Quarks and leptons as quasi Nambu-Goldstone Fermions

Volume 124B. number 1,2 PHYSICS LETTERS 21 April 1983

QUARKS AND LEPTONS AS QUASI NAMBU-GOLDSTONE FERMIONS

W. BUCHMI]LLER, R.D. PECCEI and T. YANAGIDA 1 Max-Planck-Institut fiir Physik und Astrophysik - Werner Heisenberg Institut fi~r Physik - Munich, Fed. Rep. Germany

Received 15 December 1982

D e d i c a t e d to the m e m o r y o f J.J. Sakura i

We discuss a new idea for constructing composite quarks and leptons which have (approximately) vanishing mass. They are associated with fermionic partners of Goldstone bosons arising from the spontaneous breakdown of an internal symmetry Gf in a supersymmetric preon theory. For Gf = SU(5) being broken to SU(3) X U(1)em there arise as quasi Goldstone fermions, naturally and unequivocally, precisely the quarks and leptons of one family. The dynamics of these quasi Gold- stone fermions is explored by constructing a general supersymmetric nonlinear effective lagrangian. By means of a reduced model, we show that the first nontrivial interactions of the quasi Goldstone fermions can give rise, in an effective way, to the weak interactions. Issues connected with the incorporation of fatuities in the scheme and the generation of masses, as well as the possible structure of the underlying preon theory are briefly discussed.

At present we have no evidence that quarks and leptons are anything but elementary excitations. Yet there has been a growing interest recently in considering models of composite quarks and leptons [ 1 ]. This is moti- vated principally by the feeling that, unless quarks and leptons are composite, there is no hope in ever under- standing the complicated patterns of their masses. Composite models of quarks and leptons (preon theories) face an immediate challenge: they must explain why the resulting bound states have sizes, (r) ~ l /A, which are so much smaller than their Compton wavelengths, 1/m. The strong inequality A >> m is only understandable dynamically if there exists an, approximate, symmetry which forces certain bound states to have zero mass. It was originally suggested by 't Hooft [2] that such a symmetry might be chiral symmetry. However, the construction of realistic models of quarks and leptons based on this idea is made difficult by the necessity of matching chiral anomalies at the preon and composite level. Although models exist in which these anomalies do match [3], these models tend to be rather complicated, often involving more preons than quarks and leptons.

The purpose of this note is to discuss an alternative possibility for obtaining light composite quarks and leptons. The dynamical mechanism we suggest to produce (approximately) massless bound states makes use of supersymmetry and of the spontaneous breakdown of an internal symmetry, rather than of chiral symmetry. Our mechanism is the following: imagine a preon theory which is supersymmetric and which has an overall global internal symmetry group Gf. If this global symmetry is spontaneously broken down to a subgroup Hf, then there will appear in the theory Goldstone bosons - one for each of the generators in Gf/Hf. However, because of the supersymmetry, these Goldstone bosons are accompanied by fermionic partners - which in an earlier publication with Love [4] we called quasi Goldstone fermions - as well as an additional set of bosonic partners. Our suggestion is then simply that quarks and leptons are quasi Goldstone fermionic bound states of a confining supersymmetric theory *

1 On leave of absence from the Department of Physics, College of General Education, Tohoku University, Sendai, Japan. 4-1 Our asking that quark and leptons are bound states of a confining theory, rather than elementary excitations is tied to our de-

sire of having the possibility of eventually calculating their masses. The confinement requirement for the preon theory follows only because it is hard to imagine confined quarks made out of unconfined preons!

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Volume 124B, number 1,2 PHYSICS LETTERS 21 April 1983

This idea has two apparent drawbacks, which at first sight may appear to render it phenomenologically use- less:

(1) The quasi Goldstone femfions are accompanied by partner scalar excitations. One knows experimentally that the scalar partners of quarks and leptons, if they exist, must be considerably heavier than the fermions.

(2) Quasi Goldstone fermions transform as real or as pairs of complex conjugate representations under the unbroken group Hf. We know, however, that with respect to the weak interactions the quarks and leptons are complex and not paired.

