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Z7 phenomenology
Volume 247, number 1
PHYSICS LETTERS B
6 September 1990
Z7 phenomenology J.A. Casas a, A. de la Macorra b, M. Mondragbn b and C. Muiioz b a Instituto de Estructura de la Materia. CSIC, Serrano 119, E-28006 Madrid, Spain ’ Department of Theoretical Physics, University of Oxford, Oxford OXI 3NP, UK Received 6 June 1990
We consider the possibility of constructing phenomenologically viable vacua based on the Z, orbifold. The acceptable ones are classified and drastically reduced by the use of equivalence operations between them. The results are applicable for SU (3) x SU (2) X U ( 1)” and grand unification schemes. Finally we discuss some physically relevant properties of the 2, orbifold, in particular the possibility of having just one (3,2) quark representation in the untwisted sector, which allows for a natural explanation of the Cabbibo angle.
In the last years different four-dimensional string backgrounds have been considered extensively for phenomenology. In particular, orbifold compactilications [ 1 ] have proved to be very interesting vacua. However, among the abelian orbifolds preserving N= 1 supersymmetry only the Z3 orbifold has been studied in depth [ l-9 1. In this letter we consider the possibility of constructing phenomenologically viable models based on the 2, orbifold. An aspect on which we focus is the classification and reduction of the possible vacua, since (as it usually happens in superstring theories) there is an enormous proliferation of spurious constructions. Concerning this point we give a set of equivalence operations between models which allow to discard most of them. The results are applicable for SU ( 3 ) x SU (2 ) x U ( 1)’ and grand unification schemes, although we have paid more attention to the former ones with three generations of matter. Finally, we sketch some physically relevant properties of the 2, orbifold, especially the possibility of having two (3,2) quark representations in the twisted sectors and the third one in the untwisted sector. The phenomenological consequences of this fact in connection with the Cabbibo angle are discussed below. Let us review briefly the construction of 2, orbifold models with Wilson lines [ 91. They are built upon compactification of the heterotic string on a Z7 orbifold and are characterized by the shift v associated with the twist &P (P=point group of the orbifold), and the Wilson line a associated with a translation eEI (F=lattice of the 6-torus). For the 2, orbifold F must be the lattice of SU( 7) (up to deformations), which implies that there exists just one independent Wilson line. The gauge bosons are obtained by projecting the E8 x E’s roots (p* = 2 ) onto those which are shift and Wilson line singlets, i.e. pb=Omod
1,
pa=Omod
1.
(12)
The matter fields coming from the untwisted pv=$, $,: mod 1 .
sector are obtained
from the Es X EL roots satisfying
(2) and (3)
The twisted matter comes in 2 1 sectors associated with the 7 fixed points of the orbifold (each fixed point is left invariant by 19,e2, o4 which gives the twisted particles, and by @, e5, e3 which gives the respective antiparticles). The vectors with the same chirality as the previous (untwisted) ones, satisfy the masslessness condition (d)2+-2NL,
Nc=O, +, 3, +,;,
(4)
where 0 are the shifted vectors 50
0370-2693/90/$ 03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
Volume 247, number 1
PHYSICS LETTERSB
m=1,2,4,
~=p+mv+na,
n=0,+1,_+2,_+3.
6 September 1990 (5)
For the Z7 orbifold all the twisted fields satisfying (4) also survive the projection onto the shift and the Wilson lines, so they are present in the four-dimensional theory [ 2 ]. Point group and modular invariance impose [ 1 ] 7v, 7a~Es ×E~ lattice,
(mv+na)Z=ZN,
(6,7)
Ne~q.
The first eight entries of v and a (from now on, v and a stand only for the first eight entries of these vectors) can be taken, without loss of generality, satisfying [ 1 ] (8)
v2, a2<~l.
It is straightforward to construct from ( 6 ), ( 8 ) the patterns of shifts and Wilson lines (observable part ) which can be used to build up Z7 models +(0,0,0,0,0,0,0,0),
}(1,1,0,0,0,0,0,0),-~½(1,1,1,1,1,1,1,1),
+(1,1,1,1,0,0,0,0),
+(2,0,0,0,0,0,0,0),
etc.,
(9)
where it is understood that in each pattern all the permutations and allowed changes of sign (according to the form of the Es roots) are considered. The second E8 part is also filled using eq. (9), in such a way that the modular invariance condition (7) is satisfied. The number of patterns for v and a obtained in this way is 189 (vector-like) plus 179 (spinor-like). There is a huge number of models that can be constructed in this way. However most of them are in fact equivalent. To be precise, we can pass from a model to an equivalent one using any of the operations listed below. Operation (i) v ~ v+ul ,
(10a,b)
a ~ a-~u2,
with u~, Uz~E8lattice. Operation (ii) v~mv,
a~,na
m,n=l,2,3.
(lla,b)
Operation (iii) v~ -v.
(12a)
Operation (iv) a ~ -a.
(12b)
Operation (v) v~,v-(ej.v)
ej,
a~,a-(ej.a)
ej,
(13)
with ej a root of E8. Operation (vi) v~v_+na
n=1,2,3,
a~a.
(14)
Operation (v) corresponds to the application o f a Weyl reflection on the shift vector and the Wilson line simultaneously. Operations (ii), (iii), (iv), (vi) are associated to relabellings of the twisted sectors. The previous equivalence relations represent a very handy set of rules and can be checked using eqs. ( 1 ) - (4) for the final spectrum. A similar study for the Z3 orbifold was performed in ref. [6]. Application of operation (i) to the previous 189 + 179 patterns for v and a, reduces the number to 139 (vector-like) plus 53 (vector-like with 51
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Table 1 Patterns for 7v and 7a in the 2, orbifold. 11000000 20000000 11110000 21100000 11111100 21111000 11111111 22110000 31000000 21111110 31110000 22200000 22111100 32 100000
31111100 22211000 22111111 32111000 31111111 22211110 32111110 32210000 22221100 41100000 41111000 32211100 22222000 22221111
Table 2 Entries of 7v and 7a in the first five positions
41111110 32211111 22222110 42111100 3321 1000 33111111 32221110 42111111 33211110 51110000 42211110 33221 100 32222111 22222220
51111100 42221 100 33311100 33222000 33221111 51111111 42221111 33311111 332221 10 52111110 43211111 52211111 43311110 33322111
that preserve the SU (3 ) x SU (2) gauge group.
