton scattering and deeply virtual
meson production. Their forward limit coincides with the
usual parton distributions and, using Ji’s sum rule [15],
allows one to determine the contribution of a specific
parton to the nucleon spin. In the context of the ‘‘proton
spin puzzle,’’ which refers to the unexpectedly small frac-
tion of the total spin of the nucleon carried by quarks, this
has triggered intense experimental activity [16–20].
II. LATTICE EVALUATION
In this work we consider the nucleon matrix elements of
the vector and axial-vector operators
O
1
...
n
V
a
¼
c
f
1
iD
$
2
...iD
$
n
g
a
2
c
; (2.1)
O
1
...
n
A
a
¼
c
f
1
iD
$
2
...iD
$
n
g
5
a
2
c
; (2.2)
where
a
are the Pauli matrices acting in flavor space,
c
denotes the two-component quark field (up and down). In
this work we consider the isovector combination by taking
a ¼ 3, except when we discuss the spin fraction carried by
each quark. Furthermore, we limit ourselves to n ¼ 1 and
n ¼ 2. The case n ¼ 1 reduces to the nucleon form factors
of the vector and axial-vector currents, while n ¼ 2 corre-
sponds to matrix elements of operators with a single
derivative. The curly brackets represent a symmetrization
over indices and subtraction of traces, only applicable to
the operators with derivatives. There are well-developed
methods to compute the so-called connected diagram,
depicted in Fig. 1, contributing to the matrix elements of
PHYSICAL REVIEW D 88, 014509 (2013)
1550-7998=2013 =88(1)=014509(22) 014509-1 Ó 2013 American Physical Society
these operators in LQCD. Each operator can be decom-
posed in terms of generalized form factors (GFFs) as
follows: The matrix element of the local vector current,
O
V
3
, is expressed as a function of the Dirac and Pauli form
factors
hNðp
0
;s
0
ÞjO
V
3
jNðp; sÞi
¼
u
N
ðp
0
;s
0
Þ
F
1
ðq
2
Þþ
i
q
2m
N
F
2
ðq
2
Þ
1
2
u
N
ðp; sÞ;
where u
N
ðp; sÞ denotes the nucleon spinors of a given
momentum p and spin s. F
1
ð0Þ measures the nucleon
charge while F
2
ð0Þ measures the anomalous magnetic mo-
ment. They are connected to the electric, G
E
, and mag-
netic, G
M
, Sachs form factors by the relations
G
E
ðq
2
Þ¼F
1
ðq
2
Þþ
q
2
ð2m
N
Þ
2
F
2
ðq
2
Þ;
G
M
ðq
2
Þ¼F
1
ðq
2
ÞþF
2
ðq
2
Þ:
(2.3)
The local axial current matrix element of the nucleon
hNðp
0
;s
0
ÞjO
A
3
jNðp; sÞi can be expressed in terms of the
form factors G
A
and G
p
as
hNðp
0
;s
0
ÞjO
A
3
jNðp;sÞi¼
u
N
ðp
0
;s
0
Þ
G
A
ðq
2
Þ
5
þ
q
5
2m
N
G
p
ðq
2
Þ
1
2
u
N
ðp;sÞ: (2.4)
The matrix elements of the one-derivative operators are
parametrized in terms of the GFFs A
20
ðq
2
Þ, B
20
ðq
2
Þ,
C
20
ðq
2
Þ, and
~
A
20
ðq
2
Þ and
~
B
20
ðq
2
Þ for the vector and axial-
vector operators, respectively, according to
hNðp
0
;s
0
ÞjO
V
3
jNðp; sÞi
¼
u
N
ðp
0
;s
0
Þ
A
20
ðq
2
Þ
f
P
g
þ B
20
ðq
2
Þ
i
f
q
P
g
2m
þ C
20
ðq
2
Þ
1
m
q
f
q
g
1
2
u
N
ðp; sÞ; (2.5)
hNðp
0
;s
0
ÞjO
A
3
jNðp; sÞi
¼
u
N
ðp
0
;s
0
Þ
~
A
20
ðq
2
Þ
f
P
g
5
þ
~
B
20
ðq
2
Þ
q
f
P
g
2m
5
1
2
u
N
ðp; sÞ: (2.6)
Note that the GFFs depend only on the momentum transfer
squared, q
2
¼ðp
0
pÞ
2
; p
0
is the final and p the initial
momentum. The isospin limit corresponds to taking
3
=2
in Eq. (2.2) and gives the form factor of the proton minus
the form factors of the neutron. In the forward limit we thus
have G
E
ð0Þ¼1 and G
M
ð0Þ¼
p
n
1 ¼ 4:71 [21],
which is the isovector anomalous magnetic moment.
