Canonical Gauge Coupling Unification in the Standard Model with High-Scale Supersymmetry Breaking

# Canonical Gauge Coupling Unification in the Standard Model with High-Scale Supersymmetry Breaking

## Abstract

Inspired by the string landscape and the unified gauge coupling relation in the F-theory Grand Unified Theories (GUTs) and GUTs with suitable high-dimensional operators, we study the canonical gauge coupling unification and Higgs boson mass in the Standard Model (SM) with high-scale supersymmetry breaking. In the SM with GUT-scale supersymmetry breaking, we achieve the gauge coupling unification at about , and the Higgs boson mass is predicted to range from 130 GeV to 147 GeV. In the SM with supersymmetry breaking scale from to , gauge coupling unification can always be realized and the corresponding GUT scale is from to , respectively. Also, we obtain the Higgs boson mass from 114.4 GeV to GeV. Moreover, the discrepancies among the SM gauge couplings at the GUT scale are less than about 4-6%. Furthermore, we present the and models from the F-theory model building and orbifold constructions, and show that we do not have the dimension-five and dimension-six proton decay problems even if .

###### pacs:
11.10.Kk, 11.25.Mj, 11.25.-w, 12.60.Jv
1

## I Introduction

It is well-known that there might exist an enormous “landscape” for long-lived metastable string/M theory vacua where the moduli can be stabilized and supersymmetry may be broken in the string models with flux compactifications String (). Applying the “weak anthropic principle” Weinberg (), the string landscape proposal might provide the first concrete solution to the cosmological constant problem, and it may address the gauge hierarchy problem in the Standard Model (SM). Notably, the supersymmetry breaking scale can be high if there exist many supersymmetry breaking parameters or many hidden sectors HSUSY (); NASD (). Although there is no definite conclusion whether the string landscape predicts high-scale or TeV-scale supersymmetry breaking HSUSY (), it is interesting to study the models with high-scale supersymmetry breaking due to the turn on of the Large Hadron Collider (LHC) NASD (); Barger:2004sf (); Barger:2005gn (); Barger:2005qy (); Hall:2009nd ().

Assuming that supersymmetry is indeed broken at a high scale, we can classify the supersymmetry breaking scale as follows Barger:2004sf (): (1) the string scale or grand unification scale; (2) an intermediate scale; and (3) the TeV scale. We do not consider the TeV-scale supersymmetry here since it has been studied extensively during the last thirty years. However, we would like to emphasize that for high-scale supersymmetry breaking, most of the problems associated with some low energy supersymmetric models, for example, excessive flavor and CP violations, dimension-five fast proton decay and the stringent constraints on the lightest CP-even neutral Higgs boson mass, may be solved automatically.

If supersymmetry is broken at the high scale, the minimal model at the low energy is the Standard model. The SM explains existing experimental data very well, including electroweak precision tests. Moreover, we can easily incorporate aspects of physics beyond the SM through small variations, for example, dark matter, dark energy, atmospheric and solar neutrino oscillations, baryon asymmetry, and inflation Davoudiasl:2004be (). Also, the SM fermion masses and mixings can be explained via the Froggatt-Nielsen mechanism FN (). However, there are still some limitations of the SM, for example, the lack of explanation of gauge coupling unification and charge quantization Barger:2005gn (); Barger:2005qy ().

Charge quantization can easily be realized by embedding the SM into the Grand Unified Theories (GUTs). Anticipating that the Higgs particle might be the only new physics observed at the LHC, thus confirming the SM as the low energy effective theory, we should reconsider gauge coupling unification in the SM. Previously, the generic gauge coupling unification can be defined by

 kYg2Y=g22=g23 , (1)

where is the normalization constant for the hypercharge interaction, and , , and are the gauge couplings for the , , and gauge groups, respectively. However, it is well-known that gauge coupling unification cannot be achieved in the SM with canonical normalization, i.e., the Georgi-Glashow normalization with  Ellis:1990zq (). Interestingly, it was shown that gauge coupling unification can be realized in the non-canonical normalization with  Barger:2005gn (); Barger:2005qy (). The orbifold GUTs with such normalization have been constructed as well. The key question remains: can we realize the gauge coupling unification in the SM with canonical normalization?