00000 00111 00222 00333
11000 I l&(1 1 I)
1 lk(222) 11?(333) 22000 2f(l 11) 22&(222) 22?(333)
33000 33?(1 11) 33&(222) 2 -5*(1 11) 3-4k(lll) 1 l&(22-5) 1 l&(33-4)
modulus bigger than one). The subsequent application of (ii) reduces this set to the 56 patterns (with modulus smaller than f,,/%) listed in table 1. They can be understood as the initial inputs for 2, model-building. Let us now look for the inequivalent models with SU (3 ) x SU (2) contained in the gauge group. The following steps are valid for SU ( 3 ) x SU (2 ) x U ( 1) ’ and grand unification schemes, although we will focus on the former ones. We take the eight non-zero SU ( 3 ) x SU (2) roots to be #l (O,O,l,-1,0,0,0,0),
(15a,b)
(~,O,O,O,O,O,O).
Any other possibility of choosing the roots is equivalent to ( 15) performing a Weyl reflection, so we are already eliminating many equivalent models. On the other hand only Weyl reflections that leave ( 15 ) invariant can be considered from now on. It should be noticed here that when the gauge group that is being searched, is bigger than SU( 3) x SU( 2) xU( 1 )5, there are additional conditions coming from the situation of the extra gauge bosons. From table 1 and with the help of conditions ( 1) and (2) we can classify the possible values of the first five entries of 7v and 7a for which the SU ( 3 ) x SU (2) gauge bosons, as given by eq. ( 15 ), survive. There are 3 1 different possibilities, listed in table 2. To write down this list we have eliminated many possibilities by using the equivalence operations (i), (iii), (iv). The last three entries are completed in a trivial way with the use of table 1. This leaves 2885 different inputs for v and a after a new application of operations (i), (iii), (iv). Our next step is to use Weyl reflections (see eq. ( 13 ) ). If we denote a Weyl reflection by R, then (Rv, a) - (v, R - ‘a). This means that we can use Weyl reflections to reduce the previous inputs, but in principle only for the v’s Weyl reflections with respect to ( 1, - 1, 0, 0, 0, 0, 0,O) generate the complete group of permutations of the eight entries. Analogously, Weyl reflections with respect to ( 1, 1, 0, 0, 0, 0, 0,O) roots, combined with the pre#’ In this notation
52
the underlining
means that all different
permutations
are taken into account.
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PHYSICS LETTERS B
Volume 247, number 1
vious ones, generate the group of changes in the sign of an even number of entries. The application of these two kinds of operations, taking care of leaving ( 15) invariant, reduces the number of v’s from 2885 to 148. There are further Weyi reflections that leave ( 15) invariant, thus having to be considered. They correspond to the EB roots i(l,
1, -1,
f(l,l,l,
-1,
-1,
1, l,l,
-1,
1,l))
-1, f(l,l,l,
1,1,-l,
The use of these Weyl reflections, choices for v: 7v,=(0,0,1,1,1,1,0,0), 7v,(O,O,
l,l,
1, 1, -1,
4(1,1,-l,
-1))
-1,
-l,l).
(16)
with the help of the operations
l),
7vg=(0,0,1,1,1,3,1,-l),
7~,0=(0,0,1,1,1,3,2,0),
7~,,=(0,0,1,1,1,4,1,0),
28
7~~~=(0,0,1,1,1,3,2,2),
7v,3=(0,0,
1, 1, 1, 3,2, -2))
7v,4=(0,0,
~v,,=(o,o,
1, 1,1,4,2,
7~,~=(0,0,1,1,1,5,0,0),
-1))
0, 1, 1, 1, 5, 1, 1) ,
1, 1, 1,4,2,
1))
7v*s(O. 0, 1, 1, 1, 5, 1, -1)
I
7v,g=(O,O,l,l,l,3,3,3),
7%0=(0,0,
7vz, =(l,
1, 1, 1, 1,2, l,O),
7v**=(l,
1, 1, 1, 1, 1, 1, 1) >
7vz3=(1,
l,l,
7vz4=(l,
l,l,
7v2,=(1,1,2,2,2,3,2,-l),
possible
721~=(0,0,1,1,1,2,2,1), 7v*=(0,0,1,1,1,3,1,1),
7~25=(1,1,1,1,1,5,0,0),
gives the following
1, 1, 1,2, 170) >
7va=(O,O,
7v,=(0,0,1,1,1,2,2,-l),
1,1,2,2,1),
(i)-(iv),
7v~=(0,0,1,1,1,1,1,1),
7u~=(0,0,1,1,1,3,0,0),
7v,,=(O,
-l,-l,l,l),
1, 1, 1,393, -3)
,
1,1,3,2,0),
7026 = (0, 0,2,2,2,
2, 1, - 1) ,
7v28=(0,0,0,0,0,0,0,0).
(17)
This is actually the most tedious part of the work. When the previous v’s are combined with the 2885 a’s that can be formed from table 2 and table 1, it turns out to be 20 007 SU( 3) x SU( 2) xU( 1)5 models (this has to be done with the help of the computer). This number admits further reduction. Let us take, for instance, v3= f (0, 0, 1, 1, 1, 1, - 1, 1) . When this is combined with the a’s there appear 84 models. An inspection of them, using the set of equivalence operations (i)-(v) (which are non-commutative) reduces this number. It should be noticed here that even though the form of v and the position of SU( 3) xSU(2) have been fixed, there remain many Weyl reflections which leave them invariant and can be used. Finally there survive just three models with the following Wilson lines: 7~=(1,1,1,1,1,3,3,-l),
(3,3,0,0,0,2,1,1),
(2,2,1,1,1,-3,2,2).
