Leptophobic Z' in Heterotic-String Derived Models
aa r X i v : . [ h e p - ph ] J un LTH–917
Leptophobic Z ′ inHeterotic–String Derived Models Alon E. Faraggi and Viraf M. Mehta Department of Mathematical SciencesUniversity of Liverpool, Liverpool, L69 7ZL, United Kingdom
Abstract
The CDF collaboration’s recent observation of an excess of events in the
W jj channel may be attributed to a new Abelian vector boson with sup-pressed couplings to leptons. While D0 finds no evidence of an excess, theCDF data provide an opportunity to revisit an old result on leptophobic Z ′ in heterotic–string derived models. We re-examine the conditions for the exis-tence of a leptophobic U (1) symmetry, which arises from a combination of the U (1) B − L symmetry and the horizontal flavour symmetries, to form a universal U (1) symmetry. While the conditions for the existence of a leptophobic com-bination are not generic, we show that the left–right symmetric free fermionicheterotic–string models also admit a leptophobic combination. In some casesthe leptophobic U (1) is augmented by the enhancement of the colour group,along the lines of models proposed by Foot and Hernandez. E-mail address: [email protected] E-mail address: [email protected] he discrepancy between the recent CDF [1] and D0 [2] results suggests consider-able ambiguity as to whether there is an excess of
W jj events in the
M jj ∼ Z ′ have tobe suppressed.Additional Abelian space–time vector bosons beyond those that mediate the SU (3) × SU (2) × U (1) Y subatomic interactions are abundant in extensions of theStandard Model [6]. Indeed, they arise in Grand Unified theories, based on SO (10)and E gauge extensions of the Standard Model gauge group, which are well moti-vated by the Standard Model matter states and charges. Similarly, Abelian extensionsof the Standard Model are common in string theories. However, most of these exten-sions will produce extra bosons with unsuppressed coupling to leptons. It is thereforeof interest to examine how a leptophobic Z ′ can arise [7, 5]. Obviously, one can sim-ply gauge the baryon number U (1) B , and this exercise has been undertaken, [8], andwithin type I string theories a gauged U (1) B may indeed arise. However, in GrandUnified theories, as well as in an heterotic–string theory that accommodates them,Abelian extensions of the Standard Model typically have unsuppressed couplings toleptons.One exception to this generic expectation was the heterotic–string model of ref.[7]. The recent CDF data provide an opportune moment to re-examine how a lep-tophobic Z ′ can arise in heterotic–string models. In this respect the type I andheterotic–string cases imply different phenomenological signatures beyond the lepto-phobic Z ′ that will be instrumental in distinguishing between the two cases. Whilethe heterotic–string maintains the Grand Unified embedding of the Standard Modelstates, the type I string does not. In particular, the heterotic–string can still preservethe embedding of the Standard Model matter states in spinorial 16 representations of SO (10), which is well motivated by the Standard Model data. In the type I scenariothe string scale is lowered to the TeV scale, which will be signalled by the emergenceof Regge recurrences at parton collision energies √ ˆ s ∼ M s ≡ string scale. In theheterotic case the string scale is still at the Planck scale. The big desert between theweak and Planck scales is preserved, albeit with an unexpected oasis in between.In this paper we therefore re-examine the ingredients that produced the leptopho-bic Z ′ model of ref. [7]. The main feature of this model is that the U (1) B − L gaugesymmetry, which is embedded in SO (10), plus a combination of the flavour U (1)symmetry produces a family universal, leptophobic U (1) symmetry. The additional U (1) symmetries compensate for the lepton number in U (1) B − L and the resulting U (1) therefore becomes a gauged baryon number. In the specific model of ref. [7]2he colour gauge symmetry is enhanced from SU (3) C × U (1) B to SU (4) C , due tospace–time vector bosons that arise from twisted sectors. We