We describe an experimental test of whether particle decay causes wave
function collapse. The test uses interference between two well separated, but
coherent, sources of vector mesons. The short-lived mesons decay before their
wave functions can overlap, so any interference must involve identical final
states. Unlike previous tests of nonlocality, the interference involves
continuous variables, momentum and position. Interference can only occur if the
wave function retains amplitudes for all possible decays. The interference can
be studied through the transverse momentum spectrum of the reconstructed
mesons.Comment: Slightly revised version, to appear in Phys. Lett. A. 11 pgs.,
  including 2 figure

Adler

Apostolakis

Baltz

Bauer

Bell

Bertlmann

Einstein

Grangier

Horne

Jackson

Joakim Nystrand

Katzenstein

Klein

Meissner

Müller

Parkins

Spencer R. Klein

Stabler

Vidovic

English

arXiv

Spencer R.  Klein

Crossref

Does particle decay cause wave function collapse: an experimental test

Klein, Spencer R.

Nystrand, Joakim

eScholarship - University of California

Lawrence Berkeley National LaboratoryRecent WorkTitleDoes particle decay cause wave function collapse: An experimental testPermalinkhttps://escholarship.org/uc/item/2mq2g0f2JournalPhysics Letters A, 308(5/6/2008)AuthorsKlein, Spencer R.Nystrand, JoakimPublication Date2003eScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaarXiv:quant-ph/0206060 v2   8 Jan 2003Does particle decay cause wave function collapse:An experimental testSpencer R. Klein1 and Joakim Nystrand21 Lawrence Berkeley National Laboratory, Berkeley, CA 947202 Department of Physics, Lund University, Lund, SwedenAbstractWe describe an experimental test of whether particle decay causes wavefunction collapse. The test uses interference between two well separated, butcoherent, sources of vector mesons. The short-lived mesons decay before theirwave functions can overlap, so any interference must involve identical finalstates. Unlike previous tests of nonlocality, the interference involves contin-uous variables, momentum and position. Interference can only occur if thewave function retains amplitudes for all possible decays. The interference canbe studied through the transverse momentum spectrum of the reconstructedmesons.Typeset using REVTEX1In 1935, Einstein, Podolsky and Rosen (EPR) showed that quantum mechanics requiredthat wave functions can be non-local [1]. When a system is observed, the wave functioncollapses from one which contains amplitudes for a host of possible outcomes to smallerset of possibilities, in accord with the measurement. This collapse is instantaneous; muchhas been written about its superluminous nature. Most studies of the EPR paradox havetested Bell’s inequality [2] using spin correlations, usually with photons produced in pairs[3]. Experimenters measure the spin correlations using two polarizers with a varying anglebetween them. Bell found that models with non-local wave functions and models with hiddenvariables produced different angular correlation spectra. Previous tests of non-locality useddiscrete variables like ’pseudo-spin’ for CP violation, as with studies using the reactionΦ → K+K− [4].We describe a very different system that, in contrast to the KLKS system, is sensitiveto the collapse of continuous variables in a wave function [5]. Short-lived vector mesons(VMs) are produced with a fixed phase relationship at two separated sources. Even thoughthe mesons do not come from a single source, and, in fact, share no common history [6],the system acts as an interferometer. The meson lifetimes are short compared to the sourceseparation, so the mesons decay before their wave functions can spatially overlap.Any interference between the two sources must involve the decay products. Interferenceis only possible between identical final states. With the large phase space for final states,interference can only occur if the wave functions retain amplitudes for all possible decaychannels and angular distributions long after