Quark transverse charge densities in the $Δ(1232)$ from lattice QCD

0.77 MB·English

by  Constantia Alexandrou

Quark transverse charge densities in the ∆(1232) from
lattice QCD
Constantia Alexandroua, Tomasz Korzeca, Giannis Koutsoua, C´edric Lorc´eb,
John W. Negelec, Vladimir Pascalutsab, Antonios Tsapalisd, Marc
9 Vanderhaeghenb
0
0 aDepartment of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
2 bInstitut fu¨rKernphysik, Johannes Gutenberg-Universita¨t,D-55099 Mainz, Germany
cCenterforTheoretical Physics, Laboratory for Nuclear Science and Department of
n
Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.
a dInstitute of Accelerating Systemsand Applications, Universityof Athens, Athens, Greece
J

2
2

]
h Abstract
p
- Weextendtheformalismrelatingelectromagneticformfactorstotransverse
p
quark charge densities in the light-front frame to the case of a spin-3/2 baryon
e
and calculate these transverse densities for the ∆(1232) isobar using lattice
h
[ QCD.The transversechargedensitiesfor atransverselypolarizedspin-3/2par-

ticle are characterized by monopole, dipole, quadrupole, and octupole patterns
1
representingthestructurebeyondthatofapurepoint-likespin-3/2particle. We
v
7 present lattice QCD results for the ∆-isobar electromagnetic form factors for
5 pionmassesdowntoapproximatively350MeVforthreecases: quenchedQCD,
4 two-degenerate flavorsof dynamical Wilson quarks, and three flavors of quarks
3
using a mixed action that combines domain wallvalence quarks and dynamical
.
1 staggered sea quarks. We extract transverse quark charge densities from these
0 lattice results and find that the ∆ is prolately deformed, as indicated by the
9
factthatthe quadrupole momentG (0)is largerthanthe value 3character-
0 E2 −
izing apoint particleandthe factthat the transversechargedensity in a ∆+ of
:
v
maximal transverse spin projection is elongated along the axis of the spin.
i
X
Key words: Electromagnetic form factors, Electric and Magnetic Moments,
r
a ∆-resonance
PACS: 13.40.Gp, 13.40.Em,14.20.Gk
1. Introduction
The question of how the structure of mesons and baryons arises from the
interactionamongtheir quark and gluonconstituents is at the forefrontof con-
temporary research in hadron physics. Key observables in this field are the
electromagnetic (e.m.) form factors (FFs), which yield the distribution of the
quark charges in a hadron. The nucleon e.m. FFs have been especially thor-
oughly studied recently at electron-beamfacilities, such as JeffersonLab, MIT-
Bates, and MAMI, see Refs. [1, 2, 3] for latest reviews.
Preprint submitted toNuclear Physics B January 22, 2009
In systems like nuclei or atoms, in which the spatial size R is large com-
pared to the system’s Compton wavelength 1/M, electromagnetic form factors
are essentially three dimensional Fourier transforms (or distorted wave analogs
thereof)of groundstate chargedensity distributions, andtherebyprovidevalu-
able physical insight into the structure of the ground state of the system in the
lab frame. In hadrons, however, where R 1/M, the Fourier transform argu-
∼
mentisinapplicable,andthereisnoknownwaytorelateformfactorstocharge
densities in the lab frame. However, when the hadron is viewed from a light
front, there is a simple and consistent field-theoretic density interpretation of
FFs as the Fourier transform of the spatial distribution of the quark charge in
theplanetransversetotheline-of-sight. Equivalently,intheinfinitemomentum
frame,the quantity that plays the role of M inthe lab frame Fouriertransform
argument is the light-front momentum p+, which goes to infinity, so the form
factor is just the two-dimensional Fourier transform of the transverse density
in the infinite momentum frame [4]. In this way, the transverse quark charge
densities have been mapped out in the nucleon [5, 6], deuteron [7], and pion [8]
based on empirical FFs. It is important to note, however, that physics in the
labandinfinitemomentumframesissignificantlydifferentandhencethereisno
simple relation between the two-dimensional transverse densities in the infinite
momentum (or light front) frame and the three-dimensional density in the lab
frame, and we will return to this issue in the discussion of deformation.
