Electromagnetic Properties for Arbitrary Spin Particles: Part 1 $-$ Electromagnetic Current and Multipole Decomposition

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by  Cédric Lorcé

Electromagnetic Properties for Arbitrary Spin Particles:
Part 1 Electromagnetic Current and Multipole Decomposition
−
C´edric Lorc´e
Institut fu¨r Kernphysik, Johannes Gutenberg-Universita¨t, D-55099 Mainz, Germany
E-mail: [email protected]
9
0
0 Inasetoftwopapers,weproposetostudyanold-standingproblem,namelytheelectromagneticinteraction
2 for particles of arbitrary spin. Based on the assumption that light-cone helicity at tree level and Q2 = 0
n should beconserved non-trivially bytheelectromagnetic interaction, weare able toderiveall thenatural
a electromagneticmomentsforapointlikeparticleofany spin. Inthisfirstpaper,weproposeatransparent
J decomposition of the electromagnetic current in terms of covariant vertex functions. We also define in a

7 general way theelectromagnetic multipole form factors, and show their relation with the electromagnetic
2 moments. Finally, by considering the Breit frame, we relate the covariant vertex functions to multipole
form factors.

]
h
1 Introduction
p
-
p
High-energy physics involves high-spin particles for many reasons. Among those, let us mention that:
e
h experimentally, more than fifty baryons with spins ranging from 3/2 to 15/2 have been observed [1];
[ •

these resonancesenteras intermediatestates inthe descriptionofthe photo-andelectro-pionproduction
1 •
off protons, which are a main focus of experiments at electron facilities such as Jefferson Lab, ELSA,
v
MAMI [2];
9
9
in proposals for physics beyond the Standard Model based on supersymmetry, which will be explored
1 •
4 soon by the LHC, elementary particles with high spin (e.g. the gravitino) are required.
.
1 For further motivations to study high-spin particles, see e.g. [3] and references therein.
0 Formalisms to study high-spinparticles have been proposeda long time ago. Dirac,Fierz and Pauli(DFP)
9 were the first to develop a general theory of particles with arbitrary spin [4], but this formalism is quite
0
complicated. Later, Rarita and Schwinger (RS) proposed a more convenient formalism for arbitrary half-
:
v integer spin particles [5], which is still the most often used to date in the literature. In the case of integer spin
Xi particles, the Klein-Gordon (KG) equation with subsidiary conditions is more suited. Actually, it has been
shown by Moldauer and Case that both RS and KG approaches can be derived from the DFP formalism [6].
r
a There exists another formalism due to Bargmann and Wigner [7] from which one can also derive the RS and
KG equations together with their subsidiary conditions [8].
Theorieswithspin>1arehoweverplaguedbyarbitraryparametersinboththeirLagrangianandpropagator
[6, 9]. This is due to the fact that the corresponding fields contain extra degrees of freedom related to lower
spins. To eliminate these extra degrees of freedom, one imposes constraints (the subsidiary conditions in RS
and KG formalisms). Unfortunately, interactions usually break these constraints, generating many problems.
Among the various difficulties [10], let us mention an important result of Velo and Zwanziger [11] that showed
that a massive c-number spin-3/2 field minimally coupled to an external electromagnetic field will lead to the
existence of tachyons, and therefore to non-causality. The causality issue has been shown to be related to the
value of the gyromagnetic factor at tree level in the case of elementary spin-3/2 particles [3, 12].
The gyromagneticfactor ofelementary particles has alsoa long history. Basedonthe minimal substitution
pµ πµ pµ eAµ, it was argued that the gyromagnetic factor at tree level depends on the spin j of the
→ ≡ −
particle [6, 13] as
1
g = , (1)
j
j
1
i.e. the magnetic momentdepends only onthe massandelectricchargeofthe particle. This resultagreeswith
the case j = 1/2 but seems to disagree with higher-spin cases. Many reasons suggest that the gyromagnetic
factor at tree level is in fact independent of the spin [14]
g =2, (2)
j
i.e. the magnetic moment is directly proportionalto the spin. Among the various reasons,let us mention that
the only higher-spin and charged elementary particle observed is the W boson. At tree level, it has
•
g =2 [15] and not g =1 as suggested by (1) for spin-1 particles;
W W
the relativistic equation of motion of the polarization four-vector S in a homogeneous external electro-
µ
•
magnetic field is simplified for g =2 [16];
j
in the supersymmetric sum rules framework, when all magnetic-moment matrix elements are diagonal,
•
the gyromagnetic factor of arbitrary-spinsupersymmetric particles must be equal to 2 [17];
string theory also suggests that g =2 [18];
j
•
by requiringthatforwardComptonscatteringamplitudesofphysicaltheoriespossessagoodhigh-energy
•
behavior, Weinberg showed [19] that this implies g 2 for any non-strongly interacting spin-j particle.
