IL NUOVO CIMENTO

V'OL. XXXVII, N. 3

1o Giugno 1965

Some Problems of Symmetry Breaking (*). A. B S ~ C) International Atomic Energy Agency International Centre ]or Theoretiea[ Physics . Trieste (ricevuto il 12 Dieembre 1964)

Summary. - - A possible explanation of the problem of equidistance rule for the masses of the baryon resonances posed recently by 0akes and Yang is given. The same method is also applied to explain the discrepancy between the Okubo-Sakurai-value of the r angle and the value calculated by Kim Oneda and Pati.

1. - I n spite of the g r e a t successes of the SUa-theory in its usual form, the notion of s y m m e t r y b r e a k i n g involves some essential difficulties as, e.g., pointed out b y OAKES a n d YA~G (1). Therefore it seems to be necessury to find an a l t e r n a t i v e a p p r o a c h to the usual p e r t u r b a t i o n - t h e o r e t i c a l t r e a t m e n t of the s y m m e t r y breaking, which simultaneously m i g h t give a deeper e x p l a n a t i o n for it. This m i g h t be achieved b y assuming the existence of a larger exact 9s y m m e t r y group ~r for the strongly i n t e r a c t i n g particles (SIP) (2-e), which con(') Supported in part by the Deutsche Forschungsgemeinsehaft and the International Atomic Energy Agency. (") On leave from the Institut fiir Theoretischo Physik Universit~t Marburg. (1) R. C. OAK]~S and C. N. YANC~: Phys. Rev. Lett., 11, 174 (1963). C. N. YANk: Proceedings of the Argonne User's Group. (2) a) A. O. BARUT: University of Colorado preprint (1963); b) Conference on Symmetry Principles at High Energy, (1964); c) Journ. Math. Phys., to be published; d) Phys. Rev., 135, B 839 (1964); e) Lorentz Group Symposium, (Boulder, Colo., 1964). (~) B. KURSUSOGLU: Phys. Rev., t35, B 761 (1964). (4) W. D. McGLINN: Phys. Rev. Zett., 12, 467 (1964); F. CO]~ST~R, M. HA~]~RM]~SH and W. D. M c G L I ~ : Phys. Bey., 135, B 451 (1964). (5) M. E. MAYER, H. J. SCH~ITZ~R, E. C. G. SUI)V.RSI~AN, R. ACHA~YA and M. Y. HA~: preprint (1964). (e) O. W. GRE]~Nm~G: Phys. Bey., 135, B 1447 (1964).

1072

A. B~HM

tains the intrinsic and space-time s y m m e t r y group. F r o m several theorems t h a t have been p r o v e d (4-7), one already knows some limitations on this group. F o r our purpose, to give an explanation for the validity of the equidistanee rule for b a r y o n resonances and for the discrepancy between the Okubo-Sakurai value (8) of the ~-r angle and the value calculated b y KI~, OSIEDA and PATI (9), a closer knowledge of this group ~ is not necessary. Rather, these considerations based on experimental facts could give some indications a b o u t the structure of such a group. 2. - The strongly interacting particle (SIP) should be associated with one or the smallest possible n u m b e r of irreducible representations of ~, which will be c h a r a c t e r i z e d - - a m o n g o t h e r s - - b y an eigenvalue of the second-order Casimir operator ~ of ~. This Casimir operator should contain the mass operator ~g/---- P~ P z and the (exact) mass formula will be given b y the form of ~ (*). As we want to give some explanations of difficulties connected with the S U3t h e o r y we assume t h a t SUB/Z3 is contained in ~ as the intrinsic s y m m e t r y group (or as a subgroup of it). I n particular we assume t h a t ~ has the prop e r t y t h a t the b a r y o n resonances A~ (1238), Y* (1385), 7Z* (1530) and ~ - will be associated with an irreducible representation space ~B~ of ~ and the vectormeson resonances (p (1019), r (782), p (750), K* (888) with a representation space ~v~. The group (g is certainly only a v e r y crude a p p r o x i m a t i o n to the final group, which contains the correct intrinsic s y m m e t r y group, and to the irreducible representation spaces of which belong a larger n u m b e r of particles. The mass-formula given b y the Casimir operator of this ~ - - f o r which we take as empirically given the Gelld~Iann-Okubo f o r m u l a - - w i l l therefore be only a mutilation of the final one (lO). 3. We can construct a basis in the irreducible representation space according to various complete sets of commuting observables (CSCO). Especially we m a y choose a basis in which S Ua is diagonal, the CSCO of which will be C~, C~, 12, HI, H~, A~, ..., A., where A~, ..., A~ contain elements of the enveloping algebra of ~, which are not elements of the enveloping algebra

