Flavor structure in D-brane models: Majorana neutrino masses

We study the flavor structure in intersecting D-brane models. We study anomalies of the discrete flavor symmetries. We analyze the Majorana neutrino masses, which can be generated by D-brane instanton effects. It is found that a certain pattern of mass matrix is obtained and the cyclic permutation symmetry remains unbroken. As a result, trimaximal mixing matrix can be realized if Dirac neutrino mass and charged lepton mass matrices are diagonal.


Introduction
The Standard Model has been confirmed by the discovery of the Higgs scalar and other precision measurements. However, it has various mysteries still. One of them is the mystery on the flavor structure. Why are there three generations ? Why are quark and lepton masses hierarchical ? Which mechanism determines their mixing angles ? Indeed, the Yukawa sector has most of free parameters in the Standard model. Discrete flavor symmetries would be important to understand fermion masses and mixing angles [1,2,3]. For example, the mixing matrix in the lepton sector, the PMNS matrix, can be approximated by the tri-bimaximal mixing matrix in the limit θ 13 = 0 [4]. In fieldtheoretical model building, one starts with a large flavor symmetry. Then, one assumes that the flavor symmetry breaks properly into Z 3 and Z 2 subsymmetries in the charged lepton or the neutrino masses, such that the tri-bimaximal mixing can be realized.
Superstring theory is a promising candidate for unified theory of all of the interactions including gravity and all of the matter fields and Higgs field(s) (see for a review [5]). It is found that superstring theory on six-dimensional compact space leads to interesting flavor structures. In particular, certain types of four-dimensional superstring models with rather simple six-dimensional compact spaces such as tori and orbifolds lead to definite discrete flavor symmetries. For example, intersecting D-brane models and magnetized D-brane models are among interesting model building in superstring theory [6,7,8,9,10,11] (see for review [12,5] and references therein). These intersecting/magnetized D-brane models can lead to discrete flavor symmetries such as D 4 , ∆ (27), ∆(54) [13,14,15]. 1 Similar discrete flavor symmetries can be derived in heterotic string theory on orbifolds [16]. 2 In these models, we can calculate explicitly Yukawa couplings and higher order couplings [18,19,20] However, such discrete flavor symmetries may be broken by non-perturbative effects. From such a viewpoint, anomalies of discrete symmetries [29,30,27,28,25] are important because anomalous symmetries may be broken by non-perturbative effects. Even anomalyfree U(1) gauge symmetries can be broken when axions couple with U(1) gauge bosons and they become massive. Furthermore, as concrete non-perturbative effects, D-brane instanton effects have been studied [31] (see also for a review [32] and references therein). From the viewpoint of flavor physics, one of important points is that D-brane instanton effects can generate right-handed Majorana neutrino masses [33,34,35]. Then, it is also important to investigate patterns of right-handed Majorana neutrino mass matrices derived by D-brane instanton effects and study whether such effects break some or all of discrete flavor symmetries and which symmetries remain unbroken.
In this paper, we study the flavor structure in intersecting D-brane models as well as magnetized D-brane models. We study anomalies of discrete flavor symmetries derived in intersecting D-brane models. We also study right-handed Majorana neutrino mass matrices, which can be generated by D-brane instanton effects. We show which types of Majorana mass matrices can be derived and which flavor symmetries remain unbroken 1 See also [17]. 2 See for recent works on other discrete stringy symmetries, e.g. [21,22,23,24,25,26,27,28] . even with right-handed Majorana neutrino mass matrices generated by D-brane instanton effects.
This paper is organized as follows. In section 2, we review briefly the discrete flavor symmetries derived from intersecting D-brane models as well as magnetized D-brane models. In section 3, we study anomalies of these discrete flavor symmetries. In section 4, we study right-handed Majorana masses generated by D-brane instanton effects. Section 5 is devoted to conclusion and discussion.

