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Intermediate scale symmetry breaking in superconformal theories
Volume 251, number 1
PHYSICS LETTERSB
8 November 1990
Intermediate scale symmetry breaking in superconformal theories D. Bailin, E.K. K a t e c h o u School of Mathematical and Physical Sciences, Universityof Sussex, Brighton BN1 9QH, UK and A. L o v e Department of Physics, Royal Holloway and Bedford New College, University of London, Egham, Surrey TW20 OEX, UK Received 25 June 1990
The possibility of theories with intermediate scale breaking of the flipped SU ( 5 ) of SO(6) × SO(4) subgroups of SO( 10) is studied in the frameworkof tensoring of N= 2 superconformalminimal models.
Four dimensional heterotic string theories with space-time supersymmetry are obtained by arranging for N = 2 world sheet supersymmetry and for the U ( 1 ) charge associated with the N = 2 superconformal algebra to take only odd integral values. A method [ 1 ] for achieving this in a way consistent with modular invariance has been given by Gepner and generalised by Kazama and Suzuki [ 2 ]. His approach employs a conformal anomaly C = 9 realisation of the N = 2 superconformal algebra for the right mover internal degrees of freedom, obtained by tensoring N = 2 minimal models [ 3 ], a replica of this C = 9 realisation for the left movers [so that, in the first instance, the theory is a (2, 2 ) theory ], and a realisation of the remaining C = 13 for the left mover internal degrees of freedom by 13 free bosons compactified on the SO (10) × E8 torus. We shall focus attention on the simplest examples of such models where all minimal model factors are at the same level. There are four possibilities, namely 19, 2 6, 35 and 64, where k r denotes r minimal model factors at level k. For the 19 case, models with SU(3) × S U ( 2 ) X U ( 1 ) gauge group constructed directly have already been obtained [4,5 ]. It is our purpose here to study instead the possibility of obtaining the flipped SU ( 5 ) × U ( 1 ) or SO (6) X SO (4) subgroup of SO (10) at an inter-
mediate scale with spontaneous symmetry breaking to SU(3 ) × SU(2) X U ( 1 ). The models we are considering are obtained by tensoring N = 2 minimal models at level k. The right movers of such a minimal model [ 3 ] are characterised by quantum numbers l, q and s with O<~l<~k, O<<,lq-sl<~l,
l+q+s=Omod2,
(1)
where q is defined mod 2 ( k + 2), and s takes the values 0, 2 for the Neveu-Schwarz (NS) sector, and + 1 for the Ramond (R) sector. The conformal dimension h and the superconformal U ( 1 ) charge Q of the right mover (l, q, s) are given by 1
h= - [ l ( 1 + 2 ) - q 2 1 + Is 2, 4(k+2)
(2)
and Q=-~+
q
½s.
(3)
Similarly, the left movers are characterised by quantum numbers, ~ q and g, which determine the conformal dimension h - a n d the superconformal U( 1 ) quantum number (~. The states (/, q, s ) ( ~ q, g) and ( k - l , q + k + 2 , s + 2 ) ( k - ~ q + k + 2 , g + 2 ) are equivalent.
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
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PHYSICS LETTERS B
The masses mR and m L for right and left movers are given by !s m R2= N R
(4)
+~wE+h--½ ,
l 2 =ArE + ½02+ ½p 2 + / / _ 1 , ~mL
(5)
where NR and NL are the oscillator contributions, h and h-are the conformal dimensions for the states obtained by tensoring the N = 2 minimal models, and w, and p are the weights of SO (2), SO (10) and Es describing the space-time properties of the state and the representation of SO (10) × Es. The Gepner construction [ 1 ] employs generalised GSO projections to ensure odd integral total superconformal U( 1 ) charges and also to ensure alignment of the boundary conditions of the fermions in the various sectors, so that there is a well-defined world sheet supercharge. These projections are conveniently described by associating a vector Va with a right mover state where VR=(W;ql,q2
(6)
. . . . , q , ; Sl , S2 . . . . , S t )
for a model obtained by tensoring r minimal models, and similarly for the left movers. A scalar product is defined by ,
i=l
,
sisi- ~
i=l k i + 2
.
(7)
8 November 1990
Modular invariance correlates the choice of right and left movers for a physical state. We shall restrict attention to the standard modular invariant, in which case
l,=~
( i = 1 , 2 .... , r ) .
