Soft spin 32 fermions require gravity and supersymmetry

Volume 67B, number 3

PHYSICS LETTERS

SOFT SPIN 3/2 FERMIONS

REQUIRE

11 April 1977

GRAVITY

AND SUPERSYMMETRY

*

M.T. GRISARU and H.N. PENDLETON

Physics Department, Brandeis University, Waltham, Massachusetts 02154, USA Received 21 February 1977 If massless fermions of spin 3/2 have non-vanishing low-energy couplings, the fermions must have massless partners of spin 2, and all particles to which the fermions couple must display supersymmetry.

A number o f years ago Weinberg pointed out that the interactions of photons and gravitons with other particles are to a large extent determined b y their masslessness and by Lorentz invariance [1 ]. In particular from the Lorentz invariance of the S-matrix for emission o f soft photons or gravitons he was able to derive charge conservation and the equality o f gravitational and inertial mass. He also concluded that Lorentz invariance forbids the existence o f massless particles of integral spin s ~> 3 having low frequency interactions with other particles. In this note we apply Weinberg's method to the case o f spin 3/2 massless Majorana fermions (hemitrions). We derive relations between S-matrix elements which are known to hold in theories with global supersymmetry [2]. We obtain the equivalent o f the supersymmetry charge commutation relations and show that if the hemitrion interacts with a pair of massless l particles of spins s, s - - i g r a v i t o n s must be present also. We also find that massless fermions of spin 5/2 cannot have low-frequency couplings. The method relies on the fact that for an on-sheU spin 3[2 massless particle described by a wavefunction uu(p) = eu(p) u(p) Lorentz invariance requires that the S-matrix vanish under the replacement eu(p) ~Pu" (This on-shell gauge invariance was used by Ferrara et al. [3] to determine some of the couplings for supergravity [ 4 - 6 ] interacting with matter.) We first treat the case of a spin-2, spin-3/2 system, and obtain the S-matrix relations for pure supergravity [2 ]. Consider for instance fig. la, representing the scattering o f several hard hemitrions and gravitons (with momenta qi, q] and chiralities oi, o]) and one soft * Work supported in part by NSF Grant PHY 76-02054.

/ I

(a)

~

/ I ~ I

(b)

Fig. 1. (a) Typical process with one soft hemitrion. (b) Diagrams dominant in the soft hemitrion limit. hemitrion with momentum p and chirality o. We take all particles as ingoing. We shall consider the limit p ~ 0. In this limit the process is dominated by a sum of diagrams where the soft fermion is attached to external lines in all possible ways as depicted in fig. 1. The scattering amplitude can be written as a sum o f terms which are products of a vertex function, a propagator, and a residual amplitude,

F= ~" M ~• Pxu,up E(~,oi)(p, qi) ~-~ , S~u(qj) + ZJN~ ~ U(~°'°/)(p,q/), j

z p - q/

(1)

where

Pxu,vp =½ (Sxv6up + 5xpSuv - 5xuS~p),

(2)

s..(q) =~I "r. "r'q ~..

(3)

The amplitudes M~u, N/ become physical when contracted with graviton or hemitrion wavefunctions (ex(qi) eu(qi ) or u u(qi ) or uu(qj)) and so satisfy i i = '4u "iN./"u = 0 . qxMxu

(4) 323

For convenience we are using conventional forms for the spin 2 and spin 3/2 propagators, although we have obtained the same results by analyzing the pole structure of the S-matrix. We also find it convenient to obtain our vertex functions from products of local fields. These vertex functions must contain a factor e(p)" qi" Under the replacement e(p) ~ p this factor will cancel the pole (p. qi) -1 and make it possible for the whole amplitude to vanish. Therefore the two fermion-one graviton couplings must contain one derivative (we shall discuss later the case with several derivatives). In the p -~ 0 limit all nonvanishing couplings are equivalent to

v3(~,, ~, h) -- ~ htav,p % v,,~,.

(5)

where htav is the graviton field, ffta the Majorana field and ~ has the dimensions of (mass) -I. After some algebra we find

P~ta,vp Eu(;'°i) = -- ~ o oi A(O)(P, qt') qi e (p) eXai(qi) etaai (qi) .

