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Testing Quantum Mechanics in the Neutral Kaon System

arXiv:hep-ph/9207268v1 29 Jul 1992CERN-TH.6596/92ACT-18/92CTP-TAMU-59/92Testing Quantum Mechanics in the Neutral Kaon SystemJohn Ellis, N.E. Mavromatos and D.V.

Nanopoulos†Theory Division, CERN, CH-1211, Geneva 23, Switzerland.AbstractThe neutral kaon system is a sensitive probe of quantum mechanics.We revive aparametrization of non-quantum-mechanical effects that is motivated by considerations ofthe nature of space-time foam, and show how it can be constrained by new measurementsof KL →2π and KL,S semileptonic decays at LEAR or a φ factory.CERN-TH.6596/92ACT-18/92CTP-TAMU-59/92July 1992† Permanent address: Center for Theoretical Physics, Dept. of Physics,Texas A & M University, College Station, TX 77843-4242, USA, andAstroparticle Physics Group, Houston Advanced Research Center (HARC), TheWoodlands, TX 77381, USA

1Introduction and SummaryThe neutral kaon system is a textbook example of a microscopic quantum-mechanicalsystem that exhibits a rich variety of physical phenomena. Its resolution of the tau-theta puzzle was a manifestation of parity violation [1].

It is still the only placewhere CP violation has been observed in the laboratory [2], and the suppression ofKL →µ+µ−decays was one of the primary motivations for charm [3]. It offers oneof the most sensitive available tests of the CPT invariance that is inherent to localquantum field theory [4].

It has provided very elegant quantum-mechanical inter-ference effects [5]. Indeed, several years ago, two of us (J.E.

and D.V.N.) argued ina paper with Hagelin and Srednicki [6] that the neutral kaon system was, togetherwith long-range neutron interferometry, one of the two most sensitive probes of apossible breakdown of conventional quantum mechanics suggested by investigationsof local field theory in the presence of microscopic event horizons.It was observed some time ago [7] that black holes apparently required a mixedstatistical description, understood intuitively as being due to the loss of informationacross the event horizon.

Hawking later suggested [8] that pure initial states couldevolve into mixed final states in the presence of a microscopic event horizon. Heproposed a density matrix formalism in which ρin and ρout were linearly related by a/S-matrix that could not be factorized as the product of S- and S†-matrix elementsas expected in conventional local quantum field theory:ρout = /Sρin:/S ̸= SS†(1)Ref.

[6] then pointed out that in this case the normal Liouville equation that de-scribes the time-evolution of the quantum-mechanical density matrix would alsorequire modification by the addition of an extra linear term:∂tρ = i[ρ, H] + /δHρ(2)It was shown explicitly that addition of the /δH term would allow an initially purestate to evolve into a mixed state with positive entropy. The extra term in equation(2) is characteristic of open quantum mechanical systems [9].

However, we regardedit as necessary because of the intrinsic impossibility of measurements within theevent horizon.Bounds on this type of non-quantum-mechanical behaviour werederived from the agreement of observations of neutral kaons and of neutrons withconventional quantum mechanics.Both systems were used [6] to obtain similarupper limits on the hadronic matrix elements of /δH of order 10−20 GeV .Although very small in a microscopic system, such effects would be magnified ina macroscopic system with Avogadro’s number of elementary particles [10]. Theycould even engender the transition from quantum-mechanical to classical behaviourin large systems [10].

Indeed, the modification (2) has a form similar to that pos-tulated for this purpose in ref. [11] without any microscopic justification.

Thus the

modification (2) constitutes a possible realization of the idea, advocated more re-cently by Penrose [12], that quantum gravity might explain the classical behaviourof large systems.It could be interesting to test this possibility in macroscopicquantum-mechanical laboratories such as SQUIDs [10].Lately, the possibility of a microscopic violation of the laws of quantum mechan-ics has been re-examined in the context of string theory [13]. Specifically, studiesof scattering and decay processs in a spherically-symmetric string black hole back-ground have not revealed any loss of quantum coherence [14].

We have attributedthis to the presence in string theory of an infinite set of local symmetries that in-clude a W1+∞-algebra [13]. This in turn contains an infinite-dimensional Cartansubalgebra of charges that are in involution with the Hamiltonian, and hence con-served.

