Volume 181, number 3,4 PHYSICS LETTERS B 4 December 1986

I N T E R N A L S U P E R G R O U P P R E D I C T I O N F O R T H E G O L D S T O N E - - H I G G S P A R T I C L E M A S S *

Yuval N E ' E M A N t

Sackler Faculty of Exact Sciences, Tel Aviv Universit). Tel Aviv. Israel and Center for Particle Theory, University of Texas, AtLs'tin, TX 78712, USA

Received 19 September 1986

Using a seemingly ad hoc but phenomen01ogically fitting global internal supersymmetry SU(2/1) ~ SU(2) X U(1) con- straining the weak-electromagnetic otherwise arbitrary parameters, we predict for the Goldstone-Higgs particle a mass m H = 2mw, or 160-170 GeV.

Weak-electromagnetic unification [1 ] is based on the non-simple gauge group SU(2) X U(1), with two independent couplings g and g' , uncorrelated weak- hypercharge quantum numbers for the left- and right- chiral fermions,

Uw(v[, ) = Uw(eL) = - 1 , Uw(dR) = - 2 , (1)

Uw(u2/3) = a r t 2/3~_ Vw(dL1/3) 5", t-'wtUL 5 - = ~

Uw(d 1/3) = - 2 . (2)

The weak-isospin multiplet selection Iw = ~ for (L,V e e~), (u 2/3-, d~l/3), i.e., the left-chiral states, and

Iw = 0 for e~, u 2/3, d~ 1/3, i.e., the right-chiral states, appears ad hoc, and so is the adjunction of a Lorentz- scalar Goldstone-Higgs multiplet ~bn(x ) with Ivg = ~-,

+ 4 2 Uw = - I . The additional couplings ~-$H and -o~'q~l are not constrained by the gauge theory.

Some time ago, we pointed out [2,3] that the in- clusion of the gauge group in a simple supergroup con- straining the low-energy parameters,

SU(2) X U ( 1 ) C SO(2 /1) , (3)

removes most of the arbitrariness. The couplings have to obey

* Supported in part by the US DOE Grant DE-FG05- 85ER40200 and by the US-Israel Binational Science Foun- dation. Wolfson Chair Extraordinary in Theoretical Physics.

308

tan 0 w =g'/g = 1 /V~, or sin20w = 0.25 . (4)

Note that the experimental value has come closer to this value recently [4]. The quarks in (2) all fit pre- cisely into one four-component irreducible multiplet of SU(2/1) with eigenvalues

iw,4): (u°,o,O;.o-l,L½;uo-l,½,-½;uo-2,o,o). (5)

Thus u0 = 4 for the quarks. Moreover, for integer u 0 this four-multiplet becomes reducible and 4 -+ 1 + 3, the upper state disconnects and we get triplets fitting the leptons (1) precisely (u0 = 0)! The supergroup grading follows the chirality assignments.

The quantum numbers of ¢H (x)become correlated with those of the gauge mesons (W~, Z°,Au) within an SU(2/1) octet. Note that the non-vanishing vacuum expectation value for the K°-likeq~ O (in the/.t 6 direc- tion [3] of the superalgebra) indeed leaves only electric charge Q invariant,

a = - ~ t t 3 - ½x/~tz8,

[/16,x} = 0 , x E s u ( 2 / 1 ) ~ x = Q (6)

(where [ , ) is the Lie superbracket) as against su(3), where ?'6 itself would also have been a solution.

We do not claim to have a good understanding of the gauging of a supergroup [5], though much pro- gress has been achieved [6,7] in this direction. In such

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Volume 181, number 3,4 PHYSICS LETTERS B 4 December 1986

a (high-energy limit) ,fundamental" interpretation [5-7] , the conventional implementation of a super- group action as relating fermions to bosons (and vice versa) results in a doubling of the multiplets, with half the states representing ghost fields appearing in that formalism [7].

For example, the four (virtual) ~ states make up the boson components of a Lorentz-scalar octet (the adjoint representation of SU(2/1))whose fermionic states correspond to g(W±), g(Z0), g(Au), the four (virtual) Feynman-DeWitt-Faddeev-Popov "ghost" fields of the gauge mesons, as essential to the gauge- theory formalism as the Goldstone-Higgs field itself. However, the more interesting features o f SU(2/1} derive from its direct f i t to the phenomenological in- ternal quantum numbers and low-energy parameters. It is thus plausible that SU(2/1), rather than represent- ing directly some fundamental (high-energy)structure, could emerge "accidentally" as a result, for instance, of the de-facto supertracelessness of the su (2) X u ( I ) algebra matrices, when the grading Z(2) is selected so as to follow chirality [8].

This ad-hoc aspect, with no indication of a theoreti- cal foundation was precisely the situation with "fla- vour" SU(3), which appeared to clash with the spin- statistics theorem (before the introduction of "colour" SU(3)) and whose good fit went beyond what could be expected from a perturbative treatment of the sym- metry breaking.

Indeed, the analogy with the static regularities of the Gursey-Radicati [9] SU(6) is even closer. That symmetry violated both Lorentz invariance and uni- rarity [ 10], and yet its predictions were astonishingly accurate, a fact now attributed to QCD. Indeed these SU(6) results were among the strongest clues that led to the definition of the requirements from a theory of the strong interactions, present in QCD as asymptotic freedom [11].

