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SU(5) × SU(3) × U(1)
Volume 134B, number 5
PHYSICS LETTERS
19 January 1984
UNIFICATION OF FAMILIES BASED ON A COSET SPACE E7/SU(5 ) × SU(3) × U(1) T. KUGO 1 and T. YANAGIDA 2 Max-Planck-lnstltut fur Phystk und Astrophystk - Werner-Hetsenberg-Institut fur Physik Munich, Fed. Rep Germany
Received 6 October 1983
We show that the usual three famlhes of quarks and leptons are identifiable with quasi Nambu-Goldstone fermlons in a supersymmetrlc non-hnear realization of E 7 corresponding to the Kahler manifold ET/SU(5) × SU(3) × U(1) So the tnphcatlon of famdles suggests the underlying preon theory reahzlng the global E 7 linearly. Possible connections with N = 8 supergravity are also discussed
I n t r o d u c n o n . After the discovery of the hidden symmetry SU(8)local X E7(+7)globa1 by Cremmer and Juha [ 1], many physicists have come to expect that N = 8 supergravity, i.e., maximally local-supersymmetrlc theory, might provide the ultimate theory that describes umfiedly all the fundamental matter fields and Interactions of Nature. Advancing a step m that direction, Ellis et al. [2] proposed a possible scenario how the three famllaes of quarks and leptons could emerge as bound states of "preons", the fields appearing in the N = 8 supergravlty lagranglan. Although Interesting, their arguments are, however, a bit arbitrary and unsatisfactory that is, m their scheme, the dynamacal mechanism to guarantee the e x i s t e n c e and masslessness of those bound-state fermions is left obscure and the determination of their interaction forms would be even further difficult. We pursue in this letter an alternative approach to those problems of families, namely the "quasi N a m b u Goldstone fermxon" approach based on the coset ET/SU(5 ) X SU(3) × U(1), which might be related to the N = 8 supergravlty theory. Here, however, we propose our model as regarding It quite an independent scheme and we defer the d~scussion of possible connections with N = 8 supergrawty to the last part of this letter. A dynamical scheme which is able to generate massless composite fermlons has recently been suggested [3], based on the supersymmetrlc preon theory, where quarks and leptons are fermlon superpartners of Nambu Goldstone bosons (called quasi Nambu-Goldstone fermlons) arising from the spontaneous breakdown of a global symmetry. In this scheme, the number and particular quantum numbers of massless fermlons are determined by the well-defined geometrical properties of the coset space G/H [4,5]. It has been pointed out [5,6] that E 7 is a minimal candidate for G leading to a Kahler mamfold G/H which may accommodate the three famihes of quarks and leptons. In this letter we explicitly show that the supermultiplets of quarks and leptons indeed hve m the Kahler manifold E7/SU(5 ) × SU(3) × U(1) together with a Hlggs mulnplet 5, and construct the effective lagrangian of the Nambu-Goldstone supermultiplets. It is surprising that the quasi Nambu-Goldstone fermions of E7/SU(5 ) × SU(3) × U(1) have precisely the required SU(5) quantum numbers 3 × (5* + 10) for three famihes of quarks and leptons and 5 for a hlggsino. Notice here that the uniqueness of the coordinate of the Kahler manifold (or of the complex mamfold, more precisely) enforces the combination 3(5* + 10) but not 3(5 + 10) for quarks and leptons. Therefore, this model may answer the fundamental questions m particle physics why the observed quarks and leptons transform as 5* + 10 under the grand unified SU(5) [7] and
1 On leave of absence from Department of Physics, Kyoto Umverslty, Kyoto 606, Japan 2 On leave of absence from Department of Physics, College of General Education, Tohoku University, Sendal 980, Japan 0.370-2693/84/$ 03.00 © Elsevier Soence Publishers B.V. (North-Holland Physics Pubhshing Division)
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19 January 1984
why the repetitive structure of families exists. However, there are SU(5) gauge anomalies if only the quasi NambuGoldstone fermions are light particles. We will show later a way to solve this problem which seems natural if there exists an underlying E 7 preon theory. Let us discuss the geometry of the coset space ET/SU(5 ) × SU(3) × U(1).
The coset space ET/SU(5 ) × SU(3) × U(1) E 7 has 133 generators, traceless T/J of the SU(8) subgroup and totally anti-symmetric EIJKL (/ ..... L = 1-8) subject to a reahty constraint (EIJKL)* = (1/4!)C-IJKLMNOPEMNoP. They satisfy the E 7 L e algebra [8] ; -glT~2,
=
[T/, EKLMN] = gKEILMN J J J 1 J EKLMN ' + gLJ EKIMN + ~MEKLIN + gNEKLMI -- ~gl [EIJKL, EMNOP] = 12(T? eQJKLMNOP + TjQ elQKLMNOP + TQKelJQLMNOP + TQLeIJKQMNOP)
-- ½(T~e, JKLQNOP + TQNeuKLMQOP + TQoVKLMNQe + TQ eIJKLMNOQ) .
