Charge renormalization and high spin fields

Volume 102B, number 1

PHYSICS LETTERS

4 June 1981

CHARGE RENORMALIZATION AND HIGH SPIN FIELDS Thomas L. CURTRIGHT

Department of Physics, Universityof Florida, Gainesville, FL 32611, USA Received 21 January 1981

A simple physical interpretation of charge renormalization in non-abelian theories permits the rapid determination of 13(one-loop) in models containing high spins (S > 1). For any possible choice of spins, all N > 4 extended supersymmetry multiplets are "doubly-finite" and give t3 (one-loop) = 0 for either their SO(N) or SU(N) internal symmetry groups.

Introduction. Recently, Hughes [1 ] presented a simple physical interpretation of the charge renormalization effects occurring in non-abelian gauge theories. He performed a one-loop calculation in which the infinities appearing in go = Zgg, and hence the corresponding contributions to the Gell-Mann-Low function, separated into two distinct components arising from "convective charge" and "polarization" interactions. Consequently, one was finally able to understand both the sign and the magnitude of the one-loop charge renormalization occurring in Yang-Mills theory as due to the dominance o f the negative polarization contribution from the vector field. In this paper, we apply the same interpretation to theories containing high spin fields (S >1). This enables us to rapidly reproduce the conclusions of Christensen et al. [2] concerning one-loop charge renormalization in extended supergravity theories with N fermionic transformations. For N = 2 to 4, the divergent part of Zg is positive, while for N = 5 to 8, Zg is finite and the GeU-Mann-Low function vanishes. These results hold when the gauge group is either the natural SO(N) or the SU(N) internal symmetry of the multiplet. In fact, we find N > 4 supergravity theories are "doubly-Finite" in the sense that the two ultraviolet divergent contributions to Zg, i.e. the convective charge and the polarization contributions, each vanish individually. These features are special cases of spinmoment sum rules for extended supermultiplets,

which may be connected with the finiteness o f supergravities at the ttighe/-loop level. We also consider other extended supermultiplets w i t h N > 4 , including those with spins S >2. We show that these too are doubly-finite and hence give /3(SO(N) or SU(N) one-loop) = 0. Indeed, all N > 4 particle supermultiplets give/3 (one-loop) = 0, even those constructed from maximum helicity states which are not group singlets. In performing our analysis, we simply make a direct extension of the results in ref. [1 ] and do not concern ourselves with the inconsistencies of.minimal gauge coupling to high spins * 1 or the lack of renormalizability of such models. Such issues are important, but simply beyond the one-loop charge renormalization effects considered here. Finally, we note that our results apply not only to the fundamental massless multiplet o f N = 8 supergravity, but also to each of the massless supermultiplets which compose the massive N = 8 supercurrent multiplet as proposed by Ellis et al. [4]. These multiplets all contribute finite one-loop charge renormalization when considered as particle representations of either an orthogonal SO(8) or a unitary SU(8) gauge group.

Basic results. By considering quantum corrections to the interaction energy of fixed external sources ,1 These and other problems associated with high spins are reviewed in ref. [3]. 17

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coupled to non-abelian gauge fields fluctuating about a specified coulombic background, it is possible [ 1] to separate the one-loop diagrams relevant for charge renormalization into two sets. Those in the first set depend only on "convective" (electric type) charges and contribute equally to the divergences in Zg for each degree of freedom appearing in the quantum corrections, regardless of the spin carried by that particle (degree of freedom). Those diagrams in the second set, however, are due to "polarization" effects and depend simply but crucially on the helicities carried by the contributing degrees of freedom. As explicitly observed in ref. [1] for helicities 0, 1/2, and 1, once such a separation is realized to occur, one can immediately write the general contribution to the oneloop charge renormalization factor from a physical particle carrying helicity S as/3(S) = (A +BS 2) × C ( - 1 ) 2S, where C is the appropriate (positive) quadratic invariant for the gauge group representation also carried by that particle. The phase ( - 1 ) 2S corresponds to the usual particle statistics. Note that the one-loop divergent part [i.e. the coefficient of 1/e, e = (4 - D ) / 2 ] of Zg is just/3/2. Now the coefficients of the convective charge and polarization contributions, A and B, can be determined by comparing with the results known for S = 0, 1/2, and -1/2. Thus the one-loop charge renormalization contributed by a physical degree of freedom carrying helicity S is simply given by

