Experimental Characteristics of Dynamical Pseudo Goldstone Bosons

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    3S.-S. Shei, Phys. Rev. D 14, 535 (1976). 4E. Witten, Harvard University Report No. HUTP-

    78/A027 (unpublished). 5S. Coleman, Commun. Math. Phys. 31, 259 (1973). GA. Neveu and N. Papanicolaou, to be published. 7H. Bethe, Z. Phys. 71, 205 (1931); H. B. Thacker,

    Phys. Rev. D 17, 1031 (1978). 8H. Bergknoff and H. B. Thacker, Phys. Rev. Lett. 42,

    135 (197.9), and Phys. Rev. D 19, 3666 (1979). We would like to thank Dr. Thacker for sending us the sec-

    The usual implementat ion of the Goldstone -Higgs mechan i sm of spontaneous s y m m e t r y breaking, via e lementary sp in-0 fields, i s one of the l e s s a t t r ac t ive fea tu res of conventional quan-tum flavor dynamics (QFD). And the si tuat ion is acute in a t tempts to unify QFD and quantum chromodynamics (QCD) into a s ingle gauge theory . In such grand unified theo r i e s , one is obliged to introduce a multi tude of Higgs fields with jud ic i -ously contr ived couplings; the essen t ia l s i m p l i c -ity of the gauge- theore t ic approach i s , thereby, i r r e t r i evab ly lost . It has been suggested, 1" 4

    the re fore , that one d i s ca rd e lementa ry Higgs fields a l together , and seek a dynamical mechan-i s m for s y m m e t r y breakdown.

    In the s imp les t dynamical mechanism, the r e -quisi te Goldstone bosons , which furnish the longi-tudinal deg rees of f reedom for m a s s i v e gauge fields, a r e bound s t a t e s of a new spec i e s of quark, whose supe r s t rong gauge in te rac t ions (generated by gauging a c o l o r ' degree of freedom and d e -sc r ibed by a theory here inaf ter called Q C D ) spontaneously b reak ch i r a l s y m m e t r y .

    The c o l o r ' quarks a r e likely to come in s e v e r a l f lavors , in which case the re will be s e v e r a l light pseudo Goldstone bosons a s well . In this Le t t e r

    ond reference prior to circulation. 9C. N. Yang, Phys. Rev. Lett. 19, 1312 (1967). B. Klaiber, in Lectures in Theoretical Physics:

    Quantum Theroy and Statistical Theory, edited by A. O. Barut (Gordon and Breach, New York, 1968), Vol. XA, p. 141.

    n M. Takahashi, Prog. Theor. Phys. 46, 401 (1971); C. K. Lai, Phys. Rev. Lett. 26, 1472 (1971); Phys. Rev. A 8 , 2567 (1973).

    12B. Sutherland, Phys. Rev. Lett. 20, 98 (1967).

    we obse rve that these pa r t i c l e s may be a s light a s 10 GeV, that they will be re la t ive ly pointl ike (of s ize 1 TeV"1) , and will have production and d e -cay modes that a r e de te rmined by p a r t i a l con-serva t ion of a x i a l - v e c t o r - c u r r e n t a rgumen t s and hence a r e fairly model independent. Thei r s igna-t u r e s a r e , crudely speaking, s i m i l a r to those of e lementa ry Higgs p a r t i c l e s ; consequently, we s t r e s s the di f ferences . If sp in-0 weakly i n t e r a c t -ing p a r t i c l e s a r e d iscovered with m a s s e s of tens of GeV, the question of composi te v e r s u s e l emen-t a r y need not wait unti l ene rg ie s of 1 TeV probe the possible bound-s ta te s t r u c t u r e .

    The c o l o r ' deg ree of freedom, f i r s t introduced by Weinberg, 3 may be dubbed "hypercolor . " 4 The terminology is convenient, with words such a s hyper quark, hyperpion, and h y p e r - a having an obvious meaning. We take the weak and e l e c t r o -magnetic in te rac t ions of the hyperquarks to be i somorphic to those of o rd inary quarks so that a flavor doublet such a s (u', d') t r a n s f o r m s a s (u, d) under the e lec t roweak group.

    The exis tence of hyperquarks i s , of course , logically independent of the exis tence of e l e m e n -t a r y Higgs p a r t i c l e s ; we do not ru l e out the p o s s i -bil i ty that t he r e may exis t both hyperpions and

    Experimental Characteristics of Dynamical Pseudo Goldstone Bosons M. A. B. Bg

    The Rockefeller University, New York, New York 10021


    H. D. Po l i t ze r and P . Ramond California Institute of Technology, Pasadena, California 91125

    (Received 15 October 1979)

    The hypothetical existence of new color interactions, which participate in the spon-taneous breaking of the weak-interaction group, will in general lead to relatively light composite pseudo Goldstone bosons. Their production and decay characteristics are analyzed to be close to, yet actually distinguishable from, those of the elementary Higgs bosons of the Weinberg-Salam model.

