G=2 As the Natural Value of the Tree-Level Gyromagnetic Ratio Of

Home , Gyromagnetic ratio

CERN-TH.6432/92

UCLA/92/TEP/7

g=2 as the Natural Value of the Tree-Level Gyromagnetic Ratio

of Elementary Particles

1 2

Sergio Ferrara , Massimo Porrati

Theory Division, CERN, CH-1211 Geneva 23, Switzerland

and

Valentine L. Telegdi

California Institute of Technology, Pasadena, CA 91125, USA

ABSTRACT

We prop ose a novel \natural" electromagnetic coupling prescription, di erent from

the minimal one, which yields, for elementary particles of arbitrary spin, a gyro-

magnetic ratio g = 2 at the tree level. This value is already known to arise in

renormalizable theories for spin 1/2 and spin 1, is suggested by the classical, rel-

ativistic equations of the spin p olarization, and is also found for arbitrary spin,

charged excitations of the op en string.

CERN-TH.6432/92

UCLA-TEP-92/XX

March 1992

and DepartmentofPhysics, UCLA, Los Angeles CA 90024, USA. S.F. work is supp orted in part by the

Department of Energy of the U.S.A. under contract No. DOE-AT03-88ER40384, Task E.

On leave of absence from I.N.F.N., sez. di Pisa, Pisa, Italy.

Visiting Professor, current address: CERN, PPE Division, CH-1211 Geneva 23, Switzerland.

1 Intro duction

Classically, the ratio b etween the total angular momentum L, and the magnetic moment ~ of a

uniformly charged rotating b o dy is Q=2M , where Q is the total charge and M the mass of the

body. Indeed

Z Z

1 Q Q

3 3

~ = d x (~x)~x ^ ~v (~x)= d x(~x)~x ^ ~v (~x)= L: (1:1)

2 2M 2M

Here (~x) is the mass density, (~x) is the charge density, and ~v (~x) is the velo city of the b o dy

at the p oint ~x. Equivalently, setting ~ g (Q=2M )L , one has g = 1 whenever the ratio of

the charge to the mass density is a constant. Thus, when spin 1/2 was intro duced, g =2 was

anomalous.

Quantum Mechanics and Field Theory change this result signi cantly. In the context of

Field Theory, indeed, the most natural assumption ab out e.m. interactions would b e that of

Minimal Coupling according to which all derivatives of charged elds are replaced bycovariant

ones:

@ ! @ + ieA D ; (1:2)

where e is the charge of the eld . This pro cedure is unambiguously de ned only for fermionic

elds, whose Lagrangian is rst-order in the derivatives. To de ne the substitution (1.2) on

b osonic elds uniquely, one must rst rewrite their Lagrangian in a rst-order formalism [1].

This pro cedure gives a de nite answer for the gyromagnetic ratio g of elementary particles [1,

2, 3, 4 , 5 ]:

g = ; (1:3)

where s denotes the spin of the particle.

This result, a triumph for s =1=2, is in con ict, for higher spin, with a b o dy of (scattered)

evidence that favors another answer, namely that g = 2 for any spin. Let us review that

evidence:

a) Up to now, the only higher-spin, charged, elementary particle that has b een found in nature

is the W b oson. At the tree level it has g = 2, instead of g = 1, as predicted by (1.3).

b) The relativistic (classical) equation of motion of the p olarization 4-vector S in a homogeneous

external e.m. eld is [6] (the classical equation of motion used here had b een anticipated by

Frenkel and Thomas):

dS e e

= gF S + (g 2)U S F U ;

d 2m 2m

U = ; = prop er time: (1.4)

This equation simpli es for g = 2, irresp ective of the spin of the particle.

c) The only known example of a completely consistent theory of interacting particles with spin

> 2 is string theory [7]. For op en (b osonic and sup ersymmetric) strings, it is p ossible to obtain

the exact equations of motion for (massive) charged particles of arbitrary spin, moving in a

constant, external e.m. background. This has b een carried out explicitly for spin 2 in ref. [8]. 1

The equations of motion give g = 2 for al l spins. (This p oint will b e dealt with in greater detail

b elow, see Section 2.)