j
≈
The way to reconcile Lagrangiantheories with g =2 is to allow non-minimal coupling [20]. Indeed, using the
j
minimal substitution is ambiguous for spins j >1/2 since one has [πµ,πν]= ieFµν while [pµ,pν]=0. Note
−
that a non-minimal coupling FµνW+W− is already present in the Standard Model [21].
µ ν
In order to characterize completely the electromagnetic interaction of an elementary particle with spin
j >1/2,theelectricchargeandthe magneticdipolemomentarenotsufficient. Ingeneral,aspin-j particlehas
2j+1electromagneticmultipolesifLorentz,parityandtime-reversalsymmetriesarerespected. Theknowledge
of these “natural” electromagnetic moments is obviously very important since they
constrain the construction of a consistent higher-spin electromagnetic interaction theory;
•
shedlightontherelationbetweencausalityandunitarityintheultrarelativisticlimit,assuggestedalready
•
by the gyromagnetic factor;
allow a better understanding of internal structure of composite particles with spin j >1/2;
•
...
•
In a set of two papers, we present our results concerning the electromagnetic interaction for particles with
arbitraryspin. Basedonthe simple assumptionthatQEDconservesnon-triviallythe light-cone helicity ofany
elementaryparticleattreelevelandrealphotonpointQ2 =0,wewereabletoderiveall naturalelectromagnetic
moments for particles of any spin.
This paper contains the first part of our investigations. In Section 2, we give the general form of the
electromagnetic current in terms of covariant vertex functions. An explicit multipole decomposition is then
performed in Section 3 together with a transparent relation to electromagnetic moments. In Section 4, we use
the Breit frame to obtain a general relation between multipole form factors and covariantvertex functions.
2 Electromagnetic Current for Arbitrary Spin
The interaction of a particle with the electromagnetic field can be described in terms of matrix elements, also
known as on-shell vertex functions, of the following form
Jµ p′,λ′ Jµ (0)p,λ , (3)
≡h | EM | i
where p (resp. p′) is the initial (resp. final) four-momentum of the particle, and p,λ is the spin-j single-
| i
particlestatewithfour-momentumpandpolarizationλ. Forfurtherconvenience,wedefine thescalarquantity
Q2 q2 as minus the square of the four-momentum transfer q =p′ p due to the photon.
≡− −
2
If the particle respects parity and time-reversal symmetries, its matrix elements can in principle be conve-
niently written in terms of 2j +1 independent covariant vertex functions F (Q2) [22], also often called form
k
factors. Note however that the decomposition is not unique. In this sense, the matrix elements are more
fundamental than the covariant vertex functions.
Since any spin representation can be constructed from spin-1/2 and spin-1 representations, it is rather
straightforwardto obtain the general structure of the matrix elements satisfying Lorentz covariance,gauge in-
variance,togetherwithparityandtime-reversalsymmetries. WewillusetheconvenientRSandKGformalisms
for the description of the particle polarizations.
2.1 Arbitrary Integer Spin Current
Aparticlewithintegerspinj andmassM canbedescribedintermsofapolarizationtensorε (p,λ). This
α1···αj
tensor is completely symmetric and satisfies subsidiary conditions, namely transversity and tracelessness
pµε (p,λ)=0,
µα2···αj (4)
εµ (p,λ)=0,
µα3···αj
in order to insure the correct number of degrees of freedom 2j+1.
The electromagnetic interaction current for integer spin particles has the following general simple form
Jµ =(−1)jε∗α′1···α′j(p′,λ′)Pµ F2k+1(Q2)+ gµαjqα′j −gα′jµqαj F2k+2(Q2)εα1···αj(p,λ), (5)
(Xk,j) (cid:16) (cid:17)(kX,j−1)
 
where
j k qα′iqαi j gα′iαi , (6)
≡ " − 2M2 ! #
(Xk,j) Xk=0 Yi=1 i=Yk+1
and P =p′+p is twice the averagedfour-momentum of the particle.