(~) L. MICHEL: preprinr (s) j . j . S ~ U R ~ : Phys. Rev., 132, 343 (1963); S. 0KU~O: Phys. Lett., 5, 165 (1963); S. COLW~, S. L. GLASHOWand D. KL~IT~A~: preprinL (9):y. S. KI~, S. ON~DA and J. C. PATI: Phys. _Rev., 135, B 1076 (1964). (')'Such (~dynamical groups, (2) for simple models are under investigation (A. O. B~mUT and A. BS~M: t,o be published.) (10) A more detailed explanation can be found in A. BSu~: University of Marburg report (August 1964).

SOME I'ROBL:~MS OF SYMMETRY BREAKING

1073

of SUa and complete the set Cz, C~, F', Hx, Ha (*) of commuting observables; and a second basis, in which ~ is diagonal (**), the CSCO of which we denote b y ~Z/, B~, ..., B,~ (j~ and j~ might be among B~, ..., B,,,). Let Ibm, ..., b,n, m> denote the eigenvectors of the set ~.$', B~, .., B~ with the eigenvalues m, bx, ..., b~ respectively and ]a~, ..., a~, h~, ha, I , (2~, Xa)> the eigcnvectors of the set (7,2, C~, I ~, H~, H.,, A~, ..., A~, where (X~, 2~) denotes the eigenvalue of (71, and Ca (***). Then we have:

Ibm,..., b,~, m} = ~ [ a ~ , ..., a . , h~, ha, I , (2~, 2a)>" "<(~, 2a), I, he, h~, a,,, ..., a~]b~, ..., b ~ nt> and vice versa:

[a~, ..., a,,, h~, hz, I , ()~, ~)> = = ~ Ibm, ..., b~, m>, where the sum (respectively the integral for the continuous spectrum) is ext e n d e d over all state vectors of the irreducible representation space ~ of (if one of the B~, ..., B~ should be among C2, Cx, I a, Hx, H~, Ax, ..., A~ the dependence of the transition coefficients <(2~,2a), I, ha, h~, an, ..., a~ ]b~, ..., b,~, m> on its eigenvalues is only through a ~-funetion of a K r o n e c k e r ~.)

4. - In this language the question of OAKES and YANG (~) why the masses of the b a r y o n resonances obey the equidistanee rule, might be f o r m u l a t e d in the following way: as the masses are supposed to be the eigenvalues m of the mass operator (.%) dr', we have

ra :

~m b,,,.bl [~g[ Ibl, ..., bm m> =

= ~: ~: <(~, ~ ) , I~, h~, h~, ~., ..., ~, l" ~m,...,~><(~'~,

Z:), r , h~, ' h~, ' ~,,' ..., ~'~ [~1 ~ , . . . , ~,,, h~, ha, I, (a~, ~a)>,

(') C 2, C 1 denote the two Casimir operators of ,~U3, I ttle isospin, H 1 and lf~ are related with isospin component and hypereharge. ('*) As [dt', C1] # 0 Jr' will not be among A 1..... A,,. 1 ['4 ~ 1~21 ~- ~,2 -[- 21~2) + 4(~1+ 22)] ('.') Eigonvaluo of C1 = i-~-L~ Eigenvaluo of C~ = ~ ( ~ - - 22) [w(2~ + 2~)a+ ~ 2~2~ + 2~ + ~ + 1]. (~ For baryons one conventionally uses dr'= (P~Pg)89instead of PgPg; as long as P~Pg is positive the square root is well defined [see e.g. ref. (~) AI Sect. 7]. But one could equally well use a quadratical mass-formula for the baryons IS. e.g. BARUT (2) and OKUBO, R Y A N (12)].

