VOLUME 14, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 1 MARCH 1965

dieted to be independent of the Treiman-Yang angle (p^, and to vary as cos2^^ as a function of the im scattering angle. The data show a much more complicated depen­dence on (pff and cosfl^.

10The value T€ = 140 MeV corresponds to the full width of the resonance. Since the only decay modes of the e° which are expected to be significant for present purposes are e°-*7r++7r"~ and e0—27r0, with a 2:1 branch­ing ratio, the effective e07r"V~ coupling constant used in the calculation corresponds to the partial width for the 7r+7r"" decay mode, r e + _ = 93 MeV.

11 The lack of exact agreement between the present theory and experiment is hardly surprising consider­ing the uncertainties in the low partial waves in the ir~~ +p~~p° + n and ir~~ +p—*€° + n amplitudes. We have also neglected the small real T=0, J = 2+ amplitude ex­pected from the tail of the / ° resonance. Although this will have very little effect on the cross section it can change the 7r+7r~ decay distribution significantly, par­ticularly the forward-to-backward ratio. On the other hand, it cannot, by itself, explain the data. It is inter­esting also to note that the absorptive effects reduce the forward-to-backward asymmetry which would be predicted by the unmodified single-pion-exchange mod­el. The odd term in cosO^ in the angular distribution results from the interference between the €° and p° contributions to the state of the di-pion system with Jz = 0 (the z axis is taken along the direction of motion of the incoming pion as seen in the rest system of the di-pion). In the unmodified theory, the fact that the incoming pion is spinless implies that the p° is always produced in this state. This need not be the case in the modified theory because of the initial- and final-state interactions. For example, at 3 BeV/c for — t < lOra^2, the modified theory predicts that the p± and p° spins are 60% in the state with Jz = 0, and 40% in the state with Jz =±1. As we have seen, this predic­

tion yields a good fit to the p""-decay angular distribu­tion, and we expect it to be essentially correct for the p° as well. The amplitude of the odd term in cosfl^ is thus smaller by roughly 0.78 = (0.6)1/2 than is the case for the unmodified model, and the forward-to-back­ward asymmetry is reduced accordingly. On this ba­sis, it seems unlikely that one can account for the asymmetry in the modified theory using a constant J = 0 phase shift of ~60°, as was proposed in the first paper of reference 4.

12R. W. Birge, R. P. Ely, J r . , T. Schumann, Z. G. T. Guiragossian, and M. N. Whitehead, Proceed­ings of the International Conference on High Energy Physics, Dubna, 1964 (to be published). The authors are greatly indebted to Dr. Guiragossian and Dr. Birge for supplying their data prior to publication, and for several comments thereon.

13A missing-mass spectrum for the reaction IT +d -—p +p +MM has been reported by N. Gelfand, G. Ltit-jens, M. Nussbaum, J. Steinberger, H. O. Conn, W. M. Buss, and G. T. Condo, Phys. Rev. Letters 12., 568 (1964). The e° would appear in this experiment as a bump with a width of ~140 MeV containing ~25 events. The situation is confused by the strong con­tributions to the missing-mass spectrum from the neutral decay modes of the a>° and r}° mesons. Al­though these contributions could, in principle, be r e ­moved by measuring the w° and 17° components of the reaction TT+ +d-~p +p +7r +7r~ +7r°, and using the known branching ratios for the neutral and charged decay modes, this subtraction was apparently not attempted. The results are thus inconclusive for present pur­poses. We note only that the neutral background in the neighborhood of the p° or CJ° mass appears to be rather high, and that the apparent C*J° peak is shifted to­ward the high-mass region.,

INTERNAL SYMMETRY AND LORENTZ INVARIANCE

L. O'Raifeartaigh*! Department of Physics, Syracuse University, Syracuse, New York

(Received 27 January 1965)

Recently a number of papers1 have appeared in which the possibility of combining internal symmetry and Lorentz invariance is discussed. Most of these papers have discussed the pos­sibility within the framework of some specific model, or under some specific assumptions. It may be of interest, therefore, to consider the question from a general point of view and to try to determine the most general conditions under which a combination is possible. The purpose of the present note is to consider this problem. The point of departure for our dis­cussion is the general result2 that any Lie al­gebra E can be expressed as the semidirect sum of a semisimple subalgebra G (the Levi

factor) and an invariant solvable subalgebra S (the radical), i.e.,

E = G®S. (1)

Using this result, we establish the following theorem:

Theorem. - Let E be any Lie algebra with Levi factor G and radical S. Let L be the Lie algebra of the inhomogeneous Lorentz group, consisting of the homogeneous part M and trans­lation part P . If L is a subalgebra of E, then either

(a) Mc.G:P<zS, (2)

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or

(b) L 0 S = 0, (3)

where c denotes " is a subalgebra of" and 0 denotes in te rsec t ion .