During the last a few years, GUTs have been constructed locally in the F-theory model building Vafa:1996xn (); Donagi:2008ca (); Beasley:2008dc (); Beasley:2008kw (); Donagi:2008kj (); Font:2008id (); Jiang:2009zza (); Blumenhagen:2008aw (); Jiang:2009za (); Li:2009cy (). A brand new feature is that the gauge symmetry can be broken down to the SM gauge symmetry by turning on flux Beasley:2008dc (); Beasley:2008kw (); Li:2009cy (), and the gauge symmetry can be broken down to the and gauge symmetries by turning on the and fluxes, respectively Beasley:2008dc (); Beasley:2008kw (); Jiang:2009zza (); Jiang:2009za (); Font:2008id (); Li:2009cy (). It has been shown that the gauge kinetic functions receive the corrections from fluxes Donagi:2008kj (); Blumenhagen:2008aw (); Jiang:2009za (); Li:2009cy (). In particular, in the models with flux Donagi:2008kj (); Blumenhagen:2008aw () and in the models with flux Li:2009cy (), the SM gauge couplings at the GUT scale satisfy the following condition

 1α2−1α3 = 53(1α1−1α3) , (2)

where , , and for . In other words, the gauge coupling unification scale is defined by Eq. (2). Especially, we have canonical normalization here. Moreover, the above gauge coupling relation at the GUT scale can be realized in the four-dimensional GUTs with suitable high-dimensional operators Hill:1983xh (); Shafi:1983gz (); Ellis:1985jn (); Li:2010xr () and in the orbifold GUTs kawa (); GAFF (); LHYN (); AHJMR (); Li:2001qs (); Dermisek:2001hp (); Li:2001tx () with similar high-dimensional operators on the 3-branes at the fixed points where the complete GUT gauge symmetries are preserved. We emphasize that the above gauge coupling relation at the GUT scale was first given in Ref. Ellis:1985jn ().

In this paper, considering high-scale supersymmetry breaking inspired by the string landscape, we shall study the gauge coupling unification in the SM where the GUT-scale gauge coupling relation is given by Eq. (2). In the SM with GUT-scale supersymmetry breaking, the SM gauge couplings are unified at about . In the SM with supersymmetry breaking scale from to , gauge coupling unification can always be realized, and we obtain the corresponding GUT scale from to , respectively. Also, the discrepancies among the SM gauge couplings at the GUT scale are less than about 4-6%. Moreover, we calculate the SM Higgs boson mass. In the SM with GUT-scale supersymmetry breaking, the Higgs boson mass is predicted to range from 130 GeV to 147 GeV. And in the SM with supersymmetry breaking scale from to , we obtain the Higgs boson mass from 114.4 GeV to GeV where the low bound on the SM Higgs boson mass from the LEP experiment Barate:2003sz () has been included. Furthermore, we present the and models from the F-theory model building and orbifold constructions, and show that there are no dimension-five and dimension-six proton decay problems even if .

This paper is organized as follows. In Section II, we study the gauge coupling unification in the SM with high-scale supersymmetry breaking. In Section III, we consider the Higgs boson masses. We present the concrete and models without proton decay problems 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 the universal supersymmetry breaking scale , we consider the supersymmetric SM. Following the procedures in Ref. Barger:2005qy () where all the relevant renormalization group equations (RGEs) are given, we consider the two-loop RGE running for the SM gauge couplings, and one-loop RGE running for the SM fermion Yukawa couplings.

In numerical calculations, we choose the top quark pole mass GeV :2009ec (), and the strong coupling constant  Nakamura:2010zzi (), where is the boson mass. Also, the fine structure constant , weak mixing angle and Higgs vacuum expectation value (VEV) at are taken as follows Nakamura:2010zzi ()

 α−1EM(MZ)=128.91,   sin2θW(MZ)=0.23116,   v=174.10GeV. (3)

First, we consider the GUT-scale universal supersymmetry breaking, i.e., we only have the SM below the GUT scale. With the GUT-scale gauge coupling relation in Eq. (2), we present the gauge coupling unification in Fig. 1, and find that the unification scale is about . Next, we consider the intermediate-scale universal supersymmetry breaking. Interestingly, gauge coupling unification can always be realized. In Fig. 2, we present the GUT scale for the universal supersymmetry breaking scale from to . The GUT scale decreases when the supersymmetry breaking scale increases. Moreover, the GUT scale varies from to for the supersymmetry breaking scale from to , respectively. Moreover, the GUT scale is almost independent on the mixing parameter , which is defined in the first paragraph in the next Section.

