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Physics Letters B

www.elsevier.com/locate/physletb

Gauge origin of discrete flavor symmetries in heterotic orbifolds

Florian Beyea, Tatsuo Kobayashib, Shogo Kuwakinoc*

a Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan b Department of Physics, Hokkaido University, Sapporo 060-0810, Japan

c Department of Physics, Chung-Yuan Christian University, 200 Chung-Pei Rd., Chung-Li 320, Taiwan

CrossMark

A R T I C L E I N F 0

Article history:

Received 1 July 2014

Received in revised form 28 July 2014

Accepted 29 July 2014

Available online 4 August 2014

Editor: M. cvetic

A B S T R A C T

We show that non-Abelian discrete symmetries in orbifold string models have a gauge origin. This can be understood when looking at the vicinity of a symmetry enhanced point in moduli space. At such an enhanced point, orbifold fixed points are characterized by an enhanced gauge symmetry. This gauge symmetry can be broken to a discrete subgroup by a nontrivial vacuum expectation value of the Kahler modulus T. Using this mechanism it is shown that the A(54) non-Abelian discrete symmetry group originates from a SU(3) gauge symmetry, whereas the D4 symmetry group is obtained from a SU(2) gauge symmetry.

© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

1. Introduction

It is important to understand the flavor structure of the standard model of particle physics. Quark and lepton masses are hierarchical. Two of the mixing angles in the lepton sector are large, while the mixing angles in the quark sector are suppressed, except for the Cabibbo angle. Non-Abelian discrete flavor symmetries may be useful to understand this flavor structure. Indeed, many works have considered field-theoretical model building with various non-Abelian discrete flavor symmetries (see [1-3] for reviews).

Understanding the origin of non-Abelian flavor symmetries is an important issue we have to address. It is known that several phenomenologically interesting non-Abelian discrete symmetries can be derived from string models.1 In intersecting and magnetized D-brane models, the non-Abelian discrete symmetries D4, A(27) and A(54) can be realized [5-8]. Also, their gauge origins have been studied [6]. In heterotic orbifold compactifications [9-17] (also see a review [18]), non-Abelian discrete symmetries appear due to geometrical properties of orbifold fixed points and certain properties of closed string interactions [19]. First, there are permutation symmetries of orbifold fixed points. Then, there are string selection rules which determine interactions between orb-ifold sectors. The combination of these two kinds of discrete sym-

* Corresponding author.

E-mail addresses: fbeye@eken.phys.nagoya-u.ac.jp (F. Beye), kobayashi@particle.sci.hokudai.ac.jp (T. Kobayashi), kuwakino@cycu.edu.tw (S. Kuwakino).

1 In [4], field theoretical models where non-Abelian discrete groups are embed-

ded into non-Abelian gauge groups are considered.

metries leads to a non-Abelian discrete symmetry. In particular, it is known that the D4 group emerges from the one-dimensional orbifold S1/Z2, and that the A(54) group is obtained from the two-dimensional orbifold T2 / Z3. The phenomenological applications of the string-derived non-Abelian discrete symmetries are analyzed e.g. in [20].

In this paper we point out that these non-Abelian discrete flavor symmetries originate from a gauge symmetry. To see this, we consider a heterotic orbifold model compactified on some six-dimensional orbifold. The gauge symmetry Ggauge of this orbifold model is, if we do not turn on any Wilson lines, a subgroup of E8 x E8 which survives the orbifold projection. In addition, from the argument in [19], we can derive a non-Abelian discrete symmetry G discrete- Then, the effective action of this model can be derived from Ggauge x Gdiscrete symmetry invariance.2 However, this situation slightly changes if we set the model to be at a symmetry enhanced point in moduli space. At that special point, the gauge symmetry of the model is enlarged to Ggauge x Genhanced, where Genhanced is a gauge symmetry group. The maximal rank of the enhanced gauge symmetry Genhanced is six, because we compactify six internal dimensions. At this specific point in moduli space, orbifold fixed points are characterized by gauge charges of G enhanced, and the spectrum is extended by additional massless fields charged under Genhanced- Furthermore, the Kahler moduli fields T in the untwisted sector obtain Genhanced-charges and a non-zero vacuum expectation value (VEV) of T corresponds to a movement away

2 Here we do not consider the ft-charge invariance since this is not relevant to our discussion.

http://dx.doi.org/10.1016Zj.physletb.2014.07.058

0370-2693/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

from the enhanced point. This argument suggests the possibility that the non-Abelian discrete symmetry Gdiscrete is enlarged to a continuous gauge symmetry Genhanced at the symmetry enhanced point. In other words, it suggests a gauge origin of the non-Abelian discrete symmetry. Moreover, the group Genhanced originates from a larger non-Abelian gauge symmetry that exists before the orb-ifolding. We will show this explicitly in the following.

