Constituents which produce an interfamily unified model

Volume 105B, number 2,3

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1 October 1981

CONSTITUENTS WHICH PRODUCE AN INTERFAMILY UNIFIED MODEL "~ Kenneth I. MACRAE Department of Physics, B-019, University of California, San Diego, La Jolia, CA 92093, USA Received 15 December 1980 Revised manuscript received 8 June 1981

Spinor components for a four-family unified model are placed in the 28 component fundamental spinor representation for a single gamma matrix algebra which contains the subalgebras Spin (14) [covering SO(14)] plus Spin (1,3) ~ SL(2, C) as well as all Higgs representations needed to break Spin(14). The algebra is constructed from tensor products of two-dimensional algebras (related to Pauli matrices). This leads to a natural constituent model which produces standard local field theory S-matrix elements by starting from only four four-state (or eight two-state) constituents.

We unify all the components of all the families into a single spinor which transforms as the irreducible representation of a spin group [1] , 1 . Spin groups are the covering groups of rotations and are naturally contained in the gamma matrix algebras for the spinors. The gamma matrices can be constructed from tensor products of Pauli matrices, in the complex irreducible case, or the two-dimensional algebras p0,2 ~_ Q and F2, 0 "~ R [ 1], in the real irreducible case [ 1]. Noting the relationship between fermionic integration and derivation, we show that the spinors can be replaced by four.state constituents [2] appropriate to this ten. sor product structure, and yet the S-matrix elements are conventional. In addition to the three known families (e,/~, r), see refs. [3] and [4] *2, we assume there is a fourth, the X family [5], which is yet to be found. Each family is organized to form the sixteen-component fundamental spinor representation (fsr) of Spin(10). We are assuming the reality of ~R c. Of the candidates for grand

Work supported in part by the United States Department of Energy. *1 This unpublished report has substantial overlap with this paper but does not include the constituents. Here, as there, Spin(N) is the group; Spin(N) is the algebra, r°, 2 , r2, ° , Q; RIll are explained in the text. 42 This paper shows that solar neutrino fluxes are best fit ff there are between 2.6 and 4.0 families. 140

unity algebras only Spin(10) has exactly sixteen fundamental spinors * 3. To unify the families we repeat the process of composing leptocolor Spin(6) [~-SU(4)] with L - R c symmetrized weak Spin(4) [~-SU(2) + SU(2)] to form Spin(10) [ 6 - 9 ] . That is, we introduce another Spin(4) of interfamily symmetry [ 1 ]. The unification algebra Spin(14) is the simplest candidate for unifying the families' symmetries. It has a complex sixty-four ((264) component fundamental spinor which splits into four sixteen-dimensional Spin(10) spinors. Two are in conjugate representations with respect to the other two. This is the precise analog of the manner in which a Spin(10) C 16 splits into two pairs of Spin(6) C 4 spinors [in fact this process can be repeated for Spin (6) going to Spin(2)] [ 6 - 8 , 1 0 ] *3. These splittings are not only physically significant but, most interestingly, they correspond to the underlying mathematical structure of gamma matrix algebras; to the fact that F's (of definite signature) are naturally built up using tensor products of the four-dimensional Clifford matrices. The algebra of gamma matrices can be described as the complete set of 2 fi × 2 ~ matrices which can couple to pairs of 2 fi component spinors. Given h and the re*a "Grand Unification" includes a vast number of topics. A large b~liography is in Ellis, ref. [6], see also ref. [7], and ref. [8] with lots of detail and references.

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ality or complexity (or nature) of the spinors, there will be N vector 2 ~ × 2 ~ gamma matrices F I which generate the algebra (I ~< ! ~< N, we relate N to h below). All matrices which induce spinor splittings, all mass matrices, all interactions of fermions with other fields are therefore included in that algebra. Furthermore, every 2 ~ × 2 ~ matrix (of fields) ¢ has a unique [11 ] decomposition in terms of the N matrices FI and their ordered (antisymmetrized) products. Thus introduce the (ordered) p-vector basis ¢4 FIII 2"''Ip = I'll PI2 ...FIp and find (implied ordered summation)

cb = ~0I + ~IFI + ¢IlI2 Pil12 + ... + d~II""INI~IlI2 ...I N .

