Gauge coupling unification in SO(32) heterotic string theory with magnetic fluxes

We study $SO(32)$ heterotic string theory on torus with magnetic fluxes. Non-vanishing fluxes can lead to non-universal gauge kinetic functions for $SU(3) \times SU(2) \times U(1)_Y$ which is the important features of $SO(32)$ heterotic string theory in contrast to the $E_8\times E_8$ theory. It is found that the experimental values of gauge couplings are realized with ${\cal O}(1)$ values of moduli fields based on the realistic models with the $SU(3) \times SU(2) \times U(1)_Y$ gauge symmetry and three chiral generations of quarks and leptons without chiral exotics.


Introduction
Gauge coupling unification is a familiar tool to search for the underling theory of the standard model such as the Grand unified theory (GUT) or string theory.For example, in the light of the observed values of gauge couplings, the three gauge couplings are unified with the GUT normalization of U(1) Y at the so-called GUT scale, 2×10 16 GeV in the low-energy or multi-TeV scale minimal supersymmetric standard model (MSSM).On the other hand, the three gauge couplings are not unified with the GUT normalization of U(1) Y in the standard model (SM).
Superstring theory also has a certain prediction.In particular, in 4D low-energy effective field theory derived from heterotic string theory, the gauge couplings at tree level are unified up to Kač-Moody levels κ a at the string scale [1], which is of O(10 17 ) GeV [2].This prediction is very strong.In order to explain the experimental values, we may need some corrections, e.g.stringy threshold corrections [3,4,5].(See for numerical studies e.g.Refs.[6,7].)Then, we may need O(10) of moduli values. 1 When some moduli values are of O (10) or larger, the string coupling may become strong and peturbative description may not be valid.
On the other hand, in D-brane models, gauge couplings seem to be independent of each other for gauge sectors, which originate from different sets of D-branes.Thus, one may be able to fit parameters such as moduli values in order to explain the experimental values of three gauge couplings, although there may appear some relations and/or constraints [9,10] in a certain type of models.
Here, we study another possible correction within the framework of heterotic string theory.Recently, we carried out systematic analysis towards realistic models within the framework of SO(32) heterotic string theory on the toroidal compactification with magnetic fluxes [11].In such a type of model building, we have constructed the models which have the gauge symmetry including SU(3)×SU(2)×U(1) Y and three generations of quarks and leptons as chiral massless spectra.Furthermore, in this paper we show that the gauge couplings depend on magnetic fluxes in this type of SO(32) heterotic string theory.That is, gauge couplings can be non-universal at the string scale and non-universal parts depend on the Kähler moduli, although E 8 × E 8 heterotic string theory with magnetic fluxes can not lead to non-universal gauge couplings between SU(3) and SU(2) appearing in one E 8 .2Such non-universal corrections can make the gauge coupling prediction consistent with experimental values.Although such possibilities are proposed in Ref. [13], we study numerically the gauge couplings of This paper is organized as follows.In section 2, we review 4D low-energy effective field theory derived from SO(32) heterotic string theory with magnetic fluxes.We also review on our model building towards realistic models, which have the gauge symmetry including SU(3) × SU(2) × U(1) Y and three chiral generations of quarks and leptons as well as vector-like matter fields in massless spectra.In section 3, we study the gauge couplings and show how non-universal gauge couplings appear in our models.In section 4, we study numerically the gauge couplings in explicit models.Section 5 is devoted to conclusions and discussions.

SO(32) heterotic string theory on tori with U (1) magnetic fluxes
In this section, we give a review on the low-energy effective field theory derived from SO(32) heterotic string theory on factorizable tori with U(1) magnetic fluxes.We also show the consistency conditions for U(1) magnetic fluxes which give the constraints for the heterotic string models.(See for details of model construction e.g., Ref. [11].)

