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Dynamical calculation of bound-state supermultiplets in N = 8 supergravity
Volume 107B, number 3
PHYSICS LETTERS
10 December 1981
DYNAMICAL CALCULATION OF BOUND-STATE SUPERMULTIPLETS IN N = 8 SUPERGRAVITY Marcus T. GRISARU
t
and Howard
J. S C H N I T Z E R
2
Department of Physics, Brandeis University, Waltham, MA 02254, USA
Received 30 July 1981
We have studied the existence of bound states in N = 8 supergravity using S-matrix methods. We find evidence for several supermultiplets.
Considerable effort has been expended in recent years in search for a grand unified theory (GUT) which reduces to the observed SU(3)c X SU(2) × U(1) at low energies [ 1]. Several candidate theories have been offered such as SU(5), O(10), and others. Such theories certainly allow one to understand the "standard model" as an effective field theory at energies below a grand unification mass (~1015-1016 GeV say), but they are clearly incomplete. One plausible view is that the candidate grand unified theories are themselves only phenomenological theories, and that a truly urfitied theory should incorporate quantum gravity, which suggests that supergravity theory might be the unified theory being sought. However the fundamental fields of maximally extended supergravity, i.e. N = 8, cannot even accomodate the particles of SU(3)c × SU(2)
x u0). Recently Ellis, Gaillard, Maiani and Zumino [2] (EGMZ) have proposed a scenario according to which N = 8 supergravity describes physics in the Planek mass energy region. They postulate the existence of massless bound states, transforming according to certain representations of the local SU(8) group discovered by Cremmer and Julia [3]. Through a sequence of symmetry breakings from SU(8) to SU(5), in the course of which many of the original massless bound states would become very massive or otherwise decoupled in the "low energy" (below Planck mass) region, their 1 Supported in part by NSF Grant No. PHY 79-20801. 2 Supported in part by the Dept. of Energy under Contract DE-AC03-76ER03230-A005.
196
picture eventually "predicts" an SU(5) GUT model with three families, as a massless effective theory at energy scales ~1015 GeV. The set of bound states postulated by EGMZ can be described as the massless SU(8) supermultiplet ,
:[ABI., .. • ;dh
, ~--21C
+(TCeconjugate),
A=l
(1)
...... 8.
They further assume that the "trace" multiplets in eq. (1) do not bind, and that the 63 vector mesons gauge the local SU(8). If this scenario is viable it requires several modifications [4-5]. Even leaving aside problems involving the mechanism and sequence of symmetry breakings, there is the question of the number and representation content of the bound-state massless multiplets. It has even been argued [4] that Lorentz invariance and group theory restrictions require a tower consisting of an infinite number of such supermultiplets. However, at an even more fundamental level one can ask whether one has any reason to believe (other than analogy with two-dimensional CP(n) models) that the forces o f N = 8 supergravity produce any bound states at all, and if so which bound multiplets are present. In this letter we present evidence based on a study of the dynamics o f N = 8 supergravity that bound states may be present. We find evidence for six massless supermultiplets including the one postulated by EGMZ, and we argue that many more may exist. Our approach consists in studying the complex angular
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PHYSICS LETTERS
momentum plane properties of the two-body S-matrix in N = 8 supergravity and looking for Regge pole trajectories which might correspond to the bound states. Roughly speaking, if the scattering amplitude as a function of energy and complex angular momentum exhibits a Regge pole trajectory f (s, J ) ~ ~(s)/ [J - a(s)]
( - 2 ) ; (__~)A ; (_I)[AB] ;... ({)A ; ( 2 ) .
