delta-baryon electromagnetic form factors in lattice QCD
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Alexandrou, C. et al. "delta-baryon electromagnetic form factors
in lattice QCD." Physical Review D 79.1 (2009): 014507.
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http://dx.doi.org/10.1103/PhysRevD.79.014507
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American Physical Society
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PHYSICAL REVIEW D 79, 014507 (2009)
-baryon electromagnetic form factors in lattice QCD
C. Alexandrou,1 T. Korzec,1 G. Koutsou,1 Th. Leontiou,1 C. Lorcé,2 J. W. Negele,3 V. Pascalutsa,2
A. Tsapalis,4 and M. Vanderhaeghen2
1
Department of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
Institut für Kernphysik, Johannes Gutenberg-Universität, D-55099 Mainz, Germany
3
Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
4
Institute of Accelerating Systems and Applications, University of Athens P.O. Box 17214, GR-10024, Athens, Greece
(Received 22 October 2008; published 21 January 2009)
2
We develop techniques to calculate the four electromagnetic form factors using lattice QCD, with
particular emphasis on the subdominant electric quadrupole form factor that probes deformation of the .
Results are presented for pion masses down to approximately 350 MeV for three cases: quenched QCD,
two flavors of dynamical Wilson quarks, and three flavors of quarks described by a mixed action
combining domain-wall valence quarks and dynamical staggered sea quarks. The magnetic moment of
the is compared with chiral effective field theory calculations and the charge density distributions are
discussed.
DOI: 10.1103/PhysRevD.79.014507
PACS numbers: 11.15.Ha, 12.38.Aw, 12.38.Gc
I. INTRODUCTION
Lattice quantum chromodynamics (QCD) provides a
well-defined framework to directly calculate hadron form
factors from the fundamental theory of strong interactions.
Form factors characterize the internal structure of hadrons,
including their magnetic moment, their size, and their
charge density distribution. Since the ð1232Þ decays
strongly, experiments [1,2] to measure its form factors
are harder and yield less precise results than for nucleons
[3,4]. In this work, we compute form factors using lattice
QCD more accurately than can be currently obtained from
experiment.
A primary motivation for this work is to understand the
role of deformation in baryon structure: whether any of the
low-lying baryons have deformed intrinsic states and if so,
why. Thus, a major achievement of this work is the development of lattice methods with sufficient precision to
show, for the first time, that the electric quadrupole form
factor is nonzero and hence the has a nonvanishing
quadrupole moment and an associated deformed shape.
Unlike the , the spin-1=2 nucleon cannot have a quadrupole moment, so the experiment of choice to explore its
deformation has been measurement of the nucleon to electric and Coulomb quadrupole transition form factors.
Major experiments [5–7] have shown that these transition
form factors are indeed nonzero, confirming the presence
of deformation in either the nucleon , or both [8,9], and
lattice QCD yields comparable nonzero results [10,11].
Our new calculation of the quadrupole form factor,
coupled with the nucleon to transition form factors,
should in turn shed light on the deformation of the nucleon.
In order to evaluate the electromagnetic (EM) form
factors to the required accuracy, we isolate the two dominant form factors and the subdominant electric quadrupole
form factor. This is particularly crucial for the latter since
1550-7998= 2009=79(1)=014507(5)
without the construction of an optimized source for the
sequential propagator it can not be extracted with the
required precision. We note that this increases the computational cost since additional inversions are needed. Our
techniques are first tested in quenched QCD [12]. We then
calculate form factors using two degenerate flavors of
dynamical Wilson fermions, denoted by NF ¼ 2, with
pion masses in the range of 700 MeV to 380 MeV
[13,14]. Finally, we use a mixed action with chirally symmetric domain-wall valence quarks and staggered sea
quarks with two degenerate light flavors and one strange
flavor [15], denoted by NF ¼ 2 þ 1, at a pion mass of
353 MeV. Using the results obtained with dynamical
quarks, we extrapolate the magnetic moment to the physical point. We extract the quark charge distributions in the
, and discuss their quadrupole moment.
