Decay correlations in e+e− → τ+τ− as a test of quantum mechanics

Physics Letters B 275 (1992) 172-180 North-Holland PHYSICS LETTERS B

D e c a y c o r r e l a t i o n s i n e + e - - as a t e s t o f q u a n t u m m e c h a n i c s

Paolo Privitera lnstitut.~r Experimentelle Kernphysik, Universitdt Karlsruhe, 14~ 7500 Karlsruhe. FRG

Received 17 October 1991

Tau leptons which are pair produced in e+e annihilation show a strong correlation in their spin directions. The process is interpreted as an Einstein-Podolski-Rosen type of experiment. The induced correlation in the "~ decay products is analyzed in terms of possible tests of quantum mechanics versus local hidden variables theories, studying ~ pair production at energies between threshold and the Z ° mass.

1. Introduction

In 1935 Einstein, Podolski, and Rosen (EPR) produced a famous argument [ 1 ] for the incompleteness of the Copenhagen interpreta t ion of quantum mechanics. This work greatly contr ibuted to the devel- opment of hidden variables theories, where the introduction of "h idden" parameters is used to restore determinism and causality in the microphysics world. In 1965 Bell, using Bohm's [2] reformulat ion of the EPR argument in terms of spin components , proved the fundamental result [3] that a local hidden vari- able theory cannot reproduce all statistical predictions of quantum mechanics.

Several experimental tests have been performed since that t ime, mainly studying the polar izat ion correlation of photons emitted in an atomic cascade [4]. Almost all the exper iments agree with the quantum mechanical predictions, but they always tested Bell's inequalit ies deduced from local realism and some addi t ional assumptions [5]. The search for complementary tests has s t imulated the discussion on some examples of the EPR problem in particle physics [ 6 ].

In this paper, the process e+e - ~ + ~ - is proposed as a possible EPR-like experiment , and the experimental study of l imits from Bell-type inequalit ies at different product ion energies is investigated. In section 2, the EPR argument is presented, and the Bell inequalit ies are reviewed in section 3. The spin correlation in z pair product ion and decay, and its inter- pretat ion in the f ramework of the EPR problem is

discussed in section 4. Possible exper imental tests of quantum mechanics versus local hidden variables theories are discussed in section 5, studying z decay correlat ion at product ion energies between threshold and the Z ° mass. The conclusions are summar ized in section 6.

2. The EPR argument

" I f without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value o f a physical quantity', then there ex- ists an element q f physical realio, corresponding to this physical quantity". On the basis of this famous "criterion of physical reality", Einstein, Podolski and Rosen in 1935 [ 1 ], discussing the reality of the position and momen tum of two correlated particles, ar- gued that quantum mechanics was not a complete theory.

A reformulat ion of the EPR argument, involving spin functions, was proposed by Bohm [ 2 ]. Consider two spin one-half particles in a singlet state, and let them move freely in opposi te directions. The system state is described by the following function:

1 ~ = ~ [ ~ , + ( 1 ) ~ , ( 2 ) - ~ ' ( 1 ) ~ u + ( 2 ) ] , (1)

where ~u+ ( i ) is the wave function of the state in which particle i has a spin _+ h/2 in the direct ion in which is measured. Once the particles have separated, any spin

172 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

Volume 275, number 1,2 PHYSICS LETTERS B 23 January 1992

component of particle 1 can be measured. Since the total spin of the system is zero, one knows, without having interfered with particle 2, that its spin component along any axis is opposite to that of particle 1. According to the EPR criterion of physical reality, the spin component of particle 2 represents an element of physical reality, and thus existed even before the measurement was performed. But since one could have equally chosen any other direction for measuring the spin of particle 1, all three spin components of particle 2 must have simultaneously definite values after the two particles have separated. This de- duction is in clear contradiction with quantum mechanics which allows only one spin component at a time to have definite values, since spin operators do not commute. Thus quantum mechanics does not provide a complete description of physical reality in the EPR sense: "Ever), element of physical reality at- tributable to a certain physical system in a given state must have a counterpart in the mathematical description provided by the theory for that physical system in the state considered".