P r o o f . - B y the M a r c e v - H a r i s h - C h a n d r a the ­orem, 2 and the semis impl ic i ty of M, we can define G so that, in any case ,

MCG. (4)

On the other hand, from the Lorentz re la t ion

[M,P] = P, (5)

and the invariance of S, we have

[M,pns] = pns. (6)

Thus, with r e spec t to M, Pf)S is an invar ­iant subspace of the space P . But P is i r r e ­ducible with r e spec t to M. Hence

P n S = P o r P O S = 0. (7)

This es tab l i shes the theo rem. The re levance of this theorem is that it a l ­

lows us to classify the ways in which L can be imbedded as a subalgebra of a l a rge r Lie a l ­gebra E. To make this classif icat ion, it is convenient to subdivide case (a) above into the th ree c a s e s (i) S = P, (ii) S Abelian, but l a rge r than and containing P, (iii) S solvable, but not Abelian, and containing P . If we add to this the other possibi l i ty, (b) above, or (iv) S O P = 0, we see that we have, in al l , four c l a s s e s . P rac t i ca l ly all of the proposa ls made so far for combining L and internal s y m m e t r y belong to c l a s s e s (i) and (ii). All four c l a s s e s will be d i scussed in more detai l in a forthcoming paper . Here we shal l mere ly s u m m a r i z e some of the more outstanding r e s u l t s .

In case (i), the re la t ion

[G,S] = S (8)

impl ies that S is a represen ta t ion space for G. But s ince the space S = P is four d imension­al , and G contains M, it can be shown that G can then be wri t ten in the form

G=Gn®G , 0 r

(9)

where © h e r e denotes d i r ec t sum; G0, the com­plex extension of G0, is i somorphic to A3, B2, or AX®AX (in the Car tan notation); the t r a n s ­format ions

[G0,P] = P (10)

contain the Lorentz t rans format ions (5); and Gr (r = r emainder ) is a semis imple a lgebra satisfying

ior,P]- 0. (11)

On account of the low o rde r of G0, which a l ­ready contains the Lorentz t r ans fo rmat ions , any nontr ivial i n t e r n a l - s y m m e t r y a lgebra must be contained in G r , which commutes with G0

and P . In this s ense , case (i) i s t r i v i a l . This is one of the r ea sons for the negative r e s u l t s obtained e a r l i e r by other au tho r s .

Case (ii) is not t r iv ia l in the s a m e sense , s ince in this case the dimensionali ty of S is not r e s t r i c t e d , but it has the disadvantage of introducing a t rans la t ion group of m ^ r e than four d imens ions . This is not easy to in te rp re t physical ly . Case (iii) appea r s to be r a the r un-physical , as solvable a lgebras a r e not usually cons idered by phys ic i s t s . This is because , for solvable a lgeb ras , Hermi t ian conjugation cannot be defined for all e l ements , at l eas t for f ini te-dimensional r e p r e s e n t a t i o n s . Case (iv) is poss ib le , although it involves imbedding an a lgebra i somorphic to L in a s imple a lge ­b r a . However, the s imple a lgebra is n e c e s ­sa r i ly ex t remely noncompact, which may lead to difficulties in defining mul t ip le t s .

With r e g a r d to the question of m a s s split t ing, which is one of the r e a s o n s for at tempting to imbed L in a l a r g e r Lie a lgebra E, we can make the following s ta tement : In c a s e s (i) and (ii) of our classif icat ion, t he re can be no m a s s spli t t ing. More specifically, we es tabl i sh the following theorem:

T h e o r e m . - L e t R be the vec tor space of any i r reduc ib le r ep resen ta t ion of E, and Rm any subspace of R which is an e igenspace of the m a s s opera to r p^P^ belonging to the d i s c r e t e eigenvalue m 2 . Then in ca se s (i) and (ii) above, the space Rm i s invariant with r e s p e c t to E. Hence, R =Rm, and t he r e i s no m a s s spl i t t ing.