1074

A. BOHra

where the sum extends over all laa, ..., a., ha, h2, I, (Jta, 2~)~E~Ba. Of all the m a t r i x elements of the mass operator in the above sum only those obey the equidistanco rule for which (~a, Jt2)=(2~a, 2~)----(3n, 0) or (0, 3n), n integer (triangle representations). B u t as S U3 s y m m e t r y is strongly broken the contributions f r o m the other terms in the sum are essential, so t h a t the whole sum should not obey the equidistanee rule. W h y do the experimental masses in spite of this obey the equidistanee rule? To answer this question, we want to look a little closer at the measuring process of the resonance masses. The systems in which one measures the mass h a v e undergone previously certain preparations and are in a definite state (*) described b y the statistical operator (density operator) W (~). The mass is t h e n given b y ra~ ----Trace ( J / W ) . Thus the entire problem lies in the d e t e r m i n a t i o n of W. This would require the knowledge of all relevant conditions in the preparation of the state, which in t u r n would require the knowledge of the physical law, which we do not know. E x p e r i m e n t a l facts however suggest the conjecture, t h a t the law is such, t h a t it is possible to prepare the systems at which the resonance masses are measured to have definite values of isospin, hypercharge, charge an4 perhaps also some other q u a n t u m numbers. This s t a t e m e n t will be best explained b y an example. L e t us take the q~-meson: 5.

-

K-+p-+A+q

I---,K + ~ . Certain m a n i p u l a t i o n s - - c o n s t i t u t i n g a part of the prepuration:--bring K and p to interaction. The physical law is such t h a t a I = 0, 1r - - 0, K K - s t a t e can be sorted out; and the mass t h a t is determined is t h a t of a I----0, :Y----0 state. Taking only this information into consideration, we have for the statistical operator: W=P~_o.r~o, where P~=0.r=o is the projection operator on a subspace ~ o . r~o of the irreducible representation space ~2w, to which the vector-mesons belong. 6. W i t h this consideration we can explain the equidistance rule for the b a r y o n resonance masses. The systems, at which the resonance masses are measured~ have definite isospin I, hypercharge h,, and charge, so t h a t the statistical operator is given b y W-~ Pnlh,1, where Phil, ~ is the projection oper a t o r on the subspace rn&1 of ~Ba, in which H1H~, I S have the value ha, h~, / ( I + 1 ) respectively. The nonexistence of b a r y o n resonances with the same q u a n t u m numbers suggests t h a t rn,h. x is one-dimensional. (If ~Ba would contain

(') If W2=W, a s~ate is called a pure s~a~e; if W2< W, it is called a ml,r~ure. (11) G. LUDWlO: Grundlagen der Quantenmeehanik (1954).

SOME

PROBLEMS

OF

SYMMETRY

1075

BREAKING

several states with the same h~, h2, I these would differ b y at least one eigenvalue of a n o t h e r operator, say A1, and in the whole consideration we would hawe to replace hi, h2, I by hl, h2, I, ~1 and instead of just the decuplet b a r y o n resonances we should take into consideration a larger class of S I P ; b u t the essential arguments would remain the same.) The experimentally measured mass of the b a r y o n resonances is therefore m = Tr ( d / P h , h , , )

Q

Calculating the trace we find

m ---- X

a;a: v . . .

? ! ! t <~'1,]ta)hlhzI ...Ixr~,,,II...I',

r ? h~, h~, (~,!

Z~)>,

(~iape~tBa

'l~ =

~

<(at1, .~12), hl, h2, i, ... ].~r ]... I hi, h29 (a:, ~t:)> ,

and with the empirical assumption that rh,~ is one-dimensional and belongs to the (3, O)-representation of SU3: m = ((3, 0)

hlh~, I...I,~l...I, h2, hi, (3,0)>.