The aim of this pap er is to provide a simple physical requirement implying g = 2 for all

\truly elementary" (p oint-like) particles of any spin .

Avery imp ortant requirement that one imp oses on theories of spin 1 is that they p ossess

a go o d high-energy b ehavior. More sp eci cally, one imp oses the requirement that all tree-level

amplitudes satisfy unitarityuptoacenter-of-mass energy E much larger than the masses of all

the particles involved. For particles of spin 0, 1/2 and 1, this requirement has b een shown to

b e equivalent to the statement that the most general theory is a sp ontaneously broken gauge

theory plus an (optional) massive U (1) eld [10 ].

Among the tree-level graphs having a (p otentially) bad high-energy b ehavior, there are

+ +

those involving only W , W , and photons, and describing the coupling of a W W pair to

a high-energy photon pair ( g. 1). Notice that these graphs dep end only on the form of the

e.m. current of the W 's, plus a \seagull" term ( in g. 1), which do es not a ect the leading

high-energy b ehavior of the scattering amplitude. The W propagator is, in the unitary gauge,

p p 1

(p)= g : (1:5)

2 2 2

M p M

The term prop ortional to M is precisely the one resp onsible for the bad high-energy b ehavior

of the graphs in g. 1, when the coupling of the W , W to photons is the minimal one. In

fact, it is to cancel the divergences pro duced by the p p =M term, that one must mo dify the

minimal e.m. coupling of the W 's byintro ducing the term

F W W ; (1:6)

yielding g =2. This term, naturally, is present in the Standard Mo del (Yang-Mills) La-

grangian, which indeed gives the following electromagnetic coupling of the W 's:

[ ] 2 + + +

D W +M W W ; L = ieF W W +2D W

[

]

1 e 1 e

+ + + + +

= D W @ W +i A W @ W i A W : (1.7)

[

]

2 2 2 2

The presence of a non-minimal term in gauge theories had already b een noticed in ref. [11].

The ab ove discussion, b esides clarifying why the Standard-Mo del value of g is not the

minimal one, suggests the general requirement that one should imp ose on any theory describing

the interaction of arbitrary-spin massive particles with photons: its tree-level scattering ampli-

tudes should not violate unitarity up to c.m. energies E M=e. In particular, the amplitudes

corresp onding to the graphs of g. 2 must have a go o d high-energy b ehavior.

In the main b o dy of this pap er we shall show that this requirement will indeed imply that

the coupling of higher-spin massive particles to the e.m. eld cannot b e minimal in the usual

sense, and that the non-minimal terms that must b e intro duced lead to g =2 for any spin s.

It is curious to notice that in (unbroken) sup ersymmetric electro dynamics the electron has g =2 even after

radiative corrections, so that g 2 is a measure of sup ersymmetry-breaking e ects [9]. 2

It is useful to note that our requirement can b e cast in the following equivalent form:

\Any theory of massive particles interacting with photons must p ossess a smo oth M ! 0

xed-charge limit."

We remark that requiring a smo oth zero-mass xed-charge limit, and thereby that tree-level

amplitudes saturate the unitarity b ounds only at energies E M=e,isanecessary condition

for having a sensible p erturbation expansion in e (in other words, a sensible electro dynamics).

A theory not satisfying our requirementwould indeed b ecome strongly interacting at E M=e.

It is also imp ortant to remark that this argument ab out go o d high-energy b ehavior of scattering

amplitudes has b een invoked a long time ago byS.Weinb erg, to show that g 2 for any spin,

when a theory is p erturbative, by using general prop erties of the S -matrix and unsubtracted

disp ersion relations [12 ]. This recondite publication came to our attention only after the present

pap er was completed.

Three supplementary remarks are in order:

a) Sup ergravity allows for a spin 3/2 particle, the gravitino, interacting with Ab elian gauge elds,

for example in N = 2 mo dels [13] or in N = 8 mo dels [14] broken alaScherk-Schwarz [15].