_

(6)

a

sta~ ~J~°,°/) = ~ o o~ a <°)(p, qi) qi" e°(P) u°i(qJ )

(7)

where A(+) (P, qi) = I'* (7/, qi),

A(-)(p, qi) = I" (rl, qi), (Sa, b)

zi ei~'/2 + r72Yi e- i¢,//2).

['(r~, qi) = ~ ( ~ 1

(9)

We have written q = E ( I , sin 0 cos 4, sin 0 sin 4, cos 0), z =cos0/2,

11 April 1977

PHYSICS LETTERS

Volume 67B, number 3

y=sin0/2,

(1o) ( l l a , b)

where (see fig. 1)M iis a physical amplitude in which the line p has been removed and the fermion line qi has been replaced by a graviton line of the same momentum and chirality while N i is an amplitude in which the graviton qj has been replaced by a fermion of the same momentum and chirality. Note that in eq. (13) certain minus signs appear corresponding to Fermi antisymmetry. These signs can be generated by making r/l, r/2 anticommute with Fermi operators. We note that a soft fermion of given chirality o has a nonvanishing coupling only to an incoming graviton of the same chirality or to a fermion of opposite chirality so that some of the lines are unaffected. Thus taking o = +1 or - 1 yields two separate relations between amplitudes with ng gravitons, n f - 2 fermions and amplitudes with n g - 2 gravitons and nf fermions. These relations are precisely what one derives in supergravity by using the commutativity of the S-matrix with the supersymmetry charge [2]. We return now to fig. 1 and consider the possibility of a second soft hemitrion of momentum p' and chirality o' = - o . For definiteness let us take o = +1. Some of the diagrams that are dominant are depicted in fig. 2. Note that when both fermions are attached to the same line as in fig. 2a or 2d the fermion with momentum p is on the outside (we are taking the limits p -+ O, p' -+ 0 in this order). Again the only lines affected are graviton lines of the same chirality as the soft fermions and fermion lines of opposite chirality. In (c) the graviton to which the fermions attach is itself soft, and it attaches universally to all lines. When one sumes over all attachments the diagrams of fig. 2c add up to zero (this is the Weinberg result for soft gravitons: one obtains a factor 2; qi = 0).

and

u(+)(p)

*

,

•

•

= (-r/2, r/l, -- 7/2, 7/1 ) ,

(12a)

u(-)(P) = ( - rh, -7/2, r/l, 7/2),

(12b)

in terms of two complex numbers 7/1, r/2 (up to a normalization factor this is the usual form for spinors for massless particles). We insert these results into eq. (1), contract the factors exeta or uta with M x ta or Nta and make the substitution eta(p) ~ pta. We derive the relation

~16o_oiA(°)(p, qi)Mi+ ~/ 6oo/A(°)(p, qj)N]=O,(13) 324

qj (o)

s~j,PP

Fig. 2. Dominant diagrams with two soft hemitrions.

Volume 67B, number 3

PHYSICS LETTERS

If we now reverse the order of the p,p' limits and subtract the two results diagrams such as fig. 2b cancel and we obtain, after the replacements e(p) -+ p,

e' (p')-~ p' [~i F*(~',qi) F(~,qi)l ~'=O,

(14)

where P is the original amplitude with the two soft fermions removed and the sum runs over all particles. It is not difficult to check that 1-'*(~', qi)r(n, qi) = - (i/2) u(-)(p')3"qi u(-)(p).(l 5) Thus eq. (14) is satisfied by virtue of overall momentum conservation. Note that it is crucial that the sum go over all hard particles. This was achieved by choosing o' = - o. The lines p, p' attach to every hard line in one order or another. We also note that if instead o f using definite chirality spinors for the soft hemitrions as in eqs. (12a,b) we use a Majorana spinor

u(p) = rI = 2 (r/1 + 7"/2,-~'/1 + r/2, -7"/I + r/2, -'ql - r/2) (16)

