P A R T I A L L Y C O N S E R V E D A X I A L - V E C T O R C U R R E N T . . .

*Work supported by the National Science Council, I reland.

'A. P a i s and S. B. Tre iman, Phys. Rev. Le t te rs 2, 975 (1970).

2 ~ e e , for example, S. J. Brodsky, T. Kinoshita, and H. Terazawa , Phys. Rev. Le t te rs 25, 972 (1970); Phys. Rev. D 4, 1532 (1971).

3 ~ a i s and Tre iman (Ref. 1 ) suggested an experimental t es t on this problem involving the detection of neutral soft pions.

4 ~ . Fubini and G. Fur lan , Ann. Phys. (N.Y.) 48, 322 (1969); M. Ademollo, G. Denardo, and G. Furlan, Nuovo Cimento 5&, 1 (1968).

5 ~ . J. Gilman and H. Harar i , Phys. Rev. 165, 1803 (1968); J. Ballam et a l . , Phys. Rev. Le t te rs 21, 934 (1968).

$see , f o r example, M. A. B. BBg et a1 ., Phys. Rev. Le t te rs 3, 1231 (1970).

7S. Weinberg, Phys. Rev. Le t te rs 3, 1688 (1971); T. D. Lee, Phys. Rev. D 3, 801 (1971); J. Schechter and

Y. Ueda, ibid. 2, 736 (1970). 8 ~ . Bacci et a1 . , Phys. Letters m, 551 (1972). 's. Weinberg, Phys. Rev. Letters 18, 507 (1967).

'Osee, for example, R. E. Marshak, Riazuddin, and C. P. Ryan, Theory of Weak Interactions in Par t i c l e Phys ics (Wiley-Interscience, New York, 1969), p . 723. his observation has been made by severa l authors,

most notably by L.-F. Li and E . A. Paschos [Phys. Rev. D 3, 1178 (1971)l. See a l so J. D. Bjorken, Phys. Rev. 148, 1467 (1966), footnote 34. However, the accuracy of - colliding-beam experiments has recently been improved (Ref. 8). We have not considered the i sosca la r p a r t of the spec t ra l function, which i s expected to be about a third of the isovector piece, s ince e r r o r s in the experi- mental relat ion (25) and the assumption c" c' may be comparable with this omission. The IVB decay width into specific hadronic channels may be smal l (Ref. l o ) , but f o r v e r y la rge m , many such channels a r e open and the total hadronic decay width becomes la rge .

P H Y S I C A L R E V I E W D V O L U M E 7 , N U M B E R 11 1 J U N E 1 9 7 3

Symmetry-Breaking Effects on Singularities of Multichannel Amplitudes near Thresholds*

J. Curiale and S. W. MacDowell Physics Department, Yale University, New Haven, Connecticut 06520

(Received 29 November 1972)

The effects of the breaking of an internal symmetry on the singularities of the scattering matrix, especially near thresholds, is discussed in terms of the K-matrix formalism. It is shown that, in first order in the symmetry-breaking coupling constant, the K matrix transforms like the symmetry-breaking Hamiltonian. This result provides a justification for the Gell-Mann-Okubo mass formula for narrow resonances with nonzero orbital angular momentum. The position of poles in different sheets of the energy Riemann surface and their displacement with the symmetry-breaking parameter is investigated in detail, particularly for s waves. The specific case of baryon-baryon scattering in the triplet s -d states is discussed using a potential model with broken SU, symmetry. On the basis of information from low-energy n p scattering and Ap final-state interaction we suggest that a resonance probably exists in the I = 1, E N system, below the 2 A threshold. The Gell-Mann-Okubo mass formula would have placed the resonance close to the ZZ threshold.

INTRODUCTION

An analysis of the effects of a symmetry-break- ing interaction on bound-state poles and resonances in multichannel amplitudes was car r ied out by Yang and Oakes.' They investigated the splitting and displacement relative to thresholds of these singularities a s the symmetry-breaking parameter X i s gradually turned on. Their analysis cas t doubt on the validity o r , more precisely, on the theoret- ical foundation of the Gell-Mann-Okubo mass for- mula. The basic objection i s a s follows. Let us assume that in the exact symmetry limit the scat- tering amplitude for transitions in a given partial wave and irreducible representation of the sym-

metry group has a bound-state pole (or an anti- bound state). Now, in order for such a pole to change sheet o r to become a resonance a s the sym- metry-breaking interaction is gradually turned on, i t has either to c ros s a threshold o r meet with an- other pole (a shadow pole) o r both. In each case the position of the pole a s a function of A has a branch point a t the value of h for which this occurs. There- fore, the linear approximation which is the basis for the Gell-Mann-Okubo mass formula is unwar- ranted. Moreover, they pointed out that for some range of values of the symmetry-breaking param- e ter , a resonance pole might move into a Riemann sheet fa r removed from the physical one. If this should occur for the actual strength of the inter-

3462 J . C U R I A L E AND S . W . M a c D O W E L L

action, the effect of the pole on the physical scat- tering amplitude would not be easily observable. As a result, some members of a multiplet of poles would appear to be missing. This work aroused a great deal of interest, prompting several authors to discussion of the Owing to centri- fugal-barrier effects, there is an essential differ- ence between the singularity structure associated with resonance in s waves and in higher partial waves, particularly near thresholds.