M(d/dM)g =/3(S) = (h/961r2)g3C(1 - 12S2)(-1) 2S • (1) One immediately understands the asymptotic free-

4 June 1981

dom (/3 < 0) of pure Yang-Mills theory (S = 1 and - 1 ) as due to a large polarization effect [1 ]. One should also understand that in determining the total charge renormalization, it is sufficient to count only the physical degrees of freedom carried by a field, e.g. only the extreme helicities representing a massless particle of spin S. Superfluous field components and ghost fields must cancel against one another, if the latter are properly specified.

Application to extended supergravities. It is straightforward to apply eq. (1) to any spin, but it is most compelling to apply the formula to those models which naturally contain high spins, i.e. extended supergravities, where previous results have been obtained for/3(one-loop) by other methods [2]. In table 1, we list the helicities, dimensions, and quadratic invariants [5] for extended supergravity multiplets for N = 2 to 8. The gauge group is either SO(N) or SU(N) in table 1. Note that the C's of SO(N) are proportional to those of SU(N) for the representations in the table, and a universal rescaling of the C's in the table is necessary to obtain the conventional quadratic invariants that usually appear in eq. (1) [e.g. multiply by 1/2 for SU(N)]. Also note that adding CPT conjugate multiplets, where necessary, will only double the value of 13obtained. Note that both the dimensions and the invariants are simple binomial coefficients for the group representations in the table. Furthermore, one obtains C(j) for a given value of N from D ( j - 1) for N - 2 . Also note that the quadratic invariant for S = 3/2 is used to fix the normalization for other helicities.

Table 1 Helicities (S), representation dimensions (D), quadratic invariants (6"), and j3 (one-loop) for extended supergravities.

D(C) S

N=2

3

4

5

6

7

8

2 3/2 1 1/2 0 -1/2 -1

1(0) 2(1) 1(0)

1(0) 3(1) 3(1) 1(0)

1(0) 4(1) 6(2) 4(1) 1(0)

1(0) 5(1) 10(3) 10(3) 5(1) 1(0)

1(0) 6(1) 15(4) 20(6) 15(4) 6(1) 1(0)

1(0) 7(1) 21(5) 35(10) 35(10) 21(5) 7(1)

1(0) 8(1) 28(6) 56(15) 70(20) 56(15) 28(6)

1(0)

8(1)

-3/2

-2 × 96w2/(hg 3) =

18

1(o) 26

15

6

0

0

0

0

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PHYSICS LETTERS

Simple arithmetic using table 1 shows that the total convective charge contribution to/3 (one-loop) vanishes for N >2. Thus for N = 3 and 4, the positive contributions to/3 are entirely due to polarization effects, i.e. the S 2 terms in eq. (1). Similarly, f o r N > 4 we see that both the convective charge and the polarization contributions individually vanish when summed over the supermultiplet, which is a somewhat stronger statement than the previously observed/3 (one-loop) = 0. Consequently, we say that the N > 4 models have the property of being "doubly-finite". We conjecture this property is necessary for/3 = 0 in all higher orders. In this regard, note that a Y a n g Mills theory coupled to 22 S = 0 adjoint fields has t3(one-loop) = 0, but 13(two-loop) > 0. The vanishing one-loop result is due to a cancellation between nonzero convective and nonzero polarization contributions in such a model ,2 It is instructive to compare the above derivation of/3 (one-loop) = 0 for N > 4 supergravities with that of ref. [2]. Christensen et al. compute the divergences in A, the cosmological constant, for N extended supergravities modified to contain such a term. These divergences are then related to those in the natural SO(N) gauge coupling constant by supersymmetry. Finite A corrections then imply finite g corrections, but only in supergravity theories. For such an indirect calculation of 13(one-loop), the S = 2 graviton plays a major role, contributing heavily to the infinities in the cosmological constant. In our direct calculation of/3 (one-loop), as evident in table 1, the S = 2 state contributes nothing since it is a group singlet with C = 0. The compatibility of the two calculations must ultimately be due to the supersymmetry of the theories. Generalization to all extended supermultipIets. We now generalize the preceding discussion of doublyfinite supergravities by presenting a collection of simple mathematical results dealing with "spinmoment" sum rules for extended supermultiplets. For a massless N extended supermultiplet, {SO") =