(3)
The supermultiplet appears as sense choosing Regge poles corresponding to physical particles. Therefore the basic states of the theory do not disappear from the spectrum in our approximation but may be thought of as simultaneously elementary and composite. (ii) A massless multiplet with the quantum number of the supersymmetry currents as Regge poles in nonsense choosing (in our approximation), right signature amplitudes to be defined below. This multiplet is (-~)A ; 1 ( - 1 ~ ® (-1)1; [(-½)[CAB] ® (_½)A ] ; 5
nonsense choosing, wrong signature trajectories. These are (--3); (--S)A ; (--2)[AB ] ; .... ;(1) + (TCP conjugate)
(4)
• ""; (g),4 + (TCP conjugate). Notice that for example in addition to the 63 lefthanded helicity 1 states, we also find the trace, a singlet. In general, we find that the trace multiplets appear on equal-footing with the other states in a supermultiplet as required by supersymmetry. There is no evidence that the traces are unbound if a given supermultiplet is bound. (iii) Four massless supermultiplets associated with
(5)
(--S)A ; [(--2~ ® (--2)]". . . . . . ;(~)A 3 + (TCP conjugate)
(2)
and if for some value af s = So, a(So) = Jo an integer or half-integer, with/3(s0) :~ 0, then the spectrum of the theory contains a particle of mass x/~0 and spin J0, coupling to the initial or final particles, with a coupling strength x/~(s0). If/300) ve 0 (=0) one says that the trajectory chooses sense (nonsense). An exception is that/3(s0) could vanish for kinematical reasons and still a particle would exist. We now summarize our findings. First we note that a supermultiplet o f N = 8 supergravity can be described in terms of a global SU(8), constructed from the supersymmetry charge QA, a helicity rasing operator, as well as QA, a helicity lowering operator. The supermultiplets suggested by our findings can be divided into three classes: (i) The preon multiplet, described by the massless helicity states
10 December 1981
(6)
3A (--2)[AB] ; [(--~)[BC] ® (--{)A ] ; .... ; (2)lAB] + (TCP conjugate)
(7)
and ( _ I ) ; ( _ I ) A ; (0)[ABI ; ...... ;(3) (a) + (TCP conjugate). We now sketch how these results were obtained. Over the past few years we have developed an approach for examining the behavior of scattering amplitudes near small values (0, ~,1 1) of angular momentum [7], which is related to the fundamental work of GellMann et al. [8]. The approach uses the unitarity and analyticity properties of the scattering amplitude, and within the approximation scheme that we must employ, exhibits Regge poles (but not cuts) in agreement with the high-energy behavior found by summing Feynman graphs in leading logarithm approximation. For unrenormalizable theories such as gravity, graph summation is not possible at the present time but it can be argued that in addition to certain fixed singularities such theories do have moving Regge poles and that our techniques can exhibit them. Indeed it has been shown that in ordinary quantum gravity a Regge trajectory exists which passes through J = 2 at s = 0 (i.e., the graviton is a moving Regge pole), and the relation of fixed singularities to counterterms has been discussed [9]. The unitarity-analyticity approach requires the existence of "nonsense" states, which for theories of interest is possible only if particles with spin are present At a given integral or half integral value J0 of total angular momentum, a two-particle state [Xl, 2`2) (with 2`i the helicity of the ith particle) is called "nonsense" if 12`1 - 2,21 > J. This concept can be extended to many particle states. Since 2`1 - 2`2 is the projection of the angular momentum along the direction of motion (in the center of mass) the above state is unphysical. Nevertheless the scattering amplitude (2`3,2`4 I S J -- 112`1, 197
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)`2 ) for such states can be obtained by analytic continuation from larger values of J, keeping the X's fixed. A feature of such amplitudes is that they already contain much information about the Regge trajectories a(s) [but not the residues 3(s)]. In particular the Born (tree) approximation to the nonsense-nonsense amplitudes at J0 usually gives the singularity structure of f(s, J ) i n the vicinity o f J 0. The precise statement is the following: Consider the matrix of nonsense amplitudes (since there may be several coupled channels) fn'n (s, J) computed in Born approximation near J = J0. It has the form Un,n/(J - JO)" Then the amplitude for physical ("sense") helicity states is _
K(s)
fs,s(S,J)-os,n(S)[j_jo_~(s)K(S)ln,nUn,s(S)
(9)
where K(s) is a phase-space integral and usn , Vns are "sense-nonsense" transition amplitudes [7], In particular the Regge trajectory functions a(s) are given by zeroes of the denominator, i.e., the eigenvahies of the matrix [JoSn,n + On,n(S ) K(s)]. Their number is given by the rank of the On,n matrix. Roughly speaking v n,n is the one-particle exchange potential primarily responsible for binding, and as such our method has some relationship to Bethe-Salpeter methods for finding bound states. In N = 8 supergravity, which contains particles with spins from 0 to 2, two-body states can occur in nonsense combinations for values o f J ~< 3 and a priori we may expect Regge poles in the vicinity of each J up to that value. If they correspond to bound states their spins will not exceed 3. Three-body states could lead to spins as high as 4, but our methods have not been sufficiently developed to handle these or other multiparticle states (the general rule is s - 1 ~ Ni(Si -
1).
In order to determine On,n we must compute the Born approximation to a number of scattering amplitudes. Our task is facilitated by the following considerations: a) The graviton-graviton Born amplitudes are known [10] and in particular F ( 2 , - 2 ; 2, - 2 ) = K2u3/st.