The discussion of Yang and Oakes,' although ad- dressed to the meson-baryon resonances in a p wave, i s actually more pertinent to s waves. Eden and ~ a y l o ? described in detail a n s-wave model in which a number of poles appear close together, but in different sheets of the scattering amplitude. Then i t might occur that, a s the strength of the coupling constant i s varied, one pole always would be on a sheet adjacent to the physical sheet, pro- ducing a resonance. However, for this to happen, one would need a certain condition on the coupling constants of the model. As they pointed out, if that condition is not satisfied, the preceding result does not obtain. In addition, in the symmetry limit this model would actually have two dynamical poles close to the physical region. A similar model was discussed by Nauenberg and Nearing.3

Ross4 called attention to the difference in the structure of resonances in s waves and in higher partial waves, but only the lat ter case was dis- cussed in detail. The main point i s that for p and higher waves, resonance poles near a threshold always occur in pairs in sheets that a r e adjacent to the physical sheet on the left-hand and the right- hand side of that threshold. Then a s one varies the symmetry-breaking parameter, there is always a pole adjacent to the physical region. The reso- nance pole does not change sheet a s i t moves past a threshold, but is replaced by i t s pair on another sheet. This is not generally the case for s-wave resonance poles.

In this paper we shall discuss the displacement of poles in multi-channel scattering amplitudes a s a function of an internal-symmetry-breaking cou- pling constant A. In Sec. I a general discussion of the problem i s given based on the K-matrix for - malism. A theorem i s proved stating in what inter- vals of the rea l axis between two consecutive two- particle branch points a pole i s allowed to c ros s from one sheet to another. If follows from two- particle unitarity generalized to unphysical sheets. In Sec. I1 the displacement of poles for partial waves with angular momentum 12 1 is reviewed. We supplement the analysis of Ross4 by showing that at h = 0, a K / a ~ t ransforms under the symmetry group as matrix elements of the symmetry-break- ing Hamiltonian. In the narrow-resonance limit

the real part of the position of a resonance pole i s given by the zeros of detK. To f i r s t order in A it will obey a Gell-Mann-Okubo mass formula. Here however, there is, in principle, no objection to the linear approximation, a t least in so far a s pro- duction thresholds may be neglected. In Sec. III we discuss the particular case of s waves. It i s shown that the location of shadow poles in the sheets of the energy Riemann surface depend qual- itatively on the Clebsch-Gordan coefficients for the decomposition of the product of the represen- tations of the scattering particles, into irreducible representations of the symmetry group. This i s essentially a kinematical threshold effect. As the symmetry i s broken the poles a r e displaced and may move into different sheets. The displacement of the poles also depends on the amplitudes for transitions in states belonging to representations that, through symmetry breaking, can couple to that to which the pole belongs. This happens even if dynamical representation mixing, a s given by the nondiagonal matrix elements of the K matrix, is neglected. It results from kinematical repre- sentation mixing. An application of this discussion to low-energy baryon-baryon scattering in the triplet s and d states i s car r ied out in Sec. IV. There we use dynamical parameters taken from a broken-SU, model of baryon-baryon scattering whose details will appear in a separate publica- tion.' A summary of results and conclusions i s presented in Sec. V.

I. THE K-MATRIX FORMALlSM

Let us consider a two-body transition in a defi- nite state of total angular momentum J and energy square s. We denote by o! a se t of conserved in- ternal quantum numbers, such a s isospin and hypercharge, specifying a state of a pair b of had- rons and by I the orbital angular momentum. We shall re fer to this specification of the states a s the particle basis. The matrix element for a transi- tion (b , 2 ) - (b', 1') of states with quantum numbers (a, J) will be denoted by T(s; J, . The K matrix will be defined by

where k, is the center-of-mass momentum of the pair b, given by

1 2 - - k b - 4s [S - (mi + mj)2] [ s - (mi - m,)" , (1.2)

and mi,m, a r e the masses of the particles in the

7 - S Y M M E T R Y - B R E A K I N G E F F E C T S ON S I N G U L A R I T I E S O F . . . 3463

pair b. The elements of the K matrix have the same sin-

gularities in s a s those of the elements of the T matrix except for the two-particle unitarity branch points. In addition the K matrix will have poles at the positions of zeros of detT. We shall assume invariance under time reversal which implies that the K matrix i s symmetric. We also assume an underlying internal symmetry which i s broken by an interaction whose strength i s given by a param- eter A . In the limit of exact symmetry, i.e., for X = 0, baryons and mesons belong to irreducible representations of the symmetry group (e.g., oc- tets in SU,). We shall also use the basis of i r - reducible representations B, of the symmetry group, which i s related to the particle basis in the following way:

I s ; ~ , ~ ~ ; B , l ) = ~ C , , ( c r ) I s ; ~ , o l ; b , l ) . (1.3) b

The C,,(cr)'s a r e Clebsch-Gordan coefficients for the decomposition of products of representations of the symmetry group. They form an orthogonal matrix. For A = 0 the T and K matrices a r e diago- nal in the B basis, i f each irreducible representa- tion of the symmetry group appears only once in the decomposition of the states b, o r block diago- nal i f they appear more than once (such as, for instance, in the decomposition of two octets, where the octet representation appears twice). At A = 0 all the k's a r e identical. As the symmetry i s broken, the masses and thresholds split and for a given s the k's will be different. The k matrix, which is diagonal in the particle basis, will take on, in the B basis, the form

the location of poles in other sheets, on the real axis above threshold. In fact unitarity in an arbi- t rary sheet gives

T- Tt =2iTtkOT , (1.6)

where O is a diagonal matrix given by

Qb, , = d,,,e(s - s,) , (1.7)

where s, i s the energy at the threshold b. If there is a pole of T on the real axis at s = so, s, <so < s,+ ,, then (1.6) gives

R ~ ~ , Q R = 0, (1.8)

where R is the matrix of the residues of the T ma- trix. If in this sheet all the k, 's have the same sign for b ' s b , then (1.8) can be written a s

(kO1/' OR) ( k O 1 l Z e ~ ) = o , (1.9)

which implies OR = 0. Hence, we have the following result:

Theorem 1. A pole of the T matrix between thresholds b and (b + 1), in a sheet such that the signs E, , a re the same for all b' G b, decouples from the open channels. (The thresholds are la- beled in ascending order from left to right.)

Such a situation cannot generally happen and should be regarded as anomalous. Barring this, i t follows from the above theorem, that there can- not be a real pole between the first and second threshold in any sheet.

Suppose now that there is a pole of the T matrix very near a threshold b. By this we mean that the distance of the pole to this threshold i s much smaller than the distance to any other threshold. Then near the pole we can expand A in power se-

(1.4) r ies of k,. The expansion will be of the form

kBtB= C C,fb(~)~Ba(a )kb . b

~ = a , + a , k ; + * . . -ik;'+'(a;+a:k,2+...) , We shall consider the Riemann sheets that a re reached from the physical sheet by crossing the (1.10) two-channel unitarity cuts in arbitrary order. They a re characterized by the set {E,} of signs of Irnk, in the complex s plane or, in a given k,-Rie- mann surface, by the set of signs { 7 7 b , } of Imk,, relative to the sign of Irnk,. If the number of two- body channels i s n, the number of sheets so defined i s 2" in the s-Riemann surface and 2" -' in a k, - Riemann surface. The physical sheet is defined by the condition Irnk, 2 0 for all b. The poles of the T matrix a re given by the roots of the equation

A =det(K,,,,:,, - i k ~ ~ + 1 6 b 8 b 6 1 t z )

= O . (1.5)

In the physical sheet of the variable s, the require- ment of causality precludes the existence of poles, except along the real axis below the lowest thresh- old. In addition, unitarity places restrictions on

where the coefficients a re in general complex, and I i s the lowest value of the orbital angular momen- tum. If 12 1, the roots of A = 0 near the threshold b will be approximately given by k, = *(-ao/a,)1/2, that is, there are , in general, two nearly symmet- r ic roots on some sheet of the k, plane. In the s plane these roots a re reached from one another by going around the branch point at s,. If one root i s on the sheet adjacent to the physical sheet below s,, the other is on the sheet adjacent to the physi- cal sheet above s,. If b i s the lowest threshold, the coefficients in (1.10) a re real; then for a d a , negative, the poles will be on the same sheet (the second sheet) corresponding to a resonance just above threshold, whereas for ao/al positive, the poles will be on the real axis of the s plane, below threshold, one on the physical sheet being a bound

3464 J . C U R I A L E AND S . W . M a c D O W E L L

state, the other on the second sheet being an anti- bound state.

In the case of s waves, I = 0, this situation does not generally occur; one might have just one pole a t ik, =i ao/aA .

11. DISPLACEMENT OF POLES WITH THE SYMMETRY -BREAKING PARAMETER

Let us study the poles of the T matrix a s func- tions of the parameter A. Let us assume that for

= 0 there i s a pole just below threshold in the i r - reducible amplitudes B. For 13 1 the pole occurs in both the physical and second sheets. For I = 0 there may just be ei ther a bound state o r an anti- bound state. As the symmetry-breaking interac- tion i s turned on, the thresholds split and a mani- fold of new sheets i s coupled to the physical sheet. It i s clear that, for 12 1, a pair of poles similarly located appear in all the sheets of the k plane. For I =0, if lala,/ <c in (1.10), there will be only one pole in every sheet of the k plane whose posi- tion i s given by

If for some (71~) we have C b~Bob(cr)2qb= 0, then on that sheet there will be a pair of poles symmetri- cally disposed with respect to the origin.

