Canonical Gauge Coupling Unification in the Standard Model with High-Scale Supersymmetry Breaking
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Canonical Gauge Coupling Unification in the Standard Modelwith High-Scale Supersymmetry Breaking
Yun-Jie Huo, Tianjun Li,
1, 2 and Dimitri V. Nanopoulos
2, 3, 4 Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, P. R. China George P. and Cynthia W. Mitchell Institute for Fundamental Physics,Texas A & M University, College Station, TX 77843, USA Astroparticle Physics Group, Houston Advanced Research Center (HARC),Mitchell Campus, Woodlands, TX 77381, USA Academy of Athens, Division of Natural Sciences,28 Panepistimiou Avenue, Athens 10679, Greece
Abstract
Inspired by the string landscape and the unified gauge coupling relation in the F-theory GrandUnified Theories (GUTs) and GUTs with suitable high-dimensional operators, we study the canon-ical gauge coupling unification and Higgs boson mass in the Standard Model (SM) with high-scalesupersymmetry breaking. In the SM with GUT-scale supersymmetry breaking, we achieve thegauge coupling unification at about 5 . × GeV, and the Higgs boson mass is predicted torange from 130 GeV to 147 GeV. In the SM with supersymmetry breaking scale from 10 GeVto 5 . × GeV, gauge coupling unification can always be realized and the corresponding GUTscale M U is from 10 GeV to 5 . × GeV, respectively. Also, we obtain the Higgs boson massfrom 114.4 GeV to 147 GeV. Moreover, the discrepancies among the SM gauge couplings at theGUT scale are less than about 4-6%. Furthermore, we present the SU (5) and SO (10) modelsfrom the F-theory model building and orbifold constructions, and show that we do not have thedimension-five and dimension-six proton decay problems even if M U ≤ × GeV.
PACS numbers: 11.10.Kk, 11.25.Mj, 11.25.-w, 12.60.Jv . INTRODUCTION It is well-known that there might exist an enormous “landscape” for long-lived metastablestring/M theory vacua where the moduli can be stabilized and supersymmetry may be bro-ken in the string models with flux compactifications [1]. Applying the “weak anthropicprinciple” [2], the string landscape proposal might provide the first concrete solution tothe cosmological constant problem, and it may address the gauge hierarchy problem in theStandard Model (SM). Notably, the supersymmetry breaking scale can be high if there existmany supersymmetry breaking parameters or many hidden sectors [3, 4]. Although there isno definite conclusion whether the string landscape predicts high-scale or TeV-scale super-symmetry breaking [3], it is interesting to study the models with high-scale supersymmetrybreaking due to the turn on of the Large Hadron Collider (LHC) [4–8].Assuming that supersymmetry is indeed broken at a high scale, we can classify the super-symmetry breaking scale as follows [5]: (1) the string scale or grand unification scale; (2) anintermediate scale; and (3) the TeV scale. We do not consider the TeV-scale supersym-metry here since it has been studied extensively during the last thirty years. However, wewould like to emphasize that for high-scale supersymmetry breaking, most of the problemsassociated with some low energy supersymmetric models, for example, excessive flavor andCP violations, dimension-five fast proton decay and the stringent constraints on the lightestCP-even neutral Higgs boson mass, may be solved automatically.If supersymmetry is broken at the high scale, the minimal model at the low energyis the Standard model. The SM explains existing experimental data very well, includingelectroweak precision tests. Moreover, we can easily incorporate aspects of physics beyondthe SM through small variations, for example, dark matter, dark energy, atmospheric andsolar neutrino oscillations, baryon asymmetry, and inflation [9]. Also, the SM fermion massesand mixings can be explained via the Froggatt-Nielsen mechanism [10]. However, there arestill some limitations of the SM, for example, the lack of explanation of gauge couplingunification and charge quantization [6, 7].Charge quantization can easily be realized by embedding the SM into the Grand UnifiedTheories (GUTs). Anticipating that the Higgs particle might be the only new physicsobserved at the LHC, thus confirming the SM as the low energy effective theory, we shouldreconsider gauge coupling unification in the SM. Previously, the