P H Y S I C A L R E V I E W V O L U M E 1 3 8 , N U M B E R I B 12 A P R I L 1 9 6 5

Spontaneous Mass Splitting in a Bootstrap Model* DAVID Y. WoNGf

University of California, San Diego, La Jolla, California (Received 30 November 1964)

A model is constructed in which all members of the vector octet (<pr,K*,p) appear as two-body bound states of themselves, i.e., <p' -> (K*K*), K* -» (<p'K*,f>K*), p -> (PP,K*K*). The two-body interaction is represented by a fixed pole of the T matrix with residues related by Clebsch-Gordan coefficients derived from the ® representation of SUs. For a given position of the pole, the over-all strength of the residue is ad­justed so that mv

r=mK*—/mp is a solution of the problem (m stands for the masses of the bound states as well as the masses of the free particles). Keeping the position and residues of the "potential" poles unchanged, the existence of a self-consistent solution with mass splitting is investigated. It is found that within a certain range of the pole position, there exists one and only one solution with nondegenerate masses, and outside of that range of the pole position, there exists none. A comparison between the nondegenerate solution and the Gell-Mann-Okubo mass formula is made. It is also noted that the deviation of the coupling constants from the SUS values is very small (< 2%) even when the mass splitting is quite sizable.

IN the study of broken symmetry, particularly the broken SUz, there are two general directions of

approach. One involves the assumption of a symmetry-breaking interaction1,2 while the other seeks a self-consistent solution of some nonlinear equations without introducing a symmetry-breaking interaction.3-5 Evi­dently, exact symmetry is possible in the latter case. The interesting question is whether there exist solutions which do not possess exact symmetry. In this paper, we construct a simple model of the T matrix in which the following points can be examined in a nonperturbative way6: (i) The existence of solutions with spontaneous splitting of the masses, (ii) The number of such solu­tions. (hi) The magnitude of the splitting, (iv) Com­parison with the first-order Gell-Mann-Okubo (GMO) mass formula.1,2 (v) The deviation of coupling constants from exact symmetry.

In what follows we assume the vector octet (</,iT*,p) to be the bound state of a pair of the same vector octets, _ i.e., / -> Z*i?*, K* -> (pK*, <p'K*\ p -» (pp,K*K*). The symbol <pr is used to distinguish it from the physical <p meson which is supposed to be a mixture of <pf and an unphysical unitary singlet co'.7,8 The T matrix is written in the form BD~X where the "poten­tial" B is assumed to be a simple pole with residues transforming exactly like the ® representation of SUz.

* Work supported in part by U. S. Atomic Energy Commission. t Alfred P. Sloan Fellow. 1M. Gell-Mann, Phys. Rev. 125, 1067 (1962). 2 S. Okubo, Progr. Theoret. Phys. (Kyoto) 27, 949 (1962). 3 S. L. Glashow and M. Baker, Phys. Rev. 128, 2462 (1962). 4 R. E. Cutkowsky and P. Tarjanne, Phys. Rev. 132, 1354

(1963). 6 R. H. Capps, Phys. Rev. 134, B460 (1964). 6 D. Y. Wong, Bull. Am. Phys. Soc. 9, 642 (1964). 7 S. L. Glashow, Phys. Rev. Letters 11, 48 (1963). 8 F. Gursey, T. D. Lee, and M. Nauenberg, Phys. Rev. 135,

B467 (1964); F. Gursey and L. A. Radicati, Phys. Rev. Letters 13, 173 (1964).

We write

1 = 0, (K*K*)->(K*K*):

Bi*'>(s) = T/(s+so).

W , (j>K*,<p'K')->{pK?,<p'K*):

£<**>(*) = i—f *)• \s+SoJ\i \) \s+s0J

1 = 1 , {PP,K*K*)^{pp,K*K*):

r \ / f b/2\

s+s0J

/ r \ / f jvZ\

\s+sJ\WZ i I

(1)

(2)

(3)

where s is the center-of-mass energy squared. The D matrix can be evaluated using the unitarity

condition and the dispersion relation.9 One obtains

#(*') (s) = l-TF(s,mK*2,mK*2,s0),

Dn^(s) = l-irF(s,mf?,fnK*2,s0),

Z)i2(JP) CO = -^TF&m^niK*2^),

Ai ( JC* ) (s) = - iTF(s y m^,mK^,So) ,

Z>22(JC*)(s) = l-^TF(s7m^,mK^so),

Dn{p)(J)= l-iYF{s,mp\mp\s,),

Dn{p) (s) = -^lYF{s,mp\mp\s,),

£21CP> (S) = - J\5rF(j,wx*2,wjc*Vo),

D22W(s)=l-irF(s,rnK*2,niK*2,s0),

where

F(s,mi2,m22,So)

(s+so) ds'

