VOLUME 43, NUMBER 23 PHYSICAL REVIEW LETTERS 3 DECEMBER 1979

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78/A027 (unpublished). 5S.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 quan­tum 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 mechan­i 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 longi­tudinal 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 con­serva 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 igna­t 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 emen­t 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. B£g

The Rockefeller University, New York, New York 10021

and

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 spon­taneous 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. How­ever, careful attention must be paid to mixing to identify the physical states and analyze their de­cays.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 appears—at least in the sim­plest models for generating fermion mass—to 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 magni­tude 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) /ff8

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 » - <0lOTt0) NC ,A'3 NCA3 (4)

Equations (2)-(4) yield

m1 m 2 mQ' A ' / 3 \ i / 2

J 7T mn (5)

using three colors for QCD and assuming four for QC'D, respectively. For mq,~mq, e.g., if the current quark masses are of a similar origin, Eq. (5) implies m1T'~rl GeV. This estimate is to be contrasted with the naive expectation for pseudo-Goldstone-boson9 masses arising at the one-loop level: ra^-a^2/^=25 GeV. (Later, where a specific numerical value is required, we take mff/~15 GeV.)

As pseudo Goldstone bosons, the hyperpions have fairly model-independent decay modes since they couple primarily to the divergences of cur­rents. To the extent that these currents contain contributions from the known quarks and leptons, these divergences will be of the form

• +(mi + mj)fiy5fj9

9^' / i = . . . + ( ^ i ~ ^ ^ ) / i / < ; ,

(6)

(7)

where m{ is the mass of the fermion/ i # The hyperpion couplings are to be contrasted with those of the left-over elementary Higgs scalar cp of the Weinberg-Salam model:

£ = . '(l/{<p))<pmtfiU (8)

Thus both types of bosons have the strongest couplings to the heaviest fermion available, but there are measurable differences. For instance, a flavor-diagonal decay of the n1 is S wave, while that of cp is P wave leading to the difference in branching ratios:

Ttcp~— bW) /™- \2s™ 2 — A™ 2 \ 2

r( cp -* TT)

4m£ 4ra r

2

rpr ' - f tS) ^ Jm r(7r /-Tf) ~ \m

5 W m / - 4 m f c2 \ l / 2

T) W i r ' 2 - 4 m T 2 /

(9)

(10)

Furthermore, if we look at exclusive channels, processes such as y~DD and IT' ^DDTI are a l ­lowed, whereas cp^DDn and n' -~DD are for­bidden.

The 2y decay of neutral hyperpions can be com­puted, as for 7T°, from the triangle anomaly:

r(7r '°~2 r) JhX/m^ 4 r 0 r ° - 2 y ) \frsj\m*) 3

(ID

modulo possible differences in q' and q charges. However, the branching ratio to 2y is likely to be very small compared to the decay to massive

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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 liberal­ly 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 de­duced using an effective-Lagrangian approach, as long as the Ttns are soft, on the scale set by /„>. For simplicity we imagine using linear represen­tations 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 fur­ther 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 <p and r\ not only have masses of

doublets, 4>'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

<a') = / v * 3 0 0 GeV. (16)

From this standpoint, the "soft" couplings of 77' and a' resemble a model with several elementary Higgs fields, where the physical spin-0 particles include not only the scalar Higgs particle of the minimal Weinberg-Salam model but also addition­al scalars and pseudo-Goldstone pseudoscalars. The distinction lies in the expected masses. The masses of fundamental spinless particles are typically comparable to each other. In contrast, a dynamical o' would have a mass of order A', ~ 1-3 TeV by analogy with QCD, and its decays into light 77"s would be so fast as to make it vir­tually unrecognizable as a distinct particle. Hence, the prominent light particles are all pseudoscalars.

We exhibit the contrasts in some simple models. The coupling of 77/,s to a lepton or quark doublet (<z, 6), mediated by W± and Z, are described by the effective interaction

order 1 TeV but also manage to decay strongly into pairs of pseudoscalar mesons.

We have discussed some experimental conse­quences of hyper color interactions that would be manifest at energies well below typical hyper-hadron masses; these low-energy consequences, follow from partial conservation of axial-vector-

£ e f f = (G F / /2 ) / 7 r , /2^[ (m f l -m,) -y 5 (m a + m & ) j6( -z)77 ' + + H.c.

