Nuclear Physics B339 (1990) 67—78 NorthHolland
THREEGENERATION SUPERSTRING MODELS WITH DISCRETE R SYMMETRIES B. ANANTHANARAYAN and
Q. SHAFI*
Bartol Research Institute, University of Delaware, Newark, DE 19716, USA
G. LAZARIDES Physics Division, School of Technology, University of Thessaloniki, GR54006 Thessaloniki, Greece Received 6 February 1990
Compactification of the tendimensional heterotic E
8 x F8 superstring theory on certain Calabi—Yau manifolds yields threegeneration models possessing discrete R symmetries. We consider an example based on Z2 (ordinary symmetry) x Z3 (R symmetry) with fourdimensional gauge symmetry SU(3)~x SU(3)1 x SU(3)R. We investigate how features such as adequate suppression of proton decay, appearance of a pair of electroweak scale Higgs doublets, and perturbative unification of the standard model gauge couplings at the compactification scale can emerge from such models. Matter parity and R symmetry, especially the latter, which are related to Z2 and Z3 respectively, play essential roles in our considerations.
1. Introduction Attempts to construct realistic threegeneration superstring models based on compactification of the tendimensional heterotic E8 x E5 superstring theory [1—3] face a number of important experimental and phenomenological constraints. For instance, how to eliminate rapid proton decay, adequately suppress proton decay through dimensionfive operators, the gauge hierarchy problem,remain correctly 2O~,and ensure that resolve the standardmodel gauge couplings in “predict” sin the perturbative regime [4] till the compactification scale. Rapid proton decay can be suppressed through a discrete (say Z 2 or Z3) matter parity, and it is quite remarkable that several Calabi—Yau (C—Y) compactifications give rise to this desirable symmetry. For examples based on the Z2 matter parity see Greene et al., Kalara and Mohapatra, and Arnowitt and Nath in ref. [5]. For Z3 parity see Lazarides et al. [51.Proton decay through dimensionfive operators imposes the requirement that the particles which mediate this process be superheavy *
Supported in part by Department of Energy Contract #DEACO278ER05007.
05503213/90/$03.50 © Elsevier Science Publishers B.V. (NorthHolland)
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(
1014_1016 GeV) [61. Ordinary discrete symmetries help, in this regard, but only to a limited extent and it seems that additional ingredient(s) are needed. The challenge of preserving a pair of light (~1 TeV) Higgs doublets in the presence of superheavy scales can be met provided some suitable additional symmetry, besides matter parity, is present. The task of predicting sin2 O~is closely linked to whether or not the gauge couplings remain in the perturbative regime as one approaches the compactification scale. It seems fair to assert that threegeneration superstring models based on C—Y compactification and containing only ordinary discrete symmetries have so far not yielded a fully satisfactory description of the observed lowenergy physics. In this paper we wish to consider prospects for threegeneration superstring models derived from C—Y compactification which, in addition to the ordinary discrete symmetries, also possess discrete R symmetries [7, 81. Many such models exist and a listing can be found in refs. [7] and [91.We have chosen to consider one of the simplest ones with fourdimensional gauge group SU(3)~x SU(3)L x SU(3)R and discrete symmetry Z 2 x Z3, where the R symmetry is Z3. The presence of Z2 (ordinary symmetry) ensures that rapid proton decay is avoided. The simultaneous presence of Z3 and Z2 enables one to have sufficiently flat potentials that can give rise to large vacuum expectation values (vevs), thereby ensuring that proton decay mediated by dimensionfive operators is adequately suppressed. This last feature is usually hard to achieve in models without R symmetries. When superheavy vevs develop along directions that preserve the R symmetry (redefined by including an appropriate E6 element) and matter parity, some SU(2) doublets are prevented from acquiring masses. In order to generate a realistic phenomenology, including presence of electroweak scale Higgs doublets, the R symmetry must be spontaneously broken at an intermediate scale 10h1_1012 GeV. The mechanism which