Delta-baryon electromagnetic form factors in lattice QCD

Abstract

We develop techniques to calculate the four Delta electromagnetic form factors using lattice QCD, with particular emphasis on the sub-dominant electric quadrupole form factor that probes deformation of the Delta. 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 Delta is chirally extrapolated to the physical point and the Delta charge density distributions are discussed. Comment: 4 pages, 5 figures

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arXiv:0810.3976v1 [hep-lat] 22 Oct 2008
∆-baryon electromagnetic form factors in lattice QCD
C. Alexandrou(a), T. Korzec(a), G. Koutsou(a), Th. Leontiou(a), C. Lorc´ e(b),
J. W. Negele(c), V. Pascalutsa(b), A. Tsapalis(d), M. Vanderhaeghen(b)
(a)Department of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
(c)Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.
(b)Institut f¨ ur Kernphysik, Johannes Gutenberg-Universit¨ at, D-55099 Mainz, Germany
(d)Institute of Accelerating Systems and Applications, University of Athens, Athens
We develop techniques to calculate the four ∆ electromagnetic form factors using lattice QCD,
with particular emphasis on the sub-dominant electric quadrupole form factor that probes defor-
mation 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 chirally extrapolated to the physical point and the ∆ charge density
distributions are discussed.
PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.Aw, 12.38.-t, 14.70.Dj
Lattice Quantum Chromodynamics (QCD) provides a
well-defined framework to directly calculate hadron form
factors from the fundamental theory of strong interac-
tions. Form factors characterize the internal structure
of hadrons, including their magnetic moment, their size,
and their charge density distribution. Since the ∆(1232)
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 suffi-
cient precision to show, for the first time, that the elec-
tric quadrupole form factor is non-zero and hence the ∆
has a non-vanishing quadrupole moment and an associ-
ated 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, 6, 7] have
shown that these transition form factors are indeed non-
zero, confirming the presence of deformation in either the
nucleon, ∆, or both [8, 9], and lattice QCD yields com-
parable non-zero 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 sub-dominant electric
quadrupole form factor. This is particularly crucial for
the latter since it can be extracted with greater precision,
although it increases the computational cost. Our tech-
niques are first tested in quenched QCD [12]. We then
calculate form factors using two degenerate flavors of dy-
namical Wilson fermions, denoted by NF= 2, with pion
masses in the range of 700 MeV to 380 MeV [13, 14]. Fi-
nally, we use a mixed action with chirally symmetric do-
main wall valence quarks and staggered sea quarks with
two degenerate light flavors and one strange flavor [15],
denoted by NF = 2 + 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.
The∆matrixelement
where jµ
EMis the electromagnetic current, can be
parametrized in terms of four multipole form factors that
depend only on the momentum transfer q2≡ −Q2=
(pf−pi)2[16]. The decomposition for the on shell γ∗∆∆
matrix element is given by
?∆(pf,sf)|jµ
EM|∆(pi,si)?,
?∆(pf,sf)|jµ
EM|∆(pi,si)? = A ¯ uσ(pf,sf)Oσµτuτ(pi,si)
?
−qσqτ
4m2
∆
2m∆
Oσµτ= −gστ
a1(q2)γµ+a2(q2)
2m∆
?
pµ
f+ pµ
i
??
?
c1(q2)γµ+c2(q2)
?
pµ
f+ pµ
i
??
,(1)
where a1(q2), a2(q2), c1(q2), and c2(q2) are known linear
combinations of the electric charge form factor GE0(q2),
the magnetic dipole form factor GM1(q2), the electric
quadrupole form factor GE2(q2), and the magnetic oc-
tupole form factor GM3(q2) [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:
G(t,? q) =
?
?
? xf
3
?
ei? x·? qΓν
j=1
e−i? xf·? qΓ4
αβ?Jjβ(xf)Jjα(0)?
Gµ
σ τ(Γν,t,? q) =
? xf? x
αβ?Jσβ(xf)jµ(x)Jτα(0)?, (2)