generic gauge coupling2nification can be defined by k Y g Y = g = g , (1)where k Y is the normalization constant for the U (1) Y hypercharge interaction, and g Y , g ,and g are the gauge couplings for the U (1) Y , SU (2) L , and SU (3) C gauge groups, respec-tively. However, it is well-known that gauge coupling unification cannot be achieved in theSM with canonical U (1) Y normalization, i.e. , the Georgi-Glashow SU (5) normalization with k Y = 5 / U (1) Y normalization with k Y = 4 / U (1) Y normalization have been constructed as well. The key question remains: can werealize the gauge coupling unification in the SM with canonical U (1) Y normalization?During the last a few years, GUTs have been constructed locally in the F-theory modelbuilding [12–21]. A brand new feature is that the SU (5) gauge symmetry can be brokendown to the SM gauge symmetry by turning on U (1) Y flux [14, 15, 21], and the SO (10) gaugesymmetry can be broken down to the SU (5) × U (1) X and SU (3) C × SU (2) L × SU (2) R × U (1) B − L gauge symmetries by turning on the U (1) X and U (1) B − L fluxes, respectively [14,15, 17, 18, 20, 21]. It has been shown that the gauge kinetic functions receive the correctionsfrom U (1) fluxes [16, 19–21]. In particular, in the SU (5) models with U (1) Y flux [16, 19]and in the SO (10) models with U (1) B − L flux [21], the SM gauge couplings at the GUT scalesatisfy the following condition1 α − α = 53 (cid:18) α − α (cid:19) , (2)where α = 5 α Y / α Y = g Y / π , and α j = g j / π for j = 2 ,
3. In other words, the gaugecoupling unification scale M U is defined by Eq. (2). Especially, we have canonical U (1) Y normalization here. Moreover, the above gauge coupling relation at the GUT scale can berealized in the four-dimensional GUTs with suitable high-dimensional operators [22–25] andin the orbifold GUTs [26–32] with similar high-dimensional operators on the 3-branes at thefixed points where the complete GUT gauge symmetries are preserved. We emphasize thatthe above gauge coupling relation at the GUT scale was first given in Ref. [24].In this paper, considering high-scale supersymmetry breaking inspired by the string land-scape, we shall study the gauge coupling unification in the SM where the GUT-scale gaugecoupling relation is given by Eq. (2). In the SM with GUT-scale supersymmetry breaking,3he SM gauge couplings are unified at about 5 . × GeV. In the SM with supersymmetrybreaking scale from 10 GeV to 5 . × GeV, gauge coupling unification can always berealized, and we obtain the corresponding GUT scale M U from 10 GeV to 5 . × GeV,respectively. Also, the discrepancies among the SM gauge couplings at the GUT scale areless than about 4-6%. Moreover, we calculate the SM Higgs boson mass. In the SM withGUT-scale supersymmetry breaking, the Higgs boson mass is predicted to range from 130GeV to 147 GeV. And in the SM with supersymmetry breaking scale from 10 GeV to5 . × GeV, we obtain the Higgs boson mass from 114.4 GeV to 147 GeV where thelow bound on the SM Higgs boson mass from the LEP experiment [33] has been included.Furthermore, we present the SU (5) and SO (10) models from the F-theory model buildingand orbifold constructions, and show that there are no dimension-five and dimension-sixproton decay problems even if M U ≤ × GeV.This paper is organized as follows. In Section II, we study the gauge coupling unificationin the SM with high-scale supersymmetry breaking. In Section III, we consider the Higgsboson masses. We present the concrete SU (5) and SO (10) models without proton decayproblems in Section IV. And our conclusion is given in Section V. II. GAUGE COUPLING UNIFICATION
For simplicity, we consider the universal high-scale supersymmetry breaking. Above theuniversal supersymmetry breaking scale M S , we consider the supersymmetric SM. Followingthe procedures in Ref. [7] where all the relevant renormalization group equations (RGEs)are given, we consider the two-loop RGE running for the SM gauge couplings, and one-loopRGE running for the SM fermion Yukawa couplings.In numerical calculations, we choose the top quark pole mass M t = 173 . ± . α ( M Z ) = 0 . ± . M Z is the Z bosonmass. Also, the fine structure constant α EM , weak mixing angle θ W and Higgs vacuumexpectation value (VEV) v at M Z are taken as follows [35] α − EM ( M Z ) = 128 . , sin θ W ( M Z ) = 0 . , v = 174 .