(14)
Moreover, for states in the Ramond sector
VL--VRmnoflo +
~ nifli
(15)
i=1
for some integers no and nt, and for states in the Neveu-Schwarz sector VL + (V; 0, 0, ..., 0; 0, 0, ..., 0)-- Vg =noflo+ ~ n~fl~,
(16)
i=l
where the SO (2) and SO (10) weight vectors are to be added in the way conjugacy classes add for either group. The individual N = 2 minimal models admit a Zk+ 2 discrete symmetry whose action on the primary states is to multiply them a phase exp [2ni(q~ + ~ ) / 2 (k~ + 2) ]. It is possible to quotient the theory by ZM subgroups of I], Zk,+2 generated by ~exp[2niyg × ( q i + ( l i ) / 2 ( k i + 2 ) ], where the y~ are integers, and M is the least integer for which
The generalised GSO projections are then associated with the vectors
M ( y l , Yz, ..., Y,) = (0, 0 ..... 0)mod(kl +2, ..., kr + 2 ) •
f l o - - ( s ; 1 , 1 .... , 1 ; 1 , 1 ..... 1 ) ,
(8)
fl,= (~ 0, 0, ..., 0; 0 .... ,0, 2,0 ..... 0 ) ,
(9)
A modular invariant partition function with ZM twisted sectors may be constructed [4,5]. These twisted sectors are conveniently described by the introduction of the vector
where the 2 occurs in the ith position, and s and v denote the highest weights of the spinor and vector representations of SO(2) [or SO(10), for the left movers ]. The corresponding projections are QTOV= 2,80" VR = odd integer,
( 10)
2fl~. VR = even integer,
( 1 1)
where QTOTis the total superconformal U ( 1 ) charge given by QTOT= Q + 2 w ' S .
(12)
For the left movers, similar formulae apply with VR replaced by VL = (~, #~, q2 .... gt,;gl,gZ, ...,g,) • 90
(13)
(17)
F = (0; 71, ~,2, ..., 9,; 0, 0, ..., 0 ) ,
(18)
in terms of which the twisted sectors are given by VL--VR=n0fl0+
~ nifli-2xr i=l
,
(19)
with x = 0, 1..... M - 1. [Additionally for the NeveuSchwarz sector we have to add (14,0, ..., 0; 0, ..., 0) to VL. ] For theories with space-time supersymmetry surviving the modding [ 5 ] the generalised GSO projection associated with the modding of the theory by Fis / " (2VR+ 2noflo +2xF)e71,
(20)
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PHYSICS LETTERSB
or equivalently
~,
Yi
i=l k - - ~
(q~+no+xTi)eZ.
(21)
The gauge fields which enlarge the SO(10) gauge group to E 6 survive the projection, and so a (2, 2) theory survives, when 2#o . r ~ z
(22)
,
or equivalently i=t k - ~
eZ.
(23)
We shall refer to such moddings as (2, 2) moddings. Breaking of the gauge group to a desired intermediate scale symmetry is to be achieved by making an embeddingd of the discrete symmetry ZM in the gauge group. This embedding is a shift on the SO(10) ×Es weight lattice which must be consistent with the homomorphism condition
MA~A,
(24)
where A is the SO (10) × Es root lattice, and with the modular invariance condition M A 2 = 0 mod 2.
(25)
The mass formula (5) is modified by the replacement of O by pC+xA [SO(10) ] a n d p b y p + x A (Es), and the GSO projection (21 ) becomes [ 4, 5 ]
-- ~
i=1
Y~ (q~+no+xy~)+A.(P+xA)_½xA2
eZ,
(26)
where P is a weight on the SO (10) × Es weight lattice (with components ~ and p on the SO(10) and Es weight lattices, respectively). More generally, the theory may be constructed by using more than one modding by a discrete symmetry. For instance, for two moddings y and 6 with associated embeddings A and B, the GSO projection associated with y on the xy +):6 twisted sector is i=1 ~
( q~ +no + x),, +),,~) +.4. (P+xA +yB)
- ½A-(xA+yB)~Z.
(27)
For the 19, 2 4, 3 s and 6 4 tensorings of minimal
8 November 1990
models being considered here the possible choices of discrete symmetries ZM are Z3, Z2 or Z4, Z5, and Z2, 7/4 or Z8, respectively. We shall assume that the standard model S U ( 3 ) × S U ( 2 ) X U ( 1 ) gauge group occurs as a subgroup of the original SO (10), and we shall focus attention on embeddings A with components in the SO(10) factor of SO(10) XEs only, in order to be sure of avoiding the introduction of states in the twisted sectors with non-trivial quantum numbers under both SO (10) and Es and a consequent enlargement of the observable gauge group. From (26), the untwisted sector gauge fields in the adjoint of SO (10) have to satisfy P..4~Z.