(18)
For a SU( 3) xSU( 2) xU( 1 )5 to be realistic it has to contain three (3,2) representations. Furthermore it is highly desirable that at least two of them belong to twisted sectors, otherwise the observed family mass hierarchy cannot be achieved [ 81. A twisted sector containing a (3,2) can always be carried to a trivial twisted sector (i.e. shift = v, 2v, 3v ) by using operation (vi), the new shift being equivalent to one of the 28 v’s listed in eq. ( 17 ). Therefore it is sufficient to study embeddings that have at least one (3,2) in the trivial twisted sector. On the other hand the matter of trivial twisted sectors, has been classified in ref. [ lo] for vanishing Wilson line. In the present case the representations of this matter are broken under SU (3 ) x SU (2 ) x U ( 1) 5. A brief glance reveals 53
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that only for 15 kinds ofv's we can expect ( 3,2 )'s in these sectors. Three of these cases are particularly interesting since they contain either two or three (3,2)'s, consequently they can work well for a wide class of Wilson lines. Of course they can be shown to be equivalent to three of the 28 v's listed in eq. ( 17 ), more precisely to v6, Vs, v23 with the appropriate second E8 part 7 t 9 6 = ( 0 , 0 , 1, 1, 1,2,2, 1)(1, 1 , 0 , 0 , 0 , 0 , 0 , 0 )
(2),
(19a)
7v8 = (0, 0, 1, 1, 1, 3, 1, 1)(0, 0, 0, 0, 0, 0, 0, 0)
(2),
(19b)
7v23=(1,1,1,1,1,2,2,1)(0,0,0,0,0,0,0,0)
(3),
(19c)
where the figure within parentheses indicates the number of (3,2)'s. Although there can appear new (3,2)'s in other twisted sectors, this is quite infrequent as can be easily checked. Of course when the twisted sector contains two (3,2)'s, the presence of an additional (3,2) in the untwisted sector is mandatory. This can be achieved, since in the Z7 orbifold the untwisted matter does not come in three identical copies, as it happens in the Z3 orbifold. This is a very important characteristic of the Z7 orbifold that allows for a quite interesting situation. If all the standard matter, but one (3,2), belong to twisted sectors, then the corresponding generation of quarks remains massless at tree level because of the point group selection rule [ 11 ]. Actually this is very attractive since a vanishing value in the first diagonal entry of the quark mass matrices has been shown to be extremely convenient to get a realistic Cabibbo angle [ 8,12,13]. The masses for this generation of quarks come from non-vanishing off-diagonal terms originated by higher order operators [ 8 ]. In order to illustrate the type of constructions that can be built up from (19) we give below three interesting examples. Example 1. 7v=(0,0,1,1,1,2,2,1)(1,1,0,0,0,0,0,0), 7a= (1, 1, 1, 1, 1, 3, - 2 , 2)(3, - 3 , - 4 , 0 , 0, 0, 0 , 0 ) ,
(20a)
gauge group: [SU(3) × SU(2) × U ( 1 )5] × [SO(10) × U ( 1 )3], ,
(20b)
spectrum: 3{ (3, 2) + 2 ( 3 , 1 ) + ( 1 , 2 ) + 1 } + 1 2 ( 3 , 1 ) + 1 2 ( 3 , 1 ) + 1 8 ( 1 , 2 ) + ( 1 0 ) ' + 1 3 0 1 .
(20c)
Example 2. 7v=(0,0,1,1,1,3,1,1)(0,0,0,0,0,0,0,0), 7a= (3, 3, - 1 , - 1 , - 1 , - 2 , - 3 , - 2 ) ( 5 , 1, 1, 1, 1, l, 1, 1),
(21a)
gauge group: [SU(3) × S U ( 2 ) X U ( 1 ) 5 ] × [SU(8) × U ( 1 ) ] ' ,
(21b)
spectrum: 3{(3, 2) + 2 ( 3 , 1) + (1, 2) + 1 } + 9 ( 3 , 1) + 9 ( 3 , 1) +22(1, 2) + 4 ( 8 , 1)' + 4 ( 8 , 1)' +781.
(21c)
Example 3. 7v=(1,1,1,1,1,2,2,1)(0,0,0,0,0,0,0,0), 7a= (0, 0, l, 1, 1 , 3 , - 2 , 2 ) ( 4 , 2 , 2 , 2 , 2 , 2 , 0 , 0 ) ,
(22a)
gauge group: [SU(3) × SU(2) × U ( 1 )5] × [SU(5) × S U ( 3 ) × SU(2) XU( 1 ) ] ' ,
(22b)
spectrum: 3{(3, 2) + 2 ( 3 , 1 ) + (1, 2) + 1 } + 5 ( 3 , 1) + 5 ( 3 , 1) +9(1, 2) +3(5, 1, 1)' +3(3, 1, 1)' +5(1,3, 1)'+7(1,3, 1)'+(1,2)(1,3, 1)'+2(1,2)(1, 1,2)'+12(1, 1,2)'+461.
(22c)
An examination of the interactions of these models shows that examples 1 and 2 are especially interesting to implement the previous mechanism for the Cabibbo angle. Moreover the hidden gauge group of example 3 has 54
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two non-abelian sectors of the appropriate size to realize the supersymmetry breaking mechanism explained in ref. [14]. A list of inequivalent SU ( 3 ) x SU (2 ) X U ( 1) 5based on the shift v6 (see eq. ( 19a) ) is given in the first column of table 3. All of them possess a single (3,2) in the untwisted sector, as is required. The second Es part of a is easily filled up according to ( 19 ) and the modular invariant condition ( 7 ). For models based on us and v23 (see eqs. ( 19b), ( 1SC) ) the modular invariance condition turns out to be quite restrictive, leaving 24 models from V, (with the required untwisted (3,2) ) and 12 ones from vz3 (with no untwisted (3,2)‘s, as is wanted here). They are listed in the second and third column of table 3. An example of our SU (3) x SU (2) x U ( 1) 5 model was recently given in ref. [ 15 1. It can be shown to be equivalent to a model based on one of the shift vectors of ( 19)) to be precise 7v=(O,O,
l,l,
1,2,2,
l)(l,
>
l,O,O,O,O,O,O)
7a=(3,3,1,1,1,-2,-l,-2)(1,-3,4,0,0,0,0,0).