Similarly, we obtain the nucleon axial charge, G
A
ð0Þ
g
A
, the isovector momentum fraction, A
20
ð0Þhxi
ud
,
and the moment of the polarized quark distribution,
~
A
20
ð0Þhxi
ud
. In order to find the spin and angular
momentum carried by each quark individually in the
nucleon, we need the isoscalar axial charge and the iso-
scalar one-derivative matrix elements of the vector opera-
tor. Unlike the isovector combinations, where disconnected
fermion loops vanish in the continuum limit, the isoscalar
cases receive contributions from disconnected fermion
loops. The evaluation of the disconnected contributions is
difficult due to the computational cost, but techniques are
being developed to compute them. Recent results on
nucleon form factors show that they are small or consistent
with zero [22–24]. The disconnected contribution to the
isoscalar axial charge has been computed and was found to
be nonzero, but it is an order of magnitude smaller than the
connected one [25]. Therefore, in most nucleon structure
calculations they are neglected. In this work we will as-
sume that the disconnected contributions are small, in
which case it is straightforward to evaluate the isoscalar
matrix elements taking into account only the connected
part depicted in Fig. 1. The quark contribution to the
nucleon spin is obtained using Ji’s sum rule: J
q
¼
1
2
½A
q
20
ð0ÞþB
q
20
ð0Þ. Moreover, using the axial charge for
each quark, g
q
A
, we obtain the intrinsic spin of each quark,
q
¼ g
q
A
, and via the decomposition J
q
¼
1
2
q
þ L
q
we can extract the quark orbital angular momentum L
q
.
In the present work we employ the twisted mass fermion
(TMF) action [26] and the Iwasaki improved gauge action
[27]. Twisted mass fermions provide an attractive formu-
lation of lattice QCD that allows for automatic OðaÞ
improvement, infrared regularization of small eigenvalues
and fast dynamical simulations [28]. In the computation of
GFFs the automatic OðaÞ improvement is particularly
relevant since it is achieved by tuning only one parameter
in the action, requiring no further improvements on the
operator level.
We use the twisted mass Wilson action for the light
doublet of quarks,
S
l
¼
X
x
l
ðxÞ½D
W
þ m
ð0;lÞ
þ i
5
3
l
l
ðxÞ; (2.7)
where D
W
is the Wilson Dirac operator, m
ð0;lÞ
is the un-
twisted bare quark mass, and
l
is the bare light twisted
mass. The quark fields
l
are in the so-called ‘‘twisted
basis’’ obtained from the ‘‘physical basis’’ at maximal
twist by the transformation
FIG. 1 (color online). Connected nucleon three-point function.
C. ALEXANDROU et al. PHYSICAL REVIEW D 88, 014509 (2013)
014509-2
c
¼
1
ffiffiffi
2
p
½1þi
3
5
l
and
c
¼
l
1
ffiffiffi
2
p
½1þi
3
5
: (2.8)
In addition to the light sector, we introduce a twisted heavy
mass-split doublet
h
¼ð
c
;
s
Þ for the strange and charm
quarks, described by the action
S
h
¼
X
x
h
ðxÞ½D
W
þ m
ð0;hÞ
þ i
5
1
þ
3
h
ðxÞ;
(2.9)
where m
ð0;hÞ
is the untwisted bare quark mass for the heavy
doublet,
is the bare twisted mass along the
1
direction
and
is the mass splitting in the
3
direction. The quark
mass m
ð0;hÞ
is set equal to m
ð0;lÞ
in the simulations, thus
ensuring OðaÞ improvement also in the heavy quark sector.
The chiral rotation for the heavy quarks from the twisted to
the physical basis is
c
¼
1
ffiffiffi
2
p
½1 þ i
1
5
h
and
c
¼
h
1
ffiffiffi
2
p
½1 þ i
1
5
:
(2.10)
The reader can find more details on the twisted mass
fermion action in Ref. [29]. Simulating a charm quark
may give rise to concerns regarding cutoff effects. The
observables of this work cannot be used to check for
such an effect. However, an analysis in Ref. [30] shows
that they are surprisingly small.