(1)
The broken generators corresponding to the coset space E7/SU(5 ) × SU(3) × U(1) are defined by
X~ = T~,
Xza = T~a, Xzab= --(1/3[)eabCdeEcdet,
X'ab =(I/2!)eO~Eabjk,
Xa =-(l/4!)eabcdeEacde,
X,, =(1/3!)d/kEa,/k,
(2)
where the indices a ..... e and i, .., k denote the SU(5) and SU(3) indices running over 1-5 and 6-8, respectively. From eq. (1) we immediately find the algebra of the broken generators: [x~tb, XJed ] [~b, ~l
I a b ---g,(gcrS+gb a -b
b ~abt.,rt --gcT~d - g ea Tbc)+ oc ,,j +
b -
=e
[X~, ~ b ] = g ~ T b -- g ba ~ ,
t --z , [Xa,-~b]: --Xab
~ [Xa, Xbl = - r ~ +~6bY,
[x~,x b] = 0 .
el/kAe,
tl
Y) ,
[xtab,x;l=(1/2!)eabcdeeqk2k e
[ X ab, Xlc] = g t1( g cbX a _ g a x b ) ,
[~b,x
c] = O,
i l = [X a, t xb] = O, [X~,Xbl
(3)
where gab =gc6 dab _ 6~g b and Yis the U(1) hypercharge defined by Y= 2Y~a= 1 5 Tad= _2N8=6 T]. Since the group SU(5) × SU(3) × U(1) is the centralizer of a torus U(1) generated by Y in E7, E7/SU(5) × SU(3) × U(1) is a Kahler manifold [9]. This manifold has the unique set of 50 complex coordinates {z a°, zia, z a} a corresponding to the broken generators X s, as is shown explicitly in the following. Thus another choice {zzab , z~, za}, for instance, Is not allowed by the E 7 symmetry. This is important, because otherwise we had wrong quarks and leptons such as 5 + 10 in the low energy spectrum. Corresponding to the Kahler coordinates we have 50 Nambu-Goldstone chlral superInultlplets ~bab, qS~ and q~a (or equivalently qSatb,¢ a and qSa)for the supersymmetric nonlinear realization of E 7. These supermultiplets cab, ~la and ca transform as (10, 3", 1), (5", 3, 2) and (5, 1, 3) under the unbroken SU(5) × SU(3) × U(1), respectively. The last numbers in the parentheses are the U(1) hypercharges. Assuming the SU(5) to be the standard grand unified gauge group, we can easily identify the fermlon components of cab and q~a ~ with the usual three families of quarks and leptons. Let us show that the E 7 is indeed reahzed on our choice of the Nambu-Goldstone supermultiplets qS. Since the knowledge of exphcit parametrlzatlon of the coset manifold is lacking to us, we proceed as follows a la Welnberg [ 10]. The direct study of the "Jacob1 xdentitles",
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19 January 1984
[XA, [XB, ¢1 ] -- [XB, [XA, ¢11 = [[XA, XB], ¢1,
(4)
(with X A,B being the broken generators) leads to an exact solution of the nonlinear realization of E 7 on the Nambu-Goldstone supermultlplets. The result as.
(i) Transformatton laws under the X's. --t cd xcd~t [X~b ' ~)j ] = Uab u I ,
t [X~b ' ~]c] = - - 41 e ,]k eabcd e ~)fe,
= a b a ,,
[X;,¢b]
--~
I{xc.u
c
t
[X~b, eft] = ~WaW, - ~b¢~)
:,~, ,
[X a , ¢ b c l = [ X a , ~ ] = 0 ,
+ lt.~loV.~
m--~-.~
tlk
x.cdx.ef
~bd~fW~ ~'k ,
[Xa,¢b I =6 ab .
(5)
(ii) Transformation laws under the X's. [xlab, ¢;d] = eqkeabcdec#ke + ½(d)fb~;d _ o;b ¢ ~ ) _ ½(~)fc4d + dpbdd);c d)bcc~;d_ ¢fd4c), [xa.b, ¢1c] = ~]xab ()d + 1r ,sab,s] ~l.aab~k xab.~de sl , --UtUcd 2t.--s't We + UiWk Wc -- UcdWt We)
+ 1~ ((1/2!)2dkl~c~8 ~ % Y ¢ 7 ~ + (1/2')du~c~ •
,),
[xa~b, oc]:--~[Oacob--(a~b)]+(--~6¢abo;d+~6~o;b
8 [¢~%~%7 8 - (a ~ b)l) +[(~¢ac4d--~oad4c)--(aob)]}O
~.
(6)
The transformation laws under the Xa~ and X a are automatically obtamed by substituting eq. (6) with the following formula:
[xA, ¢1 = (1/4!)dJk%b~de[X~ c, [X~ ~, ~]],
IX ~, ~] = I([X~, IXb~, ~]] - IX) ~, [X~, ~]])
(7)
The non-linear transformation laws given by eqs. (4) and (7) are unique up to possible coordinate transformations which, m this case, necessarily take the form ¢a _~ ca + x~)ab¢ib, d)ab ~ c#ab and ¢~ ~ ¢i with a parameter x aside from trivial scale transformations. Once the coordinates are fixed, the invarlant lagrangian of the N a m b u Goldstone modes is uniquely determined and given in terms of the Kahler potential K(~, ¢), the presence of which is guaranteed by the Kahler property of our coset.
£ = f d 4 0 K(¢, ¢).