In these cases, the charge of the gravitino is prop ortional to the gravitino mass; therefore, the

M ! 0 limit implies e ! 0. Thus, the gravitino charge cannot b e identi ed with the electric

charge: it rather describ es the coupling to the \graviphoton" [13, 14 , 15 ], and a go o d high-

energy b ehavior of the tree-level amplitudes in g. 2 is compatible with a g (prop ortionality

constantbetween the total spin and the dip ole magnetic moment of the graviphoton) di erent

from 2.

b) Interacting higher-spin Lagrangians may give rise to non-causal propagation in an exter-

nal eld [16, 17 ], as in the case of a massive spin-3/2 eld minimal ly coupled to electromag-

netism [16 ]. We shall not deal with this problem in general, since it manifests itself only at O (e )

and therefore can b e solved only with a detailed knowledge of all (sub-leading) contact terms

and gravitational interactions . This knowledge is b eyond the scop e (and against the spirit) of

the present pap er, whose aim is rather to get results that dep end as little as p ossible on the

exact form of the higher-spin Lagrangian. We shall anyhow show in the App endix that, at least

for the case of spin 3/2, appropriate non-minimal interactions (in particular those giving g =2)

eliminate non-causal propagation.

c) In General Relativityitisknown that a charged, rotating black hole has g = 2 [18 ]. Our

argument, tailored for elementary particles, cannot explain, to our knowledge, whether this

result is a mere accident or is due to some deep er reason.

The present pap er is organized as follows: Section 2 sketches the derivation of the gyro-

magnetic ratio in op en-string theory; it is self-contained and can b e skipp ed in a rst reading,

or by the reader not interested in the details of the construction. Section 3 deals with the

consequence of imp osing a smo oth xed-charge M ! 0 limit on massive b oson elds coupled to

electromagnetism, and it is shown there that g must equal 2. Section 4 treats the same problem

for fermion elds; conclusions are given in Section 5. The App endix shows how non-minimal

interaction terms ensure causal propagation of spin-3/2 elds in an external e.m. background.

5 2

We recall that the gravitational background is O (e )owing to Einstein's equations R 1=2g = T .

2 Consistent E.M. Interactions of Higher-Spin Parti-

cles: the Op en-String Example

The op en b osonic string propagating in a constant e.m. background has b een considered in

ref. [8], where it was prop osed to use it as a means of constructing consistentinteracting higher-

spin equations of motion. The case of spin 2 was worked out in detail.

For our purp oses, we need only to work at linear order in the external e.m. eld F .To

this order the b osonic string is almost identical to the free string. In particular, the Virasoro

op erator L [7] reads, as for the free string [8]

L = + : (2:1)

0 m 0

m m

m=1

Here =1; ::; D , with D equal to the dimensionality of the space which the string moves in.

In op en strings, charges are attached to the two end p oints of the string [7, 8 ]. Let us call

these charges e and e . The charged states have a total charge e equal to e + e . This charge

0 0

app ears in the commutation relations of the string oscillators, which read [8] :

[ ; ]= (mg + ieF )+O(F ): (2:2)

m+n;0

m n

This equation tells us, in particular, that the zero mo des ob ey (up to terms O (F )) the

commutation relations of the covariant derivatives D de ned in eq. (1.2), and can therefore b e

identi ed with them.

The physical states of the op en string are built in terms of a ground state j0i ob eying

j0i = @ j0i =0; 8m> 0; (2:3)

and a variable x , conjugate to @ . They read

ik x

= e j0i; m>0: (2:4)

i=1

These states are constrained by a set of auxiliary conditions

L =0; n>0; (L 1) =0; (2:5)

n 0

where L are the Virasoro op erators (see e.g. [7]). Here we need only the equation involving L .

n 0

Recalling that = D , this equation turns out to b e

D D + N +1 =0; N = : (2:6)

m=1

The op erator N satis es the commutation relation

[N; ]= (mg + ieF ): (2:7)

6 0

Here we put the string tension =1. 4

Applying this equation to (2.6), and recalling the form of the state given in (2.4), one nds

N N

X X

::: :: ::

1 1 i1 i+1

N N

D D m +1 + ie F

=0: (2:8)

i=1 i=1

This equation, where wehave written out explicitly the vector indices of the state , implies

1=2

that the mass of the state is (2 m 2) , and that the gyromagnetic ratio g is equal to

i=1

2, as claimed.