(1)

The (N) coefficients 4011...Ip are called the components of a p-vector [1 1 ]. While it is true that FIjI2 has N ( N - 1)/2 components and forms a representation of Spin(N) it is always possible to form a closed algebra out of FI and FIj. They represent Spin(N + l) and generate what is called the fundamental representation [1 1 ]. When there are p positive square and n negative square vector Fi's (p + n = N), we shall denote the Clifford algebra they generate as FP, n. The v e c t o r tensor or spin subalgebra generate Spin(p, n + 1), and we see that Spin(N+ 1) = S p i n ( 0 , N + 1). It is worth noting that because of the antisymmetry of fermions not all p-vectors can couple to bilinears (although they can couple to pairs). In fact, since (xpTE~)T = ~ T ( _ ~ T ) ~ , only the skew-symmetric (or skew-hermition, if ~ is complex) matrices can occur in the action. If X is a real p-vector basis matrix and Z = _ z T , then X2 is negative. Knowing the squares of the FI, the anticommutation relations allow one to find the squares of the p-vectors by multiplying by an overall (anticommutation) factor which repeats for p ' = p + 4 starting with p = 0. This multiplicative factor lists as + + - - + + - - etc. Thus, e.g., in F 0,8 the pattern of squares of p-forms is + - - + + - + + because all FI have negative squares. So, only 1,2, 5 and 6 vectors can couple. Of course there are further restrictions due to the R c versus L split [8]. We assume that only Higgs which can cause splittings of the spinors, i.e. those which can act (via the gamma matrices) on the spinors, are needed. Thus all -

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Computing the effect of these on the Spin(N + 1) subalgebra is especially easy because of the vector interpretation of the F I. For example, Spin (10)is contained in F0, 9. It is generated by F 1 and FjK = F j P K ; J ~ K ; 1 < I , J , K < , 9 . To break o f f a Spin(4) one can introduce [11 ] a four vector ~(4) = dplJKLFIJK L (or, dually, a five vector $(5) = i(~(4)). By rotation [1 1] we select the "last" four directions for the vacuum expectation direction (ved) $~4). We will not detail the field theory and potential here since these are known [ 6 - 8 , 1 3 ] *s Split the index I into 1 ~< i ~< 5 and 6 ~
1:(10 01) p0,2: 1,

P1 = i,

F 2 = j,

F12 = - k ,

(2)

F 0,2 = Q, which is the algebra of quaternions. Note only real coefficients are allowed. Let R IN] - R(2 N) denote the set of 2N X 2N matrices with real coefficients; define a IN] and Q IN] analogously. Q can be represented using Q C C [1] so that i = iF 1 and j = iF 2 with F 1 and F 2 in F 2,0. The subalgebra F 1,0 with p-bases I and F 1 is the same as two diagonal copies of R. We denote it by R+ (correlate F l's signs).

necessary Higgs are included in the Clifford algebra. . 4 No two indices are equal.

*s Especially Wheeler and Feynman [14] for action at a distance. 141

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The two + spinors are called semi-spinors [11]. The subalgebra p0,1 withp-bases 1 and i is just C and not C± as one might think from the C[1] matrix representation. Indeed, this is the distinction between real and complex irreducible cases. By interchanging P12 with F 2 in F 2,0 one sees that p2,0 ~ ~1,1 ~ R [1].. The product of tensored pairs of matrices is the tensor of the product. Thus i f A , B E P ' and M, N E F", then P "~ P ' ® P", the tensor product algebra, has multiplication defined by A ®M • B ® N = A" B ®M" N. E.g. if P ' = R [n] and P" = R [rn], F = R [nm]. The F spinors are in R n x m , but they are viewed as being in an n X m block ( ~ 0 ) acted upon as

R[m]]\0 t 0 1 \

I R[n]/"

