4 - S C A L I N G , L I G H T - C O N E E X P A N S I O N , . . . 1239

*Supported in part by the National Science Foundation. i ~ . D. Bjorken, Phys. Rev. 2, 1547 (1969). 'K. G. Wilson, Phys. Rev. 179, 1499 (1969). 3 ~ . Frishman, Phys. Rev. Letters 2, 966 (1971). 4 ~ . A. Brandt and G. Preparata, Nucl. Phys. z 7 , 568

(1971). 5 ~ . Gell-Mann, talks given at the Second Conference

on Fundamental Interactions at High Energy, Coral Ga- bles, Fla ., 1971 (unpublished), and International Symposi- um on Symmetry and Duality in Hadron Physics, Tel-Aviv,

Israel, 1971 (unpublished); M. Gell-Mann and H. Fritzsch, CalTech report (unpublished).

6 ~ . Abarbanel, M. Goldberger, and S. Treiman, Phys. Rev. Letters 22, 500 (1969).

'Report by R. E. Taylor to the Fifteenth International Conference on High Energy Physics, Kiev, USSR, 1970 (unpublished).

'L. Van Hove, Phys. Letters X B , 183 (1967); L. Du- rand 111, Phys. Rev. x, 1610 (1967).

PHYSICAL REVIEW D VOLUME 4 , NUMBER 4 1 5 AUGUST 1 9 7 1

Second-Class Currents, Neutral Currents, and the Strangeness-Conserving Nonleptonic Weak Interactions*

P. Herczeg Department of Physics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213

(Received 26 April 1971)

The isospin structure of the strangeness-conserving nonleptonic weak Hamiltonian con- structed from products of octets of vector and axial-vector currents is investigated in the presence of charged and neutral currents of both regular and irregular behavior under charge conjugation. It is shown that in a CP-invariant theory an isospin-invariant Hamilton- ian is possible only in the absence of charged second-class currents.

Although the description of weak interact ions is fair ly successful in t e r m s of the p resen t theory, t h e r e a r e ce r ta in deficiencies, along with some other open problems, in our knowledge of the be- havior of the hadronic cur ren t with respec t to the symmetry t ransformations of s t rong interact ions. While the t ransformation proper t i es with respec t t o the Lorentz group, parity, SU(3), s t rangeness , and isospin a r e now r a t h e r well established, the question of whether c u r r e n t s with "irregular" be- havior under charge conjugation1 ex is t h a s not been yet a n s ~ e r e d . ~ Similarly, not much is known about the possibility of neutral c u r r e n t s par t ic i- pating in the weak interactions. ' In the c a s e of charge-conjugation t ransformation propert ies , th i s lack of information is due t o the circumstance that in the experimentally bes t accessible semi- leptonic react ions the contributions of i r r e g u l a r cur ren ts , even if permit ted by selection rules , is kinematically s ~ p p r e s s e d . ~ The s e a r c h for possi- b le neutral hadronic c u r r e n t s is even m o r e diffi- cult in view of the g rea t var iety of ways i n which such cur ren ts can be incorporated into the theory. Certain types of neutral c u r r e n t s a r e already s e - verely constrained by the available data; o thers a r e s t i l l quite unrestr ic ted by them .3 The la t t e r is t r u e in part icular fo r models where the neutral hadronic c u r r e n t s a r e not accompanied by leptonic neutral cur ren ts o r in which the neutral leptonic c u r r e n t s a r e decoupled from the hadronic ones.5

In such a c a s e the nonleptonic p r o c e s s e s constitute the only source of information.

The nonleptonic weak interactions, kinematical limitations being h e r e absent, could a l s o be of i m - portance for the study of the question of the exis- tence of i r regu la r currents. ' T o t r a c e the conse- quences of the p resence of a part icular type of c u r - ren t in the nonleptonic weak interact ions is, of course, an extremely difficult, if not impossible task. The exceptional situation is the one in which the presence of such c u r r e n t leads to consequences that a r e independent of the dynamics of s t rong in- teract ions. The strangeness-changing nonleptonic weak p r o c e s s e s a r e not par t icular ly useful in th i s respect : The empir ical ly well obeyed A T = sum rules , f o r example, can be sat isf ied with any Ham- iltonian, by invoking octet dominance if necessary . However, in the A S = 0 nonleptonic weak interac- tions7 the isospin s t ruc ture is r icher , in the c u r - ren t x c u r r e n t theory A T = 0, l, 2, i n general, and even if the octet par t of the Hamiltonian dominates, the various models can s t i l l differ in the content of the corresponding i sosca la r and isovector par t s . An example is provided by the effects of charged second-class c u r r e n t s introduced into the s tandard CP-invariant cur ren t X c u r r e n t Hamiltonian: In the absence of second-class c u r r e n t s the effective coupling constant of the isovector interaction is G sin2B ( 0 ~ C a b i b b o angle); interference between f i r s t - and second-class c u r r e n t s introduces an ad-