In this work we address the e.m. FFs and transverse charge densities of the
∆(1232) isobar, the lightest nucleon excitation. The nucleon-to-∆ transition
hasbeenwellmeasuredexperimentallyoveralargerangeofphotonvirtualities,
seeRef.[9]forarecentreview. Itis dominatedby amagneticdipole transition,
while the electric and Coulomb quadrupole transitions were found to be small
(inthefewpercentrangeascomparedtothemagneticdipoletransition). These
measurementshavemade itpossible to quantifythe deformationoftheN ∆
→
transition charge distribution [6].
Onthe otherhand,theinformationonthe e.m.FFs ofthe ∆itselfisscarce.
Becauseofthetinylifetimeofthe∆,itisofcourseverydifficultifnotimpossible
toaccessthesequantitiesdirectlyinexperiment. Fortunately,however,asinthe
case of the N ∆ FFs, it is now feasible to calculate the ∆ FFs using lattice
→
QCD. First lattice QCD results in the quenched approximationwere presented
in Ref. [10], and first results on the ∆ FFs using dynamical quarks were briefly
reportedin[11]. Inthismoreextendedpublicationweprovidefurtherdetailson
thelatticeevaluationoftheseFFsaswellastheinterpretationofthecalculated
FFs in terms of the transverse charge densities.
Theoutlineofthepaperisasfollows: Section2,containsageneraldiscussion
of the e.m. interaction of a spin-3/2 system along with the definition of its
e.m. moments and FFs. In Section 3, we find the specific (‘natural’) values for
electromagneticmomentsthatanelementary(pointlike)spin-3/2particlewould
possess. In Section 4, we introduce the light-front helicity amplitudes for the
e.m. vertexofaspin-3/2system,andexpressthemintermsofthecorresponding
e.m. FFs. InSection5,wecalculatequarktransversechargedensitiesofaspin-
3/2 system in terms of the light-front helicity amplitudes. We also calculate
2
the values of the electric dipole, quadrupole and octupole moments of these
transversechargedensities, andshow that for a pointlike spin-3/2particle they
vanish. InSection6,wedescribeourthreedifferenttypesoflatticecalculations
of the ∆(1232) e.m. FFs and compare their results. In the first type, we use
Wilson fermions in the quenched approximation. The second type uses two-
degenerateflavorsofdynamicalWilsonquarks(N =2). Thethirdtype usesa
F
mixed action combining domain wall valence quarks with dynamical staggered
sea quarks including light degenerate up and down quarks and heavier strange
quarks (N =2+1). In Section 7, we use these lattice QCD results to extract
F
the transverse quark charge densities in the ∆ resonance. We summarize our
results,discusstheirrelationtomodelsandpresentourconclusionsinSection8.
The lattice results for the ∆ e.m. FFs are tabulated in an Appendix.
2. The γ∗∆∆ vertex and form factors
Letusconsiderthecouplingofaphotontoa∆,showninFig.1. Thematrix
elementofthe electromagneticcurrentoperatorJµ betweenspin-3/2statescan
be decomposed into four multipole transitions: a Coulomb monopole (E0), a
magnetic dipole (M1), a Coulomb quadrupole (E2) and a magnetic octupole
(M3). We first write a Lorentz-covariant decomposition for the on-shell γ ∆∆
∗
vertex, which exhibits manifest electromagnetic gauge-invariance[9, 12]:
∆(p,λ) Jµ(0) ∆(p,λ)
′ ′
h | | i
qαqβ
=−u¯α(p′,λ′) F1∗(Q2)gαβ +F3∗(Q2)(2M )2 γµ
(cid:26)(cid:20) ∆ (cid:21)
qαqβ iσµνq
+ F (Q2)gαβ +F (Q2) ν u (p,λ), (1)
2∗ 4∗ (2M )2 2M β
(cid:20) ∆ (cid:21) ∆ (cid:27)
where M = 1.232 GeV is the ∆ mass, u is the Rarita-Schwinger spinor for
∆ α
a spin-3/2 state, and λ (λ) are the initial (final) ∆ helicities. Furthermore,
′
F are the γ ∆∆ form factors, and F (0)= e is the ∆ electric charge in
1∗,2,3,4 ∗ 1∗ ∆
units of e (e.g., e = +1). For future reference, we also define the quantity
∆+
τ Q2/(4M2).
