Do cosmic strings determine the number of generations?

Physics Letters B 271 ( 1991 ) 377-382 North-Holland PHYSICS LETTERS B

Do cosmic strings determine the number of generations?

M e h m e t K o c a Cukurova University, Department of Physics, TR-O13 30 Adana. Turkey and Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606, Japan

Received 29 July 1991

What distinguishes the generations above the electroweak scale? We propose a model where three families of leptons and the electroweak bosons are assigned to the irreducible representations of the local binary tetrahedral group of order 24, a discrete subgroup of SU(2), and the local discrete symmetry Z3. The scattering cross sections of leptons offthe cosmic strings associated with the elements of the local discrete symmetry distinguish the members of the lepton families. The possible relevance of the cosmic strings to Es unification and the families of quarks are discussed.

1. Introduction

Recent LEP data [ 1 ] and the cosmological calcu- lations [ 2 ] agree almost unanimously in the same re- sult that there exist three generations o f leptons and quarks. Although the standard model, with the symmetry group S U ( 2 ) L × U ( 1 ), describes the low-energy electroweak phenomena with almost no contra- diction, it does not, however, tell anything about the number o f generations. In other words, the electroweak bosons and the photon do not distinguish the family structure. Predictions of the number of generations based on the compactifications o f the heterotic string model via the Calabi-Yau manifolds are far from conclusive as there are many different Calabi-Yau manifolds leading to three generation models [ 3 ].

Below the electroweak scale, an electron, for example, differs f rom a muon because they gain different masses via spontaneous symmetry breaking. Their left-handed neutrinos can simply be identified since they form doublets with their companions. It is then natural to ask the following question: What distinguishes the leptons above the electroweak scale where they are all massless?

~r Work supported by the Scientific and Technical Research Council of Turkey. E-mail address: Koca@trcuniv.

In what follows we address to this question and propose an alternative model for the solution o f the generation problem. A number o f papers [4] have discussed the issue of cosmic strings from the point of view of local discrete symmetries. In the context o f local discrete symmetries cosmic strings are associated with the group elements and the irreducible representations can be used for the classification of fundamental particles [ 5,6 ]. The Aharonov-Bohm scattering of the particles, assigned to the irreducible representations of the local discrete symmetry, off the cosmic strings described by the group elements lead to the identifications of the particles.

2. Local discrete symmetry

We assume that, above the electroweak scale, the scatterings of lepton families off the cosmic strings can be described by the local discrete symmetry < 3, 3, 2> ®Z3, a finite subgroup of SU(2 ) X U ( 1 ). Here the < 3, 3, 2 > (this notation is introduced by Coxeter [ 7 ] ) denotes the binary tetrahedral group of order 24, a discrete subgroup of SU (2). I f A, B denote the generators o f the group, the generation relation is given by

.4 3---B3 = (AB)2 = - 4 , ( 1 )

which also justifies the use of the notation < 3, 3, 2 >

0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved. 377

Volume 271, number 3,4 PHYSICS LETTERS B 21 November 1991

for the binary tetrahedral group. It is the double-cover of the tetrahedral group describing the symmetry of the tetrahedron. The surprising fact is that the character table of the ( 3, 3, 2) can be written as the columns of eigenvectors of the Cartan matrix of the affine E~ ~) [8]. Thus the group consists of seven conjugacy classes and has accordingly seven irreducible representations of dimensions 1, 2; It, 2~; 1~, 2¢ and 3 which can be easily recognized from the Dyn- kin diagram of E~ ~) (see fig. I ). Here c stands for complex and an overbar for complex conjugation. The pseudoreal representation 2 can be generated by the 2 × 2 matrices

So= ½ (1 +eL +e2 +e3) ,

$1 =½(1 - -e l +ez +e3) , (2)

satisfying the relations

So~=~,~ = ( S o ~ , ) 2 = - ~ .

Here, e,, e2, and e3 are the imaginary quaternionic units given by the Pauli matrices e~=ia~, e2=i~2, e3 = - ia3, and 1 is the 2 X 2 unit matrix. An explicit for of the ( 3, 3, 2) in the pseudoreal representation 2 is given by the units of Hurwitz integers

+ 1, +el , +e2, +e3, +So, q-S1 ,

+-$2, -+$3, ---So,---S, , +-~:2, +$3, (3)

where So= ½ ( 1 +e~ +e2 + e3), S~ = ½ ( 1 +e, - e 2 - e3), S2=½(1-el+e2-e3), S3=½(1-el-e2+e3). This set of unit quaternions describes also the root system of SO (8) [9] provided they are multiplied by x/~. The matrices corresponding to the complex representation 2c are generated by S0 = wSo and S] = ~gj where ~ is the complex conjugate of w given by

w=exp(-]ni), w + ~ + l = 0 .

Similarly the complex conjugate representation 2¢ is generated by the matrices S~ = v~So and S'{ = wS~ which satisfy the relations

= - ~ . (4)

To clarify a confusion in the literature we should em- phasize that the (3, 3, 2) is a discrete subgroup of U (2), a fact which explains the existence of complex representations. The triplet representation can be obtained by the actions of the elements given by (3) on the imaginary units of quaternions. Then the char-

1 2 3 2c lc 0 0 0 0

lc

E ~ : < 3 , 3, 2 >

1

A2 : Z3

Fig. 1. Extended Dynkin diagrams of E6 and A2 with the irreducible representations of the related finite subgroups of SU (2).