1240 P . H E R C Z E G

ditional isovector te rm which i s proportional t o G C O S ~ O . ~ This could provide a way of testing the existence of second-class currents . Such a test i s not unambiguous, however, since certain possible theories that include neutral hadronic currents also contain an isovector te rm of the same order of magnitude: In the absence of second-class cur- rents, an isovector interaction with an effective coupling constant of the order of Gcos2Q is evi- dence for the existence of neutral currents.' The converse of this statement does not hold: There a r e possible theories involving neutral currents which do not contain a large isovector interaction. As shown in a recent study by Albright and Oakes1° of the isospin structure of a general nonleptonic weak Hamiltonian constructed from products of regular currents, a suitable choice of neutral cur- ren ts can eliminate the isovector part altogether. At the same time it is possible to modify the Ham- iltonian further so a s to cancel also the isotensor part, obtaining thus an isospin invariant AS = 0 non- leptonic weak Hamiltonian."

In the present note we investigate the isospin structure of the strangeness-conserving nonlep- tonic weak Hamiltonian under more general c i r - cumstances, namely, when both regular and i r reg- ular, charged a s well a s neutral vector and axial- vector octet currents participate.

Let u s assume that the existing differences in the f t values of mi r ro r ,L3 decays2 a r e not explica- ble in t e rms of isospin-conservation-violating ef- fects . The strangeness-conserving hadronic cur- rent must be then of the form12

where a i s a complex number with a nonvanishing r ea l part and JT1,, J:,, satisfy13

More generally, we introduce a regular and an ir- regular octet1 JfE,= v ~ E , -A$,) and J&,= vf,,-Afu, satisfying

and take the A&= 1 hadronic weak current to be14

where

It is plausible to assume that the nonleptonic Ham- iltonian in this case consists of a current x cur- rent interaction built up from JAQ'l, to which we add a te rm constructed from the neutral compo- nents of the regular and irregular octets:

Hermiticity, CPT invariance, and C P invariance impose the following conditions on the coefficients h(mn):

ii

hen) * = h(nm) - j - i , (8a)

h(mn) * = h(_m;lj, i j (8b)

kgn) * = c ( rn )~ (n ) h(?:Lj, ( 8 ~ )

where (-i) = k for i =r, ~ ( m ) = 1 for m = 1, 3,4,6, 8, and c(m)= -1 for m = 2, 5, 7.'

The SU(3) structure of (7) i s a mixture of {I), (8.~1, (8a), (101, {n), and {27), that is , of all representations contained in the direct product (8) x (8) .I5 The isospin content of H;;=O is T = 0, 1, 2.

The conditions for H$;=' to be free of an isoten- sor , isovector, o r isoscalar part a r e a s follows.

No AT = 2:

hy_) = -ii Rea ' sin2%,

and, by definition,

4 - S E C O N D - C L A S S C U R R E N T S , N E U T R A L C U R R E N T S . . .

As seen from the last relation in (lo), the neutral currents do not cancel the isovector te rm intro- duced by the charged second-class currents . Con- sequently, in addition to the nonexistence, in the presence of charged currents, of a AS = 0 Hamilto- nian satisfying a A T = 1 or a AT = 2 selection rule,'' a pure isoscalar AS = 0 Hamiltonian is not possible either if charged second-class currents partici- pate.

Next we consider a more general Hamiltonian in which V+A a s well a s V - A currents a r e involved for both the regular and the irregular octets:

where

and

vR*AR, vr*A'

satisfy the relations (4) and (5). We assume H , to be CPT-invariant, but allow for possible C P vi- olation.

It is convenient to introduce the quantities

(mn) = (mn)-- - hy;)+ + h + + 9

The conditions for the absence of various isospin components of can be written in the fol- lowing concise form:

In (14), (15), and (16), l? stands for ReM, ImM, ReN, and ImL and 51 stands for ReQ, ImQ, ReL, and ImN.