The op en sup erstring to o can b e solved in the presence of a constant background e.m. eld.

This string is interesting b ecause its sp ectrum do es not contain tachyons (unlike the b osonic

string) and it contains fermions as well.

The world-sheet action of the op en sup erstring in a constant e.m. eld reads [7]

d d @ X @ X i @ + S =

1 1

[e ( )+ e ( )] F X X F : (2.9)

2 4

Here and are the world-sheet co ordinates and ranges from 0 to ; the dot denotes derivation

with resp ect to the time co ordinate . The , =1;2, are two-dimensional gamma-matrices

! !

0 i 0 i

1 0

; = =

i 0 i 0

1 0

= ; f ; g = 2 : (2.10)

0 1

The elds are two-dimensional Ma jorana fermions, and action (2.9) is invariant under the

standard 2-dimensional sup ersymmetry transformation

X = ; = i @ x : (2:11)

The equations of motion following from eq. (2.11) are

@ @ X =0; @ =0; (2:12)

supplemented by the b oundary conditions

i( ) = e F ( + ); =0; (2.13)

+ 0

i( ) = e F ( ); = : (2.14)

Here are the twochiral comp onents of . In eq. (2.14) the + or signs dep end on whether

one cho oses p erio dic (Ramond) or antip erio dic (Neveu-Schwarz) b oundary conditions for the

variation of

= (R); = (NS); = : (2:15)

+ +

The expansion in oscillators of the eld reads

(g + e F )d ; =0 =

0 n

= (g e F )d ; =0: (2.16)

0 n

n 5

Here n 2 Z for Ramond b oundary conditions and n 2 Z +1=2 for the Neveu-Schwarz ones,

while

fd ;d g = g + O (F ): (2:17)

n+m;0

n m

Physical states are de ned in terms of a vacuum state j0i (NS) or jAi (R), of the op erators X

and (identical with those of the b osonic string), and of the d (n; m < 0). The NS and R

m n

vacua ob ey

j0i = d j0i = @ j0i =0; m>0;

m1=2

jAi = d jAi=@ jAi=0; m>0: (2.18)

m m

The R vacuum is degenerate, and forms a spinor of the D -dimensional target space, of index A.

Therefore, all NS states are D -dim b osons, and all R states are fermions.

The equivalent of eq. (2.6) assumes now a di erent form in the NS and R sectors:

2 3

1 1

X X

1 1

4 5

(mg + ieF )d d D D + + +

m m

2 2

m=1

m=1=2

:::

= 0 (NS);

# "

1 1

X X

1 1

D D + + ieF d d + (mg + ieF )d d

m 0 m

2 2

m=1 m=1

:::

= 0 (R): (2.19)

:::

::: N

The states and have the form

::: ik x

a j0i; = e

i=1

:::

ik x

i 1

a jAi; m>0: (2.20) = e

i=1

Here a denotes or d oscillators. Substituting expressions (2.20) into eqs. (2.19), and using

m m m

the commutation relations (2.17) and (2.2), we nd again that, for all spins, g =2.

3 g=2 for All Spins: the Integer-Spin Case

String theory provided us with an example of a consistent theory describing higher-spin, charged,

massive particles interacting with an e.m. eld, and in that theory we found that all charged

particles have g = 2. In this section we shall show that this result holds for all integer spins,

and in any theory satisfying the general requirement stated in the intro duction, namely that its

tree-level scattering amplitudes p ossess a smo oth zero-mass xed-charge limit.

A metho d to derive free Lagrangians for massive particles of any spin was prop osed by

Fierz and Pauli [19 ]. For integer spin, explicit Lagrangians implementing that prop osal were 6

given in ref. [1]. Those Lagrangians are written in terms of a set of elds that, for spin s, are

symmetric-traceless tensors of rank s; s 2;s 3; ::; 0:

s s2 s3 0 p

; ; ; :::; ; =0: (3:1)

::: ::: ::: :::

s p

1 1 s2 1 s3

The Lagrangian reads [1]

s 2 s s s

L = (@ @ M ) + s@ @

s(s 1) s

s2 s s2 2 s2

+ @ @ (@ @ M )

2s 1 s 1

(s 1)

s2 s2 s s2

@ @ + ::: : (3.2) + @ @ +

2s 1

p :::

Here @ @ etc. The complete system of equations of motion derived from (3.2) is

p 2 s s

=0; p

In what follows, we shall not need the explicit dep endence of the Lagrangian (3.2) on etc.