(8)
It is, however, difficult to find out the exact form of K. So we construct the non-linear lagrangian up to quartlc terms of ¢ and ¢, which is rather interesting because it gives us the first non-trivial four-Fermi interactions of quarks and leptons'
3--a
]
- ~(~ ~)(¢i
--b
t
I --
a 2
Cb) - 2(~.¢ )
3 --i
~(%b~,
I --t ab --c - ~(¢~¢~ )(~; %)] - ( ¢--~~ c ~ ;a)b( ~ b-¢-
c
ab
--c
J
)(*i ec)
+3--i
~(¢~b*]
ab
-c
]
)Or ¢c) +
3-z ab - c l 2 ( ¢ a c ¢ i )(q~/Cb)
) - (~,%~)(~b¢ b) + ( ~ , % ~ ) ( ~ ¢ b )
-- ~V/~ eqk eabcde(~ab 01)(-¢d eke) -- ~X//-~eqkeabcde(-¢; cfb)(-~; ~bd)}.
(9)
1 Here, we have redefined new fields ~bN's by ~N ab -- (v/x/~)cab, ¢N ai -- VX/~Cia and ( N a = u(Ca + $~ab Cb ) with v being an energy scale which characterizes the breaking E 7 -+ SU(5) X SU(3) X U(1), and we omitted the superscript N for simplicity. We have shown that the supersymmetric extension of the Kahler manifold E7/SU(5 ) X SU(3) X U(1) contains
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the N a m b u - G o l d s t o n e supermultlplets, q~ab, ~b~ and q5a. Here, we gauge the unbroken SU(5) and consider it as the standard unified gauge group Then, we can Identify ~btab and 4~a ~ with the usual three families of quarks and lepton chiralmultiplets. However, this model has SU(5) gauge anomalies because of the presence of q5a * 1. The simplest way to cancel the anomalies is to introduce a conjugate chiral multlplet of q~a as a matter field in the sense of the formulation of Coleman et al. [ 11 ]. Nothing is wrong with such a doubhng for the non-hnear reahzatlon of E 7 [5]. Furthermore, if there exists an underlying E 7 preon theory, there are no SU(5) gauge anomalies at the preon level since E 7 is an anomaly free group. Therefore, the conjugate chlral multlplet ~b'a is required at the composite level to satisfy the 't Hooft consistency condition [ 12]. If it is the case, there is an interesting possibility to Identify q~a and qS' a with a pair of Hlggs chiral multlplets (5 + 5*) [ 13]. However, we should note that the Hlggs 4~a has no Yukawa couphng, since the scalar components of q5a are N a m b u - G o l d s t o n e bosons. As long as the supersymmetry is unbroken, it seems impossible to generate Yukawa couphngs because of the non-renormahzatlon theorem [ 14] * 2. We need either a supersymmetry breaking or some exphclt breaking of E 7 to make our model more reahstlc. We hope to come back to this problem m future publications. It should also be noted here that the scalar superpartners of quarks and leptons are, in fact, not true N a m b u Goldstone bosons but pseudo ones, since the E 7 is broken exphcitly by the SU(5) gauge interactions. Therefore, once we switch on the supersymmetry breaking terms, such pseudo Nambu Goldstone bosons will get masses ~c~A2usY through the radiative corrections. However, any mass term of quark and leptons is forbidden by the SU(5) chlral symmetry and thus they will still be massless as far as SU(2)L × U(1) [a subgroup of SU(5)] symmetry is kept unbroken, even after switching on the supersymmetry breaking. The unbroken SU(3) and U(1) also have anomalies. So it is necessary to introduce suitable SU(5)-slnglet matter multiplets in order to meet the 't Hooft condition ,3. Then the SU(3) may be identified with local or global horizontal symmetry [15]. Once SU(3) × U(1) is broken, such additional multiplets will get huge masses of the order of that breaking scale. Another option for this SU(3) × U(1) sector IS to break them at the composite level, I.e., to take a coset space E7/SU(5 ) instead of E7/SU(5 ) × SU(3) × U(1). Although E7/SU(5 ) is not a Kahler manifold, it would be possible to construct a supersymmetrlc nonlinear o model of E7/SU(5 ) by introducing six massless chlral multlplets as the N a m b u - G o l d s t o n e modes, in addition to q~tab, 4~a ~ and q5a. However, the nonlinear lagranglan of those N a m b u - G o l d s t o n e supermultiplets would not be unique since there are three extra quasi N a m b u - G o l d s t o n e bosons which are required to extend our manifold E7/SU(5) to the complex one [5]. Finally we would hke to discuss the problems concerning the possible connections of our approach with the N = 8 supergravlty. Because of the apparent similarity of the groups appearing in the present approach and m N = 8 supergravity, it is tempting to suspect that our model may correspond to a "phenomenologlcal" lagrangian realized dynamically in N = 8 supergravlty. However the following differences immediately come into notice. (1) the E 7 group in our case is compact and an off-shell symmetry while in N = 8 supergravIty it is non-compact (E7(+7)) and merely an on-shell symmetry, i e., vahd only at the level of equations of motion. (i0 Second, the hnearly realized group SU(5) is regarded as a global symmetry (to be gaugized later) in our case whereas it is usually expected to be a local symmetry contained in the hidden SU(8)loc in N = 8 supergravity. The first