(10)
b) The other amplitudes can be obtained by supersymmetry transformations: The one-particle states of N = 8 supergravity can be given global SU(8) labels according to their 0(8) representations and handedness 198
10 December 1981
(this is not the local SU(8) of Cremmer and Julia), e.g., (-2); (--3/2)A;(-1)[ ABI ;(--1/2)[ABC] ; ... ;(2), and their "in" or "out" operators satisfy commutation relations with the eight supersymmetry generators [O, a2] = FAbA/2 , (11)
[Q, bA/2] = r*Aa 2 + i r s c i 4a ,
where for example a2, bA3/2 , d ,18 annihilate positive helicity states 2, 3/2, 1, etc. We have written Q -- 23A= leA QA where eA are Majorana spinors depending on two arbitrary complex parameters r~l , r/2. The I"s depend on the rl's and the momenta carried by the operators. From these commutation relations and the invariance of the S-matrix under supersymmetry transformations one can deduce relations between Smatrix elements [11,12] and we find for example 1
F ( l A B , --2 CDH ; ~ E , - - 1 F G ) ~--St
11/2 I ~AB ~ FG ~AB ~FG ~AB~ FG UDEUHC + ~"CE"*DH + •HEU CD
= i~2u2 1--~-]
t
~AB~FG+~AB~FG ~AB~FGn + uHCUDE UDHUCE + uCDUHE |
12
AB
AB
A
(12)
AB
where 8CD = 8C~5 D -- 8D8 C. c) It turns out that the amplitudes satisfy a "pseudohelicity" rule which is related to global SU(8) invariance of the S-matrix. For the two-body preen sector this rule is the conservation law F()`3, )`4 ; )`1, )`2) = 0 unless )`1 + )`2 = )`3 + )'4. This rule is distinct from the usual helicity conservation rules which follow from supersymmetry alone [ 11 ]. For reasons of technical simplicity we have analyzed the situation nearJ = ~,1, 1 3 .... etc., but not near J = 0. • Using the procedure outlined in b), which is much ~.asier than computing the amplitudes from Feynman graphs, we obtain all the relevant amplitudes, and remove kinematical singularities. A typical amplitude free of kinematical singularities has the form in Born approximation
I 7= a/t + b/u. We associate with it a second amplitude
(13)
if'
(14)
= a/t
-
b/u
.
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They both produce Regge trajectories, which we refer to as "right signature" and "wrong signature" respectively (deviating slightly from the more technically precise definition [13]). In general, wrong signature trajectories do not correspond to physical particles. Our findings, eqs. ( 3 ) - ( 8 ) are summmarized in figs. 1a, b, where we have described trajectories which pass through massless states, plotting helicity versus SU(8) content. We have not plotted the TCP conjugate multiplets. The SU(8) representations are not irreducible but include the traces. We observe that if the points in fig. 1 actually correspond to particles, they will form supersymmetric multiplets. In particular the right signature set which consists of ( - 2 ) ; (_~)A ; ... clearly corresponds to the left handed preons. However, we wish to emphasize the following: Supersymmetry and SU(8) invarianee, which of course are part of our input, do not require that one have sets o]" trajectories forming supermultiplets. A single trajectory appearing in various amplitudes related by supersymmetry would be perfectly consistent. On the other hand ira point on a trajectory actually corresponds to a physical boundstate, supersymmetry requires that it be part o f a supersymmetric N = 8 supermultiplet and it would be inconsistent not to have the other trajectories present. Thus our findings are encouraging in indicating that at least one requirement for the presence of bound states is met. It should be emphasized that not all states of a given ~J -3 -
- ~3
-31
x
• i
x
-Z
•
X
-~
o
•
-3.
x 1
-4'
.X
_1 2.
×
Q m N
x
@ •
,8 Ak A;~D
A~ A;c ABCD '
(a)
(b)
Fig. 1. Helicities of massless states on Regge trajectories versus their global SU(8) content. The abscissa is labelled by upper SU(8) indices. The lower SU(8) indices of the appropriate representations are indicated as follows: x = (h)"'; • = (X)~'; • = (X)FG. ~ and o indicate states not accessible to twopreon nonsense amplitudes, a) right signature, b) wrong signature.
