Download pdf - Broken SU(6) predictions for baryon production with polarized vector mesons

Volume 36B, number 4 P H Y S I C S L E T T E R S 20 September 1971

B R O K E N S U ( 6 ) P R E D I C T I O N S F O R

B A R Y O N P R O D U C T I O N W I T H P O L A R I Z E D V E C T O R M E S O N S

E. HIRSCH, U. KARSHON and H. J. LIPKIN~

Weizmann Institute of Science, Department of Physics, Rehovot, Israel

Received 29 June 1971

Quark model and SU(6) predictions for branching ratios of A, ~ and Y* (1385) production in strangeness exchange reactions disagree with experiment. New predictions for vector meson polarization density matrix elements are presented which seem to agree with experiment. A "broken SU(6)" description which includes configuration mixing in the ~ and Y* wave functions is consistent with the success of the polarization predictions and the failure of the branching ratio predictions of unbroken SU(6).

The quark model makes definite predic t ions for the b ranch ing ra t ios of the f inal s ta tes in reac t ions of vec tor meson product ion by s t r ange - ness exchange [1-3]

K ' + p - " (pO, a~,~) + ( A , ~ o , y *°) (la)

7r" + p -~ K *° + (A, ~o, y*O) (lb)

where y*O denotes y*o (1385). For any given baryon B, the pO and w c ross sec t ions a r e predic ted to be equal [1]

5 ( K - p - ' Bp )= 5(K-p--* Bw) . (2a)

where ~ is the c ross sect ion co r rec ted for k inemat ic fac tors such as phase space [3]. For any given meson , the ba ryon c ross sec t ions a r e pred ic ted to be in the ra t ios [2]:

~(A)/ff(~°)/~Y(Y *°) = 27 /1 /8 if the re is baryon spin flip (2b)

= 3 /1 /0 with no baryon spin flip (2c)

where flip (AS=l) and non-f l ip (AS=0) t r a n s i - t ions a r e defined by analogy with GamowoTel le r and F e r m i beta decay t r ans i t i ons , and not by analogy with hel ic i ty flip (AS z -- 1) and non-f l ip (AS z = 0) ampli tudes . Eqs. (2b) and (2c) can be combined to give the sum ru le

~(A) = 315(Z °) + ~(y*O)] in all cases (2d)

and the inequalities

This work was performed in part under the sponsor- ship of the US National Bureau of Standards.

27~(~ °) >i ~(A) > / ~ ( y * o ) in a l l cases . (2e)

Exper imen ta l r e su l t s for reac t ions (la) show good ag reemen t [4, 5] for re la t ions (2a), some d i sag reemen t for re la t ion (2c) and ser ious d i s - ag reemen t [5, 6] for the spin re la t ion (2b). The quark model der iva t ion is also express ib le in t e r m s of SU(3) and SU(6) s y m m e t r i e s , without specif ic r e f e r e n c e to quarks. It i s the re fore of pa r t i cu l a r i n t e r e s t to ~Uunine these predic t ions in deta i l to u=ders tand why some of them work while o thers do not, and to de t e r mi ne which a s - sumpt ions a r e r e spons ib le for the d isagreements .

Po la r i za t ion m e a s u r e m e n t s provide addit ional in format ion for compar i son with theore t ica l predic t ions . Relat ions (2) hold separa te ly for each polar iza t ion. Thus m e a s u r e m e n t of vector meson po la r iza t ion densi ty ma t r i ce s allow the c ros s sec t ions to be separa ted into th ree components , each of which should sat isfy re la t ions (2). Baryon polar iza t ion informat ion is not easi ly avai lable , s ince it r equ i r e s m e a s u r e m e n t s on both in i t ia l and final s tates . However, pa r t i a l in format ion on baryon polar iza t ion is obtained from the meson m e a s u r e m e n t s by use of con- s t r a in t s imposed by angular momentum and par i ty conserva t ion [2].

