R-parity from string compactification

In this paper, we embed the $\Z_{4R}$ parity as a discrete subgroup of a global symmetry \UoR\,obtained from $\Z_{12-I}$ compactification of heterotic string $\EE8$. A part of \UoR\,transformation is the shift of the anticommuting variable $\vartheta$ to $e^{i\alpha}\vartheta$ which necessarily incoorporate the transformation of internal space coordinate. Out of six internal spaces, we identify three U(1)'s whose charges are denoted as $Q_{18},Q_{20}$, and $Q_{22}$. The \UoR~is defined as \UEE$\times$\UKK~where \UEE~is the part from $\EE8$ and \UKK~is the part generated by $Q_{18},Q_{20}$, and $Q_{22}$. We propose a method to define a \UoR~direction. The needed vacuum expectation values for breaking gauge U(1)'s except U(1)$_Y$ of the standard model carry \UoR~charge 4 modulo 4 such that \UoR~is broken down to $\Z_{4R}$ at the grand unification scale. $\Z_{4R}$ is broken to $\Z_{2R}$ between the intermediate ($\sim 10^{11\,}\gev$) and the electroweak scales ($100\,\gev\sim 1\,\tev$). The conditions we impose are proton longevity, a large top quark mass, and acceptable magnitudes for the $\mu$ term and neutrino masses.


I. INTRODUCTION
Grand unified theories (GUTs) attracted a great deal of attention ethetically because they provided unification of gauge couplings and charge quantization [1][2][3]. But there seems to be a fundamental reason leading to GUTs even at the standard model (SM) level. With the electromagnetic and charged currents (CCs), the leptons need representations which are a doublet or bigger. A left-handed (L-handed) lepton doublet (ν e , e) alone is not free of gauge anomalies because the observed electromagnetic charges are not ± 1 2 . The anomalies from the fractional electromagnetic charges of the u and d quarks are needed to make the total anomaly to vanish [4,5]. In view of this necessity for jointly using both leptons and quarks to cancel gauge anomalies even in the SM, we can view that unification of leptons and quarks is fundamentally needed in addition to the esthetic view.
In the SM, the largest number of parameters arises from the Yukawa couplings which form the bases of the family structure. Repetition of fermion families in 4-dimensional (4D) field theory or family-unified GUT (family-GUT) was formulated by Georgi [6], requiring un-repeated chiral representations while not allowing gauge anomalies. Some interesting family-GUT models are the spinor representation of SO (14) [7,8] and 84 ⊕ 9 · 9 of SU(9) [9]. 1 While Refs. [7][8][9] do not provide interesting non-vanishing flavor quantum number, the SU(11) model [6] allows a possibility for non-vanishing flavor quantum number such as U(1) µ−τ or U(1) B−L [10].
On the other hand, the standard-like models from string have been the main focus of phenomenological activities for the ultraviolet completion of the SM toward the minimal supersymmetric standard model (MSSM) in the last several decades . These models use the chiral specrum from the level-1 construction which leads to unification of gauge couplings [36]. So, the standard-like models from string compactification achieved the goal of gauge coupling unification and GUT theories from string have not attracted much attention. Nevertheless, GUTs from strings [37,38] have been discussed sporadically for anti-SU(5) [39] (or flipped SU(5) [40]), dynamical symmetry breaking [41,42], and family unification [43]. In fact, family-GUTs are much easier in discussing the family problem, in particular on the origin of the mixing between quarks/leptons because the number of representations in family-GUTs is generally much smaller than in their (standard-like model) subgroups.
In this paper we study the R-parity assignment for a family-GUT from string compactification. So far, most string compactification models used the E 8 × E ′ 8 heterotic string in which a GUT with rank greater than 8 is impossible. The group SU (11) has rank 10 which cannot arise from compactification of E 8 × E ′ 8 . Therefore, firstly we fomulate the orbifold compactification [45,46] of SO(32) heterotic string [11] whose rank is 16. From the compactification of the SO(32) heterotic string, however, we cannot derive the SU(11) model and the largest possible non-abelian gauge group we obtain is SU(9) [44].
To discuss the R-parity from string compactification, we should consider a specific model. Among compactification schemes, we adopt the orbifold method. Among 13 possibilities listed in Ref. [45,46], we employ Z 12−I orbifold because it has the simplest twisted sectors. Twisted sectors are distinguished by Wilson lines [47]. The Wilson line in Z 12−I distinguishes three fixed points at a twisted sector. Therefore, it suffices to consider only three cases at a twisted sector. In all the other orbifolds of Ref. [45], consideration of various possibilities of Wilson lines and the accompanying consistency conditions are much more involved.
In Sec. II, we recapitulate the orbifold methods used in this paper for an easy reference to Sec. III. In Sec. III, we construct a specific specific supersymmetric model possessing nonabelian groups SU(9) and SU (5). In Sec. IV, we assign Z 4R quantum numbers to the massless fields obtained in Sec. III. Section V is a conclusion.