The intimate connection between the EPR reason- ing and the nonfactorizable form of the state function ( 1 ) was discussed for the first time by Furry [7]. The fact that the measurement of the spin component of particle 1 along a certain direction determines with certainty the result of the measurement along the same direction for particle 2, has to be interpreted as the spin components of the two particles along the chosen direction have definite values before any measurement is actually performed, if one wants the EPR locality principle ("since at the time of the measurement the two systems no longer interact. no real change can take place in the second system in consequence of anything that may be done to the first system") to hold when the two particles are far apart. The two particles, thus, are in a definite quantum state qJ,( 1 )~'_,(2) ( i = + , - ). In quantum mechanics, the necessary and sufficient condition for two quantum systems to be in a definite quantum state is that the system state is factorizable. The comparison with eq. ( 1 ) gives paradoxical results analogous to the EPR argument.

Bohm and Aharonov [8] interpreted Furry's conclusions providing a possible interpretation of the EPR argument. One can make the hypothesis that, after the two particles separate, the wave function for

the system is no longer given by eq. ( 1 ), and the spins of the two particles becomes definite in some direction. The wave function will be given by the product

~ = ~u+0.o( 1 )~,_o.o(2) , (2)

where qJ+o.~( 1 ) is a wave function of particle 1 whose spin is positive in the direction given by 0 and 0, while the spin of particle 2 is negative in the same direction. Assuming a uniform probability for any direction of 0 and 0, this model predicts the experimentally observed fact of conservation of the angular momentum on the average. Since there is no precise correlation of an arbitrary component of the spin of each particle in every individual case, the EPR "paradox" is solved. Bohm and Aharonov could show that, for the case of positronium annihilation, such a model can be experimentally tested, since it gives predictions for the polarization correlation of the two photons different from that of quantum mechanics.

3. The Bell inequalities

A reinterpretation of quantum mechanics in terms of a statistical account of an underlying hidden variables theory was proposed as a possible solution to the EPR "paradox". The introduction of some supplementary parameters can complete the quantum mechanical state specification, and a classical-looking picture of the EPR correlations is recovered. The quantum mechanical predictions are, then, obtained when averaging over the supplementary parameters.

In this context, Bell [3] made a fundamental con- tribution, showing that for the Gedankenexperiment of Bohm, no deterministic hidden variables theory can reproduce all the statistical predictions of quantum mechanics. Let An(Bg) be the result of a measurement of the spin of particle 1 (2) in the direction fi(/~). The product of Aa and Bb is an observable of the two particle system, and its expectation value, according to quantum mechanics, is

E(( l , b )=P++( f~ ,b ) -P+ (ft, b)

- P _ + (ti,/~) + P (a,/~)

= - t i . /~, (3)

where P,j(d, b) is the joint probability of having the result i in the spin measurement of particle 1 along

173


the direction ti and the result j in the spin measurement of particle 2 along the direction/~, with i , j= +.

In a deterministic hidden variables theory, the observable An "Bt; has a definite value (Aa'Bg)~. for the state 2, where the symbol 2 denotes a complete specification of the system state, without any assumption on its possible complexity. Bell introduced a definition of locality,

(An . Bd ) ~ = Aa( 2 ) 'Bt;(2), (4)

which implies that once the state 2 is specified and the two particles separate, the measurement Aa(Bi) depends only on 2 and ti(/~). The expectation value of An .B~; for such theories is

E(&/~) = f A ~ . B i d p , (5) A

where p is the distribution function for the states 2 in the space A. Using eq. (5) and the perfect anticorrelation in the case of parallel directions of the analyzers (E(& li) = - 1 ), Bell could write an inequality for all deterministic local hidden variables theories involving the expectation value E(&/~) which was violated by the quantum mechanical prediction for some particular directions of the analyzers.