Notice that the minimal coupling de ned by eq. (1.2) is ambiguous. For instance, integrating by

parts the second term in eq. (3.2) and then making the substitution (1.2), one nds a Lagrangian

which di ers from the one obtained by making substitution (1.2) directly in (3.2) by

:::

F : (3:4) ies

:::

(From nowonwe suppress the index s whenever no ambiguity arises.) The on-shell e.m. current

generated by the minimal substitution (1.2) into (3.2) reads

(0) :::

J = ie ; (3:5)

:::

whereas byintegration by part of (3.2) one nds

:::

(p) ::: s

J = ie iep@ ( ); (3:6)

:::

] :::

1 s

[

where the integer p can assume all values b etween 0 and s. In the non-relativistic limit all elds

with time-like indices can b e neglected, and the current (3.6) gives rise to the following term

in the Hamiltonian of the system

(0) jj :::j kj :::j

s s

2 2

J A iep F ; j =1;2;3: (3:7)

jk a

By noticing that in the non-relativistic limit

i i

j j :::j j j :::j j j :::j

s s s

1 2 1 2 1 2

= = ; (3:8)

M M

j j :::j j j :::j

s s

1 2 1 2

where is the canonical momentum conjugate to , one discovers that the Hamil-

tonian contains the term

(0)

~ ~

SB; (3:9) J A +

2Ms 7

i ij k i

with B =(1=2) F the magnetic eld, and S the spin op erator. Equation (3.9) says that

g = p=s. The value g =1=s is recovered when Lagrangian (3.2) is rewritten in rst-order

formalism, intro ducing the eld-strengths [1]:

(s 1)

s::: ::: ( s ::: )

s s s

2 2 2 3

H = @ @ $ ; etc: (3:10)

Notice that any de nition of minimal coupling gives always g 1.

If we carry out the minimal substitution on the Lagrangian (3.2), we nd that the scattering

amplitudes in g. 2 do not have a smo oth zero-mass xed-charge limit. This result comes ab out

b ecause the propagator of the spin-s eld b ecomes singular in the M ! 0 limit. This singularity

stems from the existence of null vectors of the kinetic op erator of (3.2). In other words, the

propagator is singular due to the existence of gauge invariances at M =0.To derive a result

on the gyromagnetic ratio it suces to nd one of these null vectors; for instance

s 1

= @ g @ ;

:::

( ::: ) ( ::: )

1 s s

1 2 1 2 3

2s 1

= @ ; =0; p

::: :::

1 s2 1 s2

@ @ = 0: (3.11)

:::

The round bracket means total symmetrization.

s s2

Let us couple now the spin-s eld to an external source J ;J ; :::; by adding to the La-

grangian (3.2) the following term

::: :::

1 s2 1

+ ::: : (3:12) + J J

::: :::

1 s2 1

The existence of the null vector (3.11), together with the requirement of a smo oth zero-mass

limit, leads to the following constraint on the source

2s 1

@ J + @ J = MX : (3:13)

::: :::

( ::: )

1 s1 1 s1

1 2 s1

The lab el ST means that only the symmetric traceless part of the tensor has to b e kept; X

:::

1 s1

denotes any op erator with a smo oth zero-mass limit.

Minimal coupling gives, up to terms vanishing on-shell

J = ie@ (A )+ieA @

::: ::: :::

s s s

1 1 1

s 1

@ A ; (A )+ie g ies@

::: ) ::: ) ( (

s s

3 2 1 2 1

s(s 1)

J = ie @ A : (3.14)

::: :::

1 s2 1 s2

2s 1

Working for convenience in the gauge @ A = 0, and recalling that the elds , A are

:::

on-shell (see g. 2), we nd that the left-hand side of eq. (3.13) equals

2ieF @ ieM A : (3:15)

::: :::

1 s1 1 s1 8

The rst term of eq. (3.15) do es not have the prescrib ed b ehavior for M ! 0. In order to cancel

it, one must add a non-minimal term to the interacting Lagrangian :

:::

i F : (3:16)

:::

This term gives the following contribution to the l.h.s. of eq. (3.13)

1 s 1

i F @ i @ F : (3:17)

(

::: 1 ::: )

1 s1 2 s1

s s

Notice that the term prop ortional to F cancels for

=2se: (3:18)

This value gives g =2!

