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Letters B 533 (2002) 121–125 www.elsevier.com/locate/npe

Should E8 SUSY Yang–Mills be reconsidered as a family unification model?

Stephen L. Adler

Institute for Advanced Study, Princeton, NJ 08540, USA Received 2 February 2002; accepted 20 March 2002 Editor: H. Georgi

Abstract

We review earlier proposals for E8 family unification, and discuss why recent work of Kovner and Shifman on condensates in supersymmetric Yang–Mills theories suggests the reconsideration of E8 supersymmetric Yang–Mills as a family unification theory.  2002 Elsevier Science B.V. All rights reserved.

One of the outstanding mysteries of the current (or if right handed neutrinos are included) 16 Weyl standard model is the triple repetition of fundamental fields in each family, a group representation fermions. Many different ideas1 have been proposed of at least 45 or 48 is required. So we are to explain why there are three (or in some models necessarily considering a large group representation, more) families; here we focus on the possibility, and if we invoke naturalness to require that it be a already addressed in the earlier literature, that the low-lying representation of its or algebra, family structure has a group theoretic origin, with then we are necessarily considering a large group. all three families embedded in a large representation A particularly interesting candidate is the group E8, of a family unification group. Since there are 15 which has a 248-dimensional and, as the largest exceptional group, a unique position in the standard Cartan classification of Lie groups. Our aim E-mail address: [email protected] (S.L. Adler). in this Letter, which is unapologetically speculative 1 Three prominent types of ideas that have been advanced to and programmatic, is to review earlier work on E8 explain family structure are (1) composite models, in which the unification, to explain difficulties encountered, and to quarks and leptons are not elementary, and the different families are argue that recent developments suggest that there may different internal symmetry states of their constituents, (2) - be mechanisms that can overcome these difficulties. inspired models, in which the different families correspond to string excitations with different topological quantum numbers of the Thus the time may be ripe to reconsider E8,and background manifold on which the string spacetime is compactified, specifically E8 supersymmetric Yang–Mills theory, as and (3) as discussed here, group theoretic family unification models, a family unification model. in which the quarks and leptons are elementary (at least below the Unification theories based on simple Lie groups Planck scale) and the family structure arises because the fermions follow a basic paradigm established by the SU(5) lie in a large group representation. For early discussions of family groups, see Ref. [1]. and SO(10) models. The gauge bosons are as usual