Since Clifford algebras can be generated by the vector matrices (FI) which all anticommute, it is enough to f'md these generators. These algebras can therefore be constructed iteratively using the following rules: one: I~P,n+ 2 ,-.-, t o , 2 ® F n , p , t w o : [,p+ 2,n ~ 172,0 ® 1`n,p, and three: I-'P+1,n+1 ~- pl,1 ® I',p,n. In terms of 1`], generating F°, 2 , and F a' generating rm,P, rule one implies the relations P i = P i ® I and F a = F12 ® F a . These all anticommute. (ri) 2 and (F[) 2 have the same signs, (I'a)2 and (Pa') 2 have opposite signs since (r~2) 2 = - I . The same holds for rule two. But for rule three the matrix (I'~2)2 = I, and so (I'a)2 and (Pa') 2 have the same sign. Thus in order to build the algebras p0,N for internal symmetry spin algebras, we use both rules one and two. •

.

f

I

rt

1,0,N+4 ~ 1-0,2 ® I-N+2,0 ~ 1-0,2 ® 1-,2,0 ® 1,0,N _~ F0,4 ® F0, N . In order to know the structure of FP ,n, we merely need to know what the tensor products of F0,2, p2,0 and their subalgebras are equal to. In fact since tensoring matrices in R or even R IN] with those in an algebra F ' will preserve the structure (complex, quaternionic or +-) of F ' , while merely increasing the dimension, it is enough to understand the C and Q parts of table 1. C ® C is C± + depending on whether one is complex conjugated with respect to the other [8]. C ® Q "~ C[1], the complex 2 X 2 matrices. Just view O C C[1] as a set of four matrices;then give them coefficients in C and find any C [1 ] matrix. The last re142

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Table 1 ®products.

® IR C

C

I~

C+

C [13

C lationship follows from the fact that any tensor product algebra representation can be viewed as acting on a spinor by left multiplication for the first algebra and right transpose (conjugate) multiplication for the second. That is, the algebras act as L

0

•

L

t

To emphasize this write QL ® Q~," It acts on spinors q E O as q L q q ~ = ql. Let qi be a basis for Q. Use the standard quaternionic multiplication rules and find that q = xiqi.and q' = x'iqi in Q are related by mii x i = x 'i with m l i any real 4 X 4 matrix. Thus Q ® Q ~R [2] = R [4]. Also we see that F0,4 ~ F2,0 ® p0,2 ~ R [1] ® O " O[1] , and that to,8 ~- rO,4 ® t o , 4 ~- Q [ I ] ® O [ 1 ] ~- R[4]. The results of these calculations are in table 2. It is worth noting that the F algebras with complex coefficients, Fc, are more easily computed. Thus r n,p = c ® r n , P , and so PcN "~ -cl~n'P, if N = n + p . For this reason one only needs N even or odd [8,11,12]. Using table 1, find p2n ~_ C [nl and F 2n+l ~- C± In]. For these algebras only one of the three rules suffices. For the interfamily algebra Spin(14) C F0,13,we observe that p0,13 ~ 1`0,4 @ p0,4 @1`0,4 ® r0,1

is thus C[6] since P 0,1 ~- C. But C[2] "~ F0, 5 is not split to F 0,4 ® F 0,1 ~ Q [2] ® C, physically. There is a suggestive parallel between the mathemat. ical construction of the interfamily unity group (using the algebras F 0,4) and the observed family and eL--VL--eRc --vRC subfamily splittings. Note that the reality of the algebra F 0,8 ~ R [4] preserves the structure (complex, etc.) of I ~p,n . In going to

e

I

2

3

4

5

6

7

8

9

IO

II

12

13

14

15

16

17

18

1 :;T’

n+&O

\n

%q-1 for s pin In,q)

-

Symmetry Axis

C r””

spinor

Q2N

CzN

*.

(n.D)

spinor ECliff

with Spin(n,ptl)

with

C [N]=Mat(EN,C) P [N]EMat(2N

,Q)

(f.s.rJ

If.s.r.)

Table 2 Gamma matrix (Clifford) algebras for ail metrics. Note that Spin@, 9) c rp~4-~ 9 > 1. To find an algebra count vertically up p + 9 and horizontally in from the left p + 1 spaces. The spinors have the number of components as a column vector for the algebra does. E.g. Spin(lO,4) C J?l”13 is on the 10 + 4 = 14th line and Z6. The curves enclose algebras with spinors having equal numbers of real compoisthe10+1=Ilthalgebrafromtheleft.Thus~’0~3.~C[6].SpinorsareinC nents (labelled by the log base two of the number of real components, by number of bits) *6. In the case of algebras having semi-spinors (* subscript) we enclose them with the same dimension of spinor. E.g. R+_[8] with R[8], i.e. R$* spinors with Rz8.