P r o o f . - L e t \Rm) be any vector in Rm, E any element of E, and consider the vector

\ip) = E\R ) . (12) m

Since

(/> / > M - m 2 ) | f l w > = 0, (13)

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V O L U M E 14, N U M B E R 9 P H Y S I C A L R E V I E W L E T T E R S 1 MARCH 1965

we have

{p p^-m2)2\4>)

= [p^,[p/,E}}\Rm)

= [ y M , / > a [ / , E]+ [pa, E]pa}\Rm). (14)

Using the invariance of S and the fact that for cases (i) and (ii) S contains P, we have

{ p p i l - m 2 ) 2 \ i P ) = [ t > P t J - , p S 0 + S p 0 ] \ R ) , ( 15 )

JU LX o o m

where SQ is some element of S. But every ele­ment in the commutator on the right-hand side of (15) is contained in S, which in cases (i) and (ii) is Abelian. Hence, this commutator is zero. Thus

(/> />M-m2)2|^> = 0. (16)

Multiplying this equation to the left by (ip I, we have, using the Hermiticity of p^,

(<p\<p) = 0, (17)

where

\<p) = (p p^-m2)\ip). (18)

Hence, from the positive definiteness of the matrix, we have

(p /> M -w 2 ) l$ = 0, (19)

whence

\ip)=E\R ) c R . (20) m m

Since R is irreducible, this implies that R

To explain the decay KT°-~ 27r observed recent­ly,1 a very attractive possibility has been sug­gested by Bell and Perring, and by Bernstein, Cabibbo, and Lee2: The apparent CP violation may be a consequence of the CP asymmetry of the state of our world. Starting with the CP-

= Rm; hence, p^p^ is equal t o w 2 throughout R, and there is no mass splitting.

In conclusion, I should like to express my deepest thanks to Dr. E. C. G. Sudarshan and Dr. M. Y. Han for very many stimulating and fruitful discussions. I am further indebted to the high-energy group at the Argonne Nation­al Laboratory for some very stimulating and helpful comments.

*Supported in pa r t by the U. S. Atomic Energy Com­miss ion .

fOn leave of absence from Dublin Inst i tute for Ad­vanced Studies, Dublin, I re land.

1 F . Lurca t and L. Michel , Nuovo Cimento 21 , 574 (1961); Cora l Gables Conference on Symmetry P r i n ­c ip les at High E n e r g i e s , Univers i ty of Miami , 1964, edited by B. Kur§unoglu and A. P e r l m u t t e r (W. H. F r e e m a n & Co. , San F r a n c i s c o , 1964); A. O. Barut , Nuovo Cimento 32, 234(1964); B. Kursunoglu, Phys . Rev. 135, B761 (1964); W. D. McGlinn, Phys . Rev. L e t t e r s 2 1 , 467 (1964); F . Coes t e r , M. H a m m e r m e s h , and W. D. McGlinn, Phys . Rev. 135., B451 (1964); M. E . Mayer , H. J . Schni tzer , E . C. G. Sudarshan, R. A c h a r -ya, and M. Y. Han, Phys . Rev. 136, B888 (1964); A. Beskow and U. Ottoson, Nuovo Cimento 34, 248 (1964); L. Michel , Phys . Rev. 137, B405 (1965); E. C. G. Sudarshan ,"Concern ing Space-Time Symme­t r y Groups and Charge Conservat ion" (to be published); U. Ottoson, A. Kihlberg, and J . Nilsson, Phys . Rev. 137, B658 (1965); M. Y. Han, Phys . Rev. (to be pub­l i shed) ; L. Michel and B. Sakita, "Group of Invariance of a Rela t iv is t ic Supermult iplet Theory" (unpublished); S. Okubo and R. E. Marshak , Phys . Rev. Le t t e r s 14, 156 (1965); R. Delbourgo, A. Salam, and J . Stra thdee, "U(12) and the Baryon F o r m F a c t o r " (to be published); C. W. Gard iner , Phys . L e t t e r s JUL, 258 (1964).

2N. Jacobson, Lie Algebras ( Interscience Pub l i she r s , Inc. , New York, 1962), p . 24-25 .

3The definitions of a s emis imp le Lie a lgebra as a Lie a lgebra which does not contain an Abelian invariant subalgebra o r a Lie a lgebra which does not contain a solvable invariant subalgebra a r e equivalent.2

symmetric interaction

H. =igcp (K°d K°-K°d K°), (1)

and supposing that in the local vacuum state

(01 <p | 0) # 0, (01 <p. I 0) = 0 for 3 = 1, 2, 3, (2)

DECAY KL°~ 2?r AND SPONTANEOUS BREAKDOWN OF THE CP SYMMETRY

G. Marx* Institute of Theore t ica l Phys ics , Depar tment of Phys i c s , Stanford Univers i ty , Stanford, California

(Received 28 Janua ry 1965)

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