II. LATTICE EVALUATION
The matrix element of the EM current j
EM ,
hðpf ; sf Þjj
jðp
;
s
Þi,
can
be
parametrized
in
terms
of
i i
EM
four multipole form factors that depend only on the momentum transfer q2 Q2 ¼ ðpf pi Þ2 [16] The decomposition for the on shell matrix element is
given by
ðpf ; sf ÞO u ðpi ; si Þ
hðpf ; sf Þjj
EM jðpi ; si Þi ¼ Au
a ðq2 Þ ðpf þ p
Þ
O ¼ g a1 ðq2 Þ þ 2
i
2m
2
q q
2 Þ þ c2 ðq Þ ðp þ p Þ ;
ðq
(1)
c
1
i
f
2m
4m2
where a1 ðq2 Þ, a2 ðq2 Þ, c1 ðq2 Þ, and c2 ðq2 Þ are known linear
combinations of the electric charge form factor GE0 ðq2 Þ,
the magnetic dipole form factor GM1 ðq2 Þ, the electric
014507-1
Ó 2009 The American Physical Society
PHYSICAL REVIEW D 79, 014507 (2009)
C. ALEXANDROU et al.
2
quadrupole form factor GE2 ðq Þ, and the magnetic octupole form factor GM3 ðq2 Þ [17], and A is a known factor
depending on the normalization of hadron states. These
form factors can be extracted from correlation functions
calculated in lattice QCD [17]. We calculate in Euclidean
time the two- and three-point correlation functions in a
frame where the final state is at rest:
~ ¼
Gðt; qÞ
3
XX
eix~f q~ 4 hJj ðxf ÞJj ð0Þi
x~ f j¼1
X
~ ¼ eix~ q~ hJ ðxf Þj ðxÞJ ð0Þi;
G ð ; t; qÞ
(2)
x~ f x~
where j is the electromagnetic current on the lattice, J
and J are the þ interpolating fields constructed from
smeared quarks [12], 4 ¼ 14 ð1 þ 4 Þ, and k ¼
i4 5 k . The form factors can then be extracted from
ratios of three- and two-point functions in which unknown
normalization constants and the leading time dependence
cancel
v
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
u
ð; t; qÞ
~
~
Gðtf t; p~ i ÞGðt; 0ÞGðt
~
G
u
f ; 0Þ
t
:
R ¼ ~
~
Gðtf t; 0ÞGðt;
p~ i ÞGðtf ; p~ i Þ
Gðtf ; 0Þ
(3)
For sufficiently large tf t and t ti , this ratio exhibits a
~ ! ð; qÞ,
~ from which the form factors
plateau Rð; t; qÞ
are extracted, and we use the particular combinations
3
X
~ ¼ K1 GE0 ðQ2 Þ þ K2 GE2 ðQ2 Þ;
k k ð4 ; qÞ
The connected part of each combination of three-point
functions can be calculated efficiently using the method
of sequential inversions [18]. These yield the isovector
form factors. At present, it is not yet computationally
feasible to calculate the contributions arising from disconnected diagrams. A calculation of disconnected contributions in the case of the electromagnetic form factors have
shown that these are consistent with zero [19]. We note that
these disconnected contributions are particularly hard to
calculate not just because they require the all-to-all propagators but also because they are noise dominated [20]. We
expect that, like in the case of nucleon electromagnetic
form factors, the disconnected contributions to the electromagnetic form factors are small. The known kinematical
coefficients K1 , K2 , K3 , K4 are functions of the mass and
~ The combinations above are
energy as well as of and q.
chosen such that all possible directions of and q~ contribute symmetrically to the form factors at a given Q2
[21]. The over-constrained system of Eqs. (4)–(6) is solved
by a least-squares analysis, and GE2 ðQ2 Þ can also be isolated separately from Eq. (6).
