Volume 119, number 1 PHYSICS LETTERS A 24 November 1986

Q U A N T U M N O N - S E P A R A B I L I T Y VERSUS L O C A L REALISM: A N E W TEST U S I N G T H E B°B ° SYSTEM

Amitava DATTA Department of Physics, Jadavpur University, Calcutta 700 032, India

and

Dipankar HOME Theoretical Nuclear Physics Division, Saha Institute of Nuclear Physics, Calcutta 700 009, India

Received 18 August 1986; accepted for publication 29 September 1986

We point out that the decay of the spin 1 vector meson ~/'(4s) into a pair of neutral pseudo-scalar mesons B°l) ° provides a new arena for exploring an EPR-type example amenable to experimental studies concerning the incompatibility between quantum. mechanics and local realism.

I f two particles interact and then get spatially sep- arated, the two-particle quantum wavefunction retains its non-separable character even if the parti- cles are space-like separated. This feature (referred to as "quan tum wholeness" by Bohm [ l ]) leads to a non-local correlation between the observable prop- erties o f the two particles concerned. Its puzzling conceptual aspects were highlighted by Einstein, Podolsky, and Rosen (EPR) [2] in their celebrated paper, which evoked animated deliberations over the past fifty years. Apart f r o m the epistemological implications, the EPR debate stimulated studies on viable experimental tests o f the non-separability o f the two-particle quantum wavefunction and its incompatibility with various local realist models. The photon-polarization correlation measurements using radiative atomic cascade transitions by Aspect et al. [ 3 ] mark a significant effort in this direction. How- ever, the interpretation o f these experimental results has been subjected to considerable controversy due to low efficiency of the photomultiplier detectors and uncertainties in the estimation of the background accidental coincidence counting rates [ 4 ]. In view of the importance o f the issues involved, it seems nec- essary to explore the possibility o f new types o f

experimental tests related to the EPR-type situation. Six [5] and Selleri [6] have pointed out that the

decay of the j P c = 1 - - state (for example, the spin l resonance) into a pair o f neutral kaons K°I( ° pro-

vides an interesting EPR-type example. Here, because o f the odd relative orbital angular momentum, the space part o f the wavefunction o f the K°I( ° pair is antisymmetric. Hence the strangeness part o f the wavefunction for this pair o f bosons can be written as

~o = [K°(1 )K°(2) - K ° ( 1 )K°(E) ] /x /~

= [ K s ( l ) K L ( 2 ) - - K L ( i ) K s ( 2 ) ] / x / ~ , ( l )

where l (2) refers to the left ( r ight ) hemisphere, K ° = (KL +Ks) /v /2 , K ~° = (KL - K s ) / v / 2 ; KL, Ks are respectively the long-lived ( ~ 10 - s s) and short-lived ( ~ 10-io s) mass eigenstates o f the kaons. Through- out this paper, we ignore the effects o f CP noninvariance ,1

The non-separable form of the wavefunction ~o

:t The effect of CP violation in the EPR-type example involving the K°I( ° system has been recently studied and some intriguing results have been obtained by A. Datta, D. Home and A. Ray- chaudhuri (to be published).

0375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 119, number 1 PHYSICS LETTERS A 24 November 1986

given by (1) implies the EPR-type non-local corre- lation. I f the left (right) kaon is observed to be a K ° (strangeness S = + 1 ) at a certain instant then the right (left) kaon can be predicted with certainty to be observed as a I( ° (S= - 1 ) at that same instant. Alternatively, if the left (right) kaon decays in the Ks mode ( C P = + I ) then the right (left) kaon is bound to decay as a KL at any future instant [ 6 ].