points (1), the non-compactness of E7(+7 ), m particular, are serious: Indeed if we constructed the theory based on the coset E7(+7)/SU(5) × SU(3) × U(1), which is not Kahler, m the same way as above, we would have obtained a negative metric supermultlplet. So It is clearly impossible to regard our lagrangian as a phenomenological lagrangian o f N = 8 supergravity. But there still remains a possiblhty that our spectrum of the N a m b u ,1 If we take a coset space Es/SO10, we do not need any additional massless field beside the Nambu-Goldstone supermultiplets, since SO10 is an anomaly free group In this case, four famdxes of quarks and leptons and three Hxggs 10, aside from several smglets, are reqmred for the supersymmetrlc non-hnear realization of Es/SOlo ,2 The E 7 is, in fact, exphcltly broken by the SU(5) gauge interactions which we have introduced. However, any Yukawa couphng of q5a is not reduced even after breaking the SU(3) × U(1) symmetry, by the ra&atlve corrections as long as the supersymmetry is kept unbroken [14] ,3 The authors thank P. Ramond for the discussion on this point 316
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19 January 1984
Goldstone supermultiplet itself, 1.e. 3(5* + 10) for quarks and leptons, is realized also in N = 8 supergravity, aside from the dynamics those particles obey. The reason is as follows. As being shown in another paper [16], it IS not the lagranglan symmetry but merely the superpotential symmetry that is required to conclude the existence of Nambu Goldstone supermultIplets. So suppose that the N = 8 supergravity lagranglan is w m t e n in terms o f N = 1 superflelds and the superpotentlal term xs found to have an (of course off-shell) lnvarlance under E7(+7 ). Then it is necessarily invarlant under the complex form E~ (because the real transformation parameters can always be enlarged into complex numbers in the superpotentlal term). So at this stage the difference of the compact form E 7 and the non-compact form E7(+7 ) disappears. Further there seems to exist a certain (33 + 50)-damenslonal complex subgroup H e of E~, which contains SL(5, C) × SL(3, C) × GL(1, C) [complex form of SU(5) X SU(3) × U(1)] as ItS subgroup, such that the N a m b u - G o l d s t o n e supermultaplets for the spontaneous breaking of E~ into H e just fits with our 50 chlral multiplets q~ab, ~az and ca. Next as for the point (11), we should recall the interplay of the global SU(8)glo b C E7(+7 ) and the local SU(8)loc as a hidden symmetry. The 70 spin-zero fields o f N = 8 supergravlty fall into the coordinate of the coset manifold E7(+7)/SU(8 ) after the gauge fixing of the SU(8)loc symmetry. Then the SU(8)loc lnvariance is lost, and the remaining linearly realized SU(8), denoted by the denominator of E7(+v)/SU(8), is neither the SU(8)loc nor the SU(8)glo b but a global lnvariance given by combining those two in such a way that the imposed gauge conditions remain intact. So there IS no apparent contradiction in identifying the SU(5) group in our approach with the subgroup of that SU(8). Further xf we are allowed to assume that our Nambu Goldstone supermultiplets fields are all slnglets under the SU(8)glo b in the stage of before the SU(8)loe gauge fbxing, the SU(5) quantum numbers of these NambuGoldstone modes are identical with those under the SU(5)loc subgroup of the hidden SU(8)loc symmetry. Then the SU(5) gauge Interaction which we have introduced by hand by gaugizing the denominator group SU(5) at the final stage could be regarded as having its dynamical origin in the hidden SU(8)loc symmetry. The authors are deeply indebted to R.D. Peccel for the valuable suggestions, discussions and continuous encouragements on this work. One of them (T Y.) thanks also W. Buchmuller and H.B. Nielsen for the helpful discussions
References [1] E Cremmer and B Juha, Nucl Phys B159 (1979)141 [2] J EUls,M K Galllard and B Zumino, Phys Lett 94B (1980) 343. [3] W. Buchmuller, R D Peccel and T Yanagida, Phys Lett 124B (1983) 67, W. Buchmuller, S T Love, R D Pecce~and T Yanagxda, Phys Lett llSB (1982) 233 [4] B Zumano, Phys Lett 87B(1979) 203, L Alvarez-Gaum~and D Z Freedman, Commun. Math Phys 80 (1981) 443, see also P Fayet and S Ferrara, Phys Rep 32C (1977) 249 [5] W Buchmuller, R D Peccel and T Yanagida, MPI-PAE/PTh 28/83, to be pubhshed in Nucl Phys. [6] C LOng, SLAC-PUB 3056 (1983) [7] H GeorgxandSL Glashow, Phys Rev Lett 32 (1974) 438 [8] E g, R Gilmore, Lie Groups, Lie algebra and some of their apphcatlons (Wiley, New York, 1974) [9] A Borel, Proc Natl Acad Scl 40 (1954)1147. [10] S Welnberg, Phys. Rev. 166 (1968) 1568 [11] S Coleman, J Wessand B Zumlno, Phys Rev 177 (1969) 2139 [12] G 't Hooft, in Recent developments m gauge theories (Carg~se, 1979), eds G 't Hooft et al (Plenum, New York) p 135 [13] S DlmopoulosandH Georgl, Nucl Phys B193 (1981) 150, N Sakal, Z Phys Cll (1981) 153 [14] J. Ihopoulos and B Zumlno, Nucl Phys B76 (1974) 310, S. Ferrara, J lllopoulos and B Zumlno, Nucl Phys B77 (1974) 413 [15] DB Rexss,Phys. Lett 115B(1982) 217, G B Gelmlm, S Nussmovand T. Yanaglda, Nucl Phys B219 (1983) 31, F Wllczek, Phys Rev Lett 49 (1982)1549 [16] T Kugo, I OjImaand T. Yanagada, in preparation, see also W Lerche, preprlnt MPI-PAE/PTh 59/83, C Lee and H S. Sharatchandra, preprlnt MPI-PAE/PTh 54/83 317