To illustrate this, let us consider for example the case j =1. The explicit expression of (5) becomes
Jµ = ε∗ (p′,λ′) Pµgα′αF (Q2)+ gµαqα′ gα′µqα F (Q2) Pµ qα′qα F (Q2) ε (p,λ),
(1) − α′ " 1 − 2 − 2M2 3 # α
(cid:16) (cid:17)
which coincides with the well-known expression for the electromagnetic interaction of vector particles [23]
provided that
F (Q2)=G (q2), k =1,2,3.
k k
2.2 Arbitrary Half-Integer Spin Current
A particle with half-integerspin j =n+1 andmass M canbe describedin terms of a polarizationspin-tensor
2
u (p,λ). This spin-tensor,besides being completely symmetric inits Lorentzindices, has to fulfill a setof
α1···αn
equations1 [5]
(/p M)u (p,λ)=0,
− α1···αn (7)
γµu (p,λ)=0,
µα2···αn
in order to insure the correct number of degrees of freedom 2j+1.
The electromagneticinteractioncurrentfor half-integerspinparticles has the followinggeneralsimple form
iσµνq
Jµ =(−1)nu¯α′1···α′n(p′,λ′) γµF2k+1(Q2)+ 2Mν F2k+2(Q2) uα1···αn(p,λ). (8)
(k,n)(cid:20) (cid:21)
X
1Note that the usual subsidiaryconditions of integer spin (4) applied to the spin-tensor arejust consequences of (7). Indeed,
theycanbeobtainedbymultiplyingthetwoequations in(7)ontheleftbyγα1 andγα2,respectively.
3
To illustrate this, we will consider two examples. Let us consider first the case j = 1/2. The explicit
expression of (8) becomes
iσµνq
Jµ =u¯(p′,λ′) γµF (Q2)+ ν F (Q2) u(p,λ),
(1/2) 1 2M 2
(cid:20) (cid:21)
which coincides with the well-known expression for the electromagnetic interaction of spin-1/2 particles. The
covariantvertex functions F (Q2) and F (Q2) are known in the literature as the Dirac and Pauliform factors,
1 2
respectively. Let us consider now the case j =3/2. The explicit expression of (8) becomes
J(µ3/2) =−u¯α′(p′,λ′)(gα′α(cid:20)γµF1(Q2)+ iσ2µMνqν F2(Q2)(cid:21)− q2αM′q2α (cid:20)γµF3(Q2)+ iσ2µMνqν F4(Q2)(cid:21)) uα(p,λ),
which coincides with the well-known expression for the electromagnetic interaction of spin-3/2 particles [24]
provided that
F (Q2)= F∗(Q2)=a (q2)+a (q2),
1 1 1 2
F (Q2)= F∗(Q2)= a (q2),
2 2 − 2
1 1
F (Q2)= F∗(Q2)= c (q2)+c (q2) ,
3 −2 3 −2 1 2
F (Q2)= 1F∗(Q2)= 1c (cid:2)(q2). (cid:3)
4 −2 4 2 2
As it will become obvious in the following, the decompositions (5) and (8) we propose are such that spurious
( 1) factors are avoided. Note also that the electromagnetic gauge invariance is explicit, i.e. one can easily
−2
see that q Jµ =0.
