Symmetric quark model analysis of the masses and decay rates for (70, 1−)-states

IL NUOVO CIMENTO VOL. 19 A, N. 3 1 Febbraio 1974

Symmetric Quark Model Analysis of Rates for (70, 1-)-States (').

i~[. J o N e s and R. LEVI SETTI

E~rico Fermi Insbitute and Department o/ Physics University of Chicago - Chicago, Ill .

T. LASINSK] (**)

Laurrenee Berkeley Laboratory - Berkeley, Cal.

(ricevuto il 9 Luglio 1973)

the Masses and Decay

Summary. - - We have re-analyzed the (70, 1-)-representation of S U e • Oa using the symmetric quark model of Greonberg and Resnikoff and the most recent experimental information. The eleven parameters of the model mass formu~ are determined by z~-min~mization, and predictions are obtained for the masses and mixings of all (70, 1-)-states. With these predictions and the assumption that decays are inv.~iant under S Uew • 02L z, partial widths for all decays of (70, 1-)-states into baryons in (56, 0 +) and pseudosealar mesons in (35, 0-) are computed and compared to experimental values. The predictions for S-wave decays are in serious disagreement with experiment but the predictions for D-wave decays, with one exception, are a reasonable first approximation in an average sense t~ the observed values. We could not, however, obtain a good fit to the masses and D-wave decay rates simultaneously. Therefore, either the mass formula does not correctly give the mixing of the states or SU6w• is violated in the decays.

l . - I n t r o d u c t i o n .

As a guide in the search for new Y* resonances of mass below 2 GeV/e 2,

it would be ex t r eme ly helpful to h a v e a reliable mode l which pred ic ted no t on ly

(~ Research supported by the National Science Foundation. (**) Also supported in part by the U.S. Atomic Energy Commission.

365

366 M. JONES, R. LEVI SETTI and T. LASINSKI

the quan tum numbers bu t also the masses and decay rates of the undiscovered resonances. For a model of baryons, we have taken the symmetr ic quark model as formulated by GRE~NB]~]~G and RES~IKOFF (1). Choosing only l -body and 2-body operators, this model gives an eleven-parameter mass formula which predicts the masses and mixings of all particles in the (56, 0+), (70, 1~) and (20, 1 +) representat ions of ~ U6 • 03. With this mass formula, we have re-analysed the (70, 1-) using the most recent experimental da ta (2) and have determined the parameters and their errors by minimizing the function

(t .1) I M o ~ - - M~r.d ~ ffi

Previous authors (1.,-6) determined the mass formula parameters by minimizing

a (~figure of m e r i t , , ~ (Mob . - -M,, .d) 2, which gives all resonances an equal weight. Since the masses of some resonances in the (70, 1-) are much bet ter known than those of others, we feel tha t our determinat ion of the mass formula parameters is an improvement over previous work.

In addition, we have used the masses and mixing given by the mass formula to predict the partial widths for decays of (70, ]-)-s tates into pseudoscalar mesons in (35, 0-) and baryons in the (56, 0+). To do this, we have assumed (6) tha t the decays are invariant under ,~Uew• and have used the H-wave and D-wave coupling constants determined by FA~AN and PL_r (7). Thus, the determinat ion of the eleven parameters of the mass formula in conjunction with two additional decay parameters allows one to predict not only the masses

but also the decay rates for all states in the (70, 1-). Finally, we have performed a joint fit to the masses and D-wave decay

rates in order to determine if the mass formula can provide a good quant i ta t ive description of these quantit ies for the (70, 1-)-states. This joint fit serves as a str ingent test of the mixing ~mplitudes given by the mass formulu and also pro- vides a method of searching for other solutions for the mass formula parameters. Failure to obtaih a good fit would indicate tha t e i ther the mass formula does not correct ly give the mixing amplitudes or tha t 8Uew • O~z is violated in the decays.

(1) O. W. GREENBERG and M. RESNIKOFF: Phys. Rev., lb3, 1844 (1967). (2) PARTICLE DATA GROUP: P. SODING, J. BARTELS, A. BARBARo-G~LLTIERI, J. E. ENSTROM, T. A. L&SINSKI, A. R ITTENBERG, A. H. ROSENFELD, T. G. TRIFPE, N. BARASH- SCHMIDT, C. ]~RICMAN, V. CHALOUPKA and M. Roos: Phys. LeU., 39 B, 1 (1972). (3) D. R. DIVOI: Ph. D. Thesis, University of Maryland (1970). (~) D. R. DIVGI and O. W. GREENEERO: Phys. Rev., 175, 2024 (1968). (s) M. RESNIKO~'F: Phys. Rev. D, 3, 199 (1971). (~) J. ROS~ER: Proveedings o] XVI In~rnational Cwa]ere~we wa Hiyh-Energy Physics, Vol. 3 (1972), p. 149. (7) D. FAIMAN and D. E. PLANE: Nucl. Phys., 5 0 B , 379 (1972).

SYMM~BTRIC QUARK MODEL ANALYSIS OF THE MASSES AND DECAY RATES ETC. 3 6 7

2 . - T h e m o d e l .

The work of GREENBERG and ]~ESNIKOFF (1) and references cited therein give a detailed description of the symmetric quark model. We summarize here for completeness the basic assumptions. This model assumes tha t baryons are composed of three spin-�89 quarks, with spin-orbit coupling between quark spin and orbital angular momentum between quarks, and tha t the baryon wave functions are total ly symmetric. In the absence of symmet ry breaking, the wave functions are a product of irreducible representations of SUe and irreducible representations of 03. Considering SUe alone, a three-quark state is represented by the product 6 • 6 • 6, which, when decomposed into irreducible representations of SUe, yields

(2.1) 6 • 56 + 70 + 7 0 + 2 0 .

The 56-dimensional representation is totally symmetric, the 20-dimensional representation is total ly antisymmetrie and the two 70-dimensional representations (they are equivalent) have mixed symmetry. The symmetric quark model requires that , if there is no orbital angular momentum between the quarks, the SUB wave function must be in the 56-dimensional representation. A 70- dimensional representation requires at least one unit of orbital angular momentum between some pair of quarks to make the baryon wave function symmetric. We shall consider here only the (56, 0 +) and (70, 1-) representations of S Ue • 03.

Decomposing the SUB representations into products of representations of SUa and SU2 yields 5 6 = ( 8 , 2 ) + ( 1 0 , 4 ) and 70 = (1, 2) + (8, 2) + (10, 2) + + (8, 4). That is, the (56, 0+)-representation of SUe • 0.~ is composed of an SU3 octet of spin pari ty �89 and a decuplet of spin pari ty {+. Coupling an angular momentum L = 1 to the 70-dimensional representation of SUe yields

SU~ singlets of spin pari ty �89 and ~- respectively,

2 ~ octets ~ ~ ~ �89

2 ~) octets ~ * ~ ~-,

1 * octet , ,~ , ~-,

�9 decuplets , ~) ~ �89 and ~- respectively.

As SU6 symmetry-breaking operators GREENBERG and RESNIKOFF (1) have chosen all 1-body and 2-body operators which are either singlets or octets

3 6 8 M. JONES, R. LEVI 8ETTI and T. LASINSKI

under SU3 (octet dominance) and which have quark spin S----0. There are

four such operators which can act on symmetr ic 2-body states ZtT~, z t ~ , 2tT~os, and ~T~405 , and four which can act on ant isymmetr ic 2-body states X~T 1, ~6~5, ~T~ss, ~ . Spin-orbit operators can act only on ant i symmetr ic 2-body states; therefore, ~ T ~ , ~T~ and X~T~,~ are the only quark spin S = 1 operators tha t can contribute. The use of these operators leads to an eleven- parameter mass formula for the (56, 0 +) and (70, 1-) representations. Since only the first four operators act on the (56, 0+), four of the parameters can be determined from fitting the masses of the particles in the (56, 0+). The mass formula

TA.BIm I. -- Matrix elements o/ (L.S)-operators TSz.s.3~ and ~.a.zsr

Basis J = 5 / 2 J = 3 / 2 J = l / 2 states S U 2

Matrix ~ements of ~/~vs.~ 1 TsL.s.~= 0 on A and ~ states. ~ . s . ~ = (1/3)Tvs.~ on d~ states