10 GeV . (3)First, we consider the GUT-scale universal supersymmetry breaking, i.e. , we only havethe SM below the GUT scale. With the GUT-scale gauge coupling relation in Eq. (2),4 - Α - Α - M U = ´ GeV2 4 6 8 10 12 14 160102030405060 Log H Μ (cid:144) L Α - FIG. 1: Canonical gauge coupling unification in the SM where the gauge coupling unification scale M U is defined by Eq. (2). we present the gauge coupling unification in Fig. 1, and find that the unification scale isabout 5 . × GeV. Next, we consider the intermediate-scale universal supersymmetrybreaking. Interestingly, gauge coupling unification can always be realized. In Fig. 2, wepresent the GUT scale for the universal supersymmetry breaking scale M S from 10 GeV to5 . × GeV. The GUT scale decreases when the supersymmetry breaking scale increases.Moreover, the GUT scale varies from 10 GeV to 5 . × GeV for the supersymmetrybreaking scale from 10 GeV to 5 . × GeV, respectively. Moreover, the GUT scale isalmost independent on the mixing parameter tan β , which is defined in the first paragraphin the next Section.To demonstrate that the deviations from the complete gauge coupling universality arestill modest, we study the discrepancies among the SM gauge couplings at the GUT scaleby defining two parameters δ + and δ − at the GUT scale δ + = α − − α − α − , δ − = α − − α − α − . (4)In Fig. 3, we present δ + and δ − for the supersymmetry breaking scale from 10 GeV to5 . × GeV. We find that δ + and | δ − | increase when the supersymmetry breaking scale M S increases. Also, δ + and | δ − | are smaller than 4% and 6%, respectively. Similar to the5 ´ ´ ´ ´ ´ ´ ´ Log H M S (cid:144) L M U H G e V L FIG. 2: The GUT scale M U versus the universal supersymmetry breaking scale M S . We considertan β = 3 (dotted line) and 35 (solid line), and M t = 171.8 GeV, 173.1 GeV, 174.4 GeV. Theresults for different cases are roughly the same. GUT scale, δ + and δ − are almost independent on tan β as well. Thus, these discrepanciesamong the SM gauge couplings at the GUT scale are indeed small. III. HIGGS BOSON MASS
If the Higgs particle is the only new physics discovered at the LHC and then the SMis confirmed as the low energy effective theory, the Higgs boson mass is one of the mostimportant parameters. Above the supersymmetry breaking scale, we have supersymmetricSMs. There generically exists one pair of Higgs doublets H u and H d , which give masses tothe up-type quarks and down-type quarks/charged leptons, respectively. Below the super-symmetry breaking scale, we only have the SM. Let us define the SM Higgs doublet H as H ≡ − cos βiσ H ∗ d + sin βH u , where σ is the second Pauli matrix and tan β is a mixingparameter [4–6]. For simplicity, we assume the gauginos, squarks, Higgsinos, and the othercombination of the scalar Higgs doublets sin βiσ H ∗ d + cos βH u have the universal super-symmetry breaking soft mass M S . We first assume that supersymmetry is broken at theGUT scale M U , i.e. , M S ≃ M U . And then we assume that supersymmetry is broken at the6 + ∆ - - - - H M S (cid:144) L ∆ FIG. 3: δ + and δ − versus the universal supersymmetry breaking scale M S . We consider tan β =3 (dotted line) and 35 (solid line), and M t = 171.8 GeV, 173.1 GeV, and 174.4 GeV. The resultsfor different cases are roughly the same. intermediate scale, i.e. , below the GUT scale but higher than the electroweak scale, such asbetween 10 GeV and M U .We consider the supersymmetry breaking scale M S from 10 GeV to the SM unificationscale 5 . × GeV. At the supersymmetry breaking scale, we can calculate the Higgsboson quartic coupling λ [4–6] λ ( M S ) = g ( M S ) + k Y g ( M S )4 k Y cos β, (5)where k Y = 5 /