µ
2.3 Polarization Tensor and Spin-Tensor
In order to work out the vertex function in specific cases, we need an explicit form for the polarization
(spin-)tensor. The easiest way has been proposed a long time ago by Auvil and Brehm [8, 25]. By always
coupling the maximum possible spin of a lower-order polarization tensor and a polarization four-vector, the
product will satisfy KG or RS equations together with the subsidiary conditions. One therefore obtains the
following recursion formula
T (p,λ)= 1m,(j 1)m′ jλ ε (p,m)T (p,m′), (9)
α1···αn h − | i α1 α2···αn
m,m′
X
where j m ,j m jm represents the Clebsch-Gordan coefficient in the Condon-Shortley phase convention,
1 1 2 2
h | i
and T (p,λ) stands for the polarization spin-tensor u (p,λ) with j = n+ 1 (half-integer spin case)
α1···αn α1···αn 2
or the polarization tensor ε (p,λ) with j =n (integer spin case). For the spin-tensor of half-integer spin,
α1···αn
it is however more convenient to consider
1m
u (p,λ)= ,nm′ jλ u(p,m)ε (p,m′),
α1···αn h2 2 | i α1···αn
m,m′
X
which is naturally equivalent to (9). More explicitly,
j+λ 1 j λ 1
u (p,λ)= u(p,+)ε (p,λ )+ − u(p, )ε (p,λ+ ). (10)
α1···αn s 2j α1···αn − 2 s 2j − α1···αn 2
We can therefore concentrate on the polarization tensor of integer spin. By interating (9), one obtains
j−1
ε (p,λ)= 1m lm′ (l+1)m′ ε (p,m ) ε (p,m ),
α1···αj " h l l| l+1i αl l # αj j
mlX=0,±1 Yl=1
4
where the sum is implicitly restricted to configurations such that j m =λ, and where
l=1 l
l−1 P
m′ =m + m .
l j k
k=1
X
The Clebsch-Gordan coefficients can be written as [26]
C1+mlCl+m′l
1m lm′ (l+1)(m +m′) = 2 2l ,
h l l| l l i vuCl+ml+m′l+1
u 2l+2
t
where Ck is the binomial function
n
n n! , n k 0
Ck = k!(n−k)! ≥ ≥ . (11)
n k ≡ 0, otherwise
(cid:18) (cid:19) (cid:26)
One easily obtains
j C1+mlε (p,m )
l=1 2 αl l
ε (p,λ)= ,
α1···αj mlX=0,±1Q q C2jj+λ
q
which can be rewritten more conveniently as [27]
m/2 k ε (p, ) m−k ε (p,0) j ε (p,+)
P l=1 αP(l) − l=k+1 αP(l) l=m−k+1 αP(l)
ε (p,j m)= , (12)
α1···αj − k=0 P hQ 2k−m/2ikh!Q(m−2k)!(j−m+ihkQ)! C2mj i
X
p
where stands for a permutation of 1, ,j . The presence of factorials in the denominator is due to the
P { ··· }
factthatpermuting the indices oftwopolarizationfour-vectorswiththe samepolarizationdoes notgiveanew
contribution.
3 Electromagnetic Moments
Our final aim is to obtain the natural values for the electromagnetic moments of a particle of arbitrary spin.
These moments are related to the so-calledmultipole form factors G (Q2) and G (Q2) at real photon point
El Ml
Q2 = 0. Thanks to an explicit multipole decomposition of a four-current jµ, we define all the magnetic and
electric multipole form factors and present the explicit relation with the (cartesian) electromagnetic moments.
For further convenience, we denote by e the electric charge of an electron, and we introduce the notation
−
Q2
τ . (13)
≡ 4M2
3.1 Multipole Decomposition
The zerocomponentµ=0ofafour-currentjµ correspondstothe chargedensitywhile the spatialcomponents
µ= i are related to some kind of “magnetic density” [28]. Let us refer, for the moment, to these densities by
means of the generic density
ρ(~q)= d3reiq~·~rρ(~r). (14)
Z
The multipole decomposition of this density can be written in the following form (Q ~q )
≡| |
+∞ l
ρ(~q)= QlF (Q2)Y (Ω ), (15)
lm lm q
l=0m=−l
X X
5
where Y (Ω) are the usual spherical harmonics, Ω stands for the couple (θ,φ). Thanks to the identity [26]
lm
+∞ l
eiq~·~r =4π ilj(Qr)Y (Ω )Y∗ (Ω ),
l lm q lm r
l=0m=−l
X X
where j(x) are the spherical Besselfunctions, and using orthonormalrelations among the spherical harmonics
l
dΩYl∗′m′(Ω)Ylm(Ω)=δll′δmm′,
Z
we find
4πil
F (Q2)= d3rj (Qr)Y (Ω )ρ(~r).
lm Ql l lm r
Z
Remembering that we are interested in the electromagnetic properties of particles, one can naturally consider
that the electric and magnetic densities exhibit an azimuthal (or cylindrical) symmetry with respect to the
quantization axis. Let us identify the spin quantization axis with the z-axis. Due to this symmetry, the
multipole decomposition (15) reduces to m=0 components only
+∞
ρ(~q)= QlF (Q2)Y (Ω ). (16)
l0 l0 q
l=0
X
The spherical harmonics Y (Ω ) are related to cartesian multipole structures C (~r) as follows
l0 r l
4π
C (~r)=l!rl Y (Ω )=l!rlP (cosθ ),
l l0 r l0 r
2l+1
r
i.e. more explicitly
C (~r)= 1,
0
C (~r)= z,
1
C (~r)= 3z2 r2,
2
−
C (~r)= 15z3 9r2z,
3
−
.