A~ 1 - - 2 / 3 - - ( 2 / 3 ) ~ ] O - - ( I / 3 ) ~ / F O - - 5 ] 3 - - 4 /3 - - 2/3

A~ - - 2 / 3 5/3 4/3 --10/3

A~ 0 0

~ss --1 2/3 (2/3)v/YO --~/i-O 5/3 4/3 -- 2

~ 2/3 - - 1 - - 4 /3 2

~o o o

- - 1 3 /2 - - ( l / 3 ) v / i - O - - ~ / F O 5/3 - - 2 /3 - - 2

Z~ - - 4 1 3 - - I 813 2

s~o o o

Matrix elements of T~.a.zs 8

2~z.a.lsg= 0 on A and ~ states, ~ . sasg= (I/6)T~.z.~ on 3~ states

A~ 1/2 --1/3 (2/3)v/i-~ --(1/6) v/FO --5/6 4/3 -- 1/3

A~ 5/3 --2/3 --10/3 -- 4/3

A~ 0 0

X~s --1/2 1 --(2/3)v/i-0 (3/2)v/i-0 5]6 -- 4/3 3

2:~ --5/3 3/2 10/3 -- 3

X~o o o

--1/2 1/3 (5/6)V/i-0 (3 /2)V~ 5/6 5/3 3

~ 4/3 3/2 - - 8 /3 - - 3

S~o o o

S Y M M E T R I C Q U A R K M O D E L A N A L Y S I S O F T H E M A S S E S A N D D E C A Y R A T E S E T C . 369

TABLE II. - Q~an~um ~mbers for all 8U, states i~ the (8U3, 8U~s}-basis. Quantum numbers in the (8174, SU2s~}-basis are given only for those states which are not simultaneously diagonal in the (8U3, 8UsH)-basis. The relationship between these state~ in different bases is given at the end of the Table.

state s v . r ~3, s I s~ s~ ~" ~') 8U3

Quantum numbers for states in (SUa, S U2s}-basis.

56 N~ 1 6 1/2 1/2 1/2 0 45/2 63/4

A~ o 6 1/2 o o 1/2 45/2 9

~ 0 6 1/2 1 1 1/2 45/2 9

z - - 1 6 1/2 1/2 1/2 1 45/2 ]5/4 8

A~o 1 12 3/2 3/2 3/2 0 45/2 63/4

~v~l o 0 12 3/2 1 1 1/2 4512 9

~'~1o - - ] 12 3/2 1/2 1/2 1 45/2 15/4

"041o -- 2 12 3/2 0 0 3[2 45/2 0

70 A~ 0 0 1/2 0 - - - - 33/2 - -

.N~ 1 6 1/2 1/2 1/2 0 33/2 39/4

A~ 0 6 1/2 0 - - - - 33/2 - -

~2 s 0 6 1 / 2 1 - - - - 33/2 - -

~2 - - 1 6 1/2 1/2 33/2 - - 8 - - d

A~o 1 12 1/2 3/2 1/2 0 33/2 39/4

4 o 0 12 1/2 1 -- -- 33/2 - -

~,2 - - 1 12 1/2 1/2 33/2 -- ~ 1 0 - - d

~ o - - 2 12 1/2 0 0 1/2 33/2 0

N~ 1 6 3/2 1/2 3/2 0 33/2 39/4

A~ 0 6 3/2 0 1 1/2 33/2 5

~s 0 6 3/2 1 1 1/2 33/2 9

- - 1 6 3/2 1/2 1/2 1 33/2 15/4

20 /12 0 0 3/2 O 1 1/2 21/2 5

1 6 1/2 1/2 1/2 0 21/2 15/4

As z 0 6 I/2 0 1 1/2 21/2 5

2~ 0 6 1/2 1 0 1/2 21/2 5

~2 - - 1 6 1/2 1/2 1/2 0 21/2 15/4 8

370

TABLE II (co~inued).

M. JONES, R. LEVI SETTI and T. LASINSKI

state sv~ r c; 3, s ~ s . sa ~" c: ~

Quantum numbers in {8 Ui, S U3sa}-basis for states which are not diagonal simultaneously in {SU_,, SU,~}-basis.

~ o 0 - - 1/2 0 0 1/2 33/2 9

0 - - 1/2 0 1 1/2 33/2 5

~ o 0 - - 1/2 1 1 1/2 33/2 9

0 - - 1/2 1 0 1/2 33/2 5

- - 1 - - 1/2 1/2 1/2 0 33/2 15/4

~ - - 1 - - 1/2 1/2 1/2 1 33/2 15/4

Relation between states in {SU~, 2U~s}-basis and states in {2U~, SU~a}-basis.

( A~I= 1 ( 1 1~[,~1o / [ , ~ I = 1 ( 1 :)[~!o~

is written

(2.2)

+ N3[I(x + ~ ) - (~/4} r~ + s~(s . + ~) - ~a(s~ + 1)] + N , [v~"- 45/2] +

§ .N3[Y-- 3/4 § I(I + 1 ) - (1/4)17'-- S,(S, § 1) § S~(Sa § 1)] +

§ ~Ve[~ 3 '- 2S(S § 1 ) - 9/2] §

+ ~ , [ ~ " - 2s~(s~ + ~ ) - (U4) ~ ' - s t - ~5/2] +

+ NsT~z.,.~ § N, TS,..,.33§ N, oTSL.,.,,,

where 17 is hypercharge, S is total quark spin, I is isospin, S. is the total spin of the nucleonlike quarks, S a is the total spin of the A-like quarks, and ~s~, C~ ('~ and ~'~ are Casimir operators for S U3, SUe and S U4 respectively.

The parameters N o - hrs are determined by fitting the masses of particles in the (56, 0 +) and N , - N~o are determined by fitting the masses of particles in the (70, 1-). The (L.S)-operators cannot be put in terms of SU2, SU3 and SU~ quantum numbers since they break the SUe • 03 symmetry. We have used the matrix elements of the operators T~.s. ~ and 2~L.s.~ calculated by GREEN- BERG and RESNIKOFF (~) and those of the operator ~.s.~, calculated by DZVGI (3). An omission to Table X of ref. (1) and Table II-2 c) of ref. (3) has been pointed

SYMM~ETRIC QUARK MODI~.L ANALYSIS OF THE MASSES AND DECAY RATIOS ETC. 371

out to us by GREENBERG. Table I contanis the complete set of matr ix elements.

Table I I contains quan tum numbers for SUe states in the {SU~, SU2a}-basis

a n d in the (SUd, S U~sa}-basis. A sample calculation using the quan tum num-

bers and the mass formula is given in the Appendix.

3. - F i t to t h e (56 , O+).

All of the particles in the (56, 0 +) are well established and the masses of most

are known very accurately. In fact, the electromagnetic sphttings of all isospin

multiplets in the octet and of the ~* multiplet in the decuplet are quite well

known, and there is some information on the splittings of Z* and A multiplets (2).

Since the mass formula eq. (2.2) does not take electromagnetic splitting into

account, it can only predict the degenerate mass of an isospin multiplet. There-

fore, in order to apply the mass formula to the (56, 0+), we have taken the un-

split mass of a multiplet to be the average of the spht masses and have assigned

an error equal to half the width of the multiplet splitting. Because the errors

thus assigned are not statistical, one should not expect the X 2 to have its usual

statistical interpretation. However, this quant i ty can still be regarded as a measure of the goodness of fit and as a useful criterion for minimization.

We have, therefore, determined N o - N3 by minimizing the ;~2-function eq. (].1). The input data and results are shown in Table I I I and Fig. 1. The

TABL]~ III . - Observed and predicted masses /or pwdicles in the (56, O+)-vepresentation. For isospin singlets the observed mass and error are from reL (3). For other isospin multiplets, the observed mass is the average of the masses (from ref. (~)) of particles in the multiplet and the assigned error is half the splitting of the multiplet. The values of the mass formula parameters N 0 - N 3 and their errors, determined from the fit to the (56, 0+), are also given.

