The effective interactions of chiral families in four-dimensional superstrings

Volume 194, number 3

PHYSICS LETTERS B

13 August 1987

THE EFFECTIVE INTERACTIONS OF CHIRAL FAMILIES IN FOUR-DIMENSIONAL SUPERSTRINGS ~ S. F E R R A R A CERN, CH-1211 Geneva 23, Switzerland and Physics Department, UCLA, Los Angeles, CA 90024, USA L. G I R A R D E L L O Physics Department, UCLA, Los Angeles, CA 90024, USA Dipartimento di Fisica, Universith di Milano, and INFN, Sezione di Milano, 1-20133 Milan, Italy C. K O U N N A S l Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA and Department of Physics, UCB, Berkeley, CA 94720, USA

and M. P O R R A T I 2 Physics Department, UCLX, Los dngeles, CA 90024, USA

Received 18 May 1987

The low-energyeffective interactions in four-dimensional superstrings with N=2 and N= 1 space-time supersymmetry and massless twisted (family) sector are obtained. Our results rely on some general symmetry properties of superstring particle states and on tensor-calculustechniques for supergravity couplings.The novel feature is that the N= 2 quaternionic manifold and N= 1 K~ihlerspace of the scalar superpartners of family multiplets are non-symmetric spaces whose structure can be obtained by "integrating out" the massive superstring modes.

Recently superstring theories in four-dimensional space-time [ 1-5] have been constructed using different versions of the underlying two-dimensional superconformal field theory. A particularly convenient way of constructing these "~This work was suported in part by the Director, Officeof Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the US Department of Energy under Contract DE-AC03-76SF00098 and in part by DOE under grant number DE-AA03-76SF00034,NSF, under grant number PHY8118547, and Ministero della Pubblica Istruzione, Italy. On leave of absence from LPT, Ecole Normale Sup6rieure, F-75231 Paris, France. 2 Address after October 1, 1987: Department of Physics, University of California, Berkeley, CA 94720, USA. 358

models makes use of two-dimensional free fermions [ 2 - 4 ] to describe the internal degrees of freedom of the string ( f e r m i o n i z a t i o n of the internal coordinates), carrying a non-linear realization of the world sheet supersymmetry [2]. Consistency of the models severely restricts the choices of the fermionic spin structures. As a result of (e.g. ref. [4]) modular invariance, absence of local anomalies and factorization define a class of superstring theories with gauge groups not necessarily contained in SO(32) or E8×E~ [6,7] a n d with one or more u n b r o k e n s p a c e - t i m e supersymmetries. For physical reasons the interesting models are those with N = 1 target space-time supersymmetry and families chiral under some gauge group

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G = SU(3) × S U ( 2 ) XU(1 ). The existence of chiral families further selects the possible choice of consistent theories, since their presence is related to the possibility of having a "massless twisted sector" in these models [3-5]. If we consider N = 4 , 4D superstring models defined by n fermionic number projections Z~ × . . . x Z ~ , N = 2 and N = 1 models can be obtained by some extra Z2 and Z2 × Z~ projections [4,8,9]. The theory defined in this way has the following characteristics: a sector of the N = 2 and N = 1 models can be obtained by Z2 and Z 2 × Z ; truncations of the corresponding N = 4 model. This we call "untwisted sector" [ 8]. Another sector of the N = 2 and N = 1 models is separately Z2 or Z2NZI invariant, and cannot be obtained by N = 4 truncations. This sector will be hereafter called "twisted". If we confine ourselves, for simplicity, to the N = 4 SO (44) superstring model, the corresponding N = 2 and N = 1 models have gauge groups respectively given by N=2

SO(nA)×SO(nB),

/7A = 2 r +

nA+nB=44,

8m,

(la)

3

N=I

IqSO(n,),

no=2+4k,

13 August 1987

SO(10) × S O ( 6 ) ×SO(14) × S O ( 1 4 ) .

