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i 2 dnR (3, 1, 1, 1, 3 ), enR (1, 1, 1, 1, 2), (1) after spontaneous symmetry breaking Σ splits into a sin- ∼ − ∼ − glet and a triplet. Each row and each column of Σ is a where n denotes the three generations, qnL = (unL, dnL) doublet under SU(2)qℓ˜ and SU(2)qℓ, respectively. Under and ℓ = (ν ,e ) are the SU(2) doublets, i denotes ˜ nL nL nL P˜ symmetry φ φ and Σ Σ. The equality of the the color degree of freedom, and the numbers in brackets q q˜ bare VEVs of φ ↔and φ is broken↔ by radiative corrections describe the transformation properties under the unified q q˜ due to the hierarchy of Yukawa couplings responsible for gauge group. The ν ’s are introduced because of the R masses, so that the renormalized quantities are experimental signatures for masses [3]. actually equal up to finite radiative corrections. We re- The quark and lepton assignments in Eq. (1), however, mark that this type of Higgs multiplets is also present introduce axial anomalies of the U(1) [SU(2)]2 type. Y in ununified models, but without any restriction on the They are associated with the nonvanishing trace of Y VEVs of Higgs doublets. for quarks and for leptons due to the separation of the The gauge symmetry breaking follows the pattern weak gauge group. To achieve their cancellation we in- clude exotic fermions as done in the so-called ununified SU(2) ˜ SU(2) U(1) P˜ models [4]. We consider the exotic quarks and leptons qℓ × qℓ˜ × Y ×

i 1 ↓ q˜ (1, 3, 1, 2, ), ℓ˜nL (1, 1, 2, 1, 1), SU(2) U(1) nL ∼ 3 ∼ − L × Y i 4 u˜ (1, 3, 1, 1, ), ν˜nR (1, 1, 1, 1, 0), nR ∼ 3 ∼ ↓ i 2 U(1) . (6) d˜ (1, 3, 1, 1, ), e˜nR (1, 1, 1, 1, 2). (2) em nR ∼ − 3 ∼ − The unbroken electric charge generator is given by Under the exotic symmetry P,˜ ordinary and exotic fermions are transformed as follows: 1 1 Q = I3L + Y = I3q + I3˜q + Y, (7) i i i i i ˜i 2 2 qnL q˜nL, unR u˜nR, dnR dnR, ↔ ↔ ↔ and the electroweak couplings of the model, in terms ℓ˜nL ℓnL, ν˜nR νnR, e˜nR enR, (3) ↔ ↔ ↔ of the standard electromagnetic coupling constant and a a b b a a Weinberg angle, take the form and Wq Wq˜ , Gq Gq˜, where Wq , Wq˜ are the weak ↔ ↔ b b gauge bosons of SU(2) ˜, SU(2) , and Gq, Gq are the qℓ qℓ˜ ˜ √2e ′ e gluons of SU(3) , SU(3) , respectively. Besides, the cor- g = gq = gq˜ = , g = . (8) q q˜ sin θ cos θ responding bare gauge couplings are required to be equal. W W Thus the whole Lagrangian of electroweak and strong in- The two physical sets of left-handed weak bosons and teractions, with the Higgs fields defined below, becomes the get the following linear combinations of the ˜ invariant under P symmetry. This leads to a great sim- electroweak gauge bosons Wq, Wq˜ and A: plification of many results which cannot be achieved for ± 1 ± ununified models due to current structures; phenomeno- W = (W ± + W ), logical implications are therefore different. 1 √2 q q˜ In this work we assume that the weak gauge sym- 1 3 3 ˜ Z1 = cos θW (Wq + Wq ) sin θW A, metry and exotic symmetry P are spontaneously bro- √2 ˜ − ken at one stage, which happens with our choice of 1 3 3 Higgs multiplets. The minimal set of colorless scalar Aem = sin θW (W + W )+cos θW A, √ q q˜ multiplets that breaks the electroweak gauge symmetry 2 ± 1 ± ± SU(2) ˜ SU(2) U(1) P˜ down to U(1) and gives qℓ qℓ˜ Y em W2 = (Wq Wq ), mass to× all fermions× of the× model includes √2 − ˜ 1 3 3 φq (2, 1, +1), φq (1, 2, +1), Z2 = (Wq Wq˜ ). (9) ∼ ˜ ∼ √2 − Σ= σ + iτ π (2, 2, 0), (4) · ∼ They pick up the mass with vacuum expectation values (VEVs) 2 2 1 2 2 2 1 2 ′2 2 mW1 0 0 mW1 = g v , mZ1 = (g +2g )v = 2 , 4 4 cos θW φq =   , φq˜ =   , h i h i 2 1 2 2 2 v/√2 v/√2 mAem =0, mW2 = g (v +2u ),     4 u 0 mZ2 = mW2 . (10) Σ =   . (5) h i W W W 0 u The gauge fields 1 = ( q + q˜)/√2 are identified   with the SM weak bosons, while the orthogonal combi- The Pauli matrices τ in Eq. (4) belong to the standard nation W = (Wq Wq)/√2 represents the nonstan- 2 − ˜ gauge group SU(2)L and they are there to indicate that dard ones. We note that the standard mass relation 3

