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 3 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 2 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. 2 On leave of absence from I.N.F.N., sez. di Pisa, Pisa, Italy. 3 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) c 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 c ~ 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 ]: 1 g = ; (1:3) s 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 dx U = ; = prop er time: (1.4) d 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 4 \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 2 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 2 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- W 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 W 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. 4 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 2 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- 0 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- 2 netism [16 ].