(3)

The chiral families are in the (16, 4) and (16, 4) representations of SO(10) × SO(6) and are singlets under the SO(14) ×SO(14) gauge symmetry. In this particular model there are 16 families whose number can be reduced by further Z2 projections acting on the gauge group, down to 8, 4 or 2 families. Another interesting model is obtained by considering Z2 truncations of the N = 4 model based on the gauge group SO (28) × Es. We will discuss some features of this model at the end of this paper. Our basicpoint is to derive the N = 2 lagrangian [ 11,12] in the presence of a massless "twisted" sector and perform a Zz truncation in order to get the final N = 1 effective lagrangian. Since the N = 2 massless twisted sector cannot be obtained by an N = 4 truncation we have to use other methods to derive the effective N = 2 theory. As it was already explained in ref. [ 8 ], in order to specify the complete N = 2 theory, one needs only the structure of the quaternionic manifold for the "matter" hypermultiplets. In the absence of a twisted sector, the matter hypermultiplets belong to the quaternionic manifold [8,9],

i 0

yN°,= no + n , = 0 mod 8,

no+nl+n2+n3=44.

(lb)

Chiral families, due to the presence of a massless twisted sector, only exist if nA=8or

16,

no+ni=8orl6

forN=2 for at least one i, f o r N = l .

(2)

In recent papers, effective interactions for N = 2 and N = 1 theories without massless twisted sector were obtained [8,9] by suitable truncations of the N = 4 low-energy effective theory, which was completely specified because of N = 4 local space-time supersymmetry [ 10 ]. The aim of the present work is to describe the lowenergy effective lagrangian in the most relevant case of a massless twisted sector. We will derive this lagrangian in the case of the "minimal" N = 1 models obtained by a Z; truncation of the SO (16) × SO (28) N = 2 model [8], containing a massless twisted sector. Requiring one of the SO(n,) to admit complex representations we are led to choose the group

SO(4, Nm) SO(4) × SO(Nm) '

(4)

where Nu, are the untwisted hypermultiplet states. To obtain the twisted sector we will make the only assumption that the entire quaternionic manifold could be obtained as a quaternionic quotient [ 12,13 ] by quaternionic isometries of a non compact version of a quaternionic projective space Hpn=

Sp(1 + n ) Sp(1) × S p ( n ) "

(5)

It is known, for example, that y N c a n be obtained as an SU(2) quotient of HP x+3 [13]. This result is automatic for the untwisted states, by performing a Z2 truncation of the SO(6, n)/[SO(6)×SO(N)] manifold of the N = 4 theory. However, it is most important to observe that the same result can be obtained by considering the "effective" lagrangian coming from the superstring interaction, after integration of the string massive modes [ 14]. As it is well known, ~-model interactions can be obtained by integrating over "auxiliary" (non-prop359

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agating) vector fields in a massless field theory. However, the same result can be obtained if massive fields are coupled to the massless modes, in the limit of infinite mass. This is precisely what happens in string theories. The a-model interactions only arise after integration over "massive" spin-one fields which have precisely th e quantum numbers of the quaternionic quotient. This is the analogon for internal symmetries, of the mechanism which restores general coordinate transformation invariance in string theories [14]. There, also, four-graviton coupling, and in general all the couplings necessary in order to recover the nonpolynominal structure of the Einstein equations, were obtained by integrating over massive intermediate states [ 14]. With this technique we are able to extend the amodel structure to the massless twisted sector, and to determine the precise interference terms between twisted and untwisted states. Before derivation we anticipate the results: the quaternionic manifold of the N = 2 model is a nonsymmetric, non-homogeneous quaternionic a-model, obtained as a specific quaternionic quotient of HP". The N = 1 truncation leads to a k~ihlerian a-model with manifold

SU(1, 1 )

-

-

U(I)

x

SO(2, no) SO(2) x S O ( n o )

X.W,

(6)

where J{ is a non symmetric non homogeneous K~ihler space. If we drop the twisted states this manifold reduces correctly to a product of two SO(2, n)/ [ SO (2) X SO (n) ] manifolds, while, by dropping the untwisted states, it reduces to the irreducible CP ~ non-compact manifold suggested in ref. [8]. We pass now to the derivation of the above results. As it is well known, the massive fields, whose integration gives rise to modifications of the kinetic terms of the effective action, and thus changes the geometry of the low energy theory, are those only of spin one. The geometry of the massless scalar manifold is, more precisely, determined by the knowledge of its local "ungauged" symmetries, i.e. whose gauge vectors are composite non propagating fields. In string theories those vectors are physical states: they are massive spin one in the limit of infinite mass. They are equivalent to nonpropagating fields in the region where the exchanged m o m e n t u m p 2 << M 2, so 1/(pZ-M 2) ... 1/MZ+O(p2/M4). (Here M2--360