hoc mW1 = mZ1 cos θW is preserved. Also significant, the global symmetry. The left-handed bidoublet Σ then heavier nonstandard neutral and charged weak bosons has no Yukawa couplings to fermions. Substitution of acquire the same mass. All of these only occurs within VEVs leads to mass matrices that can be made real and the context of the unified model of ordinary and exotic diagonal by suitable unitary transformations of the left- quarks and leptons in which the exotic symmetry P˜ is and right-handed components of the pertinent fermion essential, making the difference from other extensions of fields. In this mass eigenstate basis, Z1 and Z2 have the SM. flavor-conserving couplings. ± The charged-current interactions involving the W The equality between Yukawa couplings of ordinary ± 1 and W2 left-handed bosons can be written as and exotic particles yields, however, equal mass values at tree level. These mass relationships can easily be g + −µ + −µ ′ CC = (W J + W J + h.c.), (11) avoided if two new Higgs doublets φ (2, 1, 1) and 2 1µ 1L 2µ 2L q L ′ ∼ ˜ − φq˜ (1, 2, 1) are added, which under P symmetry with weak currents defined, at the ordinary and exotic ∼ − ′ c ′ c transform as φq φq˜ and φq˜ φq . It can be seen quark and lepton level, by that in this case the↔ new Yukawa↔ Lagrangian leads to no ± ± ± ± ± J = J + J + J + J , standard and exotic mass relations if the VEVs of the 1L qL ℓL ℓL˜ qL˜ ± ± ± ± ± new Higgs doublets are equal to each other but different J = J J + J J . (12) from those in Eq. (5). This complication in the Higgs 2L qL − ℓL ℓL˜ − qL˜ sector has no effect on the properties of the gauge bosons As for the neutral-current interactions involving the Z1 discussed in this paper, except that their masses now re- and Z2 left-handed bosons, we have ceive contributions from the new doublets (see Eq. (10)). g oµ 2 µ As mentioned above, all of this happens because of the P˜ NC = Z1µ(J1L sin θW Jem) L √2cos θW − symmetry for the full Lagrangian. Clearly, this scenario, g oµ rendering the model rather uneconomic but richer phe- + Z2µJ , (13) √2 2L nomenologically, favors the SM with two Higgs doublets. Unfortunately, the systematics of the masses of the where now known quarks and leptons are unexplained, so that there is no reliable extrapolation to masses of exotic genera- J o = J 3 + J 3 + J 3 + J 3 , 1L qL ℓL ℓL˜ qL˜ tions. However, strict constraints on the masses of the J o = J 3 J 3 + J 3 J 3 , new fermions are obtained from bounds placed by cos- 2L qL − ℓL ℓL˜ − qL˜ 3 3 3 3 mology and precision electroweak experiments. Jem = J + J + J + J qL ℓL ℓL˜ qL˜ On the one side, cosmological evidences on the ordi- +JqY + JℓY + JℓY˜ + JqY˜ . (14) nary asymmetry and absence at present of exotic ± made of exotic quarks require that exotic mat- We see, from Eqs. (8)–(14), that W and Z have the 1 1 ter be unstable, decaying rapidly enough into observable SM mass and coupling relationships with a universality ordinary matter [3]. As argued below, this is a strong that comprises exotic fermions, and no mixing with the ± motivation to embed the duplicated SM into a duplicated nonstandard W and Z exists, which exhibit no quark- 2 2 GUT such that exotic quarks and ordinary leptons trans- lepton universality and also the breaking of exotic sym- form under one GUT gauge group, and exotic leptons and metry. At the mass scale of the SM, this corresponds to ordinary quarks do under the other. Besides of allowing adding three new generations of quarks and leptons. the decay of , such a duplicated GUT would We next write the most general renormalizable, gauge indicate the ordinary proton stability. In this scenario, invariant and P˜ symmetric Lagrangian for the Yukawa the cosmological ordinary baryon (lepton) asymmetry re- couplings of Higgs bosons with fermions, consistent with quires the exotic lepton (baryon) asymmetry. Thus, the separate ordinary and exotic baryon number conserva- problem of ordinary baryogenesis (leptogenesis) is also tion, as well as separate ordinary and exotic lepton num- that of exotic leptogenesis (baryogenesis) [5]. ber conservation, which are associated with U(1) global On the other side, there are constraints on the masses vector symmetries: and mass-splittings of exotic fermions from the standard d ˜ Z and W gauge