2. Gauge origin of non-Abelian discrete symmetry

Table 1

Field contents of U (1) * Z2 model from Z2 orbifold. U (1) charges are shown. Charges under the Z4 unbroken subgroup of the U (1) group are also shown.

Sector Field U (1) charge Z4 charge

U U 0 0

U U1 a 0

U U 2 —a 0

T M1 a 4 1 4

T M2 a 4 1 4

In this section we demonstrate the gauge origin of non-Abelian discrete symmetries in heterotic orbifold models. We concentrate on the phenomenologically interesting non-Abelian discrete symmetries D4 and A(54) which are known to arise from orbifold models.

2.1. D4 non-Abelian discrete symmetry

First, we study a possible gauge origin of the D4 non-Abelian discrete symmetry. This symmetry is associated with the one-dimensional S1/Z2 orbifold. Here, we consider the heterotic string on a S1 /Z2 orbifold, but it is straightforward to extend our argument to T2/Z2 or T6/(Z2 x Z2). The coordinate corresponding to the one dimension of S1 is denoted by X. It suffices to discuss only the left-movers in order to develop our argument. Let us start with the discussion on S1 without the Z2 orbifold. There is always a U(1) symmetry associated with the current H = idX. At a specific point in the moduli space, i.e. at a certain radius of S1, two other massless vector bosons appear and the gauge symmetry is enhanced from U(1) to SU(2). Their currents are written as

E± = e

±ia X

where a = 42 is a simple root of the SU(2) group. These currents, H and E±, satisfy the su(2) Kac-Moody algebra.

Now, let us study the Z2 orbifolding X ^ —X. The current H = idX is not invariant under this reflection and the corresponding U (1) symmetry is broken. However, the linear combination E++ E— is Z2-invariant and the corresponding U(1) symmetry remains on S1/Z2. Thus, the SU(2) group breaks down to U(1) by orbifolding. Note that the rank is not reduced by this kind of orb-ifolding. It is convenient to use the following basis,

H ' = id X ' = -= ( E+ + E —), V 2

E±= e±iaX' = -= H t 1 (E + — E—).

(2) (3)

The introduction of the boson field X' is justified because H' and E± satisfy the same operator product expansions (OPEs) as the original currents H and E±. The invariant current H' corresponds to the U (1) gauge boson. The E± transform as

under the Z2 reflection and correspond to untwisted matter fields Ui and U2 with U(1) charges ±a. In addition, there are other untwisted matter fields U which have vanishing U (1) charge, but are charged under an unbroken subgroup of E8 x E8.

From (4), it turns out that the Z2 reflection is represented by a shift action in the X' coordinate,

XX' + 2n—, 2

where w = 1/V2 is the fundamental weight of SU(2). That is, the Z2-twisted orbifold on X is equivalent to a Z2-shifted orbifold on

X' with the shift vector s = w/2 (see e.g., [21]). In the twist representation, there are two fixed points on the Z2 orbifold, to each of which corresponds a twisted state. Note that the one-dimensional bosonic string X with the Z2-twisted boundary condition has a contribution of h = 1/16 to the conformal dimension. In the shift representation, the two twisted states can be understood as follows. Before the shifting, X' also represents a coordinate on S1 at the enhanced point, so the left-mover momenta pL lie on the momentum lattice

rSU(2) u (rsU(2) + w),

where rSU(2) is the SU(2) root lattice, rSU(2) = na with integer n. Then, the left-mover momenta in the Z2-shifted sector lie on the original momentum lattice shifted by the shift vector s = w/2, i.e.

rSU(2) + 2 J U [PSU(2) + ~Y

Thus, the shifted vacuum is degenerate and the ground states have momenta pL = ±a/4. These states correspond to charged matter fields M1 and M2. Note that p2L/2 = 1/16, which is exactly the same as the conformal dimension h = 1 /16 of the twisted vacuum in the twist representation. Indeed, the twisted states can be related to the shifted states by a change of basis [21]. Notice that the twisted states have no definite U (1) charge, but the shifted states do. Table 1 shows corresponding matter fields and their U (1) charges.

From Table 1, we find that there is an additional Z2 symmetry of the matter contents at the lowest mass level (in a complete model, these can correspond to massless states): Transforming the U(1)-charges q as

q ^ —q,

while at the same time permuting the fields as U1 ^ U2 and M1 ^ M2 maps the spectrum onto itself. The action on the Ui and Mi fields is described by the 2 x 2 matrix

0 1 1 0

This Z2 symmetry does not commute with the U (1) gauge symmetry and it turns out that one obtains a symmetry of semi-direct product structure, U (1) x Z2.