Low-energy effective action of SO(32) heterotic string theory
First of all, we show the bosonic part of 10D effective supergravity action derived from the SO(32) heterotic string theory on a general complex manifold M with multiple U(1) magnetic fluxes.By calculating the relevant scattering amplitudes on the worldsheet up to of order O(α ′ ), we obtain the string-frame bosonic action in the notation of Refs.[13,14,15], where the gauge and gravitational couplings are set by g 2 10 = 2(2π) 7 (α ′ ) 3 and 2κ 2 10 = (2π) 7 (α ′ ) 4 , respectively.The string coupling is determined by the vacuum expectation value of the tendimensional dilaton φ 10 , that is, g s = e φ 10 .The field-strength of SO(32) gauge group F has the index of vector-representation, which can be normalized as tr v (T a T b ) = 2δ ab .In addition, the heterotic three-form field strength H is defined by where the part of α ′ corrections are characterized by the gauge and gravitational Chern-Simons three-forms, w YM and w L , respectively.By the ten-dimensional Hodge duality, the Kalb-Ramond two-form B (2) and its dual six-form B (6) are related as, * dB (2) = e 2φ 10 dB (6) . ( Then, from the 10D bosonic action (1), we can extract the kinetic term of Kalb-Ramond field, (10)   e 2φ 10 dB (6) ∧ * dB (6)   + α ′ 8κ 2 10 M (10) where we have added the Wess-Zumino terms induced by the magnetic sources for the Kalb-Ramond field B (6) , i.e., stacks of N a five-branes with their tension being T 5 = ((2π) 5 (α Note that these heterotic five-branes wrap the holomorphic two-cycles Γ a and their Poincáre dual four-forms are denoted by δ(Γ a ).When we study SO(32) heterotic string theory on three 2-tori, M = (T 2 ) 1 × (T 2 ) 2 × (T 2 ) 3 , the tadpole cancellation is obtained by solving the equation of motion for the Kalb-Ramond field, where F represents for the internal U(1) gauge field strengths.These conditions should be satisfied on any four-cycles (T 2 ) i ×(T 2 ) j with i = j, i, j = 1, 2, 3.However, if the nonvanishing fluxes are not canceled by themselves, five-branes would contribute to the tadpole cancellation [16].

Axionic coupling through the Green-Schwarz term
In addition to the effective action (1), the loop effects induce the Green-Schwarz term at the string frame [17,18], whose normalization factor is determined by its S-dual type I theory as shown in [19] and the anomaly eight-form X 8 reads, where "Tr" stands for the trace in the adjoint representations of the SO(32) gauge group.As pointed out in Refs.[20,13], the gauge and gravitational anomalies for the (non-)Abelian gauge groups are canceled by the above Green-Schwarz term (6) and the tadpole condition (5).It is remarkable that even if the Abelian gauge symmetries are anomaly-free, the Abelian gauge bosons may become massive due to the Green-Schwarz coupling given by Eq. ( 6).Therefore, in order to ensure that the hypercharge gauge boson is massless, they should not couple to the axions which are hodge dual to the Kalb-Ramond fields.
Let us study the couplings between the hypercharge U(1) Y gauge boson and the axions, explicitly.First of all, we decompose the SO(32) gauge group into the standard-like model gauge group, which can be realized by inserting all the multiple U(1) constant magnetic fluxes.Within the 16 Cartan elements, 2) is taken as H 5 − H 6 , whereas the other Cartan directions of SO(32) are defined as, in the basis H i .Furthermore, when these U(1) fluxes are inserted along the Cartan direction of SO(32), the field strengths of U(1)'s are also decomposed into the four-and extra-dimensional parts f , f , respectively.Then we can dimensionally reduce the one-loop Green-Schwarz term (6) to where From here, we write the Kalb-Ramond field B (2) and internal U(1) a field strengths fa , where d a are normalization factors appeared in the basis H i (9) and m which implies that one of the multiple U(1) gauge fields absorbs the universal axion and become massive.As shown in Sec. 3, the hypercharge U(1) Y is identified with the linear combinations of multiple U(1)'s, i.e., U(1 , where the summation over c depends on the concrete models.In such cases, the U(1) Y gauge field becomes massless under 6tr(T 4 3 )m (2) 3 m where d ijk denotes the intersection number and there appear the non-vanishing intersection numbers of 2-tori, In addition to the universal axion, other axions also appear from the associated internal two-cycles, which are known as the Kähler axions.When the dual field B (6) is expanded as where ŵk are the Hodge dual four-forms of the Kähler forms, we can extract the axionic couplings between Kähler axions and the U(1) gauge bosons through Eq. ( 4), which lead to the following U(1) Y massless condition, with i = 1, 2, 3.