(28)
T h e 7/2 case cannot produce a flipped SU(5) or SO (6) × SO (4) intermediate scale gauge group consistently with (24) and (25). However, in the Z3 case, the SO(6) × S O ( 4 ) subgroup U ( 3 ) × S O ( 4 ) arises, for the Za and Zs cases S O ( 6 ) × S O ( 4 ) occurs, and for the Z5 case SU (5) × U ( 1 ) occurs, which may be identified with flipped SU(5). The corresponding choices of A are given in table 1. Other choices of A, for a given intermediate scale gauge group, other than multiples of these are also possible, which differ from these in the order of the entries, by replacing some of the entries ai by ai+ 1, or by replacing some of the entries by 1-a~. (Replacing an entry by a~+2 is equivalent to the original choice because this amounts to adding a root lattice vector.) Cases where some entries are replaced by 1 - a i are equivalent to a case given in table 1 but with a different construction of the quantum numbers associated with the Cartan subalgebra of SO ( 10 ). For the k r model, (2, 2 ) moddings, in the sense of (23), are those for which
Table 1 Embeddings of the discrete group and intermediate scale gauge groups. Model
Discrete group
A
Surviving SO(10) subgroup
19
Z3
( ~3 02)
U(3 ) × SO(4)
26 35 64
7]4 Z~ Z4 or Zs
(½2 0 3 )
SO(6) ×SO(4) SU(5)XU(1)
(25) (½2 03) or (02 ½3)
SO(6) ×SO(4)
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i=1
y~= 0 mod ( k + 2 ) .
PHYSICS LETTERS B (29)
A necessary condition to obtain one of the desired intermediate scale SO (10) subgroups is that the gauge fields enlarging SO(10) to E 6 should be deleted by the generalised GSO projections. The E 6 gauge fields in 16_3/2 and 163/2 o f S O ( 1 0 ) × U ( I ) occur as the states (/'= 0, # = ~ = -T-1 )', for the k ' model. In no case can E 6 be reduced to S O ( 1 0 ) X U ( 1 ) using (2, 2) moddings [in the sense of (29) ] even when A v~0. In all cases except the 19 model, it is possible to delete the 16_3/2 and 163/2 gauge fields using a single (0, 2 ) modding with .4 # 0, by a suitable choice of ~y~. For the 26 model, with A = (½2 03), the requirement is Z y~= 1 or 3 (mod 4 ), for the 35 model with A = ( ] 5) it is ~y~= 1 or 2 (mod 5), and for the 64 model with Zs point group it is ~ y ~ 0 or 4 ( m o d 8 ) when .4=(½203 ) and E y ~ 2 or 6 ( m o d 8 ) when .4= (02 ½3). For the 19 case, a single modding with A # 0 will n o t do the trick, and it is necessary to use at least two moddings, at least one of which is (0, 2). [Of course, a single (0, 2 ) modding with A = 0 will always delete the gauge fields enhancing SO (10) to E6, but then there is n o breaking of the SO (10) gauge group. ] For breaking of the intermediate scale group to be able to occur along a fiat direction, it is necessary for appropriate massless scalars with quantum numbers which occur in 16 + 16 of SO (10) to be present in the theory. Before modding the theory with any discrete group, the 19, 26 , 35 and 64 models are known (see e.g. ref. [6] ) to have 84, 90, 101 and 149 copies of 27 o f E6, respectively. The 35 and 64 models also have a single 27, whereas the 19 and 26 models have none. Thus, in the case of the 35 and 64 models it appears possible, a priori, for the required massless scalars in 10+1--0 of SU(5) or (4, (1, 2 ) ) + ( 4 , (1, 2 ) ) of SO(6) × S O ( 4 ) [ - S O ( 6 ) XSUL(2) × S U R ( 2 ) l to arise from massless states already present in the unmodded theories, and so present in the untwisted sector of the modded theory without using the e x t e n d e d f l - l a t t i c e to link left and right movers. However, if we construct a theory with a single modding (with embedding .4 ) the requirement that this modding removes the gauge fields enhancing SO (10) to E6, while breaking SO (10) to the required intermediate scale gauge group, results in it also deleting the scalars in 16 of SO (10) needed for intermediate scale breaking. 92
8 November 1990
Thus, if we wish to construct a theory with a single modding, it will be necessary for some scalars with quantum numbers in 16 of SO(10) to occur in a twisted sector. The 19 models already need more than one modding to delete all extra E 6 gauge fields enhancing SO(10) to E6. In the 26 models, with A = (½2 03), the x = 2 twisted sector contains a 16 of SO(10) only when y = ( 1 2 0 4 ) . However, for this choice of y n o t all gauge fields enlarging SO (10) to E 6 are deleted. For the 35 models, with A = ( ] 5), states in 10 of SU(5) occur in both the x = l and x = 2 twisted sectors, but n o t with the correct U ( 1 ) quantum numbers to have an electrically neutral component which can be used in intermediate scale