*2 Incidentally, the model with 22 (pseudo) scalars can be obtained by dimensionally reducing the Veneziano model from its critical dimension, D = 26, just as the N = 4 supersymmetric Yang-Mills theory can be obtained by reducing the spinning string model from its critical dimension, D = 10. This suggests some connection between /3(one-loop) = 0 and dual model consistency conditions.

4 June 1981

S - f/2;/= 0, 1 ..... N}, we define the ~ and e k-th spin-moments as "r(k) = ~ ( /

1)2Sq) SO')k DO') ,

o(k) = ~ ( - 1)2S(/)SQ)kC0'), /

(2)

where DO') (CO')) is the dimension (quadratic invariant) for the internal SU(N) or SO(N) symmetry group representation at the helicity SO'). For massless multiplets whose maximum helicity, S, is an arbitrary SU(N) (or SO(N)) representation, one can prove the following statements.

Lemma. If o(0) = 0, then o(1) = 0, and if 0(2) = 0 as well, then E [ ( - 1 ) 2s(/) [S(j) + t]2C(j) = 0, for any translation, t, of the helicities in the multiplet. Theorem 1. The dimension (D) and the quadratic invariant (C) of the group representation at helicity S - ]/2 are given in terms of binomial coefficients. In general these representations are reducible.

o0 :o<1,

)

where D(0) (C(0)) is the dimension (quadratic invariant) for the representation at maximum helicity, S.

Theorem 2. For any N extended supermultiplet, regardless of the value of the maximum helicity, ~-(k) = 0 ,

for k = 0, 1 ..... N - 1 ,

o(k) = 0 ,

for k = 0, 1 ..... N -

3.

For those unwilling to make an inductive leap on the basis of the results in table 1, we remark that the preceding theorems can be proved using simple properties o f representation indices as given in ref. [5]. The preceding results immediately lead to the following physically interesting conclusion.

Corollary. Any N > 4 supermultiplet gives vanishing convective charge and polarization contributions to the Gell-Mann-kow function for its internal SU(N) or SO(N) symmetry group: o(0) = 0, o(2) = 0, and hence 13(one-loop) = 0.

19

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PHYSICS LETTERS

This corollary is the essential physical result o f this paper * 3 Note that we have always taken the supermultiplets to be massless in stating the above results. This is n o t a necessary restriction, as can be easily understood if one observes that massive multiplets may be obtained through a Higgs-type procedure implemented on massless multiplets, either singly or in combinations. For such massive supermultiplets, the preceding spinmoment sum rules also hold. Also note that the spin-moment relations in theorem 2 are similar to the mass-moment relations found in spontaneously broken supergravities [6]. Both sets o f moment relations are connected to broken scale invariance, the absence o f ultraviolet divergences in the cosmological constant, and the general softening of radiative corrections. Thus we firmly suspect that such moment relations are closely related manifestations o f a deeper infra-structure. Unfortunately, we cannot yet state the precise relationship. One possibility is that the spin- and mass-moments b o t h derive from higher indices [5] of supergroup representations, including those based on superalgebras with central charges ,4 Another possibility is that de Sitter space provides the key to understanding connections between moment relations. Not only would this naturally involve the cosmological constant, but it might also provide the connection between translations and rotations needed to understand the simultaneous vanishing o f the convective ("translational") and polarization ("spinning") contributions to the charge renormaliza-