To demonstrate that the deviations from the complete gauge coupling universality are still modest, we study the discrepancies among the SM gauge couplings at the GUT scale by defining two parameters and at the GUT scale

 δ+=α−12−α−11α−11 ,            δ−=α−13−α−11α−11 . (4)

In Fig. 3, we present and for the supersymmetry breaking scale from to . We find that and increase when the supersymmetry breaking scale increases. Also, and are smaller than 4% and 6%, respectively. Similar to the GUT scale, and are almost independent on as well. Thus, these discrepancies among 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 SM is confirmed as the low energy effective theory, the Higgs boson mass is one of the most important parameters. Above the supersymmetry breaking scale, we have supersymmetric SMs. There generically exists one pair of Higgs doublets and , which give masses to the up-type quarks and down-type quarks/charged leptons, respectively. Below the supersymmetry breaking scale, we only have the SM. Let us define the SM Higgs doublet as , where is the second Pauli matrix and is a mixing parameter NASD (); Barger:2004sf (); Barger:2005gn (). For simplicity, we assume the gauginos, squarks, Higgsinos, and the other combination of the scalar Higgs doublets have the universal supersymmetry breaking soft mass . We first assume that supersymmetry is broken at the GUT scale , i.e., . And then we assume that supersymmetry is broken at the intermediate scale, i.e., below the GUT scale but higher than the electroweak scale, such as between GeV and .

We consider the supersymmetry breaking scale from to the SM unification scale . At the supersymmetry breaking scale, we can calculate the Higgs boson quartic coupling  NASD (); Barger:2004sf (); Barger:2005gn ()

 λ(MS)=g21(MS)+kYg22(MS)4kYcos22β, (5)

where , and then evolve it down to the Higgs boson mass scale. The one-loop RGE for the quartic coupling is given in Ref. Barger:2005qy () as well. To predict the SM Higgs boson mass, we consider the two-loop RGE running for the SM gauge couplings, and one-loop RGE running 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 Higgs boson mass by minimizing the effective potential

 Veff=m2hH†H+λ2!(H†H)2−316π2h4t(H†H)2[logh2t(H†H)Q2−32], (6)

where is the squared Higgs boson mass, is the top quark Yukawa coupling from , and the scale is chosen to be at the Higgs boson mass. For the top quark mass , we use the two-loop corrected value, which is related to the top quark pole mass by Melnikov:2000qh ()

 Mt = mt(mt)⎧⎨⎩1+4α3(mt)3π+[13.4434−1.04145∑k=1(1−43mkmt)][α3(mt)π]2⎫⎬⎭ , (7)

where denotes the other quark mass. Also, the two-loop RGE running for has been used.

For the SM with GUT-scale supersymmetry breaking, the predicted Higgs boson mass is shown as a function of for different and in Fig. 4. When we increase top quark mass or decrease strong coupling, the predicted Higgs boson mass will increase. If we vary and within their range, and from 1 to 60, the predicted Higgs boson mass will range from to . Moreover, focussing on the high-scale supersymmetry breaking around , Hall and Nomura made a very fine prediction for the Higgs boson mass from to  Hall:2009nd (). Thus, our predicted Higgs boson masses are a little bit larger than their results. Concretely speaking, the discrepancy between our low bound and their low bound is about 1.5% while the discrepancy between our upper bound and their upper bound is about 4%. Because the inputs for the top quark mass are the same, it seems to us that these discrepancies may be due to the following three reasons: (1) Our supersymmetry breaking scale is while their supersymmetry breaking scale is , thus, the boundary conditions are different. (2) For the SM fermion Yukawa couplings and Higgs quartic coupling, we consider the one-loop RGE running while they considered the two-loop RGE running. (3) We consider from 1 to 60 while they considered from 1 to 10. Although each of these effects is small, we may understand the discrepancies by summing up all these effects.

In Fig. 5, we present the Higgs boson mass for the intermediate-scale supersymmetry breaking. Generically, the predicted Higgs boson mass will increase when supersymmetry breaking scale increases. For supersymmetry breaking scale varying from to , and between 3 and 35, within its range, the predicted Higgs boson mass will range from to , where the low bound on the SM Higgs boson mass from the LEP experiment Barate:2003sz () has been included. If we also vary within its range, the predicted Higgs boson mass will range from to .

## Iv F-Theory GUTs and Orbifold GUTs

Because the GUT scale in our models can be as small as , we might have dimension-five and dimension-six proton decay problems. In this paper, we shall consider the and models from the local F-theory constructions and the orbifold constructions, where these proton decay problems can be solved. In particular, the GUT-scale gauge coupling relation given by Eq. (2) can be realized.

Let us explain our convention. In the supersymmetric SMs, we denote the left-handed quark doublets, right-handed up-type quarks, right-handed down-type quarks, left-handed lepton doublets, right-handed neutrinos, and right-handed charged leptons as , , , , , and , respectively. In the models, the SM fermions form and and representations. The Higgs fields form and representations, where and are the colored Higgs fields. In the models, one family of the SM fermions form a spinor representation, and all the Higgs fields form a representation.