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where jµis the electromagnetic current on the lattice, J
and J are the ∆+interpolating fields constructed from
smeared quarks [12], Γ4=1
The form factors can then be extracted from ratios of
three- and two-point functions in which unknown nor-
malization constants and the leading time dependence
cancel
4(1+γ4), and Γk= iΓ4γ5γk.
Rµ
σ τ=Gµ
σ τ(Γ,t,? q)
G(tf,?0)
?
G(tf− t,? pi)G(t,?0)G(tf,?0)
G(tf− t,?0)G(t,? pi)G(tf,? pi).
(3)
For sufficiently large tf−t and t−ti, this ratio exhibits a
plateau R(Γ,t,? q) → Π(Γ,? q), from which the form factors
are extracted, and we use the particular combinations
3
?
k=1
Πµ
k k(Γ4,? q) = K1GE0(Q2) + K2GE2(Q2) (4)
3
?
3
?
j,k,l=1
ǫjklΠµ
j k(Γ4,? q) = K3GM1(Q2) (5)
j,k,l
ǫjklΠ4
j k(Γj,? q) = K4GE2(Q2).(6)
The connected part of each combination of three-point
functions can be calculated efficiently using the method
of sequential inversions [18]. At present, it is not yet
computationally feasible to calculate the small correc-
tions arising from disconnected diagrams. The known
kinematical coefficients K1,K2,K3,K4 are functions of
the ∆ mass and energy as well as of µ and ? q. The combi-
nations above are chosen such that all possible directions
of µ and ? q contribute symmetrically to the form factors at
a given Q2[19]. The over-constrained system of Eqs. (4-
6) is solved by a least-squares analysis, and GE2(Q2) can
also be isolated separately from Eq. (6).
The details of the simulations are summarized in Ta-
ble I.In each case, the separation between the final
and initial time is tf− ti ? 1fm and Gaussian smear-
ing is applied to both source and sink to produce ade-
quate plateaus by suppressing contamination from higher
states having the quantum numbers of the ∆(1232). For
the mixed-action calculation, the domain-wall valence
quark mass was chosen to reproduce the lightest pion
mass obtained using NF = 2 + 1 improved staggered
quarks [19, 20].
The results for GE0(Q2) are shown in Fig. 1 as a func-
tion of Q2at the lightest pion mass for each of the three
actions. For Wilson fermions, we use the conserved lat-
tice current requiring no renormalization. The local cur-
rent is used for the mixed action, and the renormaliza-
tion constant, ZV = 1.0992(32), is determined by the
condition that GE0(0) equals the charge of the ∆ in
units of e. As can be seen, all three calculations yield
consistent results. The momentum dependence of the
charge form factor is described well by a dipole form
?
GE0(Q2) = 1/ 1 +
Q2
Λ2
E0
?2
. To compare the slopes at
TABLE I: Lattice parameters and results. Nconf denotes the
number of lattice configurations,
dius, µ∆+ is the ∆+magnetic moment in nuclear magnetons
and Q∆
3
p?r2? gives the charge ra-
2is the ∆+quadrupole moment.
Nconf mπ [GeV] m∆ [GeV]
Quenched Wilson, 323× 64, a = 0.092 fm
2000.563(4)1.470(15) 0.6147(66) 1.720(42) 0.96(12)
2000.490(4)1.425(16) 0.6329(76) 1.763(51) 0.91(15)
2000.411(4)1.382(19) 0.6516(87) 1.811(69) 0.83(21)
NF = 2 Wilson, 243× 40(32 for lightest pion), a = 0.077 fm
185 0.691(8)1.687(15) 0.5279(61) 1.462(45) 0.80(21)
157 0.509(8)1.559(19)0.594(10) 1.642(81) 0.41(45)
2000.384(8)1.395(18) 0.611(17)
NF = 2 + 1, Mixed action, 283× 64, a = 0.124 fm [21]
3000.353(2)1.533(27)0.641(22)
p?r2? [fm] µ∆+ [µN]Q∆
3
2
1.58(11)0.46(35)
1.91(16)0.74(68)
00.511.52 2.5
0
0.2
0.4
0.6
0.8
1
Q2 in GeV2
GE0


quenched Wilson, mπ = 410 MeV
hybrid, mπ = 353 MeV
dynamical Wilson, mπ = 384 MeV
FIG. 1: The electric charge form factor versus Q2. The green
(red) line and error band show a dipole fit to the mixed action
(quenched ) results.
Q2= 0, we follow convention and show in table I the so-
called “rms radius” [17]?r2?= −6
The momentum dependence of GM1(Q2) is displayed
in Fig. 2. To extract the magnetic moment, an extrapo-
lation to zero momentum transfer is necessary. Both an
exponential form, GM1e−Q2/Λ2
the Q2-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 dy-
namical 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 = 2 Wilson fermions are within the
error band. The magnetic moment in natural units is
given by µ∆ = GM1(0)e/(2m∆), where m∆ is the ∆
mass measured on the lattice and GM1(0) is from the
exponential fits. In Table I we give the values of the
∆+magnetic moment in nuclear magnetons e/(2MN),
with MN the physical nucleon mass.
d
dQ2GE0(Q2)
???
Q2=0.
M1, and a dipole describe
The magnetic

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3
00.511.522.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Q2 in GeV2
GM1


quenched Wilson, mπ = 410 MeV
hybrid, mπ = 353 MeV
dynamical Wilson, mπ = 384 MeV
FIG. 2: The magnetic dipole form factor. The green (red) line
and error band show an exponential fit to the mixed action
(quenched )results.
FIG. 3: The magnetic dipole moment in nuclear magnetons.
The value at the physical pion mass (filled square) is shown
with statistical and systematic errors [1].
dashed curves show the chiral extrapolation and theoretical
error estimate [22] .
The solid and
moments of the ∆+and ∆++are accessible to experi-
ments [1, 2], which presently suffer from large uncertain-
ties. The magnetic moment as a function of m2
in Fig. 3, together with a chiral extrapolation to the phys-
ical point [22], which lies within the broad error band
µ∆+ = 2.7+1.0
The ∆ moments using an approachsimilar to ours are cal-
culated only in the quenched approximation [17, 23, 24].
Our magnetic moment results agree with recent back-
ground field calculations using dynamical improved Wil-
son fermions [25], which supersede previous quenched
background field results [26]. The spatial length Ls of
our lattices satisfies Lsmπ> 4 in all cases except at the
lightest pion mass with NF = 2 Wilson fermions, for
which Lsmπ = 3.6. For that point, the magnetic mo-
ment falls slightly below the error band, consistent with
the fact that Ref. [25] shows that finite volume effects
decrease the magnetic moment.
The electric quadrupole form factor is particularly in-
teresting because it can be related to the shape of a
hadron, and lattice calculations for each of the three
πis shown
−1.3(stat.) ± 1.5(syst.) ± 3.0(theory)µN [1].
00.51 1.5
−4
−3
−2
−1
0
Q2 in GeV2
GE2