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Fp, n+8 ,-~ Fp,n ® F0,8, the dimension merely increases. So FP, n+8 is said to be one period (octave) higher than 1`p,n [15]. The interfamily group is thus exactly one period higher than the leptocolor group in this model. F0,13 ~- 1316] ~- R [4]®C[2] ~ F0, 8 ®F0, 5. Physical spinors are handed. Thus instead of requiring four vectors of P 1,3 to describe the kinetic behavior, it suffices to use three. The four-component action for ~I' becomes a two-component action for 4. ,I'

=(*o) -i~+3'0(iT0O0 + i¥- 6 ) ~ = ff'+(B0 + 3'0 ¥" 8 ) ~ = ~*(a 0 + a - s)~0.

(3)

The three matrices a generate F 3,0. Since ~k carries F 0,13 [or Spin(14)] internal quantum numbers, it is actually a 64 × 2 matrix with complex entries. There are therefore 128 complex or 256 real independent spinors in our four-family model. Note, ia plus • generate Spin(3,1). While it is true that the Lorentz subalgebra is intimately related with space(time) symmetries, it plays a role identical with the other symmetries in labelling particle states. The generators for these symmetries can be included with the other internal generators. Only after this symmetry is split off and attached or glued to the space-time transformations does this subalgebra have its usual interpretation [ 16]. (Of course, the ~ matrices play a role in forming the kinetic term also.) But until the spinors are attached to space(time) (by reducing the structure group), they can be thought of as objects in the fundamental spin representation (fsr) for a single large Clifford algebra which has both Spin(14) internal and Spin(1,3) -~ Spin(3,1) subalgebras. Thus we place spinors in the fsr for 1`0,16 ~_ r0,8 ® r0,8 ~(1`0,4)~4. The spin.subalgebra is Spin(17) but the dual vector matrices can be included along with commutators to obtain Spin(I,17), or observe that p1,16 has R 2s spinors and take a choice of semi-spinors to resolve the apparent discrepancy. This is (P 0,4) semiperiodically [ 15] analogous to the fact that spinors for F 0,4 can carry a Spin(1,5) representation, if one includes (axial) dual vector matrices plus commutators. This works because p0,4 spinors are I"1,4 semi-spinors (table 2). We therefore break off (by hand) P 0,3 from 1`0,16 leaving 1`0,13 internal at the same time that we attach 144

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spinors to their space(time) sites [16]. Using periodicity [15] we see that we could examine a simpler model which has strong and spin symmetries only. That is F 0,8 ~-F0,4®F0,4 ~ O [ 1 ] ® O [ 1 ] , is broken (by hand, upon attachment) [16] to F 0,5 -~ F 0,4 ® F 0,1 , the algebra containing Spin(6) SU(4) with two complex four internal component spinors. Write the vector bases in either F0, 4 as 0 1 o q (-1 0 ) a n d ( a o); q = i ° r j ° r k ' U s e Q ® Q = ~ R [ 2 ] to see that the" real sixteen-component spinors for p0,8 can,be written as the sixteen elements occurring in Q [1], when one splits 1`0,8 into F0, 4 ® 1`0,4. These q2 algebras act by left and right multiplication on (~13 q4). When the (right conjugate) algebra 1`o,4 breaks to p0,1 only one generator is left, e.g. ( 0 01).Because its square is negative, we can replace the spinor in Q[ t] b Y (qa ql +iq2 1 (right • con+ iq4) in C 4X2 an d r eplace (-10 0) jugate multiplication) by multiplication with i. Use tl C C [1] =(A - ~ ) for each qa, 1 ~
f dxpA = ~ = AAI - ~ = AAI f dO, ~xpA with AAIAI B = 5AB. These are the same as bosonic derivation rules. But fermions anticommute. So if a