Bell's proof, of fundamental heuristic value, is not, however, experimentally testable, since it requires the existence of a pair of spin analyzer orientations with perfect anticorrelation, while, in a real experiment, weak correlation in the source emissions or poor detector efficiencies strongly limit this assumption.

A more general inequality was obtained by Bell [9 ] without the requirement of a perfect anticorrelation:

-2~<E(& b ) - E ( & b')+E(d',/~) +E(~i', b')~<2, (6)

which is maximally violated, by a factor x/2, if the quantum mechanical prediction of eq. (3) is used, choosing the four analyzing directions to be coplanar and the angle between ci and/~,/~ and ~', ~' and/~' equal to g/4.

Inequalities expressed in terms of coincidence counts between detectors, which are more suitable for a real experiment, have been derived by several au- thors [10]. Clauser and Horne proposed a general experimental set-up shown in fig. 1, where a source of coincident two particle emissions is viewed by two

Analyzer 2 Analyzer 1

Detect°r2 ~ zSOUrcy ~ D e t e c t o r l ~ - - ~ Z ( )

Y Apparatus 2 Apparatus I

Fig. 1. Experimental arrangement of the Clauser and Home proof.

analyzer-detector assemblies. Each apparatus has an adjustable parameter and let a be the value of the parameter for apparatus 1 and b that for apparatus 2, The parameters a and b could denote the angles spec- ifying the orientation of the analyzers, like the axes of linear polarizers for photons, or the directions of the field gradients of Stern-Gerlach magnets, but the specific interpretation of these adjustable parameters is not essential for the proof. With the help of a general locality definition, Clauser and Horne found the following inequality:

- 1 <~p,2(a, b) -p~2(a, b' ) +p.2(a' , b) +Pl2(a ' , b' )

- p , ( a ' ) - p 2 ( b ) <~ 0 , (7)

where p~2(a, b) is the probability of coincidence counts in detector 1 and 2 with the analyzers in position a and b, and P L (a), P2 (b) are the probabilities of single counts respectively in detector 1 with the analyzer in position a, and in detector 2 with the analyzer in position b. Inequality (7) is also maximally violated by a factor x~2 for some specific values a, b, a ' , b ' .

4. Decay correlations in e+e - -~+~-

Decay correlations in heavy leptons production through electron-positron annihilation were discussed for the first time in 1971 by Tsai [ 11 ], before the ~ lepton was actually discovered [ 12]. The process is described in fig. 2. In the limit me/Ec-,O, the total angular momentum of the electron-positron system is unity, and ils direction is either parallel or antiparallel to the direction of the incident electron. This specific initial state spin configuration results in a strong correlation in the spins of ~+ and z - , produced in the annihilation o fe+e - into a virtual photon (or a Z ° boson ). Near the threshold ( m ~ / E ~ 1 )

174


e + T+

T

Fig. 2. Feynman diagram of the process e+e--~x+x -.

the direct ions o f spins o fx + and z - prefer to be parallel to each other, and their total spin prefers to be ei ther parallel or ant iparal lel to the direct ion o f the incident electron, whereas far above the threshold (m J E t - - O ) , the belicities o f z + and z - prefer to be opposi te to each other. A general expression for the spin dependent ~ pair product ion cross section [ 13 ] can be writ ten as

d a = d a o ( 1 +az, s I q.-bl, s 2 +c~usis2) , (8)

where s{'(s~) is the spin o f the z + ( x - ) in the z rest frame and the coefficients a, b and c are functions of the momenta of the incoming and outgoing fer- mions. The terms involving single powers of st ~ or s~ result in net polar izat ion of the ~ (coming from Z ° exchange and 7 - Z ° interference) while the terms involving two powers result in spin correlat ion between the two z's.