Thus they provide an infinite set of W-hair that characterizes the blackhole state, preserves information, and hence maintains quantum coherence. Thuswe find no evidence for the modifications (1,2) of conventional quantum mechanicsin scattering offone particular topologically non-trivial space-time background.However, this does not mean that the S-matrix of quantum field theory andconventional quantum mechanics are sacrosanct.The symmetries that preservequantum coherence relate states with different masses [13] : in particular, theyrelate the light particles that appear in laboratory experiments to Planck massstring states.

Since realistic measurements are conducted with a truncation of thefull physical string spectrum, they do not include all observables. The connectionsbetween light and massive states mean that the former should be considered as anopen system as in equation (2), with the possibility of apparent information loss[15].

Thus it is relevant to test the general formalism (2) also in the context ofstring theory.Two new experimental tools to do this in the neutral kaon system have becomeavailable since ref. [6] was written.

One is the CP-LEAR experiment [16], in whichcopious tagged K0 decays are available, and the other is the DAφNE φ-factory nowunder construction [17], which will provide copious coherent K-K pairs. In bothexperiments, it will be possible to observe CP-violating asymmetries in KS decays,and hence new tests of CPT invariance can be made [18].

The purpose of this paperis to point out that the types of measurements proposed as tests of CPT invariancealso serve as probes for violations of quantum mechanics.In section 2 we remind the reader of basic features of the modification (2) ofquantum mechanics, with reference to the neutral kaon system in which /δH hasthree possible matrix elements to be bounded by experiment [6]. Then, in section3 we show explicitly how two of them can be disentangled by measurements of CP-violating KL,S semileptonic decay asymmetries and KL →2π decays.

One of theseparameters bears a phenomenological resemblance to the CPT-violating parameter

introduced in ref. [18], but the other appears in a different way.

Finally, in section4 we comment on the outlook for such probes of quantum mechanics.2Formalism for the Violation of Quantum Me-chanics in the Neutral kaon System.This is described in the usual quantum-mechanical framework [5] by a phenomeno-logical Hamiltonian with hermitian (mass) and antihermitian (decay) components:H = M −12iΓM∗12 −12iΓ12M12 −12iΓ12M −12iΓ! (3)in the (K0, K0) basis.

When H is not hermitian, the time-evolution of the densitymatrix ρ is ordinarily given by∂tρ = −i(Hρ −ρH†)(4)and the state is pure if Trρ2 =(Trρ)2, which it remains forever it started pure. Wedefine components of ρ and H byρ≡12ρασαH≡12hβσβ(5)where we use the Pauli σ-matrix basis, and the ρα are real but the hβ are complex.It is convenient for our subsequent discussion to use the CP eigenstate basisK1,2 =1√2(K0 ± K0), in which we can represent the ordinary evolution (4) by∂tρα = hαβρβ wherehαβ =−Γ−ReΓ12ImΓ120−ReΓ12−Γ0−2ImM12ImΓ120−Γ−2ReM1202ImM122ReM12−Γ(6)At large t, ρ decays exponentially toρ ∝ 1ǫ∗ǫ|ǫ|2!

(7)which corresponds to the usual pure long-lived mass eigenstate KL, with the CPimpurity parameter ǫ given byǫ =12iImΓ12 −ImM1212∆Γ −i∆M(8)where ∆M ≡ML −MS is positive and ∆Γ ≡ΓL −ΓS is negative.

We now consider the possible addition to hαβ (6) of a modification of the form (2),which we parametrize as /hαβ. As discussed in ref.

[6], we assume that the dominantviolations of quantum mechanics conserve strangeness, in which case /h1α = 0, andtherefore that /h0α = 0 to conserve probability. One can show that /hαβ must be anegative matrix, and hence in turn that /hα1 = /hα0 = 0.