discuss how leptopho-bic U (1) symmetries may arise in this class of superstring compactifications withoutenhancement of the gauge group. In particular, we show that the class of left–rightsymmetric models of ref. [9] reproduces the conditions that admits a leptophobic U (1) combination without gauge enhancement.The superstring models that we discuss are constructed in the free fermionicformulation [10]. In this formulation a model is constructed by choosing a consistentset of boundary condition basis vectors. The basis vectors, b k , span a finite additivegroup Ξ = P k n k b k where n k = 0 , · · · , N z k −
1. The physical massless states in theHilbert space of a given sector α ∈ Ξ, are obtained by acting on the vacuum withbosonic and fermionic operators and by applying the generalised GSO projections.The U (1) charges, Q ( f ), with respect to the unbroken Cartan generators of the fourdimensional gauge group, which are in one to one correspondence with the U (1)currents f ∗ f for each complex fermion f, are given by: Q ( f ) = 12 α ( f ) + F ( f ) , (1)where α ( f ) is the boundary condition of the world–sheet fermion f in the sector α ,and F α ( f ) is a fermion number operator counting each mode of f once (and if f iscomplex, f ∗ minus once). For periodic fermions, α ( f ) = 1, the vacuum is a spinorin order to represent the Clifford algebra of the corresponding zero modes. For eachperiodic complex fermion f there are two degenerate vacua, | + i , |−i , annihilated bythe zero modes f and f ∗ and with fermion numbers F ( f ) = 0 , − { , S, b , b , b } . Thegauge group after the NAHE set is SO (10) × SO (6) × E with N = 1 space–timesupersymmetry. The space–time vector bosons that generate the gauge group arisefrom the Neveu–Schwarz (NS) sector and from the sector 1 + b + b + b . The Neveu–Schwarz sector produces the generators of SO (10) × SO (6) × SO (16). The sector1 + b + b + b produces the spinorial 128 of SO (16) and completes the hidden gaugegroup to E . The vectors b , b and b produce 48 spinorial 16 of SO (10), sixteenfrom each sector b , b and b . The vacuum of these sectors contains eight right–moving periodic fermions. Five of those periodic fermions produce the charges underthe SO (10) group, while the remaining three periodic fermions generate charges withrespect to the flavour symmetries. Each of the sectors b , b and b is charged withrespect to a different set of flavour quantum numbers, SO (6) , , .The NAHE set divides the 44 right–moving and 20 left–moving real internalfermions in the following way: ¯ ψ , ··· , are complex and produce the observable SO (10)symmetry; ¯ φ , ··· , are complex and produce the hidden E gauge group; { ¯ η , ¯ y , ··· , } ,3 ¯ η , ¯ y , , ¯ ω , } , { ¯ η , ¯ ω , ··· , } give rise to the three horizontal SO (6) symmetries. Theleft–moving { y, ω } states are divided into, { y , ··· , } , { y , , ω , } , { ω , ··· , } . The left–moving χ , χ , χ states carry the supersymmetry charges. Each sector b , b and b carries periodic boundary conditions under ( ψ µ | ¯ ψ , ··· , ) and one of the three groups:( χ , { y , ··· , | ¯ y , ··· } , ¯ η ), ( χ , { y , , ω , | ¯ y , ¯ ω , } , ¯ η ), ( χ , { ω , ··· , | ¯ ω , ··· } , ¯ η ).The division of the internal fermions is a reflection of the underlying Z × Z orb-ifold compactification [18]. The Neveu–Schwarz sector corresponds to the untwistedsector and the sectors b , b and b correspond to the three twisted sectors of the Z × Z orbifold models. At this level there is a discrete S permutation symmetrybetween the three sectors b , b and b . This permutation symmetry arises due tothe symmetry of the NAHE set and may be essential for the universality of the lep-tophobic U (1) symmetry. Due to the underlying Z × Z orbifold compactification,each of the chiral generations from the sectors b , b and b is charged with respectto a different set of flavour charges.The second stage of the basis construction consists of adding three additional basisvectors to the NAHE set. Three additional vectors are needed to reduce the numberof generations to three; one from each sector b , b and b . One specific example isgiven in table 1. The choice of boundary conditions to the set of real internal fermions { y, ω | ¯ y, ¯ ω } , ··· , determines the low energy properties, like the number of generations,Higgs doublet–triplet splitting and Yukawa couplings [19].The final gauge group in the free fermionic standard–like models arises as follows.The Neveu–Schwarz sector produces the generators of SU (3) C × SU (2) L × U (1) C × U (1) L × U (1) , , × U (1) , , × hidden , where the hidden gauge group arises fromthe hidden E gauge group of the heterotic–string in ten dimensions. The SO (10)symmetry is broken to SU (3) C × U (1) C × SU (2) L × U (1) L , where U (1) C = Tr U (3) C ⇒ Q C = X i =1 Q ( ¯ ψ i ) ,U (1) L = Tr U (2) L ⇒ Q L = X i =4 Q ( ¯ ψ i ) . (2)The flavour SO (6) symmetries are broken to U (1) n with ( n = 0 , · · · , U (1) j , arise from the world–sheet currents ¯ η j ¯ η j ∗ ( j = 1 , , U (1) symmetries are present in all the three generation free fermionicmodels which use the NAHE set. Additional horizontal U (1) symmetries, denotedby U (1) j ( j = 4 , , ... ), arise by pairing two real fermions from the sets { ¯ y , ··· , } , { ¯ y , , ¯ ω , } , and { ¯ ω , ··· , } . The final observable gauge group depends on the numberof such pairings. In the model of ref. [7] there are three such pairings, ¯ y ¯ y , ¯ y ¯ ω and ¯ ω ¯ ω , which generate three additional U (1) symmetries, denoted by U (1) , , .It is important to note that the existence of these three additional U (1) currents is U (1) C = U (1) B − L and U (1) L = 2 U (1) T R . U (1) symmetries, the colour triplets from the NS sector are projected out ofthe spectrum by the GSO projections while the electroweak doublets remain in thelight spectrum.The key to understanding how the leptophobic U (1) arises in the model of ref. [7]are the charges of the matter states from the sectors b , b and b under the flavour U (1) j with j = 4 , ,
6. For example, the charges of the states from the sector b are:( e cL + u cL ) , , , , , +( d cL + N cL ) , , , , , +( L ) , , , − , , + ( Q ) , , , − , , , (3)and similarly for the states from the sectors b and b . With these charge assignments,the quarks are charged with respect to the following combination U (1) B = 13 U C − ( U r + U r + U r ) , (4)whereas the leptons are neutral with respect to it. Hence, this combination is afamily universal, leptophobic U (1) symmetry. In the model of ref. [7] additionalspace–time vector bosons arise from the sector X = 1 + α + 2 γ in which X L · X L = 0and X R × X R = 8. The additional vector bosons transform as triplets of SU (3) C andenhance it to SU (4) C , where the U (1) combination given by U (1) B ′ = U (1) B − U + U , is the U (1) generator of the enhanced SU (4) symmetry. Here, U and U arisefrom the world–sheet complex fermions ¯ φ and ¯ φ . The full massless spectrum andcharges of this model were given in ref. [7]. In this model the U (1) , , symmetries areanomalous with Tr U = 24, Tr U = 24 and Tr U = 24. Hence, the family universalcombination of these three U (1) is anomalous, whereas the two family non–universalcombinations are anomaly free. The U (1) , , , , , as well as U (1) B − L , are, however,anomaly free. Hence, the leptophobic U (1) combination is anomaly free and canremain, in principle, unbroken down to low scales.The existence of a leptophobic, family universal and anomaly free U (1) is highlynon–trivial and not generic in phenomenological heterotic–string models. To demon-strate that this is indeed the case, we examine the model of [14]. The sectors b , , produce the three chiral generations that are charged with respect to the same flavoursymmetries, but differ from the corresponding charges in the model of ref. [7]. Forexample, the states from the sector b carry the following charges:( e cL + u cL ) , , , , , +( d cL + N cL ) , , , − , , +( L ) , , , , , + ( Q ) , , , − , , . (5)5e observe that e cL ad L have like–sign charges under U (1) . Since they carry oppositesign charges under U (1) C , U (1) cannot be used to cancel the B − L charge for boththese states. Since they