A. Correlation functions
The GFFs are extracted from dimensionless ratios of
correlation functions, involving two-point and three-point
functions that are defined by
Gð
~
q; t
f
t
i
Þ¼
X
~
x
f
e
ið
~
x
f
~
x
i
Þ
~
q
0
hJ
ðt
f
;
~
x
f
Þ
J
ðt
i
;
~
x
i
Þi;
(2.11)
G
1
...
n
ð
;
~
q; tÞ¼
X
~
x;
~
x
f
e
ið
~
x
~
x
i
Þ
~
q
hJ
ðt
f
;
~
x
f
Þ
O
1
...
n
ðt;
~
xÞ
J
ðt
i
;
~
x
i
Þi: (2.12)
For the insertion O
1
...
n
, we employ the vector (
c
c
),
the axial-vector (
c
5
c
), the one-derivative vector
(
c
f
1
D
2
g
c
) and the one-derivative axial-vector
(
c
5
f
1
D
2
g
c
) operators. We consider kinematics for
which the final momentum
~
p
0
¼ 0, and in our approach we
fix the time separation between sink and source, t
f
t
i
.
The projection matrices
0
and
k
are given by
0
¼
1
4
ð1 þ
0
Þ;
k
¼
0
i
5
k
: (2.13)
The proton interpolating field written in terms of the quark
fields in the twisted basis (
~
u and
~
d) at maximal twist is
given by
JðxÞ¼
1
ffiffiffi
2
p
½1 þ i
5
abc
½
~
u
a>
ðxÞC
5
~
d
b
ðxÞ
~
u
c
ðxÞ; (2.14)
where C is the charge conjugation matrix. We use Gaussian
smeared quark fields [31,32] to increase the overlap with
the proton state and decrease the overlap with excited
states. The smeared interpolating fields are given by
q
a
smear
ðt;
~
xÞ¼
X
~
y
F
ab
ð
~
x;
~
y;UðtÞÞq
b
ðt;
~
yÞ;
F¼ð1þa
G
HÞ
N
G
;
Hð
~
x;
~
y;UðtÞÞ¼
X
3
i¼1
½U
i
ðxÞ
x;y
^
{
þU
y
i
ðx
^
{Þ
x;yþ
^
{
:
(2.15)
We also apply APE smearing to the gauge fields U
enter-
ing the hopping matrix H. The parameters for the Gaussian
smearing, a
G
and N
G
, are optimized using the nucleon
ground state [33]. Different combinations of Gaussian
parameters, N
G
and a
G
, have been tested, and it was found
that combinations of N
G
and a
G
that give a root mean
square radius of about 0.5 fm are optimal for suppressing
excited states. The results of this work have been produced
with
¼ 1:95: N
G
¼ 50;a
G
¼ 4;N
APE
¼ 20;a
APE
¼ 0:5;
¼ 2:10: N
G
¼ 110;a
G
¼ 4;N
APE
¼ 50;a
APE
¼ 0:5:
As already point out, in correlators of isovector operators
the disconnected diagrams are zero up to lattice artifacts,
and can be safely neglected as we approach the continuum
limit. Thus, these correlators can be calculated by evaluat-
ing the connected diagram of Fig. 1 for which we employ
sequential inversions through the sink [34]. The creation
operator is taken at a fixed position
~
x
i
¼
~
0 (source). The
annihilation operator at a later time t
f
(sink) carries
momentum
~
p
0
¼ 0. The current couples to a quark at an
intermediate time t and carries momentum
~
q. Translation
invariance enforces
~
q ¼
~
p for our kinematics. At a fixed
sink-source time separation we obtain results for all pos-
sible momentum transfers and insertion times as well as for
any operator O
f
1
...
n
g
, with one set of sequential inver-
sions per choice of the sink. We perform separate inver-
sions for the two projection matrices
0
and
P
k
k
given in
Eq. (2.13). An alternative approach that computes the
spatial all-to-all propagator stochastically has shown to
be suitable for the evaluation of nucleon three-point
functions [35]. Within this approach one can include any
projection without needing additional inversions.
