a family dependent u(1) charge in algebraic unification

Volume 152B, number 5,6 PHYSICS LETTERS 14 March 1985

A FAMILY DEPENDENT U(1) CHARGE IN ALGEBRAIC UNIFICATION

Geoffrey DIXON Mathematical Institute, University of Kent, Canterbury, Kent CT2 7NF, England

Received 22 October 1984 Revised manuscript received 16 November 1984

It is shown that the theory of algebraic unification can give rise to a U(1 ) charge V the value of which is family dependent. In particular, V = 0 for all leptons and quarks in the electron family, and V = -1 on the muon family.

Algebraic unification [1] (AU) is the name given to an approach to s t rong-weak-elect romagnet ic unification founded upon the notion that space- t ime is a derived geometry with a fermionic (symplectic) sub- structure [2], that this fermionic space has an associated geometric algebra (in much the same way that the Dirac algebra is the geometric algebra of bosonic space- t ime) , and that the physics of this space is most naturally expressed as a field theory on its geometric algebra (as the classical field equations of bosonic electromagnetism are most naturally expressed on the Dirac algebra). The particle spectrum of AU does not depend on its symmetries as in the grand unified [3] and supersymmetry [4] theories, but arises in a natural way from the underlying geometric algebra. Likewise, the symmetries of the lagrangian are derived from this same mathematical machinery. The particle spectrum consists o f four families o f leptons and quarks replete with completely chargeless righthanded neutrinos which provide Dirac masses for these particles. Among the symmetries are a chiral, broken SU(2), and a nonchiral, exact SU(3).

AU is a completely hypercomplexified theory making use of all o f the Hurwitz algebras [5] (reals FI, complexes C, quatemions O, and octonions O). The foundation o f the theory is an algebra M iso- morphid to FI(3) ® C ® O ® O (as in refs. [1,2] we are here using a restricted form of this algebra [6] with only half the full spectrum). The fermion hyper- field q~a of AU resides in M. It is 128-dimensional and

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consists of one complete family of leptons and quarks, and one antifamily.

Roughly speaking the quaternions and octonions give rise to the respective internal symmetries SU(2) and SU(3). These are then expanded to U(2) and U(3), thereby fixing the U(1) charges, and this com- pletes the standard symmetry.

In constructing a lagrangian for AU an octonian unit, which we choose to be e 7 (see ref. [1]), is singled out. As a necessary consequence both the U(1) c U(2) and U(1) C U(3) exploit e 7 in their generation rather than the complex unit i. However, in order to assign these symmetries charges we must somehow change the noncommuting e 7 into a commuting i. This is done via the idempotent operators PO = ~(1 + ie7) and P2 = ~(1 - ie7) which satisfy

e7p 0 = - i p o , eTp 2 = +i,o 2 .

The decomposition of the field xpa begins with PO and/9 2 which project out from xI ,a its family and antifamily parts. In particular, pO q,a is identified with the electron family, and p2 ~a with the muon antifamily. Lefthanded (LI-I) and righthanded (RH) components of qza are obtained via the projection operators L = ~-( eex + 75), R = 21-( eex - 75), where eeX,'called the external identity, satisfies

(eex)2 = eex, eexkoa = qza, ~aeex = 0 .

Finally, in the spinor space of M there are four ele- ments Fa,a = 0, 1 ,2 , 3, with which we can complete the decomposition. A complete particle decomposi-

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Table 1 Independent U(1) charges.

Subfield V1 V2 //'3 V

poR~aFo 0 0 0 poRq, aF1 1 -1 0 poRg, ar2 0 1 -1 poRq, aF3 1 0 -1 poL~aFo 1/2 -1/2 0 poL~I, aF 1 1/2 -1/2 0 poL,~aF2 1/2 1/2 -1 poLq, aF3 1/2 1/2 -1 p2Lq, aFo 0 0 1 pzL~aF1 1 -1 1 p2L~aF2 0 1 0 p2L~aF3 1 0 0 p2R,~aF o 1/2 -1/2 1 p2R~ap 1 1/2 -1/2 1 p2Rq, ar2 1/2 1/2 0 o2RkvaF 3 1/2 1/2 0

0 electron-neutrino RH 0 electron RH 0 up-quark RH 0 down-quark RH 0 electron-neutrino LH 0 electron LH 0 up-quark LH 0 down-quark LH 1 antistrange-quark LH 1 antieharm-quark LH 1 antimuon LH 1 antimuon-neutrino LH 1 antistrange-quark RH 1 anticharm-quark RH 1 antimuon RH 1 antimuon-neutrino RH

tion of q~a is listed in table 1. We now want to create a new internal U(1) symmetry

exploiting the generators o f U(1) C U(2) and U(1) C U(3), but with charges now dissociated from the SU(2) and SU(3) charges (in refs. [1,2] we linked the magnitudes of the charges of U(1) C U(2) and U(1) C U(3) with those of SU(2) and SU(3) in the most ob- vious way by imposing the condition that the sum of the U(1) charges in the respective fundamental repre- sentations should be equal to one), together with a U(1) similar to the U(1) C U(2) except that it uses the complex unit i instead of the octonion unit e 7 in its generation. The generators of these three U(1) groups each satisfy A 3 = - A , and we will link their charges (modulo this equation) to form a single new U(1) by generating each with the same arithmetic strength, which is deffmed by the map A -* exp(~A0) (0 is a real parameter).