The additional term in (3.17), whichvanishes for spin 1 (the W case), and in a constant

e.m. eld, can b e cancelled by adding the following term to the Lagrangian:

::: :::

s s

3 3

+i F @ i @ @ F @ @

::: :::

s s

3 3

s 1 2

::: :::

1 s2 1 s2

2(@ F @ )+i @ F @ i

::: :::

1 s2 1 s2

M 2s 1

+h:c: ; (3.19)

2 4

where =2s(s1)e=M , =2s(s1)e=M .

The term (3.19) intro duces contributions prop ortional to (s 2) in the l.h.s. of equa-

tion (3.13). These contributions can themselves b e cancelled by adding higher-multip ole and

o -diagonal terms, and so on. We do not carry through the pro cedure, which yields all the non-

minimal multip ole terms of the interacting Lagrangian, since it b ecomes rapidly burdensome

and is not illuminating. Rather, we conclude this section by observing that the requirementof

a smo oth M ! 0 xed-charge limit, i.e. of a go o d high-energy b ehavior, not only xes uniquely

the gyromagnetic ratio g to b e equal to 2 for al l integer spins, but also requires the presence of

ad hoc non-minimal multip ole terms, which are non-power-counting renormalizable operators.

In particular equation (3.19) implies that the ad hoc electric quadrup ole momentvanishes ,

whereas non-zero magnetic o ctup ole-moment terms must b e present in the interacting theory

of charged particles of spin larger than 1.

Next, we shall examine the extension of the general results obtained in this section to the

case of fermionic elds.

4 g=2 for All Spins: the Half-Integer-Spin Case

The free Lagrangian for particles of half-integer spin s = n +1=2 can b e written in terms of

symmetric gamma-traceless vector-spinors of rank n; n 1; ::; 0 [1].

p n n1

; ; p

::: ::: :::

p n

1 1 1 n1

The analogous term where F is replaced by F is not allowed, b ecause it breaks parity.

The total electric quadrup ole moment is already non-zero in the spin 1 case. 9

ip n n1

= = =0: (4.1)

::: ::: :::

p n

2 2 2 n1

Notice that the vector-spinors of rank lower than n 1 app ear with multiplicity 2 [1, 4].

Here we switch to the Pauli metric g = , and de ne the gamma-matrices as follows

y 5 1 2 3 4

f ; g = ; = ; = : (4:2)

The Lagrangian reads

2 2

2n 2n

n n n n1 n1 n

L = (@= M ) @ + @ +

2n +1 2n +1

2n n+1

n1 n1

@= + M + ::: (4.3)

2n+1 n

We shall not need the explicit dep endence of (4.3) on the lower-rank vector-spinors.

The minimal coupling to electromagnetism of the Lagrangian (4.3) is well de ned. Substi-

tuting (1.2) into eq. (4.3) determines the e.m. current, which, up to terms vanishing on-shell,

:::

J = ie : (4:4)

:::

Again, we shall drop the lab el n whenever unambiguous. The equations of motion derived

from (4.3) are [1]:

n n n1 ip

(@= M ) =0; @ =0; = =0; 8i; p: (4:5)

These equations allow us to rewrite the current J as (see also [5]):

e e

::: :::

n n

1 1

@ + i ); J = i @ (

::: :::

n n

1 1

2M 2M

1 1

: (4.6)

2 2

:::

In the non-relativistic limit all containing at least one time-like index vanish. The

non-relativistic Hamiltonian thus contains a term

e e 1

ij j :::j j :::j

n n

1 1

~ ~

= F B S; (4:7)

4M 2M n +1=2

which implies g =1=s.