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)01596-4 122 S.L. Adler / Physics Letters B 533 (2002) 121–125 in the of the group, and left- families into a single representation. The point that handed Weyl fermions are placed in one or more ad- E8 Yang–Mills theory can contain SU(3) as a family ditional representations, chosen to give cancellation of group was made by Bars and Günaydin [4] and anomalies together with the standard model fermion was emphasized in Slansky’s review [7] and also structure under breaking of the unification group to by Barr [10]. In the different dynamical context of SU(3) × SU(2) × U(1). Turning to E8,thisisthe supersymmetric nonlinear σ models, the point that E8 unique in which the adjoint repre- can naturally lead to three families was made in papers sentation, of dimension 248, is also the fundamental of Ong [11], Buchmüller et al. [12], Itoh et al. [13], and representation. Hence the natural implementation of Ellwanger [14]. the basic paradigm is to place left-handed Weyl fermi- Despite the attractive features of automatic super- ons in the 248 representation, giving a model in which symmetry and natural inclusion of three families, the the gauge bosons or gluons, and the fermionic matter reason that E8 has not been further pursued as a uni- fields, are both in the adjoint 248 representation. Since fication group is that in addition to three families, in four supersymmetric Yang–Mills the- it contains three mirror families. Thus, under E8 ⊃ ory can be constructed with adjoint fermions that are SU(3) × , in addition to three 27’s there are three either Majorana or Weyl [2], in this E8 model the 27’s, while under E8 ⊃ SU(3)×SO(10)×U(1),inad- fermions and gluons are in the same supermultiplet, dition to three 16’s there are three 16’s. The presence achieving a complete unification of matter fields and of mirror families leads to potential phenomenological force-carrying fields. The point that an E8 unifica- and theoretical difficulties. tion model is automatically supersymmetric was made The phenomenological difficulty is that since the independently more than twenty years ago by Baak- masses of mirror families break SU(2) × U(1) elec- lini [3], by Bars et al. [4], and by Konshtein et al. [5], troweak symmetry, they must be of order the elec- was followed up on in a paper of Koca [6], and was troweak symmetry breaking scale, at most a few hun- briefly noted in Slansky’s comprehensive review [7] dred GeV. Hence, although they need not have been of for model building. produced in current accelerator experiments, they will Another interesting feature of E8 is that it natu- manifest themselves indirectly through electroweak rally contains three families. Most of the recent dis- radiative corrections, and should be copiously pro- cussions of single family grand unification are based duced once the large hadron collider (LHC) is op- on either the group SO(10) [8] or the group E6 [9]. In erative. A detailed review of experimental signatures SO(10) unification the 16 Weyl fermions of a family for mirror fermions has been given by Maalampi (including a right handed neutrino) are placed in a 16 et al. [15] (see also Montvay [16] and Triantaphyl- representation, while in unification in the larger group lou [17]; the latter also has discussed a possible role E6,ofwhichSO(10) is a , these fermions are for mirror fermions in dynamical electroweak symme- placed in a 27 representation. Under the decomposi- try breaking). One potential phenomenological objec- tion E8 ⊃ SU(3) × E6, the 248 of E8 branches [7] as tion to mirror fermions is that under the assumptions that their masses are much larger than the Z-boson = ( , ) + ( , ) + ( , ) + ( , ), 248 8 1 1 78 3 27 3 27 (1a) mass and are degenerate within right-handed doublets, while under E8 ⊃ SU(3) × SO(10) × U(1), the 248 each family of mirror fermions would make a contri- branches [7] as bution of 2/(3π) to the electroweak S parameter (see Peskin et al. [18], and the review by Erler et al. [19]), 248 = (1, 16)(3) + (1, 16)(−3) + (3, 16)(−1) in strong disagreement with experiment. However, this + (3, 16)(1) + (3, 10)(2) is not as definitive as it seems; when the degeneracy + (3, 10)(−2) + (3, 1)(−4) + (3, 1)(4) assumption is dropped the contribution of a mirror family to S can have either sign (or be zero), and recent + ( , )( ) + ( , )( ) + ( , )( ), 8 1 0 1 45 0 1 1 0 (1b) analyses of the electroweak precision data by Novikov with the U(1) generator in parentheses. Thus, the et al. [20] and by He, Polonsky, and Su [20] conclude 248 of E8 naturally contains three 27’s of E6 and that additional chiral generations are not currently ex- three 16’s of SO(10), and so can unify the three cluded, with Novikov et al. finding a chi-squared min- S.L. Adler / Physics Letters B 533 (2002) 121–125 123 imum between one and two extra generations. A sec- velop. While the two independent arguments advanced ond analysis by Choudhury et al. [21], focusing on ad- by Kovner and Shifman to support their suggestion for ditional mirror bottom quarks, also finds an improved a new phase are now discounted (one of these was fit to the electroweak data. In an E8 unification model, based on problems with the Witten index for certain each fermion family is accompanied by a family of groups, which are now resolved [27]; the other on a vector gluons which will also, at least [22] in the case mismatch between the strong and weak coupling in- of non-mass degenerate vector doublets, make contri- stanton calculations of the gluino condensate, which butions to the S parameter, and therefore will further has been given another explanation [28]), their effec- weaken the constraints coming from the electroweak tive action argument for the existence of a phase with- data. Thus the mirror structure predicted by an E8 out a gluino condensate is still viable, and their conjec- model may well be consistent with current data. ture of a new phase for supersymmetric gluodynamics The theoretical difficulty is that under the most is open, although still debated [29,30]. attractive channel rule, a theory with equal numbers In particular, although Csáki et al. [30] have used of ordinary and mirror families would in general be discrete matching to argue against the expected to form a chiral symmetry breaking family- Kovner–Shifman vacuum, their argument assumes that mirror family condensate, and so one would naively the ground state spectrum consists of hypercolor (here expect no low energy families to survive in the low E8) singlets. Thus it does not rule out the possibil- energy effective action. This expectation has become ity that the Kovner–Shifman vacuum is in a trivial, virtual dogma in model building, where it is usually deconfined phase with the same particle spectrum as stated that in a model with nf families and nf¯ mirror the starting E8 , before symmetry break- families, the difference nf –nf¯ gives the number of ing arising from perturbations to the SUSY gluody- surviving low energy families if positive, and the namics structure is taken into account. A deconfined number of surviving low energy mirror