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Table 1 Character table of the binary tetrahedral group ( 3, 3, 2). a= 0, 1,2, 3; w=exp(]~ti); i= 1, 2, 3.

Representation Conjugacy classes

1 - 1 ±e , S~ S~ - S a - ,~a

real 1 I 1 1 1 1 1 1 pseudo real2 2 - 2 0 1 1 -1 -1 complex lc 1 1 1 w ~ w complex2, 2 - 2 0 w • - w - ~ complex conjugate ic 1 1 1 ~P w • complex conjugate 2¢ 2 - 2 0 ~¢ w - ~ - real 3 3 3 - 1 0 0 0 0

Table 2 Character table of Z3. w = exp ( ] rd).

1 C C 2

Ft 1 1 1 /'2 1 w P~ I ~ w

acter table given by table 1 follows straightforwardly. Similarly, the character table o f Z3 is given by table 2. The history of use of the finite groups in particle physics dates back to the 1950's [ 10].

resentation we assume that the Higgs particle is not fundamental and we invoke a model of t icondensate for the symmetry breaking [ 11 ]. The Z3 assignments for the leptons can be taken as follows:

eR: F2, ]~R : /m2, l"a" /'1 ,

//eL : FI, P/rE : F2 , /)rE : /~2 ,

eL:P2, #L :FI , rL: / '2 - (6)

The eigenvectors of the A~ ~) Cartan matrix consti- tute the columns of the character table of Z3. The assignment in (6) is such that there is a cyclic symmetry in each vertical and horizontal line. We have in mind a hierarchy of the symmetry breaking that at a lower energy scale the (3, 3, 2 ) breaks down to its quaternion subgroup where three doublet representations transform exactly the same way as the doublet representation of the quaternion group.

We adopt the point o f view that the (3, 3, 2 ) ®Z3 symmetry is orthogonal to the standard electroweak symmetry. However, it is perhaps not too weird to visualize the usual electroweak symmetry SU ( 2 ) × U ( 1 ) as the local ( 3, 3, 2 ) ®Z3 symmetry above the energy scale of the symmetry breaking. Either point o f view does not change what we discuss below.

The differential cross sections o f the spin ½ particles for the abelian and the non-abelian Aharonov- Bohm scattering are given by the formulae [ 5,12 ]

3. Particle a s s i g n m e n t

We assign the lepton families to the irreducible representations o f the binary tetrahedral group as follows:

l :eR, lc:/tR, 1c: ZR,

2: l/e 2 c : ( ) , 2c: • (5) e L' \ f l , ] L "t" L

The electroweak bosons are assigned to the triplet representation 3. This assignment supports an inter- esting interpretation o f the Dynkin diagram of E~ l~ in terms of the particle interactions. The nodes rep- resenting the left-handed doublets not only couple them to the vector bosons o f the (3, 3, 2 ) via the branches linked to the representation 3 but also doublets interact with their right-handed leptons via some kind of condensate. As we have no extra doublet rep-

da s in2(na) (7a) d O - 2~zk sin2(l~)

and

da T(h,R) (7b) dO - 2ztk sin2(½¢)

respectively, where the group theoretical term T(h, R_) is given by

T(h ,R_)= ~ [(vjlgin)12sinZ(notj) , (8) j = l

Here R represents the n dimensional irreducible representation o f the discrete group with the normalized incoming wavefunction ( ~//in [ ~//in ) ~--" 1. //j a n d exp (27tiaj) are respectively the eigenvectors and the eigenvalues of the group element h in the representation R.

The numerator in (7b) can be calculated using the

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matrix representations corresponding to the irreducible representations given in (5). For the singlet representations the numerator of (7a) can be obtained from the character tables of the related symmetries given in table 1 and table 2. But for the calculation of T(h, R) one should consider the explicit representation of the group element h and find the eigenvalues as well as the eigenvectors. For illustration, let us consider the group elements So and SI in the pseudoreal representation 2. Both So and $1 have the same eigenvalues - w and - ~; the eigenvectors for - v~ is given by

- 1 / 2 1 S°: ( ; ) =ei°(aWV~) ( ~ + i w ) '

S,: ( ; ) = e ~' ( 3 - v / 3 ) - 1/2( l i w ) , (9a)

and for - w it is given by

So: ( : ) = e i ' ( 3 - V/3 ) - 1/2( 1 ) - w - i ~ '

( ; ) = e 'p' (3 + x//3) - l / z ( w l i ~ ) . (9b)

We assume that the incoming left-handed neutrino and the left-handed lepton are pure eigenstates of ½ a3, i.e., for example, for the electron and its neutrino we can write

(l) (0) PeL=O 0 ' e L = O ' , 1012=10'12=1. ( 1 0 )