The details of the simulations are summarized in Table I.
In each case, the separation between the final and initial
time is tf ti * 1 fm and Gaussian smearing is applied to
both source and sink to produce adequate plateaus by
suppressing contamination from higher states having the
quantum numbers of the ð1232Þ. For the mixed-action
calculation, the domain-wall valence quark mass was
chosen to reproduce the lightest pion mass obtained using
NF ¼ 2 þ 1 improved staggered quarks [21,23].
(4)
III. RESULTS
k¼1
3
X
jkl j
k ð
4
2
~ ¼ K3 GM1 ðQ Þ;
; qÞ
(5)
~ ¼ K4 GE2 ðQ2 Þ:
jkl j 4 k ðj ; qÞ
(6)
j;k;l¼1
3
X
j;k;l¼1
The results for GE0 ðQ2 Þ are shown in Fig. 1 as a function
of Q2 at the lightest pion mass for each of the three actions.
For Wilson fermions, we use the conserved lattice current
requiring no renormalization. The local current is used for
the mixed action, and the renormalization constant ZV ¼
1:0992ð32Þ is determined by the condition that GE0 ð0Þ
equals the charge of the in units of e. As can be seen,
pffiffiffiffiffiffiffiffi
TABLE I. Lattice parameters and results. Nconf denotes the number of lattice configurations, hr2 i gives the charge radius, þ is
þ
the þ magnetic moment in nuclear magnetons and Q
3=2 is the quadrupole moment.
pffiffiffiffiffiffiffiffi
Nconf
m [GeV]
m [GeV]
hr2 i [fm]
þ [N ]
Q
3=2
200
200
200
0.563(4)
0.490(4)
0.411(4)
185
157
200
0.691(8)
0.509(8)
0.384(8)
300
0.353(2)
Quenched Wilson, 323 64, a ¼ 0:092 fm
1.470(15)
0.6147(66)
1.720(42)
1.425(16)
0.6329(76)
1.763(51)
1.382(19)
0.6516(87)
1.811(69)
NF ¼ 2 Wilson, 243 40 (32 for lightest pion), a ¼ 0:077 fm
1.687(15)
0.5279(61)
1.462(45)
1.559(19)
0.594(10)
1.642(81)
1.395(18)
0.611(17)
1.58(11)
NF ¼ 2 þ 1, Mixed action, 283 64, a ¼ 0:124 fm [22]
1.533(27)
0.641(22)
1.91(16)
014507-2
0.96(12)
0.91(15)
0.83(21)
0.80(21)
0.41(45)
0.46(35)
0.74(68)
-BARYON ELECTROMAGNETIC FORM FACTORS IN . . .
quenched Wilson, mπ = 410 MeV
hybrid, m = 353 MeV
π
1
dynamical Wilson, mπ = 384 MeV
GE0
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2
2.5
2
Q in GeV
FIG. 1 (color online). The electric charge form factor versus
Q2 . The green (red) line and error band show a dipole fit to the
mixed-action (quenched) results.
all three calculations yield consistent results. The momentum dependence of the charge form factor is described well