JP Particle Input mass-~assignederror Predicted mass Z z

1/2 + J~ 938.9 • 0.6 938.6 0.3

1/2 + A 1115.59 • 0.05 1115.59 0.0

1/2 + ~. 1193.1 4 - 4 . 0 1181.4 8.5

1/2 + ~ 1318.0 -t- 3.3 1325.5 5.2

3/2 + A 1231.0 • 4.0 1240.2 5.2

3/2 + Y.* 1384.0 • 2.0 1384.3 0.0

3 / 2 + ~ * 1533.5 -4- 2 .2 1528.4 5.5

3/2 + ~ 1672.5 • 0.5 1672.5 0.0

24.8

N o ~ (1001.1 ~ 1.0) MeV/c ~ , N z ~ (16.9 :[: 0.1) MeV/c ~ , N 1 ~ ( - - 209.9 ~ 1.2) MeV/c 2 , N s ~ (16.4 ~ 0.4) MeV/c ~ ,

3 7 O" M. J O N E S , R . L ~ V I 8 E T ~ I and T. L A S I N S K I

root-mean-square deviation of the predicted masses from the input masses is 6 MeV/cL I t is curious tha t the predictions ~or the isospin singlets (A and f~)

and for the isospin mult iplet with the smallest splitting (2~0) agree extremely

1.8

1.6

1.4 o

1.2

1.0

observecL predictec~ observec~ predicted

- - Q

•--ffiffiffiffi _ _ j ~

w

m I "

Fig. 1. - Observed and predicted masses for particles in the (56, 0+)-representation.

w~eil wi th the input masses and tha t the large disagreements occur for multiplets with larger splittings. An interesting step to take would be to add op- era tors for the electromagnetic mass splitting in the (56, 0 +) and then fit the split masses and their errors. I f such a fit were not satisfactory, one could then forsake octet dominance and add 21 7 iv~o6 or one could add the 3-body operators

56 4 ~ g 5 ' ~ 7 i~2,~, and iv~s. 5 (').

(s) H . J . LIPKIN: Seminar on Hiyh-Energy Physics and Elenwatary Particles (Trieste, 1 9 6 5 ) , p . 3 9 5 .

S Y M M E T R I C QUAI~K M O D E L A N A L Y S I S O F T H ~ M A S S E S A N D D E C A Y R A T E S E T C . 3 7 3

4. - Fit to the (70, 1-) .

Of the thirty isospin mult iplets of the (70, 1-), thirteen are well established. ~-o e lectromagnet ic splittings are k n o w n for these states, probably because the errors on the observed masses are greater than any of splittings themselves . The ass ignment of errors to the observed resonant masses is somewhat ar- bitrary for m o s t of the states because of the sys temat ic errors associated with determining the part ial -wave ampli tudes and extract ing resonance parameters from those amplitudes. Therefore, here again the ;~ as defined in eq. (1.1) is not expected to fol low a normal distribution. Nevertheless , we feel that the use of this quant i ty as a minimiz ing criterion is more appropriate than the use

2.2 preeLicte~

)bserved

, 1 2

2.0

~2

~4 c. 8

~2

1.6

1.8

u) v~

1.4

3 j = 5 J'=2 2

predicted predicted observed.

2 ~ 1 0

~2 ~4

m ~ 4 - 8

_2 m "~B

? . . . . . . . �9 ~21

150 N e V / c z

t o t o J width

Fig. 2. - Masses and total widths for observed and predicted J~', A, ~ and Q resonances in the (70, 1-)-representation. Note that the length of the horizontal bar is proportional to the total width of the resonance.

24 - I ! N t ~ v o C imen to A .

~ 7 ~ M. J O N E S , R . L E V I S E T T I a n d T . L A S I N S K I

of the (, figure of merit ,> of ref. (~.s.~) because resonances whose masses are better determined are given greater weight.

Constraining N o - 2r from the fit to the (56, 0 +) and, using as starting values for hr~ - - N9 those determined by ref. ('), we then performed a fit to thirteen well-established resonances 6f the (70, 1-). The input data and results are shown in Table IV and Fig. 2 and 3. N o t e that Table IV contains two

2.2

2 .0

I j = ~ j___~ j = 5_2

precLic reel pre~ ~c teal precL ~b tecL ~bserveoL observecL observecL

~ U

~ 1.8

d E

1.6

1.4

2 _ _ - - - 4

" - - - " - - ~Zl~ ^ ~ i. e 4 . , ' . . ' : - - - . _ " " _ _ . . . . . . . .

- - - A e

~ T e ") . . . . .

�9 - . . . . ~ v e

2 . . . . . . . . . . A 1

? - ~ Ye

. . . . . . . . . . z ,

1 so Mev/~ total wicLth

Fig. 3. - Masses and total widths for observed and predicted 17 ~ 0 resonances in the (70, 1-)-representation.

solutions. The root-mean-square deviat ion for each solution is N 20 MeV/e ~. The mos t significant difference be tween the solut ions is in tbe ~ - Z - s t a t e s . Solut ion 1 takes the )3(1670) to be the lowest -mass state; solution 2 takes the Z(1670) to be the state of intermediate mass. Figure 4 summarizes the best fit (solution 1) to the masses of p~rticles in the (56, 0 +) ~nd (70, 1-).

SYMMETRIC QUARK MODEL ANALYSIS OF THE MASSES AND DECAY RATES ETC. 375

TABLE IV. -- Observed and predicted masses /or thirteen well-established vesonances i~ the (70, 1-)-representation. For J~ and A resonances, t h e observed masses and assigned errors are f rom p. 93 of ref. (2). F o r A(1405), A(1520) and Z(1765), the values come f rom the main baryon table of ref. (3). For A(1830), the mass is taken f rom ref. (~) and the error was assigned in order to ref lect the range of values in the da t a card l is t ings of ref. (3). Fo r t he remain ing resonances, the observed mass was taken a t the cen ter of t he range of va lues in t he da ta card l is t ings and the error was taken to be half the range. The va lues of t he mass fo rmula pa rame te r s and thei r errors, de t e rmined f rom the f i t to t he (70, 1-), are also given. N o t e t h a t the values for _~r o - s are no t exac t ly t he same as those in Tab le I I I . These p a r a m e t e r s were al lowed to v a r y in t he fit to t he (70, 1-) b u t were cons t ra ined to be consis tent w i t h t he va lues ob ta ined f rom the f i t to t he (56, 0+).

JP Resonance Observed Solution 1 Solut ion 2 Solu- Solu- mass • er ror p red ic t ed p red ic ted t ion 1 tion 2

mass mass X2 X2

1/2- d~o(1535) 1531 • 27 1512 1507 0.5 0.8

1/2- J~(1700) 1700 • 29 1654 1660 2.5 1.9

3/2- J~~ 1520 • 9 1525 1518 0.3 0.1

5/2- J~(1670) 1674 • 8 1688 1689 2.9 3.4

1/2- A (1650) 1637 • 23 1671 1655 2.2 0.6

3/2- A (1670) 1685 • 24 1671 1655 0.3 1.5

1/2- A (1405) 1405 + 5 1409 1405 0.5 0.0

1/2- A (1670) 1673 :t: 10 1687 1685 1.9 1.5

3/2- A (1520) 1518 • 2 1517 1517 0.2 0.4

3/2- A (1690) 1690 ~ 10 1680 1680 1.0 0.9

5/2- A (1830) 1831 • 10 1810 1810 4.4 4.5

3/2- E (1670) 1 6 7 0 2 10 1651 1704 3.8 1].2

5/2- Z (1765) 1765 ~ 5 1770 1767 0.9 0.1

21.3 27.0

Solut ion 1 Solut ion 2

No ~ 1002.9 :~ 1.0 No = 1002.9 • 1.0

N 1 ~ - -211 .2 =t: 1.2 N 1 ---- - -211 .2 J= 1.2

N3 ~ 16.8 ~: 0.1 N3 = 16.8 d: 0.1

Na ~ 17.2 ~: 0.4 Ns = 17.2 • 0.4

N 4 ~ - - 77.8=L1.2 N 4 = - - 77 .1J=1 .5

IV5 ~ - - 12 .2 : t :0 .7 N~ = - - 1 5 . 0 •

:v+ ~_ 5.8 :~ 1.3 N , = 5.6 • 1.8

N 7 = - - 1 8 . 8 ~ 1 . 3 N 7 = - - 18 .7 :~2 .2

N s = 6.5 ~: 0.6 N s = 6.4 -~: 0.7

/V, = - - 9 . 8 + 1 . 5 N 9 = - - 1 0 . 9 : l : 1 . 9

N l o = - - 6 . 8 ~ 1 . 7 N t o = - - 5 . 6 •

(~) B. CoNceRTo, D. M. HARMSEN, T. LASINSKI, R. LEv[-SETTI, M. RAYMUI~D, E. BURK- HARDT, H. F I L T H U T H , S. KLEIN, H. OBERLACK and H. SCHLEICH: Nucl. Phys. , 34 B, 41 (1971).