spinor has 2 N (real) components, it can not be constructed from fewer fermionic constituents, since polynomials (functions) in ~,A will truncate at a lower degree than N. We propose using constituents which are bosonic except that, one, instead of functionally integrating them we must functionally differentiate them, and, two, the anticommutation properties are effected by the device of replacing the exponential o f the action by a signed (ordered) exponential *6. This can work because spinors are paired into bilinears. A simple example is free fermions on a p-site lattice. Express • as a real spinor (e.g., a Majorana spinor on a four lattice) and symbolize the action as xItlPij~J where the index I runs up to N = nn' (n internal times n ' positions). Let n be an even number (2m). PIJ is just/~ expressed as a 4n' × 4n' matrix for the example. Now expand the action exponential. It truncates at N. Order the ferrnions from 1 to N. This functional integral and all its nonvanishing (even) moments (Greens functions) can be evaluated using only bosons which are evaluated by differentiation instead of integration. We call these A-bosons. Thus replace

f dq'N ~ k-~. k

by N

a E I,1__ !

,3A 1.... 3AN k

X

n

I-I

m

1 ( 0 ]m 1~ 1._1_(_~__~)n

i=1 ~-.v k ~ - 7 ]

a = l n!

since each of these differential operators evaluates to 1 on the product of A's,

nm n rn I-I AA = H I-I (Ai) m(A'a) n , i=1 a=l

1

X (Piii2 ...Pizk_tlzk)Xlllt ... xlII2k

w=a

where e ll ...12k is the totally antisymmetric (LeviCivita) symboi in N 2k. The vacuum expectation of k pairs o f spinors is obtained by ordering them (numerically here), replacing them with the A-bosons with the same indices, multiplying these with the exponential expressed in terms o f A, taking the derivatives and dividing by W. The value in different order is obtained by signed permutation. Interactions are discussed below. Now we are in a position to introduce A-bosonic constituents. We replace the nm component AA by the pair A iA'a~ where A i has n and A 'a has m componm O/aAA nents. Instead of nm first derivatives, l'IA=I we have n m t h order and m nth order derivatives,

A=I

N

W= f d q '1 ...

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Volume 105B, number 2,3

(PItI2 ....p12k_d2k)iZa1...,a2ke/t.../2k,

¢6 Take ,I, ------5 t e a t ®a2 ®A2 w i t h a t , a 2 in R[1] and At, A2 in (1. Call the two state constituents bits (as in a computer). The bit-pairs or "quads", the four deltas, are paired into "nybbles", and then paired into the local "word" (one byte = 8 hits suffices). Words at every location anticommute with those of every other location forming a single, enormous "memory" or "program". With p sites and 2 s internal degrees of freedom let N -= p2 s. One can replace spinors by elements in F N, the program algebra (let ,1,I =- AI3"I (unsummed) with 3"I the generators of FN). The measure requires a factor of (rl~_-13"I)-t and a trace. If z~I is not composite, it can be omitted entirely. This permits using unordered exponentials but requires the enormous algebra F N. If one wishes to preserve the vanishing of squares, one is led to FN, N and taking 3,I to be the sum (difference) of a positive and a negative square FN, N generator. We prefer the ordered, signed exponential. It is simpler.

and 0 on lower order products. Other derivatives with the" same values can be used, such as n

Jill

m

a

a

i=1 a=l a[(Ai) m] a[(A'a)n]

taken at Ai = ~ and A !a = ~ . In any case one pays for decreasing the number of degrees o f freedom by decreasing the simplicity of evaluation. Introducing constituents is most worthwhile if some other advantage is gained. In the case of internal degrees of freedom the advantage is the use o f smaller rank matrices in the tensor decomposition. F o r degrees of freedom associated with number o f sites there is no other advantage. Note that the process can be iterated. Starting with the 256 spinors for 1-'0,16, we reorganize them as a 16 X 16 matrix: R 16x16 becomes R [4] - R(16). p0,16 becomes p0,8 ® F0,8, acting via left and right (transpose) multiplication. The A-boson (spinor) R [4] can then be factored into Q[1] ® O[1] with p0,16 expressed as 1-`0,4 ® po,4 ® po,4 (each "spinor" Q[1 ] having both left and right multiplications). Then again each Q[1] can be factored i n t o R [1] ® Q. Thus the "spinor" R [4] is expressed as 145