≡ ∆
Aphysicalinterpretationofthefourelectromagnetic∆ ∆transitionscan
→
beobtainedbyperformingamultipole decomposition[13,12]. Forthis purpose
it is convenient to consider the Breit frame, where p~ = p~ = ~q/2. Further-
′
− −
more, we choose ~q along the z-axis and denote the initial (final) ∆ spin projec-
tionsalongthez-axisbys(s). Inthisframe,thematrixelementsofthecharge
′
operatordefine the Coulombmonopole (charge)andCoulombquadrupoleform
factors as :
~q ~q
h2,s′|J0(0)| − 2,si ≡ (2M∆)δs′s δs±32 +δs±12 GE0(Q2)
n(cid:16) 2 (cid:17)
τ δ δ G (Q2) .(2)
−3 s±32 − s±12 E2
(cid:16) (cid:17) (cid:27)
3
m
q
b a
p p’
Figure1: Theγ∗∆∆vertex. Thefour-momentaoftheinitial(final)∆andofthephotonare
givenbyp(p′)andq respectively. Thefour-vectorindicesoftheinitial(final)spin-3/2fields
aregivenbyβ (α), andµisthefour-vectorindexofthephotonfield.
Using Eq. (1), we can express the Coulomb monopole and quadrupole form
factors in terms of F as :
1∗,2,3,4
2
G = (F τF )+ τG , (3)
E0 1∗− 2∗ 3 E2
1
G = (F τF ) (1+τ)(F τF ). (4)
E2 1∗− 2∗ − 2 3∗− 4∗
Analogously, the matrix elements of the current operator define the magnetic
dipole and magnetic octupole form factors. For the transverse spherical com-
ponent J 1 (J1+iJ2) we obtain :
+1 ≡−√2
~q ~q √τ
,s J (0) ,s ( √2)(2M )
′ +1 ∆
h2 | | − 2 i≡ − √3
2
× δs′+23 δs+21 +δs′−12 δs−23 + √3δs′+21 δs−21 GM1(Q2)
(cid:26)(cid:18) (cid:19)
4
−5τ δs′+23 δs+12 +δs′−21 δs−32 −√3δs′+12 δs−12 GM3(Q2) .(5)
(cid:16) (cid:17) (cid:27)
Using Eq.(1), we can express the magnetic dipole and octupole form factors in
terms of F as :
1∗,2,3,4
4
G = (F +F )+ τG , (6)
M1 1∗ 2∗ 5 M3
1
G = (F +F ) (1+τ)(F +F ). (7)
M3 1∗ 2∗ − 2 3∗ 4∗
At Q2 = 0, the multipole form factors define the charge (e ), the magnetic
∆
dipole moment (µ ), the electric quadrupole moment (Q ), and the magnetic
∆ ∆
4
octupole moment (O ) as :
∆
e∆ = GE0(0)=F1∗(0), (8a)
e e
µ = G (0)= [e +F (0)] , (8b)
∆ 2M M1 2M ∆ 2∗
∆ ∆
e e 1
Q = G (0)= e F (0) , (8c)
∆ M2 E2 M2 ∆− 2 3∗
∆ ∆ (cid:20) (cid:21)
e e 1
O = G (0)= e +F (0) (F (0)+F (0)) . (8d)
∆ 2M3 M3 2M3 ∆ 2∗ − 2 3∗ 4∗
∆ ∆ (cid:20) (cid:21)
In the following, we will also use the relations that express the form factors
F in terms of the multipole form factors :
1∗,2,3,4
1 2 4
F1∗ = 1+τ GE0− 3τGE2+τ GM1− 5τGM3 ,
(cid:26) h i(cid:27)
1 2 4
F = G τG G τG ,
2∗ −1+τ E0− 3 E2− M1− 5 M3
(cid:26) h i(cid:27)
2 2 4
F3∗ = (1+τ)2 GE0− 1+ 3τ GE2+τ GM1− 1+ 5τ GM3 ,
(cid:26) (cid:18) (cid:19) h (cid:18) (cid:19) i(cid:27)
2 2 4
F = G 1+ τ G G 1+ τ G .(9)
4∗ −(1+τ)2 E0− 3 E2− M1− 5 M3
(cid:26) (cid:18) (cid:19) h (cid:18) (cid:19) i(cid:27)