As A varies, the poles may approach the lowest threshold. The case I = 0 will be discussed in de- tail in Sec. III. In the case 12 1 the poles will meet at threshold and then move along the branch line in the second sheet, becoming a resonance. As they move past a threshold there will always be a pole on the sheet adjacent to the physical sheet a s described before and in Ref. 4. To f i r s t order in A the position of the pole i s given by

det[~.,, 8 ; B o , -i6,3, CCBob(n)2kb21+1] b = O .

To simplify the analysis let us assume now that the orbital angular momentum i s also a constant of motion. Then (2.2) becomes

Let so be the solution of this equation. Then the displacement of the pole for sufficiently small val- ues of A will be, to f i r s t order in A, given by

where all the derivatives a r e taken at the symme- t ry point s =so, A = 0. Now

- akb2 a m , + akb2 am, ah a m , ah Am2 ax '

where m , and m, a r e the masses of the two par- ticles in the pair b . Then in (2.4) we have

a a a k 2 a m ) X--+- - .

a m 1 am, ah

Let us consider a basis { / b ) ) for a (reducible) rep- resentation of the internal-symmetry group which i s the direct product of the irreducible vector spaces of particles 1 and 2, ( b ) = 1 I)@( 2). In this basis a m ,/ax i s a diagonal operator, transforming under the symmetry group a s the symmetry-break- ing Hamiltonian. Therefore we have

The matrix element on the left-hand side trans- forms like matrix elements of the symmetry- breaking Hamiltonian. Taking this into account i t follows that the last t e rm on the left-hand side of (2.4) also transforms in the same way, and there- fore so does

This result can be generalized to a rb i t ra ry ener- gies (within the two-particle approximation). In fact, if AXJ i s the symmetry-breaking Hamiltonian density, including all the counterterms for mass and coupling-constant renormalization to f i r s t or - der in A, then we have

where the derivative i s taken with fixed external momenta. Then the total derivative with respect to A of the two-particle amplitude Tbtb , at fixed center-of-mass energy and scattering angle, i s given by

7 - S Y M M E T R Y - B R E A K I N G E F F E C T S O N S I N G U L A R I T I E S O F . . . 3465

%=a+ +mi a + , am' a ) ,,, ( a x am, ax am; d l a x ,=I , ,

where mi and m; refer to the masses of the par- ticles in the initial and final s tates b and b', re- spectively. Now, a t h = 0, a am, and 8 T/am: a r e invariant under the symmetry group and, a s has been observed, the diagonal matrices (am, /ax),, and (am; /ax),,,, t ransform a s matrix elements of 3C1 in the product space of one-particle states. Therefore we can write

=x,,, + aT am. am' a T --+A- dX 1=1.2 c (am, ax ah am; )

where al l the derivatives a r e taken a t A = 0, and

with an analogous expression for (am;/ah),lB. Now

where

Since (am,/ax),,, a s given by (2.11) transforms like the symmetry-breaking Hamiltonian, it fol- lows that (8kZ/8h),,, and dT,,,/dX also do so. Therefore, i t follows from (2.2) and (1.1) that a t fixed energy, dK,,,/dh transforms in the same way. Thus, we have the following result:

Theorem 2. To f i r s t order in A, one can write

K =K,+XK, , (2.14)

where K, i s invariant under the symmetry group and K, transforms like the symmetry-breaking Hamiltonian.

Since the K matrix has no two-particle branch points, one might expect the linear approximation to be good a t low energies. An application of this result to a phenomenological discussion of the de- cuplet of meson-baryon resonances will appear in a separate publication.

nI. BEHAVIOR OF S-WAVE POLES

To fix ideas we consider a two-channel s wave. Let b = 1 be the channel with lowest threshold when

the symmetry i s broken. The k, plane has two sheets I and I1 defined according to whether Imk, and Imk, have the same o r opposite signs. In the symmetry limit we have k, =&k, and the two sheets disconnect from each other. In the s plane there will be four sheets defined by the signs of Imk, and Imk, a s follows: S, = (+, +), S, = (-, -), S, = (+, -), S4 = (-, +). S, and S, in the s plane corre- spond to the upper and lower half of sheet I in the k, plane, S, and S4 to the upper and lower half of sheet 11. Let B = 1 be the representation which has a pole in the symmetry limit. In the bas is of irreducible representations of the symmetry group the position of the pole in sheet I i s given by

In sheet I1 there will be a pole given by

where the angle /3 i s defined by the Clebsch- Gordan matrix C,, which i s of the form

Without loss of generality we can take - in <P < i n . We shall distinguish three cases:

(a) cosp > 1 sin@ I (cos2p >O). If the pole on the f i r s t sheet i s sufficiently close to threshold, the pole on the second sheet i s approximately given by

K,, - ik cos28 = 0 (3.4)

and i s located on the same side of the rea l axis a s the pole on the f i r s t sheet. As h varies, the poles will move and may change sheets, but they will not come together. Thus a bound state may become antibound and vice versa, but i t will not turn into a resonance [see Fig. l(a)].