3 7 6 M. J O N E S , R . L E V I S E T T I and T . L A S I N S K I

The j u d g m e n t of the qua l i ty of the fit to the (70, 1-) m u s t r emain somewha t

subjec t ive as long as the sys t ema t i c errors associa ted with the de t e rmina t ion

of r e sonan t masses are significant. There is, however , a t least one serious dif-

1.8

1.6

G)

1.4

o

1.2

1.0

-T r -----r A

~r ~ C

--4--

, , , I , o 4'0

A M(MeV/c z )

Fig. 4. - Differences of predicted and observed masses for observed states in (56, 0 +) and (70, 1-) representations. Circles represent states in SUs octets, triangles represent states in 8 Us decuplets, and asterisks represent states in S Us singlets. The solid circles and triangles are states in the (56, 0+). The horizontal bar represents the error on the observed mass.

f icul ty wi th the mass fo rmula eq. (2.2). I t predicts t h a t t he ~- and ~ - A res-

onances should be degenera te in mass ; b u t exper imenta l ly , a l though errors on indiv idual s ta tes are large, the mass of the ~ - A seems to be signif icantly

SYMMETI~IC Q U A R K MODEL ANALYSIS OF THE MASSES AND DECAY RATES ETC. 3 7 7

greater than tha t of the ~-A. In fact, ALMEHED and LOVELACE (lo) find the A's split by 80 MeV/c ~, and prel iminary results of AYED, BAREYRE and LE- MOI6NE (11) find a splitting of ~ 110 MeV/cL DIVGI and GREE~BEI~G (~) point OUt three possible solutions of this problem within the framework of the symmetric quark model. Following one of their suggestions, RES~IKO~F has analysed (5) the (70, 1-) by employing three-body operators to split the A's. Res- nikoff's analysis (5) uses a (, figure of merit ,~ as a criterion for minimization bu t rejects the best solution because it predicts Z-states of mass .-~ 1560 MeV/c 2 and there seems to be no indication from exper iment of Z-states near this mass. We think tha t what is essential for any future s tudy is a i~2-analysis of the (70, ~-) with the incorporation of some mechanism to split the A's in an a t t empt to determine, f rom the qual i ty of the fit to the well-established states, which operators are required.

5. - D e c a y predict ions for ( 7 0 , 1 - ) .

Fi t t ing the masses of known resonances in the (70, 1-) yields not only the masses of the remaining resonances bu t also the mixing parameters for resonances with the same quan tum numbers. These mixing parameters are given in Table V. Using these mixing parameters and assuming tha t the decays of the resonances are invariant under S U, , x 0,Lz, one can calculate the part ial widths for the decays of (70, 1 )-resonances into baryons in (56, 0 +) and pseudoscalar mesons in (35, 0-). This calculation needs only 2 more parameters, coupling constant for S-wave decays and a coupling constant for D-wave decays. F A~ A~ and PLANE (7) have determined these coup]ing constants by fitting the decays (70, 1-) -> (56, 0 +) ~- (35, 0-) with the mixing parameters and coupling constants as free parameters . We have used the Clebsch-Gordan coefficients given b y Tables A1 (n), A2 and A3 of ref. (7) and the coupling constants given by their solution 1. We have not a t t empted to redetermine the coupling constants because they are determined, for the most part , independent ly of whatever mixing is assumed within the (70, 1-). The part ial widths were calculated with

(5.]) { 1' -~ C z ]2p M z / M B for S-wave decays ,

_~ C 2 g~p5 M , JM,~ for D-wave decays ,

{lo) S. ALMEH~.D and C. LOVe.LACE: Nucl. Phys., 40 B, 157 (1972). (11) R. AY~..D, P. BAREYRE gnd Y. L~.MOIGI~'E: Zero st~'Wnfleness ba~lo~ st~P, es below 2.SGeV mass, paper 990, XVI International Cwaf~rence on High-Energy Physics (Batavia, 1972). (12) We have chosen the upper sign for the Clebseh-Gordan coefficients given in this Table.

3 7 8 M. J O N E S , R . LEVI SETTI and T. LASINSKI

TABLE V. - Mixing matrices predicted by solutions 1 and 2 o! the /it to the (70, 1-). Since the overall sign of a row is a rb i t ra ry , we have adopted the convention of choosing the overall sign so tha t the diagonal elements always have a posit ive sign. W e l l established resonances arc indicated using the notat ion of ref. (2). The remaining resonances are denoted by the mass predicted by the mass formula.

JP Resonance (8, 2) (8, 4)

Mixing ampli tudes for (70, 1-)-resonances (solution 1)

1/2- JW(1535) 0.9996 0.0295

1/2- J~(1700) --0.0295 0.9996

3/2- oN~ 0.9989 0.0465

3/2- JW(1667) --0.0465 0.9989

C 1, 2) (8, 2) (8, 4)

1/2- & (1405) 0.8977 --0.4391 --0.0355

1/2- A (1670) 0.4406 0.8947 0.0737

1/2- A (1742) --0.0006 --0.0818 0.9966

3/2- A (1520) 0.9254 --0.3721 --0.0725

3/2- A (1690) 0.3775 0.9219 0.0875

3/2- A (1769) 0.0343 - - 0.1084 0.9935

(8, 2) (8, 4) (10, 2)

1/2- Z (1610) 0.7635 0.6448 --0.0370

1/2- ~. (1645) --0.6435 0.7643 0.0423

1/2- E (1796) 0.0555 --0.0085 0.9984

3/2- Z (1670) 0.7112 0.7019 - - 0.0392

3/2- ~. (1705) --0.6986 0.7119 0.0724

3/2- Z (1796) 0.0787 --0.0241 0.9966

1/2- ~ (1740) 0.8775 0.4775 --0.0446

1/2- ~ (1782) --0.4763 0.8786 0.0352

1/2- .~. (1921) 0.0560 --0.0097 0.9984

3/2- ~ (!780) 0.7944 0.6056 --0.0463

3/2- .~. (1838) --0.6022 0.7953 0.0704

3/2- ~ (1921) 0.0795 - - 0.0281 0.9964

SYMMlgTRIC QUARK MODEL ANALYSIS OF THE MASSES AIqD DECAl" RATES ETC.

TABLE V (co~tin~ed).

379

J ~ Resonance (8, 2) (8, 4)

Mixing ampl i tudes for (70, 1-)-resonances (solution 2)

1/2- d~(1535) 0.9997 0.0236

1/2- .N'(1700) - - 0.0236 0.9997

3/2- ,N'(1520) 0.9993 0.0373

3/2- J~'(1671) - -0 .0373 0.9993

( l , 2) (8, 2) (8, 4)

1/2- A (1405) 0.8902 - -0 .4545 - - 0.0318

1/2- A (1670) 0.4552 0.8902 0.0172

1[2- A (1745) 0.0204 - - 0.0298 0.9993

3/2- A (1520) 0.9286 - -0 .3652 - - 0.0666

3/2- A (1690) 0.3680 0.9291 0.0365

3/2- A (1770) 0.0485 - -0 .0584 0.9971

(8, 2) (8, 4) (to, 2)

1/2- ~ (1606) 0.7516 0.6593 - -0 .0195

1/2- ~ (1645) - - 0.6596 0.7513 - - 0.0224

1/2- ~ (1787) - - 0.000l 0.0297 0.9996

3/2- ~ (1640) 0,7516 0.6556 - - 0.0724

3/2- ~ (1670) - -0 .6577 0.7532 - -0 .0070

3/2- Y. (1788) 0.0499 0.0529 0.9974

1/2- E (1738) 0.9066 0.4217 - -0 .0139

1/2- ~ (1777) - -0 .4219 0.9062 - -0 .0285

1/2- E (1919) 0.0006 0.0317 0.9995

3/2- ~ (1784) 0.7684 0.6351 - - 0.0788

3/2- ~ (1833) - - 0.6376 0.7703 - - 0.0101

3/2- E (1920) 0.0543 0.0580 0.9968

3~0 M. JONES, R. LEVI SETTI a n d T. LASINSKI

where C is the appropriate Clebsch-Gordan coefficient, ]8 is the S-wave coupling constant , g2 is the D-wave coupling constant, p is the center-of-mass momentum

for the decay~ M N is the proton mass, and M R is the mass of the decaying resonance. For decays involving A(1236) or Z(1385) in the final state, we have calculated the center-of-mass momentum by averaging over the A(1236) and Z(1385) Breit-Wigner 's in the same manner as in ref. (7). For the decay A(1520) -> --> Z(1385)= we have taken p = 25 MeV/c (18). The part ial and total widths predicted by the mass formula are given in Table VI and Fig. 5 and 6. For comparison, Table VI also gives the observed part ial and total widths when they are known (2). Note tha t the predicted total widths are really only lower bounds since we have neglected three-body decays, and decays involving vector mesons or (70, 1-) baryons in the final state. The contributions from these neglected decays are small in general because of the limited phase-space available.