breaking. For the 64 models, with A = (½2 03) or (02 ½3), massless scalars in (4, ( 1, 2 ) ) of SO ( 6 ) × SO (4) are to be found in the twisted sectors, but again not with the correct U ( 1 ) quantum numbers to have an electrically neutral component. Similar considerations apply for all choices of A consistent with the above intermediate scalar groups, because in constructing twisted sector states we have to consider all multiples of A and all choices of P on the SO(10) × E s weight lattice, so that replacing an entry at of A by at+ 1 does n o t affect the possible massless states (before generalised GSO projection but after the Gepner projections. ) On the other hand, with more than one modding theories with appropriate massless scalars for intermediate scale breaking can be constructed. For instance, by introducing a second modding d, with n o associated embedding, we may obtain new 16's of SO (10) in y6 twisted sectors. These are states which are constructed using the left movers of the original unmodded theory but with r-lattice extended by 6 so that new ways of linking left and right movers are possible. In all cases, the choices of ~ that allow the construction of new 16's in this way are (2, 2) moddings in the sense of (29). Thus, they do n o t delete any of the gauge fields enhancing SO (10) to E 6. In the 26 , 35 and 64 cases, we must therefore employ the first modding y with A ~ 0 to remove these extra gauge fields, by choosing ~ i in the ways discussed earlier. For the 26 model where the intermediate scale gauge group is SO (6) × SO (4), massless scalars in the (4, (1, 2 ) ) component of 16 of SO(10) (and corresponding components in 16) survive after all GSO projections if, for instance, we take y = (01022 ) with
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PHYSICS LETTERS B
A = (1203) and t~= ( 1402). For the 35 model, with intermediate scale gauge group flipped SU(5), massless scalars in 10 of SU(5) components of 16 of SO (10) (and corresponding components in 16) survive after all GSO projections if, for example, )'----(0310) w i t h A = (25) and 6 = - (3, 2, 03). I n t h e 64 model, massless scalars for intermediate scale breaking of the S O ( 6 ) × S O ( 4 ) gauge group can occur, for instance, when ~,= ( - 1102) with A = ( ½2 03) and 6 = ( - 1212). On the other hand, in the 19 models, a further modding E is needed to complete this construction since in this case 3' alone cannot reduce E 6 to SO (10). The modding e may conveniently be chosen to be a (0, 2) modding with n o associated Wilson line. E.g. y=(02-1120102) with A = ( 2 3 0 2 ) , t~---- ( -- 121205) and ~ = ( 0 3 2 2 0 1 0 2 ) , provides massless scalars in the U ( 3 ) × S O ( 4 ) (1, (1, 2)) components of 16 of SO(10) and corresponding components in 16 for intermediate scale breaking of the U ( 3 ) × S O ( 4 ) subgroup of SO(6) × S O ( 4 ) . In conclusion, under the assumption that the standard model SU (3) × SU (2) × U ( 1 ) gauge group occurs in the natural way in the SO (10) factor of the underlying SO (10) × E8 gauge group, we have found that it is never possible to construct 19 , 26 , 35 o r 64 models with intermediate scale flipped SU(5) or S O ( 6 ) × S O ( 4 ) (or subgroups) using only (2, 2) moddings [in the sense of (23) and (29)] with associated embeddings in the gauge group. Even allow-
8 November 1990
ing (0, 2) moddings it is not possible to construct models in which appropriate scalars for intermediate scale breaking are present using a single modding. However, when at least two moddings are employed, intermediate scale breaking scenarios become possible with intermediate scale gauge group flipped SU(5) × U ( l ) for the 35 models, SO(6) × SO(4) for t h e 26 and 64 models, and, for at least three moddings, U ( 3 ) × S O ( 4 ) for the 19 models. The search in this framework for a fully realistic three generation model with intermediate scale breaking will thus involve searching through models constructed using the large number of (0, 2) moddings of the k r models rather than just the small number of (2, 2) moddings. We would like to thank Dr. S. Gandhi for helpful discussions. This work was supported in part by SERC.
References [ 1 ] D. Gepner, Nucl. Phys. B 296 (1988) 757. [2 ] Y. Kazama and H. Suzuki, Nucl. Phys. B 321 (1989 ) 232. [3] W. Boucher, D. Friedan and A. Kent, Phys. Lett. B 172 (1986) 316. [4] A. Font, L.E. Ibfifiez, M. Mondragon, F. Quevedo and G.G. Ross, Phys. Lett. B 227 (1989) 34. [5] A. Font, L.E. Ibfifiez, F. Quevedo and A. Sierra, CERN preprint CERN-TH5577/89. [6] C.A. LiJtken and G.G. Ross, Phys. Lett. B 213 (1988) 152.