(mi+m-t)

qri 1

Vs'J(s'+s0ns'-s)

(4)

(5a)

(5b)

(5c)

(5d)

(6a)

(6b)

(6c)

(6d)

(7)

9 For example, see J. D. Bjorken, Phys. Rev. Letters 4, 473 (1960).

B246

S P O N T A N E O U S M A S S S P L I T T I N G I N A B O O T S T R A P M O D E L B 247

and qn is the center-of-mass momentum for two parti­cles with masses mi and m2 [#i22=.s/4—(wi2+m2

2)/ 2+(wi2—w2

2)2/4^]. The condition for a bound state with a given energy is that the determinant of the D matrix vanishes at that energy. In the present problem, we require

D^(m^) = 0, (8)

detD<**>(mK*2) = 0 , (9)

detD<'>(wp2) = 0. (10)

Let us first consider the degenerate system m<p>2

= niK*2=znip2=ni2. Equations (8), (9), and (10) become, in this case,

£>(*') (W2)« detD<**> (m2) - detD<"> (m2)

= l~-rF(w2,w2,mVo) = 0. (11)

For any given so, we choose

r=l/ Jp(w2 ,m2 ,mVo) (12) so that the degenerate solution is a bootstrap solution with exact SUz symmetry. Now, with this same value of P, we look for solutions of Eqs. (8), (9), and (10) with m^T^MK^T^mp2. Since the dependence of the D functions on the masses is explicitly given above, this problem can be solved by numerical methods without recourse to any perturbation expansion. We find that in fact there exists one and only one nondegenerate solution of (8), (9), and (10) for each value of SQ within the range l.lm2<so<3.0m2. Outside of this range, the degenerate solution is the only solution. Numerical results are summarized on Table I. A comparison with the Gell-Mann-Okubo formula is given in the last column, where w / ( G M O ) is evaluated using the usual mass-squared formula and our calculated value of the p and the K*.

Remarks. (1) If we were to expand the masses in the neighborhood of the degenerate value m2 and keep only the first-order term in the expansion, then Eqs. (8), (9), and (10) become a system of three homogeneous linear equations with three unknowns (tn^—m2), (mK*2—M2), and (mp

2—m2). These homogeneous equa­tions have a solution for so^l.Sm2 and the solution satisfies the GMO formula exactly. This is essentially the same result as obtained by Cutkowsky and Tar-janne4 and by Capps.5 When all the higher order terms are included in the present numerical solution, the splitting diminishes at the point s0=l.Sm2 and the GMO formula is exact in the neighborhood of that point.

TABLE I. Numerical results of nondegenerate solutions over the allowed range of SQ. All quantities are given in units of the de­generate mass squared.

So

1.1 1.47 1.75 2.05 3.00

wp2

0.526 0.819 1.008 1.197 1.728

niK*2

2.119 1.131 0.996 0.927 0.844

m<p>2

2.902 1.246 0.992 0.844 0.611

« / ( G M O )

2.65 1.235 0.992 0.837 0.549

(2) Our solution satisfies the GMO formula to within « 1 % at the value of so— 1.47 m2 where the splitting between the K* and the p is approximately equal to the observed value (m#*2—wp

2)^0.167 (wic*2+mp2). Al­

though the inclusion of higher order terms in the mass splitting is essential in obtaining the present result (the first-order equations are homogeneous and have no solution for ^0= 1.47 m2), they apparently have a very small effect in breaking the GMO relation.

(3) The coupling constants between the bound state and the two-particle system are exactly related by SUz Clebsch-Gordan coefficient in the degenerate solution. For the nondegenerate solution with mass splitting, the coupling constants obtained by evaluating the residues of the bound-state poles are no longer exactly related by SUz. However, our numerical result shows that the deviation from SUz is extremely small. For example, with SQ— 1.47 m2, the deviation is less than 1% for the ratio of any two coupling constants.10

(4) Although our numerical results are obviously model-dependent, the existence and uniqueness of a nondegenerate might have a deeper significance. I t would be of considerable interest to examine some other models, for example, Cutkowsky's model where the potential itself is produced by the exchange of the vector octet.11

(5) If one assumes the vector-nonet (<p,<a,p7K*) model,8 then it is not possible to have a complete boot­strap model involving vector mesons alone. On the other hand, by bringing in the pseudoscalar octet to complete a bootstrap system, one can use the known coupling constants (e.g., gp7r7r) to eliminate so from the problem. The existence of a nondegenerate solution would then depend on the value of the physical-coupling constant rather than the parameter so-

10 Similar results are obtained by J. R. Fulco and D. Y. Wong, Phys. Rev. 136, B198 (1964), where they consider the vector octet to be a bound state of a pair of pseudoscalar octets.

11M. Leon and R. E. Cutkowsky, Bull. Am. Phys. Soc. 9, 629 (1964).