Standard Model-like D-brane models and gauge couplings

We systematically search intersecting D-brane models, which just realize the Standard Model chiral matter contents and gauge symmetry. We construct new classes of non-supersymmetric Standard Model-like models. We also study gauge coupling constants of these models. The tree level gauge coupling is a function of compactification moduli, string scale, string coupling and winding number of D-branes. By tuning them, we examine whether the models can explain the experimental values of gauge couplings. As a result, we find that the string scale should be greater than $10^{14-15}$GeV if the compactification scale and the string scale are the same order.


Introduction
The Standard Model (SM) is one of the greatest achievement of particle physics. It is consistent with all of the experimental results by tuning about 19 free parameters and succeeded in predicting new physics. The discovery of the Higgs scalar [1,2] is the latest one. However, many questions still remain in particle physics. What is quantum gravity theory ? How does the mysterious flavor structure of the SM appear ? What is the origin of neutrino masses, inflation, dark matter and so on ?
From the viewpoint of quantum gravity, superstring theory is the most promising candidate of its theory and almost only known candidate of it. Furthermore, superstring theory is also a unified theory of other interactions and matter fields. Superstring theory naturally has gauge symmetry. There appear gravitons, gauge bosons, matter fermions, and scalars in its massless spectrum. Thus, it is important to construct stringy origin theory explaining the SM.
The intersecting D-brane models are one of the interesting techniques to realize fourdimensional (4D) chiral gauge theory as low-energy effective theory from superstring theory [3,4,5,6,7] (see for review [8,9] and references therein). In these models, chiral matter fermions are realized as the R-sector of open string stretching between D-branes at angles, while gauge bosons are realized as open strings on the same set of D-branes. It is amazing that simple compactification models realize the SM spectrum or supersymmetric SM spectrum as zero modes. For example, in [7], the intersecting D-brane model with just the SM spectrum was constructed which we call the IMR model in this paper. Similarly, supersymmetric SM-like models have been constructed (see e.g [10,11,12]).
In addition to the massless spectrum, it is quite important to explain quantitative structure of the SM, i.e. the gauge couplings, Yukawa couplings and the Higgs potential parameters as well as maybe neutrino Majorana masses. In this paper, we focus on the gauge couplings. In 4D low-energy effective field theory derived from heterotic string theory, the gauge couplings at tree level are unified up to Kac-Moody levels κ a at the string scale [13], which is of O(10 17 ) GeV [14]. This prediction is very strong. In order to explain the experimental values, we may need some corrections, e.g. stringy threshold corrections [15,16,17]. (See for numerical studies e.g. Refs. [18,19].) On the other hand, the gauge coupling is a function of D-brane volume in D-brane models. In intersecting D-brane models, gauge groups of the SM are originated from different D-branes, which have volumes independent of each other. Thus, at first sight, it seems always possible to explain the three gauge couplings of the SM by tuning volume moduli, because the number of parameters, moduli, is sufficiently larger than three. 1 However, in an explicit model, the values of volume moduli are constrained by other conditions. For example, tachyonic modes may appear for some values of moduli in nonsupersymmetric models. Also, the string coupling g s may be required to be strong for some values of moduli to derive the realistic values of the SM gauge coupings. However, our theory is reliable at the weak string coupling. Then, it is non-trivial to explain the three SM gauge couplings under the above conditions.
In this paper, we study systematically the model construction of intersecting D-brane models. We construct new classes of non-supersymmetric SM-like models, which have the same gauge symmetry and chiral matter contents as those of the SM but no exotics except right-handed neutrinos. We show three classes of the SM-like models. We study their gauge couplings as well as those of the IMR model under the above constraints.
This paper is organized as follows. In section 2, we briefly review the intersecting D-brane models. In section 3, we construct new classes of SM-like models. We calculate gauge couplings in section 4. Section 5 is our conclusion. In Appendix A, we show systematic search of the SM-like models.

