Can the squashed seven-sphere predict the standard model?

Volume 148B, number 6 PHYSICS LETTERS 6 December 1984

CAN THE SQUASHED SEVEN-SPHERE PREDICT THE STANDARD MODEL?

M.J. DUFF, I.G. KOH 1 and B.E.W. NILSSON The Blackett Laboratory, Imperial College, London SW7 2BZ, UK

Received 16 July 1984

We argue that in Kaluza-Klein supergravity the breaking of the elementary SO(8) IN = 8] symmetry of the round S 7 down to the SO(5) X SO(3) symmetry of the squashed S 7 IN = 1 or N = 0] corresponds to a breaking of a composite SU(8) down to either SU(4) X SU(2) IN = 1] or SU(5) X SU(3) X U(1) IN = 0]. We suggest that forN = 0 the SO(5) X SO(3) acts as a confining force yielding bound states of SU(5) × SU(3) × U(1). By demanding that the effective bound state theory be both anomaly free and asymptotically free (in the SU(8) sense), we find with the standard SU(5) embedding a fermion spectrum with 4 generations of (5* + 10) together with a realistic Higgs sector.

In the search for a realistic Kaluza-Klein theory, it is customary to look for the SUc(3 ) × SUw(2 ) × Uy(1) of the standard model inside the isometry group G of the extra dimensional ground state metric. However this leads to severe problems with chirality [1 ] . Alternatively one could argue that if the isometry group is non-abelian and asymptotically free, it should rather be interpreted as a confining force (i.e. a meta- color force), and the particles of the standard model are bound states formed from the Kahiza-Klein preons.

By applying this idea to the S 7 compactification of d = 11 supergravity and invoking the composite SU(8) invariance of the N = 8 theory, we obtain, under the assumptions described below, an SU(5) GUT with 4 generations of (5* + 10) together with a realistic Higgs sector.

We begin by recalling that in four dimensions the elementary SO(8) symmetry o f N = 8 supergravity does not contain the SUc(3) × SUw(2) × Uy(1) of the standard model. It has been suggested that the composite symmetry of SU(8) under which the fermions are chiral might become dynamical in the quantum theory [2]. Ellis, Gaillard, Maiani and Zumino [3] focused their attention on the ungauged

1 On leave of absence from Physics Department, Sogang University, Seoul, Korea.

0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

N = 8 theory of Cremmer and Julia [2] for which the SO(8) is global, whereas de Wit and Nicolai [4] con- sidered the gauged N = 8 theory with local SO(8) × SU(8) suggesting that the SO(8) might act as a confining force. Though attractive in many ways, neither scheme met with complete success.

An alternative Kaluza-Klein origin of the standard model was suggested by Witten [5] starting from N = 1 supergravity in d = 11 by noting that the isometry group G of the extra 7 dimensions could contain SUc(3 ) × SUw(2 ) × Uy(1) but that the fermions could never be chiral. However, this work prompted Duff and Pope [6] to observe that a gauged SO(8) N = 8 supergravity in d = 4 could be obtained by a Freund-Rubin [7] compactification of d = 11 supergravity on the seven-sphere, describing a massless N = 8 supermultiplet coupled to an infinite tower of massive N = 8 supermultiplets. They conjectured [6,8] that the sector describing only the massless supermultiplet of spins (2, 3 1 : , 1, : , 0 ÷, 0 - ) lying in the (1, 8S, 28, 56S, 35V, 35C) representation of SO(8) coincides with the de Wit-Nicolai theory and hence contains a hidden SU(8) as well. This conjectured equivalence has still not been verified beyond the linearized level [9,10], but is supported by cosmological constant cal- culations [ 11 ] and other more general arguments. The SO(8) representations of the massive fields are obtained by taking the tensor product of the massless