The implications of (14), (15), and (16) can be summarized a s follows:

(a) A CP-invariant AS = 0 nonleptonic weak inter- action satisfying a AT = 0 rule is possible only in the absence of charged second-class currents . This conclusion i s not affected by the presence of V +A currents .Ie

(b) If we allow for C P violation, the condition 5112 = 0 can be satisfied, and consequently a pure isoscalar AS = 0 nonleptonic weak Hamiltonian can be constructed, even in the presence of charged second-class currents . An example of such a theory is given by the Hamiltonian (7) built up from the hadronic current (6) in which a! is taken to be pure imaginary (in this case 1m&12 = R ~ Q " = ReL" = ImN12 = 0) .I7

To answer the question of the existence of a AT = 1 component in the parity -violating nuclear forces with a s high a s possible accuracy, i s there- fore of great importance." The absence of such interaction is evidence against the existence of charged second-class currents of the type capable of producing a difference between the matrix ele- ments of mi r ro r p transitions.'$ On the other hand, an unambiguous interpretation of a AT = 1 strange- ness-conserving weak interaction, if found, will require additional information about the currents entering the nonleptonic weak interactions.

ACKNOWLEDGMENTS

I would like to thank Professor R. E . Cutkosky, Professor L . Kisslinger, Professor P . Winternitz, and D r . C . C. Shih for stimulating conversations.

*Work supported in part by the U. S. Atomic Energy L J(TT3Y)y ( p = 1 , 2 , 3 , 4 ) which goes into itself under charge Commission. conjugation satisfies (omitting in the following the Dirac ' ~ n octet of vector or axial-vector currents J(u)s indexes)

R C Z E G 4 -

where (-v)=(T -T3-Y), Q, =T,+&Y, a n d q ( J ) = + l and i s the same for a l l components of a given octet [Y. Dot- han, Nuovo Cimento 30, 399 (1963); M. Gell-Mann, Phys. Rev. Letters 12, 155 (1964)l. J(,! =V(,)*A(,) will be called regular (JR) if q =+I , and zrregular (J') when q = -1. Under G parity

and

so that ~ i 2 , ~ ~ ) i s a f irst-class and ~ f l ~ ~ ~ ) a second-class current [S. Weinberg, Phys. Rev. 112, 1375 (19591. Re- garding SU (3) we use the notation and conventions of J. J. de Swart, Rev. Mod. Phys. 35, 916 (1963). Cartes- ian components will be denoted by Latin indexes: 4,) (m=1 ,2 ,..., 8).

2 ~ e c e n t studies of differences in theft values of mir - r o r /3 decays a r e suggestive of the possibility of the existence of second-class currents: Cf. R. J. Blin- Stoyle and M. Rosina, Nucl. Phys. 70, 321 (1965); D. H. Wilkinson, Phys. Letters G, 447 (1970); D. H. Wilkin- son and D. E. Alburger, Phys. Rev. Letters 24, 1134 (1970); Phys. Letters a, 190 (1970). See, however, J. Delorme and M. Rho, CERN report , 1970 (unpub- lished) and also D. H. Wilkinson and D. E. Alburger, Phys. Rev. Letters 3, 1127 (1971).

3A general analysis of interactions involving neutral currents i s given in C. H. Albright and R. J. Oakes, Phys. Rev. D 2, 1883 (1970).

4 ~ . Weinberg, Phys. Rev. 112, 1375 (1959). 5 ~ . L. Good, L. Michel, and E. de Rafael, Phys. Rev.

151, 1199 (1966). - 6 ~ t could, of course, happen that the irregular currents

a r e present in the semileptonic weak interactions while absent in the nonleptonic ones o r vice versa. In addition, the nonleptonic weak interactions may involve other types of currents, not present in the semileptonic weak interactions, for example, charged V + A currents o r scalar and tensor currents [cf. F. Zachariasen and G. Zweig, Phys. Rev. Letters 14, 794 (1965)l. Finally, the nonleptonic weak interactions may not be of a current x current form. An example i s the theory of Nishijima [K. Nishijima, in Proceedings of the Fifth Coral Gables Conference on Symmetry Principles a t High Energies , Universi ty of Miami, January 1968, edited by A. Perl- mutter, C. A . Hurst, and B. ~urgunoglu (Benjamin, New York, 1968), p. 1751. The standard current x current theory, applied to the nonleptonic weak interactions, i s reviewed by S. Pakvasa and S. P. Rosen, in a paper pre- sented to the symposium "The Pas t Decade in Particle Theory, " University of Texas a t Austin, 1970 (unpub- lished).