Since ~+ and z - decay via weak interact ions where pari ty conservat ion is violated maximally, the angular d is t r ibut ion of decay products depends strongly on the spin or ienta t ion of the z. Since the spins o f z + and ~- are strongly correlated in the product ion one expects the angular dis t r ibut ions o f decay products o f z + to be strongly correlated to those of z - . A general form for z - decay is

dF~ =dFo(1 +h,,s ~) , (9 )

where the polar imeter vector h u depends on the par- t icular decay mode, the z m o m e n t u m and the momenta of the decay products.

The cross section for the combined product ion and decay process will therefore result:

d a ~ ( 1 - a ~ , h ~ ' - b u h ~ + c u ~ h ~ h ~ ) d F ~ d F 2 . (10)

Notice that the correlat ion between the spins of the

x'S contained in the terms cu~ induces an analogous correlat ion between the polar imeter vectors of the final decay products , which can be s tudied experimentally looking at the final state characteristics.

In the decay z-,nv~ of a polar ized ~, the polar imeter vector h~, is s imply the pion momentum:

d / ' (~ ~ ) = d F o ( i _+ w-/J~) , ( 11 )

where w is the polar izat ion of the • in its rest frame a n d / ~ is the unit vector in the direct ion of mot ion of the pion. Since this is a two body decay, v, and x - come out back to back in the z rest frame, and the component of the orbital angular m o m e n t u m along the direct ion of v, is thus zero. Now, v, has a negative helicity, and hence prefers to be emit ted opposi te to the direct ion of the spin o f ~- . Therefore, n - prefers to be emit ted in the direct ion of the spin of the ~- . The par i ty violat ing ~ decay works thus as a sort of polar imeter , in the sense that the z can be thought o f as polar ized in the direct ion o f the pion momentum.

The other decays of the ~ have a more compl ica ted form of the polar imeter vector, which generally results in a lower sensit ivity to the x polarizat ion. For this reason, only the decay ~ nv~, which accounts for 10% of the ~ branching ratio, will be considered in the following.

The nonlocal spin correlat ion described by eq. (8) leads to an interpreta t ion of the process e+e - ~x+~ - as an EPR-l ike experiment , since the ~- spin direc- t ion is influenced by that of the x+. The same nonlocal correlat ion appears in the x decay products , in the sense that the ~- decay is influenced by how ~+ has decayed (or will decay) ~a

The highest sensit ivity to the spin correlat ion is found in the process e + e - ~ T + z - - ~ n + 9 , n - v , and since its spin dependent rate can be measured, it is interest ing to study possible l imits from Bell 's inequali t ies in a field different from the usual a tomic physics tests of the EPR problem.

5. Tests of quantum mechanics with

e+e - ~ x + x - ~ + ~ , n - v ,

A possible in terpreta t ion o f the process e + e - - ~ ~+z- -~n+9~n-v~ is shown in fig. 3. The e lec t ron-

~ Similar considerations have been made by T6rnqvist with respect to the process e+e- ~AA.~ n-pn+O [ 14].

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~: Week decoy

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A SOURCE Weak Pl

decay

T

>

T +

Fig. 3. e+e - -~x+z - ~n + 9~ rt v~ as a Clauser and Horne type of experiment.

positron annihi la t ion is seen as a source of a z+z - pair. The two z's travel in opposi te directions, and at certain t imes each of them decays into a pion and a neutrino. The pari ty violat ing weak decay of the z is interpreted as a spin analyzing device. That is, when- ever a ~+(rt ) is seen in the direction/~1(/~2) this is understood as a count in detector 1 (2) with analyzer in d i rec t ion/~ (/~2), in analogy to the Clause r -Horne experiment described in fig. 1.

Using the spin dependent rate of the jo in t pions detection, which will be explicitly given in the following, one could study a possible violat ion of Bell's inequalities, thus testing the hypothesis of h idden variables theories. However, it is easy to verify that no direct violat ion of Bell's inequali t ies is achieved with this approach. The reason stands in the fact that the pion direct ion is only an index of the actual z polarization, and neither z is forced to make a d ichotomic decision.