.
.
Moreover,to the lowest order in Q, the spherical Bessel functions j (Qr) can be written as
l
(Qr)l
j (Qr)= + (Ql+2).
l
(2l+1)!! O
From this, one can see that the electromagnetic moments are proportionalto F (0)
l0
2l+1
d3rC(~r)ρ(~r)=( i)ll!(2l 1)!! F (0). (17)
l l0
− − 4π
Z r
3.2 Electric Multipoles
Electric multipoles2 are obtained from the charge density ρ(~r) = j0(~r). We define the electric multipole form
factors G (Q2) as follows
El
(2l 1)!! (2M)l 2l+1
G (Q2) ( i)l − F (Q2), (18)
El l0
≡ − l! e 4π
r
2Ifwefollowmorerigorouslytheliteratureterminology,theyshouldbecalledCoulombmultipoles.
6
so that we get at Q2 =0
(2M)l 1
G (0)= d3rC (~r)j0(~r).
El e (l!)2 l
Z
The lth electric moment Q [28] in (natural) unit of e/Ml is therefore given by
l
(l!)2
Q d3rC (~r)j0(~r)= G (0). (19)
l ≡ l 2l El
Z
The multipole expansion(16) of the chargedensity ρ(~r)=j0(~r) in terms of the multipole form factors (18)
takes the form
+∞
1 4π
j0(~q)=e ilτl/2 G (Q2) Y (Ω ), (20)
Cl−1 El 2l+1 l0 q
l=0 2l−1 r
X
where we have defined3
en!! , n k 1
Ck k!!(n−k)!! ≥ ≥− . (21)
n ≡ 0, otherwise
(cid:26)
e
3.3 Magnetic Multipoles
Magnetic multipoles are obtained from the magnetic density ρ(~r) = ~ ~j(~r) ~r . We define the magnetic
∇· ×
multipole form factors G (Q2) as follows (cid:16) (cid:17)
Ml
(2l 1)!! (2M)l 2l+1
G (Q2) ( i)l − F (Q2), (22)
Ml l0
≡ − (l+1)! e 4π
r
so that we get at Q2 =0
(2M)l 1 ~ ~j(~r) ~r
G (0)= d3rC(~r)∇· × .
Ml e (l!)2 l (cid:16)l+1 (cid:17)
Z
The lth magnetic moment µ [28] in (natural) unit of e/2Ml is therefore given by
l
~ ~j(~r) ~r (l!)2
µ d3rC (~r)∇· × = G (0). (23)
l ≡ l (cid:16)l+1 (cid:17) 2l−1 Ml
Z
The multipole expansion (16) of the magnetic density ρ(~r)= ~ ~j(~r) ~r in terms of the multipole form
∇· ×
factors (22) takes the form (cid:16) (cid:17)
+∞
(l+1)
~ ~j(~q) ~q =e ilτl/2 G (Q2)Y (Ω ). (24)
∇· × Cl−1 Ml l0 q
(cid:16) (cid:17) Xl=0 2l−1
3.4 Multipoles and Particles e
A particle of spin j respecting parity symmetry has only even electric multipoles and odd magnetic multipoles
[22]. Moreover, the total number of multipole form factors is equal to the total number of covariant vertex
functions, namely 2j+1. The decompositions (20) and (24) take therefore the form
2j
1
j0(~q)= e ilτl/2 G (Q2)Y (Ω ), (25)
Cl−1 El l0 q
l=0 2l−1
X
leven
2j e
(l+1)
~ ~j(~q) ~q = e ilτl/2 G (Q2)Y (Ω ), (26)
∇· × Cl−1 Ml l0 q
(cid:16) (cid:17) Xl=0 2l−1
lodd
3Weremindthatbydefinition(−1)!!=1. e
7
or more explicitly
2 4π
j0(~q)= e G (Q2)√4πY (Ω ) τG (Q2) Y (Ω )+ ,
E0 00 q E2 20 q
" − 3 r 5 ···#
4π 4 4π
~ ~j(~q) ~q = e2i√τ G (Q2) Y (Ω ) τG (Q2) Y (Ω )+ .