(b) cosp = 1 sin0 1 = ($)"2 ( ~ 0 ~ 2 0 = 0). Equation (3.2) reduces to

Let a,, and a,, be the scattering lengths for chan- nels B = 1 and B = 2, respectively, and r,, and r,,, the corresponding effective ranges. We assume that I%, I << (a,, 1. Then the solutions of (3.2) a r e approximately given by

Unless r,, i s very large, these poles a r e not a s close to threshold a s that on the f i r s t sheet. If these roots a r e real , they will become complex a s we change A . For small A, (3.5) will be modi- fied in the following way:

J . C U R I A L E A N D S . W . M a c D O W E L L - 7

FIG. 1. The two sheets of the ki-Riemann surface in two coupled two-particle channels. The hatches in- dicate the physical region and the region adjacent to it. The position of poles and shadow poles and their dis- placements a r e indicated: in (a) for the case cos2p > sinZ/3; in 03) for cos2p= sin", with the original pole being a bound state; in (c) for cos2p < sin2p, with the original pole being an anti-bound-state pole; for a bound- state pole, we would have a behavior similar to that shown in (b).

In sheet 11, k, - -kl a s h -0. The imaginary part of k, will be approximately given by

and will have the sign of -all. If all <O corre- sponding to a bound state pole a t h = 0, then Imk, >O and the pole cannot become a resonance. If a,, >O corresponding to an antibound state, then Imk, <O. These poles on the lower half of sheet I1 might eventually appear a s a resonance in the channel b = 1, below the threshold for b = 2, if the position of this threshold moves past the position of the pole. On the other hand, if the roots (3.6) a r e imaginary, then i t may happen that the pole in sheet I moves towards the rea l axis crossing into the second sheet and meeting with one of the poles there. Then they will move apart alongside the interval of the real axis between thresholds. If they a r e on the lower half plane, they will pro-

duce a resonance [Fig. l (b)] . But if, on the other hand, they a r e in the upper half plane, they will not produce a physical resonance unless they turn around the second threshold and back between the thresholds. In both ca ses the resonance will ap- pear in channel b = 1 below the threshold for b = 2. Here we have a situation a s conjectured by Yang and Oakes, in which for some interval of values of h the pole moves into a sheet not adjacent to the physical sheet, i t s effect on the physical am- plitude being expected to be small . As A increas- e s , however, it may emerge a s a resonance pole below the second threshold.

(c) cosp < I s inp ( (cos2P <O). The pole on the second sheet i s given by Eq. (3.4) but i s now lo- cated on the opposite side of the rea l axis with re- spect to the pole on the f i r s t sheet. As h i s changed the pole on the f i r s t sheet may move to- wards the real axis and into the second sheet and we have a situation analogous to that discussed in case (b) when the roots (3.6) a r e imaginary [Figs. l (b) and l(c)] .

Let us investigate further under what conditions poles on sheet S, and S, can c ros s the rea l axis beyond the second threshold. The position of the pole on the rea l axis i s determined by the simul- taneous equations

Since k: 2 kZ2, Eq. (3.10) has solutions only if the following condition holds:

(K,, +Kll)[(K,, -Kll)cos2P+Kl,sin2P] <O . (3.11)

If the pole i s associated with a state with little dynamical configuration mixing, then a t the pole we should have I K,, I >> I K,, 1 and I K,, I >> (K,, 1 . Under these circumstances (3.11) would imply cos2p < E , where E i s a small number. Therefore the crossing of the rea l axis i s more likely to occur if cos2p <sinZP. In general a pole i s likely to c ros s an interval of the rea l axis between two thresholds b and b + 1, if the equation

has solutions in that interval. The signs of the k,,'s determine the sheets on which crossing may occur.

It should be pointed out that these conclusions a r e valid only if the s wave i s not coupled to an- other partial wave such a s a d wave. In the case

7 - S Y M M E T R Y - B R E A K I N G E F F E C T S O N S I N G U L A R I T I E S O F . . . 3467

of coupled waves the argument presented here no longer applies. [See, for instance, the discussion of coupled partial waves in Sec. IV and, in partic- ular , the condition (4.1).]