quenched Wilson, mπ = 410 MeV
hybrid, mπ = 353 MeV
dynamical Wilson, mπ = 384 MeV
FIG. 4: The electric quadrupole form factor. The notation is
the same as that in Fig. 1.
actions are shown in Fig. 4 with exponential fits for
the quenched and mixed action cases.
electric form factor for a spin 1/2 nucleon can be ex-
pressed precisely as the transverse Fourier transform of
the transverse quark charge density in the infinite mo-
mentum frame [27], a proper field-theoretic interpreta-
tion of the shape of the ∆(1232) can be obtained by
considering the quark transverse charge densities in this
frame [28, 29, 30]. With respect to the direction of the
average baryon momentum P, the transverse charge den-
sity in a spin-3/2 state with transverse polarization s⊥is
defined as :
Just as the
ρ∆
T s⊥(?b) ≡
?
×?P+,? q⊥
d2? q⊥
(2π)2e−i? q⊥·?b
2,s⊥|J+(0)|P+,−? q⊥
1
2P+
2,s⊥?, (7)
where the photon transverse momentum ? q⊥satisfies ? q2
Q2, J+≡ J0+ J3, and?b specifies the quark position in
the xy-plane relative to the ∆ center of mass. Choos-
ing the ∆ transverse spin vector along the x-axis, the
quadrupole moment of this two-dimensional charge dis-
tribution is defined as [19]:
⊥=
Q∆
s⊥≡ e
?
d2?b(b2
x− b2
y)ρ∆
T s⊥(?b). (8)
In terms of the ∆ EM form factors [19] ,
Q∆
3
2=1
2{2[GM1(0) − 3e∆] + [GE2(0) + 3e∆]}
e
M2
∆
. (9)
The term proportional to [GM1(0) − 3e∆] is an electric
quadrupole moment induced in the moving frame due
to the magnetic dipole moment. For a spin-3/2 parti-
cle without internal structure, GM1(0) = 3e∆, GE2(0) =
−3e∆[19, 31], and the quadrupole moment of the trans-
verse charge density vanishes. Hence Q∆
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 mo-
ments, and vanishes for a particle without internal struc-
s⊥, and thus the

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4
ture. The analogous property holds for a spin-1 parti-
cle [30], indicating the generality of this description in
terms of transverse densities.
FIG. 5: Quark transverse charge density in a ∆+polarized
along the x-axis, with s⊥ = +3/2. The light (dark) regions
correspond with the largest (smallest) values of the density.
Fig. 5 shows the transverse density ρ∆
with transverse spin s⊥= +3/2 calculated from the fit
to the quenched Wilson lattice results for the ∆ form
factors (which has the smallest statistical errors of the
three calculations). 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
are all consistently positive.
In the case of the magnetic octupole form factor [19],
T s⊥for a ∆+
2obtained from Eq. (9) and given in Table I
which is related to the magnetic octupole moment O∆=
GM3(0)e/2m3
guish the result from zero.
∆, our statistics are insufficient to distin-
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 NF = 2 and 2+1 flavors.
dius and magnetic dipole moment were determined as a
function of m2
πand the dipole moment was chirally ex-
trapolated to the physical point. The electric quadrupole
form factor was evaluated for the first time with sufficient
accuracy to distinguish it from zero. The lattice calcula-
tions show that the quark density in a ∆+of transverse
spin projection +3/2 is elongated along the spin axis.
Acknowledgments
The charge ra-
This work is supported in part by the Cyprus Re-
search Promotion Foundation (RPF) under contract
ΠENEK/ENIΣX/0505-39, the EU Integrated Infrastruc-
ture Initiative Hadron Physics (I3HP) under contract
RII3-CT-2004-506078 and the U.S. Department of En-
ergy (D.O.E.) Office of Nuclear Physics under contracts
DE-FG02-94ER40818 and DE-FG02-04ER41302. This
research used computational resources provided by RFP
under contract EPYAN/0506/08, the National Energy
Research Scientific Computing Center supported by the
Office of Science of the U.S. Department of Energy under
Contract DE-AC03-76SF00098 and the MIT Blue Gene
computer under grant DE-FG02-05ER25681. Dynami-
cal staggered quark configurations and forward domain
wall quark propagators were provided by the MILC and
LHPC collaborations respectively.
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