The reasoning used in the previous section ab out the M ! 0 limit rep eats here identically.

The kinetic term of the Lagrangian (4.3) p ossesses zero mo des (gauge invariances at M = 0).

One of these zero mo des is

n 2n(n +1)

@ @= ;

( ::: ) ( ::: ) :::

n n n

1 2 1 2 1

2n+1 2n +1

= @= ;

::: :::

1 n1 1 n1

@ = =0: (4.8)

::: :::

2 n1 2 n1 10

n n1 n n1

By coupling ; etc. to an external source J ;J etc., one nds this time that the

exsistence of a smo oth zero-mass xed-charge limit implies the following equation

2n +1

@ J + @J= = MX ; (4:9)

::: ::: :::

1 n1 1 n1 1 n1

2n(n +1)

with X any op erator having a smo oth M ! 0 limit.

:::

1 n1

Minimal coupling gives

J = ie=A ie A ;

::: :::

( ::: )

n n

1 1 n

1 2

n +1

J = ie A : (4.10)

::: :::

1 n1 1 n1

2n +1

With these values of the sources, the l.h.s. of eq. (4.9) b ecomes, up to lower-spin terms

ie F ie A : (4:11)

:::

::: 1 n1

1 n1

n +1

In order to cancel the rst term in the ab ove equations, wemust add to the Lagrangian a

non-minimal term

+ :::

F ; (4:12) i

:::

where we made use of the de nitions

F F F ; =+1: (4:13)

5 1234

Term (4.12) gives rise to the following contribution to the l.h.s. of eq. (4.9)

n 1 1

: (4:14) F i @ F i M

:::

( ::: )

1 n1

1 2 n1

n n

In deriving eq. (4.14) wehave used the following identities

F = 0

:::

1 n1

F @ = M F : (4.15)

:::

1 n1

:::

1 n1

Comparing eq. (4.14) with eq. (4.11) we nd the cancellation condition

M e =0: (4:16)

This results in = en=M , which, thanks to the identity

::: :::

n n

1 2

; (4:17) F = F

:::

gives, as exp ected, g = 2 for all n. 11

Notice that the additional term present in eq. (4.14) vanishes for n = 1, that is for spin-3/2

particles, or for constant F . This term can b e cancelled by adding the following contributions

to the action

::: + :::

1 1 n1

i @ @ F +[i @ F

::: :::

2 3 1 2 n1

::: +

1 n1

+i @@= F +h:c:]; (4.18)

:::

1 2 n1

with

2 2 2

e e e 2n(1 n ) 2n(1 n ) n(1 n )

; = ; = : (4:19) =

2 3 3

2n +1 M (2n + 1)(n +2)M n +2 M

The term (4.18) also gives additional contributions to the l.h.s. of eq. (4.9), prop ortional to

(n 2) (vanishing for spin-5/2 particles). Again, these terms can b e cancelled by adding higher

multip ole terms. We remark that, as for the integer-spin case, no explicit quadrup ole moments

are intro duced, while, for generic spin, a non-minimal magnetic-o ctup ole moment arises.

5 Conclusions

Our excursion into higher-spin elds has b een successful: we found that a very simple and natu-

ral requirement imp osed on tree-level scattering amplitudes, viz. that of a smo oth xed-charge

M ! 0 limit, is p owerful enough to x the gyromagnetic ratio g to its most natural value, in the

BMT sense [6]. Our arguments are strictly related to those of ref. [12 ], and it is encouraging to

see that an explict Lagrangian calculation like ours, and an S -matrix argument, as given in [12],

lead to the same conclusion ab out the value of g .We also discovered that our same requirement

uniquely xes higher-multip ole terms (eqs. (3.19), (4.18) and (4.19)), and implies the existence

of quasi-conserved (tensor) currents. This seems to suggest, in analogy with the spin-1 case,

that a consistent theory of massivecharged particles of arbitrary spin would require a large

sp ontaneously broken lo cal symmetry. Indeed, in the string case, there have b een sp eculations

that the in nite-slop e limit would restore an in nite dimensional symmetry [20], p ossibly related

toavanishing exp ectation value of the metric tensor hg i = 0 [21 ]. We nally note that g =2

is the only value that allows for a reliable p erturbative expansion, if a completely consistent

theory of interacting higher-spin particles is ever to b e found. A di erentvalue indeed implies

that the radiative corrections, necessary in order to restore unitarity, are large; thus, whichever

the value of g at the tree level, it would receivehuge corrections due to lo ops (see also ref. [12]).