Discrete flavor symmetries
In this section, we review briefly discrete flavor symmetries appearing in intersecting Dbrane models as well as magnetized D-brane models [13,15]. For concreteness, we consider IIA D6-brane models on That is, our setup is as follows. We consider N a stacks of D6branes, which lead to U(N a ) gauge symmetry, and they have winding numbers (n i a , m i a ) along the x i and y i directions on T 2 i , where we use orthogonal coordinates (x i , y i ) on T 2 i . When we denote the basis of one-cycles on T 2 i by [a i ] and [b i ], which correspond to the x i and y i directions, the three-cycle, along which this set of D6-brane winds, is represented by Here, we consider two sets of D-branes, one set is N a stacks of D6-branes and another is N b stacks of D6-branes. These lead to U(N a ) × U(N b ) gauge groups. Suppose that these two stacks of D6-branes intersect each other on T 2 i . Their intersecting number on T 2 i is obtained by and their total intersecting number on T 6 is obtained by Then, chiral matter fields with bi-fundamental representations (N a ,N b ) (1,−1) under U(N a )× U(N b ) appear at intersecting points on T 2 i , where the index (1, −1) denotes U(1) 2 charges inside U(N a ) and U(N b ). There appear I ab families of bi-fundamental matter fields. When I ab is negative, there appear |I ab | families of matter fields with the conjugate representation (N a , N b ) (−1,1) .
The total flavor symmetry is a direct product of flavor symmetries appearing on one of T 2 i . Thus, we concentrate on the flavor symmetry realized on one of T 2 i . Then, we denote I (i) ab = g. Theses modes on T 2 i have definite Z g charges and Z g transformation is represented by where ρ = e 2πi/g . In addition, there is a cyclic permutation symmetry Z (C) g among these modes, i.e.
Furthermore, these elements do not commute each other, Thus, this flavor symmetry includes another Z ′ g symmetry, which is represented by Then, these would generate the non-Abelian flavor symmetry, g . For example, when g = 2 and g = 3, the symmetries correspond to D 4 and ∆ (27). In addition, when the totally D-brane system has the Z 2 reflection symmetry P between i-th mode and (g − i)-th mode the ∆(27) symmetry for g = 3 is enhanced into ∆(54) [13].
Similarly, we can discuss models with more than two sets of D-branes. For example, suppose that we add N c stacks of D-branes to the above system, and that their intersecting numbers satisfy G.C.D.(I The above result is applicable to intersecting D-brane models on orientifolds through simple extension. Also, we can extend our discussions to orbifold cases [13,14,15]. Since magnetized D-brane models are T-duals to intersecting D-brane models, the magnetized D-brane models also have the same discrete flavor symmetries. For example, we start with (N a + N b ) stacks of D9-branes on T 6 . Then, we introduce the magnetic flux . The gaugino fields in the off-diagonal part correspond to the (N a ,N b ) bi-fundamental matter fields under the unbroken U(N a ) × U(N b ) gauge symmetry. Zero-modes with such representation appear in this model, and the number of zero-modes on as the above intersecting D-brane model.

Discrete anomalies
In this section, we study anomalies of discrete flavor symmetries.

U (1) anomalies
Before studying anomalies of discrete flavor symmetries, it is useful to review anomalies of U(1) gauge symmetries. In this subsection, we give a brief review on U(1) anomalies [9,36] (see also [12,5]). First of all, we consider the torus compactification. A D6-brane has a charge of RR 7-form C 7 . The total charge should vanish in a compact space. That leads to the following tadpole cancellation condition, The SU(N a ) 3 anomaly coefficient is calculated in the intersecting D-brane models by because there are I ab matter fields with (N a ,N b ) (1,−1) for I ab > 0 and |I ab | matter fields with (N a , N b ) (−1,1) for I ab < 0. However, the tadpole cancellation condition leads to That implies that A a = 0, that is, anomaly free. The U(1) a × SU(N b ) 2 mixed anomaly coefficient is obtained by This anomaly is not always vanishing. However, this anomaly can always be canceled by the Green-Schwarz mechanism, where an axion shifts under the U(1) gauge transformation and the anomalous U(1) gauge boson becomes massive. The U(1)-gravity 2 anomaly coefficient is obtained by This anomaly is always vanishing when the tadpole cancellation condition is satisfied. Next, we review on anomalies for the orientifold compactification. That is, we introduce O6-branes along the direction i [a i ]. The system must be symmetric under the Z 2 reflection, y i → −y i . In this case, we have to introduce a mirror D6 a ′ -branes with the winding number (n i a , −m i a ) corresponding to (n i a , m i a ). The O6-brane has (-4) times as RR charge as a D6-brane. Then, the RR-tadpole cancellation condition requires In addition to I ab families of (N a ,N b ) (1,−1) matter fields, there appear I ab ′ families of (N a , N b ) (1,1) matter fields. Moreover, there appear matter fields with symmetric and asymmetric representations under U(N a ) with charge 2. Their numbers are obtained by In this case, we can show that the SU(N a ) 3 anomaly coefficient always vanishes when the RR-tadpole cancellation condition is satisfied, similarly to in the torus compactification. Also, the U(1) a − SU(N b ) 2 anomaly coefficient is not always vanishing, but such anomaly can be canceled by the Green-Schwarz mechanism. Finally, the U(1) a −gravity 2 anomaly coefficient is obtained by This does not always vanish, but such anomaly can be canceled by the Green-Schwarz mechanism.