13 August 1987

O ( a ' - ~ ) . ) For this reason, it is necessary to find all the massive vectors which couple to massless scalars as gauge vectors of some symmetry group G ' . This symmetry group is, obviously, always a subgroup of the massless sector global symmetry group G. Actually, G' = G , as we will demonstrate by constructing explicitly massive physical vertices for all the generators of G. A symmetry of the massless scalar I~') indeed, is an operator M: I~/)+ 8 l ~ ' ) = I q/) + M I ~/), where I~u') is another physical massless scalar. Without loss of generality we may examine the case where M is a product of the modes F,,,

=~ dz F(z)z

m+.~--

1

( F m = ~ d~ ~-~i P(Z)Y m+~ ' )

(7)

of the various (bosonic and fermionic) operators defining the conformal two-dimensional QFT [3,4]. Here 2 is the conformal weight of F. M by definition commutes with all the fermionic number projections which specify the 4D theory. From M it is possible to obtain a two-dimensional conformal field Mh'F'(z), of left conformal weight h and right conformal weight/7, by substituting everywhere

Fm~F(z) = Z F, z-"-~, n

P~-~F(z) = E F . z - " '-.

(8)

n

MB'~(z, Z) tOO, commutes with all fermionic number projections, because they are defined in the same way for all the modes Fm of F(z). In order to build a vertex for a spin one massive physical field we can use all the conformal fields of the underlying 2D QFT. We use in particular (see for a review of the formalism e.g, ref. [14]) ~ = DX~(z) = ~'u(z) +

Oazx~d(z)

of weight (½, 0),

O:x~(z)

of weight (0, 1),

:exp{ ik, t XL, ( z ) +x~(Z)]: of weight (tk=, ½k2) .

(9)

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PHYSICS LETTERS B

With Mh'h(z, i) and the fields in eq. (9), a vertex operator for vector fields, commuting with all the fermionic number projection is V(~, k ) = f dO dzdZ~'Ft, MhY'(z, Y.)

x exp{ ik~'[ X~A(z ) +x,R(z)]}.

(10)

13 August 1987

straint (the second term on the right-hand side of eq.

(13)). The N = 2 SO(16)×SO(28) model is defined, using the notation of ref. [4], by the base, {F, S, b~} where bl = {~", Z 1, Z z, y3, y4, yS, y6, 91 ..... 916}. We remind that the matter content of the 2D heterotic type fermionic string is given, in the light cone, by the left-moving fields

¢" is the polarization vector and F u is made out of tP,, O~xR(f) and spin fields consistent with the fermionic projections, and derivatives (or spinorial derivatives) thereof, and has conformal weight

Z'(z), yZ(z), e/(z),

(m,h-ft+m+½),

m=½,1,~,...,

ifh-ft>~0,

and by the right-moving ones,

( f t - h + m - ½ , m),

m=½,1,~,...,

if/~-h>0.

O~Xf_, /t=2, 3; I= 1, 2, ..., 6,

(14)

O~xf~(f), /~=2, 3, (11) The mass shell condition is k 2 = m + h - ½. For symmetries of the massless sector o f a 4D string, an operator F u with one of the dimensions listed in eq. (11 ) can always be constructed for every generator M of G, so G ' = G . Other massive vectors, whose quantum numbers do not belong to the adjoint of G, can couple to massless scalars. However, they do not modify the structure of the kinetic term in the infinite mass limit. This is easily seen in the following example, dealing, for simplicity, with the N = 4, SO(44) theory. There [1,3,4], the massless scalars tb~ have indices I s v e c t o r SO(6) and Aeadjoint SO(44). The most general minimal couplings of a vector to 9~ are

A'~B ~tl1,4 gI~ 0~1] plJ ~ B} '

4-~

j,

~t"8 ~r l[,4 O, qb~'1

~,ulJ

(12)

where { , } means symmetrization, and [ , ] antisymmetrization. Integrating out AJL~ one ends up with four scalar (]~ [1 ~ Jl 2 interaction terms of the form (--/A~uq~BI) or (9 [.10,9~]) • Thanks to the identity d ) [1 ~¢"~¢hJ] ) 2 ~ J] ~ J] (~,~W,~.B~ =2(¢~[1 0,,94 )(9~[1 0.gB )

+ 4 ( ~ . t b J c d . , B _ ~.,g , JB)O,,OAO,9J~ , ,

9A(z),

A = l , 2 ..... 44.

