We show that it is possible to achieve one step gauge coupling unification in
a general class of non supersymmetric models which at low energies have only
the standard particle content and extra Higgs fields doublets. The constraints
are the experimental values of $\alpha_{em}$, $\alpha_s$ and $\sin^2\theta_W$
at $10^2 GeVs$, and the lower bounds for FCNC and proton decay rates. Specific
example are pointed out.Comment: 10 pages, Latex file,, uses epsf style, Two Postscript figures
  included. To appear in Europhysics Letter

A Pérez-Lorenzana

A Zepeda

For a collection of reports on Susy see:

Georgi H.

Glashow S. L.

Mohapatra R.

Mohapatra R. N.

Pati J. C.

Pirogov Y. F.

Ponce W. A.

Pérez-Lorenzana A.

Ross G. G.

W. A Ponce

English

arXiv

Crossref

One-step non-SUSY unification

ABSTRACT: We show that it is possible to achieve one-step gauge coupling unification in a general class of non-supersymmetric models which at low energies have only the standard particle content and extra Higgs fields doublets. The constraints are the experimental values of αem, αs and sin2θW at 102 GeV s, and the lower bounds for FCNC and proton decay rates. Specific examples are pointed out

Pérez Lorenzana, Abdel

Ponce Gutiérrez, William Antonio

Zepeda, Arnulfo

Universidad de Antioquia Institutional Repository

                       One-step non-SUSY unificationTo cite this article: A. Pérez-Lorenzana et al 1997 EPL 39 141 View the article online for updates and enhancements.Related contentFrom minimal to realistic supersymmetricSU(5) grand unificationGuido Altarelli, Ferruccio Feruglio andIsabella Masina-A D-brane alternative to the MSSMGerardo Aldazabal, Luis E. Ibáñez andFernando Quevedo-D-Branes at singularities: a bottom-upapproach to the string embedding of thestandard modelGerardo Aldazabal, Luis E. Ibáñez,Fernando Quevedo et al.-Recent citationsAn introduction to extra dimensionsAbdel Pérez-Lorenzana-GUTs and string-GUTsA. Pérez-Lorenzana and W. A. Ponce-Non-SUSY unification in left-right modelsWilliam A. Ponce et al-This content was downloaded from IP address 200.24.16.219 on 18/10/2017 at 21:07EUROPHYSICS LETTERS 15 July 1997Europhys. Lett., 39 (2), pp. 141-146 (1997)One-step non-SUSY unificationA. Pérez-Lorenzana1W. A. Ponce2 and A. Zepeda11 Departamento de F́ısica, Centro de Investigación y de Estudios Avanzados del IPNApdo. Post. 14-740, 07000, México, D.F., México2Departamento de F́ısica, Universidad de Antioquia - A.A. 1226, Medelĺın, Colombia(received 11 April 1997; accepted 12 June 1997)PACS. 11.10Hi – Renormalization group evolution of parameters.PACS. 12.10−g – Unified field theories and models.PACS. 12.10Kt – Unification of couplings; mass relations.Abstract. – We show that it is possible to achieve one-step gauge coupling unification in ageneral class of non-supersymmetric models which at low energies have only the standard particlecontent and extra Higgs fields doublets. The constraints are the experimental values of αem, αsand sin2 θW at 102 GeV s, and the lower bounds for FCNC and proton decay rates. Specificexamples are pointed out.Although the Standard Model (SM) is a successful theory which is in good agreement withexperiments [1], it is a common belief that there must exist a more fundamental theory, notfar away from the present experimental energies, capable to provide information on the severalaspects unanswered in the SM, especially on the so-called flavor problem which is related tothe fermion mass spectrum and mixing angles, and on the number of families in nature. Thetwo most popular trends in this direction in today’s literature are Supersymmetry (SUSY) [2],and Grand Unified Theories (GUTs) [3] with and without SUSY. The hope is that the extrasymmetry provides the lacking information.A well-known fact nowadays is that the measured values of the SM coupling constants at themZ scale and the bounds on proton lifetime rule out models like minimal SU(5) [4], and othermodels that contain SU(5) as an intermediate stage in the symmetry-breaking chain. Anotherwell-known result (somehow related to the analysis we are going to present next) is that SUSYis a sufficient ingredient in order to achieve one-step unification in GUT models [5].In what follows we are going to show that one-step unification is also possible in a class ofnon-SUSY GUT models. We restrict our analysis to models in which the low-energy matterconsists only of the standard particle content and more SM Higgs doublet fields. Our analysisexcludes at the same time some of the most popular GUT