This obeys in contrast to m---- the equidistaneo rule. 7. - This hypothesis, t h a t the experimentally measured masses are not the eigenvalues of the mass-operator, is in disagreement with the well-known ~-o)mixing t h e o r y of Ol~uno and SAKUlCAI (s), who calculate the (~mass ~) of the u n i t a r y singlet and the ~0-~o-mixing angle 0 from the assumption t h a t the experimental ~- and ~o-masses are the eigenvalues of the mass-operator. I n the f r a m e w o r k of their t h e o r y one has as m a n y equations as unknown quantities, so t h a t a check of this assumption is not possible. A calculation of the mixing angle from other independent considerations would therefore be a crucial test of this assumption. Such a calculation was done b y KIWI, O~EDA and PATI (9) (*),. who d e t e r m i n e d 0 from decay rates and obtained a value m u c h smaller t h a n the Okubo-Sakurai-value (0 ~ 2 0 if we assume P(q~-+f~~ I n the following, a possible solution of this discrepancy is given on the basis of our previous assumption, which does not disagree with the as y e t known experim e n t a l data, b u t which again cannot be checked. E x p e r i m e n t a l data (~4), in (') We would like to thank Prof. Y. S. KIM for correspondence about this point. (1~) S. 0KVBO and C. RYAN: University of Rochester preprint (1964). (la) j . M. JAUCH: preprint CERN 7664 (1964). (14) V. G. GRISHIN: Dubna.-preprint (1964); A. BERTHELOT: Sienna Conference Report (1963).

1076

~. BOn~

pa~'ticulax the small value of a(~p --> v=I)? and

a(~p -~ ::pco [_+ ~:+.-~o)

P ( ? -+ ~::) P ( ? -+ K K ) ~ 0.1 4- 0.1

suggest t h a t co and ?, i.e. the (r:+r~-=~ and the (KK)-system differ by an as y e t unknown q u a n t u m number B (~5) which does not belong to the set C~, C1, I ~, H~, //2, A~, . . . A . : B [ K K > --= blKK>,

BI~+~-=o> = bo l~vt-v:-=o>.

We make the simplest possible assumption t h a t B is commensurable with H~, H~, 12 but [ B , C ] # 0 . Then we have

(P~ is the projection operator on a subspace rb of ~ w in which B has the value b). The measurement of the co-mass is done at (r:+~:-=~ with hi = 0, h~ = 0, I = 0 and the measurement of the ?-mass is done at (KK)-systems with the same h~, h2, I. Therefore we have =

.o.o

and the measured masses axe given by = Tr

P,,),

r %2 _-- T r (

'Po.o.o

As experimentally two vector-mesons with hi = h~ = I = 0 are known, we assume t h a t va,-o.a,~o.l-o is two-dimensional (not taking into account the possible new resonances which could be included in the description by taking into consideration a new q u a n t u m number; see remark in Sect. 6). Two possible bases in ro.o.o are given by the eigenvectors of C: ]...I-~O, hl=O,h~----O, (0,0)>, ]...(1,1)> and B: I . . . I = 0 , h ~ = 0 , h ~ = 0 , b~>, ]...b,o> which are connected by [... (0, 0) =

cos 0 [... b~> + sin 01... b~>,

[... (1, 1) = -- sin 0 [... b~> -}- cos 0 l... bv>. (is) This quantum number might be identical with the A-quantum number of BRONZAZ~ and Low: Phys. Rev. Lett., 12, 522 (1964), or wi~h the sa, of LIPKIN (16). (~r LIrKIZ~: Phys. Rev. Lett., t8, 590 (1964).

SOME

PROBLEMS

OF S Y M M E T R Y

BREAKING

1077

T h e n we h a v e ~

]... (0, 0)> ----

cosOl...b~>=eos~OI...(O0)>

- - cos 0 sin 0 ] ... ( l l ) > ,

Pb~ l--. (1, 1)> = - - sin 0 I... b~> = sin ~ 0 I..- (1,1)> - - cos 0 sin 0 [... (0, 0 ) > , and

sinO[...b~>=cosOsinO[...(1,1)>+sin~OI...(O,O)>,

P ~ I . . . (0, 0)> =

(1,1)> ---- cosO[...b~>----cosOsinOl...(O,O)>+cos201...(1,1)>.