(23)
This model however has the shortcoming of not containing any anomalous U ( 1)) that is extremely convenient to trigger a breakdown of the extra U ( 1 )‘s at the right scale and to generate high masses for the extra particles [ 4,5 1. The models of eqs. (20)-( 22) do possess this anomalous U ( 1) factor. In summary, we have described how to look for phenomenologically viable models based on the 2, orbifold, eliminating equivalent constructions. The shift vector (v) should belong to the set listed in eq. ( 17) and the possible Wilson lines can be obtained from tables 1 and 2. If we require the presence of two or three (3,2) quark representations under SU (3 ) x SU (2) in the twisted sector (which seems to be convenient for a realistic phenomenology [ 8 ] ) a further reduction for the v’s occurs (see eq. ( 19) ). Incidentally it is very interesting that a (3,2) can appear alone in the untwisted sector (something impossible for the Z3 orbifold). As is discussed in the text this allows for a natural explanation of the Cabibbo angle. Although we have focused on models with Table 3 Wilson lines associated
with ~6, USand vzs giving SU (3)
11111104 111113-22 11111-313 1111 l-3 -1-3 111113-24 11-l -1-l -4 -1-2 11 -1 -1 -1 -42 1 11-l -1-1401 11-l -1-l 300 11-l -1-1212 11-l -1-I -2 1-2 11-l -1 -I -300 110002-11 22-l -1-l 102 22-1-l -1-210 22 -1-l -1-102 3322210-I
x
SU (2)
x
U ( 1)’ vacua with three (3,2) quark representations.
1111133-1 11111-24-1 11 1111 -4-2 111114-32 1111134-2 11111250 11-l -1-l -3-31 11 -1 -1 -12-41 11 -1 -1 -1 -142 11 -1 -1 -1432 1 I-l -1-l -43 -2 11-l -1-l 520 11222-l -30 11 -2-2-2130 33000-121 330001 -2-l 22111320 22111-230 22-l-1-12-30 22 -1 -1 -1 -3 -20 331110-2-l 33111232 33-1 -I -1021 33-1 -1 -1 -2-3-2
001113-22 00222-20-2 11-2-2-2-21-1 11-3-3-32-1-2 110004-l -1 11000-33-2 221115-1-1 22-l-1-1300 2-5111-110 33111-40-t 33-1-l -1-4-10 3-4111-11-2
55
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SU (3) × SU (2) × U ( 1 ) 5 as unbroken gauge group, the results are applicable also to grand unification groups. A second step of symmetry breaking is guaranteed by the presence of an "anomalous" U( 1 ) factor. Finally we give a list of SU ( 3 ) × SU ( 2 ) × U ( 1 ) 5 models with phenomenologically interesting prospects in eqs. ( 20 ) - ( 22 ) and table 3. The work of C.M. was supported in part by a CSIC (Consejo Superior de Investigaciones Cientificas) fellowship. The work of M.M. was supported by a CONACYT (Mexico) scholarship. The work of A.M. was supported by an UNAM (Mexico) scholarship. C.M. is grateful to the members of the Depto. de Fisica de Particulas de la Universidad de Santiago de Compostela, Spain, for their kind hospitality. We wish to express our thanks to L.E. Ib~ifiez for very useful conversations at the first stage of this work.
References [ 1 ] L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261 ( 1985 ) 678; B 274 (1986) 285. [2] L.E. Ib~ifiez, H.P. Nilles and F. Quevedo, Phys. Lett. B 187 (1987) 25; L.E. Ib~ifiez, J.E. Kim, H.P. Nilles and F. Quevedo, Phys. Lett. B 191 ( 1987 ) 3. [3] D. Bailin, A. Love and S. Thomas, Phys. Lett. B 188 (1987) 193; B 194 (1987) 385. [4] J.A. Casas, E.K. Katehou and C. Mufioz, Nucl. Phys. B 317 (1989) 171; J.A. Casas and C. Mufioz, Phys. Lett. B 209 ( 1988 ) 214; B 214 (1988) 63; B 212 ( 1988 ) 343; B 214 (1988) 543. [ 5 ] A. Font, L.E. Ib~ifiez, H.P. Nilles and F. Quevedo, Phys. Lett. B 210 ( 1988 ) 101; A. Font, L.E. Ib~ifiez, F. Quevedo and A. Sierra, CERN preprint CERN-TH-5326/89. [6] J.A. Casas, M. Mondrag6n and C. Mufioz, Phys. Lett. B 230 (1989) 63. [ 7 ] A. Fujitsu, T. Kitazoe, M. Tabuse and H. Nishimura, Kobe University preprint KOBE-89-05. [8] J.A. Casas and C. Mufioz, Nucl. Phys. B 332 (1990) 189. [ 9 ] L.E. Ib~ifiez, J. Mas, H.P. Nilles and F. Quevedo, Nucl. Phys. B 301 ( 1988 ) 157. [ 10] Y. Katsuki, Y. Kawamura, T. Kobayashi and N. Ohtsubo, Phys. Lett. B 212 (1988) 339; preprint DPKU-8802 (1988). [ 11 ] S. Hamidi and C. Vafa, Nucl. Phys. B 279 (1987) 465; L. Dixon, D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 282 (1987) 13. [ 12 ] J.L. Lopez and D.V. Nanopoulos, Texas A&M preprint CTP-TAMU-60/89. [ 13 ] C.H. Albright, Fermilab preprint FERMILAB-PUB-89/258-T. [ 14] J.A. Casas, Z. Lalak, C. Mufioz and C.G. Ross, Oxford preprint OUTP-90-07P (1990), Nucl. Phys. B, to appear; L. Dixon, V. Kaplunovsky, J. Louis and M. Peskin, SLAC/Texas preprint, to appear. [ 15 ] Y. Katsuki, Y. Kawamura, T. Kobayashi, N. Ohtsubo, Y. Ono and K. Tanioka, preprint DPKU-8906 ( 1989 ).