M1 10 q M3 30 q
∇· × " r 3 − 5 r 7 ···#
(cid:16) (cid:17)
Notethatthesedecompositionsdonotdependontheparticlespin. Thelatterjusttellsuswhatisthemaximum
allowed multipole in these series.
In the next section, we will establish a direct relation between covariant vertex functions F (Q2) and
k
electromagnetic multipole form factors G (Q2) and G (Q2). In order to make our job easier in establishing
El Ml
thisconnection,wewillexpandthesphericalharmonics4 Y (Ω)intermsofsinθ andcosθ [26]. Forevenvalues
l0
of l, the expansion reads
l
2l+1
Y (Ω)= ( 1)s/2CsCl−1 sinsθ, (27)
l0 4π − l l+s−1
r s=0
X
seven
e e
while for odd values of l, it reads
l−1
2l+1
Y (Ω)= cosθ ( 1)s/2Cl−s−1Cs sinsθ. (28)
l0 4π − l−1 l+s
r s=0
X
seven
e e
The expansion of the charge (25) and magnetic (26) densities becomes after insertion of (27) and (28),
respectively, and a few algebraic manipulations
j0(~q)= e n n ( 1)m+tτm (Cmt )2 sin2θ tG (Q2), (29)
− C2m+2t−1 E2m
t=0m=t 4m−1
XX (cid:0) (cid:1)
~ ~j(~q) ~q = e2i√τ cosθ n n ( e1)m+t(1 δ )τm(m+1) (Cmt )2 sin2θ tG (Q2), (30)
∇· × − − j,m C2m+2t+1 M2m+1
(cid:16) (cid:17) Xt=0mX=t 4m+1 (cid:0) (cid:1)
where j =n for integer spin and j =n+ 1 for half-integer spin. e
2
As a closing remark for this section, we would like to emphasize that only G (0), G (0) and G (0)
E0 M1 E2
can directly be interpreted as electromagnetic moments. For l>2, the proportionality factor between electro-
magnetic moments and multipole form factors at Q2 =0 differs from unity, as one can see from eqs. (19) and
(23).
4 Breit Frame
WewantnowtorelatethecovariantvertexfunctionsF (Q2)ofSection2tothemultipoleformfactorsG (Q2)
k El
and G (Q2) of Section 3. We follow the common usage by considering the so-called Breit frame to connect
Ml
the classical electromagnetic current jµ of Section 3 with the quantum-mechanical one Jµ of Section 2
e e
jµ(~q) p′,j Jµ (0)p,j = Jµ. (31)
≡ 2M h | EM | i 2M B
The Breit frame (or brick-wall frame) is the frame where no energy is transferred to the system by the
4Forthesakeofclarity,weomittheindexq attached totheangles,sincewewon’treferanymoretoconfiguration space.
8
photon5
qµ = (0,~q),
~q
pµ = (p , ),
0
−2
~q (32)
p′µ = (p , ),
0
2
~q 2
p = M2+ | | =M√1+τ.
0
4
r
In this particular frame, the current Jµ has a non-relativistic appearance once explicitly expressed in terms
B
of the rest-frame polarization vectors ~ε and rest-frame spinors χ [23]. Since we have identified the spin
λ λ
quantization axis with the z-axis, the rest-frame polarization vectors and spinors are given by
1
~ε = ( 1, i,0), ~ε =(0,0,1), (33)
± 0
√2 ∓ −
1 0
χ = , χ = . (34)
+ 0 − 1
(cid:18) (cid:19) (cid:18) (cid:19)
In the following, we will use standard polarization four-vectors
~ε ~p p~(~ε p~)
εµ(p,λ)= λ· ,~ε + λ· , (35)
λ
M M(p +M)
(cid:18) 0 (cid:19)
and standard spinors in Dirac representation
1
u(p,λ)= p +M χ . (36)
0 p~·~σ λ
(cid:18) p0+M (cid:19)
p
4.1 Bosonic Case
Considering λ=λ′ =j, the polarization tensor (12) of the particle takes a very simple form
j
ε (p,j)= ε (p,+). (37)
α1···αj αl
l=1
Y
Let us discuss separately the charge and magnetic densities.