leads to R symmetry breaking is not understood but we simply must assume that this occurs (see text). The model we consider possesses other nice features such as absence of flavor changing neutral currents and a unification scale M~ which essentially has the same magnitude as the compactification scale M~,with the standard model gauge couplings remaining perturbative up to the unification scale. We have also investigated remaining modelsconsisting which haveofthe 3 and aalldiscrete symmetry a Zfourdimensional gauge group (SU(3)) 2 or Z3 matter parity and a Z3 Rsymmetry. The Z2 x Z3 model presented in detail in this paper appears to be the best in its class. ‘~
2. A threegeneration model and its discrete symmetries The multiply connected threegeneration C—Y manifold K that gives rise to the threegeneration model is obtained by dividing a simply connected C—Y manifold K() by a freely acting Z3 symmetry. The manifold K0 is specified as the complete
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intersection of three algebraic equations in CP3 X CP3 [10]. Since a complete classification of all possible K’s, together with their corresponding discrete symmetries, obtained via this procedure is now available, we refer the reader to the original references for further details [7, 9]. In this paper we will consider model (id) of ref. [9]. The defining equations of K 0 take the form 3
3
Lx~=O,
~y~+b(y0y1y2+y0y1y3)=0,
i=0
1=~0
x1y1 +c(x2y2+x3y3) +c’(x2y3+x3y2) =0,
(1)
where x0, x1,b, x2, andthree y0, y1, y2, y3 are the homogeneous coordinates 3’s, and c, c’x3 are complex parameters. The freely acting Z of the two CP 3 symmetry 2, a, a) acting on the eightvector is given by g= ~], with g’ diag(1, a and a exp(2~i/3). The flux breaking of E 6 to SU(3)~x SU(3)L X SU(3)R is achieved through the map 3=i (2) g—*L4~,EE6, U~g
[~j
[~
=
=
and the boundary condition ~/i(z)
=
U~i(gz),z
E
K
0 on the fields of the theory. The appropriate L/~for the case under consideration is [9] SU(3)~x SU(3)L X SU(3)R. (3) 2Q, where A =(1,~,3),q =(3,3, 1) and Q(3, 1,3) Under lepton, A —*aA, Q —*a superfields respectively. denote quarkq andq,antiquark The group of honest discrete symmetries of the manifold K turns out to be Z 2 x Z3, generated by C and QBCBC respectively. It is readily checked that the R symmetries are associated with Z3, while Z2 is an ordinary symmetry. For instance, 2a, where under QBCBC the holomorphic (3,0) form o transforms as w —~a a exp(2~ri/3). In the literature the R symmetry is often defined to commute with the E 6 gauge symmetry [8, 11]. For the generator QBCBC of the (Z3) R symmetry under discussion, the 27~and of E6 transform as follows: Ug
=
(13, 13, a13)
E
—~
=
271i 27_iexP(__~_)27~~
—
271i
—
27i_~f3Jexp(___)27j~
(4)
where 13 and 13; denote the transformation properties of the corresponding monomials and (1, 1) forms. These transformation properties are listed in table 1 together with the transformation properties of the fields under C. The superpotential W transforms into aW under QBCBC.
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TABLE 1 Left: The transformation properties of the monomials representing E
6 27’s under Z2 x Z3 generated by C and QBCBC respectively. Right: The transformation properties of the (1, 1) forms corresponding to E6 27’s under Z2 x C
QBCBC
27’s
C
QBCBC
a 1 a a
A1 A2 A3 A4
1 —1 I —1
a a2 a2 a
A7
1 1 1 1 —1 1 1
q2
1 1 —1
1 1 a1
qq1 2 qq3 4 qq5 6 q7
1 —1 1 —1 1 1 —11
—1 —1 1
12 a a
~1 —11
a2 aa2
27’s
A2 A3 A A4 5
a
1 1 2 a2 a 1 12 a 1 11
q 5
q 9
—1 1
12 a a2
1 —11 1 —1
a2 aa2 a2 a2
Q 3 Q4
3. Matter parity and R symmetry Discrete symmetries that arise from the compactification process are expected to be important for model building, with some playing even a critical role. One such discrete symmetry is matter parity, whose presence can ensure that rapid proton decay does not occur. It has recently been shown [121 that in threegeneration models derived from CP3 x CP3, only Z 2 and Z3 symmetries can play the role of matter parity. Moreover, none of the R symmetries present is a viable candidate for matter parity. In the model under discussion, therefore, only Z2 is a candidate matter parity. Indeed, Z2, combined with the element 1
1 1
,
1 of SU(3)~x SU(3)L
X SU(3)R
—1 1
,
1
—1 1
generates the matter parity ZT.