For completeness we also express the form factors in terms of the covariant
vertex functions a ,a ,c and c , used for instance in references [12, 14, 11] :
1 2 1 2
F =a +a , F = a , F =c +c , F = c . (10)
1∗ 1 2 2∗ − 2 3∗ 1 2 4∗ − 2
These relations, together with the identity
u¯ (p,λ)iσµνq u (p,λ)=u¯ (p,λ)(2M γµ [p+p]µ)u (p,λ) (11)
α ′ ′ ν β α ′ ′ ∆ ′ β
−
bring Eq. (1) into the form introduced in [12, 14, 11].
The empirical knowledge of the ∆ electromagnetic moments is scarce, even
though there were several attempts to measure the magnetic moment. The
current Particle Data Group value of the ∆+ magnetic dipole moment is given
as [15]:
µ =2.7+1.0(stat.) 1.5(syst.) 3(theor.)µ , (12)
∆+ 1.3 ± ± N
−
where µ = e/2M is the nuclear magneton. This result was obtained from
N N
radiative photoproduction (γN πNγ ) of neutral pions in the ∆(1232)region
′
→
by the TAPS Collaboration at MAMI [16], using a phenomenological model of
the γp π0pγ reaction [17]. For the ∆+, Eq. (12) implies :
′
→
G (0)=3.5+1.3(stat.) 2.0(syst.) 3.9(theor.). (13)
M1 1.7 ± ±
−
Thesizeoftheerror-barisratherlargeduetobothexperimentalandtheoretical
uncertainties. Recently a dedicated experimental effort is underway at MAMI
5
usingtheCrystalBalldetector[18],aimingatimprovingonthestatisticsofthe
TAPS data by almost two orders of magnitude.
For the ∆ electric quadrupolemoment, no direct measurementsexist. How-
ever the electric quadrupole moment for the N ∆ transition has been mea-
→
sured accurately from the γN πN reaction at the ∆ resonance energy [19] :
→
Q = (0.0846 0.0033) e fm2. (14)
p ∆+
→ − ± ·
In the large-N limit of QCD, the two are related as follows [20]:
c
Q 2√2 1
∆+ = + , (15)
Q 5 O N2
p→∆+ (cid:18) c (cid:19)
which, using the empirical value for Q yields for the ∆+ quadrupole mo-
p ∆+
ment: →
Q = (0.048 0.002) e fm2, (16)
∆+
− ± ·
accurate up to corrections of order 1/N2. Note that using Eq. (8d), the value
c
in Eq. (16) implies G (0)= 1.87 0.08.
E2
− ±
3. The ‘natural’ values of the e.m. moments
As first argued by Weinberg [21], based on the Gerasimov-Drell-Hearnsum
rule, there is a ‘natural’ value for the magnetic moment of a pointlike particle
with spin, which corresponds to a gyromagnetic ratio g equal to 2. It has later
been observed that all consistent field theories of charged particles with spin
respect this value, see e.g. [22, 23]. Given this universalresult for the magnetic
moment,itisreasonabletoexpectthatallelectromagneticmomentsofpointlike
particles are fixed at ‘natural’ values. To determine the values for the spin-3/2
case, we examine the values of the e.m. moments of the gravitino in extended
supergravity [24, 25].