2
by a dipole form GE0 ðQ2 Þ ¼ 1=ð1 þ Q2 Þ2 . To compare the
E0
slopes at Q2 ¼ 0, we follow convention and show in
Table I the so-called ‘‘rms radius’’ hr2 i ¼
d
2
6 dQ
2 GE0 ðQ ÞjQ2 ¼0 [17].
The momentum dependence of GM1 ðQ2 Þ is displayed in
Fig. 2. In order to extract the magnetic moment an extrapolation to zero momentum transfer is necessary. Both an
2
2
exponential form, GM1 eQ =M1 , and a dipole describe the
4
PHYSICAL REVIEW D 79, 014507 (2009)
2
Q dependence well, and we adopt the exponential form
because of its faster decay at large Q2 , in accord with
perturbative arguments. The larger spatial volume for the
quenched and mixed-action cases yields smaller and more
densely spaced values of the lattice momenta and correspondingly more precise determination of the form factor
than for the smaller volume used with dynamical Wilson
fermions. In Fig. 2, we show the best exponential fit and
error band for the mixed action and quenched results. As
can be seen, results in the quenched theory and for NF ¼ 2
Wilson fermions are within the error band. The magnetic
moment in natural units is given by ¼
GM1 ð0Þe=ð2m Þ, where m is the mass measured on
the lattice and GM1 ð0Þ is from the exponential fits. In
Table I, we give the values of the þ magnetic moment
in nuclear magnetons e=ð2mN Þ, with mN the physical
nucleon mass. The magnetic moments of the þ and
þþ are accessible to experiments [1,2], which presently
suffer from large uncertainties.
The magnetic moment as function of m2 is shown in
Fig. 3, together with a comparison to a chiral effective field
theory (ChEFT) result [24]. The ChEFT result has one free
parameter (a low-energy constant) that has been fitted to
lattice data, shown by the central line. We also estimate the
uncertainty of the ChEFT expansion (expansion in pion
mass) by the error band in Fig. 3. The uncertainty of the
ChEFT calculation vanishes in the chiral limit because in
this limit one simply has the value of the low-energy
constant, which lies within the broad experimental error
band
þ ¼ 2:7þ1:0
1:3 ðstatÞ 1:5ðsystÞ 3:0ðtheoryÞN
[1]. In this work, we do not consider the uncertainty in
the fit value of the low-energy constant due to the lattice
errors, as the calculations are still performed for pion
masses where the is stable (on the right side of the
kink). A calculation for pion mass values where the quenched Wilson, m = 410 MeV
π
3.5
hybrid, m = 353 MeV
π
dynamical Wilson, mπ = 384 MeV
3
GM1
2.5
2
1.5
1
0.5
0
0
0.5
1
2
1.5
2
2.5
2
Q in GeV
FIG. 2 (color online). The magnetic dipole form factor. The
green (red) line and error band show an exponential fit to the
mixed-action (quenched) results.
FIG. 3 (color online). The magnetic dipole moment in nuclear
magnetons. The value at the physical pion mass (filled square) is
shown with statistical and systematic errors [1]. The solid and
dashed curves show the results of ChEFT with the theoretical
error estimate [24].
014507-3
PHYSICAL REVIEW D 79, 014507 (2009)
C. ALEXANDROU et al.
becomes unstable will be a challenge for future calculations. The moments using an approach similar to ours are
calculated only in the quenched approximation [17,25,26].
Our magnetic moment results agree with recent background field calculations using dynamical improved
Wilson fermions [27], which supersede previous quenched
background field results [28]. The spatial length Ls of our
lattices satisfies Ls m > 4 in all cases except at the lightest
pion mass with NF ¼ 2 Wilson fermions, for which
Ls m ¼ 3:6. For that point, the magnetic moment falls
slightly below the error band, consistent with the fact
that Ref. [27] shows that finite volume effects decrease
the magnetic moment.
The electric quadrupole form factor is particularly interesting because it can be related to the shape of a hadron,
and lattice calculations for each of the three actions are
shown in Fig. 4 with exponential fits for the quenched and
mixed-action cases. Just as the electric form factor for a
spin-1=2 nucleon can be expressed precisely as the transverse Fourier transform of the transverse quark charge
density in the infinite momentum frame [29], a proper
field-theoretic interpretation of the shape of the ð1232Þ
can be obtained by considering the quark transverse charge
densities in this frame [30–32]. With respect to the direction of the average baryon momentum P, the transverse
charge density in a spin-3=2 state with transverse polarization s? is defined as
~
Ts? ðbÞ Z d2 q~ ?