Using the two- and three-point functions of Eqs. (2.11)
and (2.12) and considering operators with up to one
derivative, we form the ratio
R
ð
;
~
q; tÞ¼
G
ð
;
~
q; tÞ
Gð
~
0;t
f
t
i
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Gð
~
p; t
f
tÞGð
~
0;t t
i
ÞGð
~
0;t
f
t
i
Þ
Gð
~
0;t
f
tÞGð
~
p; t t
i
ÞGð
~
p; t
f
t
i
Þ
s
;
(2.16)
NUCLEON FORM FACTORS AND MOMENTS OF ... PHYSICAL REVIEW D 88, 014509 (2013)
014509-3
which is optimized because it does not contain potentially
noisy two-point functions at large separations and because
correlations between its different factors reduce the statis-
tical noise. For sufficiently large separations t
f
t and
tt
i
this ratio becomes time independent (plateau region):
lim
t
f
t!1
lim
tt
i
!1
R
ð
;
~
q; tÞ¼
ð
;
~
qÞ: (2.17)
From the plateau values of the renormalized asymptotic
ratio ð
;
~
qÞ
R
¼ Zð
;
~
qÞ, the nucleon matrix elements
of all our operators can be extracted. The equations relating
ð
;
~
qÞ to the GFFs can be found in Refs. [1–3]. All
values of
~
q resulting in the same q
2
, the two choices of
projector matrices
0
and
P
k
k
given by Eq. (2.13) and the
relevant orientations , of the operators lead to an over-
constrained system of equations, which is solved in the
least-squares sense via a singular value decomposition of
the coefficient matrix. All quantities will be given in
Euclidean space, with Q
2
q
2
the Euclidean momen-
tum transfer squared. Both projectors
0
and
P
k
k
are
required to obtain all GFFs, except for the case of the local
axial-vector operator, for which the projection with
0
leads to zero. For the one-derivative vector operator, both
cases ¼ and are necessary to extract all three
GFFs, which on a lattice renormalize differently from each
other [36]. On the other hand, the one-derivative axial-
vector form factors can be extracted using only correlators
with , but we use all combinations of , in order to
increase statistics. In Fig. 2 we show representative pla-
teaus for the ratios of the local axial-vector and the one-
derivative vector operators at ¼ 1:95, using different
momenta, projectors, and indices , .
Since we use sequential inversions through the sink, we
need to fix the sink-source separation. Optimally, one
wants to keep the statistical errors on the ratio of
Eq. (2.16) as small as possible by using the smallest value
for the sink-source time separation that still ensures that the
excited state contributions are sufficiently suppressed.
Recent studies have shown that the optimal sink-source
separation is operator dependent [37,38]. For g
A
excited
state contamination was found to be small. We have also
tested different values of the sink-source time separation
[3] for the magnetic form factor and found consistent
results when the sink-source separation was about 1 fm
within our statistical accuracy. For the momentum fraction
one would need to reexamine the optimal sink-source
separation, which would require a dedicated high accuracy
study. Since in this work we are computing several observ-
ables, we will use t
f
t
i
1fm, which corresponds to the
following values:
¼ 1:95: ðt
f
t
i
Þ=a ¼ 12 ;
¼ 2:10: ðt
f
t
i
Þ=a ¼ 18 :
This choice allows us to compare with other lattice QCD
results where similar values were used.
B. Simulation details
In Table I we tabulate the input parameters of the
calculation, namely , L=a and the light quark mass a,
as well as the value of the pion mass in lattice units [29,39].
The strange and charm quark masses were fixed to
approximately reproduce the physical kaon and D-meson
masses, respectively [40]. The lattice spacing a given in
this table is determined from the nucleon mass as explained
in the following subsection, and it will be used for the
baryon observables discussed in this paper. We note that
the study of the systematic error in the scale setting using
the pion decay constant as compared to the value extracted
using the nucleon mass is currently being pursued. Since
the GFFs are dimensionless they are not affected by the
scale setting. However, a is needed to convert Q
2
to
physical units, and therefore it does affect quantities like
the anomalous magnetic moment and Dirac and Pauli radii
since these are dimensionful parameters that depend on
fitting the Q
2
dependence of the form factors.