The vector meson polar iza t ion is most con- venient ly specif ied in the l i nea r polar iza t ion bas i s [7] in the meson Jackson f rame. The axes a r e chosen in the conventional m a n n e r in the vec tor meson r e s t f rame. The z -ax is is in the d i rec t ion of the incident momentum, the y-ax is n o r m a l to the reac t ion plane and the x-axis in

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the plane. Let rex, my and m z denote the p r o - j ec t ions of the v e c t o r meson spin on the x, y and z axes r e spec t i ve ly . We choose a b a s i s of s t a t e s having rex=0 , my =0 and mz=O r e spec t i ve ly . These c o r r e s p o n d to the c a r t e s i a n components (x, y, z) of a vec to r , r a t h e r than to the s p h e r i c a l h a r m o n i c s IT/(0, ~b). The p o l a r i z a t i o n dens i ty m a t r i c e s in th is b a s i s a r e s imply r e l a t e d [7] to those in the convent ional J a c k s o n (J) and J a c k s o n t r a n s v e r s i t y (tr) b a s e s . Using the notat ion of Haber et al. [8], we obtain

pzz=pg '- ~ , t r t r t r t5 0 = ~ l P l , 1 + P - l , - 1 + P l , - 1 + P - , , I ~ (3a)

t r = PJ,1 + Pl/, (3b) PYY =P0, 0 -1

J J ½{p~r t r t r t r Pxx=Pl,1- P l , - 1 = . , 1 + P-l , -1 - P l , - 1 - P - l , 1 }

(3c) Pxz = - ~ P ~, 0 " (3d)

The cross sectior~ for linearly polarized final states are specified by multiplying the cross sections by the polarization density matrices.

~(i)=Pii~, i =x,y,z . (3e)

In this basis ~(x) and ~(z) include only baryon spin flip contribution~. This is most easily seen in the transversity basis, as angular momentum and parity conservation require that transversity be conserved modulo 2. A transversity flip in the meson transition requires a transversity flip in the baryon transition. The vector meson states with m x --0 and m z = 0 are linear combi- nations of the states with my =~1, since they ar~ both orthogonal to the rhy -- 0 state. Thus ~(x) and ~(z) include only contributions from states with a meson transversity flip and there- fore with a baryon transversity flip. They are predicted to satisfy the relations (2b)

0 (x) (A)/~(x) (~ o)/~-(x) (y*O) = 2 7 / l / 8 (4a)

5(z)(A)/5(z)(~°)/5(Z)(Y*°) = 2 7 / 1 / 8 . ( 4 b )

The c r o s s s ec t ions ~(Y)(A) and~(Y)(~ °) involve no meson t r a n s v e r s i t y f l ip and t h e r e f o r e no ba ryon t r a n s v e r s i t y flip. However , t h e r e a r e two independent Amy = 0 ba ryon t r an s i t i ons ; namely +1/2 ~ +1/2 and -1 /2 ~ -1 /2 . The non- f l ip AS = 0 ampl i tude p r e d i c t e d to sa t i s fy eq. (2c) is invar ian t ( sca la r ) under sp in ro ta t ions and contains the above two t r a n s i t i o n s with equal ampl i tude and positive phase . It is a Am = 0 ampl i tude for all directions of the axis of quantizat ion. T h e r e is a l so a Amy =0, AS =1 ampl i tude which t r a n s f o r m s l ike a vec to r under

sp in ro ta t ions and contains the above two t r a n s i - t ions with equal magni tude but opposite phase . This v e c t o r ampl i tude contains Am = ± 1 components with r e s p e c t to the # and z axes and is p r e d i c t e d to s a t i s fy the sp in f l ip p r e d i c t i o n s (2b). These v e c t o r and s c a l a r ampl i tudes cannot be s e p a r a t e d without p o l a r i z a t i o n m e a s u r e m e n t s on the ba ryons . Thus ~(Y) inc ludes both components which sa t i s fy eqs. (2b) and those which sa t i s fy (2c). Al l components sa t i s fy the sum ru le (2d) and the inequa l i t i e s (2e).