Discrete anomalies
In the gauge theory with the gauge group G and the Abelian discrete symmetry Z N , the Z N − G 2 mixed anomaly coefficient is calculated by [37,38,29,2], where the summation of m is taken over fermions with Z N charges q (m) and the representation R (m) under G. Here, T 2 (R (m) ) denotes the Dynkin index and we use the normalization such that T 2 = 1/2 for the fundamental representation of SU(N). When the following condition is satisfied [37,38,29,2], 12) the Z N symmetry is anomaly-free. Similarly, we can calculate the Z N -gravity 2 anomaly coefficient by Trq (m) . If Trq (m) = 0 (mod N/2), Z N is anomaly-free. For example, Z 2 symmetry is always anomaly-free. Each generator of non-Abelian discrete symmetries corresponds to an Abelian symmetry. Thus, if each Abelian generator of non-Abelian discrete flavor symmetry satisfies the above anomaly-free condition, the total non-Abelian symmetry is anomaly-free. When some discrete Abelian symmetries are anomalous, the total non-Abelian discrete symmetry is broken, and the subgroup, which does not include anomalous generators, remains unbroken.
In the non-Abelian discrete symmetry, there appear multiplets and each generator is represented by a matrix, M. When det M = 1, the corresponding Abelian discrete symmetry is always anomaly-free. Only multiplets with det M = 1 can contribute on anomalies. Since we have det Z ′ = 1, the corresponding Z ′ g symmetry is always anomalyfree. On the other hand, we find det Z = det C = 1 for g = odd and det Z = det C = −1 for g = even. That means that the discrete flavor symmetry (Z g × Z ′ g ) ⋊ Z (C) g is always anomaly-free for g = odd, but Z g and Z (C) g can be anomalous for g = even. In particular, their Z 2 parts are anomalous. One has to check the anomaly-free condition for such Z 2 part for Z g and Z (C) g . For example, the ∆(27) flavor symmetry for g = 3 is always anomaly-free. However, Z 2 subgroups of D 4 for g = 2 corresponding to the following elements, can be anomalous. First, we discuss the torus compactification. For simplicity, we concentrate on the flavor symmetry appearing the first torus T 2 1 and we assume that all of intersecting numbers on T 2 1 , I 1 ab , are even. Thus, the total flavor symmetry includes the Z 2 symmetry as well as Z (C) 2 , which can be anomalous. Also, we assume that there appears a trivial symmetry from the other T 2 2 × T 2 3 . Now, let us examine the Z 2 − SU(N a ) 2 anomaly. There are I ab bi-fundamental matter fields with the representation (N a ,N b ). A half of I ab matter fields have even Z 2 charge and the others have odd Z 2 charge. The anomaly coefficient of Z 2 − SU(N a ) 2 anomaly can be written by (3.14) It vanishes because the tadpole cancellation condition, b I ab N b = 0. Thus, this Z 2 symmetry is anomaly-free on the torus compactification. Since only this Z 2 symmetry can be anomalous and the others are always anomaly-free, the non-Abelian flavor symmetries are always anomaly-free in the torus compactification. Next, we study the orientifold compactification. Similarly, we can calculate the That is not always vanishing, but it is proportional to the U(1) a -grav 2 anomaly. Thus, this anomaly could be canceled when one requires the axion shift under the Z 2 transformation, which is related with the axion shift under U(1) a . In addition, when D6 a branes are parallel to the O6-branes, Z 2 − SU(N a ) 2 anomaly coefficient is always vanishing.