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Next we combine the R symmetry QBCBC with the
1
a 1
E6 element
b a
,
71
2
1
c
,
btc~
a
where 2~i a =aexp(_~_)~
271i b =aexp(_
271i
—~_),
c
=
exp(_
_~_).
This redefined R symmetry will be simply called symmetry or generator of Z~in the rest of the paper. For completeness, we note here the transformation properties of the various components of the quark, antiquark and lepton superfields under the “internal” (i.e., excluding the /3,’s) component of R symmetry: U( a) q=
D(a) g(1)
,
Q
Uc(a2) DC(1) gc(1)
H~1~(a2)H~2~(a) L(a) ,
Ec(1)
pc(a2)
N(a2)
(5) Here ~
H~2~ and L denote SU(2)L doublet superfields.
4. Model building The presence of discrete symmetries imposes constraints on allowable terms in the superpotential W of the model. With R symmetries present, these constraints turn out to be particularly stringent. [Recall that ordinary discrete symmetries should leave W invariant, whereas under the R symmetry W aW in the present case.] This becomes especially important when we consider the nonrenormalizable superpotential terms. Let us first consider the breaking of (SU(3))3 gauge symmetry that must occur at a superheavy scale. (Nontrivial Wilson loops on K will cause E 6 to break to this subgroup.) In order to generate this scale, the potential of the fields that acquire nonzero vacuum expectation (vevs)are must possess D and Fflat directions. It 3 andvalues A3 terms Fflat in the directions which cause is well known [4] that A (SU(3))3 to break to SU(3)~x SU(2)L x U(1)~.This also holds for the nonrenor—*
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malizable terms of the form A3F1(AA)m, 113t1(AA)m with n ~ 1. D flatness is ensured by arranging that pairs of A, A acquire vevs along N, N and ~ ~c directions such that quartic contributions to the D terms cancel. The real challenge comes when one considers the nonrenormalizable superpotential terms (AA)m. Insisting that both matter parity and R symmetry be preserved when (SU(3))3 breaks has two important consequences. First, the superpotential W is exactly F flat to all orders in these nonrenormalizable terms in directions that preserve these symmetries. (It is expected that the introduction of supersymmetry breaking, in the TeV range, caused by as yet unknown mechanism (stringy effects?), will spoil F flatness, and enable the appropriate N(N) and v’(i~)fields to acquire vevs which, in principle, could be as large as the compactification scale.) Second, an unbroken R parity protects some SU(2)L doublets from acquiring large masses despite the presence of vevs that break (SU(3))3 to the standard model. Let us now discuss how exact F flatness comes about. The requirement of unbroken Z~and Z~symmetries are satisfied by the N directions of A 1, A3 and A4, N direction of 113, r” direction of A5, and the i~direction of A4. F flatness along these directions can only be violated by terms (AA)m in the superpotential W with either all, or all but one of the fields acquiring nonzero vevs. In the former case, (AA)m is invariant under Z~and, thus, does not exist in W. ln the latter case, the field which has a zero vev must be invariant under ZT and furthermore, also must acquire an a factor under the R symmetry. It turns out that in the model under discussion, none of the fields A, A has these transformation properties along the N, N or directions. Thus, exact F flatness along the directions which respect the Z~x Z~symmetry is guaranteed. For a realistic model to emerge, however, the R symmetry has to be spontaneously broken at an intermediate scale MR. The precise value of MR will be determined by a number of important physical requirements. In particular, MR should be on the order of 101t_1012 GeV if we require that there be a pair of light (‘.