Acknowledgements

Wewould like to thank R. Finkelstein, J.H. Schwarz, R. Stora and D. Zwanziger for useful

discussions and suggestions. 12

App endix: Causal Propagation of Spin-3/2 Particles in

an External E.M. Field

In this App endix we show that the propagation of spin-3/2 particles in an external e.m. eld is

causal, if an explicit magnetic dip ole moment(Pauli term) is added to the minimal Lagrangian.

Non-causal propagation [16 ] seems therefore to b e a pathology of minimal coupling, rather than

of higher-spin equations themselves.

The spin-3/2 equations of motion, in the presence of the dip ole term

i F ; (A:1)

(see eq. (4.12)) read

2 1 1

D= D M D + D= + F = 0; (A.2)

3 2 6

D= +2M D = 0: (A.3)

In order to study causality of propagation of waves, wemust rst bring the highest derivative

term in a non-singular form [16 ]. To this end, let us de ne the elds , and by

4 1

= + D D=; = + D=; D =0: (A:4)

3 3

The system of equations (A.2, A.3) now reads

2 4 1 1 1

+ +

D= D MD + M D= + D= + F D F D= = 0; (A.5)

3 3 3 6 3

D= +2M +2MD= = 0: (A.6)

Propagation of signals in (A.5) and (A.6) is now studied with the metho d of refs. [16, 17 , 22]:

one substitutes in (A.5, A.6) the characteristic 4-vector n to the covariant derivatives, and

keeps only the leading term in n . If the resulting algebraic system admits a zero eigenvalue for

some n , then the maximum sp eed of signal propagation is given by n =j~n j [16, 17, 22 ]. Notice

that this pro cedure corresp onds to solving eqs. (A.5,A.6) in the eikonal approximation

itn x itn x itn x

^ ^

= e ; = e ; =^e ; t !1; (A:7)

^ ^

for some constantvectors , and ^. Equation (A.6) b ecomes

n=( +2M^)=0: (A:8)

2 2

This equation says that, for n 6=0, =2M^. Notice that n = 0 is the standard light cone.

Carrying on the substitution D ! in in eq. (A.5), and b ecause of eq. (A.8), one nds

4 1

+ +

n= + F n ^ F n=^ =0: (A:9)

3 3 13

By contracting this equation with n one arrives at

F n ^ =0: (A:10)

If wemultiply this equation by F n , and pro ject over the left- and right-handed comp o-

nents ^ ,^ of ^,we nd

L R

2 n n g F F + F F + F F ^ =0; (A:11)

and a similar term for ^ . The term prop ortional to the identity matrix in eq. (A.11) is

2 n n T ; (A:12)

where T is the stress-energy tensor of the e.m. eld.

Non-causal propagation would arise if a solution to eq. (A.10) would exist for some time-like

n . By going to the Lorentz frame where n =(1;0;0;0), this is easily seen to b e imp ossible,

thanks to eq. (A.12). In fact, T vanishes only for F = 0, when spin-3/2 propagation is

trivially causal; therefore, a time-like n implies ^ = 0, and eq. (A.9) reduces to

n= =0: (A:13)

This equation, for n 6= 0, admits only the trivial solution =0.

This establishes causality of the propagation of spin-3/2 waves for all 6= 0. Notice that

the minimal-coupling case = 0 represents a pathological limit.

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FIGURE CAPTIONS

Fig. 1) Wiggly lines represent photons, straight lines W and W .

Fig. 2) denotes a spin-s charged, massive eld, represented by an oriented solid line. Contact

terms are ignored b ecause sub-leading at high c.m. energies. 15 W+ W- W+ W-

+ +

γ γ γ γ γ γ γ) α) β)

Fig. 1

ψ+ ψ- ψ+ ψ-

γ γ γ γ α) β)

Fig. 2