Except for A(1690)-> J~K, the predictions for D-wave decays (Fig. 5 b)) are a reasonable approximat ion in an average sense to the observed values, bu t the predictions for S-wave decays (note tha t the scale in Fig. 5 a) is three times tha t of Fig. 5 b)) are in serious disagreement with experinlent. There are some theoretical reasons to believe tha t a break-down of the SU, w• ~ symmet ry might affect the S-wave decays more than the D-wave decays (1~.16). An interesting test of the S U6w • z symmetry , independent of whatever mixing is assumed within the (70, 1-), is provided by the decays of ~--resonances and the A-resonances since these resonances are unmixed within the (70, 1-). There is some indication tha t there is significant disagreement between some of these observed decays and the predictions. For example, the ~-A(1830) is predicted by the symmet ry to have no coupling to KJ~ bu t the observed partial width, though small, is not zero. This could be an indication of the break- down of S/Jew • 0~z symmet ry or of mixing of the (70,1-) with higher S U, • 03 mnltiplets.

The reliability of the part ial-width predictions for unobserved resonances is undoubtedly less than tha t for well-established resonances. Obviously, the predictions for S-wave decays of unobserved resonances should be viewed with considerable skepticism. However , the limited success of the predictions for well-established resonances leads one to hope tha t the predictions for unobserved resonances might give at least some indication as to why these resonan~s have so far escaped detection.

J~-resonances: The only missing J~-resonance has Je = ~-. We predict this resonance to have a mass ---1670 MeV/c 8 and to have a large total width and small coupling to the J ~ channel. There is some evidence for a state a t

(13) T. S. MAST, M. ALSTON-GARNJOST, R. 0 . BANGERTER, A. ]3ARBARO-GALTIERI, F. T. SOLM~TZ and R. D. Ts~PP: Phys. Rev. Lea., 28, 1220 (1972). (14) p. G. 0. FR~U~D: private communication. (15) D. R. DIVGX: Phys. Rev., 175, 2027 (1968).

SYMMETRIC QUARK MODEL ANALYSIS OF THE MASSES AND DECAY RATES ]~TC. 381

TABLE VI. - Predicted partial and total widths /or all resonwaces in the (70, 1-). For completeness, the masses predic ted by solutions 1 and 2 are also given. We give only the par t ia l width predicted by solution 1 unless tha t predic ted by solution 2 differs by more than 10%. For comparison, observed masses and par t ia l widths are also given. For J~ and A resonances the observed masses and to ta l widths were taken from ref. (z) (p. 93 and main baryon table). The observed masses for the IT= 0 resonances were determined in the same manner as for Table IV. The observed to ta l widths for Z(1750) and for A-resonances come from the main baryon table of ref. (3). The observed to ta l widths for Z(1670) and Z(1765) represent the range of values in the da ta card listing of ref. (3). To obtain the range of observed par t ia l widths for all resonances except the A(1520), we have calculated the lower end of the range by mult iplying the lowest branching fraction (obtained from the da ta card l ist ing of ref. (2)) by the lower end of the range of t~)tal widths. The upper end of the range of par t ia l widths was calculated by mult iplying the highest branching fraction by the higher end oI the range of to ta l widths. For the A(1520), the par t ia l widths were obtained by mult iplying the to ta l width by the branching fractions quoted in the main baryon table of ref. (~).

JP Reso- Observed Predic ted Decay Observed Predic ted nanco mass mass modes pa r t i a l par t ia l width

(MoV/c 2) (sol. l)/(sol. 2) (wave) width (sol. 1)](sol. 2) (MeV/e t) (MoV/e 2) (MeV/e 2)

1/2- ~'(1535) 1500--1600 1512/1507 J~n (B) 12 --'.114 217

**** (1531:L27) J ~ (~q) ]4 --114 19

An (D) 3.5

50 --160 240 (106 • 38)

l/Z" J~(1700) 1665.'--1765 1654/1660 J~n (8) 50 --316 52

**** (1700=}:29) EK (B) 37

J ~ (S) 1 .3§ 43

A K (8) 0 § 40 0.05/~0.05

An (D) 11

100 --400 143 (231 4- 99)

3/2- J~(1520) 1510--1540 1525/1518 f i n (D) 47 -- 90 89

**** (15204- 9) X ~ (D) 0 -- 2.1 0.08

An (8) 21 -- 60 103

An (D) 1.1

105 --150 193 (121 + 12)

5/2- J~'(1670) 1655--1680 1688[1689 aYn (D) 42 -- 88 34

**** (1674~_ 8) ,N~ (D) ~ 2 4.4

A K (D) < 0.5 0

ATr (D) 56 --128 39

105 --175 77 (143 • 26)

382

TABLE VI (c~ont/nued).

M. J O N E S , R. LEVI SETTI and T. L A S I N S K I

JP Rose- Observed Predic ted Decay Observed Predicted nance mass mass modes par t ia l par t ia l width

(MeV/e 2) (sol. 1)/(sol. 2) (wave) width (sol. 1)/(sol. 2) (MoV/e~) (MeV/e 2) (MeV/e~)

1/2- A (1405) 1405-4- 5 1409/1405 ~Tr (S) 40 • 10 141 * * $ $

1/2- A (1670) 1663--1683 1687/1685 J~K (S) 2 .1+ 15 346

**** (1673• ~ (S) 2 .8+ 24 2.4/6.0

A~ (8) 1 .5+ 18 12

~'.*r~ (D) 3.2

15 -- 38 364

3/2- A (1520) 1 5 1 8 i 2 1517/1517 v~K(D) 7 .2 • 0.9 1.2

**** ~ (D) 6 .7 • 0.9 11

Z*~ (8) 2.0

Z*~ (D) ~ 0 . 0 5

16 • 2 14

3/2- A (1690) 1680+1700 1680/1680 J~K (D) 4 .9+ 29 113

**** (1690• X~ (D) 11 + 68 0.20

A~ (D) 0.06

~*~ (8) 85 / 73

X*n (D) 1.7 / 2.0

27 + 85 200

5/2- A (1830) 1827+1840 1810/1810 J~K(D) 2 .2+ 36 0

*** (1831• 10) ~ K (D) < 0.05

Eg (D) 7 .4+ 75 60

A~ (D) 3.5

Z*~ (D) 40

74 +150 104

1/2- A (1650) 1615+1695 1671/1655 JV~ (8) 32 + 70 29

**** (1637~-23) An (D) 29

130 +200 58

(164 i 35)

3/2- A (1670) 1650+1720 1671/1655 v~r~ (D) 21 + 66 30

*** (1685• A~v (8) 180

ATr (D) 26

175 +300 236 (245 -4- 23)

SYMMETRIC QUARK M'ODEL ANALYSIS OF THE MASSES AND DECAY RATES ETC. 3 8 3

TABLE VI (eo~g/~ued).