The first point above is a common problem of any "low energy" supersymmetric theory, which may be in principle circumventable by an appropriate breaking of the supersymmetry. The second point, however, appears much more serious, and in fact impeded progress for some time.

There is an advantage of our scheme. Namely, since Gf and Hf are at our disposal, it may be possible to select them appropriately to obtain quarks and leptons with the right quantum numbers. For instance [4], with Gf = E 6 and Hf = SO(10) one finds in Gf/Hf a 16 + 16 + 1 set of massless states. Thus, the quarks and leptons of one family can be fit in the 16 but, illustrating point (2) above, there is an additional 1T of mirror fermions ,2

The impasse surrounding the mirroring of representations can, however, be overcome. Let us focus for the moment on one family of 15 quarks and leptons (15 left handed fields). Although this set of states is not real with respect to the standard model group SU(3)X SU(2)X U(1), it is real with respect to the unbroken subgroup o fSU(3) × U(1)em. That is, all the quark and lepton representations come in complex conjugate pairs, save for the neutrino which is a singlet under SU(3)X U(1)e m . Thus if Hf is SU(3)X U(1)em, it is perfectly reasonable to suppose that the 15 quarks and leptons of one generation are the quasi Goldstone fermions of the breakdown of Gf -+ Hf. Since 9+ 15 = 24, clearly the unique solution for Gf is SU(5)!

It may be useful to restate this result. If one has a supersymmetric preon theory which has an SU(5) global symmetry which is spontaneously broken down to SU(3)X U(1)em, necessarily and unequivocally, one obtains a set of 15 massless fermionic excitations with precisely the quantum numbers of the quarks and leptons of one family. We note that this SU(5) has an electromagnetic charge definition which is different from the usual Geor- gi-Glashow SU(5) [7]. Namely for our case

Q = diag(~s, 1 , ~s, ~ '--~)" (1)

Using (1) it is easy to check that the Goldstone excitations which are in SU(5)/SU(3)X U(1)e m have precisely the charges and color quantum numbers of the quarks and leptons.

If quarks and leptons are the quasi Goldstone fermions of SU(5)/SU(3)X U(1)e m it is possible to discover some of their dynamical interactions, even though we do not really know specific details of the preon theory. This has an analog in hadronic physics. Even before one knew about QCD, low energy pion physics was under- stood because the pions were the Goldstone bosons of the spontaneous breakdown of chiral SU(2)X SU(2). The Goldstone nature of the pions allowed for the construction of effective lagrangians [8] where the chirality is real- ized nonlinearly. These lagrangians embody all the low energy properties of the theory. For the case at hand, we need a supersymmetric generalization of this construction applied to the specific groups Gf and Hf in question.

Before we outline the details of this construction, it will be helpful to indicate the nature of the results obtained. We find that the dominant interaction at low energy among the quasi Goldstone fermions is a four-Fermi current-current interaction, which in general contains a series of parameters characterizing the breakdown. For the case of a particularly simple toy model - to be described below - these parameters can be chosen to reproduce precisely the current-current form o f the weak interactions. We expect that this behaviour should also per- sist in the more realistic SU(5) -+ SU(3)X U(1)e m breakdown. This result suggests - although it cannot prove - that if quarks and leptons are indeed quasi Goldstone fermions then the weak interactions may be residual. Be- fore discussing this point further, as well as possible ideas for mass generation and the inclusion of families in

:~2 For grassmanian manifolds of dimension 2M there is the intriguing possibility of getting rid of the doubling of fermions by supersymmetrizing only M complex scalar fields. See refs. [5 ] and [6 ].

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our scheme, we want to proceed to the construction of the effective lagrangian which yields the above quoted result.