The flil3 and non-f l ip components of 5(Y)(A) and of ~(Y)(T °) can be ca lcu la ted f rom the ex- p e r i m e n t a l va lue of ~(Y) (y*o) if 7(Y) (y*o) i s a s s u m e d to be pu re flip as ind ica ted by the p r e - dic t ion (2c). The r e l a t i ons (2b) o r (4a) and (4b) then d e t e r m i n e the f l ip cont r ibu t ions to ~(Y) (A) and 5(Y) (T°). The non-f l ip cont r ibu t ions a r e obta ined by sub t rac t ing the flip cont r ibut ion and can be subs t i tu ted into the non-f l ip p r e d i c t i o n (2c). F r o m eq. (2b) we obtain

~(Y)(A) - (27/8) ~(Y)(y*o)= 3[~(Y)(~o) _ ~(Y)(y*O)/8]"

From eqs. (4a) and (4b) we obtain (5a)

~(Y)(^)_ ~(i)(A) ~(Y)(y*o)= ~(i)(y*O)

=3I~(Y)(po)_ _~(i)(~°) Cy)(y,o)] (5b) ~(i)(y*O)

where i = x o r z. This can a l so be wr i t t en

Ipyy (A) OYY(Y*°)Oii(A)] - * o ~ ~ ( A ) =

P ii(Y ) J

=3[pyy(~O) OYY (Y*°)Oii(~°) *o ]~(~o) (5c) P ii (Y )

where Pii can be Pxx, Pzz, or any l inea r c o m b i - nation.

Re la t ion (5a) is jus t the sum ru le (2d). If a l l p r ed i c t i ons a r e s a t i s f i e d eqs. (Sb) and (5c) a r e iden t ica l to (Sa) and a l so add nothing new. How- ever , when some p red i c t i ons a r e b roken , (Sb) and (5c) a r e no longer equivalent to (Sa). Ea. (5a) a s s u m e s that the flip cont r ibut ions to ~'(Y) sa t i s fy the t h e o r e t i c a l p r e d i c t i o n s (2b). Eqs. (5b) and (5c) a s s u m e that they have the s a m e . r a t i o s a§ the o ther f l ip cont r ibut ions ~(x) and ~(z). If ~(x) and ~(z) sa t i s fy the p r ed i c t i ons (4a) and [4b), a l l r e l a t i ons (5) a r e equivalent . If ~(k) and ~(z) do not sa t i s fy t hese r e l a t i ons , as s e e m s to be the ca se e xpe r ime n t a l l y , they a r e no longer equivalent .

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considerably l a rge r than the e r r o r s . F u r t he r invest igat ion is n e c e s s a r y to c lear up the p case, s ince there is a poss ib i l i ty of sys temat ic e r r o r s in the data [6] and cor responding p and co ent r ies a re predic ted to be equal by the re la t ions (2a), which so far has shown good ag reemen t with experiment .

Let us now cons ider how the predic t ions (6) can hold while the predic t ions (4) a re broken.. The three polar iza t ion c ross sec t ions or(x), ~(Y) and ~(z) se lect different t - channe l exchanges. ~(z) is pure unna tu ra l par i ty exchange with no meson hel ici ty flip in the Jackson f rame, a(x') and ~(Y) a r e both Jackson hel ici ty flip ampl i tudes , but ~(Y) has no t r a n s v e r s i t y flip and ~(x) has t r a n s v e r s i t y flip. At Regge asymptot ic energies or(x) is expected to be dominated by unna tu ra l par i ty exchange and or(Y) by na tura l par i ty exchange [10]. At the energies of these exper i - ments which a r e hardly asymptot ic or(z) is p r e sumab ly dominated by pseudosca la r exchange, and ~(x) is some other exchange, not of spin zero, such as vec tor or axial vector. In ei ther case, eqs. (6) re la te two different types of exchanges in A, S and Y* product ion, provided that both a(x) and a(z) a re non-vanishing.

It is difficult to re la te a l l these c ross sect ions without some sor t of quark model or SU(6) a s - sumption. At the SU(3) level , the Y* is not r e - lated at a l l to the A and ~ and the re la t ion between A and ~ r equ i r e s the D / F ra t ios for different exchanges to be the same. We a r e the re - fore led na tura l ly to a "broken SU(6)" desc r ip - tion.