In the twist representation, this model contains the Kahler modulus field T, which corresponds to the current H and is charged under the U (1) group. In the shift representation, the field T is described by the fields Ui as

T = -L (U1 + U 2).

Now we consider the situation where our orbifold moves away from the enhanced point by taking a specific VEV of the Kahler modulus field T which corresponds to the VEV direction

(U 1> = <U 2 >.

Note that this VEV relation maintains the Z2 discrete symmetry (9). Moreover, since the fields U1 and U2 are charged under the U (1) gauge symmetry and due to the presence of the Mi fields, the VEV breaks U (1) down to a discrete subgroup Z4. The Z4 charge is 1/4 for M1 and -1/4 for M2 as listed in Table 1. Written as a 2 x 2 matrix, the Z4 action is described by

i 0 0 -i

The matrices (9) and (12) are nothing but the generators of D4 ~ Z4 x Z2. After the VEV, the field U transforms as the trivial singlet 1, and (M1, M2) forms a 2 representation under the D4 group. This reproduces the known result for a general radius of S1 [19]. The pattern of symmetry breaking we have shown here is summarized as follows:

U(1) * Z2 -- D4.

orbifolding " yLJ "2 (T)' The other VEV directions of U1 and U2 break U (1) x Z2 to Z4. However, while the VEV direction defined by Eq. (11) is D-flat, the other cases do not correspond to D-flat directions and the resulting symmetries have no geometrical interpretation.

2.2. A(54) non-Abelian discrete symmetry

Next, we consider the two-dimensional T2/Z3 orbifold case which is associated with the A(54) non-Abelian discrete symmetry. Here, we study the heterotic string on a T2/Z3 orbifold. However, our argument straightforwardly extends to orbifolds such as T6/Z3. The coordinates on T2 are denoted by X1 and X2. We start with the discussion of the two-dimensional torus, T2, without orbifolding. There is always a U(1)2 symmetry corresponding to the two currents, H1 = id X1 and H 2 = id X2. At a certain point in the moduli space of T2, there appear additional six massless gauge bosons. Then, the gauge symmetry is enhanced from U(1)2 to SU(3). The corresponding Kac-Moody currents are

£±1,0, with

E 0,±1,

±1,±1,

= eŒi=1,2(n1«i +n2«2)X' ,

where a1 and a2 denote simple roots of SU(3), i.e. a1 = (V2, 0) and a2 = (-\/2/2, V6/2). These currents, Hi and En1,n2, satisfy the su(3) Kac-Moody algebra.

Now, let us study the Z3 orbifolding,

Z ^ m—1 Z, (16)

where Z = X1 + iX2 and m = e2ni/3. The currents Hi and their linear combinations are not Z3-invariant and the corresponding gauge symmetries are broken. On the other hand, two independent linear combinations of En1,n2 are Z3-invariant and correspond to a U(1)2 symmetry that remains on the T2/Z3 orbifold. Thus, the SU(3) gauge group is broken to U(1)2 by the orbifolding. It is convenient to use the following basis,

* 1 = E - ,

H 2 = --12(E1 + E2),

E1,0 = — (iH«- + E + E)

E0,1 = — (iH«- + «E + «-1E),

E-1,-1 = ^ (iH«- + «-1E+ «E «-

E-!,0 = — ^-iHw + E« + E«'

E 0,-1 = — (-iH« + « E« + «-1E

E i,1 = — (-iHi +1-1 E « +1E l )

H«-1 = -= (H1 + i*2), H« = —= (H1 - iH2),

(21) (22)

E = -= (E10 + «kE 0,1 + «-kE-1,-^, E = -—= ( E-1,0 + «kE 0,-1 + «-kE1,1).

The Enj n2 correspond to states with charges (n1aj + n1af + n2a^) under the unbroken U(1)2. They transform under the Z3 twist action as follows:

E-1,0 ^ «E-

E 1,1 ^ «E1,V

E0,-1 ^ «E0,-l,

E 1,0 ^ « 1E 1,0,

E 0 1 ^ « 1 E;

^ «-1E

-1,-1 .

Thus, the first three E'n 1 n2 correspond to untwisted matter fields with charges —a1, —a2 and a1 + a2 under the unbroken U(1)2. We denote them as U1 U2 and U3, respectively. The other three are their CPT conjugates. In addition, there are other untwisted matter fields U which have vanishing U(1)2 charges, but are charged under an unbroken subgroup of E8 x E8.