56
PHYSICS LETTERS B
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Z7 phenomenology J.A. Casas a, A. de la Macorra b, M. Mondragbn b and C. Muiioz b a Instituto de Estructura de la Materia. CSIC, Serrano 119, E-28006 Madrid, Spain ’ Department of Theoretical Physics, University of Oxford, Oxford OXI 3NP, UK Received 6 June 1990
We consider the possibility of constructing phenomenologically viable vacua based on the Z, orbifold. The acceptable ones are classified and drastically reduced by the use of equivalence operations between them. The results are applicable for SU (3) x SU (2) X U ( 1)” and grand unification schemes. Finally we discuss some physically relevant properties of the 2, orbifold, in particular the possibility of having just one (3,2) quark representation in the untwisted sector, which allows for a natural explanation of the Cabbibo angle.
In the last years different four-dimensional string backgrounds have been considered extensively for phenomenology. In particular, orbifold compactilications [ 1 ] have proved to be very interesting vacua. However, among the abelian orbifolds preserving N= 1 supersymmetry only the Z3 orbifold has been studied in depth [ l-9 1. In this letter we consider the possibility of constructing phenomenologically viable models based on the 2, orbifold. An aspect on which we focus is the classification and reduction of the possible vacua, since (as it usually happens in superstring theories) there is an enormous proliferation of spurious constructions. Concerning this point we give a set of equivalence operations between models which allow to discard most of them. The results are applicable for SU ( 3 ) x SU (2 ) x U ( 1)’ and grand unification schemes, although we have paid more attention to the former ones with three generations of matter. Finally, we sketch some physically relevant properties of the 2, orbifold, especially the possibility of having two (3,2) quark representations in the twisted sectors and the third one in the untwisted sector. The phenomenological consequences of this fact in connection with the Cabbibo angle are discussed below. Let us review briefly the construction of 2, orbifold models with Wilson lines [ 91. They are built upon compactification of the heterotic string on a Z7 orbifold and are characterized by the shift v associated with the twist &P (P=point group of the orbifold), and the Wilson line a associated with a translation eEI (F=lattice of the 6-torus). For the 2, orbifold F must be the lattice of SU( 7) (up to deformations), which implies that there exists just one independent Wilson line. The gauge bosons are obtained by projecting the E8 x E’s roots (p* = 2 ) onto those which are shift and Wilson line singlets, i.e. pb=Omod
1,
pa=Omod
1.
(12)
The matter fields coming from the untwisted pv=$, $,: mod 1 .
sector are obtained
from the Es X EL roots satisfying
(2) and (3)
The twisted matter comes in 2 1 sectors associated with the 7 fixed points of the orbifold (each fixed point is left invariant by 19,e2, o4 which gives the twisted particles, and by @, e5, e3 which gives the respective antiparticles). The vectors with the same chirality as the previous (untwisted) ones, satisfy the masslessness condition (d)2+-2NL,
Nc=O, +, 3, +,;,
(4)
where 0 are the shifted vectors 50
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m=1,2,4,
~=p+mv+na,
n=0,+1,_+2,_+3.
6 September 1990 (5)
For the Z7 orbifold all the twisted fields satisfying (4) also survive the projection onto the shift and the Wilson lines, so they are present in the four-dimensional theory [ 2 ]. Point group and modular invariance impose [ 1 ] 7v, 7a~Es ×E~ lattice,
(mv+na)Z=ZN,
(6,7)
Ne~q.
The first eight entries of v and a (from now on, v and a stand only for the first eight entries of these vectors) can be taken, without loss of generality, satisfying [ 1 ] (8)
v2, a2<~l.
It is straightforward to construct from ( 6 ), ( 8 ) the patterns of shifts and Wilson lines (observable part ) which can be used to build up Z7 models +(0,0,0,0,0,0,0,0),
}(1,1,0,0,0,0,0,0),-~½(1,1,1,1,1,1,1,1),
+(1,1,1,1,0,0,0,0),
+(2,0,0,0,0,0,0,0),
etc.,
(9)
where it is understood that in each pattern all the permutations and allowed changes of sign (according to the form of the Es roots) are considered. The second E8 part is also filled using eq. (9), in such a way that the modular invariance condition (7) is satisfied. The number of patterns for v and a obtained in this way is 189 (vector-like) plus 179 (spinor-like). There is a huge number of models that can be constructed in this way. However most of them are in fact equivalent. To be precise, we can pass from a model to an equivalent one using any of the operations listed below. Operation (i) v ~ v+ul ,
(10a,b)
a ~ a-~u2,
with u~, Uz~E8lattice. Operation (ii) v~mv,
a~,na
m,n=l,2,3.
(lla,b)
Operation (iii) v~ -v.
(12a)
Operation (iv) a ~ -a.
(12b)
Operation (v) v~,v-(ej.v)
ej,
a~,a-(ej.a)
ej,
(13)
with ej a root of E8. Operation (vi) v~v_+na
n=1,2,3,
a~a.
(14)
Operation (v) corresponds to the application o f a Weyl reflection on the shift vector and the Wilson line simultaneously. Operations (ii), (iii), (iv), (vi) are associated to relabellings of the twisted sectors. The previous equivalence relations represent a very handy set of rules and can be checked using eqs. ( 1 ) - (4) for the final spectrum. A similar study for the Z3 orbifold was performed in ref. [6]. Application of operation (i) to the previous 189 + 179 patterns for v and a, reduces the number to 139 (vector-like) plus 53 (vector-like with 51
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Table 1 Patterns for 7v and 7a in the 2, orbifold. 11000000 20000000 11110000 21100000 11111100 21111000 11111111 22110000 31000000 21111110 31110000 22200000 22111100 32 100000
31111100 22211000 22111111 32111000 31111111 22211110 32111110 32210000 22221100 41100000 41111000 32211100 22222000 22221111
Table 2 Entries of 7v and 7a in the first five positions
41111110 32211111 22222110 42111100 3321 1000 33111111 32221110 42111111 33211110 51110000 42211110 33221 100 32222111 22222220
51111100 42221 100 33311100 33222000 33221111 51111111 42221111 33311111 332221 10 52111110 43211111 52211111 43311110 33322111
that preserve the SU (3 ) x SU (2) gauge group.