Majorana neutrino masses
In the previous section, we have studied on anomalies of discrete flavor symmetries. Certain symmetries are anomaly-free. For example, the ∆(27) flavor symmetry is anomalyfree. Anomalous symmetries can be broken by non-perturbative effects. There is no guarantee that anomaly-free symmetries are not broken by stringy non-perturbative effects.
In this section, we consider D-brane instanton effects as concrete non-perturbative effects.
We study which form of right-handed Majorana neutrino mass matrix can be generated by D-brane instanton effects. Indeed, following [31,32,33], we study the sneutrino mass matrix assuming that the neutrino mass matrix has the same form and supersymmetry breaking effects are small.

Neutrino mass matrix
Here, we study right-handed Majorana neutrino masses, which can be generated by Dbrane instanton effects. We assume that g families of right-handed neutrinos ν a R appear by intersections between D6 c -brane and D6 d branes, and that their intersecting numbers are equal to I  1, 1), where the underline denotes all the possible permutations. We consider D2-brane instanton, which wraps one-cycle of each T 2 of T 6 = T 2 × T 2 × T 2 . We call it D2 M -brane. It intersects with D6 c brane and D6 d brane. At these intersecting points, zero-modes α i and γ j appear and their numbers are obtained by I M c and I dM .
Only if there are two zero-modes for both α i and γ j the neutrino masses can be generated by D2-brane instanton effect [31,32,33], where the mass scale M would be determined by the string scale M st and the instanton world volume V as M = M st e −V . Here, d ij a is the 3-point coupling coefficient among α i , ν a R and γ j [19], which we show explicitly in the next subsection. The 3-point coupling coefficient d ij a can be written by d ij a = d ij a1 d ij a2 d ij a3 , where d ij ak for k = 1, 2, 3 is the contribution from the k-th torus. In addition, when α i , γ j , or ν a are localized at a single intersecting point on the k-th torus, we omit the indexes such as d j ak , d i ak , or d ij k . We have to take into account all of the possible D2 M -brane configurations, which can generate the above neutrino mass terms. One can obtain two zero-modes of α i and γ j for the D2 M -brane set corresponding to Sp(2) or U(2) gauge group with the intersecting numbers |I M c | = |I M d | = 1 [34] or a single D2 M -brane with the intersecting numbers, When the D2 M -brane set corresponds to the Sp(2) or U(2) brane, the zero-modes, α i and γ j , are doublets and the gauge invariance allows the certain couplings, say α i and γ i , but not α i and γ j for i = j. When I M c = I dM = 1, the following form of the Majorana mass is generated, More explicitly, the following form of mass matrix is obtained [34], (4.4) We will show this form by an explicit calculation in the next subsection. As a result, there remains the cyclic permutation symmetry, Z (C) g=3 , unbroken, but Z g=3 and Z ′ g=3 symmetries are broken by D-brane instanton effects, which generate the Majorana neutrino masses. This form also has the Z 2 reflection symmetry P . Thus, if the full D-brane system has the Z 2 reflection symmetry, the symmetry is enhanced into S 3 .
Similarly, we can study a single D2 M -brane with the intersecting numbers, |I M c | = |I M d | = 2. There are two types of D2 M -brane instanton configurations leading to |I M c | = |I M d | = 2. In one type, we have the configuration with |I However, this vanishes identically [33]. We obtain the same result for |I Also the mass matrix is symmetric, i.e. c ab = c ba . For example, we obtain for g = 2 and for g = 4. It is found that the D-brane instantons break Z ′ g into Z 2 if g is even. Otherwise, the Z ′ g symmetry as well as the Z g symmetry is completely broken. However, the cyclic permutation symmetry remains. 3 We have studied the neutrino mass matrix by assuming that the neutrino and sneutrino have the same mass matrix and supersymmetry breaking effect is small [31,32,33]. The important point to derive our result is the cyclic permutation symmetry. Thus, we would obtain the same result if the D-brane instatons do not break such a symmetry but supersymmetry is broken.