What is needed is a generalization of the Coleman, Wess and Zumino [9] construction of nonlinear lagrangians applied to the case of supersymmetric theories. Although some such constructions exist already in the literature [ 10,11], they do not really quite fit our needs. We outline therefore a general procedure which we have found ex- piditious for constructing these nonlinear supersymmetric lagrangians. For a given group Gf with generators Ta one can always construct nonlinear realization of the Goldstone fields ~i in the coset Gf/Hf by explicitly constructing the Killing vectorsAai(~) * 3. The commutators of the generators T~ with the Goldstone field ~i are then

given by

i - 1 [Ta, ~il = Aai(~), (2)

and one can check that for indices c~ corresponding to generators in Hf the transformation is linear, otherwise it is nonlinear. The construction in (2) automatically satisfies all the Jacobi identities. It is this property that makes it immediately useful for a supersymmetric generalization. Let Oi be a chiral superfield which contains the ~i as part of its scalar components. Then the equation

i -1 [Tc~ ' Oil = Aod(O) (3)

obviously provides an adequate nonlinear supersymmetric realization, since it also satisfies all the required Jacobi identities. This construction is the analog of what was done by Weinberg [8] originally for chiral symmetry.

Armed with (3), it is now straightforward to construct a nonlinear lagrangian that is invariant under these transformations. First of all we note that in (3) there will appear in general arbitrary scales V i - one such scale for each of the different representations of Hf under which the Goldstone superfields q5 i transform [ 12] * 4. Fo- cusing only on the nonlinear part (generators Ti) we may write therefore (3) in the form

i -1 [Ti, OjVI. ] = Cif + Cij k Ok/Vk + Cijkl(Ok/Vk)Ol/Vl, (4)

where the coefficients Cii, Ciik, etc. are numbers determined by group theory. If we are interested in an effective nonlinear lagrangian invariant under (4) up to and including interactions of O(1/vN), we need to compute (4) mdy up to terms of similar order. For our purposes it will suffice to stop at terms of O(1/V2).

Our lagrangian must include, of course, the generalized kinetic energy for all the Goldstone superfields, (~/74~i + -")D-term, invariant under (4). It may, furthermore, include any higher order interactions among the superfield Oi which are: (a) supersymmetric and (b) invariant themselves under (4) 4- s. It is easy to check that only D-terms are allowed. F-term interactions necessarily require that some of the scalar components of the ~b i have non-zero vacuum expectation values - thereby contradicting the Goldstone nature of these fields. We note furthermore that, given a nonlinear realization, one has the freedom to reparametrize the fields again nonlinearly without af- fecting the physics. Thus, using this freedom, it is easy to convince oneself that, up to terms of O(1/V3), the most general nonlinear supersymmetric lagrangian has the form

.1~ = ~t ~)i + Aijkl(dSi /Ve)(q57/Vi)(dPk/Vk) ~l/Vl + h.c.lo_term, (5)

where the coefficients Ai/kl have dimensions of mass squared. In general these coefficients will contain pieces pro- portional to the various scales Vi, as well as pieces corresponding to new invariant tensor structures.

#3 The paper of Boulware and Brown [ 12] presents a lucid discussion of the general construction of the Killing vectorsAa/(O and of nonlinear realizations.

4-4 For the case of chiral symmetry the pions transform as vectors under the unbroken isospin group and hence there is only one dimensionful constant fn.

#s This freedom arises because the lagrangian corresponds to the Kahler potential of a complex manifold whose metric is not uniquely determined by the Killing vectors of G/H.

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The first nonlinear correction to the Goldstone superfield kinetic energies contains precisely the four-Fermi current-current interactions discussed earlier. Using a Fierz identity and writing only the piece involving fermions (written as left-handed Dirac fields) one has that

~bj 0kq$1tD-term - - ~ 0 k i L Y ~kL)(q%Yu~lL) + terms involving bosch fields. (6)

Hence (5) does indeed contain a vector (or axial vector) interaction between currents formed out of the quasi Goldstone fermions. We should note here the difference between quasi Goldstone fermions and goldstinos. If the supersymmetry were spontaneously broken one would also obtain four-fermion interactions among the goldstinos. However, the coupling would involve not 3,u 7u but 7** 3 v 7u 3v, which vanishes as the momentum t ransfe r vanishes. Goldstinos can never reproduce the weak interactions [ 13], but quasi Goldstones may * 6 !