IV. APPLICATION TO BARYON-BARYON SCATTERING

We have studied a simple meson exchange model of baryon-baryon scattering a s applied to triplet s and d waves in SU, antisymmetric states. We used an SU,-symmetric coupling of pseudoscalar and vector mesons to baryons and an SU,-sym- metric hard core. The symmetry i s broken only through the splitting of masses of the mesons and baryons, which a r e taken to obey a Gell-Mann- Okubo type mass formula linear in a symmetry- breaking parameter A . For h=O the masses a r e degenerate within a multiplet, corresponding to exact symmetry. For h = 1 one obtains the actual physical masses . We have calculated the scat ter- ing amplitude in the triplet s and d coupled chan- nels in t e rms of the parameter A. These states, being symmetric in coordinate and spin space, belong to antisymmetric representations of SU,. (The full description of the model and results of the calculation will be given in a separate ~ a p e r . ~ )

It was found that for h = 0, the model gives a pole in the amplitude for scattering in states belonging to the decuplet representation g*. The pole i s on the second sheet very near threshold, a t ik =0.11 fm-l. We have investigated how this pole i s dis- placed in the multichannel amplitudes for different components of the decuplet, a s h var ies from zero to one.

In Table I, we l is t the components in the parti- cle basis of each member of the lo* multiplet and give the Clebsch-Gordan coefficients for the de- composition of each pair in t e rms of three anti- symmetric irreducible representations in the product 8@ 8.

The pr<ton-neutron channel with I = 0 , Y = 2, a pure decuplet, and the coupled AN, CNchannels with I =$, Y = 1, were studied in detail. In the p-n amplitude the pole moves into the f i r s t sheet to become a bound state. In fact, one of the free parameters in the model, namely the hard core radius for the decuplet, was adjusted so a s to ob- tain the correct binding energy of the deuteron at A = 1.

The I =+, Y = 1 channels a r e superpositions of the decuplet g* and the antisymmetric 8,. One can see from Table I that the ~ l e b s c h - ~ o r d a n matrix for these channels corresponds to case (b) discussed above with cosP = sin@ = 1/a. The posi- tion of the poles on the second sheet of the k, plane is given by (3.6) with a,, =au, = 8.1 fm, ru,

= 2 fm, and a,, =a, =0.58 fm. These a r e the values for these parameters a t h = 0. It was found (see Fig. 1) that the anti-bound pole moves into the upper-half of sheet I1 and meets the pole in this sheet for A =0.02. As A increases they move away from each other, staying close to the rea l axis on the upper half of sheet 11. No resonance appears in the physical amplitudes. At h = 0.6, the pole on the positive side of the plane crosses the rea l axis around the NC-threshold branch point. For 1 >0.6 the pole emerges a s a reso- nance in the NA amplitude, below the NC thresh- old. It turned out in our model that the resonance (for A = 1) i s almost purely in the d wave. This suggests that the centrifugal bar r ie r might be responsible for keeping the pole close to the rea l axis. If one neglects representation mixing, at the position where the pole c rosses the rea l axis we have

( k , + k,)Kdd + (k15 + kZ5)K,, = 0 . Notice that Kdd and K,, should have opposite signs for this equation to have a solution.

We have not done a numerical calculation of the T matrix in the Y = 0, I = 1 channels. However, from the results of the calculations for the other members of the lo* multiplet, i t will be possible to make a analysis of these channels, based on the effective-range approximation.

Let us denote by 1, 2, 3 the (antisymmetric) channels EN, CA, CC, respectively, and by Ku*, K,, K,,, the diagonal blocks of the J ~ = 1' K ma- t r ix in the basis of irreducible representations of SU,. The nondiagonal blocks a r e zero in the limit A = 0. These blocks a r e 2x2 matrices correspond- ing to transitions in s and d waves.

At very small values of h and energies near

TABLE I. Clebsch-Gordan coefficients for the reduc- tion of two-baryon antisymmetric states into irreducible representations of SU?.

Irreducible representations Y I Particle pair l o + - 8, - 1 0

2 0 NA' 1 0 0

3468 J . C U R I A L E A N D S . W . M a c D O W E L L

threshold, one can argue that the off-diagonal waves, and neglect the d-wave elements which blocks of the K matrix may be neglected. Like- a r e of 5th order in k. wise, in the matrix ik2'+l in (1.1), we shall keep Then, with these approximations, the position only the elements linear in k, corresponding to s of the pole of the T matrix will be given by

where

A,-' = (de t~ , ) /K,~ ,

=Kg.s- K B , ~ ~ ' / K , . d ' (4.3)

Since A,-' and A,,-' should be large a s compared to A,*-' and the momenta k,, one can simplify the above equation s o a s to bring it to the form

[A,,* - -' - i(+k, + i k, ++~ , ) ]A , -~A, , -~ +k(k, - k3)2~ lo -1+ (- $kl + $ k, - t k 3 ) 2 ~ , - 1 = O . (4.4)