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PHYSICS LETTERSB
8 November 1990
Intermediate scale symmetry breaking in superconformal theories D. Bailin, E.K. K a t e c h o u School of Mathematical and Physical Sciences, Universityof Sussex, Brighton BN1 9QH, UK and A. L o v e Department of Physics, Royal Holloway and Bedford New College, University of London, Egham, Surrey TW20 OEX, UK Received 25 June 1990
The possibility of theories with intermediate scale breaking of the flipped SU ( 5 ) of SO(6) × SO(4) subgroups of SO( 10) is studied in the frameworkof tensoring of N= 2 superconformalminimal models.
Four dimensional heterotic string theories with space-time supersymmetry are obtained by arranging for N = 2 world sheet supersymmetry and for the U ( 1 ) charge associated with the N = 2 superconformal algebra to take only odd integral values. A method [ 1 ] for achieving this in a way consistent with modular invariance has been given by Gepner and generalised by Kazama and Suzuki [ 2 ]. His approach employs a conformal anomaly C = 9 realisation of the N = 2 superconformal algebra for the right mover internal degrees of freedom, obtained by tensoring N = 2 minimal models [ 3 ], a replica of this C = 9 realisation for the left movers [so that, in the first instance, the theory is a (2, 2 ) theory ], and a realisation of the remaining C = 13 for the left mover internal degrees of freedom by 13 free bosons compactified on the SO (10) × E8 torus. We shall focus attention on the simplest examples of such models where all minimal model factors are at the same level. There are four possibilities, namely 19, 2 6, 35 and 64, where k r denotes r minimal model factors at level k. For the 19 case, models with SU(3) × S U ( 2 ) X U ( 1 ) gauge group constructed directly have already been obtained [4,5 ]. It is our purpose here to study instead the possibility of obtaining the flipped SU ( 5 ) × U ( 1 ) or SO (6) X SO (4) subgroup of SO (10) at an inter-
mediate scale with spontaneous symmetry breaking to SU(3 ) × SU(2) X U ( 1 ). The models we are considering are obtained by tensoring N = 2 minimal models at level k. The right movers of such a minimal model [ 3 ] are characterised by quantum numbers l, q and s with O<~l<~k, O<<,lq-sl<~l,
l+q+s=Omod2,
(1)
where q is defined mod 2 ( k + 2), and s takes the values 0, 2 for the Neveu-Schwarz (NS) sector, and + 1 for the Ramond (R) sector. The conformal dimension h and the superconformal U ( 1 ) charge Q of the right mover (l, q, s) are given by 1
h= - [ l ( 1 + 2 ) - q 2 1 + Is 2, 4(k+2)
(2)
and Q=-~+
q
½s.
(3)
Similarly, the left movers are characterised by quantum numbers, ~ q and g, which determine the conformal dimension h - a n d the superconformal U( 1 ) quantum number (~. The states (/, q, s ) ( ~ q, g) and ( k - l , q + k + 2 , s + 2 ) ( k - ~ q + k + 2 , g + 2 ) are equivalent.
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
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Volume 251, number 1
PHYSICS LETTERS B
The masses mR and m L for right and left movers are given by !s m R2= N R
(4)
+~wE+h--½ ,
l 2 =ArE + ½02+ ½p 2 + / / _ 1 , ~mL
(5)
where NR and NL are the oscillator contributions, h and h-are the conformal dimensions for the states obtained by tensoring the N = 2 minimal models, and w, and p are the weights of SO (2), SO (10) and Es describing the space-time properties of the state and the representation of SO (10) × Es. The Gepner construction [ 1 ] employs generalised GSO projections to ensure odd integral total superconformal U( 1 ) charges and also to ensure alignment of the boundary conditions of the fermions in the various sectors, so that there is a well-defined world sheet supercharge. These projections are conveniently described by associating a vector Va with a right mover state where VR=(W;ql,q2
(6)
. . . . , q , ; Sl , S2 . . . . , S t )
for a model obtained by tensoring r minimal models, and similarly for the left movers. A scalar product is defined by ,
i=l
,
sisi- ~
i=l k i + 2
.
(7)
8 November 1990
Modular invariance correlates the choice of right and left movers for a physical state. We shall restrict attention to the standard modular invariant, in which case
l,=~
( i = 1 , 2 .... , r ) .