431


ones with (n, 0, 0, 0) [12,13]. It was noted by Biran et al. [14], who computed the mass spectrum, that in contrast to the massless representations, the massive representations had dimensions not consistent with SU(8) assignments. However this does not exclude the possibility that the Kaluza-Klein theory still has the SU(8) symmetry since all fields can be assigned trans- formation rules under SO(8) × SU(8) (for example, by taking the tensor product of the SO(8) × SU(8) assignments of the massless representations with (n, O, 0, 0) of SO(8)). Indeed, we have argued elsewhere [15] that such a hidden symmetry is a natural consequence of Kaluza-Klein theories. As in the work of Cremmer and Julia [2], we note that using the d = 7 Dirac ma- trices F a one may supplement the 21 generators of the tangent space SO(7) Fab = F[aFb] by the 7 F a to form SO(8) and then by the 35 7SPabc = 75P[aPbFc] to form a chiral SU(8). Moreover, de Wit and Nicolai [10] have succeeded in assigning SU(8) labels to the d = 11 fields by making a (4 + 7)-split irrespective of the particular compactification. This relies on elevating the eight-dimensional spinor representation of the tangent space group SO(7) to an 8 of SO(8) and then to a left-handed chiral 8 of SU(8). Let us assume therefore that the d = 4 lagrangian is invariant under local SO(8) × SU(8) even though the round S 7 vacuum (i.e. the vacuum computed in perturbation theory) is only SO(8) invariant.

This suggests a new origin for the fermions and bosons of the standard model: they are the bound states formed from among the infinity of Kaluza- Klein preons with the isometry group providing the binding force. Note that this picture differs from con- ventional preon dynamics in that the gauge bosons of the group under which the bound state fermions are chiral are themselves bound states. Our next task is to identify the quantum numbers of preons, ask whether they are likely to form bound states and, if so, identify the bound state quantum numbers. In the case of the round S 7 vacuum, the elementary symmetry is SO(8) and one might conjecture SU(8) bound states. An im- portant observation in this picture is that the 8 supersymmetry generators or equivalently the 8 massless gravitinos transform as an 8 s of SO(8) in the preonic phase but as a chiral 8 of SU(8) in the bound state phase, i.e., that the spinor index has to do double duty both as a representation of the preonic symmetry and as a representation of the composite symmetry. How-

ever, it could be that bound states do not form in this symmetric phase. In fact we know that the/3-function of the massless sector vanishes to one-loop [16] and maybe to all loops [17] making it unlikely that the SO(8) confines. Indeed, this was one of the major ob- jections to the de Wit-Nicolai conjecture. (It has re- cently been shown [18,19] that to one loop the ~ = 0 result persists even when the massive states are included, as had been conjectured earlier [8] .) There are, however, two other stable * 1 S 7 vacua provided by the squashed S 7 with SO(8) broken to SO(5) X SO(3) [25,12]. These have either N = 1 with the embedding 8 s ~ (4, 2) or N --- 0 with the embedding 8 s ~ (5, 1) + (1, 3). Bearing in mind our requirement that the spinor index has to do double duty as a representation of both preonic and bound state symmetries, we must demand that whatever subgroup of SU(8) be- comes dynamical it must permit complex representations of dimension (4, 2) in the case N = 1 or (5, 1) + (1, 3) in the case N = 0. Thus the maximal possibilities are SU(4) × SU(2) for N = 1 or SU(5) × SU(3) × U(!_)_for N = 0. So although both groups contain SUc(3 ) X SUw(2 ) × Uy(1), it is only in the non-su- persymmetric phase that a standard SU(5) GUT is possible.

Is confinement more likely in these phases? Let us postpone the answer to this question and instead consider the quantum numbers of the preons. Since the d = 4 supersymmetry parameter belongs to the eight-dimensional representation of SU(8), it follows that all fermions will carry an odd number of SU(8) indices and all bosons an even number irrespective of their masses or spins. We now ask what kind of bound states are permitted subject to the following assumptions:

(1) The effective bound state theory is anomaly- free with respect to SU(8).

(2) The effective bound state theory is asymptotically free with respect to SU(8).

We shall refer to (1) and (2) as "weak" assumptions since we make no further restrictions on the nature of the confinement mechanism. In particular, we do not specify the masses, spins or SO(8) quantum

,1 Stability is proved in ref. [20]. We might also consider the squashed S T with torsion [21,12,22 ] whose stability prop- erties are unknown but which has the same SO(5) × SO(3) isometry as the N = 0 vacuum. We do not consider the un- stable [23] paraUelized S 7 [24].