00000 00111 00222 00333
11000 I l&(1 1 I)
1 lk(222) 11?(333) 22000 2f(l 11) 22&(222) 22?(333)
33000 33?(1 11) 33&(222) 2 -5*(1 11) 3-4k(lll) 1 l&(22-5) 1 l&(33-4)
modulus bigger than one). The subsequent application of (ii) reduces this set to the 56 patterns (with modulus smaller than f,,/%) listed in table 1. They can be understood as the initial inputs for 2, model-building. Let us now look for the inequivalent models with SU (3 ) x SU (2) contained in the gauge group. The following steps are valid for SU ( 3 ) x SU (2 ) x U ( 1) ’ and grand unification schemes, although we will focus on the former ones. We take the eight non-zero SU ( 3 ) x SU (2) roots to be #l (O,O,l,-1,0,0,0,0),
(15a,b)
(~,O,O,O,O,O,O).
Any other possibility of choosing the roots is equivalent to ( 15) performing a Weyl reflection, so we are already eliminating many equivalent models. On the other hand only Weyl reflections that leave ( 15 ) invariant can be considered from now on. It should be noticed here that when the gauge group that is being searched, is bigger than SU( 3) x SU( 2) xU( 1 )5, there are additional conditions coming from the situation of the extra gauge bosons. From table 1 and with the help of conditions ( 1) and (2) we can classify the possible values of the first five entries of 7v and 7a for which the SU ( 3 ) x SU (2) gauge bosons, as given by eq. ( 15 ), survive. There are 3 1 different possibilities, listed in table 2. To write down this list we have eliminated many possibilities by using the equivalence operations (i), (iii), (iv). The last three entries are completed in a trivial way with the use of table 1. This leaves 2885 different inputs for v and a after a new application of operations (i), (iii), (iv). Our next step is to use Weyl reflections (see eq. ( 13 ) ). If we denote a Weyl reflection by R, then (Rv, a) - (v, R - ‘a). This means that we can use Weyl reflections to reduce the previous inputs, but in principle only for the v’s Weyl reflections with respect to ( 1, - 1, 0, 0, 0, 0, 0,O) generate the complete group of permutations of the eight entries. Analogously, Weyl reflections with respect to ( 1, 1, 0, 0, 0, 0, 0,O) roots, combined with the pre#’ In this notation
52
the underlining
means that all different
permutations
are taken into account.
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vious ones, generate the group of changes in the sign of an even number of entries. The application of these two kinds of operations, taking care of leaving ( 15) invariant, reduces the number of v’s from 2885 to 148. There are further Weyi reflections that leave ( 15) invariant, thus having to be considered. They correspond to the EB roots i(l,
1, -1,
f(l,l,l,
-1,
-1,
1, l,l,
-1,
1,l))
-1, f(l,l,l,
1,1,-l,
The use of these Weyl reflections, choices for v: 7v,=(0,0,1,1,1,1,0,0), 7v,(O,O,
l,l,
1, 1, -1,
4(1,1,-l,
-1))
-1,
-l,l).
(16)
with the help of the operations
l),
7vg=(0,0,1,1,1,3,1,-l),
7~,0=(0,0,1,1,1,3,2,0),
7~,,=(0,0,1,1,1,4,1,0),
28
7~~~=(0,0,1,1,1,3,2,2),
7v,3=(0,0,
1, 1, 1, 3,2, -2))
7v,4=(0,0,
~v,,=(o,o,
1, 1,1,4,2,
7~,~=(0,0,1,1,1,5,0,0),
-1))
0, 1, 1, 1, 5, 1, 1) ,
1, 1, 1,4,2,
1))
7v*s(O. 0, 1, 1, 1, 5, 1, -1)
I
7v,g=(O,O,l,l,l,3,3,3),
7%0=(0,0,
7vz, =(l,
1, 1, 1, 1,2, l,O),
7v**=(l,
1, 1, 1, 1, 1, 1, 1) >
7vz3=(1,
l,l,
7vz4=(l,
l,l,
7v2,=(1,1,2,2,2,3,2,-l),
possible
721~=(0,0,1,1,1,2,2,1), 7v*=(0,0,1,1,1,3,1,1),
7~25=(1,1,1,1,1,5,0,0),
gives the following
1, 1, 1,2, 170) >
7va=(O,O,
7v,=(0,0,1,1,1,2,2,-l),
1,1,2,2,1),
(i)-(iv),
7v~=(0,0,1,1,1,1,1,1),
7u~=(0,0,1,1,1,3,0,0),
7v,,=(O,
-l,-l,l,l),
1, 1, 1,393, -3)
,
1,1,3,2,0),
7026 = (0, 0,2,2,2,
2, 1, - 1) ,
7v28=(0,0,0,0,0,0,0,0).
(17)
This is actually the most tedious part of the work. When the previous v’s are combined with the 2885 a’s that can be formed from table 2 and table 1, it turns out to be 20 007 SU( 3) x SU( 2) xU( 1)5 models (this has to be done with the help of the computer). This number admits further reduction. Let us take, for instance, v3= f (0, 0, 1, 1, 1, 1, - 1, 1) . When this is combined with the a’s there appear 84 models. An inspection of them, using the set of equivalence operations (i)-(v) (which are non-commutative) reduces this number. It should be noticed here that even though the form of v and the position of SU( 3) xSU(2) have been fixed, there remain many Weyl reflections which leave them invariant and can be used. Finally there survive just three models with the following Wilson lines: 7~=(1,1,1,1,1,3,3,-l),
(3,3,0,0,0,2,1,1),
(2,2,1,1,1,-3,2,2).