Six [5] suggested an experimental test of the quantum non-separability inbuilt in the wavefunc- tion (1) through measurement of the joint probabil- ity P__(tt, t2) of a double I( ° observation on two sides at times tl and t2; its quantum mechanical pre- diction is given by

P_ _ (tl , t2) = I (/£°( 1 )/('0(2) I ~ ( t l , t2) ) I 2 ,

where

~( t~, t2) = (1/v/2 )

× [Ks(1 )KL(2) exp( --astl --aLt2)

--KL(1 )Ks(2) exp( --OtLtl --Otst2)] . (2)

Here aL=FL/2+imL, a s = F s / 2 + i m s ; mL (ms) and FL (Fs ) are respectively the mass and the decay width of KL (Ks). However, this suggested experi- mental study has the difficulty that for meaningful results q, t2 must be shorter than the lifetime of KL, Ks; i.e., one requires tl, t2~ < 10-1o s. The uncertain- ties involved in ensuring observations at the speci- fied instants tl, tz will be quite appreciable within such a small time interval. The proposed scheme, there- fore, appears rather difficult to realise in practice. Such difficulty may be circumvented by considering the time-integrated joint probability. This aspect has not yet been probed.

Of late, the experimental studies concerning the decay of the spin 1 Y(4s) vector meson into a pair of neutral pseudo-scalar mesons B°l] ° have attracted attention in the context of the search for the evidence of B°B ° mixing [ 7 ]. In this paper we analyse the pos- sibility of experimentally investigating the EPR-type non-local correlations within the framework of the ongoing experiments on T(4s)--.B°B °. We note that the formalism for treating this B°B ° system is exactly similar to that for the K°K °, with the only difference that F L = F S for the states of the B°B ° system analo- gous to the KL, Ks states, which are denoted by BH,

BLC2. We focus our attention on the time-integrated joint probabilities remembering that B ° and ~o can be identified by their characteristic semi-leptonic mode of decays: B°~£ - pX; B ° ~ ÷ v X , where £ and X denote lepton and hadron respectively.

The present experimental arrangement to study T(4s) ~B°B ° is geared to measure the parameter R defined as follows:

R = (N++ +N__)/(N+_ +N_+), (3)

where N+ + = total number of double Bo decays (cor- responding to the observation of double £+ decay products on the both sides); N__ = total number of double B ° decays (corresponding to the observation of double ~- decay products on the both sides); N+ _ = total number of ~o decays on the left associ- ated with B ° decays on the right (corresponding to the observation of ~ ÷ decay products on the left asso- ciated with ~- decay products on the right); N_ + = total number of B ° decays on the left associ- ated with 1) ° decays on the right (corresponding to the observation o f~ - decay products on the left asso- ciated with ~+ decay products on the right). The parameter R is calculated by evaluating the quan- tities N a (i, j = + 1 ). The general expression for N a is given by

Na=2No 22 i dtl i dt2 Pa(tl,/2), (4) 0 tl

where Po(tl, t2) is the joint probability for observing the decay products ~i, ~ on two sides at times tl and t2 respectively, No is the total number of Y(4s) decays, and 2 is the semi-leptonic decay width ofB ° decaying into a ~- = the semi-leptonic decay with of ~o decay- ing into a ~+. The quantum mechanical expressions for Pij(tl, t2) are given by (derived from the non- separable form of the wavefunction (2))

P++(tl, t2)=P__(tl, t2)={2 e x p [ - f ' ( t l +t2)]

- 2 e x p [ - F ( f i +t2)] cos [Am( t2- t~ ) ]} /8 , (5)

52 KLhaS a longer lifetime compared to Ks because the phase space available in its principal decay mode KL--'~3g is smaller than that available in the decay mode Ks--, 2~. For the decay of the BH, BE states, the phase space available is more or less the same. Hence their decay widths are taken to be identical.

Volume 119, number 1 PHYSICS LETTERS A 24 November 1986

P+_( t l , t2) = P _ + ( f i , t2) ={2 exp[ - I ' ( t ~ +t2)]

+ 2 e x p [ - F ( t l +t2)] cos[Am(t2- t~) ]} /8 . (6)

Here F is the common decay width of the Bn, BL states and Am = mH-- mL where mn (mL) is the mass OfBH (BL). Using (5) and (6) we obtain from (4) the following quantum mechanical values for No:

N+ + = N_ _ =No22(1/4I "2 - 1/40t 2) , (7)

N+_ =N_+ =No22(I/41 -'2 + 1/40t 2) , (8)

where ot2=F2+ (Am) 2. This leads to the following quantum mechanical prediction ,3 for the parameter R defined by (3)

RQM = X 2 / ( 2 + X 2) , (9)

where x = A m / F . The result given by (9) hinges on the quantum non-separability inbuilt in the wave- function (1), assumed to be maintained even after the particles get well separated in space. The experi- mental verification of (9) will, therefore, constitute a crucial test for the quantum non-separability in this EPR-type situation.