(14)
Moreover, for states in the Ramond sector
VL--VRmnoflo +
~ nifli
(15)
i=1
for some integers no and nt, and for states in the Neveu-Schwarz sector VL + (V; 0, 0, ..., 0; 0, 0, ..., 0)-- Vg =noflo+ ~ n~fl~,
(16)
i=l
where the SO (2) and SO (10) weight vectors are to be added in the way conjugacy classes add for either group. The individual N = 2 minimal models admit a Zk+ 2 discrete symmetry whose action on the primary states is to multiply them a phase exp [2ni(q~ + ~ ) / 2 (k~ + 2) ]. It is possible to quotient the theory by ZM subgroups of I], Zk,+2 generated by ~exp[2niyg × ( q i + ( l i ) / 2 ( k i + 2 ) ], where the y~ are integers, and M is the least integer for which
The generalised GSO projections are then associated with the vectors
M ( y l , Yz, ..., Y,) = (0, 0 ..... 0)mod(kl +2, ..., kr + 2 ) •
f l o - - ( s ; 1 , 1 .... , 1 ; 1 , 1 ..... 1 ) ,
(8)
fl,= (~ 0, 0, ..., 0; 0 .... ,0, 2,0 ..... 0 ) ,
(9)
A modular invariant partition function with ZM twisted sectors may be constructed [4,5]. These twisted sectors are conveniently described by the introduction of the vector
where the 2 occurs in the ith position, and s and v denote the highest weights of the spinor and vector representations of SO(2) [or SO(10), for the left movers ]. The corresponding projections are QTOV= 2,80" VR = odd integer,
( 10)
2fl~. VR = even integer,
( 1 1)
where QTOTis the total superconformal U ( 1 ) charge given by QTOT= Q + 2 w ' S .
(12)
For the left movers, similar formulae apply with VR replaced by VL = (~, #~, q2 .... gt,;gl,gZ, ...,g,) • 90
(13)
(17)
F = (0; 71, ~,2, ..., 9,; 0, 0, ..., 0 ) ,
(18)
in terms of which the twisted sectors are given by VL--VR=n0fl0+
~ nifli-2xr i=l
,
(19)
with x = 0, 1..... M - 1. [Additionally for the NeveuSchwarz sector we have to add (14,0, ..., 0; 0, ..., 0) to VL. ] For theories with space-time supersymmetry surviving the modding [ 5 ] the generalised GSO projection associated with the modding of the theory by Fis / " (2VR+ 2noflo +2xF)e71,
(20)
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PHYSICS LETTERSB
or equivalently
~,
Yi
i=l k - - ~
(q~+no+xTi)eZ.
(21)
The gauge fields which enlarge the SO(10) gauge group to E 6 survive the projection, and so a (2, 2) theory survives, when 2#o . r ~ z
(22)
,
or equivalently i=t k - ~
eZ.
(23)
We shall refer to such moddings as (2, 2) moddings. Breaking of the gauge group to a desired intermediate scale symmetry is to be achieved by making an embeddingd of the discrete symmetry ZM in the gauge group. This embedding is a shift on the SO(10) ×Es weight lattice which must be consistent with the homomorphism condition
MA~A,
(24)
where A is the SO (10) × Es root lattice, and with the modular invariance condition M A 2 = 0 mod 2.
(25)
The mass formula (5) is modified by the replacement of O by pC+xA [SO(10) ] a n d p b y p + x A (Es), and the GSO projection (21 ) becomes [ 4, 5 ]
-- ~
i=1
Y~ (q~+no+xy~)+A.(P+xA)_½xA2
eZ,
(26)
where P is a weight on the SO (10) × Es weight lattice (with components ~ and p on the SO(10) and Es weight lattices, respectively). More generally, the theory may be constructed by using more than one modding by a discrete symmetry. For instance, for two moddings y and 6 with associated embeddings A and B, the GSO projection associated with y on the xy +):6 twisted sector is i=1 ~
( q~ +no + x),, +),,~) +.4. (P+xA +yB)
- ½A-(xA+yB)~Z.
(27)
For the 19, 2 4, 3 s and 6 4 tensorings of minimal
8 November 1990
models being considered here the possible choices of discrete symmetries ZM are Z3, Z2 or Z4, Z5, and Z2, 7/4 or Z8, respectively. We shall assume that the standard model S U ( 3 ) × S U ( 2 ) X U ( 1 ) gauge group occurs as a subgroup of the original SO (10), and we shall focus attention on embeddings A with components in the SO(10) factor of SO(10) XEs only, in order to be sure of avoiding the introduction of states in the twisted sectors with non-trivial quantum numbers under both SO (10) and Es and a consequent enlargement of the observable gauge group. From (26), the untwisted sector gauge fields in the adjoint of SO (10) have to satisfy P..4~Z.