Intersecting D-brane model building
In this section, we briefly review the toroidal orientifold models with intersecting D6branes. We first consider Type IIA superstring theory compactified on factorized six dimensional tori T 6 = T 2 1 × T 2 2 × T 2 3 including intersecting D6-branes, where T 2 i is the i-th two-dimensional torus. It is the two-dimensional Euclid space modded by a lattice, D6 a -branes wrap 3-cycles [Π a ] on T 6 . Here, we restrict ourselves to the D-brane system in which all D6-brane's 3-cycles [Π a ] are factorized, is a 1-cycle of T 2 i . Then we can specify the 3-cycles by using 6 integer winding numbers (n i a , m i a ). n i a is the winding number along the τ i direction and m i a is the winding number along the imaginary axis of z i . The intersecting number between the D6 a -brane and D6 b -brane is denoted by I ab which is determined by winding numbers, The open string stretching between D6 a -brane and D6 b -brane has the following boundary conditions, is the angle of D6 a -brane in the i-th torus. These boundary conditions resolve degeneracies of the ground states in the R-sector. The resultant ground state corresponds to a 4D massless chiral fermion. Scalars appear in the NS sector. The ground state in the NSsector depends on the intersecting angle θ i ab = (θ i b − θ i a )/π. Assuming 1 > θ i ab > 0, the masses squared of four candidates for the lightest state are shown in Table 1. They would be massive, massless or tachyonic depending on the angles. If there are any massless states, a part of supersymmetry is recovered. For example, when θ 1 ba + θ 2 ba − θ 3 ba = 0, the first state in Table 1 is the massless ground state and the others are massive.

State
Mass 2 1 In this way, each intersecting point has a 4D massless chiral fermion as well as scalars. Also, a stack of N a D6 a -branes has gauge symmetry U(N a ). The open strings ending at the D6 a -brane have Chan-Paton charges, which correspond to the fundamental representation of U(N a ). This class of models lead to 4D chiral U(N a ) Yang-Mills theory as low energy effective theory. This is the attractive fact to derive the SM at low energy. Now, we introduce the orientifold 2 . Torus orientifold is obtained by modding T 6 by reflection operator R, R : Imz 1,2,3 → −Imz 1,2,3 . (2.6) To define this operator R well, Imτ i in L(τ i ) must be either 0 or 1/2. The torus is rigid for Imτ i = 0, while the torus is tilted for Imτ i = 1/2. It is useful to define new "winding number" (ñ i a ,m i a ), whereñ i a = n i a andm i a = m i a + Imτ i n i a . Hereafter, we use this (ñ i a ,m i a ) as a winding number of D6 a -brane in the i-th torus.
In this setup, we can construct perturbative vacua which have several stacks of N a D6 a -branes wrapping whole 4D Minkowski spacetime and factorized 3-cycles [Π a ] of T 6 . In addition to D6 a -branes, we need their orientifold mirror D6 a * -branes such that the system is R-invariant. D6 a * -brane's winding numbers must be (ñ i a , −m i a ). In the presence of orientifold, the gauge symmetry G a which a stack of N a D-branes has depends on whether D a -branes lie on top of its orientifold mirror D a * -branes or not. If D6 a -branes are apart from D6 a * -brane, the gauge group is U(N a ). Otherwise the gauge group is Sp(2N a ) or SO(2N a ). The intersecting points between D6 a -branes and D6 bbranes have a massless 4D chiral fermion transforming as bifundamental representations The number of intersecting points I ab is written by I ab is the number of chiral fermions having same representation appearing in 4D lowenergy effective theory, which means that I ab is the generation number. Using this D-brane system, we can realize a lot of patterns of chiral (super) Yang-Mills theory as effective theory, but not all theories.
Next, let us discuss the constraints on intersecting D-brane models. D-branes have RR charges which must be canceled in compact space. This constraint is derived from D-brane kinematics, and it is the same as Gauss's law of electromagnetism in compact space. This is called the RR tadpole cancellation condition. Since the RR charge is proportional to the D-brane homology, the constraint is written by In general, the gauge symmetry includes several U(1) factors. Some of them become massive by the generalized Green-Schwartz mechanism. That is, U(1) gauge bosons have non zero couplings with RR-forms, especially C 5 and have non-perturbative Stückelberg mass. The coupling between U(1) a gauge boson and C 5 is obtained by the Chern-Simons term, (2.9) We introduce [α k ] as the basis of 3-cycles and its dual basis Then the coupling between U(1) gauge bosons and B k 2 can be written by In the next section, we will construct intersecting D-brane models which have the same gauge group as that of the SM. We will show that we can get the exact SM gauge group by using above mechanism to make extra gauge bosons massive.