3, and then evolve it down to the Higgs boson mass scale. The one-loopRGE for the quartic coupling is given in Ref. [7] as well. To predict the SM Higgs bosonmass, we consider the two-loop RGE running for the SM gauge couplings, and one-loop RGErunning for the SM fermion Yukawa couplings and Higgs quartic coupling. Using the one-loop effective Higgs potential with top quark radiative corrections, we calculate the Higgsboson mass by minimizing the effective potential V eff = m h H † H + λ
2! ( H † H ) − π h t ( H † H ) (cid:20) log h t ( H † H ) Q − (cid:21) , (6)where m h is the squared Higgs boson mass, h t is the top quark Yukawa coupling from m t = h t v , and the scale Q is chosen to be at the Higgs boson mass. For the M S top quark7ass m t , we use the two-loop corrected value, which is related to the top quark pole mass M t by [36] M t = m t ( m t ) ( α ( m t )3 π + " . − . X k =1 (1 − m k m t ) α ( m t ) π (cid:21) ) , (7)where m k denotes the other quark mass. Also, the two-loop RGE running for α has beenused. Β m h H G e V L FIG. 4: The predicted Higgs boson mass versus tan β in the SM with GUT scale supersymmetrybreaking. The top (orange) three curves are for M t + δM t , the bottom (purple) M t − δM t , and themiddle (blue) M t . The dotted curves are for α − δα , the dash ones for α + δα , and the solidones for α . Here, we choose M t = 173 . δM t = 1 . For the SM with GUT-scale supersymmetry breaking, the predicted Higgs boson mass isshown as a function of tan β for different M t and α in Fig. 4. When we increase top quarkmass or decrease strong coupling, the predicted Higgs boson mass will increase. If we vary M t and α s within their 1 σ range, and tan β from 1 to 60, the predicted Higgs boson masswill range from 130 GeV to 147 GeV. Moreover, focussing on the high-scale supersymmetrybreaking around 10 GeV, Hall and Nomura made a very fine prediction for the Higgsboson mass from 128 GeV to 141 GeV [8]. Thus, our predicted Higgs boson masses are a8ittle bit larger than their results. Concretely speaking, the discrepancy between our lowbound and their low bound is about 1.5% while the discrepancy between our upper boundand their upper bound is about 4%. Because the inputs for the top quark mass are thesame, it seems to us that these discrepancies may be due to the following three reasons: (1)Our supersymmetry breaking scale is 5 . × GeV while their supersymmetry breakingscale is 4 × GeV, thus, the boundary conditions are different. (2) For the SM fermionYukawa couplings and Higgs quartic coupling, we consider the one-loop RGE running whilethey considered the two-loop RGE running. (3) We consider tan β from 1 to 60 while theyconsidered tan β from 1 to 10. Although each of these effects is small, we may understandthe discrepancies by summing up all these effects.In Fig. 5, we present the Higgs boson mass for the intermediate-scale supersymmetrybreaking. Generically, the predicted Higgs boson mass will increase when supersymmetrybreaking scale increases. For supersymmetry breaking scale M S varying from 10 GeV to5 . × GeV, and tan β between 3 and 35, M t within its 1 σ range, the predicted Higgsboson mass will range from 114 . α within its1 σ range, the predicted Higgs boson mass will range from 114 . IV. F-THEORY GUTS AND ORBIFOLD GUTS