II. ORBIFOLD COMPACTIFICATION
A. Tensor representations of SU(N ) and SO(2N ) spinors where we defined differently for SU (9) and SU (5). It is a matter of convention to call what are 9 and 5. Here, the first (second) ones are called one-(two-)index tensors. The second ones carry two non-zero numbers. For spinor types, we also use this convection, but spinors fill all the 9 (for SU(9)) and 5 (for SU(5)) slots such as (+ + − − · · ·). One index spinor is defined to have one sign and the rest the opposite sign. In string compactification, there appear only up to two index tensors. Define 36 of the spinor type in SU (9) such that 36 · 9 · 9 is allowed if two times the sum of entries is 0 modulo 9.
Counting the dimension of SO(32) spinor, we note that both even and odd numbers of + signs should be included. This is in contrast to the E 8 spinor where only even numbers of + signs are considered.

B. Two dimensional orbifolds
In compactifying 10D string theory to 4D effective field theory, the small internal 6D is a compactified three two-tori. So, the basic is a two dimensional torus. A two dimensional orbifold is this two dimensional torus modded by discrete groups for which we adopt the discrete group Z 12−I . The shift vector we use here is similar to that of [44] but not the same, and the spectrum we obtain is a bit simpler. Since the strategy of compactifying SO(32) heterotic string is already presented there, we list key formulae used in the next chapter in Appendix A.
Most interesting spectra in the present paper are arising in the twisted sectors. Geometrically, twisted sectors correspond to fixed points. The moding vector of Z 12−I is φ s [45,46], φ s = 1 12 (5, 4, 1).
The multiplicities in the fixed points are three because φ s contains 1 3 in the second torus. This multiplicity 3 can be distinguished by Wilson lines a 3 = a 4 (3, 4 denoting two directions in the second torus) [47]. Entries of a 3 are some integer multiples of 1 3 . But 3a 3 contains only integer entries, that leads to some conditions at twisted sectors T 3 , T 6 and T 9 .
There is one point to be noted for Z 12 . If N =even, the k = 1, · · · , N 2 − 1 sectors provide the opposite chiralities in the k = N − 1, · · · , N 2 + 1 sectors, Then, the corresponding phases of Eq. (A3) compare as whose difference is e 2πi(N −2k)/12 . Thus, if 2k = N then T N −k do not provide the charge conjugated fields of T k , but they are identical. For T 3 or T 9 , therefore, we must provide the additional charge conjugated fields with an extra phase e 2πi(10/12) = e 2πi(−2/12) , the difference ofŝ · φ s forŝ = (+ + +) of R and (− − −) of L. We choose T 9 for this phase. In T 6 , however, we do not need this since the charged conjugated fields also appear there.

C. Multiplicities in the twisted sectors
In the compactification of the SO(32) heterotic string, spinors in U are not appearing because it is not possible to have P 2 = 2 from sixteen ± 1 2 's. Only vector types are possible in U . The T k twisted sector has three possibilities We select only the even lattices shifted from the untwisted lattices, therefore, we consider even numbers for the sum of absolute value of each element of P . They should be even numbers if the absolute values are added. In Appendix A, Θ k in Eq. (A3) is defined for the twisted sector T k . Since different P 's in the same twisted sector may lead to different gauge group representations, there is a need to distinguish them. So, we may use where

III. THE MODEL
The left-hand side (LHS) sector of the heterotic string is the gauge sector. The shift vector V 0 and Wilson line a 3 are restricted to satisfy the Z 12−I orbifold conditions, Here, a 3 (= a 4 ) is chosen to allow and/or forbid some spectra, and is composed of fractional numbers with the integer multiples of 1 3 because the second torus has the Z 3 symmetry. The model is where The right-hand side (RHS) sector of the heterotic string is given by the spin lattice s = (⊖ or ⊕;ŝ) with every entry being integer multiples of 1 2 , satisfying s 2 = 2. Theφ s for Z 12−I with three entries is Eq. (5), A. Untwisted sector U In U , we find the following nonvanishing roots of SU(5)×SU(9) ′ ×U(1) 4 , SU(5) gauge multiplet : P · V = 0 mod. integer and P · a 3 = 0 mod. integer SU (5) : SU(9) ′ gauge multiplet : In addition, there exists U(1) 4 symmetry. The non-singlet matter fields satisfy SU(5) and/or SU(9) ′ matter multiplet : P · V = 1, 4, 5 12 , P · a 3 = 0 mod. integer.
The conditions (16) allows the P 2 = 2 lattice shown in Table I.
• Two index vector-form from T + 7 : gives (P 9 + 7V + ) 2 = 186 144 which is short of 2 12 from the target value 210 144 , and massless fields are shown in Table XII.     • Two index spinor-form from T − 7 : The vector gives (P 5 + 7V − ) 2 = 186 144 which is short of 2 12 from the target value 210 144 , and massless fields are shown in Table XIII.
• One index spinor-form from T − 7 : The vector gives (P 5 + 7V − ) 2 = 138 144 which is short of 6 12 from the target value 210 144 , and massless fields are shown in Table XIV.  Note that k 2 (V 2 a − φ 2 s ) in Eq. (A3) is not distinguished by Wilson lines, and we just calculate the spectra from 6V 0 with multiplicity 3 (for T 0,+,− 6 ). With 3 and 0 5 in (25), we obtain only vectorlike representations. To clarify, we list all the possibilities for 5, 5, 10, and 10. All the allowed ones are spinor forms.