In SU(2/1), we may also have an ad-hoc low-ener- gy "effective" systematics, the supergroup structure appearing accidentally. In this approach we forego both the statistical correlations of the superalgebra and Lorentz invariance, focusing singly on the internal quantum numbers and low-energy parameters. All components of a multiplet now describe physical par- ticles (i.e. no ghosts). The fermion multiplets then re- late particles of opposite chiralities, and the meson oc- tet relates W ±, Z 0, T O with J = 1 to the 4'H with J = 0.

Note that SU(2/1) in either interpretation does not preclude [8] an additional high-energy embedding of SU(2) × U(I) in a GUT such as SU(5) or E(6). In the present ad-hoc low-energy approach, the analogy with SU(6) can be further extended: while the low-energy static SU(6) referred to post-renormalisation "constit- uent" quarks with masses of some 300 MeV, there was another SU(6) describing almost massless (high- energy) "current" quarks. The two systems were re- lated by the Melosh transformation [12]. A similar though probably more complicated mechanism would relate the low-energy SU(2/1) to the high-energy GUT.

We would now like to derive the SU(2/1) prediction for m H, the mass of the residual Goldstone-Higgs parti- cle. We have treated this issue previously in a "hybrid" approach [2,3,5], namely using the conventional field theory mechanism with -o2~2H + ~q~l, feeding into it the SU (2/1) parameters. Since we do not know if the supergroup gauge theory is renormalizable, this ap- pears problematic. However, in the ad-hoc low-energy global symmetry approach, mH is directly related to the masses o f the weak interactions intermediate bo- sons.

The superalgebra bracket relations are

[itA, ItS) = 2CAsEItE, str ItA = 0 , (7)

where str stands for the supertrace, and/2 A (,4 = 1 ... 8) are a basis of 3 X 3 matrices identical to the SU(3) ;k matrices except for

/aS = -~CA8 - 2 V ~ 0 ) • (8)

As a result, we have

Cabc=ifabc ( a , b , c = l , 2 , 3 ) ,

Ca8 c = ifa8 c = 0 ,

Cai / = ifaii , C/a / = i./~a / ( i , /= 4, 5, 6, 7),

1. - -1 . Cs i / = ~ l f s i j , C i s / - -~ lf/.sj ,

Ci/a = di/a , Ci/8 = -½X/~Si/ , (9)

where f and d refer to the usual SU(3) coefficients. The particle masses are generated by spontaneous

symmetry breakdown in the/z6 direction, in the same way as "flavour" SU(3) is broken in the 7,8 direction. However, It6 is not a diagonal operator, and has no matrix-elements between a meson and itself(in fer-

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Volume 181, number 3,4 PHYSICS LETTERS B 4 December 1986

mion mult iplets i t relates the two chiral components of one particle). For the mesons we are thus forced to take the squared tz 6 transitions. We shall therefore read

o f f M 2 matr ix elements [with o = (01q~)10>],

(XA IM21XA ) = B~(XA IIz61B)(BIIa61XA) °2

2 G c A =o ~CB6 a . (10) B

As a result, we find unique ly (H is the residual qS(H 0)

particle):

m (70) = 0 ,

m ( W ± ) : m ( z O ) : m ( H ) = l : 2/vr3 : 2 , ( l l )

thus predict ing

m ( H ) ~ 1 6 0 - 1 7 0 G e V . (12)

R eferen ces

[11 S. Weinberg, Phys. Rev. Lett_ 19 (1967) 1264; A. Salam, in: Elementary particle theory, Proc. VIII Nobel Symp., ed. N. Svaxtholm (Almquist and Wiksell, Stockholm, 1968) pp. 367-377.

[2] Y. Ne'eman, Phys. Lett. B 81 (1979) 190; D. Fairlie, Phys. Lett. B 82 (1979) 97.

[3] Y. Ne'eman and J. Thierry-Mieg, in: Differential geo- metrical methods in mathematical physics, Proc. Conf. Aix-en-Provence and Salamanca (1979), Lecture Notes in Mathematics, Vol. 836, eds. P.L. Garcia et al. (Springer, Berlin, 1980) pp. 318-348.

[4] P. Reutens et al., in: The Santa Fe meeting, eds. T. Goldman and M.M. Nieto (World Scientific, Singapore, 1984) p. 270.

[5] Y. Ne'eman and J. Thierry-Mieg, Proc. Nat. Acad. Sci. USA 77 (1980) 720.

[6] J. Thierry-Mieg and Y. Ne'eman, Nuovo Cimento A 71 (1982) 104.

[7] Y. Ne'eman and J. Thierry-Mieg, Proc. Nat. Acad. Sci. USA 79 (1982) 7068.

[8] Y. Ne'eman and S. Sternberg, in: Proc. XXth Intern. Conf. on High energy physics (Madison, 1980), AlP Conf. Proc. Vol. 68, Particles and fields subset. 22, eds. L. Durand and L.G. Pondrom (American Institute of Physics, New York, 1981) pp. 460-462.

[9] F. Gursey and L.A. Radicati, Phys. Rev. Lett. 13 (1964) 173.

[10] S. Coleman, Phys. Rev. B 138 (1965) 1262. [11] D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973)

1343; H.D. Politzer, Phys. Rev. Lett. 30 (1973) 1346.

[12] H.J. Melosh, Phys. Rev. D 9 (1974) 1095.

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