models.In the SM the coupling constants are defined as effective parameters which include loopcorrections in the gauge boson propagators according to the renormalization group equationsc© Les Editions de Physique142 EUROPHYSICS LETTERS(rge). They are, therefore, energy scale dependent, and to one loop they readµdαidµ' −biα2i , (1)where µ is the energy at which the coupling constants αi = g2i /4π are evaluated, with g1, g2,and g3 the coupling constants of the SM factor groups U(1)Y, SU(2)L and SU(3)c, respectively.The constants bi are completely determined by the particle content in the model by4πbi =113Ci(vectors)−23Ci(fermions)−13Ci(scalars), (2)where Ci(· · ·) is the index of the representation to which the (· · ·) particles are assigned, andwhere we are considering Weyl fermion and complex scalar fields. The boundary conditions atthe mZ ' 102 GeV scale for these equations are determined by the relationshipsα−1em = α−11 + α−12 , and tan2 θW =α1α2, (3)valid at all energy scales, and by the experimental valuesα−1em = 127.90± 0.09 [1], [6],sin2 θW = 0.2315± 0.0002 [1] and (4)α3 = αs = 0.1123± 0.006 [1], [7].The unification of the SM gauge coupling constants is achieved if they merge together intoa common value α = g2/4π at a certain energy scale M , where g is the gauge couplingconstant of the unifying group G. However, since G ⊃ Gs, the normalization of the generatorscorresponding to the subgroups U(1)Y, SU(2)L and SU(3)c is in general different for eachparticular group G, and therefore the SM coupling constants αi differ at the unification scalefrom α by numerical factors ci (αi = ciα) which are pure rational numbers satisfying ci ≤ 1 (dueto the normalization of the generators in G). For example, in SU(5), c1 =35 and c2 = c3 = 1.These values are the same in SO(10) [8] and E6 [9], but they are different for other caseswhich do not contain Gs embedded into an SU(5) subgroup [3] as is the case for E7 [10],SU(5)⊗SU(5) [11], [SU(6)]3×Z3 [12], [SU(6)]4×Z4 [13], SU(8)⊗SU(8) [14] or the Pati-Salammodels [15].The constants ci are fixed once we fix the unifying gauge structure. Then, from eq. (3) itfollows that at the unification scale the value of sin2 θW is given bysin2 θW =αemα2=c1c1 + c2. (5)In this paper we shall consider for c3 only two values, c3 = 1 for those models which containSU(3)c embedded into a simple group, or c3 =12 for those which contain SU(3)c embeddedinto the chiral color extension SU(3)cL ⊗ SU(3)cR [16].To compute the bi coefficients in the rge we will assume that only the standard particles arelight so that, according to the decoupling theorem [17], only they contribute. We obtain2π b1b2b3 = 022311−2094343F −16160H, (6)where F is the number of families and H is the number of low-energy complex Higgs doublets(whose contribution was neglected in the early analysis, see for example the first referencesa. pérez-lorenzana et al.: one-step non-SUSY unification 143in [4], [8], [9], [11]). Notice that we are not including in the former equation the normaliza-tion factor 35 into b1 coming from the SU(5) theory and wrongly included in some generaldiscussions. In the minimal SM, F = 3 and H = 1. Nevertheless, a more general model couldhave more than one low energy Higgs field doublet, then H may be taken as a free parameter.Notice also that we are including in our analysis only doublet Higgs fields, due to the factsthat singlets do not contribute to the rge, and the presence of higher multiplets may spoil the∆I = 1/2 weak isospin rule.The solutions to (1) areα−1i (mZ) =1ciα−1 − bi(F,H) ln(MmZ), (7)which for i = 1, 2, 3 constitute a system of three equations with the unification variables α,M and H as the three unknowns (for F = 3 families). The system of equations (7) may besolved for these variables as a function of the numerical factors ci and the experimental valuesfor αi at the mZ scale. The solution is unique for each set of values {c1, c2, c3} characteristicof each model. For c3 = 1 (and also for c3 =12 ), the solutions to (7) produce three familiesof curves in the c1-c2 plane defined by the equations α(c1, c2) = cα, H(c1, c2) = cH andM(c1, c2) = cM , where cα, cH and cM are arbitrary constant values. Each curve in each familyis then characterized by the numerical constant cα, cH and cM , respectively. As a consequence,for each point in the plane (c1, c2) corresponds unique values for the unification variablesassociated with those curves which intersect at that particular point.Now, there exist some experimental and theoretical bounds for the possible values of