10 December 1981
supermultiplet are obtainable from two-body preon nonsense scattering amplitudes. For example, one cannot form the bound-state (--~)A from the two-body nonsense matrix; only ( - s ) A is accessible to us. However, if any of the supermultiplets ( 4 ) - ( 8 ) appear as physical particles, then the entire supermultiplet must appear in the spectrum due to supersymmetry. These additional states would be found in the nonsense matrix of multipreon scattering. The interpretation of our results depends on the extent to which supermultiplets ( 4 ) - ( 8 ) actually produce bound states. It is possible that only the preons exist as physical particles in the theory, in which case N = 8 supergravity might not be of physical interest. Another possibility is that in addition to the preons the supermultiplet (4) (including the traces) produces physical states. The consequences of this possibility have been analyzed by Derendinger et al. [6]. They show that it has some realistic features, but the resulting model is not completely satisfactory. A third possibility requires that N = 8 supergravity exhibit exchange degeneracy: The wrong-signature trajectories of fig. lb would have right-signature partners (produced by multiparticle states) with the same quantum numbers, which would then produce physical bound states described by the supermultiplets (5)-(8). This possibility does not appear to produce a phenomenologically acceptable situation either. Is there a physically interesting set of bound-states produced by N = 8 supergravity? It has been argued that an infinite set of supermultiplets are required for an acceptable model. This view is not inconsistent with our results, since we suspect that multipreon scattering amplitudes will reveal bound supermultiplets not accessible to two-preon scattering. We have argued that the state (--S)A was not connected to a two-body nonsense state, although it must be present in the spectrum if (6) involves physical particles. A similar situation probably holds for entire supersymmetric towers which are only revealed by multipreon scattering. In our program a number of issues need further study: a) We are applying techniques which have been well tested in the case of renormalizable theories to situations where they need not apply, b) The global SU(8) we are using for classification need not have associated with it a relevant loeal SU(8). Even if 63 (or 64) vector meson bound states exist, they need not gauge anything, c) We have found no indications 199
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that N = 8 supergravity is better than any o f the other supergravity theories. Adthbugh the detailed properties and number o f trajectories are different for each N = 1 , 2 . . . . 8 there i s n o qualitatively different behavior that seems to stand out. d) However, is exchange degeneracy a property o f N = 8 supergravity only? e) Can it be proved that the Regge trajectories we have found actually correspond to bound states? The trajectory pattern we have found in fig. 1 gives us reasons for optimism. One set of trajectories corresponds to the preons and certainly the other right signature trajectories behave in all respects very much like the preon trajectories. Further cause for optimism comes from the close relationship that seems to exist between dual models and supersymmetric models. (This relationship seems to be better established for N = 4 Yang-Mills theory and N = 4 supergravity rather than N = 8, b u t that may be a technical distinction at the present time.) In fact we have applied similar methods to a study o f N = 4 Yang-MiUs theory and have found trajectories which exhibit exchange degeneracy and are in m a n y respects reminiscent o f those in the corresponding dual string model. Details of the results described above and several others will be presented in a forthcoming paper. We wish to thank Professors M.K. Gaillard, M. Gell-Mann, H. Georgi, P. F r a m p t o n , and B. Zumino for discussions.
200
10 December 1981
Reference$ [1 ] First Workshop on Grand Unification, April 1980, eds. P.H. Frampton, S.L. Glashow, and A. Yildiz (Mat. Sci. Press). [2] J. Ellis, M.K. GaiUard, L. Maiani and B. Zumino, Unification of the fundamental particle interactions, eds. S. Ferrara, J. Ellis, and P. van Nieuwenhuizen (Plenum Press, New York, 1980) p. 69, J. Ellis, M.K. Galliard and B. Zumino, Phys. Lett. 94B (1980) 343; J. Ellis, ref. [ 1 ]. [3] E. Cremmer and B. Julia, Phys. Lett. 80B (1978) 48; Nucl. Phys. B159 (1979) 141. i4] M. GeU-Mann (unpublished); B. Zumino (unpublished). [5] P. Frampton, Harvard preprints HUTP-80/A050, July 1980, and HUTP-80/A072, October 1980. [6] J.-P. Derendinger, S. Ferrara and C.A. Savoy, CERN preprints, TH3025, February 1981, and TH3052, March 1981. [7] M.T. Grisaru, H.J. Schnitzer and H.-S. Tsao, Phys. Rev. Lett. 30 (1973) 811; Phys. Rev. D8 (1973) 4498, D9 (1974) 2864; M.T. Grisaru, Phys. Rev. D13 (1976) 2916, D16 (1977) 1962; M.T. Gdsaru and H.J. Schnitzer, Phys. Rev. D20 (1979) 784, D21 (1980) 1952. [8] M. GeU-Mann, et al., Phys. Rev. 133 (1964) B145; B161. [9] M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Phys. Rev. D12 (1975) 1563. [10] M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Phys. Rev. D12 (1975) 397. [11] M.T. Grisaru and H.N. Pendleton, Nucl. Phys. B124 (1977) 81. [12] M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen, Pliys. Rev. D15 (1977) 996. [13] P.D.B. Collins and E.J. Squires, Springer Tracts in Modem Physics, Vol. 45 (Springer-Verlag, 1968).