The different spin flip exchanges a re re la ted in the quark model and in SU(6) because the i r couplings to baryons a re propor t ional to ma t r ix e lements of components of the same spin vector and there is no polar iza t ion m e a s u r e m e n t on the baryon to dis t inguish between the different components. For example, the dominant cou- pling (neglecting recoi l) of quarks to pseudosca la r , vec tor and axial vector mesons a re descr ibed by the opera tors [11]

(o .k) - pseudosca la r exchange (7a)

Combining (4a) and (4b) and the inequal i t ies (2e) gives the re la t ions

°'(X)(A) cr(X)(~O)- °'(X)(y*O) (6a) ~(Z)(A ) -~(z)(~o)- ~(Z)(y*O)

~(y)(~o) >/ cr (Y)(A) >I Jx)(~o)+ ~(z)(~o) ~(X)(A)+~(z)(A )

a (Y)(y*O) (6b) (X)(y*O) + JZ)(y*O)

These re la t ions can also be expressed ent i re ly in t e r m s of po la r iza t ion densi ty m a t r i c e s by u se of the defini t ions (3)

P xx(A)/~ z z (A) : pxx(r ,°) /~zz(~ °) - -

= pxx(Y*°) /pzz(Y *°) (6c)

pyy(~O) >I pyy(A) >~pyy(y*O) (6d)

These re la t ions (6) a r e a lso l inear combinat ions of other re la t ions and add nothing new if a l l p r e - dict ions (4) hold. However, when some p red ic - t ions a re broken they provide s ignif icant tes ts for the b reak ing of the model. All the re la t ions (4), (5) and (6) hold also when re l a t iv i s t i c Wigner rota t ion effects a re taken into account, s ince the t r a n s v e r s i t y a rgument is r e l a t iv i s t i c [9].

Exper imenta l data show that predic t ions (4a) and (4b) a r e badly broken by very la rge fac tors , but by the same factors in both cases [6]. This pecu l i a r r egu la r i ty s t imula ted this invest igat ion and suggests a compar i son with the predic t ions (6d) which a r e seen to be val id (table 1). The exper imenta l tes t of the predic t ion (6c) extended by l sospm a rgument s to :, and Y - product ion [6] is a lso shown in table 1. The p product ion r e su l t s a re inconclus ive , because the e r r o r s a r e la rge in compar i son with the effect. How- ever , s ignif icant ag reemen t with eq.(6c) is shown in the ¢o case. Both a(x) and 9(z) a r e of the same o rde r of magni tude, and the ra t ios ~(x)/~(z) a re

Table 1 Value of pyy andPxx/Pz z for various final states

Final state ~-po ApO y*-po ~-w Aw Y*-w

pyy 0.79 ± 0.06 0.68 + 0.07 0.21 + 0.06 0.65 ± 0.05 0.52 + 0.06 0.14 + 0.07

0.29 + 0.39 0.18 ~: 0.25 0.43 + 0.14 1.00 + 0.36 0.87 + 0.31 0.59 ± 0.18 Z Z

Data from the work of ref. [6].

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Volume 36B, number 4 PHYSICS L E T T E R S 20 September 1971

((~ × k. ~) - vector exchange (7b)

( u . t ) and (¢~. k) (e .k ) - axial vector exchange (7c)

where a is the quark spin, k the momentum of the exchanged boson and ~ the vec tor or axial vec tor meson polar izat ion. The only baryon dynamical va r i ab le which appears in these ope- r a to r s is the vector o.

The ma t r ix e lements for the contr ibut ions of the pa r t i cu la r exchanges (7) to the product ion of a given baryon B are al l propor t ional to the same reduced mat r ix e lement (B I I~[ ]p), and a r e otherwise independent of B. The predic t ions (4) depend on the values of these reduced ma t r ix e lements with SU(6) baryon wave functions. How- ever , the predic t ions (6) a re seen to be independent of the value of (B I ]ol [P), which cancels out. Thus, the predic t ions (6) a r e insens i t ive to the baryon wave functions, and a r e comple te - ly independent of them in the approximat ion where reco i l is neglected.

The s imples t way to break SU(6) and the r e - la t ions (4a) and (4b) while p r e s e r v i n g the r e l a - t ions (6) between different exchanges is to a l - low for SU(6) and SU(3) breaking in the baryon wave functions; i.e. configurat ion mixing. The most obvious mixing [12] would be with the L=2 baryon 56, which mixes in some S=3/2 decuplet into the ~ and some S =3/2 decuplet and S = 1/2 octet into the Y*. The nucleon and A cannot mix with this configuration which con- ta ins no (1=1/2, Y = l ) o r (I=0, Y =0) states. These admixtures could produce an enhance- ment of the ~ and Y* c ross sec t ions in spin flip t r ans i t i ons pa r t i cu la r ly for the ~ where the dominant contr ibut ion is suppressed. They should not affect the A and should have a much s m a l l e r effect on the ~ /A rat io in non-f l ip t rans i t ions .