Model building and constraints
Here we review our approach to construct realistic models.(See the detail for Ref. [11].)So far, we introduce the magnetic fluxes m a along all U(1) a for a = 1, • • • , 13.In our model, such magnetic fluxes break SO(32) into SU(3) C × SU(2) L × 13  a=1 U(1) a and a certain linear combination of 13 a=1 U(1) a corresponds to U(1) Y .However, the degenerate magnetic fluxes lead to the enhancement of gauge symmetry e.g., SU(4) × SU(2) × SU(2) in the visible sector.In such a case, we introduce Wilson lines to break this remaining large gauge group into In addition to gauge symmetry breaking, non-vanishing magnetic fluxes can realize the 4D chiral theory, where the number of zero modes is determined by their U(1) charges and magnetic fluxes.The relevant matter contents in the SM reside in the adjoint and vector representations of SO (12) in SO(32) and their generation numbers are given by where Q 1,2 are the left-handed quarks, L 1,2 , L a 3,4 are the charged leptons and/or Higgs, It is remarkable that there are constraints for these U(1) magnetic fluxes.First one is that the U(1) Y massless conditions (17) and (20) by taking account of the axionic couplings with U(1) Y gauge boson.Furthermore, there are the tadpole conditions given by Eq. ( 5).When the heterotic five-branes are absent in our system, the Eq. ( 5) is rewritten as which are required from the consistencies of heterotic string theory.Without the existence of the heterotic five-branes, it is known that U(1) magnetic fluxes satisfy the so-called K-theory constraints, e.g., Ref. [13], for i = 1, 2, 3.These K-theory constraints are discussed in the S-dual to the SO(32) heterotic string theory, that is, type I string theory.(See for instance, Ref. [21,22].)Finally, non-vanishing magnetic fluxes generically induce the non-vanishing Fayet-Illipoulous (FI) terms for U(1) a with a = 1, 2, • • • , 13.Even if such FI terms are not canceled by themselves, they may be able to be canceled by the vacuum expectation values (VEVs) of scalar fields in the hidden sector.

Gauge couplings in heterotic string
In this section, we show the formula of gauge kinetic functions in our model.After dimensionally reducing the 10D effective action (1) as well as the one-loop GS term (6), it is found that the gauge kinetic functions of SU(3) C and SU(2) L receive the different one-loop threshold corrections depending on the abelian fluxes, while that of U(1) Y do not receive such corrections due to the vanishing axionic couplings with U(1) Y gauge boson.

Gauge couplings at tree-level
After compactifying on a 6D internal manifold M with volume Vol(M), the 4D reduced Planck scale M Pl and the gauge coupling constant g 4 can be extracted as which lead to the following relation between the string scale M s = 1/l s with l s = 2π √ α ′ being the typical string length, and the Planck-scale, where α −1 4 = 4πg −2 4 .Since the four-dimensional gauge coupling is determined by the VEV of the dilaton at the tree-level, ReS = g −2  4 , where S being the universal axion, the string scale is roughly estimated as, from Eq. ( 25) by employing M Pl = 2.435 × 10 18 GeV and the four-dimensional gauge coupling constant, α −1 4 ≃ 25, implied by the renormalization group (RG) equations of the MSSM.As mentioned in the introduction, the gauge couplings of SM gauge groups are different from each other at the string scale as illustrated in Fig. 1 under the assumption that the SUSY is broken at the TeV scale and the U(1) Y gauge coupling is normalized as satisfying the so-called GUT relation.In Fig. 1, we involve two-loop effects to the RG equations.On the other hand, when the SUSY is broken at the string scale, the behavior of gauge couplings from the electroweak scale to the string scale is obeyed by the renormalization group equations of the standard model.As seen in Fig. 2, the gaps between the gauge couplings of non-abelian gauge groups are better than that of MSSM.However, it requires the slight corrections to be coincide with the unified one at the string scale.In this case, the string scale is estimated as M s ≃ 1.0 × 10 17 GeV by use of α −1 4 ≃ 45.We also employ the experimental values such as the gauge coupling of SU(3), α −1 SU (3) C ≃ 0.1184, the Weinberg angle sin 2 θ ≃ 0.231 and the fine-structure constant α ≃ 1/128 at the electroweak scale [23] in Figs. 1 and