~TeV) Higgs doublets. The R symmetry is spontaneously broken if suitable combinations of the fields N2, N6, N1, v~,i~acquire nonzero All these fields area invariant underof ZT 2 factor under the vevs. R symmetry. In addition, large number E and acquire an a 6 singlet fields are also present, some of which can spontaneously break the R symmetry. (Recent work [8] seems to indicate that the flatness arguments above remain intact despite the occurrence of these fields.) Precisely how R symmetry breaking occurs cannot be understood in the restricted fourdimensional field theoretic context we are using here. The R symmetry breaking fields tend to acquire masses of order M~,as well as 44, 4~, ....... terms in the potential. All these terms get an infinite number of contributions of the same order of magnitude. It is then not easy to see how the potential takes the form appropriate for R symmetry breaking. Understanding of this breaking appears to require a much better understanding and a more complete analysis of the fully stringy system at ~
j~C
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M~.In the present work we will not attempt to do this. We will simply assume that this breaking takes place at some appropriate intermediate scale MR. 5. Mass spectrum The spectrum of the theory is computed from the mass matrices exhibited in tables 2—6. The leptonic spectrum has to be carefully treated due to the superHiggs mechanism. This has been discussed in the context of superstring models by Arnowitt and Nath [5].When the N direction of A1, A3 and A4 and the N direction of A3 acquire2)R nonzero vacuum SU(3)L X U(l)B_L. As aexpectation result, one values, combination of xLSU(3)R breaks to SU(2)L x SU( 1, L3 and L4, L3, one combination of e~,e~and e~,as well as ë~,do not appear in the lepton spectrum. Similarly, when the vC direction and breaks direction of A4 acquire 2)RofXA5 U(1)BL to U(1)~,and e~ nonzero vacuum expectation values, SU( and ~ are not present in the spectrum. Taking all this into account, we spell out below the spectrum of the model. The three chiral families of quarks and leptons are identified by the ZT, Z~ quantum numbers as follows: j3C
SU(2)L lepton doublets l~(+, a2),
i
=
1,2,3,
SU(2)L singlet charged leptons e~(—, 1) and e~(—, a),
i
=
2,3,
SU(2)L quark doublets q 1( +, a), i 1,2,3, i 1,2,3, =
2)L singlet u~( a), —,
=
SU( SU(2)L singlet d~(—, a2),
i
=
1,2,3
TABLE 2a SU(2)L doublet lepton mass matrix from the positive matterparity sector. The known lightlepton doublets 1, arise from this sector
[1]
[a]
[a]
[a]
L
[a2]
2> 2
[a]
L6
H~
[a2]
[a2]
[a2]
L4
H~
L
2> 1
L3
L 3
[a]
j~(2)
[1]
L,
[1]
~j(2)11
[1]
H~
[a2]
Hi 1) 7
Mr
M~/MC MR
MR
M~/M~
M~
MR
M~/MC
M~
[a2] ~(1)
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Superstring models
TABLE 2b
SU(2)L doublet lepton mass matrix from the negativeparity Sector. t2~arise from this sector The pair of “light” Higgs doublets ha>, h [1]
[a]
~(1)
L
[a]
7 [a]
[a]
[a2]
2~ H~2~
L
Hi7
[a2]
[a2]
[a2]
2~ H~2~ Hf’)
2~ H 5
H1~
[a2]
3~
L 4
H~
[1]
L
[1]
~(2)
2~
[a]
M~
MR
MI~/MC
2 [1] [1]
1t Hf H~’t
[1]
H~’t
[a2]
Hi7’>
[a2]
Hg’>
M,~/MC
M~
MR
MR
MI~/MC
M~
TABLE 3
Left: SU(2)L singlet chargedlepton matrix for positiveparity sector. Right: SU(2) singlet chargedlepton matrix for negative matterparity sector. The three known singlet lepton states e~’arise from this sector