432


numbers of the preons forming the bound states, nor the number of preons from which the bound states are built. If the binding takes place at all, it must do so in a consistent fashion. So assumption (1) is mandatory, and recent work suggest that assumption (2) is also re- quired for consistency [26]. (Note that the usual anomaly matching conditions [27] of 't Hooft do not apply to this theory since the gauge bosons with respect to which the fermions are chiral are themselves bound states; the SO(8) preon theory is vectorlike.) The result is that the fermions can belong only to the (5 × 8* + 56) representation of SU(8) and this may be repeated up to l times, where 1 ~< l ~< 4. We emphasise that this is already a non-trivial consequence of octal- ity(i.e, that ' 1 spin- ~ bound states can carry only an odd number of SU(8) indices) since one can show [28] that assumptions (1) and (2) alone would permit 203 different solutions! There are also restrictions of the Higgs sector. The only allowed representations are 28 + 28", 36 + 36", 63 and 70. (Note that 420 discussed in ref. [4] is ruled out by assumption (2).) Moreover they can occur only in certain combinations. For example, with l = 4, the requirement of asymptotic freedom (i.e. fl < 0) permits only 28 + 28* + 63 or 28 + 28* + 70 or 28 + 28* + 36 + 36* or any one representation by itself (and its complex conjugate if it is complex). Furthermore this picture is consistent with a SU(8) bound state phase breaking either to SU(4) × SU(2) or SU(5) × SU(3) × U(1). Note, however, that it is inconsistent with an unbroken supersymmetry in the bound state phase so one must assume either that supersymmetry is broken by the confinement or else that bound states form only in the N --- 0, SU(5) × SU(3) × U(1) phase.

We now turn to the question of embedding SUc(3 ) × SUw(2 ) × Uy(1) in SU(8). It is quite easy to show that the SU(4) × SU(2) route can never lead to a realistic theory. There is no room in SU(4) for the SUw(2 ) and consequently the (5 × 8* + 56) does not contain SUw(2 ) singlets. We therefore turn to all embeddings of SUc(3 ) × SUw(2 ) × Uy(1) in SU(5) × SU(3) × U(1). Under SU(8) ~ SU(5) X SU(3) × U(1) we have

(5X8" + 5 6 ) ~ 5 × ( 5 " , 1 , - 3 ) + 5 X (1,3",5)

+ (10", 1, 9) + (10, 3, 1) + (S, 3", - 7 ) + (1, 1, -15).

We shall refer to the above hypercharge as Yd" For

each copy of (5 × 8* + 56) we obtain either (A) 2 standard generations of quarks and leptons, or (B) 3 generations plus exotics, or (C) only exotics. (By "exotics" we mean complex representations of SUc(3 ) × SUw(2 ) × Uy(1) which are non-standard.) In each case there are also real representations which we assume acquire heavy masses. Case (C) is clearly unrealistic and we also rule out (B) since it predicts massless leptons charged under UEM(1 ). In this way we rule out all but 6 of the 32 possible embeddings of SUc(3) × SUw(2 ) × Uy(1) in SU(5) × SU(3) × U(1). Each of these 6 describes 21 standard generations of quarks and leptons as shown below.

Embedding (i):

SO(5) ~ SUc(3 ) X SOw(2 ) X Oa(]),

5 ~ (3, 1, 2) + (1, 2, -3) ,

SU(3) ~ Ub(1 ) X Uc(1),

3-~ (1, 1)+ ( -1 , 1) + (0, -2) ,

1 y = _ ZYa.

Embedding (ii):

SU(5) -~ SU(4) X Uc(1 ) -~ SU(3) X Ub(1 ) X Uc(1 )

--> SUw(2 ) X Ua(1 ) X Ub(1 ) X Uc(1),

S-+(2, 1, 1, 1 ) + ( 1 , - 2 , 1, 1 ) + ( 1 , 0 , - 3 , 1)

+ (1,0, 0, -4) ,

SU(3) ~ SUc(3),

3 ~ 3 ,

_ 1 Y-ZYa + ~Yb + ~Yc + ~Yd.