PHYSICS LETTERS
19 January 1984
UNIFICATION OF FAMILIES BASED ON A COSET SPACE E7/SU(5 ) × SU(3) × U(1) T. KUGO 1 and T. YANAGIDA 2 Max-Planck-lnstltut fur Phystk und Astrophystk - Werner-Hetsenberg-Institut fur Physik Munich, Fed. Rep Germany
Received 6 October 1983
We show that the usual three famlhes of quarks and leptons are identifiable with quasi Nambu-Goldstone fermlons in a supersymmetrlc non-hnear realization of E 7 corresponding to the Kahler manifold ET/SU(5) × SU(3) × U(1) So the tnphcatlon of famdles suggests the underlying preon theory reahzlng the global E 7 linearly. Possible connections with N = 8 supergravity are also discussed
I n t r o d u c n o n . After the discovery of the hidden symmetry SU(8)local X E7(+7)globa1 by Cremmer and Juha [ 1], many physicists have come to expect that N = 8 supergravity, i.e., maximally local-supersymmetrlc theory, might provide the ultimate theory that describes umfiedly all the fundamental matter fields and Interactions of Nature. Advancing a step m that direction, Ellis et al. [2] proposed a possible scenario how the three famllaes of quarks and leptons could emerge as bound states of "preons", the fields appearing in the N = 8 supergravlty lagranglan. Although Interesting, their arguments are, however, a bit arbitrary and unsatisfactory that is, m their scheme, the dynamacal mechanism to guarantee the e x i s t e n c e and masslessness of those bound-state fermions is left obscure and the determination of their interaction forms would be even further difficult. We pursue in this letter an alternative approach to those problems of families, namely the "quasi N a m b u Goldstone fermxon" approach based on the coset ET/SU(5 ) X SU(3) × U(1), which might be related to the N = 8 supergravlty theory. Here, however, we propose our model as regarding It quite an independent scheme and we defer the d~scussion of possible connections with N = 8 supergrawty to the last part of this letter. A dynamical scheme which is able to generate massless composite fermlons has recently been suggested [3], based on the supersymmetrlc preon theory, where quarks and leptons are fermlon superpartners of Nambu Goldstone bosons (called quasi Nambu-Goldstone fermlons) arising from the spontaneous breakdown of a global symmetry. In this scheme, the number and particular quantum numbers of massless fermlons are determined by the well-defined geometrical properties of the coset space G/H [4,5]. It has been pointed out [5,6] that E 7 is a minimal candidate for G leading to a Kahler mamfold G/H which may accommodate the three famihes of quarks and leptons. In this letter we explicitly show that the supermultiplets of quarks and leptons indeed hve m the Kahler manifold E7/SU(5 ) × SU(3) × U(1) together with a Hlggs mulnplet 5, and construct the effective lagrangian of the Nambu-Goldstone supermultiplets. It is surprising that the quasi Nambu-Goldstone fermions of E7/SU(5 ) × SU(3) × U(1) have precisely the required SU(5) quantum numbers 3 × (5* + 10) for three famihes of quarks and leptons and 5 for a hlggsino. Notice here that the uniqueness of the coordinate of the Kahler manifold (or of the complex mamfold, more precisely) enforces the combination 3(5* + 10) but not 3(5 + 10) for quarks and leptons. Therefore, this model may answer the fundamental questions m particle physics why the observed quarks and leptons transform as 5* + 10 under the grand unified SU(5) [7] and
1 On leave of absence from Department of Physics, Kyoto Umverslty, Kyoto 606, Japan 2 On leave of absence from Department of Physics, College of General Education, Tohoku University, Sendal 980, Japan 0.370-2693/84/$ 03.00 © Elsevier Soence Publishers B.V. (North-Holland Physics Pubhshing Division)
313
Volume 134B, number 5
PHYSICS LETTERS
19 January 1984
why the repetitive structure of families exists. However, there are SU(5) gauge anomalies if only the quasi NambuGoldstone fermions are light particles. We will show later a way to solve this problem which seems natural if there exists an underlying E 7 preon theory. Let us discuss the geometry of the coset space ET/SU(5 ) × SU(3) × U(1).
The coset space ET/SU(5 ) × SU(3) × U(1) E 7 has 133 generators, traceless T/J of the SU(8) subgroup and totally anti-symmetric EIJKL (/ ..... L = 1-8) subject to a reahty constraint (EIJKL)* = (1/4!)C-IJKLMNOPEMNoP. They satisfy the E 7 L e algebra [8] ; -glT~2,
=
[T/, EKLMN] = gKEILMN J J J 1 J EKLMN ' + gLJ EKIMN + ~MEKLIN + gNEKLMI -- ~gl [EIJKL, EMNOP] = 12(T? eQJKLMNOP + TjQ elQKLMNOP + TQKelJQLMNOP + TQLeIJKQMNOP)
-- ½(T~e, JKLQNOP + TQNeuKLMQOP + TQoVKLMNQe + TQ eIJKLMNOQ) .