It should be instructive to compare eq. (9) with the corresponding prediction derived from the notion of local realism. As an example, let us consider Fur- ry's hypothesis [ 9 ] in the following form: The wave- function has the non-separable form (1) at the production of the B°B ° pair, but after spatial separa- tion between the two particles the wavefunction becomes an equal mixture (not superposition) of the two independent states BH ( 1 ) BE (2) and BE ( 1 ) BH ( 2 ). One is tempted to envisage this hypothesis because it enables to avoid the conceptual anomalies arising from the quantum non-separability presented by the EPR paradox. Einstein ,4 himself favoured such a proposal. Bohm and Aharonov [ 10] ,5 analysed the tenability of Furry's hypothesis and pointed out the significance of testing whether this hypothesis leads

:3 Eq. (9) has been derived in a different context in ref. [ 8 ]. ~4 A. Einstein, in a private communication to D. Bohm; see ref.

[10] ,p. 1071. ~5 The suggestion by Bohm and Aharonov for verifying Furry's

hypothesis through measurement of the polarization correla- tion of the two photons produced in a positron-electron anni- hilation has certain critical aspects which generated delicate controversy; see ref. [ 11 ].

to any conflict with the available experimental results. In the present EPR-type example of the B°B ° sys-

tem, applying Furry's hypothesis to evaluate the gen- eral formula given by (4) we obtain

P++(h , t E ) = P _ _ ( t l , t2)

= P + _ ( t l , t 2 ) = P _ + ( t l , t2)

-4- ! exp[ - F ( t ~ + t z ) ] ,

whence

N++ = N _ _ =N+_ =N_+

= (No 22) ( I /4F z) , (10)

which leads to the following prediction for the parameter R according to Furry's hypothesis:

R F = I . ( 1 1 )

Comparing (11 ) with (9) we observe that within the present experimental framework for the study of / ' (4s) ~B°/) ° it is possible to discriminate between the predicted values RQM and RF, unles x2>> 1 (the case of maximal B°B ° mixing corresponding to Am>>F). In this context it is interesting to note that the CLEO group have already furnished an experi- mental upper bound on R given by R < 0.3 [ 12 ]. This may appear to rule out Furry's hypothesis. However, a word of caution is necessary. This upper bound involves certain theoretical model-dependent inputs (such as the use of the spectator model of mesonic decay). Nevertheless, with better statistics it should be possible to set the empirical upper bound in a model-independent way.

As regards the quantum mechanical prediction for R given by eq. (9), we note that RQM is model- dependent. Our following remarks concerning RQM will be confined within the ambit of the Glashow-Weinberg-Salam standard model of elec- tro-weak interactions. There are two types of B ° mesons: B ° (bd quark-antiquark bound state) and B ° (b~ quark-antiquark bound state). ~'(4s) decays into the o - o o - o BdBd system only (Bs Bs channel is forbid- den by kinematic considerations). For this case, the standard model predicts A m / F <0.2fl where fl is estimated to be within the range 0.33-1.5 [13]. Therefore it follows that RQM<<I according to the standard model, suggesting that the experimental distinction between Rv and RQM based on the start-

Volume 119, number 1 PHYSICS LETTERS A 24 November 1986

dard model should be quite feasible. Finally, we make the following comment indicat-

ing further work based on this paper. Apart from cal- culating the parameter R using the various local realist models (analogous to the types used for ana- lysing the EPR atomic-cascade experiments :6), it seems important to derive general bounds on R from local realism independent of the details for any par- ticular model. This would enable decisive use of the ongoing experimental studies on R to test the notion of local realism against quantum mechanics. Such tests would constitute a valuable complement to the current EPR experiments. They would, incidentally, be also the first EPR test involving electroweak interactions.