The construction of the nonlinear lagrangian (3) is perfectly straightforward but for the case of the breakdown SU(5) -+ SU(3)X U(1)e m it is rather laborious. In this note, therefore, we will discuss only a toy example involving the breakdown of U(2) -+ U(1)e m . This "model for leptons" will illustrate most of the relevant physics. We reserve for a forthcoming publication [16] the detailed discussion of the SU(5)-+ SU(3)X U(1)e m case.

Let us denote the broken generators of U ( 2 ) b y 7"+, T_ and T O and let the corresponding Goldstone superfields be ~b+, q~ and q$0, where the subscripts give the relevant EM charges. The nonlinear realization corresponding to the U(2)/U(1)e m breakdown will give rise to the commutators

i - l [T i , cJj]=Aij(d#), i , / = + , - , 0 . (7)

A straightforward calculation, using the methods of ref. [ 12] to construct the Killing vectors, gives, up to terms of O(1/V3), the following formula for the matrix A#:

+ - 0

A i r = -

- (112i)(Vol V+) qa+ ~ ¢ +~, +lV +

+ ~4 ~o~+lV+

2V_ + (ll2i)(V_IVo)Oo 1 2 2 ~a (v-IVo)epo - ~ ~+<b_lV+

(1/2i)~+ + ~ ~+(JolVo 2 V o

~ (Vo/V+ v_ ) ¢,+¢,_ -(1/2i)4b +~ d#oqo lV 0

2V+ - (1/2i)(Vl/Vo)(PO ( l / 2 i ) ( V o / V _ ) q a -

o ~4~_~_lV - ~ (V+lV2)+~ - ~ +++_iv + 7a +o+_lV_

Having eq. (8) in hand, it is now easy but tedious to construct an effective lagrangian which is invariant under these transformations, up to terms of O(1/V3). The result of this computat ion is the following nonlinear lagrangian, in which some simplifications were made by using the freedom of arbitrarily reparametrizing the superfields q~i:

"~e ff = dS+O+ + (Pt_dP_ + dP~OdPO + {¼(Z + B - -~1 V 2 .1.._~ VO)O+(~+~)+~)+/V "f ~f 4 + ¼(h - n - ~1 V 2_ + _~ V 2) ~ t (at_ (a_ ~_ /V 4

1 2 21 + (19-C21V2)q~tO(a)o(OO(aoIV4 + [2E + ~ C - V+2(B + aVo) i(a~+~b~O(a+d#o/V2V2

+ (A BZlV20)e#t+(o* (a+(a IV2V 2 + [2E ' 1 , ,z2-~21 q~_ - _ _ - ~ c - v - _ ( z i - a - o , , ~ t o ~ - ~01 v2 v2

+ ( E - 2 ) t t t 2 BC/Vo)(~+dP dPOdP 0 + ~OdPOqg+dp_)/V+ V V O} + O(1/V3)lD_term .

:1:6 In the recent suggestion of Bardeen and Vi]nji6 [ 14] that quarks and leptons are tile goldstinos of spontaneously broken supcrsymmctric theories (sec also ref. [ 15]) the resulting nonlinear derivative interactions arc interpreted as characteristic signals of the onset of compositeness.

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(8)

(9)


Here tile constants A, B, C, D, E, with dimension of (mass) 2 are arbitrary and characterize additional invariants allowed under the transformation (7). The terms involving only the scales V i give the invariant generalization of the kinetic energy to this order.

Because of the rather large amount of freedom in eq. (9), it is not surprising that such a lagrangian can reproduce the effective current-current fom~ of the standard model. In fact, for sin 2 0 W - 1 - ~ and p = 1 for simplicity, the standard model lagrangian in units o f× /2 G F necessitates the following coefficients for the terms appearing in eq. (9): 4,1 ~,a - 1 , +1, - 3 , -~,1 0. It is easy to convince oneself that such numbers can be obtained via suitable choices of A ..... E and V+, V_ and V 0.