[a] [a2]
[1]
[a] E~
E~
M~
MR
E,f
MR
[a] [a] [a] E~ E~ E.f [a]
M,~/M~
[a2]
E~
MR
E~
M,~/MC
[1] Ei7
[1] E~ M~
MR
TABLE 4
Left: SU(2)L singlet, positiveparity U”—U” mass matrix. Right: SU(2)L singlet, positiveparity U~r~Uc mass matrix. The three u,C singlets arise from this sector [a] [a] U~ U~ [a2]
[1]
U~ U,~’
[1] U~
M,~/MC
[1]
M~
[,s2]
U~
U~
[a] [a] [a] U,’~ U~ U,’
Ml~/MC MR
Mj~/M~
[a] U~
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TABLE 5a
SU(2)L doublet, positiveparity q—~mass matrix. The three known light quark doublets q, arise from this sector
[a2] 2
[a
[1]
[1]
[a]
[a]
[a]
q
q5
q3
q6
~is
~
I
MR

q2
M~/MC
TABLE 5b
Negativeparity SU(2)L doublet q—i~mass matrix
[1]
£l~
[cr2]
[a 2 [a]
[1]
[a]
[a]
[a]
92
q4
q7
q9
M~/MC
M~
~
]
MR
‘i~
M,~/MC
i~
MR
TABLE 6a Positiveparity SU(2)L singlet DC_DC mass matrix
[a]
25C
[a]
~‘i
[1] [1]
g3 g6
[1]
g~
[a2]
~c
[a2] [a]
~ g,
[a2]
g 5
[1]
[1]
[a]
g,
g2
gi7
MR
Di7
[a2]
Di7
[a2]
[a2]
[a2]
gi7
gi7
g~
M~/MC
M~
MR
2/MC
Mr
MI~/MC
MR
[a2]
M
[a2]
g~
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/ Superstring models
TABLE 6b Negativeparity SU(2)L singlet DC_DC mass matrix. The three known light singlets di7 arise from this sector
[1] g 4 [a]
15i7
[a]
g~
[1] [1]
g4 g7
[1]
g9
[a2]
~c
[a21 2]
4C
[1]
[a]
[a]
4~
Di7
g,
[a2]
Di7
[a21 Di7
[a2]
D.~
[a2]
D~
MC
MR
M,~/MC
M~/MC
Mr
MR
MR
MI~/MC
g
[a2]
[cr2]
[a2]
gi7
gi7
.
2
[a
The SU(2)L Higgs doublets emerge from the negative matter parity sector which gives a pairthat of states masslepton ‘~M~/M~ that we identifythe as W’~(—,a)and 12~( rise a2).toNote one ofofthe families (presumably electron) does hnot have treelevel coupling to the Higgs field. In addition to the “known” states (and their SUSY partners), there appear some new ones lying between a TeV and M~.These include —,
at scale
M~/M~:two q—~pairs and one Uc_U~pair,
at scale
MR:
two q—~pairs, one Uc_Uc pair, two g—~pairs with positive matter parity, one g—~pair with negative matter parity, two L—L pairs with negative matter parity.
All other additional states are associated with the superheavy scale M~.
6. Proton decay and sin2 O~
The presence of Z 2 matter parity ensures that rapid proton decay through dimensionfour operators is absent. The couplings which give the dominant contribution to proton decay mediated by dimensionfive operators are qqG, uceeG, u’~d’~G’~, qlGc, where q, u’~,e’, dc, 1 are light fields and G(Gc) are heavy quark
/ Superstring models 77 ~, obtained as linear (antiquark) SU(2)L singlet fields with electric charge combinations of the fields g, gC, DC (gC DC, ~) in tables (6a,b). The Z X Z3 2), GC(2 +, a) symmetry ensures that the possible mediators must behave like G( +, a and G( +, 1). From table 6a, one sees that the first two fields acquire masses of order M~.Even though G( +, 1) combines with Gc( +, a2) to give states of order M~/Mc,it turns out that these fields do not couple to qq, ucdc and qi. Thus, the proton decay amplitude mediated by dimensionfive operators is suppressed by ~ Presumably M~ 10l4_b6 GeV [6] which, thanks to the R symmetry, is not hard to achieve in the present model. The calculation of sin2 O~is hampered somewhat by the fact that a large number of particles acquire masses through the nonrenormalizable interaction terms whose coefficients are unknown. At best, we can provide a rough estimate of sin2 O~ and related quantities. We present an example based on a oneloop analysis: B. Ananthanarayan et al.