theunification variables. First, the unification scale M must be lower than the Planck scaleMP ∼ G1/2N ∼ 1019 GeV, and also it must be greater than 105 GeV in order to agree withthe experimental bounds on FCNC [1]. Also, since some models predict proton decay, and theexperimental bound for the proton lifetime τp is τp→ep ∼M4 > 1032 y, then M must be greaterthan 1016 GeV if the proton is unstable in the model under consideration. Hence, in the analysiswe have to consider two different zones in the c1-c2 plane, given by 1016 GeV < M < MP and105 GeV ≤ M ≤ 1016 GeV, admitting and not admitting proton decay, respectively. Next,because b3 > 0 and b1 < 0 always, α1(mZ) < α < αs(mZ)/c3 and thus α is finite. Hence,as ln(M/mZ) is also finite, from (7) we deduce that H should be also finite and then there isan upper bound Hmax which represents the maximum number of low-energy Higgs doubletsallowed. Therefore, 0 ≤ H ≤ Hmax. These bounds limit the region in the c1-c2 plane wherethe coupling constant unification is possible and consistent with the experimental data andtheoretical requirements. Notice also that H can take only integer values.The solutions of eqs. (7) for α,H and M areα−1 = c1c2c3 ·(α−11 − α−12 )(99− 12F ) + α−13 (8F + 66)c1c2(8F + 66) + c1(c1 − c2)(12F − 99), (8)H=23·c2(α−11 c1−α−13 c3)(66−12F )+c3(α−11 c1−α−12 c2)(12F−99)+20c1(α−12 c2−α−13 c3)c1c2(α−11 −α−12 )+α−13 c3(c1 − c2), (9)ln(MmZ)= 18π ·c1c2(α−11 − α−12 ) + α−13 c3(c1 − c2)c1c2(8F + 66) + c1(c1 − c2)(12F − 99). (10)From these expressions, the limited region obtained for values of c1 and c2 that give unificationis plotted in fig. 1 for c3 = 1 and in fig. 2 for c3 =12 , where we used F = 3 for three families,and central values for αs, αem and sin2 θW. Let us see the consequences of those graphs.144 EUROPHYSICS LETTERSH=911017  GeVH=0105  GeVH=1H=2 1018  GeV1019   GeV0.1 0.3 0.5 0.7 0.90.50.60.70.80.91.01016  GeVc 3=1c2c1Fig. 10.0 0.2 0.4 0.6 0.80.30.40.50.60.70.80.91.0105  GeV1016  GeV1017  GeV1018   GeV1019   GeVH=0H=1H=28H=136c 3=½ c2c1Fig. 2Fig. 1. – Plots for some values of H and M for the non-chiral color models. The bounds in c1 and c2impose at once for α the bounds 16.5454 < α−1 < 48.4809.Fig. 2. – Plots for some values of H and M for GUTs containing the chiral color extension. In thiscase we have 8.27269 < α−1 < 26.1967.Analysis of fig. 1. – It corresponds to the case of a GUT group which does not includechiral color symmetries. The allowed region of parameters (c1, c2) lies inside the lines M =105 GeV s, H = 0 and c2 = 1. There is a maximum unification mass scale possible givenby M ≤ 1017.5 GeV s < MP and the number of Higgs field doublets allowed is such that0 < H ≤ 91 in general, but if the proton does decay in the context of the GUT model, then0 < H ≤ 2. Let us see the implications of this for some specific models:1) SU(5). For all the models in this group proton decay is always present [4], and (c1, c2) =(35 , 1) which lies inside the allowed zone, but in a region where M = 1013 GeV s. Hence theSU(5) GUT scale M is in conflict with the bounds for proton decay. Since SU(5) allows onlythe one-step symmetry-breaking chain (sbc) SU(5)M−→ SM , SU(5) is ruled out in general.That is, the experimental bounds on proton decay rule out not only minimal SU(5) but alsoall the possible extensions which include arbitrary representations of Higgs field multiplets.2) SO(10). As for the previous model, proton decay is always present for this group [8] and(c1, c2) = (35 , 1). Therefore the one-step sbc SO(10)M−→ SM is ruled out. From our analysisnothing can be said about the two-stage sbc SO(10)M−→ SU(4)c⊗SU(2)L⊗SU(2)RM′−→ SM .3) SU(4)c⊗SU(2)L⊗SU(2)R. This group can be viewed as a subgroup of SO(10) [8], or asa subgroup of the Pati-Salam model [15], or either as a non-simple unification model by itself.For this group (c1, c2) = (35 , 1) again. In this model, the proton cannot decay via leptoquarkgauge bosons (see the first paper in [15]), but it can decay via Higgs fields scalars. So, theone-stage breaking of this model is not ruled out as long as one can break the symmetry usingscalars which do not break spontaneously the baryon quantum number B.4) E6. Proton decay is always present for this group [9], and (c1, c2) = (35 , 1) also. So, theone-step sbc E6M−→ SM is ruled out. Nothing can be said for the multistage sbc.5) SU(3)L ⊗ SU(3)c ⊗ SU(3)R. This group can be viewed as a subgroup of E6 [9] or as aunification model by itself (the trinification model of Georgi-Glashow-de Rujula [18]). Again(c1, c2) = (35 , 1) and the proton