The si tuation described is s imilar to that of Kas- day, Ullman and Wu [ 15 ], who used Compton scattering of the photons coming from e lec t ron-pos i t ron annihi la t ion to measure the photon l inear polarization correlation. Using their approach, that is mak- ing the addi t ional assumptions that ( 1 ) it is possible in principle to construct an ideal x polar izat ion analyzer, and (2) the results obta ined in an exper iment with ideal analyzers and the results obta ined in a decay exper iment are correctly related by the theory ,2, Bell's inequali t ies appl ied to the ideal analyzers can be used to calculate corresponding restric- t ions on the angular d is t r ibut ion of the pions coming from the z decays.

Another interesting possibi l i ty is the test of the Bohm-Aharonov hypothesis discussed in section 2.

,2 Note that assumption (2) corresponds to the correctness of eq. (9), which has been recently tested with the measure- ments of the z polarization at the Z ° peak [16], and of eq. (10) .

When the two z's are separated by some large dis- tance, they get polar ized in a random direction, and the spontaneous decay of one z will not affect the decay of the other z, which results in a sensible decrease of the two pion correlations.

These tests can be performed for r + z - product ion energies in all the range from threshold to the Z ° mass, which provides increasing spatial separat ion of the two ~'s ( ? ,~ 1-25) , giving, thus, addi t ional interest to the measurement .

5. I. Threshold production qf z pairs

In e lec t ron-pos i t ron annihi la t ion near the z threshold (Ebcam ~ me), Z pairs are produced almost at rest. The two-body decay z-~nv~ will, thus, provide monochromat ic pions, and the z spin correlat ion re- suits in an angular correlat ion between n + and n - directions.

The expected correlat ion rate [ 17 ] is given by

N(/~, ,/~2) sc 1 t - - ~p~ ",02, ( 12 )

where /~ and ~2 are unit vectors respectively in the direct ion of mot ion o f~ + and n - .

The te rm/~ . /~2=cos 0~2 in the correlat ion rate is equivalent to the Bell inequal i ty violat ing expecta- t ion value of Bohm's Gedankenexper iment (cf. eq. ( 3 ) ) . If one writes the correlat ion rate as ( A - B COS 012 ), the experimental measurement of the slope B/A will be a test of quan tum mechanics, since B/A is l imited by Bell's inequali ty to no more than 1 /x /2 of the value predicted by quantum mechanics ( ~ ).

Taking the Bohm-A ha ronov hypothesis that the z becomes polar ized in a random direct ion .~, an addi- tional factor ½ is expected for the slope parameter , since one has to average over a term of the type

( 1 --J~'/~l) ( 1 +a?@2). A Monte Carlo [ 18] dis tr ibut ion, generated with

105 r~+r~ - at Eb~am= 1.785 GeV, is shown in fig. 4, together with the upper l imit of Bell's inequality, and the Bohm-A ha ronov prediction.

Such an experiment , as well as other independent tests ~3, could be actually realized at a z -cha rm factory [ 20 ], where one expects of the order of 107 z pairs per year. The correspondent 105 events with both z's decaying into a pion and a neutrino, would exclude

~3 A deta i led analysis is discussed in ref. [ 19].

176


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1250

1 ooo a)[ - 0 .8 - 0 .6 - 0 .4 -0 .2 0 0.2 0.4 0.6 0.8

COS 19~2

Fig, 4. ~+x- correlation rate versus c o s 012: (a) quantum mechanics prediction, (b) the Bell inequality upper limit, (c) Bohm- Aharonov hypothesis. The points are Monte Carlo data corresponding to one year of running at a z-charm factory ( 105 x+~ ).

Bell's l imit and the B o h m - A h a r o n o v hypothesis at 99.9% confidence level. A valuable test of quan tum mechanics at a z - cha rm factory is, thus, given by the exper imental s tudy of angular correlat ion in the process e+e - ~ z + z - -,rt + 9~ 7t-vx.