The gravitino, if it existed, would be a spin-3/2 particle described by a
Rarita-Schwinger field which, in the framework of =2 supergravity, couples
N
consistentlytoelectromagnetism. Wethereforeexpectthatallthee.m.moments
arising in this theory are ‘natural’. To find their values, we start from the
following Lagrangiandensity:1
= ψ¯ γµνα(i∂ eA )ψ mψ¯ γµνψ
µ α α ν µ ν
L − −
+ em 1ψ¯ (iκ Fµν κ γ F˜µν)ψ . (17)
− µ 1 2 5 ν
−
It describes the spin-3/2 Rarita-Schwinger field (ψ ) with mass m coupled to
µ
the electromagnetic field (A ) via the minimal coupling (with positive charge
µ
1Inour conventions: γµν = 1[γµ,γν], γµνα = 1{γµν,γα}, Fµν =∂µAν−∂νAµ, F˜µν =
2 2
εµν̺λ∂̺Aλ.
6
e) and two non-minimal couplings κ and κ . As shown in Ref. [26], this is
1 2
the most general Lagrangian of this type that gives the right number of spin
degrees of freedom for a spin-3/2 particle. This theory, however, would still
lead to rather subtle pathologies such as non-causal wave propagation [27, 28],
at least if no other fields are present. Adding gravity in a supersymmetric way
makes the theory fully consistent from this viewpoint, but also constrains the
non-minimal couplings as follows:
κ =κ =1. (18)
1 2
We shall refer to these values as the ‘SUGRA choice’.
The e.m. vertex stemming from Eq. (17) is
κ κ
Γαβµ(p,p)=γαβµ 1(qαgβµ qβgαµ)+i 2γ εαβµ̺q , (19)
′ 5 ̺
− m − m
whereq =p p,andε =+1. Itiseasytoverifythatthiscouplingconserves
′ 0123
−
the e.m. current:
qµu¯α(p′)Γαβµ(p′,p)uβ(p)=0, (20)
as well as, for the SUGRA choice, the supersymmetric current:
(p 1mγ )Γαβµ(p,p)u (p)ǫ (q) = 0, (21a)
′α− 2 α ′ β µ
u¯ (p)Γαβµ(p,p)(p 1mγ )ǫ (q) = 0, (21b)
α ′ ′ β − 2 β µ
where ǫ (q) is the photon polarization vector, and q ǫ=0=q2 in Eqs. (21).
µ
·
The matrix elements of this vertex can be compared with the general de-
composition of the spin-3/2 e.m. current given in Eq. (1), and in doing so we
obtain the following result:
sugra
F = 1+2(κ +κ )τ = 1+4τ, (22a)
1∗ 1 2
sugra
F = 2κ = 2, (22b)
2∗ 1
sugra
F3∗ = 4(κ1+κ2) = 8, (22c)
F = 0, (22d)
4∗
Thus, the values of gravitino’s e.m. moments in =2 supergravity are:
N
G (0)=1, G (0)=3, G (0)= 3, G (0)= 1. (23)
E0 M1 E2 M3
− −
We take these values as the ‘natural’ values characterizing a structureless spin-
3/2 particle, and interpret any deviation from them as a signature of internal
structure.
4. The γ∗∆∆ light-front helicity amplitudes
In the following, we consider the electromagnetic ∆ ∆ transition when
→
viewedfromalightfrontmovingtowardsthe ∆. Equivalently,this corresponds
7
to a frame where the baryons have a large momentum-component along the
z-axis chosen along the direction of P =(p+p)/2, where p (p) are the initial
′ ′
(final) baryon four-momenta. We indicate the baryon light-front + component
by P+ (defining a a0 a3). We can furthermore choose a symmetric frame
±
≡ ±
where the virtual photon four-momentum q has q+ = 0, and has a transverse
component (lying in the xy-plane) indicated by the transverse vector ~q , sat-
isfying q2 = ~q2 Q2. In such a symmetric frame, the virtual photo⊥n only
couplestofor−wa⊥rd≡m−ovingpartonsandthe+componentoftheelectromagnetic
current J+ has the interpretation of the quark charge density operator. It is
givenby: J+(0)=+2/3u¯(0)γ+u(0) 1/3d¯(0)γ+d(0),consideringonlyuandd
−
quarks. Eachtermin the expressionis a positive operatorsince q¯γ+q γ+q 2.