Embedding (iii):

SU(5) -~ SU(4) × Uc(1 )

SUw(2 ) × SO(2) × Ub(1 ) × Uc(1 )

-~ SUw(2 ) X Ua(1 ) X Ub(1 ) X Uc(1),

5 -> (2, 0, 2, 1) + (1, 1, -2 , 1) + (1, -1 , -2 , 1)

+ (1, O, O,-4) , 433

Volume 148B, numbei 6 PHYSICS LETTERS 6 December 1984

SU(3) --> SUc(3),

3 ~ 3 , 1 ~ y .

Y=~Yb +~Yc+lS d Embedding (iv):

SU(5) ~ SU(3) X SU(2) × uc(a )

SU(3) X Ub(1 ) X Uc(1 )

--> SUw(2 ) X Ua(1 ) × Ub(1 ) X Uc(1),

5 ~ (2, 1 ,0 , 2) + (1, - 2 , 0, 2) + (1, 0, 1, - 3 )

+ (1,0, --1, --3),

su(3) -~ SUc(3),

3 ~ 3 ,

1 + ~ Y b + ~Yd" Y=gYa

Embedding (v):

SU(5) -~ SU(4) X Ub(1 ) --> SUc(3 ) X Ua(1 ) × Ub(1),

5 ~ ( 3 , 1, 1)+ ( 1 , - 3 , 1) + (1,0,--4) ,

SU(3) ~ SUw(2 ) X Uc(1),

3 ~ (2, 1) + (1, -2) ,

Y 1 - = - - + g Yc ~ Yd,

Embedding (vi):

SU(5) -'+ SUe(3 ) X SU(2) X Ub(1 )

SUc(3 ) X Ua(1 ) X Ub(1),

5 ~ (3, 0, 2) + (1, 1, - 3 ) + (1, - 1 , - 3 ) ,

SU(3) ~ SUw(2 ) X Uc(1),

3 ~ (2, 1) + (1 , -2 ) ,

1 Y = - ~ r b + 6 Y c - ?-sYd"

The normalizations of Ya, Yb and Yc are given by the branching of the defining representation. The linear

combination of Ya, Yb, Yc and Yd is chosen so as to yield the correct hypercharge Y assignments of the standard generations. (Here Y is related to the electric charge Q and weak isospin T 3 by Q = T 3 + Y.)

All the above embeddings are in principle realizable with the Higgs assignments 63 + 28 + 28* for l ~< 4 or 63 + 36 + 36* if l ~< 3. However some intermediate steps in embeddings (ii)-(vi) for the first case and in embedding (iv) for the second case are not permitted.

It is interesting to note that if we insist on the standard GUT embedding of SUc(3 ) × SUw(2 ) X Uy(1) in SU(5) then only one of the original 32 possibilities survives, namely that given in case (i) above. It corresponds to 2l generations of (5* + 10) with 1 ~< l ~< 4. Moreover the Higgs sector contains the 24 necessary for the breaking to SUc(3 ) × SUw(2 ) × Uy(1) and the 5 + 5" favoured for the breaking to SUc(3 ) X UEM(1 ).

This summarizes the bound state spectrum obtained using only the "weak" assumptions (1) and (2) and we now ask what further restrictions are imposed if we make some stronger assumptions about the nature of

1 the preons. The spin- ~ preons in d = 4 come from the d = 11 gravitino field with its world index in the extra 7 dimensions i.e. from Fourier expanding ~ffm~i(xU,y m) in harmonics on S 7. Here we have split the d = 11 world index into/a = 1 .... ,4 and m = 1 ..... 7 and the d = 11 spinor index into (~, i) where ¢t = 1, ..., 4 is the SO(l, 3) Lorentz spinor index and i = 1, .... 8 is the SO(7) spinor index, which is then promoted to an 8 of SU(8) after taking chiral projections [2,10]. Thus the spin- 5 preons belong to SO(8) representations corresponding to all transverse-vector harmonics of SO(8), the lowest of which is a 28 corresponding to the Killing vector Krn 1J (1, J = 1 ..... 8). (The transverse as-