Therefore we arrive at thegeneral parametrization [6]hαβ =0000000000−2α−2β00−2β−2γ(9)where the negativity of /hαβ further imposes α, γ > 0 and αγ > β2. The equationsof motion for the components of ρ are∂tρ11=−(Γ + ReΓ12)ρ11 −γ(ρ11 −ρ22) −2ImM12Reρ12 −(ImΓ12 + 2β)Imρ12∂tρ12=−(Γ −2iReM12)ρ12 −2iαImρ12 + (ImM12 −12iImΓ12 −iβ)ρ11−(ImM12 + 12iImΓ12 −iβ)ρ22∂tρ22=−(Γ −ReΓ12)ρ22 + γ(ρ11 −ρ22) + 2ImM12Reρ12−(ImΓ12 −2β)Imρ12(10)and it is clearly possible in principle to determine independently the three parame-ters α, β, γ by measurements of the evolution of the density matrix over all times.3Constraining the Parameters that violate Quan-tum MechanicsSince the time-evolution (10) is described by a 4 × 4 linear matrix equation, thegeneral solution can be written in the formρα(t) =4Xj=1cαjexp(λjt)(11)where the coefficients cαj depend on the initial conditions, e.g., tagged K0 or K0beam.

However, it is clear that the large-time behaviour is dominated by the eigen-vector whose eigenvalue λj has the least negative real part, corresponding to theconventional KL component. On the other hand, the eigenvector whose eigenvaluehas the most negative real part, corresponding to the conventional KS component,can only be probed at short times.

Interference effects at intermediate times canin principle probe the other two eigenvectors.The feasibility of this possibilitydepends crucially on the nature of the experiment, and we will not discuss such

interference measurements here. Nor we will discuss measurements where the cor-relations between K0 and K0 particles emanating from φ decay play a crucial way.We will concentrate on the information that can be obtained from measurements ofindividual kaons at large and small times.It is easy to check that, for large t, ρ decays exponentially to [6]ρ ∝1−12 i(ImΓ12+2β)−ImM1212∆Γ+i∆M12 i(ImΓ12+2β)−ImM1212 ∆Γ−i∆M|ǫ|2 +γ∆Γ −4βImM12(∆M/∆Γ)+β214∆Γ2+∆M2(12)where the CP impurity parameter ǫ is given as usual by equation (8).

The densitymatrix (12) describes a mixed state with Trρ2 < 1 when ρ is normalized so thatTrρ = 1. It corresponds to a mixture of a conventional KL beam with a low-intensityKS beam.

Conversely, if we look for a solution of the time-evolution equation (10)with ρ11 << ρ12 << ρ22, corresponding to what would conventionally be a KS beam,we again find a mixed state:ρ ∝|ǫ|2 +γ|∆Γ| −4βImM12(∆M/∆Γ)+β214 ∆Γ2+∆M2ǫ −iβ∆Γ2 −i∆Mǫ∗+iβ∆Γ2 +i∆M1(13)We note that the signs of the terms in (13) that are linear in β, relative to thoseof ImM12 and ImΓ12, are reversed with respect to the corresponding terms in the“KL” density matrix (12).The experimental value of an observable O is given in this formalism by⟨O⟩= Tr(Oρ)(14)as for a conventional mixed quantum-mechanical state. The K →2π observable isrepresented in our K1,2 basis byO2π = 0001!

(15)Therefore the rate of K →2π decays is given in the long lifetime “KL” limit by|ǫ|2 + γ∆Γ −4βImM12(∆M/∆Γ) + β214∆Γ2 + ∆M2(16)and is hence not a direct measurement of the CP-violating parameter ǫ. Previously[6], we discarded the β-dependent terms in (16), assuming that β was similar inmagnitude to γ.

Here we will be more general, keeping the term in (16) that islinear in β.

Other observables that are useful in constraining theories of CP violation and, inour case, looking for a deviation from quantum mechanics, are semileptonic K0/K0decays. The K →π−l+ν observable isOπ−l+ν = 1111!

(17)whilst the K →π+l−ν observable isOπ+l−ν = 1−1−11! (18)Hence the CP-violating observable δ is given byδ ≡Γ(π−l+ν) −Γ(π+l−ν)Γ(π−l+ν) + Γ(π+l−ν)(19)in both the “KL” and “KS” limits (12) and (13).

In the usual quantum-mechanicalformalism, we would simply findδ ≃2Reǫ(20)where the phase φǫ of ǫ is determined with high precision, from measurements of∆Γ and ∆M. However, using equations (12, 13, 19) we find thatδL,S ≃2Re[ǫ(1 −iβImM12)], 2Re[ǫ(1 +iβImM12)](21)So far, there has not been any high-statistics measurement of δS.