PHYSICS LETTERS
10 December 1981
DYNAMICAL CALCULATION OF BOUND-STATE SUPERMULTIPLETS IN N = 8 SUPERGRAVITY Marcus T. GRISARU
t
and Howard
J. S C H N I T Z E R
2
Department of Physics, Brandeis University, Waltham, MA 02254, USA
Received 30 July 1981
We have studied the existence of bound states in N = 8 supergravity using S-matrix methods. We find evidence for several supermultiplets.
Considerable effort has been expended in recent years in search for a grand unified theory (GUT) which reduces to the observed SU(3)c X SU(2) × U(1) at low energies [ 1]. Several candidate theories have been offered such as SU(5), O(10), and others. Such theories certainly allow one to understand the "standard model" as an effective field theory at energies below a grand unification mass (~1015-1016 GeV say), but they are clearly incomplete. One plausible view is that the candidate grand unified theories are themselves only phenomenological theories, and that a truly urfitied theory should incorporate quantum gravity, which suggests that supergravity theory might be the unified theory being sought. However the fundamental fields of maximally extended supergravity, i.e. N = 8, cannot even accomodate the particles of SU(3)c × SU(2)
x u0). Recently Ellis, Gaillard, Maiani and Zumino [2] (EGMZ) have proposed a scenario according to which N = 8 supergravity describes physics in the Planek mass energy region. They postulate the existence of massless bound states, transforming according to certain representations of the local SU(8) group discovered by Cremmer and Julia [3]. Through a sequence of symmetry breakings from SU(8) to SU(5), in the course of which many of the original massless bound states would become very massive or otherwise decoupled in the "low energy" (below Planck mass) region, their 1 Supported in part by NSF Grant No. PHY 79-20801. 2 Supported in part by the Dept. of Energy under Contract DE-AC03-76ER03230-A005.
196
picture eventually "predicts" an SU(5) GUT model with three families, as a massless effective theory at energy scales ~1015 GeV. The set of bound states postulated by EGMZ can be described as the massless SU(8) supermultiplet ,
:[ABI., .. • ;dh
, ~--21C
+(TCeconjugate),
A=l
(1)
...... 8.
They further assume that the "trace" multiplets in eq. (1) do not bind, and that the 63 vector mesons gauge the local SU(8). If this scenario is viable it requires several modifications [4-5]. Even leaving aside problems involving the mechanism and sequence of symmetry breakings, there is the question of the number and representation content of the bound-state massless multiplets. It has even been argued [4] that Lorentz invariance and group theory restrictions require a tower consisting of an infinite number of such supermultiplets. However, at an even more fundamental level one can ask whether one has any reason to believe (other than analogy with two-dimensional CP(n) models) that the forces o f N = 8 supergravity produce any bound states at all, and if so which bound multiplets are present. In this letter we present evidence based on a study of the dynamics o f N = 8 supergravity that bound states may be present. We find evidence for six massless supermultiplets including the one postulated by EGMZ, and we argue that many more may exist. Our approach consists in studying the complex angular
Volume 107B, number 3
PHYSICS LETTERS
momentum plane properties of the two-body S-matrix in N = 8 supergravity and looking for Regge pole trajectories which might correspond to the bound states. Roughly speaking, if the scattering amplitude as a function of energy and complex angular momentum exhibits a Regge pole trajectory f (s, J ) ~ ~(s)/ [J - a(s)]
( - 2 ) ; (__~)A ; (_I)[AB] ;... ({)A ; ( 2 ) .