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(R [1] ® O) ®(R[1] ® O) and r '0,16 is resolved into

(I"0,2 ® I'2,0) ®(FO,2 @1'2,0) @(1"0,2 ® I'2,0) . (r,O,2. p2,0), with the obvious multiplications. No further (numerical) advantage is gained by replacing the four A-bosons in I:l [1] or O with pairs of two-states objects (bits) *6. In order to include the effect of gauge, Higgs or other b osonic (e.g. gravitational [ 18,16] interactions, we can begin with a conventional local field theory and convert it to the equivalent non-local field theory obtained by performing the functional integral for each of the bosonic degrees of freedom [13,14]. This gets rid of 3 X 91 Spin(14) gauge fields, 2 gravitational and a number of other (Higgs) bosons. Thus we will have an effective action expressed entirely in terms of spinors (only 28 per site). These can in turn be replaced by their A-boson constituents. We do not intend to suggest that calculation is more convenient for this form o f the action. But all S-matrix elements for q (the composite) will be the same as in the local theory, by construction. We make the proviso that only xI, correlations are directly observed. Even photons can only be observed by their effects on spinor currents. But then, we need only four four-state constituents (quads) * 6 I would like to acknowledge helpful discussions with F. Mansouri, W. Frazer, N. Kroll and C. Ong.

References [1] K.I. Macrae, Beyond grand unification, the structure of spinors, UCSD-10P10-216.

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[2] See for an incomplete list of constituent models: H. Harari, Phys. Lett. 86B, (1979) 83; M.A. Shupe, Phys. Lett. 86B (1979) 87; J.C. Patiand A. Salam, Phys. Rev. D10 (1974) 275; R. Casalbuoni and R. Gatto, Phys. Lett. 88B (1979) 306; 90B (1980) 81; R. Casalbuoni, G. Domokos and S. Kovesi-Domokos, SLAC-PUB-2585 ; S.L. Glashow, Harvard preprint HUTP-77/A005 ; F. Mansouri; Yale preprint YTP80-25. [3] R.M. Barnett, SLAC-PUB-2579; M.L. Perl et al., Phys. Rev. Lett. 35 (1975) 1489; S.L. Glashow, C.T.P. (1980) p. 886, [4] J.N. Bahcall et al., Phys. Rev. Lett. 45 (1980) 945. [5] P. Wilczek and A. Zee, Phys. Lett. 70B (1977) 418; Phys. Rev. Lett. 42 (1979) 421; C.L. Ong, Phys. Rev. D19 (1979) 2738. [6] J. Ellis, Grand unified theories, Ref, TH. 2942-CERN. [7] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438; H. Fritzsch and P. Minkowski, Ann. Phys. (NY) 93 (1957) 193. [8] R. Slansky, LA-UR-80-3495. [9] M. Yasue, INS-Rep.-391 (1980). [10] F. Giirsey, Proc. 1977 International School of Subnuclear Physics, Ettore Majorana, ed. A. Zichichi (Plenum, New York, 1979); M. Gell-Mann, P. Ramond and R. Slansky, Rev. Mod. Phys. 50 (1978) 721 ; P. Van Nieuwenhuizen and D.Z. Freedman, eds., Supergravity (North-Holland, Amsterdam, 1979). [11] E. Cartan, The theory of spinors (Hermann, Paris, 1937). [12] R. Casalbuoni and R. Gatto, Phys. Lett. 88B (1979) 306. [13] E.S. Abers and B.W. Lee, Phys. Rep. 9C (1973) 1 and references therein.~ [14] J.A. Wheeler and R.P. Feynman, Rev. Med. Phys. 17 (1945) 157. [15] M.F. Atiyah, R. Bott and A. Shapiro, Clifford modules, Topology 3 (Supp. 1) (1964) 3; W.K. Clifford, paper 43, in: Mathematical papers, ed. R. Tucker (MacMillan, London, 1882). [16] K.I. Macrae, Phys. Rev. D18 (1978) 3737, 3761, 3777. [17] F.A. Berezin, The method of second quantization (Academic Press, New York, 1966). [18] K.I. Macrae, Phys. Rev. D23 (1981) 886,893,900; UCSD 10P10-217.