observables. If any of the exotic YUK = Gnm qnLφq dmR + q˜nLφq˜dmR L   fermions is lighter than about half of the mass of the u c c +Gnm qnLφqumR + q˜nLφq˜u˜mR SM-like Z boson of the model, this will affect its decay width very strongly. For instance, if the Z can decay e ˜  +Gnm ℓnLφq˜emR + ℓnLφqe˜mR into exotic light , its width will be significantly   increased, in contradiction with experiment. In fact, the ν c ˜ c +Gnm ℓnLφq˜νmR + ℓnLφqν˜mR direct measurement of the invisible Z width gives the   +h.c. (15) bound Nν = 2.92 0.07 on the number of light neu- trino types [3]. On± the contrary, if the exotic fermions Because of gauge invariance, only the ordinary or exotic are heavy, their couplings to the lighter Z and W lepton number conservation is related to an apparently ad bosons will affect the values of the precision electroweak 4 observables S, T and U, or equivalent oblique parame- GeV, respectively, while the current direct lower bounds ters on the gauge self-energies at the loop level [6]. These on extra stable neutral and charged leptons are about parameters are defined to constrain heavy new physics, 45.0 GeV and 102.6 GeV, respectively [3]. Moreover, the vanishing for the SM. In the case of new generations ordinary and exotic fermions do couple to the degenerate of fermions, these values restrain the masses and mass- heavy nonstandard Z and W gauge bosons, so that the splittings of such fermions. For completeness, we present new physics cannot be fully parametrized in the usual S, in the following the contributions to these oblique param- T and U framework. Actually, we will use the mass ra- eters from each nondegenerate doublet (f1,f2) of extra tio between the light and heavy W (see Eq. (19) below). SM-like heavy quarks or leptons with masses (m1,m2), And to restrain this new parameter, we will consider in in the limit mZ , ∆m = m1 m2 m1,2 [6]: this paper charged-current and neutral-current data at | − | ≪ the standard matter level. N S c , The relevant charged-current effective Lagrangian for ≃ 6π the low-energy ordinary quark-lepton physics becomes N ∆m2 T c , 2 2 2 eff 4GF + −µ + −µ ≃ 12π sin θW cos θW mZ = (J J + ηJ J ), (18) LCC √2(1 + η) 1Lµ 1L 2Lµ 2L 2N ∆m2 U c , (16) ≃ 15π m2 ± 1 where J1,2L are the currents in Eq. (12) limited to stan- dard quarks and leptons, η is a parameter defined in the where Nc = 3 (1) denotes the color number of extra minimal model by quarks (leptons). We note that if m1 and m2 are non- − degenerate and not too large, additional negative correc- m2 2u2 1 tions to S may arise; this parameter in a sense measures η = W1 = 1+ , (19) m2  v2  the size of the new fermion sector, while T and U do on W2 its weak-isospin breaking. The experimental values for and G is the Fermi constant from the decay de- these observables are [3] F termined (at tree level) by

S = 0.13 0.10 ( 0.08), 2 − ± − 4GF g T = 0.17 0.12 (+0.09), = 2 (1 + η). (20) − ± √2 4mW1 U =0.22 0.13 (+0.01), (17) ± According to Eq. (18), there is no universality in the where the central values assume mH =117 GeV for the strength of leptonic and hadronic weak interactions, SM , and the values shown in the parentheses which depends on the magnitude of the η-parameter. The correspond to the changes for mH =300 GeV; the S and corresponding effective neutral-current weak interactions T parameters are highly correlated. It has been shown are given by an expression similar to Eq. (18), which is that three generations of relatively heavy extra quarks derived from Eq. (13) restricted to conventional quarks and leptons cannot fit the data with the SM Higgs dou- and leptons. blet, but they do it with a two-Higgs-doublet extension, Therefore, within our scenario the electroweak predic- although the parameter space becomes very restrictive tions of the SM also get corrections parameterized by with fermion masses significantly below 1 TeV [7]; the the quantity η defined in Eq. (19). To exhibit the degree nonstandard gauge bosons would have an effect on the of violation of the universality of weak interactions, we oblique parameters only at the two-loop level through consider the constraint dictated by low-energy charged- their