-0.9
-0.8
-0.7
-0.6
R
k
(Γ
k
,p=(0,0,0))
-0.7
-0.6
-0.5
R
k
(
Γ
k
,p=(1,0,0))
-0.7
-0.6
-0.5
R
k
(Γ
k
,p=(0,-1,0))
-0.5
-0.4
-0.3
1 2 3 4 5 6 7 8 9 10 11 12
R
k
(Γ
k
,p=(1,0,1))
t/a
-0.15
-0.1
-0.05
0
R
00
(Γ
0
,p=(0,0,0))
0.03
0.04
0.05
R
k0
(Γ
0
,p=(0,0,0))
-0.15
-0.1
-0.05
R
00
(Γ
0
,p=(1,0,0))
0.01
0.02
0.03
0.04
1 2 3 4 5 6 7 8 9 10 11 12
R
ij
(
Γ
k
,p=(1,0,0))
t/a
FIG. 2 (color online). Ratios for the matrix elements of the
local axial-vector operator (upper) and one-derivative vector
operator (lower) for a few exemplary choices of the momentum.
The solid lines with the bands indicate the fitted plateau values
with their jackknife errors. From top to bottom the momentum
takes values
~
p ¼ð0; 0; 0Þ, (1, 0, 0), ð0; 1; 0Þ and (1, 0, 1).
C. ALEXANDROU et al. PHYSICAL REVIEW D 88, 014509 (2013)
014509-4
C. Determination of lattice spacing
For the observables discussed in this work the nucleon
mass at the physical point is the most appropriate quantity
to set the scale. The values for the nucleon mass were
computed using N
f
¼2 þ1þ1 ensembles for ¼ 1:90,
¼ 1:95 and ¼ 2:10, a range of pion masses and
volumes. To extract the mass we consider the two-point
correlators defined in Eq. (2.11) and construct the
effective mass
am
eff
N
ðtÞ¼log ðCðtÞ=Cðt 1ÞÞ
¼ am
N
þ log
1 þ
P
1
j¼1
c
j
e
j
t
1 þ
P
1
j¼1
c
j
e
j
ðt1Þ
!
t!1
am
N
(2.18)
where
j
¼ E
j
m
N
is the energy difference of the ex-
cited state j with respect to the ground state mass m
N
. Our
fitting procedure to extract m
N
is as follows: The mass is
obtained from a constant fit to m
eff
N
ðtÞ for t t
1
for which
the contamination of excited states is believed to be small.
We denote the value extracted as m
ðAÞ
N
ðt
1
Þ. A second fit to
m
eff
N
ðtÞ is performed including the first excited state for
t t
0
1
, where t
0
1
is taken to be 2a or 3a. We denote the
value for the ground state mass extracted from the fit to two
exponentials by m
ðBÞ
N
. We vary t
1
such that the ratio
jam
ðAÞ
N
ðt
1
Þam
ðBÞ
N
j
am
mean
N
; where am
mean
N
¼
am
ðAÞ
N
ðt
1
Þþam
ðBÞ
N
2
(2.19)
drops below 50% of the statistical error on m
A
N
ðt
1
Þ. The
resulting values for the nucleon mass are collected in
Table II.
In Fig. 3 we show results at three values of the lattice
spacing corresponding to ¼ 1:90, ¼ 1:95 and
¼ 2:10. As can be seen, cutoff effects are negligible
and we can therefore use continuum chiral perturbation
theory to extrapolate to the physical point using all the
lattice results.
To chirally extrapolate we use the well-established
Oðp
3
Þ result of chiral perturbation theory (PT) given by
m
N
¼ m
0
N
4c
1
m
2
3g
2
A
16f
2
m
3
: (2.20)
We perform a fit to the results at the three values given in
Table II using the Oðp
3
Þ expansion of Eq. (2.20) with fit
parameters m
0
N
, c
1
and the three lattice spacings. The
resulting fit is shown in Fig. 3 and describes well our lattice
data (
2
=d:o:f:) yielding, for the lattice spacings, the values
a
¼1:90
¼ 0:0934ð13Þð35Þ fm;
a
¼1:95
¼ 0:0820ð10Þð36Þ fm;
a
¼2:10
¼ 0:0644ð7Þð25Þ fm:
(2.21)
We would like to point out that our lattice results show a
curvature supporting the m
3
term. In order to estimate the
systematic error due to the chiral extrapolation we also
perform a fit using heavy baryon (HB) PT to Oðp
4
Þ with
explicit degrees of freedom in the so-called SSE [33].