the decay takes place [7]. We have previouslycalculated the interference pattern [8]. This letter will focus on the effects of the wavefunction collapse and Bells inequality-like tests, and sketch an alternate derivation of theinterference, to emphasize the symmetries of the system.Figure 1 shows electromagnetic VM production in relativistic heavy ion collisions at largeimpact parameters, ~b. A photon from the electromagnetic field of one nucleus fluctuates toa virtual quark-anti-quark pair which elastically scatters from the other nucleus, emergingas a real vector meson [9]. Either nucleus can emit the vector meson. The momentum2transfers from the nuclei are similar, and they remain in the ground state, so it is impossibleto determine which nucleus emitted the photon and which is the target.The electromagnetic interaction (photon) has a long range, while the elastic scatteringhas a short range, around 0.6 fm [10], far smaller than the < 7 fm radius of a heavy nucleusor the typical impact parameter. So, VM production takes place essentially ‘on top of’ theemitting nucleus, and the two nuclei act as a two-source interferometer.Electromagnetic VM production is studied at the Relativistic Heavy Ion Collider (RHIC)at Brookhaven National Laboratory, where gold ions collide at center of mass energies upto 200 GeV per nucleon. Starting in 2006, the Large Hadron Collider (LHC) at CERN willcollide lead ions at a center of mass energy of 5.5 TeV per nucleon.The cross sections were previously calculated [11] using the Glauber approach [9] withthe photon spectrum given by the Weizsäcker-Williams virtual photon method [12]. Thecalculated photonuclear cross sections agree with data to within 20%. The cross sectionsare large, about 10% of the hadronic cross section at RHIC, rising to 50% at the LHC. Thecorresponding production rates, more than 100 ρ0/sec at RHIC, rising to 230,000 ρ0/sec atthe LHC, are large enough that it will be easy to collect adequate statistics to study wavefunction collapse. Already, the STAR collaboration [13] has observed more than 10,000 ρ0,which should be enough to observe the interference.The impact parameters for these interaction are large compared with the nuclear radii,RA. For ρ and ω production, the median impact parameter 〈b〉 is about 40 fm at RHIC,rising to 300 fm at the LHC; for the J/ψ, 〈b〉 rises from 23 fm at RHIC to about 50 fmat the LHC. All are much larger than RA ≈ 7 fm for heavy ions. It is possible to selectevents with smaller 〈b〉, but still with b > 2RA, by choosing events where VM production isaccompanied by nuclear breakup [14].The 〈b〉 are larger than the distance travelled by most VMs before they decay. The VMlifetimes τ range from 4 × 10−24s for the ρ up to 7.5 × 10−21s for the J/ψ. The VM areproduced with typical transverse momentum pT ≈ 2h̄/RA ≈ 60MeV/c; at mid-rapidity, thelongitudinal momentum is zero, so VM have a median decay distance d = 2h̄cτ/RAMV .3Except for the J/ψ, d 〈b〉; for the J/ψ at the LHC, d ≈ 〈b〉.The final state wave function from ion source i at a time t can be expressed schematicallyψ(t)i = exp (−t/2τ) |V > +(1 − exp (−t/2τ)) |DP > (1)where τ is the vector meson lifetime, |V > is the vector meson wave function, and |DP >is the final state. For stable particles, τ = ∞, the decay products drop out, leaving aconventional two-source interferometer.The interference can be seen by examining the symmetries of the system. The totalamplitude AT for observing the VM with momentum ~p at position ~r, and time, t, dependson the production amplitude A(~p, ~x, t′) and a propagator P (~p, ~x, t′, ~r, t) which transports themeson from ~x′, t′ to ~r, t:AT (~p,~r, t) =∫A(~p, ~x′, t′)P (~p, ~x, t′, ~r, t)d~xdt′. (2)The production amplitude A(~p,~r, t) depends on the electromagnetic field, E(~x, t′), nucleardensity ρ(~x, t) and the amplitude f(~p,~k) for a photon with momentum ~k to fluctuate to aqq pair and scatter from a nucleon, emerging as a vector meson with momentum ~p:A(~p, ~x′, t′) = f(~p,~k)ρ(~x′, t′)E(~x′, t′) (3)The electromagnetic