carry like–sign charges under U (1) , a leptophobic, familyuniversal U (1) cannot be made from these U (1) symmetries. The model of ref. [14]preserves the cyclic permutation of the NAHE set. Hence, a similar charge assignmentis obtained in the sectors b and b . In this model the flavour symmetries U (1) ,, , are anomalous. Therefore, their combination with U (1) C is not anomaly free andmust be broken.As a second negative example, we consider the model of ref. [12]. In this modelthe states from the sector b carry the following U (1) charges( e cL + u cL ) − , , , − , , +( d cL + N cL ) − , , , − , , +( L ) − , , , , , + ( Q ) − , , , , , . (6)In this sector the combination given in eq. (4) is leptophobic. However, the statesfrom the sector b have charges( e cL + u cL ) , − , , , , +( d cL + N cL ) , − , , , − , +( L ) , , , , − , + ( Q ) , , , , , , (7)Hence, in this sector the combination (4) is not leptophobic and is not family uni-versal. Furthermore, the flavour symmetries are anomalous in this model and, con-sequently, as is the combination given in eq. (4).Is the existence of a leptophobic U (1) combination therefore a peculiarity of themodel of ref. ([7])? As seen from the charge assignments in eq. (3) the key is thatthe charges of the left– and right–handed fields differ in sign with respect to U , , inthe sectors b , b and b , respectively. This model preserves the cyclic permutationsymmetry of the NAHE set and therefore, the U (1) combination in eq. (4), is familyuniversal. Furthermore, U (1) , , are anomaly free in the model of ref. [7] andtherefore, their combination with U (1) B − L is also anomaly free. In this model thegauge symmetry is enhanced by space–time vector bosons arising from the twistedsectors. However, we can envision a more systematic classification, along the linesof ref. [22, 16], and that the extra bosons can be projected out from the spectrumin vacua that resemble the properties of this model. In such a case, the leptophobic U (1) will arise without enhancement.As seen from the other two examples provided by the models in refs. [14] and[12], the existence of a family universal, anomaly free leptophobic U (1) combinationin heterotic–string vacua is highly non–trivial. A class of models that reproduces theconditions for the existence of such a U (1) combination are the left–right symmetricmodels of ref. [9]. However, in this case the U (1) symmetries that are combined with6 (1) B − L are not the flavour U (1) , , , but rather the U (1) , , . This possibility is par-ticular to the left–right symmetric heterotic–string models [9], and is not applicablein the other quasi–realistic free fermionic models, in which the SO (10) symmetry isbroken to the flipped SU (5), SO (6) × SO (4) or SU (3) × SU (2) × U (1) subgroups.The reason is that, in these cases, the charges of all the states from a given sector b j are the same with respect to U (1) j with j = 1 , ,
3. This situation arises because thestates from the sectors b j in these models preserve the E charge assignment underthe decomposition E → SO (10) × U (1). A further consequence is that the U (1)combination which arises from E becomes anomalous in these models [21].On the other hand, in the left–right symmetric models, the GSO projection thatbreaks the SO (10) symmetry to SU (3) × U (1) × SU (2) dictates that the U (1) , , charges of the left–handed fields, Q L and L L , differs in sign from those of the right–handed fields, Q R ≡ u cL + d cL and L R ≡ e cL + N cL . Their charges with respect to U (1) , , may, or may not differ in sign. Hence, for example in the first model of ref.