+ (GF/&)fir,2(maaiy5a-mbfiiy5b)Ti'0o (17)

Note that the a' does not couple directly to a or b and, therefore, it is not responsible for their mass­es . Note also the universality of ir' couplings to any doublet: Apart from the trivial mass rescaling, the coupling to (c, d) is the same as that to (a, b).

The contrast with the minimal standard model, in which

^ = + 2m^siH{m"3a+mbhb)(p0' ( 1 8 )

is discussed above [e.g., Eqs. (9) and (10)]. Consider then a variant with an additional Higgs doublet (a total of five physical spin-0 fields):

£2 = £1 + 2-l/%f1aa+f2mv° + 2-l/2(-fi^ (19)

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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 mech­anism [a' ,7r,0,7r/±] and in theories with elementary Higgs fields [^0,x0»X±]« The <P decay rates quoted are for m y= 15 GeV. Note that <x', the dynamical analog of <p°, decays rapidly to ir/+ir,m, etc. , via strong inter­actions.

PARTICLE

a'

7T'°

, +

*°

x°

+ X

EXPECTED

MASS

(in GeV)

-2000

-15

-15

>10

Could be -15

Could be -15

FEASIBLE

PRODUCTION

MECHANISM

?

Via Heavy Flavors in e+e~-*. . . or pp+ . . . .

W ~+ TTI"7T,°

e+e~-vTT,+TT,~

± .± .o

Via Heavy Flavors

and Virtual

W-Pairs in e e -*•.. or pp-»- ...

Same as for TT'0

+ Same as for TTT

2y DECAY

Rate , (in sec )

-

~2xl0 1 5

-

ao17

?

-

Photon Pols.

-

1

-

II

1

-

f f DECAYS

Universal Coupling

-

Yes

Yes

Yes

No

No

V T 1 T T T

Rate - Rate , (in sec J ( in sec )

-

-

~3xl0 1 9

-

-

?

-

-io20

-

~Axl0 1 9

?

-

T(bb)

r(Ti)

-

-15

-

-10

?

-

PSEUDOSCALAR MESON

CHANNELS

DD

-

No

Yes

Yes

No

Yes

DDTT

-

Yes

Yes

No

Yes

Yes

c u r r e n t a rgumen t s and a r e the re fore fairly in­sens i t ive to how hypercolor is implemented. We have a l so con t ras ted the exper imenta l profi le of re la t ive ly low-lying dynamical bosons , in the hypercolor scena r io , with that of e l ementa ry Higgs p a r t i c l e s . (Some of our r e s u l t s a r e s u m ­mar i zed in Table I.) It i s our hope that our ex ­pe r imen ta l col leagues will be able to shed some light on the na tu re of the Higgs mechan ism in the no t - too-d is tan t future.

This work was par t i a l ly supported by the Uc So Depar tment of Energy under Contrac t Grant Noc

EY-76-C-02-2232B.*000 and a l so by the Uc S. Depar tment of Energy under Contrac t No. D e - A c -79-ER0068.

Two of us (M.A.B.B. and P.R.) a r e indebted to the Aspen Center for Phys i c s for the hospital i ty which led to our col laborat ion; one of us (M.A.B. B.) is grateful to Brookhaven National L a b o r a ­tory for hospital i ty during the cou r se of this work. It i s a p l e a s u r e to acknowledge s t imu la t ­

ing conversa t ions with M. Gell-Mann.

1M. A. B. Beg and A. Sirlin, Annu. Rev. Nucl. Sci. 24, 379 (1974).

2M. A. B. Beg, in New Frontiers in High Energy Physics, Proceedings of Orbis Scientiae, 1978, Coral Gables, edited by A. Perlmutter and L. Scott (Plenum, New York, 1978).

3S. Weinberg, Phys. Rev. D 1£, 974 (1976). 4Compare L. Susskind, SLAC Report No. 2142, 1978

(Phys. Rev. D, to be published). 5See, for example, M. Weinstein, Phys. Rev. D S9

2511 (1973). We thank E. Eichten and K. Lane for dis­cussions on this point.

6S. Dimopoulos and L. Susskind, NucL Phys. Bl55, 237(1979).

7E. Eichten and K. Lane, to be published; M. Volo-shin and V. Zakharov, private communication.

8M. Gell-Mann, P. Ramond, and R. Slansky, to be published; E. Far hi and L. Susskind, to be published; M. A. B. Beg, to be published.

9S. Weinberg, Phys. Rev. Lett. 29, 1698 (1972).

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