The lessons to be learned from this model depend in part on the amount of speculation one allows oneself. It is clear that, although we have shown that the residual dynamics are compatible with the standard four-fermion weak interactions, it could well be that these interactions have nothing to do with weak processes at all. Indeed if the V i >> 250 GeV then these interactions could just be additional short range contributions which slightly mod- ify the normal weak interactions (tell tale signs of compositeness?). On a much more speculative level, however, is the possibility that the weak interactions are these residual interactions. Obviously, the particular structure of the coefficients A, ..., E and Vi's which give one this option must reflect particular dynamical properties of the underlying theory. If one has residual weak interactions then the Z 0 and W ± of the standard model, although probably present dynamically, would have different properties (masses, widths, number of states, etc.). In this sense this idea is very much along the lines pursued by Sakurai [ 17]. Note, in particular, that if the interactions of eq. (9) are the weak interactions, then the scale of the weak interactions and that of compositeness are essentially the same. This proposition suggests therefore a nearby compositeness with a concomitant rich and interesting physics.

The idea that quarks and leptons are quasi Goldstone fermions clearly opens up interesting vistas. However, there are many open and interesting problems which need to be urgently answered. Foremost among these are:

(1) How does one incorporate families? (2) How does one generate masses? (3) What is the structure of the underlying preon theory? We obviously do not have very precise ideas on how to answer these universal and deep questions. Neverthe-

less, we want to close this paper by outlining some of our tentative thoughts on these issues. Families have probably a deeper reason for their existence. However, if one just wanted to incorporate them in

the quasi Goldstone way as group theoretical repetitions, then there are not that many possibilities because the group theory is quite restrictive. The two more "natural" ways we have found are:

(1) Gf = (SU(5)) nf broken down to (SU(3)X U(1)) nf, with the color and U(1)em group being eventually identified with the diagonal sums; or:

(2) Gf = E 7 broken down to Hf = SU(6) × SU(3). In this last case the coset space Gf/Hf has 90 excitations which comprise 3 families of states plus their mirrors. The mirroring here may ultimately prove to be unacceptable, but the fact that E 7 is the global symmetry connected with N = 8 supergravity [18] makes this alternative worth exploring.

The issue of mass generation is most likely connected to the existence of families. Nevertheless, perhaps it is worthwhile describing here some of our intuitive ideas, even within the context of one family. The picture we have is the following: Goldstone excitations can acquire small masses when the global symmetry, whose spontaneous breakdown engendered them, is slightly broken. The classical application of this idea is in pion physics: The spontaneous breakdown of SU(2)X SU(2) gives three massless pions. However, turning on C%m breaks explicitly the SU(2) × SU(2) and the 7r ± get a mass shift relative to the 7r 0. This shift 6 m2± in QCD is obviously of O(c~A~cD). Indeed, even before the advent of QCD it was calculated [19] in terms of appropriately saturated vector and axial vector spectral functions and related to a and the P and A 1 masses.

We expect that the SU(5) global symmetry of the preon theory is explicitly broken when we turn on the SU(3) and U(1)e m coupling constants. Hence, intuitively, the electron should get a mass because it has charge; the quarks get masses because they carry color and charge, and the neutrino remains massless. This expectation,

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however, is not borne out if the supersymmetry remains exact! This is because of the, so called, nonrenormal- ization theorem [20] of supersymmetric theories, which does not allow particles which have zero mass at tree level to acquire radiative masses * 7. For mass to be generated for the Goldstone bosons and the quasi Goldstone fermions it is necessary that the supersymmetry be broken first.

It is an open question how the supersymmetry is actually to be broken. One option which might be useful to consider, at least for the purpose of mass generation, is the following. Imagine that the gauging of the SU(3)

× U(1)e m is done in a nonsupersymmetric way, by omitting altogether the gauginos ,8 . In this case, one expects

that the scalar excitations pick up a (mass) 2 of O(aA2Mc) where AMC is the typical scale associated with the compositeness. This is very much like in the pion case. The quasi Goldstone fermions cannot, however, get a mass directly in this way, since massless fermions are chiral. Their mass can only arise through graphs in which the quasi Goldstone fermions couple directly with the heavy states in the theory. In zeroth order, these graphs themselves cannot generate mass, but eventually, through the dressing with the nonsupersymmetric gauge fields some mass can be expected. It is obviously hard at this stage to give an estimate of the dynamical mass so acquired by the quasi Goldstone fermions. However, it appears certain that it will be less than that of the corresponding bosons. But will it be light enough to satisfy the necessary phenomenological requirements?