∝| |
We start by expressing the matrix elements of the J+(0) operator in the ∆
as :
P+,~q⊥,λ′ J+(0)P+, ~q⊥,λ = (2P+)ei(λ−λ′)φqAλ′λ(Q2), (24)
h 2 | | − 2 i
where λ,λ denotes the ∆ light-front helicities, and where ~q = Q(cosφ eˆ +
′ q x
sinφqeˆy). The helicity form factors Aλ′λ depend on Q2 onl⊥y and can equiva-
lently be expressed in terms of F as :
1∗,2,3,4
τ
A = A =F F ,
2332 −23−32 1∗− 2 3∗
τ1/2 1
A = A =A = A = 2F F τ F F ,
2312 − −23−12 −12−32 − 2123 √3 1∗− 2∗− 3∗− 2 4∗
(cid:20) (cid:18) (cid:19)(cid:21)
τ 1
A = A =A =A = 2F + F +τF ,
23−12 −2312 −2123 21−32 √3 − 2∗ 2 3∗ 4∗
(cid:20) (cid:21)
1
A32−32 = −A−2332 =−2τ3/2F4∗,
4 τ 1
A = A = 1 τ F + 4F 2τ F 2τF ,
2112 −21−12 − 3 1∗ 3 2∗− 2 − 3∗− 4∗
(cid:18) (cid:19) (cid:20) (cid:18) (cid:19) (cid:21)
τ1/2 1
A = A = 4F 2(1 2τ)F 2τF +τ 2τ F . (25)
12−12 − −2112 3 1∗− − 2∗− 3∗ 2 − 4∗
h (cid:18) (cid:19) i
Sincethe∆ ∆electromagnetictransitionisdescribedbyfourindependent
→
form factors, one finds two angular conditions among the helicity form factors
of Eq. (25) :
0 = (1+4τ)√3A 8τ1/2A +2A √3A ,
33 31 3 1 11
22 − 22 2−2 − 22
1 3
0 = 4τ3/2A +√3(1 2τ)A + A A . (26)
33 31 3 3 1 1
22 − 22 2 2−2 − 2 2−2
8
5. The transverse charge densities for a spin-3/2 particle
Wedefineaquarkchargedensityforaspin-3/2particle,suchasthe∆(1232),
in a state of definite light-cone helicity λ, by the Fourier transform :
d2~q 1 ~q ~q
ρ∆λ(b) ≡ (2π)⊥2 e−iq~⊥·~b 2P+hP+, 2⊥,λ|J+|P+,−2⊥,λi
Z
= ∞ dQQJ (Qb)A (Q2). (27)
0 λλ
2π
Z0
The two independent quark charge densities for a spin-3/2 state of definite
helicity are given by ρ∆(b) and ρ∆(b). Note that for a pointlike particle, the
3 1
2 2
‘natural’ values of Eq. (22) lead to A (Q2)=1, implying
33
22
ρ (~b)=δ2(~b). (28)
3
2
Theabovechargedensitiesprovideuswithtwocombinationsofthefourinde-
pendent∆FFs. Togetinformationfromthe otherFFs, we considerthe charge
densities in a spin-3/2 state with transverse spin. We denote this transverse
polarization direction by S~ = cosφ eˆ +sinφ eˆ , and the ∆ spin projection
S x S y
along the direction of S~ b⊥y s . We first express the transverse spin basis in
termsofthe helicitybasi⊥sforsp⊥in-3/2. Forthe states oftransversespins = 3
and s = 1 this yields : ⊥ 2
⊥ 2
3 1 3 1
s =+ = e−iφS λ=+ +√3 λ=+
| ⊥ 2i √8 | 2i | 2i
(cid:26)
1 3
+√3eiφS λ= +e2iφS λ= ,
| −2i | −2i
(cid:27)
1 1 3 1
s =+ = √3e−iφS λ=+ + λ=+
| ⊥ 2i √8 | 2i | 2i
(cid:26)
1 3
eiφS λ= √3e2iφS λ= , (29)
− | −2i− | −2i
(cid:27)
where the states on the rhs are the spin-3/2 helicity eigenstates.