1 signment ensures that we do not count helicity- 5 com- . 3 ponents of the spin- 5 particles.) Thus the simplest of

• 1 the spin- i preon fields is ×ill(x), a (28, 8) of SO(8) X SU(8). (Note that this differs from the de Wit- Nicolai assignment xiik(x) i.e. a (1, 56). This is because they first convert the world index on the gravitino into a tangent space index by multiplying by the siebenbein ema(x, y) which gives rise to d = 4 scalars. From our point of view, therefore, the (1, 56) is composite.) Sim- ilarly, the simplest of the spin-0 preon fields are (in agreement with de Wit and Nicolai) uiilJ(x ) and viHY(x) i.e. the (28, 28*) and (28, 28). On the grounds of sim- plicity, therefore, let us assume:

434


(3) Massless bound states are formed only from X if J, UiJJ and v ijlJ.

(4) Massless bound states are formed only from two or three preons i.e. XXX, XU, XUV etc.

Once again, we have no strong justification for assumption (4) except the intuitive feeling that it is eas- ier to bind a small number of preons.

With these "strong" assumptions (3) and (4) we find firstly that the round S 7, SO(8) confinement yields no solutions satisfying (1) and (2), and that we are forced therefore to consider the squashed S 7 SO(5) X SO(3) confinement which permits more singlets. As before, the N = 1 phase with SU(4) × SU(2) bound states is unrealistic but the N = 0 phase with SU(5) X SU(3) X U(1) bound states satisfies all four assumptions. We find the same 6 solutions as under the weak assumptions but now the maximum number of generations is 4 rather than 8. In particular the standard SU(5) embedding yields uniquely the (5* + 10) representation and would seem therefore to predict four generations of (5* + 10).

The strong assumptions thus rule out the formation of SU(8) bound states, but bound states can form in the SU(5) X SU(3) X U(1) phase. Of course, this group can never be asymptotically free so we have continued to apply assumption (2) in the SU(8) sense. Applying the anomaly-free assumption (1) to SU(5) × SU(3) X U(1) rather than SU(8) does then not change any of our conclusions.

We now return to the question of why confinement in the N = 0 phase should be favoured over confinement in the N = 1 or N = 8 phases. We have already argued that the vanishing of the N = 8/3-function makes confinement unlikely in this phase. The problem with calculating ~ in the squashed S 7 phase is knowing which of the infinity of preons to include in the calculation. If all were included, one would presumably obtain/3 = 0 again in spite of the different spectrum since it merely corresponds to a spontaneous- ly broken version o f N = 8. However this is not the correct calculation if one believes that only the "light" preons are relevant for the confinement. Unfortunate- ly, the problem of the decoupling of heavy states in Kaluza-Klein theories is very obscure and it is made more difficult by the existence of a Planck-sized cosmological constant in the preon phase. (It so happens that the states with vanishing anti-de Sitter mass yield

< 0 for the N = 0 squashed S 7 if we ignore scalars

whose masslessness is presumably an artifact of the tree approximation. This contrasts with the N = 1 squashed S 7 which yields/3 > 0. However we do not attach any great significance to this.) Thus the whole question of asymptotic freedom and confinement in Kaluza-Klein theories remains obscure. The only certain statements are that/3 = 0 to one loop on the round S 7. Note, incidentally, that the bound state picture discussed in the present paper involves the existence of Higgs fields formed from fermionic bilinears like X75X and hence is consistent with the Ricci-flattening torsion resolution of the cosmological constant problem discussed in refs. [29,15].

The bound state picture put forward in this paper resembles those presented previously [3,4,15] inas- much as we invoke bound state gauge bosons formed from the elementary scalars of supergravity. These bound states do not form an extended supersymmetry multiplet, however, so we are not obliged to invoke

3 massless bound states of spin 5 and higher in contrast to ref. [3]. Nor do we invoke a supersymmetry breaking mechanism via confinement, as was done in ref. [4], suggesting instead that the breaking to N = 0 oc- curs at the preonic level. It differs also from the squashed S 7 unification picture presented previously [15]. There we were able to exploit the breaking of N = 8 to N = 1 induced by the squashed S 7 and hence provide a resolution of the gauge hierarchy problem. Since, in the present scheme, all supersymmetries are broken at a higher mass scale the gauge hierarchy remains as enigma. However, the present scheme does provide a natural resolution of the chirality problem and, even more remarkably, admits realistic solutions.