h here i s good evidence now for the existence of AS = O nonleptonic weak interactions from observations of parity nonconservation in nuclear transitions. The present experimental situation i s reviewed by F. Boehm, in an invited paper presented a t the International Conference on Angular Correlation in Nuclear Disintegration, Delft, The Netherlands, 1970 (unpublished). Recent reviews of the theory a r e R. J. BLin-Stoyle, in Proceedings of the Topical Conference on Weak Interactions, CERN, 1969, edited by J . Prentki and J. Steinberger (CERN,

Geneva, 1969), p. 495; E. M. Henley, Ann. Rev. Nucl. Sci. 2, 367 (1969). See a lso papers presented a t the ~nte&tional Conference on High-Energy Phys ics and Nuclear Structure, Columbia Universi ty , 1969, edited by S . Devons (Plenum, New York, 1970).

8 ~ . J. Blin-Stoyla and P. Herczeg, Phys. Letters 2, 376 (1966).

9 ~ . F. Dashen, S. C. Frautschi, M. Gell-Mann, and Y. Hara, The Eighvold Way (Benjamin, New York, 1964), p. 254.

'OC. H. Albright and R. J. Oakes, Phys. Rev. D 3, 1270 (1971). l ' ~ o w e v e r , a Hamiltonian satisfying a AS = 0, AT = 1,

o r a AS =0, AT =2 selection ru le i s not possible, unless the charged currents a r e absent (Ref. 10).

%ee, for example, N. Cabibbo, in Part icle Symmetr ies , Brandeis University Summer Institute in Theoretical Phys- ics , 1965, edited by M. Chraien and S. Deser (Gordon and Breach, New Pork , 1966), p. 1.

1 3 ~ o t e that Eqs. (3) imply J?Aa),, = -J71-io)p and ~ f r ~ ~ ) ~ =Jfl-10), (J;), =JT,),, for^ = 1 , 2 , 3 and JT, )~=-J&,~;

TEHermitian conjugation), i.e., the Cartesian compon- ents of J~ and J' a r e Hermitian and anti-Hermitian, respectively. If Ima =0, the semileptonic weak Hamil- tonian corresponding to (1) i s C P -invariant.

1 4 ~ h e r e i s no information a t present on AS = 1 irregular currents. The identification of such currents i s compli- cated by the SU(3)-breaking strong interactions which a r e capable of inducing second-class form factors even in the absence of irregular currents. Cf. L. Wolfenstein, Phys. Rev. 135, B1436 (1964).

15cf. P. Herczeg, Nucl. Phys. E, 153 (1967). In the standard theory not involving irregular currents only the symmetric combinations {I}, {a,), and (27) enter.

1 6 ~ o t e also that the nonexistence in the presence of charged currents of a Hamiltonian satisfying a AT = 1 o r a AT =2 selection rulei0 pers is ts in the presence of V +A currents. The condition r11=l?33 =0, for example, re- quired to satisfy a AT =1 rule, eliminates again the corresponding charged-current products. ' I ~ o t e that for Rea= 0 the matrix elements of mi r ro r P

transitions a r e equal. This i s the theory of CP violation proposed by N. Cabibbo [Phys. Letters 12, 137 (1964)l. It could lead to a large muon polarization perpendicular to the decay plane i n K p 3 decays. Although the experi- mental results a r e consistent with zero polarization [K. K. Young et al., Phys. Rev. Letters 18, 806 (1966); D. Bartlett e t a l . , ibid. 16, 282 (1966); J. Bettels e t a l . , Nuovo Cimento %&, 1106 (1968); M. J. Longo, K. K. Young, and J. A. Helland, Phys. Rev. 181, 1808 (1969)1, this does not necessarily rule out the theory [ cf. N. Ca- bibbo, Phys. Rev. Letters 14, 965 (1965); L. Maiani, Phys. Letters B, 538 (1968); C. W. Kim and H. Pr ima- koff, Phys. Rev. =, 1502 (1969)l.