The one-loop threshold corrections
As shown in the previous section, the dilaton S gives the universal gauge kinetic function.However, the Green-Schwarz term (6), in particular ( 12) and ( 13), can lead to non-universal gauge kinetic functions [13].Indeed, we obtain the non-universal axionic couplings, which lead to the non-universal gauge kinetic functions.This is because we insert the different U(1) fluxes between U(1) 1 and U(1) 2 Cartan directions.Such structures are typical in SO(32) heterotic string theory which is expected as the S-dual of type I string theory with several D-branes.However, the non-universal gauge kinetic functions for SU(3) C and SU(2) L cannot be seen in E 8 × E 8 heterotic string theory due to the trace identities, Tr(F 4 ) = 1 100 (Tr(F 2 )) 2 , where F denotes the gauge field strength of E 8 .From this trace identities, Eq. ( 12) is regarded as the type of Eq. ( 13).Therefore, the gauge kinetic functions of SU(3) C and SU(2) L are equal to each other.It might be preferred in the non-supersymmetric theory such as the standard model from the Fig. 2, although some other threshold corrections are required to unify the gauge couplings.
When we define the Kähler moduli as where t k corresponds to the volume of (T 2 ) k , the gauge kinetic functions of the SU(3) C and SU(2) L become where with d 1 = √ 2 and d 2 = 2.Note that threshold corrections depend on the magnetic fluxes.On the other hand, since U(1) Y is defined as the linear combinations of multiple U(1)'s, the normalization of U(1) Y is then determined by in which the threshold corrections do not appear due to the vanishing axionic couplings with U(1) Y gauge boson, and the gauge kinetic function of U(1) Y is extracted as 4 Numerical studies in explicit models In this section, we show the models satisfying the several consistency conditions in section 2.3, where the chiral massless spectra in the visible sector are just three generations of quarks and leptons without chiral exotics and at the same time, the experimental values of gauge couplings are realized at the string scale.Although there are extra vector-like visible matter fields and hidden chiral and vector-like matter fields, we assume that these modes become massive around M s such that the massless spectra of the SM or the MSSM are realized around M s .
From now on, we consider two concrete scenarios.In one scenario, supersymmetry is broken at M s and below M s the massless spectrum in the visible sector is just one of the SM.In other scenario, supersymmetry remains at M s and breaks around 1 TeV, and below M s the massless spectrum in the visible sector is just one of the MSSM.We assume that the non-vanishing D-terms for extra U(1)'s generated by the generic magnetic flux background are canceled by VEVs of hidden scalar fields and N = 1 supersymmetry remains when we take low-energy SUSY breaking scenario.Since the size of SUSY breaking scale depends on the moduli stabilization scenario, we leave the details of them for future work.
First of all, the U(1) Y gauge coupling g 2 U (1) Y at M s is determined only by S and N through Eq. (34), where From the experimental values of U(1) Y gauge coupling, g −2 U (1) Y (M s ) = 4.80 for the SM and 2.44 for the MSSM, the dilaton VEV S is determined by N as shown in Table 1 for N = 5, 7, 9.In what follows, we discuss the explicit models with N = 5, 7, 9. Next, by solving Eq. (30) and two-loop renormalization group equations for SM and MSSM, we evaluate the ratio of gauge couplings at M s as for the MSSM, for the SM.Here, the experimental values such as the gauge coupling of SU(3), α −1 SU (3) C ≃ 0.1184, the Weinberg angle sin 2 θ ≃ 0.231 and the fine-structure constant α ≃ 1/128 at the electroweak scale are employed.From Eqs. (37) and (38), we estimate the following equalities in Tab. 2 for N = 5, 7, 9,  When we construct an explicit model, all U(1) magnetic fluxes as well as N appeared in the definition of U(1) Y given by Eq. ( 32) are fixed and the β k 2 and β k 3 in Eq. ( 39) are also fixed hereafter.Then, we can examine whether the O(1) values of T k are consistent with the values of B 2 and B 3 in Table 2 or not.Although it is expected that there appear one of the unfixed Kähler moduli by solving the two equations (39) under the three Kähler moduli T k , in some models there are no solutions for realistic values of T k , i.e., T k < 0, T k ≪ 1, T k ≫ 1 and so on.In addition to it, when one of m i 1 and m i 2 vanishes, all Kähler moduli are completely fixed by Eq. (39).
In the following, we show the three examples of magnetic flux configurations denoted by models 1 (N = 5), 2 (N = 7) and 3 (N = 9) which are realistic in the sense that there are the gauge symmetry including SU(3) C × SU(2) L × U(1) Y , three chiral generations of quarks and leptons without chiral exotics in the visible sector, the experimental values of gauge couplings in Eq. (39) and at the same time, they satisfy the consistency conditions in Sec.