field at a distance b from a nucleus is a Lorentz-contracted pulse witha width b/γ where γ is the Lorentz boost. When γ  1, the electric and magnetic fields areperpendicular and the overall field may be represented as a stream of almost-real photons,with energies up to h̄γ/b [12]. The photon amplitude is proportional to E(~x′, t′). The scat-tering amplitude is obtained from data; only its symmetries are important here. Absorptionof the nascent ρ0 is neglected, but could be included with an additional position-dependentvariable, effectively modifying ρ(x, t′).At large distances, the propagator for a VM with energy ω =√M2V + |~p|2 may bemodelled with a plane wave. Neglecting, for now, VM decays,P (~p, ~x, t′, ~r, t) = ei(~p·(~r−~x)−ω(t−t′)) (4)4The nuclear density is symmetric around the center of mass (origin), giving it positiveparity, while the antisymmetric electric field has negative parity): ρ(~x, t′) = ρ(−~x, t′) andE(~x, t′) = −E(−~x, t′). With this, the range of integration in Eq. (2) can be restricted to asingle nucleus:AT (~p,~r, t) =∫y>0d~xdt′ρ(~x, t′)E(~x, t′)ei(~p·~r−ω(t−t′))[f(~p,~k)ei~p·~x − f(~p,−~k)e−i~p·~x] (5)The only differences between the two nuclei are the phases ±i~p ·~x and between f(~p,−~k) andf(~p,~k). The latter is because the sign of pz reduces the symmetry of the system. Of course,interference is only significant when |f(~p,~k)| ≈ |f(~p,−~k)| which occurs near pz = 0.The equation simplifies by defining ~x = ~b/2 + ~x′. The bulk of the cross section isfrom when the photon couples coherently to the target nucleus, i.e. when ~k · ~x′  h̄, sothe exponential phase is constant over the nucleus. Then, the maximum transverse andlongitudinal momenta are h̄/RA and γh̄/RA. Emitted photons are subject to similar limits[15]. Near pz = 0, the photon momentum, and the momentum exchange due to the scatteringare very similar so it isn’t possible to determine which nucleus emitted the photon, and whichwas the scatterer; in fact, at ~p = 0, the two momentum transfers are equal and opposite.The electromagnetic pulse lasts a time, b/cγ, which may be slightly longer than onephoton period (h̄/ω). This partial temporal incoherence will reduce the overall productionamplitude. There is a pairwise cancellation between space-time volume elements d~xdt′ atpositions ~x and −~x, so the interference is not affected.With this, ~p · ~x = ~p ·~b/2 andAT (~p,~r, t) =∫y>0d~x′dt′ρ(~x′, t′)E(~x′, t′)ei(~p·~r−ω(t−t′))[f(~p,~k)ei~p·~b/2 − f(~p,−~k)e−i~p·~b/2]. (6)We now introduce a few approximations. The amplitude for production from the firstnucleus is A1(~p,~r, t) =∫y>0 d~xdt′ρ(~x, t′)E(~x, t′)f(~p,~k) and we define c to be the ratio of theamplitudes for production from the two nuclei:c(pz) =∫y>0 d~xdt′ρ(~x, t′)E(~x, t′)f(~p,−~k)∫y>0 d~xdt′ρ(~x, t′)E(~x, t′)f(~p,~k). (7)5The ratio f(~p,−~k)/f(~p,~k) does not vary significantly over the nucleus, so the single nucleusproduction amplitude factors out of the integral in Eq. (6). The transverse momenta do notaffect c, and kz = M2V /4pz. Then,AT (~p,~r, t) = A1(~p,~r, t)[ei~p·~b/2 − c(pz)e−i~p·~b/2]. (8)The amplitude factorizes into a magnitude and an interference term. The pT dependence ofA1(~p,~r, t) is dominated by the nuclear form factors, with the bulk of the production havingpT < 2h̄/RA. Most of the uncertainties discussed earlier do not affect the interference term.The time and z variation in E(~x, t′) should be largely independent of k. In the soft Pomeronmodel, the photon to VM coupling increases slowly with k and has an almost constant phase.At RHIC, a photon-meson term is also present, but the phase of c still changes only slowlywith k [9].The interference is clearest when pz=0. Then c = 1 and the approximations introducedin defining c disappear. The amplitude is AT (~p,~r, t) = 2iA0 sin (~p ·~b/2). and the crosssection isσ ∼ |AT (~p,~r, t)|2 = 2A20[1 − cos (~p ·~b)]. (9)This formula applies for stable particle emission, such as bremsstrahlung photon emissionin e−e− [16] and pp collisions [17].For short-lived particles, the situation