decay in the model is only Higgs-boson mediated. The one-stagebreaking of this model is not ruled out as long as one can break the symmetry using scalarswhich do not break B spontaneously (see the second paper in [18]).a. pérez-lorenzana et al.: one-step non-SUSY unification 1456) SO(18). Proton decay is always present for this group, and (c1, c2) = (35 , 1) [19]. Theconclusions are the same as for E6.7) [SU(6)]3 × Z3. The proton is stable in the context of this model [12]. For this group(c1, c2) = (314 ,13 ), which lies outside the allowed zone, and the one-stage sbc is ruled out (thetwo-stage sbc for the model is presented in the last paper of ref. [12]).Analysis of fig. 2. – It corresponds to the case of a GUT group which includes chiral colorsymmetries. The allowed region in the plane (c1, c2) lies inside the lines M = 105 GeV s,H = 0, M = 1019 GeV s = MP and c2 = 1. Therefore, there is no bound for a maximumunification mass scale and the allowed number of Higgs field doublets is such that 0 < H ≤ 136in general, but if the proton does decay in the context of the GUT model, then 0 < H ≤ 28.Let us see the implications of this for some specific models:1) SU(5) ⊗ SU(5). Proton decay is mediated via gauge and Higgs bosons for the modelsin this group [11]. (c1, c2) = (313 , 1), which lies inside the allowed zone but in a region whereM  1016 GeV s, in serious conflict with bounds for proton decay. The models are all ruledout.2) [SU(6)]4 × Z4. The proton is stable in the context of the model presented in ref. [13].For this group (c1, c2) = (319 ,13 ), which lies inside the allowed zone. So, the one-stage sbc forthis model is also possible, and is presented in ref. [13].We mention that our analysis has been done assuming non-supersymmetric unification. Also,we have neglected threshold effects which depend on the particular structure of each model, wehave not included second-order corrections to the rge which are typically of the order of 1 to10%, and we have not included the experimental errors of the SM gauge coupling constants.The previous analysis allows us to conclude that it is indeed possible to achieve the unificationof the coupling constants of the SM in one step in a general class of non-supersymmetric models.Two particular models with simple unifying groups were singled out: the trinification model ofGeorgy-Glashow-de Rujula [18] for GUT groups which do not include chiral color symmetry,and the model in ref. [13] for GUT models with chiral color symmetry.***The authors acknowledge P. Langacker and G. Kane for discussions and comments.This work was partially supported by CONACyT, México and COLCIENCIAS, Colombia.REFERENCES[1] Particle Data Group (Montanet L. et al.), Phys. Rev. D, 50 (1994) 1173; Barnett R. M.et al., Phys. Rev. D, 54 (1996) 1.[2] For a collection of reports on SUSY see: Supersymmetry and Supergravity, A reprint volume ofPhysics Reports, edited by M. Jacob (North Holland/World Scientific, Amsterdam/Singapore)1986.[3] For a review, see Ross G. G., Grand Unified Theories. Frontiers in Physics, No. 60 (Benjamin-Cummings Publ. Co.) 1984; Mohapatra R., Unification and Supersymmetry, second edition(Springer-Verlag, Berlin) 1992; Langacker P., Phys. Rep., 72 (1981) 185, and references therein.[4] Georgi H. and Glashow S. L., Phys. Rev. Lett., 32 (1974) 438; Georgi H., Quinn H. R. andWeimberg S., Phys. Rev. Lett., 33 (1974) 451.[5] Amaldi U., de Boer W. and Furstenau H., Phys. Lett. B, 260 (1991) 447; Langacker P.and Luo M. X., Phys. Rev. D, 44 (1991) 817; Giunti C., Kim C. W. and Lee U. W., Mod.Phys. Lett. A, 6 (1991) 1745.[6] Fanchiotti S., Kniehl B. and Sirlin A., Phys. Rev. D, 48 (1993) 307 and references therein.[7] Virchaux M. and Milsztajn A., Phys. Lett. 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EPL (Europhysics Letters)

We show that it is possible to achieve one-step gauge coupling
unification in a general class of non-supersymmetric models which at low
energies have only the standard particle content and extra Higgs fields
doublets.
The constraints are the experimental values of $\alpha_{\rm em}$,
$\alpha_s$ and $\sin^2\theta_{\rm W}$ at $10^2\,{\rm GeV\;s}$, and the lower bounds
for FCNC and proton decay rates. Specific examples are pointed out

A. Pérez-Lorenzana

W. A. Ponce

A. Zepeda

EDP Sciences OAI-PMH repository (1.2.0)

http://bibliotecadigital.udea.edu.co/bitstream/10495/8925/1/PonceWilliam_1997_OneStepNonSusy.pdf

One Step Non SUSY Unification

One Step Non SUSY Unification

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