The question of the structure of the preon theory is, obviously, the hardest to answer, and here we are likely

to be wrong. A possibility which we have been exploring is an SU(5)M C of metacolor with 5 L and 5 R flavors of preons. This theory requires a mass term for the flavor preons so as to have only a vectorial global SU(5). The natural SU(5)M C singlet condensates (53) and (55555), (55555) can break the SU(5) of flavor (via a 24 plus a 175 and 175 ) to SU(3)× U(1)e m . In this theory it is possible to identify an SU(2) which could correspond to the weak interactions among the lower two components of the 5 L preons * 9. The open question, however, is whether one should gauge these interactions then at the preonic level or make use of the bound state dynamics to generate. the weak interactions via the particular pattern of symmetry breakdown. It is clear that this and many other open questions will need to await future developments for their answer.

We have profited greatly from conversations with D. Amati, P. Breitenlohner, A. Buras, C.K. Lee, L. Girardello and D. Maison. We are also grateful to many colleagues, both at the Max-Planck-Institut as well as elsewhere, for their thoughtful criticisms. We are sorry that because of S. Love's return to the USA he could not participate in these new developments.

,7 One should perhaps worry about applying these theorems to bound states, but it appears to us that they probably should also go through in this case.

,8 Such an option is not sensible if the Goldstonc excitations are elementary, since it breaks the supersymmetry in a hard way. It might be sensible for composite Goldstonc particles, but it must be modified at the preon level. At that level it is more reasonable to add a gaugino mass.

,9 Note, however, that the mass termM 5L5R explicitly breaks this SU(2).

References

[ l ] For a review see for example: R.D. Peccei, Lectures Arctic School of Physics (,~k//slompolo, Finland) MPI-PAE/PTh 69/82. [2] G. 't Hooft, in: Recent developments in gauge theories, Cargbse Lectures (1979) (Plenmn, New York, 1980) p. 135.

13] I. Bars, Nucl. Phys. B208 (1982) 77. [4] W. Buchmiiller, S.T. Love, R.D. Peccei and T. Yanagida, Phys. Lett. 115B (1982) 233. [5] C.L. Ong, SLAC preprint SLAC-PUB-2936 (1982). [6] J. Bagger and E. Witten, Phys. Lett. l18B (1982) 103. [7] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [8] S. Weinberg,Phys. Rev. 166 (1968) 1568. [9] S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2139;

C.G. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247.

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[10] B. Zumino, Phys. Lett. 87B (1979) 203. [l 1] L. Alvarez-Gaum6 and D.Z. Freedman, Commun. Math. Phys. 80 (1981) 443. [12] D.G. Boulware and L.S. Brown, Ann. Phys. (NY) 138 (1982) 392. [13] B. de Wit and D.Z. Freedman,Phys. Rev. Lett. 35 (1975) 827;

W.A. Bardeen, unpublished. [14] W.A. Bardeen and V. Vi'~nji4, Nucl. Phys. B194 (1982) 422. [15] H. Terazawa, Prog. Theor. Phys. 64 (1980) 1763. [ 16] W. Buchmiiller, R.D. Peccei and T. Yanagida, in preparation. [ 17] J.J. Sakurai, Schladming Lectures (1982). [181 E. Cremmer and B. Julia, Phys. Lett. 80B (1978) 48; Nucl. Phys. Bl59 (1979) 141. [19] T. Das, G.S. Guralnik, V.S. Mathur, F.E. Low and J. Young, Phys. Rev. Lett. 18 (1967) 759. [20] J. Iliopoulos and B. Zumino, Nucl. Phys. B76 (1974) 310;

S. Ferrara, J. Iliopoulos and B. Zumino, Nucl. Phys. B77 (1974) 413.

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Quarks and leptons as quasi Nambu-Goldstone Fermions