is more interesting. We can express the crosssection in Eq. 9 as a sum of a coherent (interfering) and an incoherent (non-interfering)term2A20[1 − (1 − η) · cos (~p ·~b)], (10)where η measures the degree of decoherence, as has been done for the system Φ → K0K0[4,18]. Complete quantum mechanical coherence between the two sources here means η = 0.If the interference is restricted to the time the system spends in the vector meson (parent)state, then one would expect partial decoherence. According to Eq. 1, only a fraction6exp(−(MV b)/(ωτ)) of the vector mesons will have survived long enough for the amplitude topropagate the distance b between the nuclei before the decay. This scenario thus correspondstoη = 1 − exp(−MV bωτ). (11)For the ρ, ω, and φ, which all have cτ < b >, η ≈ 0 and one expects almost completedecoherence in this scenario. For the J/Ψ, on the other hand, cτ ≈< b >, so the decoherencewould be only partial. This scenario could therefore be distinguished by observing the J/Ψ,while for the lighter vector mesons it would essentially be indistinguishable from a scenariowith no interference.However, there is a broader issue. A distant observer sees the decay products from theoriginal meson, and most VM have many decay modes. The final state may be written|DP >= Σjαj |DPj >, where αj is the amplitude for final state |DPj >. For example,the J/ψ can decay to e+e− or π+π−π0, among many possibilities. The probability for anyspecific final state is small, less than 7%.If the wave function contains amplitudes for every possible decay mode, at times t τ ,the individual particles may be modelled as plane waves and the wave function isΨ(t) =∫ t0dtd2τ[ei~p·(~xd+~b/2) + cei~p·(~xd−~b/2)]e−td/2τ−iωtd |Σj√BrjΨj > (12)where the decays occur at time td and displacement ~xd = (~p/MV )td from the productionpoints. Here, Brj are the branching ratios to different final states, and Ψj is the k−particlefinal stateΨj = Σkei[~pk(~xk− ~xd)−ωj(t−td)]|Ψjk > δ(Σkωk − ω)δ(Σk~pk − ~p). (13)Here ~xk, ωk and ~pk are the particle positions, energy, and momenta, and |Ψjk > includes theparticle types and angular distribution. The δ functions impose 4-momentum conservation.If the decay occurs before the two amplitudes overlap, then either the wave function mustretain amplitudes for all final states, or the two decays will produce different final states,7and the interference must be small. Also, as Eq. (9) shows, amplitudes for different decaytimes must also be included. Thus, the presence or absence of interference tests whetherparticle decay collapses the wave function.The two source terms in Eq. (12) entangle the final state wave functions. The phasesdiffer by exp [i(~p ·~b+ δ)] where δ is the phase of c. Any measurement on one decay productat least partially collapse the wave function of the others [3]. Detectors could accuratelymeasure either the position or momentum of the k final state particles. ’Accurately’, iscompared to the relevant distance (b) or momentum (h̄/b) scales. By these metrics, currentand planned experiments measure momentum accurately, but not position.The interference pattern, Eq. (9) can be seen in the reconstructed VM pT [8]. Becauseb is not generally measurable, σ must be integrated over all b. Figure 2 shows the expectedpT spectrum for ρ0 production at mid-rapidity at RHIC [8]. The large dip for pT < h̄/〈b〉 isa distinctive experimental signature.Alternately, at least in a gedanken experiment, position sensitive detectors could be usedto localize the decay to a single nucleus, provided the ion trajectories are known. The decayJ/ψ → e+e− produces two relatistic electrons that are back-to-back in the transverse plane.For pT = 0, a line between the two measured electron positions will intersect one of the iontrajectories. When the electron ~pk are perpendicular to ~b then it is possible to determinewhich nucleus emitted the VM. The nonzero meson pT introduces some uncertainty, but notenough to encompass both ion trajectories. For detectors 500 fm from the collision point,the