Using the values of (9a), (9b) in (8) we can show that

T( So, PeL)= T(SI, V~L)= T(So, eL)= T(St, eL) __1 --~. ( l l )

This shows that the cosmic strings So and St cannot distinguish the VeL and eL states. In fact it can be shown that each cosmic string of (3, 3, 2) scatters l/eL and eL with equal probabilities. Therefore the cosmic strings of the binary tetrahedral group are blind to the upper and the lower components of the pseudoreal representation 2. However, as we shall explain below, the particles transforming as the upper and lower components of the representations 2c and 2c scatter off the cosmic strings with different

probabilities. Now, let us consider the representation 2c to which we assigned the///zL and ~tL. For simplic- ity, we take the cosmic string So where the corresponding group representation is given by S0 = WSo. We use Sa (a = 0, l, 2, 3 ) as generic element of the ( 3, 3, 2) as well as the matrix representation in the pseudoreal representation 2. When we diagonalize S0 we find the same eigenvectors but the eigenvalues are given by - 1 and - v~ respectively. In this case we ob- tain the following values:

T(So, v~L)=~(5-,~),

T(So, UL) = ~ (5 + x /~) , (12)

Therefore the So cosmic string knows the existence of two different particles in the representation 2c. More- over, comparison of (11 ) and (12) shows that the cosmic string So distinguishes the particles of two different doublets. In table 3 we have displayed all possible values of the factor T(h, R).

A close inspection of table 3 shows that the number of distinct T(h, R) values for the scattering over the ( 3, 3, 2) cosmic strings is nine, equal to the number of leptons in the singlet and doublet representations. The values of T(h, R) are classified according to the conjugacy classes. Although it is possible to see the bare differences between the singlets and doublets, to our great surprise, the cosmic strings of the binary tetrahedral group cannot identify each member of the lepton families. When we impose the additional Z3 symmetry the scatterings of the leptons off the Z3 cosmic strings lead to a complete identification of each particle in the lepton families. For example, PR and ZR can only be distinguished by the Z3 cosmic strings as much as l)eL and eL. A careful analysis of table 3 shows that all members of the lepton families can be identified one by one without any reference to a particular cosmic string associated with the symmetry ( 3, 3, 2 ) ®Z3. Moreover, once the leptons are identified, the cosmic strings, in turn, can also be identified. To achieve this, it is sufficient to identify two members of the (3, 3, 2) so that the rest will follow from the recombinations of these two cosmic strings, say So and $1. Similarly the cosmic string C generates the Z3 cosmic strings. It is amusing to no- tice that the sum of the values of T(h, R), for each

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Volume 271, number 3,4

8

I % + ,~ m I

II

%

II ~ ~ rg

+ ~ ~

~ N

. . , -

~= = +1

. . . . . . . . . . . . . . . . . .

PHYSICS LETTERS B 21 November 1991

particle besides eR, over the ( 3, 3, 2) strings are equal to 12.

4. Discussions

The character tables of the ( 3, 3, 2) and Z3 follows from the diagonalization of the Cartan matrix of the affine algebra E~) ®A~ l) where the E 6 ® A 2 is one of the maximal subalgebras of Es. In the octonionic representation of the root system of E8 we encounter the binary tetrahedral symmetries [ 13 ] in which E 6 roots can be classified as three overlapping sets of Hurwitz integers which transform into each other by an additional cyclic symmetry. It is then not too speculative to say that the local discrete symmetries, (3, 3, 2) and Z3, are some remnants of a broken E8 symmetry, possibly connected with the heterotic string. A symmetry breaking of a larger group to a discrete group line ( 3, 3, 2) could only be possible through a group which contains several reducible representations of SU (2). One possible scenario could be the breaking of E8 through its maximal subgroup SU (5) × SU (5) toge tSU(3) × SU(2) × U(1) × [SU(3) × SU(2) × U( 1 ) ]D, where D stands for discrete. This may help us introduce the quark families into scheme. However, it is very difficult to say anything without knowing all finite subgroups of SU (3). In order to incorporate the quarks into the present simple scheme we assign down quark sectors exactly in the same forms of the representations (5), (6) but invoke three more singlet representations of the ( 3, 3, 2) and Z3 for UR, CR and tR.

The electroweak symmetry is then broken via the t/-condensate. Through the same mechanism the local discrete symmetry may break down to its subgroups.

Out of all finite subgroups of SU (2) one more in- teresting symmetry is the dicyclic group (4, 2, 2 ) of order 16 which has three distinct irreducible doublet representations in addition to its four singlet representations. The character table of (4, 2, 2) is associated with the Dynkin diagram of the affine SO(12). Details of this work and some related problems will be discussed elsewhere [ 14 ].

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Acknowledgement

I wou ld like to thank Professor E. Br6zin, Professor

T. Inami , Professor R. Sasaki and Professor H.

Ver l inde for helpful c o m m e n t s . I should also thank

the col leagues at Y I T P , par t icular ly , Dr . B. Zwiebach

and Dr. H. K ikuch i for discussions. I a m grateful to

p rofessor Y. N a g a o k a and Professor Z. M a k i for the

hospi ta l i ty at the YITP .

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Do cosmic strings determine the number of generations?