This research is supported in part by KOSEF.

References

[1] E. Witten, Princeton preprint (1983). [2] E. Cremmer and B. Julia, Nucl. Phys. B159 (1979) 141. [3] J. Ellis, M. Galliard, L. Maiani and B. Zumino, in: Unifi-

cation of the fundamental interactions, eds. J. Ellis, S. Ferrara and P. van Nieuwenhuizen (Plenum, New York, 1980).

[4] B. de Wit and H. Nicolai, Nucl. Phys. B208 (1982) 323. [5] E. Witten, Nucl. Phys. B186 (1981) 412. [6] M.J. Duff and C.N. Pope, in: Supersymmetry and super-

gravity '82, eds. S. Ferrara, J.G. Taylor and P. van Nieuwenhuizen (World Scientific, Singapore, 1983).

435


[7[ P.G.O. Freund and M.A. Rubin, Phys. Lett. 97B (1980) 233.

[8] M.J. Duff, in: Supergravity '81, eds. S. Ferrara and J.G. Taylor (Cambridge U.P., London, 1982) p. 257; M.J. Duff and D.J. Toms, in: Unification of the fundamental interactions, eds. J. Ellis, S. Ferrara and P. van Nieuwenhuizen (Plenum, New York, 1982); M.J. Duff, Nucl. Phys. B219 (1983) 389.

[9] M.A. Awada, B.E.W. Nilsson and C.N. Pope, Phys. Rev. D29 (1984) 334.

[10] B. de Wit and H. Nicolai, preprint NIKHEF-H/83-21 (1983).

[11] M.J. Duff, C,N. Pope and N.P. Warner, Phys. Lett. 130B (1983) 254.

[12] MJ. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Rev. Lett. 50 (1983) 2043;51 (1983) 846 (E).

[13] D.Z. Freedman and H. Nicolai, Nucl. Phys. B237 (1984) 342.

[14] B. Biran, A. Casher, F. Englert, M. Rooman and P. Spindel, Phys. Lett. 134B (1984) 179.

[15] MJ. Duff, B.E.W. Nilsson and C.N. Pope, Nucl. Phys. B233 (1984) 433.

[16] S.M. Christensen, M.J. Duff, G.W. Gibbons and M. Ro~ek, Phys. Rev. Lett. 45 (1980) 161.

[17] K.S. SteUe and P.K. Townsend, Phys. Lett. l13B (1982) 25.

[18] G.W. Gibbons and H. Nicolai, Phys. Lett. 143B (1984) 108.

[19] T. Inami and K. Yamagishi, Phys. Lett. 143B (1984) 115. [20] M.J. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Lett.

139B (1984) 154. [21] F. Englert, M. Rooman and P. Spindel, Phys. Lett. 127B

(1983) 47. [22] F.A. Bais, H. Nicolai and P. van Nieuwenhuizen, Nucl.

Phys. B228 (1983) 333. [23] B. Biran and Ph. Spindel, Phys. Lett. 141B (1984) 181. [24] F. Englert, Phys. Lett. 119B (1982) 339. [25] M.A. Awada, M.J. Duff and C.N. Pope, Phys. Rev. Lett.

50 (1983) 294. [26] G. 't Hooft, Phys. Lett. 109B (1982) 474. [27] G. 't Hooft, in: Recent developments in gauge theories

(Carg6se, 1979), eds. G. 't Hooft et al. (Plenum, New York, 1980).

[28] E. Eichten, K. Kang and I.G. Koh, J. Math. Phys. 23 (1982) 2529.

[29] M.J. Duff and C. Orzalesi, Phys. Lett. 122B (1983) 37.

436

Documents

Can the squashed seven-sphere predict the standard model?