Because the GUT scale in our models can be as small as 5 . × GeV, we might havedimension-five and dimension-six proton decay problems. In this paper, we shall considerthe SU (5) and SO (10) models from the local F-theory constructions and the orbifold con-structions, where these proton decay problems can be solved. In particular, the GUT-scalegauge coupling relation given by Eq. (2) can be realized.Let us explain our convention. In the supersymmetric SMs, we denote the left-handedquark doublets, right-handed up-type quarks, right-handed down-type quarks, left-handedlepton doublets, right-handed neutrinos, and right-handed charged leptons as Q i , U ci , D ci , L i , N ci , and E ci , respectively. In the SU (5) models, the SM fermions form i = ( Q i , U ci , E ci )and i = ( D ci , L i ) and i = N ci representations. The Higgs fields form H = ( T u , H u ) and H = ( T d , H d ) representations, where T u and T d are the colored Higgs fields. In the SO (10)models, one family of the SM fermions form a spinor i representation, and all the Higgs9 H M S (cid:144) L m h H G e V L FIG. 5: The predicted Higgs boson mass versus M S in the SM with high-scale supersymmetrybreaking. The top (red) two curves are for M t + δM t , the bottom (green) M t − δM t , and themiddle (blue) M t . The dash curves are for tan β = 3, the solid ones for tan β = 10, and the dottedones for tan β = 35. The horizontal line is the LEP low bound 114.4 GeV. fields form a H = ( T u , H u , T d , H d ) representation.First, we briefly review the proton decay. The dimension-five proton decays arise fromthe color-Higgsino exchanges. In SU (5) and SO (10) models, we have the following super-potential in terms of the SM fermions W = y iju ( Q i Q j + 2 U ci E cj ) T u + y ijde (2 Q i L j + 2 U ci D cj ) T d + M T T u T d , (8)where y iju are the Yukawa couplings for the up-type quarks, and y ijde are the Yukawa couplingsfor the down-type quarks and charged leptons. In SO (10) models, we shall have y iju = y ijde as well. The dimension-five proton decay operators are obtained after we integrate outthe heavy colored Higgs fields T u and T d . The corresponding proton partial lifetime fromdimension-five proton decay is proportional to M T M S , and we require M T M S ≥ GeV from the current experimental bounds [37, 38].The dimension-six proton decay operators are obtained after we integrate out the heavygauge boson fields. In SU (5) models, we have two kinds of operators ∗ i i ∗ j j and ∗ i i ∗ j j . In the flipped SU (5) × U (1) X models, we also have two kinds of operators10 , ) ∗ i ( , ) i ( , ) ∗ j ( , ) j and ( , ) ∗ i ( , ) i ( , − ) ∗ j ( , − ) j . In SO (10) models,we only have one kind of operators ∗ i i ∗ j j . In terms of the SM fields, we obtainthe possible dimension-six operators which contribute to the proton decay [39] O I = g U M X,Y ) U ci γ µ Q i E cj γ µ Q j , (9) O II = g U M X,Y ) U ci γ µ Q i D cj γ µ L j , (10) O III = g U M X ′ ,Y ′ ) D ci γ µ Q i U cj γ µ L j , (11) O IV = g U M X ′ ,Y ′ ) D Ci γ µ Q i N cj L γ µ Q j , (12)where g U is the unified gauge coupling at the GUT scale, and M ( X,Y ) and M ( X ′ ,Y ′ ) are themasses of the superheavy gauge bosons in the SU (5) models and flipped SU (5) × U (1) X models, respectively. In the SU (5) models, we obtain the effective operators O I and O II respectively in Eqs. (9) and (10) after the superheavy gauge fields ( X, Y ) = ( , , / ) areintegrated out. In the flipped SU (5) × U (1) X models, we obtain the effective operators O III and O IV respectively in Eqs. (11) and (12) after the superheavy gauge fields ( X ′ , Y ′ ) =( , , − / ) are integrated out. Because both the SU (5) models and the flipped SU (5) × U (1) X models can be embedded into the SO (10) models, we have all these superheavygauge fields as well as all the above dimension-six proton decay operators. Note that thedimension-six proton decays have not been observed from the experiments, we obtain thatthe GUT scale is higher than about 5 × GeV. Because the GUT scale in our models canbe as small as 5 . × GeV, we require that the (