5.2. High energy production o f t pairs

At high energy (mJEb~am~0) the spin correlat ion in the z product ion results in a correlat ion of the z helicities: when the z+ has a negative (pos i t ive) helicity the z - has a posi t ive (negat ive) helicity. The decay z--,~tv~ is, thus, sensit ive to the z helicity:

d F ( z + ) = d F o ( 1 T - h + cos 0~) , (13)

where h+ is the z + helicity and 0~ is the angle between the pion m o m e n t u m and the z direction.

The Lorentz boost from the ~ rest frame to the laboratory determines a s imple relat ion between the angle G~ and the energy of the pion measured in the laboratory:

" 2Ebeam(m~+m'~) ~ 2 X , ~ - - l , COS O~ = 4E,~m'; - 2 2 '~ 2 2 (rnz - m ~) x/4Ebe,m --4m~

E~ X~t ~-,~- E b e a m , (14)

where the approx imat ion is val id when terms of the

order ( rn~/m,) 2 and ( mJEb . . . . ) 2 are neglected. The spin dependence of the rate for the process

e + e - - - . z + x - - , r c + % r t - v ~ results, thus, in a strong correlat ion between the pion energies [ 13 ]:

N(x~ , x 2 ) ~ 1 + (2xj - 1 ) ( 2 X 2 - - 1 ) , (15)

where x~ and x2 refer respectively to posit ive and negative pion.

The pion energy-energy correlation reflects only the longitudinal spin correlat ion of the z's, which is in a factorized form (cos O, cos ~ ) and does not violate Bell's inequality.

The complete z spin correlat ion results in an angular correlat ion between ~+ and I t - direct ions in the

rest frame. The expected correlat ion rate [17] is given by

N(/~7, /~) ~ 1 - ½/~I' "/~*, ( 16 )

where/~7 and ~ are unit vectors, defined in the z - rest frame, respectively in the direct ion of mot ion of ~+ and ~ - . In the following, the starred quanti t ies are defined in the z rest frame, with the z axis along the z direction.

The term/~T " /~ - - cos 0T2 can be expressed by ob- servables which are measurable in the laboratory through the following relations [21 ]:

COS 072 =cos 07 cos 0* + s i n 07 sin 0* cos (0T - 0 ~ ) , (17)

cos Ot cos O~ = - cos 6, cos 02,

sin 07 sin O~ cos(¢T - 0 ~ )

+

4m 2 ( = ( m ~ - m ~ ) 2 \ p ip2 cos~/

(18)

(2Eb~am El - - m ~ - - m 2 ) ( 2Eb ~ . . . . E2 - r n ~ 2 _ m 2 ) "~

4E~eam - 4 m ~ ]

(19)

where P1~2) and E~{2) are ~+ ( ~ - ) momen tum and energy and ~, is the angle between the pion direct ions in the laboratory.

Using the same arguments discussed in section 5.1, one can provide interesting tests of quantum mechanics measuring the correlat ion parameter B / A of the exper imental dis t r ibut ion ( A - B cos 01'2 ).

A Monte Carlo [ 18 ] dis tr ibut ion, generated with 1 0 4 Z + Z - events at Eb . . . . = 5 GeV, is shown in fig. 5,

177

Volume 275, number 1.2 PHYSICS LETTERS B 23 January 1992

500

~) 275

~o 250

~ 225

~" 200

175

150

125

1 O0

" +T

} "r

- o s -0 .6 -0 .4 - o ~ o o.2 0.4 o.6 o.s 1

c o s v~;,

Fig. 5. rc+n - correlation rate versus cos 0T2: (a) quantum mechanics prediction, (b) the Bell inequality upper limit, (c) Bohm- Aharonov hypothesis. The points are Monte Carlo data corresponding to the statistics collected by existing experiments ( 10 4

~ + ~ ).

together with the upper l imit of Bell's inequality, and the Bohm-Aha ronov predict ion.