pointing uncertainty is 16 fm, less than the 〈b〉 ∼ 50 fm for J/ψ at the LHC.As with existing tests of Bell’s inequality, two detectors, on opposite sides of the collisionregion could randomly measure either position or momentum. All single-detector measure-ments are insensitive to what happens in the other detector. However, when both detectorsmeasured position, the production point could be determined, while when both measuredmomentum, a null at pT = 0 would be seen, showing the interference. These two possi-bilities are only compatible if the wave function collapses when a measurement is made,and not earlier, when the meson decays. For J/ψ → e+e−, the collapse would have to be8superluminal.One possible source of decoherence is the decay timing. The VM are produced nearly atrest, but the decay products may be relativistic. The maximum flight time difference fromthe two sources to a detector is b/c. If the detectors could resolve this time difference, thiscould partially localize the production, reducing the coherence.Since the probability of producing a VM in grazing collisions (b = 2RA) is high, about 1%at RHIC and 3% at the LHC [11], multiple VM production is also observable. Multi-mesonfinal states will exhibit more complicated entanglements, with possibly new behavior.We have described a 2-source interferometer for short lived particles, and showed that itsdescription requires a non-local wave function. The observation of interference will clearlydemonstrate that, after a decay, a systems’ wave function includes amplitudes for all possibledecay modes and angular distributions, and does not collapse to a specific decay modeuntil the wave function is externally observed. Measurements of this interference should beavailable soon. The STAR detector has observed exclusive ρ0 production in gold collisionsat RHIC [13]. Current data should provide higher statistics and an accurate ρ0 pT spectrum,probing the EPR paradox for continuous variables.This work was supported by the U.S. Department of Energy under Contract No. DE-AC-03076SF00098 and by the Swedish Research Council (VR).9REFERENCES[1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).[2] J. S. Bell, Physics 1, 195 (1965).[3] M. A. Horne, A. Shimony and A. Zeilinger, Phys. Rev. Lett. 62, 2209 (1989).[4] A. Apostolakis et al., Phys. Lett. B422, 339 (1998).[5] A. S. Parkins and H. J. Kimble, Phys. Rev. A61, 052104 (2000).[6] P. Grangier, Nature 419, 577 (2002)[7] T. Sudbery, in Quantum Concepts in Space and Time, ed. R. Penrose and C. J. Isham,(Oxford, 1986).[8] S. R. Klein and J. Nystrand, Phys. Rev. Lett. 84, 2330 (2000).[9] T. H. Bauer, R. D. Spital, D. R. Yennie and F. M. Pipkin, Rev. Mod. Phys. 50, 261(1978).[10] B. Müller and A. J. Schramm, Nucl. Phys. A523, 677 (1991).[11] S. R. Klein and J. Nystrand, Phys. Rev. C60, 014903 (1999).[12] J. D. Jackson, Electrodynamics, 2nd edition, Wiley, 1975.[13] C. Adler et al. (STAR Collaboration), nucl-ex/0206004, submitted to Phys. Rev. Lett.;F. Meissner, nucl-ex/0210028.[14] A. Baltz, S. Klein and J. Nystrand, Phys. Rev. Lett. 89, 012301 (2002).[15] M. Vidovic, M. Greiner, C. Best and G. Soff, Phys. Rev. C47, 2308 (1993).[16] J. Katzenstein, Phys. Rev. 78, 161 (1950).[17] R. C. Stabler, Nature 206, 922 (1965).[18] R. A. Bertlmann, W. Grimus and B. C. Hiesmayr, Phys. Rev. D60, 114032 (1999).10FIGURESγγAuAuxzyρ−> ππFIG. 1. Diagram showing ultra-peripheral ρ0 production and decay in heavy ion collisions.The nuclear momenta follow the z axis, and come closest at z = 0, when their separation (impactparameter), ~b follows the y axis.11p⊥ [GeV/c]dn/dp ⊥2Au+Auρ00.20.40.60.810 0.05 0.1 0.15 0.2 0.25FIG. 2. Perpendicular momentum spectra for ρ0 production at RHIC, at pz = 0, for gold ongold collisions at a center of mass energy of 200 GeV per nucleon. Plotted are dN/dpT , withand without interference. The curves are normalized to 1 for pT = 0 and no interference. Thecalculation assumes that the impact parameter is not measured, so the interference is washed out,except for pT < 25 MeV/c.12