Detailed calculations" of the effect of mixing a re not poss ib le without a specific model to give the t r ans i t ion ma t r ix e lement to the L=2 configuration. However, extensive exper imenta l t es t s of these ideas a re poss ib le by looking at the genera l iza t ion of the predic t ions (4) and (6) for al l cases of vector meson product ion with octet and decuplet baryons. Eqs. (4) apply to a l l s t r angeness exchange reac t ions (1), and can be extended by i sospin to a l l charge s tates .

S imi la r re la t ions hold for vec tor meson product ion with charge exchange product ing nucleon and A final baryons [13]. The baryon flip contr ibut ions sat isfy the re la t ions

~(i)(N)/~(i)(A):25/24, i=x,z . (8a)

388

There is no non-f l ip contr ibut ion to ~(A). Thus we have the inequali ty

~Y(N)/~Y(A) >! 25/24 (8b)

The analogs of eqs. (6c) and (6d) a re

pxx(N)/pzz iN) = pxx(A)/pzz (~ (8c)

pyy (N) >~ pyy (A) . (8d)

If sys temat ic d i sagreements a re found for p r e - dict ions (4) and (8a), while non - t r i v i a l ag ree - ments a re found for re la t ions (8c) and (8d) this would be in te res t ing evidence favoring the con- f igurat ion mixing hypothesis.

Fu r the r genera l iza t ion of the re la t ions (6) and (8c) a re poss ib le if the dominant flip exchanges a re descr ibed by the express ions (7) and recoi l effects a r e negligible. The ra t io Pxx/Pzz should be equal for al l s t r ange baryons produced in the reac t ions (1), not only A, ~ and Y*(1385). The same would hold for a l l non - s t r a nge baryons produced with vector mesons in charge exchange reac t ions , not only N and A (1238). These p r e - dict ions can be violated by exchanges descr ibed by opera tors other than (7), which involve spa- t ia l degrees of freedom as well as spin.

Discuss ions with Y. Avni, U. Maor~ M. Milgrom and A. Shapira a r e grateful ly ac- knowledged. We also wish to thank A. Rouge for in te res t ing comments on our manuscr ip t .

References [ 1] G. Alexander, H.J. Lipkin and F. Scheck, Phys.

Rev. Letters 17 (1966) 412. [2] H.J. Lipkin and F.Scheck, Phys. Rev. Letters 18

(1967) 347. [3] H.J.Lipkin, Nuclear Physics B7 (1968) 321. [4] J.Mott et al.,Phys. Rev. 177 (1969) 1966;

F.A.DiBianca et al. , Nuclear Physics B16 (1970) 69.

[5] SABRE Collaboration (U.Karshon et al.), Nuclear Physics B29 (1971) 557.

[6] E. Hirsch, U.Karshon, H.J. Lipkin, Y. Eisenberg, A. Shapira, G.Yekutieli and J.Goldberg, Physics Letters 36B (1971) 139.

[7] H. Frass and D. Schildknecht, Phys. Letters 35B (1971) 72.

[8] SABRE Collaboration (B.Haber et al.) Nuclear Physics B17 (1970) 289.

[9] H.J. Lipkin, Phys. Rev. 183 (1969) 1189; Nuclear Physics B20 (1970) 652.

[ 10] J. P. Ader, M. Capdeville, G. Cohen-Tannoudji and Ph. Salin, Nuovo Cimento 56A (1968) 952.

[11] H.J.Lipkin, Phys. Rev. 159 (1967) 1303. [12] S. Meshkov, in Hyperon Resonance-70. Edited by

C. Fowler, (Moore Publishing Co., Durham, North Carolina, 1970) p.471 H. J. Lipkin, Configuration mixing in the baryon octet, to be published.

[13] T. Hofmokl and M. Szeptycka, Nuclear Physics Bt3 (1969) 53.