Explicit computation
Here, we discuss the Majorana neutrino mass matrix by computing explicitly the threegeneration models. We consider the D2-brane instanton corresponding to Sp(2) or U(2) gauge symmetry. Suppose that D6 c and D6 d branes have the intersecting number, I cd = 3, and at three intersecting points there appear three generations of right-handed neutrinos. We set (I There are three generations of ν a and we here label their flavor index as a = 0, 1, 2. Also there are two-zero modes, α i and γ j (i, j = 1, 2), but note that these indexes i, j correspond to the doublets under Sp(2) or U(2) and the intersecting numbers, I cM , and I M d , are equal to one, I cM = I M d = 1.
Suppose that there are three fields φ a , χ i ′ and χ j ′ with the "flavor numbers", a = 0, · · · , I cd − 1, i ′ = 0, · · · , I dM − 1, and j ′ = 0, · · · , I M c − 1, where I cd , I dM , and I M c are the corresponding intersecting numbers on the torus. In this case, the 3-point couplings d i ′ j ′ a among three fields can be calculated by [19] where C is a flavor-independent constant due to quantum contributions and and A denotes the area of the first torus. Here, ε denotes the position of D2 M -brane on the first torus and we normalize ε such that ε varies [0, 1] on the torus. Note that this coupling corresponds to the contribution on the first torus, which determines the flavor structure, but we have omitted the index corresponding to the first torus. By using the ϑ-function, we can write Our model corresponds to a = 0, 1, 2, I cd = 3, i ′ = j ′ = 0, I dM = I M c = 1. In the above model, the 3-point couplings among ν a , α i , and γ j are written by Recall again that the indexes i and j of α i and γ j are doublet indexes under Sp(2) or U (2). Using this, the matrix c ab is written by the integration of the position ε over [0, 1], We obtain Using it, the matrix elements c ab can be computed as follows. It is found that the diagonal elements c aa do not depend on a and they are written by Similarly, the off-diagonal elements are written by However, we have the following formula of the ϑ-function Then, we see that all of the off-diagonal elements are the same, That is, we can realize the form (4.4) by explicit calculations. Figure 1 shows the ratio B/A = c 12 /c aa in (4.4) by varying the area 3A/2π 2 α ′ .

Phenomenological implication
Here we discuss phenomenological implication of our result. The Majorana mass matrix with the form (4.4) can be diagonalized by the following matrix, where c = cos θ and s = sin θ, and the eigenvalues are A − B, A + 2B and A − B. That is, two eigenvalues are degenerate. This is because the mass matrix (4.4) has the additional Z 2 reflection symmetry P and the symmetry is enhanced into S 3 . At any rate, this form of the mixing matrix is interesting, although the mass eigenvalues may be not completely realistic.
Suppose that the Dirac neutrino Yukawa couplings and charged lepton mass matrix are almost diagonal. 4 Then, the lepton mixing matrix is obtained as the above matrix (4.23). That is the trimaximal matrix.
When s = 0, the above matrix becomes the tri-bimaximal mixing matrix. In fieldtheoretical model building, the tri-bimaximal mixing matrix can be obtained as follows [1,2,3]. We start with a larger flavor symmetry and break by vacuum expectation values of scalar fields. However, one assumes that Z 3 and Z 2 subsymmetries remain in the charged lepton or neutrino mass terms. Then, the tri-bimaximal mixing matrix can be realized. In our string theory, such a Z 3 symmetry is realized by geometrical symmetry of the cyclic permutation Z C 3 , which can not be broken by the D-brane instanton effects, although other symmetries are broken.
We may need some corrections to realize the experimental values of neutrino masses. 5 At least, the above results show that we can realize non-trivial mixing in the lepton sector even though our assumption above the Dirac masses can not be realized.

Conclusion and discussion
We have studied the flavor structure in intersecting D-brane models. We have discussed the anomalies of flavor symmetries. Certain symmetries are anomaly-free, and anomaly coefficients of discrete symmetries have the specific feature. We have studied the Majorana neutrino masses, which can be generated by D-brane instanton effects. It is found that the mass matrix form with the cyclic permutation symmetry can be realized by integrating over the position of D-brane instanton. That would lead to the interesting form of mixing angles. It is interesting to apply our results for more concrete models. We would study numerical analyses elsewhere.
In some models, there appear more than one pair of Higgs fields. Their masses would be generated by D-brane instanton effects. It would be important to study the form of such Higgs mass matrix. Also, some of Yukawa couplings may be generated by D-brane instanton effects. Thus, it would be important to extend our analysis to Higgs mass matrix and Yukawa matrices.