(15)

After the fermionic number projections defined by {F, S, b~} the massless physical hypermultiplet scalars are ZL,,2941/z9 B_1,210),

A=1,...,16, B = 1 7 .... , 4 4 , bl" S ) , i,

i~2ofSU(2),

(16a)

a~spinSO(20) +

(twisted sector).

(16b)

The index i is in the 2 of SU(2) contained in S U ( 2 ) x S U ( 2 ) ' = S O ( 4 ) , and moving the indices of the ;(3, Z4, Zs, Z6 fields. Spin SO(20) + under the gauge group SO(16) and SU(2)" x S U ( 2 ) " acting on y~, ye, y3, y4, decomposes as spin SO(20)+ = ( 2 , 1 , 1 2 8 + ) + ( 1 , 2, 1 2 8 - ) , (17) 128 + and 128 are the positive and negative chirality spinorial representations of SO(16). In this specific model the operators Mh'r'(z, Z) in eq. (10) are, for the untwisted sector,

m~l.°)(z, .¢)=9(A(z)gB] (Z), (13)

and to the similar one with A, B and I, J interchanged, all the contact interactions among scalars are of the form arising from integration of the "gauge" fields A[, A'I'II and Al,[l,j ] and corrections to the kinetic terms due to the presence of a scalar con-

M(°'2)( z, ~) =g[4( ~)gR( z)gC( z)oD]( z) , M(J'°)(z, ,f) =X[l(z)xJ](z) ,

(18)

generating the symmetry SO (4) X SO (498), as it is expected from ref. [8]; and for the twisted sector 361

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PHYSICS LETTERS B

M("/2"/2) ( z, z ) : y [ 1 ' ( z ) . . . y l ' " ( z ) O J ' ( z ) . . . ~ J " ] ( Z )

,

M ( l ' ° ) ( Z, "2)=Z[J(Z))~JI(z) ,

n+m=even, L e ( 3 , 4, 5, 6),

m=0,1,2,

n=0,1,...,16,

Jes (1, 2,..., 16).

(19)

The operators in eq. (19) generate the symmetry of the twisted sector. Applying M('"/2,"/21(z, ~) and M(~'°)(z, ~) to the twisted states Ib~'S> one sees that this symmetry is Sp(512) xSp(1 ). From the above discussion we are led to consider an N = 2 model in which the basic fields are N = 2 vector multiplets, whose scalars z ~ are complex coordinates of a K~ihler manifold [ 8,9 ] SU(1, 1 ) - - X U(1)

SO(2, 498) SO(2) x S O ( 4 9 8 ) '

4 9 8 = d i m SO(16) + d i m S O ( 2 8 ) ,

(20)

while the hypermultiplets divide in two categories. In the first are the hypermultiplets belonging to the untwisted sector. They transform in the (16, 28) representation of S O ( 1 6 ) x S O ( 2 8 ) . The second consists of hypermultiplets in the (128++ 128-, 1) of the same group, and forms the twisted states. The only assumption we need, in order to derive the N = 2 low energy theory, is that its effective lagrangian could be written with in the N = 2 superconformal tensor calculus [ 12]. With this information, the interactions of the hypermultiplets are obtained by assigning them to (a non compact version of) the projective quaternionic space S p ( 4 + N ) / [ S p ( 1 ) x S p ( 3 + N ) ] ( N = 4 4 8 + 5 1 2 ) and by coupling them to a non-propagating SU(2) N = 2 vector multiplet. The SU (2) couples here only to the untwisted states. In this way we recover all the local, non-gauge symmetries we found from string theory. There are indeed, because S U ( 2 ) ~ S p ( I ) , non propagating composite gauge vectors in the adjoint of SO(4) x S O ( 4 4 8 ) x S p ( 5 1 2 ) . If we introduce the compensator fields AJ,~ [12], where (i, e¢) are S U ( 2 ) c x S U ( 2 ) indices, SU(2)~ begin the group rotating the two gravitini, the quaternionic manifold is defined by the following constraints: IA~ 12_ jAR.~1 2 362