Does particle decay cause wave function collapse: An experimental test

UNT (University of North Texas) Digital Library

arXiv:quant-ph/0206060 v2   8 Jan 2003Does particle decay cause wave function collapse:An experimental testSpencer R. Klein1 and Joakim Nystrand21 Lawrence Berkeley National Laboratory, Berkeley, CA 947202 Department of Physics, Lund University, Lund, SwedenAbstractWe describe an experimental test of whether particle decay causes wavefunction collapse. The test uses interference between two well separated, butcoherent, sources of vector mesons. The short-lived mesons decay before theirwave functions can overlap, so any interference must involve identical finalstates. Unlike previous tests of nonlocality, the interference involves contin-uous variables, momentum and position. Interference can only occur if thewave function retains amplitudes for all possible decays. The interference canbe studied through the transverse momentum spectrum of the reconstructedmesons.Typeset using REVTEX1In 1935, Einstein, Podolsky and Rosen (EPR) showed that quantum mechanics requiredthat wave functions can be non-local [1]. When a system is observed, the wave functioncollapses from one which contains amplitudes for a host of possible outcomes to smallerset of possibilities, in accord with the measurement. This collapse is instantaneous; muchhas been written about its superluminous nature. Most studies of the EPR paradox havetested Bell’s inequality [2] using spin correlations, usually with photons produced in pairs[3]. Experimenters measure the spin correlations using two polarizers with a varying anglebetween them. Bell found that models with non-local wave functions and models with hiddenvariables produced different angular correlation spectra. Previous tests of non-locality useddiscrete variables like ’pseudo-spin’ for CP violation, as with studies using the reactionΦ → K+K− [4].We describe a very different system that, in contrast to the KLKS system, is sensitiveto the collapse of continuous variables in a wave function [5]. Short-lived vector mesons(VMs) are produced with a fixed phase relationship at two separated sources. Even thoughthe mesons do not come from a single source, and, in fact, share no common history [6],the system acts as an interferometer. The meson lifetimes are short compared to the sourceseparation, so the mesons decay before their wave functions can spatially overlap.Any interference between the two sources must involve the decay products. Interferenceis only possible between identical final states. With the large phase space for final states,interference can only occur if the wave functions retain amplitudes for all possible decaychannels and angular distributions long after the decay takes place [7]. We have previouslycalculated the interference pattern [8]. This letter will focus on the effects of the wavefunction collapse and Bells inequality-like tests, and sketch an alternate derivation of theinterference, to emphasize the symmetries of the system.Figure 1 shows electromagnetic VM production in relativistic heavy ion collisions at largeimpact parameters, ~b. A photon from the electromagnetic field of one nucleus fluctuates toa virtual quark-anti-quark pair which elastically scatters from the other nucleus, emergingas a real vector meson [9]. Either nucleus can emit the vector meson. The momentum2transfers from the nuclei are similar, and they remain in the ground state, so it is impossibleto determine which nucleus emitted the photon and which is the target.The electromagnetic interaction (photon) has a long range, while the elastic scatteringhas a short range, around 0.6 fm [10], far smaller than the < 7 fm radius of a heavy nucleusor the typical impact parameter. So, VM production takes place essentially ‘on top of’ theemitting nucleus, and the two nuclei act as a two-source interferometer.Electromagnetic VM production is studied at the Relativistic Heavy Ion Collider (RHIC)at Brookhaven National Laboratory, where gold ions collide at center of mass energies upto 200 GeV per nucleon. Starting in 2006, the Large Hadron Collider (LHC) at CERN willcollide lead ions at a center of mass energy of 5.5 TeV per nucleon.The cross sections were previously calculated [11] using the Glauber approach [9] withthe photon spectrum given by the Weizsäcker-Williams virtual photon method [12]. Thecalculated photonuclear cross sections agree with data to within 20%. The cross sectionsare large, about 10% of the hadronic cross section at RHIC, rising to 50% at the LHC. Thecorresponding production rates, more than 100 ρ0/sec at RHIC, rising to 230,000 ρ0/sec atthe LHC, are large enough that it will be easy to collect adequate