SM-like model
Our aim is to construct perturbative vacua which have the SM-like effective theory by using type IIA orientifold. For such a purpose, we systematically search vacua satisfying the following conditions: • Gauge symmetry is the same as that of the SM up to the hidden sector, • The chiral massless spectrum is the same as that of the SM with three right-handed neutrinos up to the hidden sector.
For the RR tadpole cancellation, we need right-handed neutrinos and the G hidden sector. The matter fields in the hidden sector are singlets under the SM gauge. There are two methods to realize the SU(2) gauge symmetry. One is to use two D6 abranes separating from their orientifold mirror D6 a * -branes. The theory in D6 a -brane worldvolume is U(2) Yang-Mills theory which contains SU(2) group as subgroup. We call this class of models SU(2) models. In this scenario, we must use tilted tori to cancel U(2) anomaly. There are many models using SU(2) method, see for the model satisfying the above condition, e.g. [7]. The other is to use one D6 a -brane whose orinetifold mirror D6 a *brane is coincident with the D6 a -brane. In this case, the gauge group can be enhanced from U(1) to Sp(2). Sp (2) is homomorphic to SU(2) as Lie algebra. Then, we can get the SU(2) gauge symmetry. We call this class of models Sp(2) models.
We concentrate on the latter models in the following: • We construct Sp(2) models where SU(2) gauge symmetry is realized by one brane and its orientifold mirror.
We can satisfy these conditions by using four stacks of branes, D6 a,b,c,d -branes. The multiplicity of D6 a -branes N a is equal to three, and the others are one. D b -brane is parallel (or perpendicular) to O6-planes and gauge symmetry is enhanced to Sp(2). The intersection numbers of these branes are required as follows, I ab = 3; I ac = −3; I ac * = −3; I ad = 0; I ad * = 0, such that the chiral spectrum of this model realizes the SM matter contents. For desired zero mode, we require D6 a,c,d -branes to be parallel to O-plane in at least one torus, too. The hypercharge U(1) Y corresponds to the following linear combination of U(1)s, There are some arbitrariness of the definition of U(1) Y , but we can absorb it by renaming branes. In Table 2, we summarize the chiral spectrum of this model, quantum numbers of non-Abelian and Abelian gauge symmetries, and their names in the SM. We carry out a systematic analysis on all the possible D-brane configurations, see Appendix A for the details. As a result, it is found that general solutions realizing Eq.(3.1) are classified into two classes of models.   There are no other solutions satisfying the conditions. Note that gauginos and adjoint scalars appear in the gauge sector of our models which would be massive by loop correction [7].
For later calculation, we classify further the class of models to three new classes as Table 4, Table 5 and Table 6. We refer to the class of models in Table 4 as 0til-SM, because they have no tilted torus. Also we refer the class of models in Table 5 and Table 6 as 1til-SM and 2til-SM, respectively. As we show in Table 3, we can not construct the SM-like models using three tilted tori since they always lead to even number of generations.
The Higgs bosons correspond to the open string in the NS sector stretching between D6 b -branes and D6 c -brane. These branes are parallel in T 2 2 and T 2 3 . This situation is the same as that in the IMR model [7]. The Higgs mass is determined by the distance of D6-branes and the intersecting angle. Note that we need fine tuning to get light Higgs D-brane Table 4: 0til-SM models. All of the tori T 2 i are rigid. The integer parameters satisfy

mass.
The D-brane configurations in Tables 4, 5, and 6 do not satisfy the RR tadpole condition yet, but it is always possible by adding extra D6-branes which are parallel to O6planes. Since D6 a,b,c,d -branes and its orientifold mirrors have no intersecting point with O6-plane, there are no intersecting points between extra D-branes and D6 a,b,c,d -branes. Thus, the introduction of these extra D6-branes does not change the chiral spectrum in the visible sector. In this sense, extra D6-branes correspond to the completely hidden sector.
These models have characteristic winding numbers. D6 b -brane and D6 c -brane are parallel to the O6-plane in T 2 2 and T 2 3 . D6 a -brane and D6 d -brane are parallel to the O6plane in T 1 2 . The charge of U(1) a is 3 times baryon number and the U(1) d is the lepton number. The intersecting numbers of D6 a,c -brane and D6 b,c -brane in T 2 2,3 are same. Thus, the flavor structure of the quarks and leptons are exactly the same at perturbative level. (See for discrete flavor symmetries [21,22].) 3 However, if we take non-perturbative effects into account, these structure must be broken and, for example, right-handed Majorana neutrino masses might be generated [25,26,27]. At any rate, the study on the flavor sector is beyond our scope, and we would study elsewhere. Table 6: 2til-SM models. 4 Gauge couplings