J P Rose- Observed Predicted Decay Observed Predicted nance mass mass modes partial partial width

(1KeV/c 2) (sol. 1)/(sol. 2) (wave) width (sol. 1)/(sol. 2) (MoV/o ~) (MeV/c 2) (MoV/c z)

3/2- Z (1670) 1660+1680 1651/1704 ,N'~(D) 2 .1+ 5.4 <0.05/ 5.2 **** (1670+10) Zn (D) 7 .5+ 38 15 / 22

An (D) 2 .1+ 23 6.5 / 0.14 Z*n (8) 47 / 4.2

2:*7: (D) < 0.05/ 1.0

30 + 60 69 / 33

5/2- Z (1765) 1765-t- 5 1770/1767 ,N'K,(D) 40 + 65 47 **** Z~ (D) 0 .9+ 28 8.3

x~ (D) < o.o5

An (D) 13 + 31 21

AK (D) 0.06

E*n (D) 3 .8+ 35 4.0

110 --145 80

3/2- J~(1700) 1667/1671 J~n (D) 2.7 / 3.3 * ZK (D) <0 .05

oN~ (D) 0.87

AK (D) <0 .05

Arc (S) 328

An (D) 29 361

1/2- A(1870) 1742/1745 J~K(S) 1.5 / 0.02 ** Zn (S) 122

A~ (8) 22 / 25

Z*n (D) 2.1 / 2.9

148

3/2- A ( 1769/1770 J~K (D) 0.75/ 0.01

2:n (D) 2.8 A~ (D) 0.17/ 0.23

Z*n (S) 109

X*n (D) 17

220

5/2- ~ ( ) 1912/1909 .~.n (D) 59

ZK (D) 6.7

~.~ (D) 0

AK (D) 15 Z*K(D) 0.08/ 0.06

E*n (D) 6.0 87

384

TABLE VI (contimw~t).

M. JONES~ R. LEVI SETTI a n d T. L A S I N S K I

JP Rose- Observed Predicted nanee mass mass

(MeV/e 2) (sol. 1)/(sol. 2) (MeV/e ~)

Decay Observed Predicted modes partial partial width (wave) width (sol. 1)/(8ol. 2)

(MeV/c 2) (MeV/e 2)

1/2- ~ ( ) 2045/2051 EK (8) 37

E*~(D) 0.10/ 0.23

37

3/2- G ( ) 2045/205l EK (D) 13

E*~(8) 37 / 43

E*E(D) O.O5/ 0.11

5o / 56

1/2- ~ (1620) 1610/1606 J~'K, (8) 20

** Z~ (8) 38

Azt (8) 47

Z*rr (D) 0.15

105

1/2- Z ( ) 1645/1645 .N'K (8) 96

ZT: (8) 144

Az: (8) 8.o

Z*~ (D) < 0.05

248

1/2- ~ (1750) 1730--1790 1796/1787 J~~ 6 -- 64 7.0 / 10

*** Y.~ (8) 13 / 6.9

zn (8) 5.3

A~ (,.~) 0.5-- 42 12.0

AK (D) 0.50/ 0.32

Y.*~ (D) 24 / 21

50 -- 80 62 / 56

3/2- E ( ) 1705/1640 ~,CK (D) s.o / < 0.o5

~.~ (D) 32 / 12

An (D) 0.61/ 5.5

X*~ (,.~) 5.7 / 37

Y.*~ (D) 1.6 / < 0.05

48 / 55

SYMMETRIC QUAR~ MODEL ANALYSIS OF THE MASSES AND DECAY RATES ~TC.

TABLE VI (co~/~med).

385

JP Rose- Observed Predicted Decay Observed Predicted nanee mass mass modes part ial part ial width

(MoV/e 2) (sol. 1)/(sol. 2) (wave) width (sol. l)/(sol. 2) (MoV/o ~) (MeV/e ~) (Merle ~)

3/2- ~ ( ) 1796/1788 JY'K (D) 4.9

r,~ (D) 8.6 / 6.O

Y,~ (D) 0.10/ 0.06

A~ (D) 9.4

~g, (8) 38 / 28

A~ (D) 0.22/ 0.18

Y~*~ (8) 91

Y~*~ (D) 13 10

165 /15o

1/2- .~. ( ) 1740/1738 E~ (8) lO / 5.5

Z ~ (8) 94

AK (S) 5.5 7.4

E*~(D) 0.14

110

1/2- ~ ( ) 1782/1777 E~ (8) 147

zg , {8) o.o8/ 0.06

AK.(8) 66 / 59

E*~ (D) <0 .05

213

1/2- ~ ( ) 1921/1919 ~ : (8) 13 / 8.5

Y.K (8) 14 / 11

.~.~ (S) 3.2 / 4.6

A~, (8) 8.4 / 13

Y.* K,(D) 0.65

�9 ~.*~ (D) 7.1

46

3/2- .~. ( ) 1780/1784 F~n (D) < 0.05

Y.K. (D) 3.4

AK. (D) 2.1

E*rc (S) 54

E*~ (D) < 0.05

6O

386

TABLE VI (co~timued).

M. JONES, R. LEVI SETTI a n d T. LASINSKI

JP Reso - I l a i l e e

Observed mass (MoV/o 2)

Predicted m a s s

(sol. 1)/(sol. 2) (MeV/e ~)

Decay Observed Predicted modes partial partial width (wave) width (sol. 1)/(sol. 2)

(MeV/e 2) (MeV/e ~)

3/2- ~ ( 1838/1833 ='r: (D) 8.0

ZK (D) 3.3 3.9

AK. (D) 9.1

-~-*~ (S) 12 7.5

E*~ (D) 2.1

34

3/2- ~ ( 1921/1909 ~ (D) 8.1 6 .2

Zg. (D) 5.2

E~ (D) O.O5

AK (D) 4.4 / 5.6

Z 'g48) 55 / 48

Z*K.(D) 0.31

~*~ (S) 31 [ 38

E*~ (D) 4.0 / 3.0

110

mass ~ 1700 MeV/c 2 with an elastici ty of ~ 0.10 and to ta l width --~ 100 MeV/c 2 in some par t ia l -wave analyses (~.11.~6).

~- resonances : I t should be no surprise to anyone t ha t these have not been s e e n .

E-resonances: The exper imenta l s i tuat ion concerning E-s ta tes is confused because of the lack of spin and par i ty informat ion for repor ted resonances. P a r t of the confusion m a y also be a result of several s tates occurring with abou t the same mass. Perhaps the predictions for the E-s ta tes can be of some help to exper imental is ts in deciding how m a n y s tates are present in the mass region (1750-- 2000) MeV/c 2.

Z-resonances: We predict ~- Z-resonances a t ~ 1 6 1 0 , ~ 1 6 4 5 and 1790MeV/C. There is some evidence for �89 Z-resonances a t mass

~-~ 1620 MeV/c ~ (17.~s) and (1750-- 1790)MeV/c ~ (..s, tT-~s). This leaves the ~- Z(1645)

( i s ) D. HERNDON et al.: A partial wave analysis o! =3W--*wTvoV ir the c.m. energy range (1300+2000)IKeV (LBL-SLAC), paper 289, X V I International Con]ore'nee on High-Energy Physics (Batavia, 1972). (1~) j . K. KIM: Phys. Rev. I~ett., 27, 356 (1971). Qs) W. LANGB•IN and F. WAGNER: Nucl. Phys., 47 B, 477 (1972).

S Y M M E T R I C Q U A R K M O D E L A N A L Y S I S O F T H E M A S S E S A N D D E C A Y R A T E S E T C . ~ 7

f

I o l I I

i

7

f

I I I I I I I

".I"

I::

i i o

N ( ~/Aai, i ) W77o!A4

1 ] i

7D!7,~Z)d l~aTo/poJd

'1

0

1

l

t t D

E

o

o ~

q)

v

99

0~ o9

3 ~ M. JONES~ R. LEVI SETTI &nd T. LASINSKI

360' ^(1670)

300

24oL 3[ /

180

13

120

60

A(1690)

J~(1535)

,N'(1520)

& (1670)

A(1405) , / ,N'(I?00) , , - -O - - /

/ A (1830)

T(1765)

J~(1670)

&(1650)

/-o- A (1520)

' 6'o ' ' , ; o ' 2 .o observecL t o t a l wielth (MeV/2)

Fig. 6. - Comparison of observed and predicted total widths for ,well-established resonances in the (70, 1-). The two predictions for the E(1670) correspond to solutions 1 and 2 described in the text.

ye t unobserved. But we predict this resonance to be very wide, so it could easi ly have escaped detection. The si tuat ion with the ~- Z-resonances is un- cer ta in because it is not clear f rom our analysis whether the observed ~ Z(1670) is the lowest mass ~ - Z - s t a t e or the s ta te with in termediate mass. In ei ther case, the ~ - Z - s t a t e with highest mass ( ~ 1 7 9 0 MeV/c 2) is predicted to have a ra ther small coupling to 2~K and could have escaped detection. I f one assumes tha t the Z(1670) is the s ta te with the lowest mass, the s tate of inter- media te mass is predicted to have a mass ,-~ 1705 MeV/c% a substant ia l coupling to ~ K and reasonably small width. But , if one assumes t ha t the Z(1670) is

the s tate with the in termedia te mass, the s ta te of lowest mass is predicted to have an ex t remely small coupling to J~K. The exper imenta l s i tuat ion is con-

SYMMETRIC QUARK MODEL ANALYSIS OF THE MASSES AND DECAY RATES ETC. ~ 9

fused h e r e , b u t p r o d u c t i o n e x p e r i m e n t s sugges t m o r e t h a n one .~- Z - s t a t e of

m a s s ,-~ 1670 MeV/c 2 whi le f o r m a t i o n e x p e r i m e n t s f ind o n l y one s t a t e ( ' ) .