The high statistics collected by the A R G U S and CLEO Collaborations [22 ] (around 106 r + r - ) could already exclude the B o h m - A h a r o n o v hypothesis at 95% confidence level, but more statistics is needed to exclude Bell's limit. It could also provide the first evidence for spin correlat ion in r pair product ion.

A future B factory could produce of the order of 107 r pairs per year, with a sensit ivity to the correlation parameter B/A equal to that of a z -charm factory.

5. 3. Production q f r pairs at the Z ° peak

At center of mass energies a round the Z ° mass ( ~ 91.17 GeV) , e lec t ron-pos i t ron annihi la t ion is domina ted by the Z ° resonance. The process Z ° ~ z + z - produces a polar izat ion [23] of the r lepton, due to the difference in the coupling of the Z ° to r ight-handed and left-handed ~+:

P~±~+_2 a~u~ a ~ + v { . ~ + 2 ( l - 4 s i n 0w), (20)

where a~ and v~ are the x axial and vector-axial coupling constants, and 0,~ is the electroweak mixing angle.

The unique possibil i ty to determine final state hel-

icities using the spin analyzing decay of the r allows the measurement of the small polar izat ion P~ ( P ~ - 0 . 1 6 for sin 2 0w=0.23).

The measurement of P: at the Z ° peak has been recently performed by LEP detectors [16], providing evidence for a small negative value of the r polarizat ion.

The existence of a net polar izat ion results in some addit ional terms, linear in the polarimeter vectors h ~, in the differential cross section (cf. eq. (10) ). Thus, the n+n energy-energy correlat ion function contains also a term propor t ional to P#

N ( & , x 2 ) ~ I + ( 2 & - l ) ( 2 x 2 - 1 )

- 2 P , ( x , + x 2 - 1 ) , (21)

where .v~ and x2 are the posit ive and negative pion normal ized energies defined in eq. (14) .

The pion energy-energy correlat ion contains only the longitudinal spin correlat ion of the r 's, whose factorized form (cos 0j cos ~ ) does not violate Bell's inequality.

The = + ~ - correlat ion rate with respect to cos OT2 as defined in eq. (17) recuperates the full x spin correlation. Due to the Z ° exchange, the transverse spin correlat ion term sin 0T sin 0* c o s ( ~ T - ¢ ~ ) contains a factor

a~ --v~ to= a~ +u~ (22)

However, since ~c~ I (~c=0.987 for sin20w=0.23), the effect is negligible for the specific tests discussed here, and the n+n - correlat ion rate is still given by eq. ( 16 ). This is shown by the Monte Carlo [ 24 ] distri- but ion of fig. 6, generated with 10 ~ n+n - events at Eb ..... = Mzo/2 .

The spin correlat ion in the process e + e - - , Z ° - ~ r+z - ~ = + % n-v~ provides many interesting physics possibil i t ies [21 ]: the measurement of P~ and a test for maximal P, maximal C violat ion are amongst them.

Following the discussion of section 5.1, interesting tests of quantum mechanics can be performed by ex- tracting the correlat ion parameter B/A of the experimental dis t r ibut ion ( A - B cos 0T2) and comparing the result with the Bell inequali ty upper l imit and the Bohm-A ha ronov hypothesis.

The high luminosi ty option [25] at the LEP colli-

178


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2000 Z

1750 £

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- - o . s - o . s - 0 . 4 o.2 0 0,2 0.4 o.s os c o s ~ ; :

Fig. 6. s+~ - correlation rate versus cos 072: (a) quantum mechanics prediction, (b) the Bell inequality upper limit, (c) Bohm- Aharonov hypothesis. The points represent the Monte Carlo prediction for 105 x+Tt- generated with sin 2 0w=0.23.

der could produce a round 0.3 × l 0 7 T pairs per year, giving a sensit ivity comparable with a z -cha rm or a B factory.