First, we briefly review the proton decay. The dimension-five proton decays arise from the color-Higgsino exchanges. In and models, we have the following superpotential in terms of the SM fermions

 W = yiju(QiQj+2UciEcj)Tu+yijde(2QiLj+2UciDcj)Td+MTTuTd , (8)

where are the Yukawa couplings for the up-type quarks, and are the Yukawa couplings for the down-type quarks and charged leptons. In models, we shall have as well. The dimension-five proton decay operators are obtained after we integrate out the heavy colored Higgs fields and . The corresponding proton partial lifetime from dimension-five proton decay is proportional to , and we require from the current experimental bounds Hisano:1992jj (); Murayama:2001ur ().

The dimension-six proton decay operators are obtained after we integrate out the heavy gauge boson fields. In models, we have two kinds of operators and . In the flipped models, we also have two kinds of operators and . In models, we only have one kind of operators . In terms of the SM fields, we obtain the possible dimension-six operators which contribute to the proton decay Nath:2006ut ()

 OI = g2U2M2(X,Y) ¯¯¯¯¯¯¯Uci γμ Qi ¯¯¯¯¯¯EcjγμQj , (9) OII = g2U2M2(X,Y) ¯¯¯¯¯¯¯Uci γμ Qi ¯¯¯¯¯¯Dcj γμ Lj , (10) OIII = g2U2M2(X′,Y′) ¯¯¯¯¯¯Dci γμ Qi ¯¯¯¯¯¯¯Ucj γμ Lj , (11) OIV = g2U2M2(X′,Y′) ¯¯¯¯¯¯¯¯DCi γμ Qi ¯¯¯¯¯¯¯NcjL γμ Qj , (12)

where is the unified gauge coupling at the GUT scale, and and are the masses of the superheavy gauge bosons in the models and flipped models, respectively. In the models, we obtain the effective operators and respectively in Eqs. (9) and (10) after the superheavy gauge fields are integrated out. In the flipped models, we obtain the effective operators and respectively in Eqs. (11) and (12) after the superheavy gauge fields are integrated out. Because both the models and the flipped models can be embedded into the models, we have all these superheavy gauge fields as well as all the above dimension-six proton decay operators. Note that the dimension-six proton decays have not been observed from the experiments, we obtain that the GUT scale is higher than about GeV. Because the GUT scale in our models can be as small as , we require that the gauge bosons in the models and the and gauge bosons in the models do not generate the above dimension-six proton decay operators. Therefore, we need to forbid at least some of the couplings 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 model proposed in Ref. Li:2009cy (), the Higgs fields and are on the different Higgs curves, and and do not have zero modes by choosing proper fluxes. And then the KK modes of and do not form vector-like particles, i.e., the third term in Eq. (8) does not exist. The mass terms between the KK modes of and arise from the usual term. So the proton partial lifetime via the dimension-five proton decay is proportional to . In generic GUTs with high-scale supersymmetry breaking, we have , and . Thus, the proton partial lifetime via the dimension-five proton decay is proportional to , which is much larger than . And then we do not have the dimension-five proton decay problem. Moreover, the SM quarks and are on different matter curves. And then the and gauge bosons can not couple to both and . Therefore, we do not have the dimension-six proton decay problem via superheavy gauge boson exchanges.

In the Type I and Type II F-theory models proposed in Ref. Li:2009cy () where the gauge symmetry is broken down to the gauge symmetry, the SM fermions , , and are on one matter curve, while , , and are on the other matter curve. On the Higgs curve, and do not have zero modes by choosing proper fluxes, and the KK modes of and do not form vector-like particles. Thus, similar to the discussions in the above F-theory models, we do not have the dimension-five proton decay problem. Moreover, the SM quarks and / are on different matter curves. So the and gauge bosons can not couple to both and , and the and gauge bosons can not couple to both and , Therefore, we do not have the dimension-six proton decay problem via superheavy gauge boson exchanges.

Third, we consider the five-dimensional orbifold and models on where the proton decay problems can be solved as well kawa (); GAFF (); LHYN (); AHJMR (); Li:2001qs (); Dermisek:2001hp (); Li:2001tx (). We assume that the fifth dimension is a circle with coordinate and radius . The orbifold is obtained by the circle moduloing the following equivalent classes

 P:   y∼−y ,          P′:   y′∼−y′ , (13)

where . There are two inequivalent 3-branes located at the fixed points and , which are denoted by and , respectively. In particular, the zero modes of the SM fermions in the bulk do not form the complete GUT representations due to the orbifold gauge symmetry breaking Li:2001tx ().