A - r e s o n a n c e s : The re a re two miss ing A- resonances . T h e r e is ev idence for

a } - A - s t a t e of mass ( 1 7 8 0 - - 1 8 7 0 ) MeY/c ~ f rom p a r t i a l - w a v e a n a l y s e s (z.~,).

W e p r e d i c t t h a t b o t h t h e mi s s ing �89 a n d mi s s ing ~ - A - s t a t e s shou ld h a v e v e r y

sma l l coup l ing to J ~ K a n d t h u s cou ld h a v e e s c a p e d de t e c t i on . The ~ - A - s t a t e

is a lso p r e d i c t e d to be q u i t e wide .

I n conc lus ion , t he p r e d i c t i o n s for u n o b s e r v e d r e sonances e i the r a re qua l i -

t a t i v e l y c o n s i s t e n t w i th s t a t e s for which the re is some ev idence or p r o v i d e

poss ib le r easons w h y the r e sonances h a v e no t been obse rved .

6. - Fit to D - w a v e decays and masses .

I n a n a t t e m p t to i m p r o v e t h e p a r t i a l - w i d t h p r e d i c t i o n s a n d to s e a r c h for

o t h e r so lu t ions , we have also p e r f o r m e d a fi t to t h e d e c a y s of o b s e r v e d r e sona nc e s

in t he (70, t - ) us ing the mass f o r m u l a to p a r a m e t r i z e the m i x i n g b e t w e e n

s t a t e s . S ince the S - w a v e d e c a y p r e d i c t i o n s were v e r y bad , we f i t t ed on ly t h e

TAnLE VII . - Observed and predicted masses and observed and predicted D-wave decay rates lot solutions A and B o/ the j o i n t / i t to masses and D-wave decays. The observed masses and errors are the same as in Table IV. Except for A(1520) decays, the observed par t ia l widths are taken a t the center of the range oi values and the errors are taken to be half the range.

JP Resonance Observed Solution A, Solution B, Solu- 8o111- mass • error predic ted predicted tion A tion B (MeV/c 2) mass mas~ X z ~t

1/2- ,N'(1535) 1531 ~ 27 1532 1514 0.0 0.4

1/2- ,}~('(1700) 1700 • 29 1667 1672 1.3 0.9

3/2- oN'(1520) 1520 • 9 1529 1514 1.1 0.5

5/2- ~'(1670) 1674 • 8 1659 1672 3.5 0.1

1/2- A (1650) 1637 + 23 1704 1660 8.4 1.0

3[2- A (1670) 1685 -4- 24 1704 1660 0.6 1.1

1/2- A (1405) 1405-4- 5 1406 1404 0.0 0.1

1/2- A (1670) 1673 + 10 1687 1658 2.1 2.2

3/2- A (1520) 1518 • 2 1521 1521 2.1 2.9

3/2- A (1690) 1690 • 10 1684 1680 0.4 1.0

5/2- A (1830) 1831 i 10 1790 1794 17.2 13.4

3/2- Y~ (1670) 1670 • 10 1627 1713 18.4 18.6

5/2- Z (1765) 1765 • 5 1766 1754 0.0 4.7

55.2 46.8

25 - I/ Nuovo Cimento A.

390

TABLE VII (continued).

M. J O N E S , R. LEVI SETTI and T. L A S I N S K I

Decay Par t i a l Solution A, Solution B, Sol- Solu- width -t- error predic ted predicted tion A tion B (MoV/c ~) width width g2 X~

J ~ ( 1 5 2 0 ) - ~ J ~ 69 -{- 22 88 88 0.7 0.7

-*J~'~ 1.0 ~ 1.0 0.09 0.09 0.8 0.8

2~ ' (1670)-+J~ 65 -t- 23 34 34 1.8 1.8

- -~J~ 0.0 t 2.0 4.4 4.4 4.8 4.8

- ~ A K 0.0 • 0.5 0 0 0.0 0.0

-~A~ 92 -!-. 36 39 39 2.2 2.2

A (1670)-+J~'Tr 43 • 22 30 30 0.3 0.3

A (1520)-+J~K 7.2-~ 0.9 2.1 2.3 32.6 29.9

--~Y.~ 6.7 • 0.9 11 10 18.2 16.6

A (1690)-+,N'K 17 ~ 12 92 79 39.6 26.7

-~Z~ 40 -I- 29 0.28 0.16 1.9 1.9

A (1830)-+J~K. 19 • 17 0 0 1.2 1.2

-~E~ 41 -~ 34 60 60 0.3 0.3

Y, ( 1 6 7 0 ) - - ~ K 3.7 • 1.6 0.002 2.4 5.3 0.6

- ~ 23 • 16 12 28 0.5 0.1

-~A~ 12 -t- 10 8.2 0.01 0.1 1.4

Z (1765)-~J~'K 52 -t- 12 47 47 0.2 0.2

-+Zr~ 14 • 14 8.3 8.3 0.2 0.2

->AT: 22 -i- 9 21 21 0.0 0.0

-~Z*r: 20 • 16 4.0 4.0 1.0 1.0

X ~ on par t ia l widths 111.7 90.7 Total g 8 166.9 137.5

D - w a v e decays . H a v i n g f o u n d a so lu t ion b y f i t t i ng the D - w a v e de c a ys , we

t h e n p r o c e e d e d t o f i t t h e masses a n d D - w a v e d e c a y s j o i n t l y . A g a i n we f o u n d

t w o so lu t ions . S o l u t i o n A is a n a l o g o u s to s o l u t i o n 1, t h a t is, i t a s sume s t h a t

t h e Y.(1670) is t ho lowes t mass 3 - Z - s t a t e . S o l u t i o n B is s i m i l a r l y a n a l o g o u s

to so lu t i on 2. T a b l e V I I c o m p a r e s t h e o b s e r v e d masses a n d D - w a v e d e c a y s

to t h e p r e d i c t i o n s of so lu t ions A a n d B. A l t h o u g h the e~Tors on p a r t i a l w id th s

a rc diff icul t to d e t e r m i n e , we h a v e p r o b a b l y o v e r e s t i m a t e d t h e m in an a t t e m p t

to inc lude s y s t e m a t i c u n c e r t a i n t i e s . Th is m a k e s t he h igh Z 2 for so lu t ions A

a n d B r a t h e r d i s t u r b i n g . Therefore , i t seems t h a t e i t he r t he mass f o r m u l a

does n o t c o r r e c t l y g ive t h e m i x i n g of t he s t a t e s or t h a t t h e S U6w • 02Lz sym-

m e t r y is v i o l a t e d in t h e decays .

SYM'IKETRIC QUARK MODEL ANALYSIS OF THE MASSES AND DECAY RATES ETC. 391

7. - Conelus ion .

The results of this analysis are encouraging because we have obtained a reasonable first approximation to a quantitative description of the masses and decay rates (except for S-wave decays) for all (70, 1-)-states. FArMAN and PLANE (7) have shown that it is possible to get a good fit to both S-wave and D-wave decays if the mixing parameters are left free. I t is conceivable, therefore, that some mass formula might yield correct predictions for both S-wave and D-wave decays. Important considerations for future analyses of masses and decay rates should be the incorporation of some mechanism to split the �89 and the ~-A 's and an at tempt to determine if S Usw • OzL ~ is violated in the decays. The discovery and confirmation of missing (70, 1-)-states in addition to the better determination of the resonance parameters of the well- established states would be of great help. Hopefully, these analyses can serve not only as a guide for resonance hunting but also as an indicator of which symmetry-breaking operators are required in the mass formula and as a test

of the S Usw • O2z ~ symmetry.