It is relevant to note here that Selleri [6] derived an upper bound on P+ + (t~, t2) for the K°IZ ° system using a general argument based on local realism; interestingly, this bound coincides with the value obtained from Furry's hypothesis. It needs to be explored whether Selleri's treatment can be extended for the B°l) ° example to obtain general local-realist bounds on the parameter R. Then the empirical probing can be concentrated on the domain of incompatibility between such bounds and the quan- tum mechanical prediction for R. This possibility is presently under study.

D.H. is grateful to Franco Selleri for encourage- ment and helpful comments. We thank Amitava Raychaudhuri for fruitful discussions.

Note added

It was observed in ref. [ 12] that the then data could accommodate R = 1 i fa large departure (> 40%) from the spectator model predictions for B-decay was allowed. More recent results [15] reveal that the above departure should have to be even larger (> 50%) in order to accommodate R = 1. The limits on B +, B °

,'6 For an up-to-date review of such local realist models see ref. [14].

meson life time also presented in the above preprint "are consistent with the spectator model predictions and begin to rule out dominant contributions from non-spectator diagrams in B decay". Theoretical arguments also strengthen the view that the above deviations, if any, should not be more than 30% (see, e.g. ref. [ 16]).

References

[ 1 ] D. Bohm, in: Quantum theory and beyond, eds. B.J. Hiley and F.D. Peat (Routledge and Kegan Paul, London, 1986).

[2] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47 (1935) 777.

[3] A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 47 (1981) 460; 49 (1982) 91; A. Aspect, J. Dalibard and G. Roger, Phys. Rev. Lett. 49 (1982) 1804; A. Aspect and P. Grangier, Lett. Nuovo Cimento 43 (1985) 345.

[4] T.W. Marshall, E. Santos and F. Selleri, Phys. Lett. A 98 (1983) 5; T.W. Marshall, Phys. Lett. A 99 (1983) 163; A. Garuccio and F. Selleri, Phys. Lett. A 103 (1984) 99; T.W. Marshall and E. Santos, Phys. Len. A 108 (1985) 373; D. Home and T.W. Marshall, Phys. Len. A 113 ( 1985 ) 183; M. Ferrero and E. Santos, Phys. Lett. A, to be published.

[5] J. Six, Phys. Lett. B 114 (1982) 200. [6] F. Selleri, Lett. Nuovo Cimento 36 (1983) 521. [7] AJ. Buras, W. Slominski and H. Steger, Nucl. Phys. B 245

(1984) 369. [ 8 ] L.B. Okun, V.I. Zakharov and B.M. Pontecorvo, Lett. Nuovo

Cimento 13 (1975) 218. W.H. Furry, Phys. Rev. 49 (1936) 393. D. Bohm and Y. Aharonov, Phys. Rev. 108 (1957) 1070. A. Peres and P. Singer, Nuovo Cimento 15 (1960) 907; D. Bohm and Y. Aharonov, Nuovo Cimento 17 (1960) 964; M. Home, Ph.D. Thesis (Boston University, 1970) pp. 82-85. P. Avery et al., Phys. Rev. Lett. 53 (1984) 1309. A.J. Buras, W. Slominski and H. Steger, Nucl. Phys. B 238 (1984) 529. T.W. Marshall, E. Santos and F. Selleri, in: Open questions in quantum physics, eds. G. Tarozzi and A. Van der Merwe (Reidel, Dordrecht, 1985 ); T.W. Marshall, in: Microphysical reality and quantum for- malism, eds. G. Tarozzi and A. Van der Merwe (Reidel, Dordrecht, 1986). A. Bean et al., CLNS 86/741, CLEO-86-11 (1986). I.I. Bigi, Phys. Lett. B 169 (1986) 101.

[91 [10] [11]

[121 [131

[141

[15] [161