(1)
The broken generators corresponding to the coset space E7/SU(5 ) × SU(3) × U(1) are defined by
X~ = T~,
Xza = T~a, Xzab= --(1/3[)eabCdeEcdet,
X'ab =(I/2!)eO~Eabjk,
Xa =-(l/4!)eabcdeEacde,
X,, =(1/3!)d/kEa,/k,
(2)
where the indices a ..... e and i, .., k denote the SU(5) and SU(3) indices running over 1-5 and 6-8, respectively. From eq. (1) we immediately find the algebra of the broken generators: [x~tb, XJed ] [~b, ~l
I a b ---g,(gcrS+gb a -b
b ~abt.,rt --gcT~d - g ea Tbc)+ oc ,,j +
b -
=e
[X~, ~ b ] = g ~ T b -- g ba ~ ,
t --z , [Xa,-~b]: --Xab
~ [Xa, Xbl = - r ~ +~6bY,
[x~,x b] = 0 .
el/kAe,
tl
Y) ,
[xtab,x;l=(1/2!)eabcdeeqk2k e
[ X ab, Xlc] = g t1( g cbX a _ g a x b ) ,
[~b,x
c] = O,
i l = [X a, t xb] = O, [X~,Xbl
(3)
where gab =gc6 dab _ 6~g b and Yis the U(1) hypercharge defined by Y= 2Y~a= 1 5 Tad= _2N8=6 T]. Since the group SU(5) × SU(3) × U(1) is the centralizer of a torus U(1) generated by Y in E7, E7/SU(5) × SU(3) × U(1) is a Kahler manifold [9]. This manifold has the unique set of 50 complex coordinates {z a°, zia, z a} a corresponding to the broken generators X s, as is shown explicitly in the following. Thus another choice {zzab , z~, za}, for instance, Is not allowed by the E 7 symmetry. This is important, because otherwise we had wrong quarks and leptons such as 5 + 10 in the low energy spectrum. Corresponding to the Kahler coordinates we have 50 Nambu-Goldstone chlral superInultlplets ~bab, qS~ and q~a (or equivalently qSatb,¢ a and qSa)for the supersymmetric nonlinear realization of E 7. These supermultiplets cab, ~la and ca transform as (10, 3", 1), (5", 3, 2) and (5, 1, 3) under the unbroken SU(5) × SU(3) × U(1), respectively. The last numbers in the parentheses are the U(1) hypercharges. Assuming the SU(5) to be the standard grand unified gauge group, we can easily identify the fermlon components of cab and q~a ~ with the usual three families of quarks and leptons. Let us show that the E 7 is indeed reahzed on our choice of the Nambu-Goldstone supermultiplets qS. Since the knowledge of exphcit parametrlzatlon of the coset manifold is lacking to us, we proceed as follows a la Welnberg [ 10]. The direct study of the "Jacob1 xdentitles",
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[XA, [XB, ¢1 ] -- [XB, [XA, ¢11 = [[XA, XB], ¢1,
(4)
(with X A,B being the broken generators) leads to an exact solution of the nonlinear realization of E 7 on the Nambu-Goldstone supermultlplets. The result as.
(i) Transformatton laws under the X's. --t cd xcd~t [X~b ' ~)j ] = Uab u I ,
t [X~b ' ~]c] = - - 41 e ,]k eabcd e ~)fe,
= a b a ,,
[X;,¢b]
--~
I{xc.u
c
t
[X~b, eft] = ~WaW, - ~b¢~)
:,~, ,
[X a , ¢ b c l = [ X a , ~ ] = 0 ,
+ lt.~loV.~
m--~-.~
tlk
x.cdx.ef
~bd~fW~ ~'k ,
[Xa,¢b I =6 ab .
(5)
(ii) Transformation laws under the X's. [xlab, ¢;d] = eqkeabcdec#ke + ½(d)fb~;d _ o;b ¢ ~ ) _ ½(~)fc4d + dpbdd);c d)bcc~;d_ ¢fd4c), [xa.b, ¢1c] = ~]xab ()d + 1r ,sab,s] ~l.aab~k xab.~de sl , --UtUcd 2t.--s't We + UiWk Wc -- UcdWt We)
+ 1~ ((1/2!)2dkl~c~8 ~ % Y ¢ 7 ~ + (1/2')du~c~ •
,),
[xa~b, oc]:--~[Oacob--(a~b)]+(--~6¢abo;d+~6~o;b
8 [¢~%~%7 8 - (a ~ b)l) +[(~¢ac4d--~oad4c)--(aob)]}O
~.
(6)
The transformation laws under the Xa~ and X a are automatically obtamed by substituting eq. (6) with the following formula:
[xA, ¢1 = (1/4!)dJk%b~de[X~ c, [X~ ~, ~]],
IX ~, ~] = I([X~, IXb~, ~]] - IX) ~, [X~, ~]])
(7)
The non-linear transformation laws given by eqs. (4) and (7) are unique up to possible coordinate transformations which, m this case, necessarily take the form ¢a _~ ca + x~)ab¢ib, d)ab ~ c#ab and ¢~ ~ ¢i with a parameter x aside from trivial scale transformations. Once the coordinates are fixed, the invarlant lagrangian of the N a m b u Goldstone modes is uniquely determined and given in terms of the Kahler potential K(~, ¢), the presence of which is guaranteed by the Kahler property of our coset.
£ = f d 4 0 K(¢, ¢).