couplings to fermions, because as exhibited in current data. At the low-energy level of ordinary quarks Eq. (9) they do not mix with the standard ones [8]. This and leptons, charged-current interactions are governed motivates even more, within the context of our model, by the effective Lagrangian of Eq. (18), which can be the duplication of the SM forces with a two-Higgs-doublet rewritten as follows: extension included. Hence, all of these can be met if the exotic fermions eff 4GF + −µ + −µ CC = JqLµJqL + JℓLµJℓL are massive and nondegenerate enough to decay into or- L √2 h dinary fermions in a duplicated GUT scenario, with a 1 η − − + − (J + J µ + J J +µ) . (21) generic lower mass bound about the standard mZ /2. By 1+ η qLµ ℓL qLµ ℓL contrast, if the masses are significantly greater than the i generic bound H 174 GeV, the Higgs VEV caus- Thus, η makes the strength of the leptonic charged- ing the SM-likeh electroweaki ≃ symmetry breaking, then the current weak interactions different from the strength of weak interaction among heavy fermions becomes strong the semileptonic weak interactions, so that semileptonic and perturbation theory breaks down, so eluding any processes can be considered to constrain its value. We mass bound depending on the perturbative assumption. will do this by adapting a calculation by Rajpoot [4] Collider searches for new long-lived up and down quarks based on preserving Cabbibo universality and Cabbibo- give by now the lower mass bounds of 220 GeV and 190 Kobayashi-Maskawa (CKM) unitarity. The relevant four 5 fermion effective Lagrangian is given by We approximately find

eff GF 1 η ′ µ ∆mW1 1 (νq)= − Vqq′ q γµ(1 + γ5)q ℓγ (1 + γ5)ν, CC η =2 ( 2 2), (25) L √2 1+ η mW1 sin θW − (22) SM where ∆mW1 = mW1 mW1 , obtaining forthwith the where V ′ are the elements of the CKM mixing matrix. − qq bounds η < 0.0013 and mZ2(W2) > 2.2 TeV. Equa- Typically these are determined from leptonic decays of tion (25) shows how, in the case of the W1-mass, a small hadrons. The data from high precision measurements deviation of the experimental result from the SM predic- yields [3] tion defines the η-parameter of the new physics implied by the P-symmetric˜ model. The model favors an exper- 1 η − Vud = 0.9738 0.0005, imental value of mW1 below the SM prediction. Lower 1+ η | | ± mass bounds coming from other observables can be com- 1 η puted as well. − Vus = 0.2200 0.0026, 1+ η | | ± Thus, more experimental accuracy is needed to have 1 η a signal of the new physics from precision electroweak − Vub = 0.00367 0.00047. (23) 1+ η | | ± measurements at high energies, where the SM exhibits a very good agreement with data. We know that the SM 2 2 Unitarity of the mixing matrix requires Vud + Vus + cannot be complete, so eventually quantitative disagree- 2 | | | | Vub = 1 . This constraint immediately implies η = ments between data and the SM predictions shall occur. | | 0.00083 0.00037 and mW2 = 2.8 0.9 TeV, although We expect that all the magnitudes will be accounted for the experimental± results quoted in Eq.± (23) are still con- in part by a simple, consistent value of η, which in turn troversial [9]. It is worth emphasizing that there is no will give the mass scale for the new left-handed weak analogous result in ununified models because the low- bosons. In a sense, the η parameter plays a role similar energy charged-current structure differs from that of the to that of the mixing angle θW in the SM prior to the unified variant. LEP era; its value obtained from a fit to neutral-current Additional limits can be obtained from constraints at data made possible to predict the mass of weak bosons. higher energies coming from precision electroweak mea- In conclusion, we have analyzed the unified model ˜ surements. With the increasing accuracy of data, a very SU(3)q SU(3)q˜ SU(2)qℓ˜ SU(2)qℓ˜ U(1)Y P in convenient choice of free parameters to use in the gauge × × × × × which the Z2 exotic symmetry defined by P˜ is spon- sector of the extended electroweak model is given by αem, taneously broken. It may be viewed as an important GF , mZ1 and mW1 , since mZ1 and mW1 are directly stage in the process of full understanding