We are greatly indebted to Prof. O. W. GREE1NBERG for his suggestions and criticisms and for supplying us with the quantum numbers in Tables I and II.

APPENDIX

Calculation o f m a s s m a t r i x for �89 A-states .

The mass matrix for the three ~- A-states of the (70, 1-) is a symmetric matrix whose elements are given by

Mn----- <A~[M[A,S>,

<A;IMIAI> ,

M,, = <A:IJClZ:>,

M3,= M , s= <AIIMiAI>,

M . = <A:I tlA:>.

Matrix elements involving As' are easily evaluated since A~ is diagonal in both {SU,, ~qU.} and {SU,, 8U2a~} bases. Using the mass formula eq. (2.2),

399- M. JONES~ R. LEVI SETTI and T. LASINSKI

Tables I and II, and Table IX of reference (') yields

M~=No+Nx(O)+N. 6 + 2 ~ 2 + 1 +

+ N 3 [ ( 0 ) ( 0 + l ) - - l ( 0 ) t + ( 1 ) ( l + l ) - - ( ~ ) ( ~ + l ) ] +N,[~---3---~]+

1 + N 5 1 0 - - - 3 4 + ( 0 ) ( 0 + 1 ) - - ] ( 0 ) ' - - ( 1 ) ( 1 + 1 ) + ( ~ ) ( ~ + 1 ) ] +

+ N,, [6--2 (~)(~ + 1 ) - - : ] + 3", [5--2 (})(~ + 1 ) - -4 (0)'--8(0)--~2~ ] +

+ N s ( - - 1 0 ) + N , -- +N,o --~ =

5 N 5N, 5 = N o + ~ N . + ~ s -- 6N, -- 2N, -- 6N, -- 4N, -- ION, -- ~ , - -~N,o ,

4 4 M,8-------2N8-- S No + S Nx.,

M,, = _ 2 N. 1 3 -- ~ N1o.

The other matrix elements are somewhat more difficult to evaluate because A~ and AI axe not diagonal in {SU,, SU, sa}-basis. For SU, operators one must replace A~ and A8 z by the appropriate {SU,, ~U,n~} eigenstates.

For example

<A;]C:'IA]>= <~-2 (./i:0 + ,;I]) C~ I ~(./I;0 + 2I:)>=

= - ' (" ' " '" ' = { 9 + 5 } - - - - 7 1 {<A,oIC, IA,.> + <&It. IA.>} 1 2 2 '

<A;IS.(S . + I)[A;> = ~ {<./I;olS.(S. + l)l./i;o> + <./i;[S.(S. + 1)VI:>} =

= ~ {o(o + I ) + 1(1 + 1)} = 1,

<A~'IOTIAI>= < ~ (AIo + AI) c',~)l 1 ' ~__~ (-A~o + All> =

1 ~ <,) , I{__9+5}=__ 2 = ~ { - <&olC, i.,I,o> + <AllOi"l.,tl>} = ~

8YMKETRIC QUARK MODEL ANALYSIS OF THE MASSES AND DECAY RATES ETC, 3 9 3

Similar calculations for the other S U, operators yield

1

J----�89 3 ]

No -~ ~ N2 -~ ~ N3 - - 6 N . - - Ns - - 6N6 - - 2N7 - - 8 N 8 ,

\ \ / J = /

= N~[1-o] + ar~[- ~ + o] + ~ r , [ - 2 - 2 ( o 1 1 - ~ ar . -~ ~r,~ 4

= N3 - - N5 - - 2N7 - - ~ N~ - - g Nlo,

Disgonalization of the mass matr ix gives the eigenvalues (masses of the physical states) and the eigenveetors (physical states represented as vectors in the { S U3, S U2s} basis).

The calculation of the mass matrix for ~- A-states is the same except one needs to use the J ~ ~ matrix elements of the L-S-operators. ~ince the ~- A is unmixed, its mass is given by M~ but with the J: ~ matrix elements of the L-S-operators instead of the J-~ �89 elements.

�9 RIASSUNTO (*)

Si esogue una nuova analisi della rappresentazione (70, 1-) di S U o x O s impiegando il modello a quark simmetrico di Greenberg e Resnikoff ed i pih recenti dati sperimentaii. Gli undiei parametri della formula delle masse del modeUo si determinano mediante

(*) Traduzione a cura della Redazione.

~94 M. JONES, R. LEVI 8ETTI and T. LASINSKI

la min imizzaz ione di Z~, e si o t tengono previs ioni delle masse e delle miscele di t u t t i gl i st~tti (70, 1-). Facendo use di ques te previs ioni e suppoueudo ehe i deead imen t i siano inva r i an t i r i spe t to a S U s w • si ealcolano le ampiezze parzial i pe r t u t t i i deead imen t i degli s t a t i (70, 1-) in barioni nena rappresentaz ione (56, 0 § e quelle in mesoni pseudosc, alari nel la rappresentaz ione (35, 0-), e si fa i1 confronto con i va lor i sper imenta l i . Le previs ioni per i decad iment i in onda 8 r isul tano in g rave disaccordo con gli esporimont i , t u t t a v i a le previs ioni per i deead imen t i in onda D, con una sola eccezione, rappresen tano una ragionevole p r ima appross imazione, in medi~, ai va lor i osservat i . Non sarebbe t u t t a v i a possibile o t t ene re con t emporaneamen te un buon a d a t t a m e n t o per le masse e per i r appor t i di dcoad imento in onda D. Perci6, o la fo rmula deUe masse non fornisee e o r r e t t a m e n t e la miscele degli s t a t i oppure la sim- me t r i a d i ~ U~w • 0 ~ z ~ v io la ta nei deeadiment i .

Ana~m~ ~acc s c~copocre~ p a c n a ~ s naa (70, 1 "l) c o c r e - - - ~ : s c x o ~ s3 Cm~Merim, mo~

IO~lpKOBOl~ MO,~...,'IH.

Pe3mMe (*). - - Mht 3aHono aKanx3HpyeM n p e ~ c r a s a e m i e (70, 1-) a a s 8Us x 08, H c n o n ~ y a CHMMCTp~tHyIO KBapKonylo MO~enb VpnH6epra n PeaH~OBa ~ n o c n e ~ o I o 3Kcnepn~e~- TaHJbHylO H H ~ b o p M a ~ . C HOMOHIBIO Z 2 ~ M . u ~ m ~ one]Icn~OTC~ O mmm~a~IaTb napa- MeTpOB MO~eJII~HO~ MaCCOBO~ r ~eJIaIOTCR I I p e ~ c ~ l : r ~ q ~JI~I Mace H CMOIII~[ = BaHla~ BOeX (70, 1-) COCTOHHH~. HCROnJ~aya 3TH npe~cKa3am~ n npe~nonaraa , ~TO p a c n a ~ aBn~OTCa mmapHanTm,iMH OTHOCHTem, HO 8Uew| z, nbI~HCaa~TCa napima.u~ HI, xc nmpHH~ h a s vcvx pacnanos (70, 1-) COCTOs--~ aa 6apnom,I n (56, 0 § H Ha nccv~ocxaaspm, lc Me3om, i B (35, 0-). Floayqvmv, ir peaym,Ta~i cpammBa~oTcs c 3xcne- p ~ e H T a n 6 m ~ a nenHqmmM~. I Ipe~cKa .~m~ ~ n s 8-BOnuOnb~x pacna~on CymccTn~HHO pacxoRaTC~ c 3KCIIepHMeHTOM, He Hpc~cga3aHH~ ~Jlg D-BOJIHOBbIX pacna~Ion, 3a o ~ m ~ HCK~Ho~em~CM, Hp~cTaBJI~HOT pa3yMHOe nepsoe n p ~ n m ~ e ~ m e ~aa na6mo~aeM~iX ncrm- q][]:H. O~[~IaEo ~ He MOdeM IIOJIy~IHTB x o p o m ~ COOT][~TCTBHC ~JL~[ Macc H CKOpOCTCi~

])-BOYIHOBIdX pacnanoB O~HOBpeMeHHO. Cne~onaTCnbHO, nn6o MaccoBa~[ r He OHHCbIBaeT IIpaBHJII~HO CMettmmlHHe COCTO~IHH~, JI"H60 ~q Uew | O~z HapyllIaCTC~l B pacna~ax.

(*) llepeaebeno pedwcttuef~.

Documents

Symmetric quark model analysis of the masses and decay rates for (70, 1−)-states