Conclusion
We have studied on SO(32) heterotic models with U(1) magnetic fluxes, which have the gauge symmetry including SU(3) C × SU(2) L × U(1) Y and three chiral generations of quarks and leptons as well as vector-like matter fields.In contrast to E 8 × E 8 heterotic string theory, there is the non-universality among the gauge couplings of standard model at the string scale and they depend on magnetic fluxes as well as the VEVs of dilaton and Kähler moduli.Although there are several approaches to realize the gauge couplings consistent with their experimental values, they require the large stringy threshold corrections by employing the large field values of Kähler moduli [3,4,5] which implies the large string coupling at the vacuum.In this paper, we have considered the two SUSY breaking scenarios.One of them is that the SUSY is broken at the string scale, whereas the other model is the TeV SUSY breaking scenario.In both scenarios, it was found that certain explicit models can lead to the gauge couplings consistent with the experimental values even if the values of Kähler moduli are of order unity.Thus, we have constructed the realistic models from both viewpoints of massless spectra and the gauge couplings.
What is important for a next study would be Yukawa couplings.The zero-mode profiles of quarks and leptons as well as higgs fields are non-trivial because of introducing magnetic fluxes.That would lead to non-trivial Yukawa matrices. 3.Also, in Ref. [11], it was shown that the models with N = 9 have SU(3) flavor symmetry.Such a flavor symmetry might be useful to realize the realistic values of fermion masses and mixing angles.We would study this issue elsewhere.
So far, we have taken the dilaton and Kähler moduli VEVs as free parameters in order to obtain the gauge couplings consistent with the experimental values.It is also next issue to study moduli stabilization at proper values of them.

a
are the U(1) a fluxes constrained by the Dirac quantization condition.From the Eqs.(10) and (11), we can extract the Stueckelberg couplings between the U(1) gauge fields and the universal axion b (0) S which is the hodge dual of the tensor field b (2)

R 2 are
the charge conjugate of right-handed up type quarks, d c R 1,2 , d c a R 3 are the charge conjugate of right-handed down type quarks, e c a R 1 are the charge conjugate of right-handed leptons, n 1 , n a 2 are the singlets in the standard model gauge groups. 2.

Figure 1 :
Figure 1: RG flow of the gauge couplings in the MSSM at the two-loop level.These lines show the gauge coupling of U(1) Y (thick line), SU(2) L (dashed line) and SU(3) C (dotted line), respectively.Here, we include the error bar associated with the QCD coupling α −1 SU (3) C [23].

1 Figure 2 :
Figure 2: RG flow of the gauge couplings in the SM at the two-loop level.These lines show the gauge coupling of U(1) Y (thick line), SU(2) L (dashed line) and SU(3) C (dotted line), respectively.
) Tab. 2 shows that for N = 5, the values of |B 2 | and |B 3 | are much smaller than O(1).In order to realize T k = O(1), i.e., the small value of string coupling, it requires the certain cancellations within the VEVs of Kähler moduli T k and U(1) 1,2 fluxes in Eq. (39).A similar behavior on B 2 and m i 1 would be required for N = 7.On the other hand, for N = 9, the large B 3 requires the large U(1) 2 fluxes m i 2 to obtain T k = O(1).

1 Figure 4 : 1 Figure 5 :
Figure 4: The VEVs of moduli, T 2 (red dashed curve), T 3 (blue dotdashed curve) and the volume of three-tori Vol(M) = T 1 T 2 T 3 (black thick curve) as a function of T 1 in model 1.The left and right panels show those of SM and MSSM, respectively.

1 Figure 6 :
Figure 6: The VEVs of moduli, T 2 (red dashed curve), T 3 (blue dotdashed curve) and the volume of three-tori Vol(M) = T 1 T 2 T 3 (black thick curve) as a function of T 1 in model 3.The left and right panels show those of SM and MSSM, respectively.

Table 1 :
The VEV of dilaton S for the SM and MSSM.

Table 2 :
The values of B 2 and B 3 for the SM and MSSM in the case of N = 5, 7, 9.By increasing the number of N appeared in the definition of U(1) Y (32), the values of B i become larger in both cases of SM and MSSM.