Now, since the primed currents fulfill the same OPEs as their unprimed counterparts, it is justified to introduce bosons X'i, so that

H/i = id X'i

p' — eiEi=12 (n 1 a i +n2«2 )X"

n1 n2 =

The Z3 twist action on Xi can then be realized as a shift action on X'i as

X'i ^ X/i + 2n-31.

In the twist representation there are three fixed points on the T2/Z3 orbifold, to each of which corresponds a twisted state. The two-dimensional bosonic string with the Z3 boundary condition has a contribution of h = 1/9 to the conformal dimension. As in the previous one-dimensional case, the twisted states can be described in the shift representation as follows. The left-moving momentum modes pL of the torus-compactified SU(3) model lie on the momentum lattice

rsu(3) U (rsu(3) + w 1 ) U (rsu(3) - W1),

where rSU(3) denotes the SU(3) root lattice which is spanned by the simple roots of SU(3), rSU(3) = n1a1 + n2a2, and w1 = (V2/2, Ve/6) is the fundamental weight corresponding to a1. Then, the momenta pL in the fc-shifted sector lie on the momentum lattice shifted by the Z3 shift vector s = a1/3,

Table 2

Field contents of U (1)2 x S3 model from Z3 orbifold. U (1)2 charges are shown. Charges under the Z3 unbroken subgroup of the U (1)2 group are also shown.

<U 1> = <U 2 > = <U 3>.

Sector Field U (1)2 charge Z32 charge

U U (0, 0) (0,0)

U U1 —a (0, 0)

U U2 —a 2 (0, 0)

U U3 a1 + a.2 (0, 0)

T M1 a1 3 ( 1 1 )

T M2 a2 (— 3, 0)

T M3 «1 +a2 3 (0,— 3 )

rsu(3) + k— U rsu(3) + w i + k

U I rsu(3) - w i + k-3-

For k = 1, there are three ground states with pL e {ai/3, a2/3, —(ai + a2)/3}. They correspond to (would-be-massless) matter fields which we denote by M1, M2 and M3, respectively. These matter fields are shown in Table 2. The states for k =—1 correspond to CPT-conjugates. As expected, the shifted ground states have conformal dimension h = p2/2 = 1/9, which coincides with the twisted ground states. Indeed, the shifted states are related to the twisted states by a change of basis [21]. The shifted states have definite U(1)2 charges.

From Table 2, it turns out that the matter contents at the lowest mass level possess a S3 permutation symmetry (in a complete model, these can correspond to massless states). Let S3 be generated by a and b, with a3 = b2 = (ab)2 = 1. Then, for a point (q1t q2) on the two-dimensional U(1)2 charge plane, a and b shall act as

1 0 0 - 1

The action of a is equivalent to the replacement a1 ? a2 ? —(a1 + a2) ? a1. Then, the spectrum is left invariant if at the same time we transform the fields Fi = (Ui, Mi) as F1 ? F2 ? F3 ? F1. The action of a on the Fi is described by the 3 x 3 matrix

'0 0 1\ 1 0 0 ,0 1 0/

The action of b corresponds to a1 ^ a1 and a2 ^ — (a1 + a2), so simultaneously transforming F1 ^ F1 and F2 ^ F3 results in a symmetry of the spectrum. This transformation corresponds to the matrix

'1 0 0\ 0 0 1 ,0 1 0/

The S3 symmetry just shown does not commute with U(1)2. Rather, S3 and U(1)2 combine to semi-direct product U(1)2 x S3.

Next we shall consider the situation where our orbifold moves away from the enhanced point by taking a certain VEV of the Kahler modulus field T, which corresponds to H m. The Kahler modulus can be described by the Uj fields as

T = -L (U1 + U 2 + U 3).

The deformation is realized by the following VEV direction,

Note that this VEV relation preserves the S3 discrete symmetry generated by (36) and (37). However, the U(1)2 gauge symmetry breaks down to a discrete Z| subgroup due to the presence of the Mi fields. The two Z3 charges (z1, z2) are determined by U(1)2 charges (U1,u2) as Z1 = q1/V2 — q2/V6, Z2 = q1/V2 + q2/V6. The Z32 charges are listed in Table 2. The Z3 actions are described by

The matrices (36), (37), (40) and (41) are nothing but the generators of A(54) ~ (Z3 x Z3) x S3 in the 31(1) representation [22]. Thus, the fields (M1, M2, M3) transform as the 31(1) under A(54), and the field U is the A(54) trivial singlet 1. This reproduces the known properties of ordinary Z3 orbifold models at a general point in moduli space [19]. Summarizing, the origin of the A(54) discrete symmetry in orbifold models can be explained as follows:

O) 0 0

0 o—1 0

0 0 o—1

? U (1)2 x S3

A(54).

orbifolding v ' 3 ——) There are other VEV directions that one might consider. For (U1) = <U2) = (U3) = 0 the U(1)2 x S3 symmetry is broken to (U(1) x Z2) x Z6. In the case where (U1) = <U2) = <U3) = 0 one obtains Z3 x S3. Finally, when all VEVs are different, i.e. (U1) = (U2) = (U3) = <U 1) the symmetry is broken to Z3 x Z3. However, while the VEV direction defined by (39) is D-flat, the other directions are not D-flat and do not allow for a geometrical interpretation.