We can then define the charge densities in a spin-3/2 state with transverse
spin s as :
⊥
d2~q 1 ~q ~q
ρ∆Ts⊥(~b) ≡ (2π)⊥2 e−iq~⊥·~b 2P+hP+, 2⊥,s⊥|J+(0)|P+,− 2⊥,s⊥i.(30)
Z
ByworkingouttheFouriertransforminEq.(30)forthetwocaseswheres = 3
⊥ 2
9
and s = 1, using the ∆ helicity form factors of Eq. (25), one obtains :
⊥ 2
ρ∆ (~b)= +∞ dQQ J (Qb)1 A +3A
T 23 Z0 2π h 0 4(cid:16) 3223 2112(cid:17)
1
sin(φ φ )J (Qb) 2√3A +3A
b S 1 31 1 1
− − 4 22 2−2
(cid:16)√3 (cid:17)
cos[2(φ φ )]J (Qb) A
b S 2 3 1
− − 2 2−2
1
+ sin[3(φ φ )]J (Qb) A , (31)
b S 3 3 3
− 4 2−2
i
and
ρ∆ (~b)= +∞ dQQ J (Qb)1 3A +A
T 12 Z0 2π h 0 4(cid:16) 2323 1212(cid:17)
1
sin(φ φ )J (Qb) 2√3A A
b S 1 31 1 1
− − 4 22 − 2−2
(cid:16)√3 (cid:17)
+ cos[2(φ φ )]J (Qb) A
b S 2 3 1
− 2 2−2
3
sin[3(φ φ )]J (Qb) A , (32)
b S 3 3 3
− − 4 2−2
i
where we defined the angle φ in the transverse plane as, ~b = b(cosφ eˆ +
b b x
sinφ eˆ ). One notices from Eqs. (31,32) that the transverse charge densities
b y
display monopole, dipole, quadrupole, and octupole field patterns, which are
determined by the helicity form factors with zero, one, two, or three units of
helicity flip respectively between the initial and final ∆ states.
Itisinstructivetoevaluatetheelectricdipolemoment(EDM)corresponding
to the transverse charge densities ρ∆ , which is defined as :
Ts⊥
d~∆ e d2~b~bρ∆ (~b). (33)
s⊥ ≡ Ts⊥
Z
Eqs. (31,32) yield :
e
d~∆ =3d~∆ = S~ eˆ G (0) 3e . (34)
32 21 −(cid:16) ⊥× z(cid:17) { M1 − ∆} (cid:18)2M∆(cid:19)
Expressingthespin-3/2magneticmomentintermsoftheg-factorgivesG (0)=
M1
g3e , so that the induced EDM d~∆ is proportional to g 2. The same result
2 ∆ s⊥ −
was found before for the case of spin-1/2 particles in [6] and spin-1 particles in
Ref. [7]. One thus observes as a universal feature that for a particle without
internalstructure(correspondingwithg =2[22,23]),thereisnoinducedEDM.
Wenextevaluatetheelectricquadrupolemomentcorrespondingtothetrans-
versechargedensitiesρ∆ . ChoosingS~ =eˆ ,theelectricquadrupolemoment
Ts⊥ ⊥ x
can be defined as :
Q∆ e d2~b(b2 b2)ρ∆ (~b). (35)
s⊥ ≡ x− y Ts⊥
Z
10