11 April 1977

line in fig. 2d must have either spin 1 or spin 2 (see remark (c) below). However a spin one boson has no gauge invariant couplings to a Majorana fermion. We conclude that if spin 3/2 Majorana fermions have soft interactions with other particles gravitons must be present. Furthermore, for eq. (14) to be satisfied the st rength of their coupling must be related to that of the hemit rions. We conclude with some comments: (a) We need consider only couplings containing exactly one derivative. Higher derivative couplings vanish in the soft fermion limit. (b) We can look at the possibility of higher spin fermions. Consider the spin 5/2 case with the particles described by a wave function eu(p) ev(P)U (p). The vertex function in the p ~ 0 limit must contain a factor [q. e(p)] 2 (so as to cancel the ( q . p ) - l pole when either e u is replaced by pu). However one runs into the same difficulty Weinberg encounters with high spin bosons. After the substitution e ~ p one finds equations of the form

we derive, instead o f e q . (15), r*(r/',q) l"(r~, q) - F*(~,q) r'(~',q) = - 2 i ~ ' 7 . q r l .

/

(17)

which can be satisfied only if We have shown in ref. [2] that eq. (16) is equivalent to the supersymmetry charge anticommutation relations. We consider now a situation similar to that o f fig. 1 but where, except for the soft hemitrion, the particles i have spins s and s - 7. Again in the p -+ 0 limit the vertex functions are unique and one obtains relations like (13) which hold for any supersymmetric (s, s --~) pair. Here the hemitrion ~lays the role of a probe. It interacts with the (s, s - ~ ) pair but apparently need not have other interactions. However let us assume that we look at an amplitude where in addition to (s, s -½) particles we have two soft hemitrions and one or more hard hemitrions. We shall conclude that gravitons must be present. We note first that the diagram o f fig. 2c does not affect our argument. The important fact is that while the diagram o f fig. 2d will be present if the qi line is a fermion in the (s, s - ½ ) pair it will be absent if the qi line is a hard hemitrion and the hemitrions don't interact with each other. But then the sum in eq. (14) will not run over all hard particles and the equation will not be satisfied unless P = 0. We conclude that the hemitrions must interact with each other. Furthermore the wiggly

A(p, qi)M i= A(p,q/)N y,

(19)

for all i,/. But since p (or r/1,r/2 ) is arbitrary this equation can be satisfied only ifM i = N] = 0. Therefore if spin s ~> 5]2 particles can be coupled in a Lorentz invariant way their coupling must vanish in the soft particle limit. (c) We may consider coupling a spin 3/2 fermion 1 to an (s, s - 3) paxr. As in case (b) above we obtain equations which can be satisfied only if all the scattering amplitudes vanish. (d) If several types o f hemitrions are present as in the O(N) models [ 7 - 9 ] each type of hemitrion generates its own supersymmetry and for that supersymmetry has the graviton as its partner while all the other types of hemitrions have massless vectors as partners.

Note added. A.O. Barut (Acta Physica Austriaca, Suppl. VI, (1969), 1) has examined, by methods similar to ours, the coupling of a massless spin 3/2 fermion to a (0, 1/2) pair. However, he assumed m 0 :~ ml/2 and concluded that there are difficulties with such coupling. 325

Volume 67B, number 3

PHYSICS LETTERS

We n o t e also t h a t B a r u t ' s c o n c l u s i o n s , t o g e t h e r w i t h our results, i n d i c a t e t h a t if b r e a k d o w n o f s u p e r s y m m e try o c c u r s so as to p r o d u c e m~ 4~ mj_l/2 t h e h e m i t r i o n e i t h e r d e c o u p l e s or must b e c o m e massive.

References [1 ] S. Weinberg, Phys. Rev. 135 (1964) B1049; in: Lectures on particles and field theory, eds. S. Deser and K. Ford (Prentice Hall, 1965). [21 M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen, Phys. Rev. D, to be published; M.T. Grisaru and H.N. Pendleton, to be published.

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11 April 1977

[3] S. Ferrara, F. Gliozzi, J. Scherk and P. van Nieuwenhuizen, Nucl. Phys. lI7B (1976) 333. [4] D.Z. Freedman, P. van Nieuwcnhuizen and S. Ferrara, Phys. Rev. DI3 (1976) 3214. [5 ] S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335. [6] D.Z. Freedman and P. van Nieuwenhuizen, Phys. Rev. DI4 (1976) 912. [7] S. Ferrara and P. van Nieuwenhuizen, Phys. Rev. Left. 37 (1976) 1669. [8] S. Ferrara, J. Scherk and B. Zumino, LPTENS 76/23 preprint. [9] A. Das, ITP-SB-77-4 preprint.