Z. Physik C, Particles and Fields 4, 53-54 (1980) t0r ~ysik C and FL:4Os ~) by Springer-Verlag 1980 Fourth Generation of Quarks and Leptons* S. Pakvasa, H. Sugawara,** and S. F. Tuan Department of Physics, University of Hawaii at Manoa, Honolulu, HI 96822, USA Received 13 August 1979 Abstract. Permutation Symmetry suggests an inter- generation quark-lepton mass formula m/mJm L = 9 /.t . mJmb/m b, = mc/mt/mt,. We consider several possi- bilities for the fourth generation masses. Recently it was pointed out by two of the authors (S.P. and H.S.) [1] that by applying permutation symmetry to Higgs couplings of quarks and leptons, striking relations amongst quark and lepton masses are obtained. In particular for six quarks and six leptons S4 | SU 2 | U 1 gave the mass formula [1] ira! m2 m2 mc 2 m2 = 0 (1) im 2 m2 m2 leading to mt = 26 GeV for the current quark mass of the t-quark. In the limit of me, mu, and ma being negligibly small, we obtained the simple formula [1-4] mu/m ~ = rnc/m t = ms/m b. (2) Renormalization effects are expected to bring down the mass of the physical (t ~) bound state vector meson (topsilon) and it could be as low as 40 GeV [3, 4]. If there are more than three quark and lepton doublets, we suggested [1] a generalization of Eq. (2) should hold, viz. mu/m~/m L = rnc/mt/m ,, = ms/mb/m b, (3) in the limit that m e, mu, and rn a are negligible as before. Equation (3) arises as follows. Consider S 4 | SU E @ U 1, where Higgses transform as 3, left- handed doublets and right-handed singlets as 3_ 9 1 under S 4. Now e, u, and d get masses only through mixing and are decoupled in the limit of m e, m u, and ma going to zero. Then, since 3 | 3_| 3 contains 1 * Work supported in part by the U.S. Department of Energy under Contract DE-AC03-76-ER00511. ** Permanent address: KEK, National Laboratory for High Energy Physics, Tsukuba-gun, Ibaraki-ken, Japan only once, the mass-squared matrices of leptons, +2/3 quarks and -1/3 quarks are proportional. Hence their eigenvalues are proportional and Eq. (3) follows. This argument can be trivially generalized to n generations. Such formulas can also be obtained by using SU (n) as a horizontal gauge symmetry [4]. Equation (3) tells us the mass of the next quark doublet if m L is known: m b, = (mL/m~)m b, m t, = (mL/m~)m t. (4) One possibility mentioned earlier [1] is that lepton masses scale geometrically as once speculated [5], then 2 m L = m~/m u - 30 GeV, and ms, ~ 90 GeV. (5) Such heavy quarks have very interesting decay properties (e.g., as W-factories) which have been discussed already [1, 2, 6, 7]. Another possibility is that rn L ~ 10 GeV as sug- gested recently [2, 8]. In this case m b, "~ 28 GeV and b' is nearly degenerate with t t It is attractive to suppose that this degeneracy is reflected in large mixing between t and b' leading to "flavor flip" so that (t, b') nearly form a weak doublet after mixing (i.e. t--, b' transition is large), and b couples mostly to the heavier t'-quark. A third possibility of more immediate experimental relevance is the following. Suppose the un-understood lepton-hadron degeneracy (m, ~ m~, rn~ ,-~ too) is repeated [9] and mL ~ mB ~ 5 GeV, (6a) then m b, ~ 14 GeV, m t, ,-~ 72 GeV. (6b) Then the mass of the (b'/~') bound state can be as low as ,-- 24 GeV (due to renormalization) and the fourth 0170-9739/80/0004/0053/$01.00