(18)
For a SU( 3) xSU( 2) xU( 1 )5 to be realistic it has to contain three (3,2) representations. Furthermore it is highly desirable that at least two of them belong to twisted sectors, otherwise the observed family mass hierarchy cannot be achieved [ 81. A twisted sector containing a (3,2) can always be carried to a trivial twisted sector (i.e. shift = v, 2v, 3v ) by using operation (vi), the new shift being equivalent to one of the 28 v’s listed in eq. ( 17 ). Therefore it is sufficient to study embeddings that have at least one (3,2) in the trivial twisted sector. On the other hand the matter of trivial twisted sectors, has been classified in ref. [ lo] for vanishing Wilson line. In the present case the representations of this matter are broken under SU (3 ) x SU (2 ) x U ( 1) 5. A brief glance reveals 53
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that only for 15 kinds ofv's we can expect ( 3,2 )'s in these sectors. Three of these cases are particularly interesting since they contain either two or three (3,2)'s, consequently they can work well for a wide class of Wilson lines. Of course they can be shown to be equivalent to three of the 28 v's listed in eq. ( 17 ), more precisely to v6, Vs, v23 with the appropriate second E8 part 7 t 9 6 = ( 0 , 0 , 1, 1, 1,2,2, 1)(1, 1 , 0 , 0 , 0 , 0 , 0 , 0 )
(2),
(19a)
7v8 = (0, 0, 1, 1, 1, 3, 1, 1)(0, 0, 0, 0, 0, 0, 0, 0)
(2),
(19b)
7v23=(1,1,1,1,1,2,2,1)(0,0,0,0,0,0,0,0)
(3),
(19c)
where the figure within parentheses indicates the number of (3,2)'s. Although there can appear new (3,2)'s in other twisted sectors, this is quite infrequent as can be easily checked. Of course when the twisted sector contains two (3,2)'s, the presence of an additional (3,2) in the untwisted sector is mandatory. This can be achieved, since in the Z7 orbifold the untwisted matter does not come in three identical copies, as it happens in the Z3 orbifold. This is a very important characteristic of the Z7 orbifold that allows for a quite interesting situation. If all the standard matter, but one (3,2), belong to twisted sectors, then the corresponding generation of quarks remains massless at tree level because of the point group selection rule [ 11 ]. Actually this is very attractive since a vanishing value in the first diagonal entry of the quark mass matrices has been shown to be extremely convenient to get a realistic Cabibbo angle [ 8,12,13]. The masses for this generation of quarks come from non-vanishing off-diagonal terms originated by higher order operators [ 8 ]. In order to illustrate the type of constructions that can be built up from (19) we give below three interesting examples. Example 1. 7v=(0,0,1,1,1,2,2,1)(1,1,0,0,0,0,0,0), 7a= (1, 1, 1, 1, 1, 3, - 2 , 2)(3, - 3 , - 4 , 0 , 0, 0, 0 , 0 ) ,
(20a)
gauge group: [SU(3) × SU(2) × U ( 1 )5] × [SO(10) × U ( 1 )3], ,
(20b)
spectrum: 3{ (3, 2) + 2 ( 3 , 1 ) + ( 1 , 2 ) + 1 } + 1 2 ( 3 , 1 ) + 1 2 ( 3 , 1 ) + 1 8 ( 1 , 2 ) + ( 1 0 ) ' + 1 3 0 1 .
(20c)
Example 2. 7v=(0,0,1,1,1,3,1,1)(0,0,0,0,0,0,0,0), 7a= (3, 3, - 1 , - 1 , - 1 , - 2 , - 3 , - 2 ) ( 5 , 1, 1, 1, 1, l, 1, 1),
(21a)
gauge group: [SU(3) × S U ( 2 ) X U ( 1 ) 5 ] × [SU(8) × U ( 1 ) ] ' ,
(21b)
spectrum: 3{(3, 2) + 2 ( 3 , 1) + (1, 2) + 1 } + 9 ( 3 , 1) + 9 ( 3 , 1) +22(1, 2) + 4 ( 8 , 1)' + 4 ( 8 , 1)' +781.
(21c)
Example 3. 7v=(1,1,1,1,1,2,2,1)(0,0,0,0,0,0,0,0), 7a= (0, 0, l, 1, 1 , 3 , - 2 , 2 ) ( 4 , 2 , 2 , 2 , 2 , 2 , 0 , 0 ) ,
(22a)
gauge group: [SU(3) × SU(2) × U ( 1 )5] × [SU(5) × S U ( 3 ) × SU(2) XU( 1 ) ] ' ,
(22b)
spectrum: 3{(3, 2) + 2 ( 3 , 1 ) + (1, 2) + 1 } + 5 ( 3 , 1) + 5 ( 3 , 1) +9(1, 2) +3(5, 1, 1)' +3(3, 1, 1)' +5(1,3, 1)'+7(1,3, 1)'+(1,2)(1,3, 1)'+2(1,2)(1, 1,2)'+12(1, 1,2)'+461.
(22c)
An examination of the interactions of these models shows that examples 1 and 2 are especially interesting to implement the previous mechanism for the Cabibbo angle. Moreover the hidden gauge group of example 3 has 54
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two non-abelian sectors of the appropriate size to realize the supersymmetry breaking mechanism explained in ref. [14]. A list of inequivalent SU ( 3 ) x SU (2 ) X U ( 1) 5based on the shift v6 (see eq. ( 19a) ) is given in the first column of table 3. All of them possess a single (3,2) in the untwisted sector, as is required. The second Es part of a is easily filled up according to ( 19 ) and the modular invariant condition ( 7 ). For models based on us and v23 (see eqs. ( 19b), ( 1SC) ) the modular invariance condition turns out to be quite restrictive, leaving 24 models from V, (with the required untwisted (3,2) ) and 12 ones from vz3 (with no untwisted (3,2)‘s, as is wanted here). They are listed in the second and third column of table 3. An example of our SU (3) x SU (2) x U ( 1) 5 model was recently given in ref. [ 15 1. It can be shown to be equivalent to a model based on one of the shift vectors of ( 19)) to be precise 7v=(O,O,
l,l,
1,2,2,
l)(l,
>
l,O,O,O,O,O,O)
7a=(3,3,1,1,1,-2,-l,-2)(1,-3,4,0,0,0,0,0).