I8In the experiments performed so far , the AT =1 channel is not isolated (Ref. 7). Suggestions of experi- ments in which the isovector component can be sepa- rated from the isoscalar and isotensor par ts have been made [ cf. Ref. 9 and G. S. Danilov, Phys. Letters 18, 40 (1965); D. Tadib, Phys. Rev. 174, 1694 (1968); E. M. Henley, Phys. Letters a, 1 (1968)l. The existence of an isoscalar component in the parity-nonconserving nu - clear force has been recently established in studies of parity-forbidden a decays IH. HAttig, K. Hhchen, and

4 - S E C O N D - C L A S S C U R R E N T S , N E U T R A L C U R R E N T S . . . 1243

W. WBffler, Phys. Rev. Letters 25, 941 (1970); E. L. H. Primakoff, D. Tadi%, and K. Trabert, Phys. Rev. D 3, SprenkelSegel, R. E. Segel, and R. H. Siemssen, in 2289 (1971)l. Proceedings of the International Conference on High- 19As seen from the relations (15), neutral currents Energy Physics and Nuclear Structure, Columbia Uni- must in this case also exist, unless the charged AS = 1 versity, 1969, edited by S. Devons (Plenum, New York, regular currents are absent in the nonleptonic weak 19701, p. 763; Wen-Kwei Cheng, E. Fischbach, interactions.

PHYSICAL REVIEW D VOLUME 4 , NUMBER 4 1 5 AUGUST 1 9 7 1

Proposed Experiment to Test Local Hidden-Variable Theories

R. Fox and B. Rosner Department of Physics, Technion-Israel Institute of Technology, Haifa, fa,ssral

(Received 8 February 1971)

It is shown that by making a plausible assumption concerning the behavior of linear polar- izers, it is possible to relax the experimental condition of Clauser , Horne, Shimony, and Holt for testing local hidden-variable theories.

Bell recently proved that any local hidden-vari- able theory cannot contain a l l of the predictions of quantum mechanics.' The proof i s based on a Bohm2 variant of the Einstein-Podolsky-Rosen ex- ~ e r i m e n t . ~ As a result of this proof, in a recent let ter Clauser, Horne, Shimony, and Holt (CHSH) proposed an experimental test of local hidden-vari- able t h e ~ r i e s . ~ The proposed experiment i s that of Kocher and Commins5 - requiring, however, high- sensitivity high-transmission polarizers a t appro- priate relative orientations. We show that the high- transmission condition, which makes an experi- mental test extremely difficult, is unnecessary.

In the Kocher and Commins's experiment the lin- e a r polarization correlation was examined of two successive photons emitted in the cascade 6 'So - 4 'PI- 4 'So in calcium. Each of the two detectors consisted of a linear polarizer, followed by a wave- length filter and photomultiplier. We define E: as the transmission of the polarizers, where i = M , m designates light polarized parallel o r perpendicu- l a r to the polarizer axis and p = I, I1 designates the polarizer station. We will examine, a s do CHSH, the case for E: >> 6 ; . We define tP a s the product of the transmissivity of the filter and the photo- multiplier efficiency.

The quantity of interest for local hidden-vari- able experiments is the correlation function P(a, b), defined a s l s4

where a and b a r e the angle of the polarizers a t stations I and 11, respectively. The variables A, B = k l designate the detection (+I) or nondetec - tion (-1) of photons a t stations I, 11. Since tP is

relatively small, A and B were reinterpreted by CHSH a s the emergence o r nonemergence of pho- tons from the polarizers, which defines a new cor- relation function P'(a, 6).

When E;>> E:, each polarizer can be described a s a perfect polarizer (E, = 0, E, = 1) followed by a gray filter of transmissivity E;. ReinterpretingA and B now a s the emergence o r nonemergence of photons from the respective perfect polarizers, we obtain another correlation function PU(a , b).

Evaluating (1) we obtain for P f ( a , b ) and P"(a, b )

and

where R(a, b) i s the emergence coincidence ra te from the perfect polarizers. The relation between R(a, b) and the measured coincidence ra te from the photomultipliers, R(a, b), i s

E(a, b) = R (a, ~)/E;E; t1 tn . In examining experimental tes t s of local hidden-

variable theories, we obtain relationships between different correlation function^."^ CHSH using Pf(a , b) obtained a minimum condition for 6 ; = E:

= E,. They obtained

~ ~ > 2 / ( 1 + a ) = 8 3 % (5)

[ ~ q . 4, Ref. 4, F ( 0 ) = 11. Since Pn(a , b) could just a s well have been used, comparing (2) and (3) we obtain

1 > 2/(1 +JZ-) (6)

which is always t rue . From the above we observe that an experimental