families if phase would obey all anomaly matching constraints, negative. However, this dogma must be treated with and even if not generic for SUSY Yang–Mills gluody- some skepticism, since there are known instances (see namics, its presence just in special cases including E8 Seiberg [23] and Holdom et al. [24]) where the most would suffice for the arguments we are making. attractive channel rule breaks down. In the specific If supersymmetric Yang–Mills for the E8 group context of supersymmetric E8 Yang–Mills theory, the is in the Kovner–Shifman vacuum, then the principal issue is whether an E8 singlet gluino condensate λλ theoretical objection to E8 as a unification group dis- forms, as suggested by an effective action argument appears, since the theory in isolation would remain of Veneziano et al. [25]. The presence of such a a supersymmetric theory (as assured by Witten in- condensate would prevent the appearance of fermions dex arguments [27]2) with massless gluinos in the (which are the E8 gluinos) in the low-energy effective Kovner–Shifman phase. Of course, to get a realistic action. theory breaking of both E8 symmetry and supersym- Recently, in a very interesting paper, Kovner et metry is needed. As noted by Shifman et al. [29] (in al. [26] have argued that the Veneziano–Yankielowicz the course of a discussion of the Witten index, but effective action must be modified so as to explicitly ex- their remark is more generally relevant) the Kovner– hibit the Z2T(G) discrete chiral symmetry, which is the Shifman vacuum is “potentially unstable under various non-anomalous remnant of the anomalous U(1) axial deformations”. One obvious deformation that could be symmetry generated by phase rotations of the gluino relevant is the embedding of supersymmetric E8 in su- fields. (Here  = 2T(g)is the Dynkin index of the ad- pergravity. When the gravitino and fields are joint representation, which equals 60 for the adjoint integrated out at tree level, one obtains [32] a super- 248 representation of E8.) They show that there is a symmetric four-gluino effective action that could be simple modification of the Veneziano–Yankielowicz the trigger for dynamical symmetry breaking of either action which has the required discrete symmetry, and that this action predicts that there is a phase in which the discrete chiral symmetry is unbroken, and thus in 2 For a survey of the Witten index analysis of supersymmetric which the usual singlet gluino condensate does not de- gauge theories, see Sections 29.1 and 29.4 in [31]. 124 S.L. Adler / Physics Letters B 533 (2002) 121–125 or both the E8 internal symmetry and supersymme- are contained in the 248 × 248 of E8, and so could try. breaking could also arise from su- be generated by the formation of non-singlet gluino– persymmetry breaking in another sector of the theory gluino condensates. As a final remark on symmetry (such as the second E8 expected in ) com- breaking, we mention that a much studied alternative municated by the interaction between the to dynamical generation of Higgs condensates is their two; a general review of this approach is given in generation by dimensional reduction from a higher- Weinberg [33], and an application of gravity-mediated dimensional gauge theory; for a recent discussion of supersymmetry breaking to sequential breaking of E8 this mechanism as applied to E8 and three family to E6 and then to SO(10) is discussed by Mahapatra et unification, in the context-dimensional reduction over al. [34]. coset spaces, see Manousselis et al. [37]. As noted by Bars et al. [4], an E8 unification the- The phenomenologyof a supersymmetric E8-based ory cannot have elementary Higgs scalars without los- grand unification and family unification model will ing the property of asymptotic freedom, because the differ significantly from that expected in the minimal Dynkin index of the smallest candidate Higgs repre- supersymmetric standard model (MSSM) and its ex- sentation (the 3875) is already too large. Hence in an tensions. As in the MSSM, the superpartners for the asymptotically free E8 theory, all symmetry breaking gauge bosons in the E8 model are spin-1/2 fermions, (other than that communicated by gravity mediation and R-parity conservation3 implies that the lightest su- from another sector) must be dynamical, through the perpartner will be stable. However, in the E8 theory, formation of suitable condensates of the gluinos (and in addition to there being mirror fermions, the super- of gluinos and gluons as well, if condensate formation partners for the quarks and leptons are vectors rather preserves supersymmetry). Chiral symmetry breaking than scalars. Thus there are potentially observable sig- by condensate formation was reviewed some time ago natures for E8 unification at the LHC and other future by Peskin [35], and recently there has been much in- facilities. terest in the role of non-singlet condensates that break gauge symmetry, in the context of “color superconduc- tivity” in high density QCD [36]. In order to give the Acknowledgements mirror fermions larger masses than the top quark, there must be a condensate that introduces an asymmetry This work was supported in part by the Depart- between the fermions and their mirror partners. One ment of Energy under Grant #DE-FG02-90ER40542. candidate arises from the fact that in SU(3) × E one 6 I wish to thank Csaba Csáki, Maurice Goldhaber, Paul has (3, 27) × (3, 27) ⊃ (6 , 27 ). Since under the de- s s Langacker, Michael Peskin, , Jacob composition E ⊃ SO(10) × U(1) one has 27 ⊃ 1(4), 6 Sonnenschein, , and Ren-Jie Zhang for a gluino–gluino condensate with non-vanishing vac- helpful discussions or e-mail correspondence. uum expectation of the 1(4) would preserve SO(10) symmetry, while breaking the U(1) factor and intro- ducing an asymmetry between the three fermion fam- ilies and their mirror families. Moreover, since un- References der the family group decomposition SU(3) ⊃ SU(2) × [1] P. Ramond, invited talk at the Sanibel Symposia, 1979, hep- U(1) the 6s contains a singlet of SU(2), this expec- ph/9809459; tation would split two degenerate families apart from A. Zee, Unity of Forces in the Universe, Vol. I, World a third, approximating what is observed. (From the Scientific, Singapore, 1982; viewpoint of E8, the condensate we are proposing is contained in 3875s ⊂ 248 × 248, which is the second 3 = − F − 3(B−L) most attractive symmetric channel according to the The R-parity is defined by R ( 1) ( 1) , with − F + most attractive channel rule.) To break SO(10) down ( 1) the fermion parity which is 1 for all bosons (irrespective of whether vector or scalar) and −1 for all fermions. Hence in the E8 to the standard model further condensates would be theory, as in the MSSM, all standard model particles have R = 1and needed; we note that all of the Higgs representations all superpartners have R =−1. For further details and references, used in models for the breaking of SO(10) unification see Weinberg, Ref. [31], p. 184 and Mohapatra, Ref. [1], p. 277. S.L. Adler / Physics Letters B 533 (2002) 121–125 125

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