In discussing the roots of this equation near k: = 0 and for small values of A , i t i s sufficient to use an ef- fective-range formula for A,*'' and to take A,-' and A,,-' a s constants. So we write

1 1 A,,*-'=- - + $Y,* klo*2, A," =-

ag* - a, ' where the variable kStz i s the average center-of-mass momentum square in the g* representation,

2 1 2 k ,+2=~k12+$k2 + 6 k 3 , (4.6)

and the a ' s a r e the scattering lengths of the irreducible amplitudes a t A = O . To f i r s t order in A the param- e t e r s of the K matrix, according to Theorem 2 of Sec. 11, a r e proportional to Y. Since Y = 0 in the chan- nels we a r e considering here, these parameters a r e constant to f i r s t order in A. According to the results (2.7) and (2.8) of Sec. 11, ak,o*2/aA, - for fixed energy and X =0, transforms in the same way a s the symme- try-breaking Hamiltonian. In the limit A - 0 the solutions of Eq. (4.2) in the different sheets of the k, plane a r e approximately given by

2 I(++) ik, = - [ I + (1 + 2~,*/a,*)'/~]-' ,

alp*

6 II(- - ) ik, = - - (1 + [I + 2(9rU* + 8a, + 8alO)/a,*] 1'2)-1 ,

a,*

3 III(+ -) ik, = - (1 + [ I + $(9rU* + 8aa + 2a,,)/a,*] '/')-' ,

a u * (4.9)

In Appendix I we discuss the change in position of the poles in the different sheets a s X varies. The pole on the f i r s t sheet moves up into the sec- ond sheet, where it meets a shadow pole. They then move away from each other and eventually turn around the branch point corresponding to the ZA threshold, into the fourth sheet far removed f rom the physical region [Fig. 2(a)]. At the same time the shadow pole on the third sheet moves up- wards, whereas the shadow pole on the upper half of the second sheet moves down. They come to-

gether for a small value of h. (A ~ 0 . 0 7 ) depending on the value of the scattering length a,, a t A = 0. If a,, <4.65 fm, the two poles meet on the upper half of the second sheet and move in a similar way a s that of the f irst pair of poles. But a s they turn around the CA branch point they move into the lower half of the second sheet adjacent to the phys- ical region in the interval below the CA threshold [Fig. 2(b)]. Therefore this pole becomes a reso- nance pole just below the CA threshold. [Notice that (3.13) has, in this case, solutions in the inter-

S Y M M E T R Y - B R E A K I N G E F F E C T S O N S I N G U L A R I T I E S O F . . .

FIG. 2. Position of poles of the scattering matrix for Y = 0, I=1 members of the baryon-baryons* decuplet, in dif- ferent sheets of the kz,Riemann surface. Conjectured displacement of poles with SUB-breaking parameter h i s shown: in (a) for the poles in sheets I and 11; in @) for the poles in sheets I11 and fir, assuming that they meet in sheet IV and go on to sheet I1 to become a resonance between the E N and ZA thresholds; in (c) for the poles in sheets I11 and IV, assuming that they meet in sheet I11 and become a resonance between the 8 A and Z Z thresholds.

val between the AC and CC thresholds.] If, on the other hand, a,, >4.65 fm, the two poles

meet in the lower half of the third sheet, and, a s they move along the rea l axis, will become a reso- nance in the interval between the CA and CC thresholds [Fig. 2(c)]. Based on the position of the poles in the Y = 2 and Y = 1 states, namely the deuteron bound state and the AN resonance below the CN threshold, the Gell-Mann-Okubo mass for- mula would predict a resonance pole in the Y = 0 states E N and CA, just below the CC threshold. Since nothing is known about the scattering ampli- tude in states with Y=O, I = 1, our analysis shows that no definite prediction can be made. It indi- cates that there can be a resonance pole below the CA threshold in violation of the Gell-Mann-Okubo mass formula, o r below the Z;C threshold, in ac- cordance with that formula. In the lat ter case a rather large scattering length in the 10 amplitude, a,, >4.65 fm, i s required. A third po~sib i l i ty i s that these poles will not end up in a resonance po- sition. Which of these alternatives i s actually realized depends on the dynamical features of the interactions not only in the g * decuplet but in the octet and 10 decuplet a s well. It should be pointed out that this dependence on the dynamics i s in no way taken into account in the derivation of the Gell-Mann-Okubo mass for mula.

In the fourth member of the decuplet the Y = - 1, I = $, B C states the anti-bound pole moves away f rom threshold. This result i s not just specific to the model. In fact, defining k,,* a s the average -

momentum for each member of the g * multiplet, then for small X's the position of the pole i s dis- placed according to the formula

Therefore the f i r s t and fourth members of the multiplet move in opposite directions. The effect of the anti-bound pole i s to produce a rather large positive scattering length in the Y = - 1 amplitude in contrast with the large negative scattering length in the triplet s-wave proton-neutron ampli- tude.