statistics to study wavefunction collapse. Already, the STAR collaboration [13] has observed more than 10,000 ρ0,which should be enough to observe the interference.The impact parameters for these interaction are large compared with the nuclear radii,RA. For ρ and ω production, the median impact parameter 〈b〉 is about 40 fm at RHIC,rising to 300 fm at the LHC; for the J/ψ, 〈b〉 rises from 23 fm at RHIC to about 50 fmat the LHC. All are much larger than RA ≈ 7 fm for heavy ions. It is possible to selectevents with smaller 〈b〉, but still with b > 2RA, by choosing events where VM production isaccompanied by nuclear breakup [14].The 〈b〉 are larger than the distance travelled by most VMs before they decay. The VMlifetimes τ range from 4 × 10−24s for the ρ up to 7.5 × 10−21s for the J/ψ. The VM areproduced with typical transverse momentum pT ≈ 2h̄/RA ≈ 60MeV/c; at mid-rapidity, thelongitudinal momentum is zero, so VM have a median decay distance d = 2h̄cτ/RAMV .3Except for the J/ψ, d 〈b〉; for the J/ψ at the LHC, d ≈ 〈b〉.The final state wave function from ion source i at a time t can be expressed schematicallyψ(t)i = exp (−t/2τ) |V > +(1 − exp (−t/2τ)) |DP > (1)where τ is the vector meson lifetime, |V > is the vector meson wave function, and |DP >is the final state. For stable particles, τ = ∞, the decay products drop out, leaving aconventional two-source interferometer.The interference can be seen by examining the symmetries of the system. The totalamplitude AT for observing the VM with momentum ~p at position ~r, and time, t, dependson the production amplitude A(~p, ~x, t′) and a propagator P (~p, ~x, t′, ~r, t) which transports themeson from ~x′, t′ to ~r, t:AT (~p,~r, t) =∫A(~p, ~x′, t′)P (~p, ~x, t′, ~r, t)d~xdt′. (2)The production amplitude A(~p,~r, t) depends on the electromagnetic field, E(~x, t′), nucleardensity ρ(~x, t) and the amplitude f(~p,~k) for a photon with momentum ~k to fluctuate to aqq pair and scatter from a nucleon, emerging as a vector meson with momentum ~p:A(~p, ~x′, t′) = f(~p,~k)ρ(~x′, t′)E(~x′, t′) (3)The electromagnetic field at a distance b from a nucleus is a Lorentz-contracted pulse witha width b/γ where γ is the Lorentz boost. When γ  1, the electric and magnetic fields areperpendicular and the overall field may be represented as a stream of almost-real photons,with energies up to h̄γ/b [12]. The photon amplitude is proportional to E(~x′, t′). The scat-tering amplitude is obtained from data; only its symmetries are important here. Absorptionof the nascent ρ0 is neglected, but could be included with an additional position-dependentvariable, effectively modifying ρ(x, t′).At large distances, the propagator for a VM with energy ω =√M2V + |~p|2 may bemodelled with a plane wave. Neglecting, for now, VM decays,P (~p, ~x, t′, ~r, t) = ei(~p·(~r−~x)−ω(t−t′)) (4)4The nuclear density is symmetric around the center of mass (origin), giving it positiveparity, while the antisymmetric electric field has negative parity): ρ(~x, t′) = ρ(−~x, t′) andE(~x, t′) = −E(−~x, t′). With this, the range of integration in Eq. (2) can be restricted to asingle nucleus:AT (~p,~r, t) =∫y>0d~xdt′ρ(~x, t′)E(~x, t′)ei(~p·~r−ω(t−t′))[f(~p,~k)ei~p·~x − f(~p,−~k)e−i~p·~x] (5)The only differences between the two nuclei are the phases ±i~p ·~x and between f(~p,−~k) andf(~p,~k). The latter is because the sign of pz reduces the symmetry of the system. Of course,interference is only significant when |f(~p,~k)| ≈ |f(~p,−~k)| which occurs near pz = 0.The equation simplifies by defining ~x = ~b/2 + ~x′. The bulk of the cross section isfrom when the photon couples coherently to the target nucleus, i.e. when ~k · ~x′  h̄, sothe exponential phase is constant over the nucleus. Then, the maximum transverse andlongitudinal momenta are h̄/RA and γh̄/RA. Emitted photons are subject to similar limits[15]. Near pz = 0, the photon momentum, and the momentum exchange due to the scatteringare very similar so it isn’t possible to determine which nucleus emitted the photon, and whichwas the scatterer; in fact, at ~p = 0, the two momentum transfers are equal and opposite.The electromagnetic pulse lasts a time, b/cγ, which may be slightly longer than onephoton period (h̄/ω). This partial temporal incoherence will reduce the overall productionamplitude. There is a pairwise cancellation between space-time volume elements d~xdt′ atpositions ~x and −~x, so the interference is not affected.With this, ~p · ~x = ~p ·~b/2 andAT (~p,~r, t) =∫y>0d~x′dt′ρ(~x′, t′)E(~x′, t′)ei(~p·~r−ω(t−t′))[f(~p,~k)ei~p·~b/2 − f(~p,−~k)e−i~p·~b/2]. (6)We now introduce a few approximations. The amplitude for production from the firstnucleus is A1(~p,~r, t) =∫y>0 d~xdt′ρ(~x, t′)E(~x, t′)f(~p,~k) and