X, Y ) gauge bosons in the SU (5) modelsand the ( X, Y ) and ( X ′ , Y ′ ) gauge bosons in the SO (10) models do not generate the abovedimension-six proton decay operators. Therefore, we need to forbid at least some of thecouplings between the superheavy gauge fields and the SM fermions in the model building.Second, let us consider the F-theory GUTs which do not have proton decay problem.In the F-theory SU (5) model proposed in Ref. [21], the Higgs fields H = ( T u , H u ) and H = ( T d , H d ) are on the different Higgs curves, and T u and T d do not have zero modes bychoosing proper U (1) fluxes. And then the KK modes of T u and T d do not form vector-likeparticles, i.e. , the third term in Eq. (8) does not exist. The mass terms between the KKmodes of T u and T d arise from the usual µ term. So the proton partial lifetime via thedimension-five proton decay is proportional to M T u M T d M S /µ . In generic GUTs with high-11cale supersymmetry breaking, we have M S ≃ µ , and M T u ∼ M T d ∼ M U . Thus, the protonpartial lifetime via the dimension-five proton decay is proportional to M U ≥ GeV ,which is much larger than 10 GeV . And then we do not have the dimension-five protondecay problem. Moreover, the SM quarks Q i and U ci are on different matter curves. Andthen the X and Y gauge bosons can not couple to both Q i and U ci . Therefore, we do nothave the dimension-six proton decay problem via superheavy gauge boson exchanges.In the Type I and Type II F-theory SO (10) models proposed in Ref. [21] where the SO (10) gauge symmetry is broken down to the SU (3) C × SU (2) L × SU (2) R × U (1) B − L gauge symmetry, the SM fermions Q i , E ci , and N ci are on one matter curve, while U ci , D ci ,and L i are on the other matter curve. On the Higgs H = ( T u , H u , T d , H d ) curve, T u and T d do not have zero modes by choosing proper U (1) fluxes, and the KK modes of T u and T d do not form vector-like particles. Thus, similar to the discussions in the above F-theory SU (5) models, we do not have the dimension-five proton decay problem. Moreover, the SMquarks Q i and U ci / D ci are on different matter curves. So the X and Y gauge bosons cannot couple to both Q i and U ci , and the X ′ and Y ′ gauge bosons can not couple to both Q i and D ci , Therefore, we do not have the dimension-six proton decay problem via superheavygauge boson exchanges.Third, we consider the five-dimensional orbifold SU (5) and SO (10) models on S / ( Z × Z ′ ) where the proton decay problems can be solved as well [26–32]. We assume that thefifth dimension is a circle S with coordinate y and radius R . The orbifold S / ( Z × Z ′ ) isobtained by the circle S moduloing the following equivalent classes P : y ∼ − y , P ′ : y ′ ∼ − y ′ , (13)where y ′ = y + πR/
2. There are two inequivalent 3-branes located at the fixed points y = 0and y = πR/
2, which are denoted by O B and O ′ B , respectively. In particular, the zero modesof the SM fermions in the bulk do not form the complete GUT representations due to theorbifold gauge symmetry breaking [32].In the orbifold SU (5) models (for a concrete example, see Ref. [28]), the SU (5) gaugesymmetry is broken down to the SM gauge symmetry via orbifold projections. With suitablerepresentations for the Z and Z ′ parities, the SU (5) gauge symmetry is preserved on the O B O ′ B O B H = ( T u , H u ) and H = ( T d , H d ) in the bulk, and then T u and T d do not have zero modes due to the orbifold projections. In particular, the KK modesfor T u and T d only have vector-like mass term via µ term. Thus, similar to the discussionsin the above F-theory GUTs, we do not have the dimension-five proton decay problem. Toforbid the dimension-six proton decay, we put the SM fermion superfields i and ′ i inthe bulk with suitable Z and Z ′ parity assignments where i = 1 , ,