(28)
T h e 7/2 case cannot produce a flipped SU(5) or SO (6) × SO (4) intermediate scale gauge group consistently with (24) and (25). However, in the Z3 case, the SO(6) × S O ( 4 ) subgroup U ( 3 ) × S O ( 4 ) arises, for the Za and Zs cases S O ( 6 ) × S O ( 4 ) occurs, and for the Z5 case SU (5) × U ( 1 ) occurs, which may be identified with flipped SU(5). The corresponding choices of A are given in table 1. Other choices of A, for a given intermediate scale gauge group, other than multiples of these are also possible, which differ from these in the order of the entries, by replacing some of the entries ai by ai+ 1, or by replacing some of the entries by 1-a~. (Replacing an entry by a~+2 is equivalent to the original choice because this amounts to adding a root lattice vector.) Cases where some entries are replaced by 1 - a i are equivalent to a case given in table 1 but with a different construction of the quantum numbers associated with the Cartan subalgebra of SO ( 10 ). For the k r model, (2, 2 ) moddings, in the sense of (23), are those for which
Table 1 Embeddings of the discrete group and intermediate scale gauge groups. Model
Discrete group
A
Surviving SO(10) subgroup
19
Z3
( ~3 02)
U(3 ) × SO(4)
26 35 64
7]4 Z~ Z4 or Zs
(½2 0 3 )
SO(6) ×SO(4) SU(5)XU(1)
(25) (½2 03) or (02 ½3)
SO(6) ×SO(4)
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Volume 251, number 1
i=1
y~= 0 mod ( k + 2 ) .
PHYSICS LETTERS B (29)
A necessary condition to obtain one of the desired intermediate scale SO (10) subgroups is that the gauge fields enlarging SO(10) to E 6 should be deleted by the generalised GSO projections. The E 6 gauge fields in 16_3/2 and 163/2 o f S O ( 1 0 ) × U ( I ) occur as the states (/'= 0, # = ~ = -T-1 )', for the k ' model. In no case can E 6 be reduced to S O ( 1 0 ) X U ( 1 ) using (2, 2) moddings [in the sense of (29) ] even when A v~0. In all cases except the 19 model, it is possible to delete the 16_3/2 and 163/2 gauge fields using a single (0, 2 ) modding with .4 # 0, by a suitable choice of ~y~. For the 26 model, with A = (½2 03), the requirement is Z y~= 1 or 3 (mod 4 ), for the 35 model with A = ( ] 5) it is ~y~= 1 or 2 (mod 5), and for the 64 model with Zs point group it is ~ y ~ 0 or 4 ( m o d 8 ) when .4=(½203 ) and E y ~ 2 or 6 ( m o d 8 ) when .4= (02 ½3). For the 19 case, a single modding with A # 0 will n o t do the trick, and it is necessary to use at least two moddings, at least one of which is (0, 2). [Of course, a single (0, 2 ) modding with A = 0 will always delete the gauge fields enhancing SO (10) to E6, but then there is n o breaking of the SO (10) gauge group. ] For breaking of the intermediate scale group to be able to occur along a fiat direction, it is necessary for appropriate massless scalars with quantum numbers which occur in 16 + 16 of SO (10) to be present in the theory. Before modding the theory with any discrete group, the 19, 26 , 35 and 64 models are known (see e.g. ref. [6] ) to have 84, 90, 101 and 149 copies of 27 o f E6, respectively. The 35 and 64 models also have a single 27, whereas the 19 and 26 models have none. Thus, in the case of the 35 and 64 models it appears possible, a priori, for the required massless scalars in 10+1--0 of SU(5) or (4, (1, 2 ) ) + ( 4 , (1, 2 ) ) of SO(6) × S O ( 4 ) [ - S O ( 6 ) XSUL(2) × S U R ( 2 ) l to arise from massless states already present in the unmodded theories, and so present in the untwisted sector of the modded theory without using the e x t e n d e d f l - l a t t i c e to link left and right movers. However, if we construct a theory with a single modding (with embedding .4 ) the requirement that this modding removes the gauge fields enhancing SO (10) to E6, while breaking SO (10) to the required intermediate scale gauge group, results in it also deleting the scalars in 16 of SO (10) needed for intermediate scale breaking. 92
8 November 1990
Thus, if we wish to construct a theory with a single modding, it will be necessary for some scalars with quantum numbers in 16 of SO(10) to occur in a twisted sector. The 19 models already need more than one modding to delete all extra E 6 gauge fields enhancing SO(10) to E6. In the 26 models, with A = (½2 03), the x = 2 twisted sector contains a 16 of SO(10) only when y = ( 1 2 0 4 ) . However, for this choice of y n o t all gauge fields enlarging SO (10) to E 6 are deleted. For the 35 models, with A = ( ] 5), states in 10 of SU(5) occur in both the x = l and x = 2 twisted sectors, but n o t with the correct U ( 1 ) quantum numbers to have an electrically neutral component which can be used in intermediate scale