Constraints on model
We have found three classes of SM-like models in section 3. In these models, the gauge symmetry is the exactly same as that of the SM up to the hidden sector. Now, let us study the gauge sector quantitatively. That is, we study the question, whether it is possible to make all gauge couplings consistent with their experimental values. At first sight, it looks possible because there are a lot of parameters in these classes of models. For example, all classes of models have more than 3 integer winding numbers as free parameters and all models have torus moduli as free parameters 4 . However, it is not simple when we take into account other constraints. One constraint is to avoid the tachyonic configuration and the other is a constraint of the string coupling. The R-sector of the open string stretching between D6 k -brane and D6 ℓ -brane has chiral fermionic zero-mode, while the corresponding NS-sector has the light scalar spectrum of Table 1. These NS-sector modes are superpartners of chiral fermions and some of them could be tachyonic in non-supersymmetric models. If a configuration has tachyons, it is unstable and decays to other configuration quickly. We must tune parameters to avoid such tachyons. This condition constrains the parameters significantly. In Sp(2) models, there are six chiral fermion modes and each of them has superpartners at intersecting points. To make these scalars massive or massless, the models must satisfy 24 inequalities.
The other constraint is the perturbativity of theory. The tree level gauge coupling α k = g 2 k /4π at the string scale is given by [28,20], where V k denotes the D6 k -brane's 3-cycle volume in the compact space, M s is the string scale and g s is the string coupling. κ k is obtained as κ k = 1 for U(N k ) and κ k = 2 for Sp(2N k )/SO(2N k ). In this way, we can calculate all the gauge couplings, α a,b,c,d . For U(1) Y , we must normalize the gauge field and α Y is written by, On the other hands, by performing dimensional reduction of the type IIA supergravity action, one can write the Planck mass M p by string parameters as, where V 6 is the volume of the compact space. From (4.1), (4.3), we can write the string coupling in terms of gauge couplings, We have concentrated on perturbative vacua and its effective theory, but when g s >O(1), perturbative theory is broken down and our models no longer make sense. To get sufficiently small g s , there are constraints on parameters.
It is natural to assume V 6 ∼ 1/M 6 s . The α k in Eq. (4.4) is gauge coupling at the string scale, then we evaluate Naively, if M s is very small, g s is very large and perturbativity of the theory is violated. Using the renormalization group equations and the experimental values of α k (M Z ), we can evaluate α k (M s ) in Eq. (4.5). The models obtained in the previous section have almost the same field contents as those of the SM, but include gauginos and adjoint scalars in the gauge sector. We assume that such gauginos and adjoint scalars gain masses around M s and neglect their threshold corrections. Hence, we can evaluate α k (M s ) by using betafunctions of the SM. We find α 3,2 (M s ) > 1/50 for M s ≤ 10 18 GeV. Then, V a,b /(V 6 ) 1 2 must be small to get sufficiently small g s . This means that the direction which is perpendicular to the a,b-brane is large and V a,b /V 6 is suppressed. However, in our models, we have I ab = 0 and there is no direction which is perpendicular to a-brane and b-brane at the same time. Hence, generally we get V a V b /V 6 > 1. When V a V b /V 6 > 1 and α 3 , α 2 > 1/50, we obtain That is, it is required M s 10 15 GeV. When there is a large hierarchy between V 6 and 1/M 6 s , this estimation would change. For V 6 M 6 s = γ, we have the constraint M s γ 1/6 10 15 GeV. For example, we find M s 10 16 GeV for γ = O(10 6 ) and M s 10 14 GeV for γ = O(10 −6 ). We comment on the effect of the gauginos and adjoint scalars on above argument. We have assumed that all of the gauginos and adjoint scalars have masses around M s . If they are lighter, α 3 and α 2 become larger because they give positive contributions to beta-functions. Therefore, the lighter gauginos and adjoint scalars strengthen the constraint.
As mentioned above, the string scale is constrained. On the other hand, winding numbers and moduli are also constrained. As a concrete example, we study the 0til-SM models. In this class of models, the ratio of tree level gauge couplings are given by, where τ i is the T 2 i torus modulus. The renormalization group flows from the experimental values show that α 2 (µ) is similar to α 3 (µ) unless the running scale µ is very low. To realize α 2 (M s ) ∼ α 3 (M s ), it is required that |n 1 a τ 1 | is less than O(1). In this way, the winding numbers and the value of the moduli are constrained.