(23)
This model however has the shortcoming of not containing any anomalous U ( 1)) that is extremely convenient to trigger a breakdown of the extra U ( 1 )‘s at the right scale and to generate high masses for the extra particles [ 4,5 1. The models of eqs. (20)-( 22) do possess this anomalous U ( 1) factor. In summary, we have described how to look for phenomenologically viable models based on the 2, orbifold, eliminating equivalent constructions. The shift vector (v) should belong to the set listed in eq. ( 17) and the possible Wilson lines can be obtained from tables 1 and 2. If we require the presence of two or three (3,2) quark representations under SU (3 ) x SU (2) in the twisted sector (which seems to be convenient for a realistic phenomenology [ 8 ] ) a further reduction for the v’s occurs (see eq. ( 19) ). Incidentally it is very interesting that a (3,2) can appear alone in the untwisted sector (something impossible for the Z3 orbifold). As is discussed in the text this allows for a natural explanation of the Cabibbo angle. Although we have focused on models with Table 3 Wilson lines associated
with ~6, USand vzs giving SU (3)
11111104 111113-22 11111-313 1111 l-3 -1-3 111113-24 11-l -1-l -4 -1-2 11 -1 -1 -1 -42 1 11-l -1-1401 11-l -1-l 300 11-l -1-1212 11-l -1-I -2 1-2 11-l -1 -I -300 110002-11 22-l -1-l 102 22-1-l -1-210 22 -1-l -1-102 3322210-I
x
SU (2)
x
U ( 1)’ vacua with three (3,2) quark representations.
1111133-1 11111-24-1 11 1111 -4-2 111114-32 1111134-2 11111250 11-l -1-l -3-31 11 -1 -1 -12-41 11 -1 -1 -1 -142 11 -1 -1 -1432 1 I-l -1-l -43 -2 11-l -1-l 520 11222-l -30 11 -2-2-2130 33000-121 330001 -2-l 22111320 22111-230 22-l-1-12-30 22 -1 -1 -1 -3 -20 331110-2-l 33111232 33-1 -I -1021 33-1 -1 -1 -2-3-2
001113-22 00222-20-2 11-2-2-2-21-1 11-3-3-32-1-2 110004-l -1 11000-33-2 221115-1-1 22-l-1-1300 2-5111-110 33111-40-t 33-1-l -1-4-10 3-4111-11-2
55
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SU (3) × SU (2) × U ( 1 ) 5 as unbroken gauge group, the results are applicable also to grand unification groups. A second step of symmetry breaking is guaranteed by the presence of an "anomalous" U( 1 ) factor. Finally we give a list of SU ( 3 ) × SU ( 2 ) × U ( 1 ) 5 models with phenomenologically interesting prospects in eqs. ( 20 ) - ( 22 ) and table 3. The work of C.M. was supported in part by a CSIC (Consejo Superior de Investigaciones Cientificas) fellowship. The work of M.M. was supported by a CONACYT (Mexico) scholarship. The work of A.M. was supported by an UNAM (Mexico) scholarship. C.M. is grateful to the members of the Depto. de Fisica de Particulas de la Universidad de Santiago de Compostela, Spain, for their kind hospitality. We wish to express our thanks to L.E. Ib~ifiez for very useful conversations at the first stage of this work.
References [ 1 ] L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261 ( 1985 ) 678; B 274 (1986) 285. [2] L.E. Ib~ifiez, H.P. Nilles and F. Quevedo, Phys. Lett. B 187 (1987) 25; L.E. Ib~ifiez, J.E. Kim, H.P. Nilles and F. Quevedo, Phys. Lett. B 191 ( 1987 ) 3. [3] D. Bailin, A. Love and S. Thomas, Phys. Lett. B 188 (1987) 193; B 194 (1987) 385. [4] J.A. Casas, E.K. Katehou and C. Mufioz, Nucl. Phys. B 317 (1989) 171; J.A. Casas and C. Mufioz, Phys. Lett. B 209 ( 1988 ) 214; B 214 (1988) 63; B 212 ( 1988 ) 343; B 214 (1988) 543. [ 5 ] A. Font, L.E. Ib~ifiez, H.P. Nilles and F. Quevedo, Phys. Lett. B 210 ( 1988 ) 101; A. Font, L.E. Ib~ifiez, F. Quevedo and A. Sierra, CERN preprint CERN-TH-5326/89. [6] J.A. Casas, M. Mondrag6n and C. Mufioz, Phys. Lett. B 230 (1989) 63. [ 7 ] A. Fujitsu, T. Kitazoe, M. Tabuse and H. Nishimura, Kobe University preprint KOBE-89-05. [8] J.A. Casas and C. Mufioz, Nucl. Phys. B 332 (1990) 189. [ 9 ] L.E. Ib~ifiez, J. Mas, H.P. Nilles and F. Quevedo, Nucl. Phys. B 301 ( 1988 ) 157. [ 10] Y. Katsuki, Y. Kawamura, T. Kobayashi and N. Ohtsubo, Phys. Lett. B 212 (1988) 339; preprint DPKU-8802 (1988). [ 11 ] S. Hamidi and C. Vafa, Nucl. Phys. B 279 (1987) 465; L. Dixon, D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B 282 (1987) 13. [ 12 ] J.L. Lopez and D.V. Nanopoulos, Texas A&M preprint CTP-TAMU-60/89. [ 13 ] C.H. Albright, Fermilab preprint FERMILAB-PUB-89/258-T. [ 14] J.A. Casas, Z. Lalak, C. Mufioz and C.G. Ross, Oxford preprint OUTP-90-07P (1990), Nucl. Phys. B, to appear; L. Dixon, V. Kaplunovsky, J. Louis and M. Peskin, SLAC/Texas preprint, to appear. [ 15 ] Y. Katsuki, Y. Kawamura, T. Kobayashi, N. Ohtsubo, Y. Ono and K. Tanioka, preprint DPKU-8906 ( 1989 ).
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