3. Conclusion

We showed that non-Abelian discrete symmetries in heterotic orbifold models originate from a non-Abelian continuous gauge symmetry. The non-Abelian continuous gauge symmetry arises from torus-compactified extra dimensions at a special enhanced point in moduli space. In the two-dimensional orbifold case, by acting with Z3 on the torus-compactified SU(3) model, the non-Abelian gauge group SU(3) is broken to a U(1)2 subgroup. We observed that the matter contents of the orbifold model possess a S3 symmetry which is understood to act on the two-dimensional U(1)2 charge plane. The resulting orbifold model then has a symmetry of semi-direct product structure, U(1)2 x S3. In the untwisted sector, the orbifold model contains a Kahler modulus field which is charged under the unbroken Abelian gauge group. By assigning a VEV to the charged Kahler modulus field, the orbifold moves away from the enhanced point and the U(1)2 gauge symmetry breaks to a discrete Z32 subgroup. Thus, effectively the non-Abelian discrete symmetry A(54) ~ (Z3 x Z3) x S3 is realized. The other VEV directions of the untwisted scalar fields break the symmetry to (U(1) x Z2) x Z6, Z3 x S3 or Z3 x Z3. In the one-dimensional Z2 orbifold case, we showed that the non-Abelian gauge symmetry SU(2) is the origin of the discrete symmetry D4 ~ Z4 x Z2. The other VEV directions of the untwisted scalar fields break the symmetry to Z4.

The resulting non-Abelian discrete flavor symmetries are exactly those that have been obtained from heterotic string theory on symmetric orbifolds at a general point in moduli space [19]. In [19], the geometrical symmetries of orbifolds were used to derive these discrete flavor symmetries. However, in this paper, we have not used these geometrical symmetries on the surface, although obviously the gauge symmetries and geometrical symmetries are

—»■

—>■

tightly related with each other. At any rate, our results also indicate a procedure to derive non-Abelian discrete symmetries for models where there is no clear geometrical picture to begin with, such as in asymmetric orbifold models [23-26] or Gepner models

We give a comment on anomalies. Anomalies of non-Abelian discrete symmetries are an important issue to consider (see e.g.

[28]). We start with a non-Abelian (continuous) gauge symmetry and break it by orbifolding and by moduli VEVs to a non-Abelian discrete symmetry. The original non-Abelian (continuous) gauge symmetry is anomaly-free and if it were broken by the Higgs mechanism, the remaining symmetry would also be anomaly-free. That is because only pairs vector-like under the unbroken symmetry gain mass terms. But this does not hold true for orbifold breaking, as it is possible to project out chiral matter fields. Thus, in our approach the anomalies of the resulting non-Abelian discrete symmetries are a priori nontrivial. However, in our mechanism we obtain semi-direct product structures such as U(l)2 x S3. Since the corresponding U(l)2 is broken by the Higgs mechanism, the remnant Z| symmetry is expected to be anomaly-free if the original U(l)2 is anomaly-free (the semi-direct product structure automatically ensures cancellation of U(l)-gravity-gravity anomalies, but other anomalies have to be checked). Thus, the only discrete anomalies that remain to be considered are those involving S3.

We also comment on applications of our mechanism to phe-nomenological model building. In our construction the non-Abelian gauge group is broken by the orbifold action. This situation could be realized in the framework of field-theoretical higher-dimensional gauge theory with orbifold boundary conditions. Furthermore, our mechanism indicates that U (1)m x Sn or U (1)m x Zn gauge theory can be regarded as a UV completion of non-Abelian discrete symmetries.3 Thus, it may be possible to embed other phenomenologically interesting non-Abelian discrete symmetries into such a gauge theory and investigate their phenomenological properties.

Acknowledgements

F.B. was supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan (No. 23104011). T.K. was supported in part by the Grant-in-Aid for Scientific Research No. 25400252 from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. S.K. was supported by the National Science Council Taiwan under grant NSC102-2811-M-033-008.

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