(3)
The supermultiplet appears as sense choosing Regge poles corresponding to physical particles. Therefore the basic states of the theory do not disappear from the spectrum in our approximation but may be thought of as simultaneously elementary and composite. (ii) A massless multiplet with the quantum number of the supersymmetry currents as Regge poles in nonsense choosing (in our approximation), right signature amplitudes to be defined below. This multiplet is (-~)A ; 1 ( - 1 ~ ® (-1)1; [(-½)[CAB] ® (_½)A ] ; 5
nonsense choosing, wrong signature trajectories. These are (--3); (--S)A ; (--2)[AB ] ; .... ;(1) + (TCP conjugate)
(4)
• ""; (g),4 + (TCP conjugate). Notice that for example in addition to the 63 lefthanded helicity 1 states, we also find the trace, a singlet. In general, we find that the trace multiplets appear on equal-footing with the other states in a supermultiplet as required by supersymmetry. There is no evidence that the traces are unbound if a given supermultiplet is bound. (iii) Four massless supermultiplets associated with
(5)
(--S)A ; [(--2~ ® (--2)]". . . . . . ;(~)A 3 + (TCP conjugate)
(2)
and if for some value af s = So, a(So) = Jo an integer or half-integer, with/3(s0) :~ 0, then the spectrum of the theory contains a particle of mass x/~0 and spin J0, coupling to the initial or final particles, with a coupling strength x/~(s0). If/300) ve 0 (=0) one says that the trajectory chooses sense (nonsense). An exception is that/3(s0) could vanish for kinematical reasons and still a particle would exist. We now summarize our findings. First we note that a supermultiplet o f N = 8 supergravity can be described in terms of a global SU(8), constructed from the supersymmetry charge QA, a helicity rasing operator, as well as QA, a helicity lowering operator. The supermultiplets suggested by our findings can be divided into three classes: (i) The preon multiplet, described by the massless helicity states
10 December 1981
(6)
3A (--2)[AB] ; [(--~)[BC] ® (--{)A ] ; .... ; (2)lAB] + (TCP conjugate)
(7)
and ( _ I ) ; ( _ I ) A ; (0)[ABI ; ...... ;(3) (a) + (TCP conjugate). We now sketch how these results were obtained. Over the past few years we have developed an approach for examining the behavior of scattering amplitudes near small values (0, ~,1 1) of angular momentum [7], which is related to the fundamental work of GellMann et al. [8]. The approach uses the unitarity and analyticity properties of the scattering amplitude, and within the approximation scheme that we must employ, exhibits Regge poles (but not cuts) in agreement with the high-energy behavior found by summing Feynman graphs in leading logarithm approximation. For unrenormalizable theories such as gravity, graph summation is not possible at the present time but it can be argued that in addition to certain fixed singularities such theories do have moving Regge poles and that our techniques can exhibit them. Indeed it has been shown that in ordinary quantum gravity a Regge trajectory exists which passes through J = 2 at s = 0 (i.e., the graviton is a moving Regge pole), and the relation of fixed singularities to counterterms has been discussed [9]. The unitarity-analyticity approach requires the existence of "nonsense" states, which for theories of interest is possible only if particles with spin are present At a given integral or half integral value J0 of total angular momentum, a two-particle state [Xl, 2`2) (with 2`i the helicity of the ith particle) is called "nonsense" if 12`1 - 2,21 > J. This concept can be extended to many particle states. Since 2`1 - 2`2 is the projection of the angular momentum along the direction of motion (in the center of mass) the above state is unphysical. Nevertheless the scattering amplitude (2`3,2`4 I S J -- 112`1, 197
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PHYSICS LETTERS
)`2 ) for such states can be obtained by analytic continuation from larger values of J, keeping the X's fixed. A feature of such amplitudes is that they already contain much information about the Regge trajectories a(s) [but not the residues 3(s)]. In particular the Born (tree) approximation to the nonsense-nonsense amplitudes at J0 usually gives the singularity structure of f(s, J ) i n the vicinity o f J 0. The precise statement is the following: Consider the matrix of nonsense amplitudes (since there may be several coupled channels) fn'n (s, J) computed in Born approximation near J = J0. It has the form Un,n/(J - JO)" Then the amplitude for physical ("sense") helicity states is _
K(s)
fs,s(S,J)-os,n(S)[j_jo_~(s)K(S)ln,nUn,s(S)
(9)
where K(s) is a phase-space integral and usn , Vns are "sense-nonsense" transition amplitudes [7], In particular the Regge trajectory functions a(s) are given by zeroes of the denominator, i.e., the eigenvahies of the matrix [JoSn,n + On,n(S ) K(s)]. Their number is given by the rank of the On,n matrix. Roughly speaking v n,n is the one-particle exchange potential primarily responsible for binding, and as such our method has some relationship to Bethe-Salpeter methods for finding bound states. In N = 8 supergravity, which contains particles with spins from 0 to 2, two-body states can occur in nonsense combinations for values o f J ~< 3 and a priori we may expect Regge poles in the vicinity of each J up to that value. If they correspond to bound states their spins will not exceed 3. Three-body states could lead to spins as high as 4, but our methods have not been sufficiently developed to handle these or other multiparticle states (the general rule is s - 1 ~ Ni(Si -
1).
In order to determine On,n we must compute the Born approximation to a number of scattering amplitudes. Our task is facilitated by the following considerations: a) The graviton-graviton Born amplitudes are known [10] and in particular F ( 2 , - 2 ; 2, - 2 ) = K2u3/st.