,3 If the gauge group G is "external", and not the internal SU(N) or SO(N), so that all members of the supermultiplet transform as the same representation of G, as is the case for the N = 4 supersymmetric Yang-Mills theory, then the physically interesting conclusion from theorems 1 and 2 is 13(one-loop for G) = 0 for N > 2 extended supermultiplets. Thus not only the m = 0 states, but each mass level of the dimensionally reduced spinning string would contribute 13(one-loop) = 0. ,4 In fact, it is possible to use quartic indices [denoted F(f)] for the group representations contained in extended supermultiplets and obtain additional spin-moment sum rules. Explicitly we find p(k) = ~](-1)2SQ)s(I')kFQ) = 0 for k = 0, 1, ..., N - 5. The pattern apparent in the .c, a, and p sum rules suggests possible generalizations for all higher indices (i.e. the corresponding upper limit on the helicity power for an nth order index would be N - n - 1). 20

4 June 1981

tion for N > 4. We plan to further pursue these speculations in a subsequent paper. A prime example o f a massive supermultiplet is provided by the multiplet containing the supercurrents which we next discuss for N = 8. Massive N = 8 s u p e r c u r r e n t multiplet. A physically interesting application of the corollary may be found in the massive N = 8 supercurrent multiplet [4], which has been proposed as a source for all presently observed (and anticipated) quarks, leptons, and bosons. For N extended supergravities, it is natural to consider such super current multiplets in order to locate gauge fields for SU(N), and avoid being limited to the SO(N) adjoint vector fields found in the fundamental supergravity multiplet. The massive N = 8 supercurrent multiplet may be obtained by first combining antisymmetrized products of 8's and 8-'s, of SU(8) with a group singlet state at the maximum helicity S = 2, reducing S b y 1/2 for each 8 or 8- in the product. Then, in order to obtain the intermediate helicities appropriate for massive fields, one must combine this helicity tower with similar towers constructed from maximum helicities S = 5/2, 3, and 7/2, as well as with C P T conjugate towers. For all of these helicity towers, the various states may be partitioned into linear combinations of the five distinct massless supermultiplets (or their C P T conjugates) which we have listed in table 2. For each o f these five massless multiplets, one can readily use the information in table 2 to show that a ( k ) = 0 for k = 0, 1, ..., 5. This is true regardless o f the maximum helicity, S, o f the massless multiplet, in accord with theorem 2. Each of the five supermultiplets in table 2 therefore gives/3(SU(8) oneloop) = 0. Since the massive N = 8 supercurrent multiplet is a simple linear combination o f those massless multiplets, it also produces a vanishing GellM a n n - L o w function. Regarding the physics of the supercurrent multiplet, it has been suggested [4] that the SU(8) symmetry exhibited b y that multiplet may be drastically broken down to SU(5), with a natural breaking scale of Mplanck and with the SU(5) identified as the standard grand unification of SU(3)c × (SU(2) X U(1))ew. Such drastic SU(8) breaking could be a very elaborate quantum effect involving anomalies. A priori, it is then not in the least apparent that SU(8) would yield

Volume 102B, number 1

PHYSICS LETTERS

4 June 1981

Table 2 a) Helicities, SU(8) representations, and quadratic invariants for massless supermultiplets appearing as components of the massive N = 8 supercurrent multiplet. Helicity

(8/)A X 1 (8])A X 8

(8-]')A x 28

( 8 / ) n × 56

(8J)A X 70

s s - 1/2 s- 1

1(0) 8(1) ~(6)

8(1) 63(16) + 1(0) 216(75) + 8(1)

s - 3/2

56(15)

420(170) + 28(6)

s- 2

70(20)

504(215) + 56(15)

s - 5/2

56(15)

378(156) + 70(20)

s- 3

28(6)

168(61) + 56(15)

s - 7/2 s- 4

8(1) 1(0)