Checks of thestandard phenomenology of CP violation have been made by combining measure-ments of δL and K →2π decays in the long lifetime limit. In our case, comparing(16) and (21), we see that(δ24 −R2πcos2φǫ) = −γ|∆Γ|cos2φǫ −β|∆Γ|8|ǫ|cos2φǫsinφǫ(22)where R2π ≡BR(K →2π).Thus measurements of these two quantities cannotdetermine both β and γ, and the basic geometry of the problem is shown in the figure.Any discrepancy between the measurements of “ǫ” from K →2π decays and of“Re ǫ” from semileptonic K decays could be taken as evidence against conventionalquantum mechanics, but its origin would be ambiguous.

In fact, putting in the latestexperimental values [19] √R2π ≃|ǫ| = (2.265 ± 0.023) × 10−3, φǫ = 43.73 ± 0.15o,and δ = (3.27 ± 0.12) × 10−3, we find,(−0.006 ± 0.204) × 10−6 = −0.522 γ∆Γ −(6.54 × 10−3) β∆Γ(23)

and hence there is currently no evidence for a violation of quantum mechanics.If we ignore the contribution of β in (23), and use the experimental value ∆Γ =737 × 10−17GeV [19], we find thatγ = (0.01 ± 0.39) × 10−6|∆Γ| ≃(0.1 ± 3) × 10−22GeV(24)whereas we findβ = (0.01 ± 0.31) × 10−4|∆Γ| ≃(1 ± 23) × 10−20GeV(25)if we ignore the contribution of γ in (23).It is possible to disentangle the parameters β and γ by also measuring the semilep-tonic asymmetry in “KS” decays. The geometrical rˆole of this measurement is alsoshown in the figure.

Taking the difference between the two asymmetries in (21), wefind thatδL −δS =8β|∆Γ|sinφǫq1 + sin2 φǫ=8β|∆Γ|sinφǫcosφǫ(26)Therefore a measurement of the difference between the semileptonic decay asym-metries in the long- and short-lifetime limits is directly sensitive to the parameterβ that violates quantum mechanics. This measurement has also been mentionedpreviously as a way to look for a violation of CPT invariance [18].

This is hardlya coincidence, since it is known that a breakdown of quantum mechanics leads ingeneral to a weakened form of the CPT theorem of conventional local field theory[20]. On the other hand, we have seen that another quantum-mechanics-violatingparameter γ does not have the same CPT-violating signature.As mentioned above, it is in principle possible to determine also the parameterα by measurements in the intermediate time region where other eigenvectors of thetime-evolution matrix equation (11) play a rˆole.

Correlation measurements may alsobe interesting. However, we will not discuss these possibilities here.4OutlookWe have shown in this paper that the neutral kaon system is a uniquely preciseand sensitive microscopic probe for possible violations of quantum mechanics.

Itis possible to set up a theoretically-motivated and well-defined parametrization ofnon-quantum-mechanical terms in the time-evolution equation for the density ma-trix of the neutral kaon system [6]. High-precision experiments already constrainthese parameters (25), and have the exciting prospect of further constraining themin the future (26).

The interpretation of the bound (25) would benefit from a the-oretical estimate of the likely magnitude of non-Hamiltonian matrix elements thatviolate quantum mechanics. A priori, one might expect any such matrix elements

in individual hadronic states to be suppressed by some power of mproton/MP lanck, al-though we are not yet in a position to calculate them. As we have mentioned earlier,quantum coherence is maintained in string theory by virtue of symmetries linkinglight particles to massive states [13], and such apparently non-quantum-mechanicalterms can arise when unmeasured observables associated with massive string statesare summed over [15].

Therefore we consider it very important to take a phenomeno-logical attitude, and analyze this possibility from a strictly experimental point ofview. The present CP-LEAR and future DAφNE experiments are well-placed to con-tribute to this programme, since they have the possibility to measure an asymmetryin semileptonic decays in the short life-time limit, as well as examine interferenceeffects that can in principle unravel all the non-quantum-mechanical parameters.AcknowledgementsOur interest in this topic was revived by a talk given by L. Maiani at the FirstCERN-Torino Meeting on Current Trends in Particle and Condensed Matter Physics.J.E.

and N.E.M. thank L. Maiani for encouraging discussions and L. Alvarez-Gaum´eand S. Fubini for providing the stimulating environment of this meeting.