In the orbifold models (for a concrete example, see Ref. LHYN ()), the gauge symmetry is broken down to the SM gauge symmetry via orbifold projections. With suitable representations for the and parities, the gauge symmetry is preserved on the 3-brane, while it is broken down to the SM gauge symmetry on the 3-brane. To realize the gauge coupling relation in Eq. (2), we introduce the adjoint Higgs field in the representation on the 3-brane. The gauge coupling relation in Eq. (2) can be generated via the suitable dimension-five operators after the adjoint Higgs field acquires the VEV Hill:1983xh (); Shafi:1983gz (); Ellis:1985jn (); Li:2010xr (). We put the Higgs fields and in the bulk, and then and do not have zero modes due to the orbifold projections. In particular, the KK modes for and only have vector-like mass term via term. Thus, similar to the discussions in the above F-theory GUTs, we do not have the dimension-five proton decay problem. To forbid the dimension-six proton decay, we put the SM fermion superfields and in the bulk with suitable and parity assignments where . We obtain the SM fermions as zero modes from while we obtain the SM fermions and as zero modes from . Because the and gauge bosons can not couple to both and , we do not have the dimension-six proton decay problem via superheavy gauge boson exchanges.

In the orbifold models (for a concrete example, see Ref. Dermisek:2001hp ()), the gauge symmetry is broken down to the Pati-Salam gauge symmetry via orbifold projections. With suitable representations for the and parities, the gauge symmetry is preserved on the 3-brane, while it is broken down to the Pati-Salam gauge symmetry on the 3-brane. To realize the gauge coupling relation in Eq. (2), we introduce the symmetric Higgs field in the representation on the 3-brane. The gauge coupling relation in Eq. (2) can be generated via the suitable dimension-five operators after the symmetric Higgs field acquires the VEV Hill:1983xh (); Shafi:1983gz (); Ellis:1985jn (); Li:2010xr (). We put the Higgs field in the bulk, and then and do not have zero modes due to orbifold projections. In particular, the KK modes for and only have vector-like mass term via term. Thus, similar to the discussions in the above F-theory GUTs and the orbifold models, we do not have the dimension-five proton decay problem. To forbid the dimension-six proton decay, we put the SM fermion superfields and in the bulk with suitable and parity assignments where . We obtain the left-handed SM fermions and as zero modes from while we obtain the right-handed SM fermions , , and as zero modes from . Because the and gauge bosons can not couple to both and and the and gauge bosons can not couple to both and , we do not have the dimension-six proton decay problem via superheavy gauge boson exchanges.

Fourth, let us comment on the superheavy threshold corrections on the gauge coupling unification in our models. In the F-theory and models, we shall have the superheavy threshold corrections from the Kaluza-Klein (KK) modes and heavy string modes. Because our unification scale is smaller than or equal to , we do not have string theshfold corrections since the string scale is generic around . Also, the KK modes can have masses around the GUT scale or higher, and then their effects on the gauge coupling unification can be negligible as well. Moreover, in the orbifold and models, we shall have the superheavy threshold corrections from the KK modes. Because the masses of the KK modes can not be larger than the GUT scale, we might have appreciable threshold corrections on the gauge coupling unification, which definitely deserves further detailed study. Thus, for simplicity, we assume that the KK mass scale is equal to the GUT scale in this paper.

## V Conclusion

Inspired by the string landscape and the unified gauge coupling relation in the F-theory GUTs and GUTs with suitable high-dimensional operators, we studied the canonical gauge coupling unification in the SM with high-scale supersymmetry breaking. In the SM with GUT-scale supersymmetry breaking, the gauge coupling unification can be achieved at about , and the Higgs boson mass is predicted to range from 130 GeV to 147 GeV. In the SM with supersymmetry breaking scale from to , gauge coupling unification can always be realized, and the corresponding GUT scale is from to , respectively. Also, we obtained the Higgs boson mass from 114.4 GeV to GeV. Moreover, the discrepancies among the SM gauge couplings at the GUT scale are less than about 4-6%. Furthermore, we presented the and models from the F-theory model building and orbifold constructions, and showed that there are no dimension-five and dimension-six proton decay problems even if .

###### Acknowledgements.
This research was supported in part by the Natural Science Foundation of China under grant numbers 10821504 and 11075194 (TL), and by the DOE grant DE-FG03-95-Er-40917 (TL and DVN).

### Footnotes

1. preprint: ACT-13-10, MIFPA-10-49

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