4.1.1 Electric Part
Concerningthechargedensityofintegerspinparticles,wefindthatallthecomplexityreducestothreestructures
only
ε∗(p′,+) ε(p,+)= 1 τ sin2θ,
· − −
[ε∗(p′,+) q][ε(p,+) q]
· · =(1+τ)τ sin2θ, (38)
2M2
ε0(p,+)[ε∗(p′,+) q] ε0∗(p′,+)[ε(p,+) q]= 2p τ sin2θ,
0
· − ·
leading to the expression
j t
JB0 =2M (1+τ)k+12 Cjj−−kt τ sin2θ t F2k+1(Q2)− 11−+δkτ,0 F2k(Q2) .
t=0k=0 (cid:20) (cid:21)
XX (cid:0) (cid:1)
5Thereader mightbe worriedby the fact that the definition of momentum Qis different inSection 2 compared to Section 3.
Covariant vertex functions and multipoleformfactors arehowever related inthe Breitframe, whereboth definitions doactually
matchQ2≡−q2=q~2.
9
By comparison of this expression with (29) and using (31), we obtain the connection we were looking for
j (−1)m+tτm−t C(2Cmm+t2)t2−1 GE2m(Q2)= t (1+τ)k+12 Cjj−−kt F2k+1(Q2)− 11−+δkτ,0 F2k(Q2) . (39)
m=t 4m−1 k=0 (cid:20) (cid:21)
X X
To illustrate this, let us consider for example the case j =1. The explicit expression of (39) becomes
e
2
G (Q2) τG (Q2)= √1+τF (Q2),
E0 E2 1
− 3
G (Q2)= √1+τ F (Q2) F (Q2)+(1+τ)F (Q2) ,
E2 1 2 3
−
which coincides with the well-known expression for t(cid:2)he electromagnetic interaction of (cid:3)vector particles [23],
provided that
G (Q2)=√1+τG (q2), G (Q2)=√1+τG (q2).
E0 C E2 Q
We remind that √1+τ =p /M in the Breit frame.
0
4.1.2 Magnetic Part
Concerning the magnetic density of integer spin particles, the first simplification is
P~ ~q =0, (40)
×
so that only covariant vertex functions with even value of k will contribute. Moreover, we have in the Breit
frame
~ [~ε(p,λ) ~q]= 0,
∇· ×
[~ε(p,λ) ~q] ~ [ε(p,λ) q]= 0, (41)
× ·∇ ·
p
[~ε(p,λ) ~q] ~ [ε∗(p′,λ′) q]= 0 (~ε∗ ~ε ) ~q,
× ·∇ · −M λ′ × λ ·
from which we obtain
[ε∗(p′,+) q][ε(p,+) q] k
~ ~ε(p,+)[ε∗(p′,+) q] ~ε∗(p′,+)[ε(p,+) q] ~q · · f(Q2)
∇·({ · − · }× (cid:18) 2M2 (cid:19) ) (42)
= 2p (k+1)2i√τ cosθ τ sin2θ kf(Q2).
0
−
This leads to the desired expression (cid:0) (cid:1)
j−1 t
∇~ · J~B×~q =4iM√τ cosθ (1+τ)k+21 Cjj−−kt−−11 τ sin2θ t(t+1) F2k+2(Q2).
(cid:16) (cid:17) Xt=0Xk=0 (cid:0) (cid:1)
By comparison of this expression with (30) and using (31), we obtain the connection we were looking for
j−1(−1)m+tτm−t (m+1) C(2Cmm+t2)t2+1 GM2m+1(Q2)=(t+1) t (1+τ)k+12 Cjj−−kt−−11F2k+2(Q2). (43)
m=t 4m+1 k=0
X X
To illustrate this, let us consider for example the case j =1. The explicit expression of (43) becomes
e
G (Q2)=√1+τF (Q2),
M1 2
which coincides with the well-known expression for the electromagnetic interaction of vector particles [23],
provided that
G (Q2)=√1+τG (q2).
M1 M
The factor√1+τ is presentfor anyintegerspin. One is free to define new multipole formfactorswithout this
inelegantfactor,atthecostofhavingdifferentmultipoledecompositionsforfermionsandbosons. Nevertheless,
at Q2 =0 there is no difference between both definitions.
10