(8)
It is, however, difficult to find out the exact form of K. So we construct the non-linear lagrangian up to quartlc terms of ¢ and ¢, which is rather interesting because it gives us the first non-trivial four-Fermi interactions of quarks and leptons'
3--a
]
- ~(~ ~)(¢i
--b
t
I --
a 2
Cb) - 2(~.¢ )
3 --i
~(%b~,
I --t ab --c - ~(¢~¢~ )(~; %)] - ( ¢--~~ c ~ ;a)b( ~ b-¢-
c
ab
--c
J
)(*i ec)
+3--i
~(¢~b*]
ab
-c
]
)Or ¢c) +
3-z ab - c l 2 ( ¢ a c ¢ i )(q~/Cb)
) - (~,%~)(~b¢ b) + ( ~ , % ~ ) ( ~ ¢ b )
-- ~V/~ eqk eabcde(~ab 01)(-¢d eke) -- ~X//-~eqkeabcde(-¢; cfb)(-~; ~bd)}.
(9)
1 Here, we have redefined new fields ~bN's by ~N ab -- (v/x/~)cab, ¢N ai -- VX/~Cia and ( N a = u(Ca + $~ab Cb ) with v being an energy scale which characterizes the breaking E 7 -+ SU(5) X SU(3) X U(1), and we omitted the superscript N for simplicity. We have shown that the supersymmetric extension of the Kahler manifold E7/SU(5 ) X SU(3) X U(1) contains
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the N a m b u - G o l d s t o n e supermultlplets, q~ab, ~b~ and q5a. Here, we gauge the unbroken SU(5) and consider it as the standard unified gauge group Then, we can Identify ~btab and 4~a ~ with the usual three families of quarks and lepton chiralmultiplets. However, this model has SU(5) gauge anomalies because of the presence of q5a * 1. The simplest way to cancel the anomalies is to introduce a conjugate chiral multlplet of q~a as a matter field in the sense of the formulation of Coleman et al. [ 11 ]. Nothing is wrong with such a doubhng for the non-hnear reahzatlon of E 7 [5]. Furthermore, if there exists an underlying E 7 preon theory, there are no SU(5) gauge anomalies at the preon level since E 7 is an anomaly free group. Therefore, the conjugate chlral multlplet ~b'a is required at the composite level to satisfy the 't Hooft consistency condition [ 12]. If it is the case, there is an interesting possibility to Identify q~a and qS' a with a pair of Hlggs chiral multlplets (5 + 5*) [ 13]. However, we should note that the Hlggs 4~a has no Yukawa couphng, since the scalar components of q5a are N a m b u - G o l d s t o n e bosons. As long as the supersymmetry is unbroken, it seems impossible to generate Yukawa couphngs because of the non-renormahzatlon theorem [ 14] * 2. We need either a supersymmetry breaking or some exphclt breaking of E 7 to make our model more reahstlc. We hope to come back to this problem m future publications. It should also be noted here that the scalar superpartners of quarks and leptons are, in fact, not true N a m b u Goldstone bosons but pseudo ones, since the E 7 is broken exphcitly by the SU(5) gauge interactions. Therefore, once we switch on the supersymmetry breaking terms, such pseudo Nambu Goldstone bosons will get masses ~c~A2usY through the radiative corrections. However, any mass term of quark and leptons is forbidden by the SU(5) chlral symmetry and thus they will still be massless as far as SU(2)L × U(1) [a subgroup of SU(5)] symmetry is kept unbroken, even after switching on the supersymmetry breaking. The unbroken SU(3) and U(1) also have anomalies. So it is necessary to introduce suitable SU(5)-slnglet matter multiplets in order to meet the 't Hooft condition ,3. Then the SU(3) may be identified with local or global horizontal symmetry [15]. Once SU(3) × U(1) is broken, such additional multiplets will get huge masses of the order of that breaking scale. Another option for this SU(3) × U(1) sector IS to break them at the composite level, I.e., to take a coset space E7/SU(5 ) instead of E7/SU(5 ) × SU(3) × U(1). Although E7/SU(5 ) is not a Kahler manifold, it would be possible to construct a supersymmetrlc nonlinear o model of E7/SU(5 ) by introducing six massless chlral multlplets as the N a m b u - G o l d s t o n e modes, in addition to q~tab, 4~a ~ and q5a. However, the nonlinear lagranglan of those N a m b u - G o l d s t o n e supermultiplets would not be unique since there are three extra quasi N a m b u - G o l d s t o n e bosons which are required to extend our manifold E7/SU(5) to the complex one [5]. Finally we would hke to discuss the problems concerning the possible connections of our approach with the N = 8 supergravlty. Because of the apparent similarity of the groups appearing in the present approach and m N = 8 supergravity, it is tempting to suspect that our model may correspond to a "phenomenologlcal" lagrangian realized dynamically in N = 8 supergravlty. However the following differences immediately come into notice. (1) the E 7 group in our case is compact and an off-shell symmetry while in N = 8 supergravIty it is non-compact (E7(+7)) and merely an on-shell symmetry, i e., vahd only at the level of equations of motion. (i0 Second, the hnearly realized group SU(5) is regarded as a global symmetry (to be gaugized later) in our case whereas it is usually expected to be a local symmetry contained in the hidden SU(8)loc in N = 8 supergravity. The first points (1), the non-compactness of E7(+7 ), m particular, are serious: Indeed if we constructed the theory based on the coset E7(+7)/SU(5) × SU(3) × U(1), which