6. C o n c l u s i o n s

Since the original EPR argument was formulated, several tests of the incompat ib i l i ty between quantum theory and local realism have been performed, almost exclusively in the domain of a tomic physics. An analysis of the EPR "pa radox" in different fields and physical processes leads to a more critical under- standing of the EPR problem.

In this context, z pair product ion through elec- t ron -pos i t ron annihi la t ion has been proposed as an example of EPR experiment . The strong spin correlation between the two z's contains, in fact, the non-

local character of the EPR argument. The subsequent decay works as a spin analyzer, and the non loca l

correlat ion is t ransferred to the decay products , since the decay of one of the z's is influenced by how the other has decayed (or will decay) .

The possibi l i ty of using these propert ies to make a test of quantum mechanics versus local hidden vari- able theories has been investigated. In part icular, the sequential z product ion and decay e + e - ~ z + ~ -~

+ 9~ ~-v~ has been analyzed as a Clauser -Horne type

of experiment . With proper assumptions, l imits from Bell's inequalities are derived and can be easily tested. The B o h m - A h a r o n o v hypothesis of a breakdown of the system state after the two z's have separated is shown to result in a significant decrease of the predicted p ion -p ion correlation.

Such studies should be performed for z product ion energies between threshold and the Z ° mass, in order to test the non loca l z spin correlat ion for increasing spatial separat ion of the two z's.

Future accelerators ( z - cha rm factory, B factory, high luminosi ty option for LEP) will provide high statistics samples of z+z - pairs, and precision mea- surements of the degree of correlat ion will be easily achieved. Available data could already be looked into.

Valuable tests of quantum mechanics can, thus, be obta ined from the study of decay correlat ions in e + e - - , z + ~ , giving a complementary insight of tile EPR problem in the field of particle physics.

A c k n o w l e d g e m e n t

The author acknowledges useful discussions with U. Amaldi , T. Camporesi , W. de Boer, M. Feindt and J. Kirkby. Many theoretical points have been clari- fied by helpful discussions with A. de Rfijula and F. Selleri. He thanks S. Jadach and Z. Was for the ~ pair Monte Carlo programs and cl'arifying comments on spin correlations.

N o t e a d d e d in p r o o f

It should be noted that from an exper imental study of e + e - ~ z + ~ - - ~ T c + 9 ~ - v ~ which finds agreement with the quantum mechanical predict ions one cannot exclude local h idden-var iable theories in a general way. In fact, it is always possible to find a local model which reproduces all the quantum mechanical predict ions for exper iments where a secondary process (z or A decay, Compton scattering of photons) is used to measure spin correlations. The test of the Bell inequali ty with addi t ional assumptions proposed in this paper has, thus, to be unders tood as a test of classes of local h idden-var iable models. In this respect, the K O R A L Z Monte Carlo program [24], which was used to produce fig. 6 and gave a result

179


c o n s i s t e n t w i th q u a n t u m m e c h a n i c s , is a va l id ex-

ample , T h e K O R A L Z M o n t e Ca r lo desc r ibes the

z + z - pa i r p r o d u c t i o n a n d decay t a k i n g an i n c o h e r e n t

s u m of the va r i ous he l ic i ty s tates, a n d is, thus , a local

real is t ic m o d e l o f the process . H o w e v e r , t he expl ic i t

m e a s u r e m e n t o f any t r a n s v e r s e sp in c o r r e l a t i o n ( o n e

cou ld use, for example , the t r a n s v e r s e spin a s y m m e -

try de f i ned in ref. [ 13 ] ) wou ld d i s t i n g u i s h th i s par-

t i cu la r m o d e l f r o m the q u a n t u m m e c h a n i c a l p red ic -

t ions . T h e a u t h o r t h a n k s Z. W~ts for s t i m u l a t i n g

d i s cus s ions c o n c e r n i n g th i s po in t .

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