V. SUMMARY OF RESULTS

We have derived a theorem for the K matrix which states that if there i s an internal symmetry broken by an interaction proportional to a param- e ter A, (aK/aX),,, t ransforms in the same way a s the symmetry-breaking Hamiltonian. This theo- r em allows us t o justify the Gell-Mann-Okubo mass formula for compound two-particle states with orbital angular momentum 12 1. The case of s-wave amplitudes was investigated in detail. It was found that (i) the position of shadow poles in the different sheets of the energy Riemann surface depends on the Clebsch-Gordan coefficients for the reduction of the product of the representations of the scattering particles; (ii) the displacement of these poles a s h varies depend not only on the dynamics of the multiplet to which they initially

3470 J . C U R I A L E A N D S . W . M a c D O W E L L

belong, but also on the dynamics of the states to of the energy Riemann surface quite fa r from the which that multiplet couples by virtue of the sym- physical region, a s conjectured by Oakes and metry breaking. This occurs even if dynamical Yang. representation mixing i s negligible, just a s a re - sult of kinematical representation mixing due to ACKNOWLEDGMENTS the splitting of two-particle thresholds; (iii) in some members of the multiplet which has a pole The authors wish to thank Professor R. Dashen in the symmetry limit near threshold, the original for a useful discussion and Professor C. Sommer- pole and the shadow poles may move into positions field for a critical reading of the manuscript.

APPENDIX: DISPLACEMENT OF POLES FOR X SMALL

In this appendix we shall investigate the displacement of poles for small values of A in the Y = 0, I = 1 amplitudes of triplet s tates of the baryon-baryon system. The change in position of the pole with A i s given by (d/dh)(ik,)d~. This i s obtained by differentiating (4.1). Making use of the SU, mass relation 2(mZ+ m,) = 3 m , + m, , one obtains at A = 0:

Therefore, in the different sheets, the initial motion of the pole i s given by

In the f i r s t and second sheets the poles move towards the rea l axis. If a,, i s positive, the same happens in the other two sheets. We a r e interested in the displacement of the poles in the last two sheets. In par- ticular we want to determine under what conditions two poles meet in either of these two sheets. To f i r s t order in A and making use of the baryon octet mass sum rule, the relations between the momenta in the different channels a re :

K,' = K'' - +A , (A61

K~~ = K'' + A , (A7)

with K~ = ikj . At the position where the poles meet the energy i s a simultaneous solution of Eq. (4.1) and i t s derivative:

~t follows from these equations that unless a,, i s Q a 8 ~ , + ($K' - +~~)a,,, J$ . very large and negative, which would imply the (A10)

existence of a bound state in the 10 decuplet, the The upper and lower signs of inequality correspond condition for the poles to meet in either of two to positions of the double pole in the upper and adjacent sheets i s given by lower k1 plane (K, 60). Let us determine the value

7 - S Y M M E T R Y - B R E A K I N G E F F E C T S ON S I N G U L A R I T I E S O F . . . 3471

of a,, for which the poles meet at K, = 0, s o that

KZ2 = + K ~ ~

=;A . (Al l )

It will be given by solving the simultaneous equa- tions (4.1) with K, = 0, K, = ( I / ~ ) A , K,

and

The upper and lower signs in these equations cor- respond to occurence of the double pole at K, = 0 on the first two sheets or the last two sheets, respec- tively. Taking the values for a,*, r,* , and a, a s given by the model discussed in Sec. IV, one finds for the two cases the solutions,

(i) For double pole at threshold in sheets I-II:

a,,= 18.15 fm, X =0.0126 . @Is)

(ii) For double pole at threshold in sheets 111-IV:

a,, = 4.65 fm, A = 0.02 . (A 14)

Then it follows from (A10) that if a,, <4.65 fm, the

*Research supported by the U. S. Atomic Energy Commission under contract AT(11-1) 3075.

'R . J. Oakes and C. N. Yang, Phys. Rev. Lett. 11, 174 (1963).

2 ~ . J. Eden and J. R. Taylor, Phys. Rev. Lett. 2, 516 (1963).

poles will meet on sheet IV, whereas if a,,>4,65 fm, they will meet on sheet III. It should be noted that (A13) and (A14) are values for a,, at unphysi- cal values of X. However, since we a re consider- ing channels with Y = 0, the parameters of the K matrix should have no first-order terms in X and therefore might not be expected to vary much a s we go from the values at X = 0 to the actual physi- cal values at h = 1.

Assuming a, = a,, = 0, the positions of the double poles and corresponding values of X are

I-II:

111-IV:

3 ~ . Nauenberg and J. C . Nearing, Phys. Rev. Lett. 12, 63 (1964).

4 ~ a r c H. Ross, Phys. Rev. Lett. 11, 450 (1963). 'c. R. Hagen, Phys. Rev. Lett. 12, 153 (1964). 6 ~ . Curiale, Yale University report (unpublished).