~ 1
eiq~ ? b þ
2
2P
ð2Þ
~
~?
q
þ þ; q
;
s
Pþ ; ? ; s? ð0Þ
;
P
J
2
2 ?
Q2 , J þ J 0 þ J 3 , and b~ specifies the quark position in the
xy plane relative to the center of mass. Choosing the transverse spin vector along the x axis, the quadrupole
moment of this two-dimensional charge distribution is
defined as [21]:
Z
~
~ 2x b2y Þ
ðbÞ:
Q
e
d2 bðb
(8)
s?
Ts?
In terms of the EM form factors [21],
1
e
Q
3=2 ¼ f2½GM1 ð0Þ 3e þ ½GE2 ð0Þ þ 3e g 2 : (9)
2
m
The term proportional to ½GM1 ð0Þ 3e is an electric
quadrupole moment induced in the moving frame due to
the magnetic dipole moment. For a spin-3=2 particle without internal structure, GM1 ð0Þ ¼ 3e , GE2 ð0Þ ¼ 3e
[21,33], and the quadrupole moment of the transverse
charge density vanishes. Hence, Q
s? , and thus the deformation of the two-dimensional transverse charge density, is
only sensitive to the anomalous parts of the spin-3=2
magnetic dipole and electric quadrupole moments, and
vanishes for a particle without internal structure. The
analogous property holds for a spin-1 particle [32], indicating the generality of this description in terms of transverse densities.
þ
Figure 5 shows the transverse density Ts? for a with
transverse spin s? ¼ þ3=2 calculated from the fit to the
quenched Wilson lattice results for the form factors
(which has the smallest statistical errors of the three cal-
(7)
where the photon transverse momentum q~ ? satisfies q~ 2? ¼
0
GE2
−1
−2
quenched Wilson, m = 410 MeV
−3
π
hybrid, mπ = 353 MeV
−4
dynamical Wilson, m = 384 MeV
π
0
0.5
1
1.5
Q2 in GeV2
FIG. 4 (color online). The electric quadrupole form factor. The
notation is the same as that in Fig. 1. The value of GE2 , in units of
e=ð2m2 Þ, at Q2 ¼ 0 are 0:810 0:291 for the quenched
calculation, 0:87 0:67 for the dynamical Wilson case, and
2:06þ1:27
2:35 for the hybrid calculation.
FIG. 5 (color online). Quark transverse charge density in a þ
polarized along the x axis, with s? ¼ þ3=2. The light (dark)
regions correspond with the largest (smallest) values of the
density.
014507-4
-BARYON ELECTROMAGNETIC FORM FACTORS IN . . .
þ
culations). It is seen that the quark charge density is
elongated along the axis of the spin (prolate). This prolate
deformation is robust in the sense that the values for Q
3=2
obtained from Eq. (9) and given in Table I are all consistently positive. The connection between the results on
deformation in the current formulation and previous ones
will be discussed in a forthcoming publication [21]. In the
case of the magnetic octupole form factor [21], which is
related to the magnetic octupole moment O ¼
GM3 ð0Þe=2m3 , our statistics are insufficient to distinguish
the result from zero.
IV. CONCLUSIONS
In summary, a formalism for the accurate evaluation of
the electromagnetic form factors as functions of q2 has
been developed and used in quenched QCD and full QCD
with two and 2 þ 1 flavors. The charge radius and magnetic dipole moment were determined as a function of m2
and the dipole moment was compared with the result of
chiral effective theory. The electric quadrupole form factor
was evaluated for the first time with sufficient accuracy to
distinguish it from zero. The lattice calculations show that
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ACKNOWLEDGMENTS
A. T. acknowledges support by the University of
Cyprus. This work is supported in part by the Cyprus
Research Promotion Foundation (RPF) under Contract
No. ENEK/ENIX/0505-39, the EU Integrated
Infrastructure Initiative Hadron Physics (I3HP) under
Contract No. RII3-CT-2004-506078, and the U.S.
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