The workof D.V.N. is partially supported by DOE grant DE-FG05-91-ER-40633 and by agrant from Conoco Inc.References[1] T.D.

Lee and C.N. Yang, Phys.

Rev. 104 (1956), 254.

[2] J.H. Christenson, J.W.

Cronin, V.L. Fitch and R. Turlay, Phys.

Rev. Lett.

13(1964), 138. [3] S.L.

Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev.

D2 (1970), 1285. [4] G. Luders, Ann.

Phys. (N.Y.) 2, (1957), 1.

[5] J.S. Bell and J. Steinberger, Proc.

1965 Oxford Int. Conf.

on Elementary Par-ticles (Oxford, 1966), p. 165.

[6] J. Ellis, J.S. Hagelin, D.V.

Nanopoulos and M. Srednicki, Nucl. Phys.

B241(1984), 381. [7] S. Hawking, Comm.

Math. Phys.

43 (1975), 199;J. Bekenstein, Phys. Rev.

D12 (1975), 3077. [8] S. Hawking, Comm.

Math. Phys.

87 (1982), 395. [9] Y.R.

Shen, Phys. Rev.

155 (1967), 921;A.S. Davydov and A.A. Serikov, Phys. Stat.

Sol. B51 (1972), 57;

B.Ya. Zel’dovich, A.M. Perelomov, and V.S.

Popov, Sov. Phys.

JETP 28 (1969),308;For a recent review see : V. Gorini et al., Rep. Math.

Phys. Vol.

13 (1978), 149. [10] J. Ellis, S. Mohanty and D.V.

Nanopoulos, Phys. Lett.

B221 (1989), 113; ibidB235 (1990), 305. [11] G.C.

Ghirardi, A. Rimini and T. Weber, Phys. Rev.

D34 (1986), 470; G.C.Ghirardi, O. Nicrosini, A. Rimini and T. Weber, Nuov. Cim.

102B (1988), 383. [12] R. Penrose, “The Emperor’s New Mind” (Oxford University Press, 1989).

[13] J. Ellis, N.E. Mavromatos and D.V.

Nanopoulos, Phys. Lett.

B267 (1991), 465;ibid B272 (1991), 261; B284 (1992), 27, 43. [14] E. Witten, Phys.

Rev. D44 (1991), 314;K. Demeterfi, A. Jevicki and J. Rodrigues, Nucl.

Phys. B365 (1991), 499;I. Klebanov and A.M. Polyakov, Mod.

Phys. Lett.

A6 (1991), 3273;A.M. Polyakov, Princeton Univ. preprint PUPT-1289 (1991);M. Bershadsky and D. Kutasov, Phys.

Lett. B266 (1991), 345.

[15] J. Ellis, N.E. Mavromatos and D.V.

Nanopoulos, CERN and Texas A & MUniv. preprint, CERN-TH.6595/92; CTP-TAMU-58/92; ACT-17/92 (preced-ing paper).

[16] R. Adler et al., CP-LEAR Collaboration, CERN-preprints CERN-PPE-91-119(1991); CERN-PPE-92-68 (1992). [17] Daφne Physics Handbook, edited by L. Maiani, L. Pancheri and N. Paver (INFN,Frascati, 1992).

[18] C.O. Dib, R.D.

Peccei et al., UCLA preprint UCLA-91-TEP-48 (1991). [19] Review of particle properties, Particle Data Group, Phys.

Rev. D45 (No 11,part II) (1992), 1.

[20] D.N. Page, Gen. Rel.

Grav. 14 (1982), 299;L. Alvarez-Gaum´e and C. Gomez, Comm.

Math. Phys.

89 (1983), 235.

Figure CaptionThe geometry of the tests of quantum mechanics proposed in this paper. The rateR2π of KL →2π decays is not just given by the magnitude |ǫ| of the CP-violatingmass mixing parameter [see equation (16)] and the CP-violating KS,L leptonic decayasymmetries δS,L are not just 2Re ǫ [see equation (21)].

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