(10)
b) The other amplitudes can be obtained by supersymmetry transformations: The one-particle states of N = 8 supergravity can be given global SU(8) labels according to their 0(8) representations and handedness 198
10 December 1981
(this is not the local SU(8) of Cremmer and Julia), e.g., (-2); (--3/2)A;(-1)[ ABI ;(--1/2)[ABC] ; ... ;(2), and their "in" or "out" operators satisfy commutation relations with the eight supersymmetry generators [O, a2] = FAbA/2 , (11)
[Q, bA/2] = r*Aa 2 + i r s c i 4a ,
where for example a2, bA3/2 , d ,18 annihilate positive helicity states 2, 3/2, 1, etc. We have written Q -- 23A= leA QA where eA are Majorana spinors depending on two arbitrary complex parameters r~l , r/2. The I"s depend on the rl's and the momenta carried by the operators. From these commutation relations and the invariance of the S-matrix under supersymmetry transformations one can deduce relations between Smatrix elements [11,12] and we find for example 1
F ( l A B , --2 CDH ; ~ E , - - 1 F G ) ~--St
11/2 I ~AB ~ FG ~AB ~FG ~AB~ FG UDEUHC + ~"CE"*DH + •HEU CD
= i~2u2 1--~-]
t
~AB~FG+~AB~FG ~AB~FGn + uHCUDE UDHUCE + uCDUHE |
12
AB
AB
A
(12)
AB
where 8CD = 8C~5 D -- 8D8 C. c) It turns out that the amplitudes satisfy a "pseudohelicity" rule which is related to global SU(8) invariance of the S-matrix. For the two-body preen sector this rule is the conservation law F()`3, )`4 ; )`1, )`2) = 0 unless )`1 + )`2 = )`3 + )'4. This rule is distinct from the usual helicity conservation rules which follow from supersymmetry alone [ 11 ]. For reasons of technical simplicity we have analyzed the situation nearJ = ~,1, 1 3 .... etc., but not near J = 0. • Using the procedure outlined in b), which is much ~.asier than computing the amplitudes from Feynman graphs, we obtain all the relevant amplitudes, and remove kinematical singularities. A typical amplitude free of kinematical singularities has the form in Born approximation
I 7= a/t + b/u. We associate with it a second amplitude
(13)
if'
(14)
= a/t
-
b/u
.
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They both produce Regge trajectories, which we refer to as "right signature" and "wrong signature" respectively (deviating slightly from the more technically precise definition [13]). In general, wrong signature trajectories do not correspond to physical particles. Our findings, eqs. ( 3 ) - ( 8 ) are summmarized in figs. 1a, b, where we have described trajectories which pass through massless states, plotting helicity versus SU(8) content. We have not plotted the TCP conjugate multiplets. The SU(8) representations are not irreducible but include the traces. We observe that if the points in fig. 1 actually correspond to particles, they will form supersymmetric multiplets. In particular the right signature set which consists of ( - 2 ) ; (_~)A ; ... clearly corresponds to the left handed preons. However, we wish to emphasize the following: Supersymmetry and SU(8) invarianee, which of course are part of our input, do not require that one have sets o]" trajectories forming supermultiplets. A single trajectory appearing in various amplitudes related by supersymmetry would be perfectly consistent. On the other hand ira point on a trajectory actually corresponds to a physical boundstate, supersymmetry requires that it be part o f a supersymmetric N = 8 supermultiplet and it would be inconsistent not to have the other trajectories present. Thus our findings are encouraging in indicating that at least one requirement for the presence of bound states is met. It should be emphasized that not all states of a given ~J -3 -
- ~3
-31
x
• i
x
-Z
•
X
-~
o
•
-3.
x 1
-4'
.X
_1 2.
×
Q m N
x
@ •
,8 Ak A;~D
A~ A;c ABCD '
(a)
(b)
Fig. 1. Helicities of massless states on Regge trajectories versus their global SU(8) content. The abscissa is labelled by upper SU(8) indices. The lower SU(8) indices of the appropriate representations are indicated as follows: x = (h)"'; • = (X)~'; • = (X)FG. ~ and o indicate states not accessible to twopreon nonsense amplitudes, a) right signature, b) wrong signature.