We take the difference between the Oðp
3
Þ and Oðp
4
Þ mean
values as an estimate of the uncertainty due to the chiral
extrapolation. This error is given in the second set of
parentheses in Eqs. (2.21), and it is about twice the statis-
tical error. In order to assess discretization errors we per-
form a fit to Oðp
3
Þ at each value of separately. We find
a ¼ 0:0920ð21Þ, 0.0818(16), 0.0655(12) fm at ¼ 1:90,
1.95, 2.10, respectively. These values are fully consistent
with those obtained in Eq. (2.21) indicating that discretiza-
tion effects are small, confirming a posteriori the validity
TABLE I. Input parameters ð; L; aÞ of our lattice calcula-
tion with the corresponding lattice spacing a, determined from
the nucleon mass, and pion mass am
in lattice units.
¼ 1:95, a ¼ 0:0820ð10Þ fm, r
0
=a ¼ 5:66ð3Þ
32
3
64, L ¼ 2:6fm a 0.0055
Number of configs. 950
am
0.15518(21)(33)
Lm
4.97
¼ 2:10, a ¼ 0:0644ð7Þ fm, r
0
=a ¼ 7:61ð6Þ
48
3
96, L ¼ 3:1fm a 0.0015
Number of configs. 900
am
0.06975(20)
Lm
3.35
TABLE II. Values of the nucleon mass and the associated
statistical error.
aVolume am
Statistics am
N
1.90
0.003
32
3
64 0.124
740 0.524(9)
1.90
0.004
20
3
48 0.149
617 0.550(19)
1.90
0.004
24
3
48 0.145
2092 0.541(8)
1.90
0.004
32
3
64 0.141
1556 0.519(11)
1.90
0.005
32
3
64 0.158
387 0.542(6)
1.90
0.006
24
3
48 0.173
1916 0.572(5)
1.90
0.008
24
3
48 0.199
1796 0.590(5)
1.90
0.010
24
3
48 0.223
2004 0.621(4)
1.95
0.0025
32
3
64 0.107
2892 0.447(6)
1.95
0.0035
32
3
64 0.126
4204 0.478(5)
1.95
0.0055
32
3
64 0.155
18576 0.503(2)
1.95
0.0075
32
3
64 0.180
2084 0.533(4)
1.95
0.0085
24
3
48 0.194
937 0.542(5)
2.10
0.0015
48
3
96 0.070
2424 0.338(4)
2.10
0.0020
48
3
96 0.080
744 0.351(7)
2.10
0.0030
48
3
96 0.098
226 0.362(7)
2.10
0.0045
32
3
64 0.121
1905 0.394(3)
NUCLEON FORM FACTORS AND MOMENTS OF ... PHYSICAL REVIEW D 88, 014509 (2013)
014509-5
![FIG. 16 (color online). The Q2 dependence of GEðQ2Þ. We show results for Nf ¼ 2þ 1þ 1 at m ¼ 373 MeV (filled blue squares) and Nf ¼ 2 [1] at m ¼ 298 MeV (filled red circles) TMF data on a lattice with spatial length L ¼ 2:8 fm and similar lattice spacing. We also show results with Nf ¼ 2þ 1 DWF at m ¼ 297 MeV, L ¼ 2:7 fm (crosses) [4], with a hybrid action with Nf ¼ 2þ 1 staggered sea and DWF atm ¼ 293 MeV and L ¼ 2:5 fm (open orange circles) [51], and Nf ¼ 2 clover at m ¼ 290 MeV and L ¼ 3:4 fm (asterisks) [52]. The solid line is Kelly’s parametrization of the experimental data [55] from a number of experiments as given in Ref. [55].](/figures/fig-16-color-online-the-q2-dependence-of-gedq2th-we-show-5l9rkrun.webp)
![FIG. 14 (color online). Comparison of the Nf ¼ 2þ 1þ 1 twisted mass data on GEðQ2Þ (upper) and GMðQ2Þ (lower) for the two different pion masses considered. The solid lines are Kelly’s parametrization of the experimental data [55], whereas the dashed lines are dipole fits to the lattice QCD data.](/figures/fig-14-color-online-comparison-of-the-nf-1-4-2th-1th-1-22zg9q7n.webp)