    1979 The American Physical Society 1701

  • VOLUME 43, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 3 DECEMBER 1979

    elementary weak scalars. In such a case, f v, (the hyperpion decay constant) need not be as large as suggested below, and hypercolor would be even more accessible at low energies. How-ever, careful attention must be paid to mixing to identify the physical states and analyze their de-cays.5 Our discussion of hyperpions would still be applicable and would be an aid in deciphering which was which.

    Three of the hyperpions must be exactly mass-less to be absorbed by the W* and Z, and in the process they contribute to the Wand Z masses:

    m 2 cos = raK,s*!e/vsin, (1)

    where e is the unit of electric charge and is the Glashow-Weinberg-Salam angle.1 Hence f^ = 300 GeV. (Note that the relationship between Wand Z masses, embodying the so-called A/weak = | rule, emerges naturally.) In the following, we shall assume that QC'D may be regarded as scaled-up QCD, albeit with a different number of colors; the characteristic mass scales, A' and A, in the two theories may, therefore, be related via

    A y A ' 0 M 1 / 2 = / ^ A ( i v c )l / 2 , (2)

    where Nc is the number of quark colors. Since A-300 MeV, we have A'~(3/ tf c , )

    l / 2 TeV. No completely satisfactory hypercolor model

    of dynamical symmetry breaking exists as yet. There are several outstanding problems: (i) Whi l e /^ gives mass to the W's, quarks and leptons must get their masses elsewhere.6

    (ii) Lifting the mass degeneracy within weak iso-spin quark doublets appearsat least in the sim-plest models for generating fermion massto destroy the A/weak = f relation between W and Z masses.7 (iii) Unified models typically have many hyperquarks,8 and hence many massless, unab-sorbed hyperpions.

    We proceed assuming that these problems are solvable and, in particular, that the unabsorbed hyperpions acquire masses; the order of magni-tude of these masses may be estimated as follows. Current-algebra considerations imply that the near-Goldstone modes in QC'D and QCD satisfy

    m^^mf ma,(0\q'q>\0) /ff


    mq A ,2 (0\qq\0) (3)

    Also, since QC'D is presumed to be scaled-up QCD, we may equate the dimensionless ratios

    < 0 | g V l t -

  • VOLUME 43, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 3 DECEMBER 1979

    fermions / / :

    r(7T'^2 r) r(TT'~ff)

    ccgilULiL} x(phase-space ratio). (12) \mf J

    The hyperpions are of size A'"1 or/^/""1. The charged ir"sf therefore, will be produced liberal-ly in e+e" annihilation; the deviation from

    Re+e-W = iSQ,< 2 ( l - 4m^2A)3 / 2 (13) 7T '

    is of order N^'s/f^2, where N^ is the number of charged hyperpions, Q^> are their electric charges, and s = ( c m . energy)2 /7r/

    2. That the departure from the pointlike limit, Eq. (13), is of the order stated may be verified by writing an effective chiral Lagrangian3 for the 7r"s.

    In fact, all of the couplings of u,ys can be de-duced using an effective-Lagrangian approach, as long as the Ttns are soft, on the scale set by />. For simplicity we imagine using linear represen-tations with the pseudoscalar 77"s accompanied by scalar a"s . From the implicit q' content of the n' and from the W and Z masses that now come from a' vacuum-expectation values, it is evident that the chiral quadruplets transform as complex ,

    There is no universality here relating t h e / ' s of one fermion doublet to those of another. A fur-ther contrast to the dynamical scheme is the presence of two relatively light neutrals which undergo P-wave decays. Needless to say, the distinctions would be blurred if, for some reason, t h e / ' s are universal and equal to the would-be 77' couplings and if 's, under the electroweak group:

    The relevant couplings can be read off from

    eff = ( ^ V ) t ( / > p ^ ) + . . . , (15)

    where D^ is the appropriate gauge-covariant derivative. Note that now

  • VOLUME 43, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 3 DECEMBER 1979

    TABLE I. Some characteristics of spin-0 particles which occur in theories with a dynamical Higgs mech-anism [a' ,7r,0,7r/] and in theories with elementary Higgs fields [^0,x0X] The


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