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supermultiplet are obtainable from two-body preon nonsense scattering amplitudes. For example, one cannot form the bound-state (--~)A from the two-body nonsense matrix; only ( - s ) A is accessible to us. However, if any of the supermultiplets ( 4 ) - ( 8 ) appear as physical particles, then the entire supermultiplet must appear in the spectrum due to supersymmetry. These additional states would be found in the nonsense matrix of multipreon scattering. The interpretation of our results depends on the extent to which supermultiplets ( 4 ) - ( 8 ) actually produce bound states. It is possible that only the preons exist as physical particles in the theory, in which case N = 8 supergravity might not be of physical interest. Another possibility is that in addition to the preons the supermultiplet (4) (including the traces) produces physical states. The consequences of this possibility have been analyzed by Derendinger et al. [6]. They show that it has some realistic features, but the resulting model is not completely satisfactory. A third possibility requires that N = 8 supergravity exhibit exchange degeneracy: The wrong-signature trajectories of fig. lb would have right-signature partners (produced by multiparticle states) with the same quantum numbers, which would then produce physical bound states described by the supermultiplets (5)-(8). This possibility does not appear to produce a phenomenologically acceptable situation either. Is there a physically interesting set of bound-states produced by N = 8 supergravity? It has been argued that an infinite set of supermultiplets are required for an acceptable model. This view is not inconsistent with our results, since we suspect that multipreon scattering amplitudes will reveal bound supermultiplets not accessible to two-preon scattering. We have argued that the state (--S)A was not connected to a two-body nonsense state, although it must be present in the spectrum if (6) involves physical particles. A similar situation probably holds for entire supersymmetric towers which are only revealed by multipreon scattering. In our program a number of issues need further study: a) We are applying techniques which have been well tested in the case of renormalizable theories to situations where they need not apply, b) The global SU(8) we are using for classification need not have associated with it a relevant loeal SU(8). Even if 63 (or 64) vector meson bound states exist, they need not gauge anything, c) We have found no indications 199
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that N = 8 supergravity is better than any o f the other supergravity theories. Adthbugh the detailed properties and number o f trajectories are different for each N = 1 , 2 . . . . 8 there i s n o qualitatively different behavior that seems to stand out. d) However, is exchange degeneracy a property o f N = 8 supergravity only? e) Can it be proved that the Regge trajectories we have found actually correspond to bound states? The trajectory pattern we have found in fig. 1 gives us reasons for optimism. One set of trajectories corresponds to the preons and certainly the other right signature trajectories behave in all respects very much like the preon trajectories. Further cause for optimism comes from the close relationship that seems to exist between dual models and supersymmetric models. (This relationship seems to be better established for N = 4 Yang-Mills theory and N = 4 supergravity rather than N = 8, b u t that may be a technical distinction at the present time.) In fact we have applied similar methods to a study o f N = 4 Yang-MiUs theory and have found trajectories which exhibit exchange degeneracy and are in m a n y respects reminiscent o f those in the corresponding dual string model. Details of the results described above and several others will be presented in a forthcoming paper. We wish to thank Professors M.K. Gaillard, M. Gell-Mann, H. Georgi, P. F r a m p t o n , and B. Zumino for discussions.
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Reference$ [1 ] First Workshop on Grand Unification, April 1980, eds. P.H. Frampton, S.L. Glashow, and A. Yildiz (Mat. Sci. Press). [2] J. Ellis, M.K. GaiUard, L. Maiani and B. Zumino, Unification of the fundamental particle interactions, eds. S. Ferrara, J. Ellis, and P. van Nieuwenhuizen (Plenum Press, New York, 1980) p. 69, J. Ellis, M.K. Galliard and B. Zumino, Phys. Lett. 94B (1980) 343; J. Ellis, ref. [ 1 ]. [3] E. Cremmer and B. Julia, Phys. Lett. 80B (1978) 48; Nucl. Phys. B159 (1979) 141. i4] M. GeU-Mann (unpublished); B. Zumino (unpublished). [5] P. Frampton, Harvard preprints HUTP-80/A050, July 1980, and HUTP-80/A072, October 1980. [6] J.-P. Derendinger, S. Ferrara and C.A. Savoy, CERN preprints, TH3025, February 1981, and TH3052, March 1981. [7] M.T. Grisaru, H.J. Schnitzer and H.-S. Tsao, Phys. Rev. Lett. 30 (1973) 811; Phys. Rev. D8 (1973) 4498, D9 (1974) 2864; M.T. Grisaru, Phys. Rev. D13 (1976) 2916, D16 (1977) 1962; M.T. Gdsaru and H.J. Schnitzer, Phys. Rev. D20 (1979) 784, D21 (1980) 1952. [8] M. GeU-Mann, et al., Phys. Rev. 133 (1964) B145; B161. [9] M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Phys. Rev. D12 (1975) 1563. [10] M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Phys. Rev. D12 (1975) 397. [11] M.T. Grisaru and H.N. Pendleton, Nucl. Phys. B124 (1977) 81. [12] M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen, Pliys. Rev. D15 (1977) 996. [13] P.D.B. Collins and E.J. Squires, Springer Tracts in Modem Physics, Vol. 45 (Springer-Verlag, 1968).