Numerical analysis
We plot the gauge coupling ratios of our models in Figures 1, 2 and 3 for M s = 10 16 , M s = 10 15 and 10 14 GeV, respectively. For comparison, we also show the gauge coupling ratios of the IMR model in these figures. The blue dots correspond to the gauge coupling ratios, which are calculated by Eqs. (4.1) and (4.2) for the parameters to satisfy g s < 1 assuming V 6 = 1/M 6 s and to avoid tachyonic modes. We vary winding numbers from 1 to 100 and torus moduli from 10 −2 to 10 2 . We neglect stringy threshold corrections [29] 5 . Note that the ratios α k /α l given by Eqs. There are some characteristic features in these figures. In all models, the ratio of gauge couplings α 3 /α Y is less than 6. This is because U(1) Y is linear combination of U(1) a,c,d s and α Y is function of α 3 . It makes upper bound on α 3 /α Y . Sp(2) models tend to have larger α 2 than U(2) model. It is because the b-brane must be parallel or perpendicular to the O6-plane in Sp(2) and its volume can not be so large. The Sp(2) models have larger allowed region than the IMR model. This is because the Sp(2) models have more parameters than the IMR model.   (2) models. We checked that blue dots disappear in this region for M s = 10 13 GeV and we can not tune parameters to fit the gauge couplings for weak g s in all of these models. The critical string scale is 10 14−15 GeV. These results are consistent with Eq. (4.6).
We have analyzed assuming V 6 M 6 s = 1. Similarly, we can analyze gauge couplings for other values of V 6 M 6 s = γ. Unless there is a large hierarchy between them, we obtain almost the same results. Furthermore, even when γ is very small or large, we would have the lower bound on M s . In some case, the one-loop threshold corrections would be

Explicit example
In this subsection, we give an explicit example of the models. As shown in Figure 1, there are a lot of winding numbers and moduli which realize the renormalized SM gauge couplings at the string scale. Table 7 shows one example. In this model, the string scale is set to be 10 18 GeV and the ratios of the gauge couplings in the model are given as, (4.9) From the experimental values, the ratios of renormalized gauge couplings at 10 18 GeV D-brane T 2 1 (1/Reτ 1 = 10 2/3 ) T 2 2 (1/Reτ 2 = 10 14/9 ) T 2 3 (1/Reτ 3 = 10 2/3 ) a (1,0) (3, 1/2) To get the realistic gauge couplings, the string coupling should be 5 × 10 −3 , which means that the theory is weak coupling.

Conclusion and discussion
We have studied SM-like intersecting D-brane models. We have constructed and classified the simplest class of models using Sp(2) which realizes the SM gauge symmetry and chiral spectrum including three right-handed neutrinos as open string zero modes. These models are very simple and attractive. They have only four stacks of D-branes. The three generations of leptons and quarks are just realized by intersecting numbers of D-branes, and each generation is originated from the same type of the intersecting points. This is different from the IMR model, where one generation of quark doublet is originated from the intersecting point between D6 a -brane and D6 b -brane, while the other two generations are originated from the intersecting point between D6 a -brane and D6 * b -brane. Thus, our models have very large flavor symmetry. Its proper breaking might be helpful to realize the flavor structure in nature.
We have studied gauge coupling constants of our models. At first sight, it seems always possible to fit the gauge couplings to the experimental values in most of models, because there are lots of free parameters. However, it is non-trivial to reproduce the SM gauge couplings because the two conditions, the absence of the tachyon and perturbativity, put the strong constraints on model parameters. Our calculation has shown that the string scale must be greater than 10 14−15 GeV to get realistic gauge couplings when there is no large hierarchy between V 6 and M s . Low energy string is disfavored in these models. This tendency may not be model-dependent. One reason is that α Y must depend on α 3 and α 3 /α Y has some limits in intersecting D-brane models. When we try to reconstruct the SM, the values of gauge coupling constants are similar values.
In order to fit the gauge couplings to the experimental values, we have used moduli parameters as free parameters. However, moduli should be stabilized and their stabilized values are important to realize the gauge couplings. All of our models have the hidden sector. Some dynamics in the hidden sector would play a role in moduli stabilization. Also the hidden sector may include dark matter. These topics are quite interesting, but beyond our scope. We would study elsewhere.
Each of B k 2 has the coupling with U(1)s as The condition to remain the U(1) Y gauge boson massless is given by