breaking. For the 64 models, with A = (½2 03) or (02 ½3), massless scalars in (4, ( 1, 2 ) ) of SO ( 6 ) × SO (4) are to be found in the twisted sectors, but again not with the correct U ( 1 ) quantum numbers to have an electrically neutral component. Similar considerations apply for all choices of A consistent with the above intermediate scalar groups, because in constructing twisted sector states we have to consider all multiples of A and all choices of P on the SO(10) × E s weight lattice, so that replacing an entry at of A by at+ 1 does n o t affect the possible massless states (before generalised GSO projection but after the Gepner projections. ) On the other hand, with more than one modding theories with appropriate massless scalars for intermediate scale breaking can be constructed. For instance, by introducing a second modding d, with n o associated embedding, we may obtain new 16's of SO (10) in y6 twisted sectors. These are states which are constructed using the left movers of the original unmodded theory but with r-lattice extended by 6 so that new ways of linking left and right movers are possible. In all cases, the choices of ~ that allow the construction of new 16's in this way are (2, 2) moddings in the sense of (29). Thus, they do n o t delete any of the gauge fields enhancing SO (10) to E 6. In the 26 , 35 and 64 cases, we must therefore employ the first modding y with A ~ 0 to remove these extra gauge fields, by choosing ~ i in the ways discussed earlier. For the 26 model where the intermediate scale gauge group is SO (6) × SO (4), massless scalars in the (4, (1, 2 ) ) component of 16 of SO(10) (and corresponding components in 16) survive after all GSO projections if, for instance, we take y = (01022 ) with
Volume 251, number 1
PHYSICS LETTERS B
A = (1203) and t~= ( 1402). For the 35 model, with intermediate scale gauge group flipped SU(5), massless scalars in 10 of SU(5) components of 16 of SO (10) (and corresponding components in 16) survive after all GSO projections if, for example, )'----(0310) w i t h A = (25) and 6 = - (3, 2, 03). I n t h e 64 model, massless scalars for intermediate scale breaking of the S O ( 6 ) × S O ( 4 ) gauge group can occur, for instance, when ~,= ( - 1102) with A = ( ½2 03) and 6 = ( - 1212). On the other hand, in the 19 models, a further modding E is needed to complete this construction since in this case 3' alone cannot reduce E 6 to SO (10). The modding e may conveniently be chosen to be a (0, 2) modding with n o associated Wilson line. E.g. y=(02-1120102) with A = ( 2 3 0 2 ) , t~---- ( -- 121205) and ~ = ( 0 3 2 2 0 1 0 2 ) , provides massless scalars in the U ( 3 ) × S O ( 4 ) (1, (1, 2)) components of 16 of SO(10) and corresponding components in 16 for intermediate scale breaking of the U ( 3 ) × S O ( 4 ) subgroup of SO(6) × S O ( 4 ) . In conclusion, under the assumption that the standard model SU (3) × SU (2) × U ( 1 ) gauge group occurs in the natural way in the SO (10) factor of the underlying SO (10) × E8 gauge group, we have found that it is never possible to construct 19 , 26 , 35 o r 64 models with intermediate scale flipped SU(5) or S O ( 6 ) × S O ( 4 ) (or subgroups) using only (2, 2) moddings [in the sense of (23) and (29)] with associated embeddings in the gauge group. Even allow-
8 November 1990
ing (0, 2) moddings it is not possible to construct models in which appropriate scalars for intermediate scale breaking are present using a single modding. However, when at least two moddings are employed, intermediate scale breaking scenarios become possible with intermediate scale gauge group flipped SU(5) × U ( l ) for the 35 models, SO(6) × SO(4) for t h e 26 and 64 models, and, for at least three moddings, U ( 3 ) × S O ( 4 ) for the 19 models. The search in this framework for a fully realistic three generation model with intermediate scale breaking will thus involve searching through models constructed using the large number of (0, 2) moddings of the k r models rather than just the small number of (2, 2) moddings. We would like to thank Dr. S. Gandhi for helpful discussions. This work was supported in part by SERC.
References [ 1 ] D. Gepner, Nucl. Phys. B 296 (1988) 757. [2 ] Y. Kazama and H. Suzuki, Nucl. Phys. B 321 (1989 ) 232. [3] W. Boucher, D. Friedan and A. Kent, Phys. Lett. B 172 (1986) 316. [4] A. Font, L.E. Ibfifiez, M. Mondragon, F. Quevedo and G.G. Ross, Phys. Lett. B 227 (1989) 34. [5] A. Font, L.E. Ibfifiez, F. Quevedo and A. Sierra, CERN preprint CERN-TH5577/89. [6] C.A. LiJtken and G.G. Ross, Phys. Lett. B 213 (1988) 152.
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