IA;~I ~

=1,

13 August 1987

Ajp ~.,~I~.tn/p~ (A~ ~*~P"(A~) = 0

(21)

where A ,~ denotes the untwisted fields transforming as ( 2, 2 ) with respect to SU ( 2 ) c x SU ( 2 ) (R is in the (16, 28) of the gauge group), and the A s are the twisted fields which transform as (2, 1) under S U ( 2 ) c X S U ( 2 ) (S is in the (128++128 , 1) of SO (16) X SO (28)). These constraints come from an N = 2 conformal supergravity theory coupled to an additional SU(2) non-propagating gauge multiplet. At the string level, this gauge multiplet is actually a massive field which has to be integrated out. The effect of this integration is to produce the non-linear constraints we gave in eq. (21 ). Since we know the N = 2 constraints we can now perform a further Z2 truncation to get the corresponding N = 1 theory. In order to fully specify the N = 1 model we restrict ourselves to an N = 1 model with a single twisted sector, based on the SO(10) x S O ( 6 ) x S O ( 1 4 ) x S O ( 1 4 ' ) gauge group. Starting from the N = 2 theory discussed above, the N = 1 effective theory is obtained by splitting the SU (2) c x SU (2) indices ( i and a ) in + and - , and defining the Z2 action on them by I+)-o[+),

I-)-ogl-),

g~Z2.

(22)

Similarly the S O ( 1 6 ) × S O ( 2 8 ) indices are decomposed under SO(10) ×SO(6) ×SO(14) xSO(14)_' as 128+-.(16,4, 1, 1 ) + ( 1 6 , 4 , 1,1); 128 -o(16,4, 1, 1 ) + ( 16,4, 1, 1 ). Vect. SO(16)-o 10+ 6; vect. SO(28) -o 14 + 14'. The action of Z2 is 16-o16,

16-og16,

14-o14,

14'-og14',

10-ol0,

6-og6, g~Z2.

(23)

Calling now by P the indices in the (16, 4 ) + ( 1 6 , 4 ) , B + those in (10, 1 4 ) + ( 6 , 1 4 ' ) a n d B thosein (10, 14') + (6, 14), the states surviving the Z2 truncation are A +++, + A--=(A

++) ++ • , A ~ , A e = ( A ~ ) *

A + ~ , A + + = ( A + + ) *, A+8+,AzB+=(A+B+) *, A++,A+ + =(A+_7) *, A+~ ,A+~

=-(A+8

)*.

(24)

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B + and B - are in a real representation of the gauge group, while P is in a complex one. Defining A ++++=a+, A + - = b + , A ++=a , A + + = b ,eq. (21) reduces to la+ 12+ Ib+ 12+ la_ 12+ Ib_ 12

- ~ IA+B+I 2 - Z IA+z~ 12- Z I A ; I 2=1 , B+

B

P

and, from # ,

B~

+/~

P

~

The simplest way to find the K~ihler potential J, and the superpotential g, is to use the N = 1 and N = 2 formulae for the gravitino mass. In N = 1, indeed, mi3/2j=eJ/2g15, while, in N = 2 it is [ 12] -- l zlict~rRRA -1- I A t ~ S S A - - ~.cx R z~. ,CaRaj~ 2,,1SA ~tSj

(27)

X~ are the scalars of the gauge vector multiplets and xRR= TRRXz, Xss= Tssxt, where the T RR, T ss are matrices of the appropriate representations of SO ( 16 ) × SO ( 28 ). Rescaling Xz= exp ( ,lo/2)x s, where Jo is the K~ihler potential of [SU (1, 1)/U (1)] × SO (2, 498)/[SO(2) ×SO(498)]. From ref. [8] we have

IA_+B_Iz,

a2+ 4b2+_ ~, (A+~+)2=0, B+ a 2_+b 2__ ~ ( A + B ) 2 = 0 . B

- Z (/L/L)"~Iz,I

(26b cont'd)

i m3/2j

la+ 124 Ib+ 1 2 - ~ IA++8+ [z B+

=la_le4[b_[2-~

a=

13 August 1987

(25)

Eqs. (25) are solved by

Jo = - l o g ( S + ~)

--log(l--2

a+= flY~2(I+B~+Y~+ ),

~/ 'Xl] 2 4

~/ X / 2 2 ) ,

(28a)

and the kinetic gauge function is

fAS=gAsS. b+ =-½i y~2(1- ~" Y2+ )