36(10) + 28(6) 8(1)

28(6) 21-6(75) + 8(1) 720(320) + 63(16) + 1(0) 1344(680) + 216(75) + 8(i) 1512(804) + 420(170) + 2--8(6) 1008(526) + 504(215) + ~(15) 336(160) + 378(156) + 70(20) 168(61) + 56(15) 28(6)

56(15) 420(170) + 28(6) 1344(680) + 21--6(75) + 8(1) 2352(1344) + 720(320) + 63(16) + 1(0) 2352(1414) + 1344(680) + 216(75) + 8(1) 1176(700) + 1512(804) + 420(170) + 2-8(6) 1008(526) + 504(215) + ~(15) 378(156) + 70(20) 56(15)

70(20) 50--4(215) + 56(15) 1512(804) + 42--0(170) + 28(6) 2352(1414) + 1344(680) + 216(75) + 8(1) 1764(1120) + 2352(1344) + 720(320) + 63(16) + 1(0) 2352(1414) + 1344(680) + 216(75) + 8(1) 1512(804) + 420(170) + 28(6) 504(215) + 56(15) 70(20)

(s-]/2: f=O,1..... 8)

a) For the highest weight coordinates for the representations in the table: see table 3. Table 3 Highest weight coordinates for the representations listed in table 2. 8 = (1000000) 28 = (0100000) 36 = (2000000) 63 =(1000001)=6-~ 56 = (0010000) 70 =(0001000)=if'0 168 =(1100000) 216 =(1000010) 336 =(0200000) 378 =(1010000) 420 =(1000100) 504 =(1001000) 720 =(0100010)=720 1008 =(0110000) 1176 =(0020000) 1344 =(0100100) 1512 =(0101000) 1764 =(0002000)=1764 2352(1414)=(0011000) 2352(1344)=(0010100)=2352(1344)

q u a n t i t a t i v e p r e d i c t i o n s r e l a t i n g t h e various families o f q u a r k s a n d l e p t o n s . H o w e v e r , t h e p r e s e n t result o f 13 ( S U ( 8 ) o n e - l o o p ) = 0 for all N = 8 s u p e r m u l t i p l e t s suggests t h a t all radiative q u a n t u m effects p e r t a i n i n g

to S U ( 8 ) m i g h t n o t be as v i o l e n t a n d inaccessible as Mplanck w o u l d indicate. Thus t h e r e m a y i n d e e d be c o m p u t a b l e i n t e r f a m i l y relations w h i c h p e r m i t exp e r i m e n t a l tests o f S U ( 8 ) b a s e d o n N = 8 supergravity. It is a pleasure t o t h a n k Professors P. R a m o n d , C.B. T h o r n , a n d Dr. C.K. Z a c h o s for e n c o u r a g i n g discussions.

References [1] R.J. Hughes, Phys. Lett. 97B (1980) 246; Caltech preprint CALT-68-811. [2] S.M. Christensen, M.J. Duff, G.W. Gibbons and M. Ro~ek, Phys. Rev. Lett. 45 (1980) 161. [3] T.L. Curtright, High spin fields, Proc. XXth Intern. Conf. on High energy physics (Madison, WI, 1980), eds. L. Durand and L.G. Pondrom (ALP) pp. 9 8 5 - 9 8 8 . [41 J. Ellis, M.K. GaiUard, L. Maiania nd B. Zumino, Proc. Europhysics Study Conf. on Unification of fundamental interactions (Erice, 1980), to be published; J. Ellis, M.K. Gaillard and B. Zumino, Phys. Lett. 94B (1980) 343. [5] J. Patera, R.T. Sharp and P. Winternitz, J. Math. Phys. 17 (1976) 1972; 18 (1977) 1519 (E); W. McKay and J. Patera, Montreal preprint CRM-742 (1977). [6] E. Cremmer, J. Scherk and J.H. Schwarz, Phys. Lett. 84B (1979) 83. 21