[9], we find for the sector b ( u cL + d cL ) , , , , , +( e cL + N cL ) , , , − , , +( L ) − , , , , , + ( Q ) − , , , − , , , (8)with similar charges under U (1) , for the states from the sectors b and b , respec-tively. The U (1) combination given by U (1) B = 13 U C − U − U − U , (9)is family universal, anomaly free and leptophobic. In the left–right symmetric mod-els, the U (1) , , are anomaly free due to the specific symmetry breaking pattern andconsequent charge assignments, whereas U (1) , , may be anomalous or anomaly freein different models. The left–right symmetric free fermionic heterotic–string modelstherefore provide a second example that produces a potentially viable leptophobic U (1) at low scales. In both cases, it is seen that the mechanism that yields a lep-tophobic U (1) symmetry involves the existence of a combination of flavour U (1)symmetries that nullifies the lepton number component of U (1) B − L . The left–rightsymmetric models produce examples that are completely free of any gauge or grav-itational anomalies. Specifically, all U (1) symmetries in these models are anomalyfree. Hence, any combination of the U (1) symmetries, including the leptophobiccombination, is anomaly free.To guarantee that a U (1) symmetry remains viable down to low scales, we mustensure that the spectrum remains anomaly free down to these scales. If we justconsider the Standard Model states U (1) B has various mixed anomalies, which arecompensated by additional states that arise in the string models. This additionalspectrum is highly model dependent, but is constrained by the string charge assign-ments. The issue of how the U (1) symmetry can remain viable down to low scales7s, therefore, model dependent and highly non–trivial. In type I string theories, thisis solved by lowering the string scale down to the TeV scale. However, generically,one expects in this case, dangerous proton decay mediating operators to be gener-ated (see, however, [20] that suggests otherwise). This scenario, in any case, has adistinct signature in the form of recurring Regge resonances, which will be confirmedor refuted in forthcoming LHC experiments. From the point of view of a bottom–upapproach, gauging baryon number is possible by judicially adding states with appro-priate charges. However, the top–down approach relies on the states and charges thatare compatible with the string charge assignments and other constraints. Therefore,the states that are contemplated in the bottom–up approach are not likely to exist instring constructions. Furthermore, string models typically produce exotic fraction-ally charged states that are severely constrained by experiments. String models inwhich the exotic states only appear in the massive spectrum do exist [16]. However,in these models the charge assignments are mundane. Recently, we have been ableto construct effective field theories with an effective low scale U (1) that suppressesproton decay mediating U (1) [23]. However, this U (1) is not leptophobic. All in all,an interesting possibility is that the enhanced non–Abelian symmetry in the model ofref. [7] is not superfluous, but required to maintain a viable leptophobic U (1) downto the low scale. This scenario will then fall into the class of models considered inref. [24], in which the colour group is enhanced. It has also been proposed [25] thatthis class of theories may explain the top forward–backward asymmetry, which isindicated by the CDF experiment [26]. While in the leptophobic model of ref. [7] thecolour group is enhanced to SU (4), enhancement to SU (5) is also possible if SU (3) C combines with an hidden SU (2) group factor.In this paper we discussed how a leptophobic U (1) symmetry may arise inheterotic–string derived models. The examples that we considered preserve theembedding of the Standard Model matter states in spinorial 16 representations of SO (10). The leptophobic U (1) arises from a combination of the U (1) B − L symmetry,which is embedded in SO (10), and the horizontal flavour symmetries, which effec-tively cancels the lepton charge, resulting in a gauged baryon number. This may, ormay not, be augmented by additional vector bosons that enhance the colour group.If forthcoming data provides further weight to the CDF claims, rather than to D0,the focus of model building will clearly shift in that direction. Acknowledgements
AEF would to thank the theory division at CERN for hospitality. This work wassupported in part by the STFC (PP/D000416/1).8 eferences [1] T. Aaltonen et al . [CDF collaboration],
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