(I¢{(10, 6)4(14, 14')}, A, BeAdj SO(10)× SO(6) ×SO(14) ×SO(14')). From eqs. (26), (27), and from the N= I expression for the gravitino mass, m2/2=e Jig[ 2, we obtain the total J function and g. Indeed, reducing eq. (27) down to N= 1 we find

a- =½ Y~-/2(I + Z y2- ) b-=-½i y~2( 1- ~

(28b)

J=Jo +J+ 4J_ 4Jp,

)'

(29a)

with Jo as in eq. (28a) and

A +8+ = y~2yB~, A +8- = y~2yB , A~ = ( Y+ Y _ ) " 4 z e ,

J+ = - l o g (26a)

with

Y+=o~fl ,

1 - 2 ~ lYB+ 124 B+

y2+

,

(29b)

Je=-21og(1- ~ IzpfZ exp[ ½(J+ +J_ )] ) . Y =aft+, (29d)

fl+=l-2

Z8+ 128+12+ ~ y~+ z,

(26b)

Also, the superpotential g is found to be g=XIyB+

fl = l - 2 Z

]y, j2+ ~ y ~

2,

T~+B Y8

+xIZpTtep'Zp

' .

(30)

T~+8-, T~,p, are, up to normalization factors, the Clebsch-Gordan coefficients for the corresponding 363

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representations of the gauge group. The expression in eq. (30) agrees with the superpotential proposed in refs. [8,9]. Eqs. (28), (29) and (30) completely specify the N = 1 effective theory and summarize our results. The N = 1 potential is positive semi-definite because it is obtained by a truncation of an N = 2 potential which is positive because the auxiliary hypermultiplets are gauged by a non-propagating vector field only. In this case [ 12 ], the scalar potential is reduced to a sum of positive semi-definite monomials in the scalar fields. The previous construction gives information also in some cases, where m o r e twisted sectors are present, and the N = 1 theory cannot be a pure N = 2 truncation. To see this let us consider the example of an N = I model based on the S O ( 1 0 ) X S O ( 6 ) X SO(6)' × S O ( 2 8 ) gauge group. This model has two twisted sectors, one singlet under S O ( 6 ) ' and the other singlet under SO(6). It might be obtained from the N = 4 SO(44) string model in two equivalent ways. We may break indeed SO(44) down to N = 2 , SO(16) × S O ( 2 8 ) and successively to N = 1 with the following reduction: SO(16)-,SO(10) z S O ( 6 ) , S0(28)--,S0(6)' xS0(22), or choose the other pattern S O ( 4 4 ) ~ S O ( 1 6 ) ' X SO(28)', S O ( 1 6 ) ' ~ S O ( 1 0 ) x S O ( 6 ) ' , S O ( 2 8 ) ~ SO(6) x S O ( 2 2 ) . In this way the roles of SO(6) and SO( 6)' are interchanged, as well as the twisted sectors. So, by looking at the parts of the effective lagrangian obtained by truncating the two different N = 2 theories, we get information about all the twisted states. The K~ihler potential of this model for small vacuum expectation value of the fields in the twisted sectors is J = J o + J l + J 2 + J 3 + ~[ [Zp, [2 PI

exp[½(J2+J3)]

+ • Iz,,~[2exp[½(J,+J3)l+O(IzP, la), P2

(31)

where J, (i = 1, 2, 3 ) are K~ihler potentials of an SO (2, N~)/[SO(2)xSO(N,)] manifold, Jo is the K~ihler potential of the dilaton manifold SU(1, 1 )/U(I ). For a general model with three twisted sectors the previously explained interchange symmetry, allows 364

13 August 1987

us to obtain unambiguously the total K~ihler potential in the limit of small zn field fluctuations (where zp, means all the fields in the ith twisted sector) J = J o + J , ~-J2 +']3 + [Zp, I2 exp[ ½(J2 '~-J3) ]

+ Izp212 exp[ ½(J~ +J3)] +lzp312 e x p [ ½ ( J l + J 2 ) l + O ( l z p l 4 )

.