In M there is an element e in, called the internal identity, which satisfies

(ein) 2 = e in, ein@ a = 0 , ~ a e i n = ~ a .

The generator o f U(1) C U(2) is (--eTein), and we will defme its dissociated action on qsa by

x~a ~ e x p ( _ e 7 O e i n / 2 ) ~ a exp(e70ein/2 )

= qja exp(e70/2) " (1)

This is the same as the associated action, so the re- suiting charge, denoted V2, is the same as the U(1) c U(2) charge Y2 (see ref. [2]).

The generator of U(1) c U(3) is e 7, which acts just on the octonions. Its dissociated action is defined on an x in O by

x ~ exp (e70 /2 )x exp( -e70 /2 ) . (2)

Therefore the resulting charge, denoted V3, satisfies V 3 = 3Y3, where Y3 is the charge of U ( 1 ) c U(3) (see ref. [2]).

The other part of the overall independent U(1) is generated by (- ie in) . Its action on ~oa, defined in ac- cordance with (1), is

~ a ~ exp(- i0 ein/2)q ~a exp(i0ein/2)= ~ a e i ° / 2 (3)

[observe that we can combine the generators in (1) and (3) into the single generator - iP2 ein = -eTP2e!n] . We denote the resukant charge V 1 . The action (3) is enough to determine V 1 = ~- for all lefthanded particles in the electron family and righthanded anti- particles in the muon antifamily (parity nonconser- rat ion is covered in refs. [1,2]). The action of this U(1) on F 0 is

F 0 ~ e - i O / 2 F 0 .

Therefore, F 1 -~ ei°/2F1, F 2 ~ e - i ° / 2 F 2 , F 3 el0/2 F3 (see ref. [ 1 ]). With this in hand we can complete the V 1 assignments for all the particle sub- fields of ~ a . In table 1 we list corresponding values o f V 1,V2, V 3 , a n d V = V 1 + V 2 + V 3 , w h i c h i s t h e real focus of our attention.

The result is rather remarkable. We note first that V = 0 for all members of the electron family. This is consistent with the U(1) hypothesized by Georgi et al. [7], which could give rise to photon oscillations and a photon mass (ironically their idea is evidently inconsistent with the big group approach of GUTs and would imply the need to look for some other algebraic origin to the standard symmetry). However, V = 1 for all members of the muon antifamily, and therefore V = - 1 on the muon family. This should give rise to a new, possibly massive, vector boson leading to a repulsive interaction between all pairs of particles in the muon family, but an interaction to which all members of the electron family (and conse- quently all experimental apparatus) would be immune.

Conservation of V-charge is only to be expected in

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purely leptonic decays due to weak mixing among quarks. Its conservation among the leptons of the first two families is equivalent to the conservation of the so-called muon lepton number, which has been hitherto, like conservation of the electron lepton number, one of those rare conservation laws evidently unassociated with a Lie group symmetry and corresponding gauge interaction. Still, in AU charge conservation arises algebraically. For example, PoP2 = P2Po = 0 implies

~aTu ~a = U~apoTuPOqta + ~tap27up29a , (4)

a consequence of which is that the total number of leptons and quarks minus the total number of anti° leptons and antiquarks is conserved. Likewise, since the product of the lepton part of ~I ,a (linear relative to the octonions in 1 and e7) and the quark part of ~ a (linear in the octonion units ea, a = 1, ..., 6) is also linear in ea,a = 1 .. . . . 6, and therefore disappears when taking the real part of the trace of (4) (see ref. [1]), we can go further and say that lepton and quark numbers are separately conserved. And so forth. Thus in AU the (gauged symmetry) - (conserved charge) link can be viewed as an artefact in so far as algebra complicated enough to give rise to a conserved charge usually also gives rise to a symmetry from which the charge can be derived. In this light it seems irrelevant whether or not we identify V-charge with the muon lepton number.

As to the other two families of AU, it seems naive- ly likely that one would be V neutral and the other V charged. However, we have not yet exhausted the family dependent 13(1) charges of AU. U(1)s generated from ie ex and e7 eex are each individually family dependent, and depending upon how they are in- cluding, if at all, a number of family dependent U(1) charges are possible. At one extreme they could ex- actly cancel V-charge leaving all families neutral. At the other extreme each family could have a unique charge (0, - 1 , - 2 , - 3 ) by which it would be labelled. These charges would presumably descend with ascend- ing mass. This possibility is not at all unlikely and is certainly aesthetically pleasing. Work in progress on the mathematics underlying a full four family theory of AU may eventually clarify the situation.

References

[1] G.M. Dixon, Phys. Rev. D28 (1983) 833. [2] G.M. Dixon, Phys. Rev. D29 (1984) 1276. [3] P. Langacker, Phys. Rep. 72 (1981) 185. [4] P. Van Nieuwenhuizen, Phys. Rep. 68 (1981) 189. [5 ] I.R. Porteous, Topological geometry (Van Nostrand-

Reinhold, London, 1969). [6] G.M. Dixon, Lett. Math. Phys. 5 (1981) 411. [7] H. Georgi, P. Ginsberg and S.L. Glashow, Nature 306

(1983) 765.

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a family dependent u(1) charge in algebraic unification

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