3. We obtain the SMfermions Q i as zero modes from i while we obtain the SM fermions U ci and E ci as zeromodes from ′ i . Because the X and Y gauge bosons can not couple to both Q i and U ci , wedo not have the dimension-six proton decay problem via superheavy gauge boson exchanges.In the orbifold SO (10) models (for a concrete example, see Ref. [31]), the SO (10) gaugesymmetry is broken down to the Pati-Salam SU (4) C × SU (2) L × SU (2) R gauge symmetry viaorbifold projections. With suitable representations for the Z and Z ′ parities, the SO (10)gauge symmetry is preserved on the O B O ′ B representation on the O B H =( T u , H u , T d , H d ) in the bulk, and then T u and T d do not have zero modes due to orbifoldprojections. In particular, the KK modes for T u and T d only have vector-like mass termvia µ term. Thus, similar to the discussions in the above F-theory GUTs and the orbifold SU (5) models, we do not have the dimension-five proton decay problem. To forbid thedimension-six proton decay, we put the SM fermion superfields i and ′ i in the bulk withsuitable Z and Z ′ parity assignments where i = 1 , ,
3. We obtain the left-handed SMfermions Q i and L i as zero modes from i while we obtain the right-handed SM fermions U ci , D ci , N ci and E ci as zero modes from ′ i . Because the X and Y gauge bosons can notcouple to both Q i and U ci and the X ′ and Y ′ gauge bosons can not couple to both Q i and D ci , we do not have the dimension-six proton decay problem via superheavy gauge bosonexchanges.Fourth, let us comment on the superheavy threshold corrections on the gauge couplingunification in our models. In the F-theory SU (5) and SO (10) models, we shall have the su-13erheavy threshold corrections from the Kaluza-Klein (KK) modes and heavy string modes.Because our unification scale is smaller than or equal to 10 GeV, we do not have stringtheshfold corrections since the string scale is generic around 4 × GeV. Also, the KKmodes can have masses around the GUT scale or higher, and then their effects on the gaugecoupling unification can be negligible as well. Moreover, in the orbifold SU (5) and SO (10)models, we shall have the superheavy threshold corrections from the KK modes. Becausethe masses of the KK modes can not be larger than the GUT scale, we might have apprecia-ble threshold corrections on the gauge coupling unification, which definitely deserves furtherdetailed study. Thus, for simplicity, we assume that the KK mass scale is equal to the GUTscale in this paper. V. CONCLUSION
Inspired by the string landscape and the unified gauge coupling relation in the F-theoryGUTs and GUTs with suitable high-dimensional operators, we studied the canonical gaugecoupling unification in the SM with high-scale supersymmetry breaking. In the SM withGUT-scale supersymmetry breaking, the gauge coupling unification can be achieved at about5 . × GeV, and the Higgs boson mass is predicted to range from 130 GeV to 147 GeV.In the SM with supersymmetry breaking scale from 10 GeV to 5 . × GeV, gaugecoupling unification can always be realized, and the corresponding GUT scale M U is from10 GeV to 5 . × GeV, respectively. Also, we obtained the Higgs boson mass from114.4 GeV to 147 GeV. Moreover, the discrepancies among the SM gauge couplings at theGUT scale are less than about 4-6%. Furthermore, we presented the SU (5) and SO (10)models from the F-theory model building and orbifold constructions, and showed that thereare no dimension-five and dimension-six proton decay problems even if M U ≤ × GeV.
Acknowledgments
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