(32)

Eq. (32) is already sufficient to evaluate expressions, around the point Zp, ~- 0, the mass formulae in particular [ 16 ]. This is the physically relevant region because the zp are the scalar partners of the family fermions so their vacuum expectation values are always I zp, I << 1. The superpotential of the theory with more than one twisted sector is also uniquely specified. Indeed, by exploring the equivalent ways of obtaining the same N = 1 model we get information on the total superpotential. We find that g = C1,1213Yl, YI2Yn + Yl, zp, Tg, P1ZPi 12 13 + Y12Ze2 T e, e2zv2 -~ YI3ZP3T e3e~zv3 ,

(33)

where YlI,Yl2,YI3 are the untwisted fields and C~,u, are coefficients related to the structure constant of the underlying N = 4 gauge group [ 8]. The T~k,p;, are up to a normalization Clebsch-Gordan coefficients for the corresponding representations of the S O ( 1 0 ) x S O ( 6 ) i gauge groups. To be more explicit let us consider an N = 1 superstring model with three twisted sectors. For instance the one based on the S O ( 1 0 ) × S O ( 6 ) ~ X SO(6)2×SO(6)3×E8 gauge group. The untwisted sector of this model is obtained by a Z2 X Z2 2D fermion number projection of the N = 4 SO(28)XE8 string model. The latter is defined by the base [4] {F, S, ~6~}; ~6~ denotes a set of sixteen right moving 2D fermions. The additional Z2 X Z~ projections act only on ( F - ~ ~16)) and break the N = 4 down to N = I and SO(28) to S O ( 1 0 ) X S O ( 6 ) ~ x S O ( 6 ) 2 × S O ( 6 ) 3. The relevant N = 2 models are based on S O ( 1 6 ) x S O ( 1 2 ) X E 8 . There are three equivalent ways to define the N = 1 model and everyone gives us informations about a twisted sector with different SO(6) i ( i = 1, 2 or 3) quantum numbers [SO(10+6 ~) at N = 2 level]. This information uniquely specifies the kinetic function J of eq. (32) and the superpo-

Volume 194, number 3

PHYSICS LETTERS B

tential g of eq. (3 3). The untwisted fields in this case transform as members of the coset S O ( 2 8 ) / [ S O ( 1 0 ) X SO(6) ~x S O ( 6 ) 2 x S O ( 6 ) 3] and the coefficients C~,~n ofeq. (33) are just the SO(28) structure constants in the directions of the untwisted fields y~,, YI~_a n d yj~ [ 8 ]. The twisted sector contains three copies of sixteen SO(10) families with different S O ( 6 ) ' q u a n t u m numbers. This model is interesting from the phenomenological point of view in the sense that with four further 2D fermion n u m b e r projections (Z~) the families can be reduced to three. Note also, that the q u a n t u m n u m b e r of the families with respect to the S O ( 6 ) ' horizontal gauge groups ( ~ ® U ( 1 ) ' : ) are different, and also that their kinetic terms have different field d e p e n d e n t rescalings. This family asymmetry is welcome and may be necessary to u n d e r s t a n d mass hierarchies between three experimentally observed families. In this model, the Es gauge bosons and gauginos have only gravitational interactions with the relevant matter fields. However, these hidden degrees of freedom may be important in the sense that they may generate a suitable supersymmetry breakdown induced by the E8 gaugino condensates [ 17]. To summarize, in the present work we presented a way of obtaining the low energy effective lagrangians for the massless modes of physically interesting four-dimensional superstrings. In these theories chiral families can exist provided a massless twisted sector is present. This severely restricts the possible choices of consistent superstring models. When massless twisted states are present the N = 1 lagrangian cannot be obtained by a t r u n c a t i o n of an N = 4 lagrangian. In the presence of one twisted sector the N = 1 chiral theory can be deduced by a specific Z2 reduction of an N = 2 theory whose matter hypermultiplets do not live on a symmetric coset space but rather in an S U ( 2 ) quotient of a quaternionic projective space. The structure of this quaternionic space has been obtained by "integrating out" the massive spin one states coupled to the massless scalar particles. The above considerations show that in the N = 1 chiral theory, the K~ihler scalar manifold is not longer a symmetric space, but its K/ihler potential is calculable. Also the scalar potential turns out to be manifestly positive as it can be shown by using properties of the corresponding scalar potential in N = 2 supergravity theories.

13 August 1987

Two of us, L.G. and C.K., would like to thank the Physics D e p a r t m e n t of UCLA for its warm hospitality.

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