A. One pair of Higgs quintets
The Higgs quintets are α and β. Twisted sector T − 1 of α and β gives multiplicity 3. Thus, three objects at these fixed points must have permutation symmetry S 3 , the largst discrete symmetry of three objects. From three permuting elements of S 3 , {x 1 , x 2 , x 3 }, the S 3 singlet is formed as and the S 3 doublet becomes where ω is the complex cube root of unity. The tensor product of two S 3 doublets, (x 1 , x 2 ) and (y 1 , y 2 ), is [54], The Higgsino mass terms, constructed with the α and β must be an S 3 singlet 1, but not 1 ′ . With this condition, we try to estimate the unmatched pairs of 5 hR and 5 hR , i.e. the number of massless Higgsino pairs. Note that SU(9) ′ confines at a high energy scale, and we consider the condensation 36 ′ · 9 ′ · 9 ′ . The Q R charges of the hidden sector fields given by Eq. (52) are The condensation we consider is the scalar component of 36 ′ and the fermion components of 9 ′ . Then, the Q R charge of C ∝ 36 ′ ϑ 0 · 9 ′ ϑ 1 · 9 ′ ϑ 1 are either 54 (both 9 ′ from T 0 4 ), 30 (both 9 ′ from T − 1 ), or 42 (one 9 ′ from T 0 4 and the other from T − 1 ), i.e. Q R = 2 modulo 4. Thus, the coupling α 1 β 1 C is allowed and α 1 and β 1 are removed are removed at the scale where SU(9) ′ confines, C . Note, however, that Z 4R is broken to Z 2R at the scale C .
Next, let us consider doublets. Two doublets, one each from the α and β sets, i.e. α 2 and β 2 (the form given in Eq. (54)), combine to form 1: β 2 ξ 2 , as noted in Eq. (55). As above, the coupling α 2 β 2 C is allowed and one of α 2 and one of β 2 are removed at the scale where SU(9) ′ confines, C , and there remains only one pair of 5 hR and 5 hR . This is the MSSM vector-like pair of 5 ′ hR and 5 ′ hR . Summarizing, 5 hR ⊕ 5 hR : one MSSM pair, two pairs at the scale C .
From two permuting elements x 2 and x 3 , the S 2 singlet is (x 2 + x 3 )/ √ 2, and the orthogonal component to that is (x 2 − x 3 )/ √ 2. Thus, the MSSM doublets are

B. The third family members
It is natural to choose the singlet combination (x 1 + x 2 + x 3 )/ √ 3 as the third family member, and the doublet as the light family members. The quark and lepton mass matrix can be constructred based on these bases.

V. CONCLUSION
We derived a Z 4R parity in a family unification model with a GUT anti-SU(5) possessing three families and one pair of Higgs quintets. Spontaneous symmetry breaking of anti-SU(5) GUT is achieved by 10 +1 ⊕ 10 −1 . The Z 4R parity together with the permutation symmetry S 3 are useful to select the third family members and one MSSM pair of the Higgs quintets.
where s 0 corresponds to L-or R-movers. In this paper, + or − in the spinor form as in Eqs. (27) and (A2), denote + 1 2 and − 1 2 , respectively. The phase Θ k in the k th twisted sector in is Θ k = j (N j L − N j R )φ j − k 2 (V 2 a − φ 2 s ) + (P + kV a ) · V a − (s + kφ s ) · φ s + integer, = −s · φ s + ∆ k , where ∆ k is V a is the shift vector V distinguished by Wilson lines a, V 0,+,− , and andφ j is in the range 0 <φ j ≤ 1 mod integer. The oscillator contributions in ∆ N k is for N L,R ≥ 0. We satisfy the massless spectrum condition for gauge sectors by where 2c k of Z 12−I orbifold is which will lead to N = 1 supersymmetry in 4D.
We distinguish the twisted sector T k by 0, + and −. The phase ∆ 0 k of Eq. (A7) contains an extra k 2 factor, but at T 6 the factor k 2 is an integer. At T 1,2,4,5 , we distinguish three fixed points just by V 0,+,− .