is not Kahler, m the same way as above, we would have obtained a negative metric supermultlplet. So It is clearly impossible to regard our lagrangian as a phenomenological lagrangian o f N = 8 supergravity. But there still remains a possiblhty that our spectrum of the N a m b u ,1 If we take a coset space Es/SO10, we do not need any additional massless field beside the Nambu-Goldstone supermultiplets, since SO10 is an anomaly free group In this case, four famdxes of quarks and leptons and three Hxggs 10, aside from several smglets, are reqmred for the supersymmetrlc non-hnear realization of Es/SOlo ,2 The E 7 is, in fact, exphcltly broken by the SU(5) gauge interactions which we have introduced. However, any Yukawa couphng of q5a is not reduced even after breaking the SU(3) × U(1) symmetry, by the ra&atlve corrections as long as the supersymmetry is kept unbroken [14] ,3 The authors thank P. Ramond for the discussion on this point 316
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Goldstone supermultiplet itself, 1.e. 3(5* + 10) for quarks and leptons, is realized also in N = 8 supergravity, aside from the dynamics those particles obey. The reason is as follows. As being shown in another paper [16], it IS not the lagranglan symmetry but merely the superpotential symmetry that is required to conclude the existence of Nambu Goldstone supermultIplets. So suppose that the N = 8 supergravity lagranglan is w m t e n in terms o f N = 1 superflelds and the superpotentlal term xs found to have an (of course off-shell) lnvarlance under E7(+7 ). Then it is necessarily invarlant under the complex form E~ (because the real transformation parameters can always be enlarged into complex numbers in the superpotentlal term). So at this stage the difference of the compact form E 7 and the non-compact form E7(+7 ) disappears. Further there seems to exist a certain (33 + 50)-damenslonal complex subgroup H e of E~, which contains SL(5, C) × SL(3, C) × GL(1, C) [complex form of SU(5) X SU(3) × U(1)] as ItS subgroup, such that the N a m b u - G o l d s t o n e supermultaplets for the spontaneous breaking of E~ into H e just fits with our 50 chlral multiplets q~ab, ~az and ca. Next as for the point (11), we should recall the interplay of the global SU(8)glo b C E7(+7 ) and the local SU(8)loc as a hidden symmetry. The 70 spin-zero fields o f N = 8 supergravlty fall into the coordinate of the coset manifold E7(+7)/SU(8 ) after the gauge fixing of the SU(8)loc symmetry. Then the SU(8)loc lnvariance is lost, and the remaining linearly realized SU(8), denoted by the denominator of E7(+v)/SU(8), is neither the SU(8)loc nor the SU(8)glo b but a global lnvariance given by combining those two in such a way that the imposed gauge conditions remain intact. So there IS no apparent contradiction in identifying the SU(5) group in our approach with the subgroup of that SU(8). Further xf we are allowed to assume that our Nambu Goldstone supermultiplets fields are all slnglets under the SU(8)glo b in the stage of before the SU(8)loe gauge fbxing, the SU(5) quantum numbers of these NambuGoldstone modes are identical with those under the SU(5)loc subgroup of the hidden SU(8)loc symmetry. Then the SU(5) gauge Interaction which we have introduced by hand by gaugizing the denominator group SU(5) at the final stage could be regarded as having its dynamical origin in the hidden SU(8)loc symmetry. The authors are deeply indebted to R.D. Peccel for the valuable suggestions, discussions and continuous encouragements on this work. One of them (T Y.) thanks also W. Buchmuller and H.B. Nielsen for the helpful discussions
References [1] E Cremmer and B Juha, Nucl Phys B159 (1979)141 [2] J EUls,M K Galllard and B Zumino, Phys Lett 94B (1980) 343. [3] W. Buchmuller, R D Peccel and T Yanagida, Phys Lett 124B (1983) 67, W. Buchmuller, S T Love, R D Pecce~and T Yanagxda, Phys Lett llSB (1982) 233 [4] B Zumano, Phys Lett 87B(1979) 203, L Alvarez-Gaum~and D Z Freedman, Commun. Math Phys 80 (1981) 443, see also P Fayet and S Ferrara, Phys Rep 32C (1977) 249 [5] W Buchmuller, R D Peccel and T Yanagida, MPI-PAE/PTh 28/83, to be pubhshed in Nucl Phys. [6] C LOng, SLAC-PUB 3056 (1983) [7] H GeorgxandSL Glashow, Phys Rev Lett 32 (1974) 438 [8] E g, R Gilmore, Lie Groups, Lie algebra and some of their apphcatlons (Wiley, New York, 1974) [9] A Borel, Proc Natl Acad Scl 40 (1954)1147. [10] S Welnberg, Phys. Rev. 166 (1968) 1568 [11] S Coleman, J Wessand B Zumlno, Phys Rev 177 (1969) 2139 [12] G 't Hooft, in Recent developments m gauge theories (Carg~se, 1979), eds G 't Hooft et al (Plenum, New York) p 135 [13] S DlmopoulosandH Georgl, Nucl Phys B193 (1981) 150, N Sakal, Z Phys Cll (1981) 153 [14] J. Ihopoulos and B Zumlno, Nucl Phys B76 (1974) 310, S. Ferrara, J lllopoulos and B Zumlno, Nucl Phys B77 (1974) 413 [15] DB Rexss,Phys. Lett 115B(1982) 217, G B Gelmlm, S Nussmovand T. Yanaglda, Nucl Phys B219 (1983) 31, F Wllczek, Phys Rev Lett 49 (1982)1549 [16] T Kugo, I OjImaand T. Yanagada, in preparation, see also W Lerche, preprlnt MPI-PAE/PTh 59/83, C Lee and H S. Sharatchandra, preprlnt MPI-PAE/PTh 54/83 317