which can be measurable. Then the other parameters of the theory 2 tested at relatively low energies. Our aim in this paper like sin θW and η become dependent parameters defined has been limited to account for the observed universality through Eqs. (8), (10), and (20), by violation of weak interactions and consider phenomeno- logical constraints on the masses and mass-splittings of m2 2 W1 exotic generations of quarks and leptons that seem to sin θW =1 2 , − mZ1 enforce the SM with two Higgs doublets as well as fur- ther gauge extensions of our model. What is interesting √2GF 2 2 1+ η = m sin θW (1 ∆r), (24) πα W1 − is that the lighter weak bosons that arise have the same em properties as the standard ones, though all predictions where the weak radiative correction parameter ∆r has of the SM (starting at the tree level) are corrected by been included. Considering that the SM value of ∆r η, the new weak parameter introduced by the symmetric does not change at the energy scale of the model which model. The magnitude of η actually measures the no uni- produces η > 0, η and therefore mZ2 (= mW2 ) can be versality of weak interactions. We have fixed the value of predicted from mZ1 and mW1 . However, they are domi- this parameter using universality/unitarity constraints, nated by the experimental bar errors. We only get an which sets a mass of order 2.8 TeV for the nonstandard upper bound η < 0.0024 which implies a lower mass weak bosons, even though the experimental results used bound of mZ2(W2) > 1.6 TeV for the nonstandard weak are still controversial. Their existence mean to touch the bosons, consistent with the above results. The magni- idea of symmetric left-handed weak forces with experi- tude of ∆r =0.03434 0.0017 0.00014 used to obtain ment, stressing the open question which inquires for the these limits was extracted∓ from± the minimal SM in the new dynamics that may be behind or associated with ex- on-shell scheme [3]. All nonstandard particles were not otic symmetry. This work shows, however, that despite included in the radiative corrections predicted by the SM the lack of knowledge of this dynamics, much information because we search for small tree-level effects of the extra can be obtained from this symmetry. gauge bosons. As a way forward, the notion that exotic symmetry Another limit can be derived from Eq. (24) by setting extends to the forces of the SM already suggests to in- in the SM prediction for mW1 instead of the ∆r term. clude the U(1)Y interactions to have the unified gauge 6

˜ group Gqℓ˜ Gqℓ˜ P with G = SU(3) SU(2) U(1). the other hand, the well-known cosmological domain wall Next, it is× natural× to embed each G in× a GUT× to have problems associated with the spontaneous breakdown of ˜ ˜ (GUT)qℓ˜ (GUT)qℓ˜ P which unifies standard and exotic the discrete symmetry P can be solved with the mech- quarks and× leptons,× and also the strong and electroweak anism of destroying domain walls by primordial black interactions; the P˜ symmetry should be preserved down holes [2], without constraints on the parameters in the to energies where the spontaneous breaking of the elec- Lagrangian of the model. Finally, from the viewpoint troweak gauge symmetry takes place. The ordinary nu- of precision electroweak experiments, current constraints cleon does not decay in models of this type because of on the masses and mass-splittings of new fermions also the separation of GUT for ordinary quarks and leptons. compel, within the context of our model, the duplication Thus, applications of GUT techniques [1] predict tran- of the SM forces with a two-Higgs-doublet extension in- sitions between exotic and ordinary quarks and leptons, cluded. giving rise to the decay of heavy exotic matter into ordi- This work was partially supported by the Departa- nary matter, with separate conservation of ordinary and mento de Investigaciones Cient´ıficas y Tecnol´ogicas, Uni- exotic color charge. From a cosmological point of view, versidad de Santiago de Chile. this is a mandatory condition not to conflict with obser- vational evidences on the absence of exotic baryons. On

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