~1...

N o w we c:m c a l c u l a t e t h e t r a c e a n d find

m~ =

I

I

I

I

!

!

2 <(;t,,X.Dh,h~I...].A'Po.o.oP~o,[...h~, ]1/2,1, ('~l, hDLL.. <(&,~),

=

Xf2)>,

o, o . . .

{~w~De~vM ----- <(0, 0), 0, 0, 0 ... ] ~ [...0, 0, 0(0, 0)> cos20 + <(1, 1) ... I ~ 1... (~, ~)> sin~0 --

<(0, 0) ... [ ~ [... (1, 1)) cos 0 sin0

--

<(1, 1) ... I~( 1... (0, 0)> cos 0 sin 0,

or if we use t h e s e l f - e x p l a n a t o r y n o t a t i o n , we h a v e m ~ ---- cos 2 0 m ~ A- sin s 0 mss~ - - sin

20m~s.

S i m i l a r l y we o b t a i n l l t ~2 _ = s i n 2 0

" 'm l~, -~- c o s ~ Om~ss_sln2Om~ls.

I f we use for cos0.m~s t h e wflue c a l c u l a t e d b y t h e Gell-lV[ann-Okubo m a s s - f o r m u l a (according to this c o n s i d e r a t i o n we should p e r h a p s s u p p o s e - - i f we a s s u m e b~ = bp ~-- bK. (~5) __ t h a t

m*p = cos*0"<0, 1), I = :,, ~ = O l ~ l X = 1, ~ = o, (1, 1)> a n d

2

m ~ . = c o s ~ 0 . < ( 1 , 1), I = 5, Y = 1 I ~ l X

= 5, Y = 1, (1, 1)>)

a n d for 0 t h e v a l u e of KI~ et. al. for P ( 9 --> po A- =o) _ 0, w h i c h is t h e case for e x a c t B - i n v a r i a n e e : 0----20 ~ we h a v e t w o e q u a t i o n s for t h e t w o v a l u e s m n a n d m~s. I f we could t h i n k of a n e x p e r i m e n t t h a t sorts out e i g e n s t a t e s of C

1078

A. B O H M

instead of eigenstates of B we could d e t e r m i n e mii i n d e p e n d e n t l y a n d check with the above-calculated value. B u t if n a t u r e is such t h a t such a p r e p a r a t i o n is not possible~ mll is unphysical a n d this test impossible.

The a u t h o r would like to express his g r a t i t u d e to Profs. and Drs. A. O. :BARUT, H . D. DOEB1NEtt a n d G. LUDWIG for valuable c o m m e n t s on this work. This investigation was b e g u n while the a u t h o r was at the I n s t i t u t ffir Theroretische P h y s i k U n i v e r s i t ~ t ~ a r b u r g ; he wishes to express bin sincere t h a n k s to Prof. G. LUDWIG for the hospitality. The a u t h o r is especially i n d e b t e d to Prof. A. O. BAttUT for his help in bringing this work to its final f o r m a n d for m a n y helpful discussions related to the subject. The a u t h o r is grateful to Prof. A. SALAM and the I A E A for the h o s p i t a l i t y extended to h i m at the I n t e r n a t i o n a l Centre fos Theoretical Physics, Trieste.

RIASSUNT0

(*)

Si avanza una possibile spiegazione del problema della regola dell'equidistanza per le masse delle risonanze barioniche posto di rocente da Oakes e Yang. Si applica inoltre lo stesso metodo per spiegare la discrepanza f r a i l valore di Okubo-Sakur~i per l'angolo di mescolanza ~-r e i l valore calcolato da Kim, Oneda e Pati. (*) T r a d u z i o n e

a cura della Redazione.