MR
1013 GeV,
M~
1018
GeV,
aG (unified coupling at M~) 0.3,
corresponding to sin2 O~ 0.27, aQCD(MW) 0.08, a(M~) ~. We have assumed that some of the g particles associated with scale M~ actually weigh about 1016 GeV.
A few comments are in order. First, it is quite rare (!) to find superstring models in which the (SU(3))3 breaking scale M~is on the order of the compactification scale M~(—~ 1018 GeV), with aG less than unity. Second, since M~is essentially M~,there could be additional contributions to the evolution of the gauge couplings which may help bring down both sin2 O~and MR. Finally, twoloop effects also may help improve the situation.
7. Other models We have also investigated all the other models which emerge from the first realization of the threegeneration Tian—Yau manifold and possess a discrete symmetry group consisting of a Z 2 or Z3 matter are parity, a Z3 (2c), R symmetry, 3. These models (4d), (4e),and (4k)the of fourdimensional gauge group (SU(3)) ref. [5]and the models (ic), (2c), (2d), (2i), (2j) of ref. [9]. We find that model (2c) of ref. [7] as well as all the abovementioned models of ref. [9] do not possess exactly flat directions that leave the discrete symmetries unbroken. In the models (4e), (4k) of ref. [7], although flat directions can be found, proton decay is not adequately suppressed since it is mediated by states which remain massless for unbroken R symmetry. Only in the model (4d) of ref. [7] are the flat directions and proton decay intact, and one can obtain an acceptable phenomenology.
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8. Conclusion and outlook The threegeneration superstring model based on the discrete group Z2 X Z3 is just about the simplest example we could find in which both matter parity and R symmetry simultaneously appear. It has some features which make it superior to other threegeneration models that only have ordinary discrete symmetries. These include adequate suppression of proton decay mediated by dimensionfive operators and perturbative unification of the standardmodel gauge couplings at scales that are comparable to the compactification scale. It certainly forms a promising starting point for a deeper understanding of the class of models that have R symmetries. An important issue raised by our work is the need to understand the mechanism responsible for breaking R symmetry at an intermediate scale. Finally, in this work we have followed the simplest scenario in which the B6 symmetry breaks to SU(3)~X SU(2)L x U(1).,~essentially in a single step. It is certainly possible consider This variations on this,under such investigation. as E6 breaking via SU(3)~x 2)R XtoU(1)B_L. is currently SU(2)L X SU( References [1] M. Green, J.H. Schwarz and F. Witten, Superstring theory (Cambridge University Press, 1987) and references therein [2] P. Candelas, G. Horowitz, A. Strominger and E. Witten, NucI. Phys. B258 (1985) 46 [3] E. Witten, NucI. Phys. B258 (1985) 75; B268 (1986) 79 [41M. Dine, V. Kaplunovsky, M. Mangano, CR. Nappi and N. Seiberg, NucI. Phys. B259 (1985) 549 [5] BR. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, NucI. Phys. B292 (1987) 606; S. Kalara and R.N. Mohapatra, Phys. Rev. D36 (1987) 3674; R. Arnowitt and P. Nath, Phys. Rev. D40 (1989) 191; G. Lazarides, P.K. Mohapatra, C. Panagiotakopoulos and Q. Shafi, NucI. Phys. B323 (1989) 614 [61 BR. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, Phys. Lett. B180 (1986) 69; J. Ellis, K. Enqvist, D.V. Nanopoulos and K.A. Olive, NucI. Phys. B297 (1988) 103; R. Arnowitt and P. Nath, Phys. Rev. Lett. 60 (1988) 1817 [7] N. Ganoulis, G. Lazarides and Q. Shafi, Nucl. Phys. B323 (1989) 374 [81 BR. Greene, Phys. Rev. D40 (1989) 1645 [9] G. Lazarides and Q. Shafi, NucI. Phys. B329 (1990) 182 [10] S.Y. Yau, in Proc. of the Argonne Symposium on Anomalies, Geometry and Topology (1985) [11] M. Dine and N. Seiberg, NucI. Phys. B306 (1988) 137 [12] G. Lazarides and 0. Shafi, NucI. Phys. B, to appear