we define c to be the ratio of theamplitudes for production from the two nuclei:c(pz) =∫y>0 d~xdt′ρ(~x, t′)E(~x, t′)f(~p,−~k)∫y>0 d~xdt′ρ(~x, t′)E(~x, t′)f(~p,~k). (7)5The ratio f(~p,−~k)/f(~p,~k) does not vary significantly over the nucleus, so the single nucleusproduction amplitude factors out of the integral in Eq. (6). The transverse momenta do notaffect c, and kz = M2V /4pz. Then,AT (~p,~r, t) = A1(~p,~r, t)[ei~p·~b/2 − c(pz)e−i~p·~b/2]. (8)The amplitude factorizes into a magnitude and an interference term. The pT dependence ofA1(~p,~r, t) is dominated by the nuclear form factors, with the bulk of the production havingpT < 2h̄/RA. Most of the uncertainties discussed earlier do not affect the interference term.The time and z variation in E(~x, t′) should be largely independent of k. In the soft Pomeronmodel, the photon to VM coupling increases slowly with k and has an almost constant phase.At RHIC, a photon-meson term is also present, but the phase of c still changes only slowlywith k [9].The interference is clearest when pz=0. Then c = 1 and the approximations introducedin defining c disappear. The amplitude is AT (~p,~r, t) = 2iA0 sin (~p ·~b/2). and the crosssection isσ ∼ |AT (~p,~r, t)|2 = 2A20[1 − cos (~p ·~b)]. (9)This formula applies for stable particle emission, such as bremsstrahlung photon emissionin e−e− [16] and pp collisions [17].For short-lived particles, the situation is more interesting. We can express the crosssection in Eq. 9 as a sum of a coherent (interfering) and an incoherent (non-interfering)term2A20[1 − (1 − η) · cos (~p ·~b)], (10)where η measures the degree of decoherence, as has been done for the system Φ → K0K0[4,18]. Complete quantum mechanical coherence between the two sources here means η = 0.If the interference is restricted to the time the system spends in the vector meson (parent)state, then one would expect partial decoherence. According to Eq. 1, only a fraction6exp(−(MV b)/(ωτ)) of the vector mesons will have survived long enough for the amplitude topropagate the distance b between the nuclei before the decay. This scenario thus correspondstoη = 1 − exp(−MV bωτ). (11)For the ρ, ω, and φ, which all have cτ < b >, η ≈ 0 and one expects almost completedecoherence in this scenario. For the J/Ψ, on the other hand, cτ ≈< b >, so the decoherencewould be only partial. This scenario could therefore be distinguished by observing the J/Ψ,while for the lighter vector mesons it would essentially be indistinguishable from a scenariowith no interference.However, there is a broader issue. A distant observer sees the decay products from theoriginal meson, and most VM have many decay modes. The final state may be written|DP >= Σjαj |DPj >, where αj is the amplitude for final state |DPj >. For example,the J/ψ can decay to e+e− or π+π−π0, among many possibilities. The probability for anyspecific final state is small, less than 7%.If the wave function contains amplitudes for every possible decay mode, at times t τ ,the individual particles may be modelled as plane waves and the wave function isΨ(t) =∫ t0dtd2τ[ei~p·(~xd+~b/2) + cei~p·(~xd−~b/2)]e−td/2τ−iωtd |Σj√BrjΨj > (12)where the decays occur at time td and displacement ~xd = (~p/MV )td from the productionpoints. Here, Brj are the branching ratios to different final states, and Ψj is the k−particlefinal stateΨj = Σkei[~pk(~xk− ~xd)−ωj(t−td)]|Ψjk > δ(Σkωk − ω)δ(Σk~pk − ~p). (13)Here ~xk, ωk and ~pk are the particle positions, energy, and momenta, and |Ψjk > includes theparticle types and angular distribution. The δ functions impose 4-momentum conservation.If the decay occurs before the two amplitudes overlap, then either the wave function mustretain amplitudes for all final states, or the two decays will produce different final states,7and the interference must be small. Also, as Eq. (9) shows, amplitudes for different decaytimes must also be included. Thus, the presence or absence of interference tests whetherparticle decay collapses the wave function.The two source terms in Eq. (12) entangle the final state wave functions. The phasesdiffer by exp [i(~p ·~b+ δ)] where δ is the phase of c. Any measurement on one decay productat least partially collapse the wave function of the others [3]. Detectors could accuratelymeasure either the position or momentum of the k final state particles. ’Accurately’, iscompared to the relevant distance (b) or momentum (h̄/b) scales. By these metrics, currentand planned experiments measure momentum accurately, but not position.The interference pattern, Eq. (9) can be seen in the reconstructed VM pT [8]. Becauseb is not generally measurable, σ must be integrated over all b. Figure 2 shows the expectedpT spectrum for ρ0 production at mid-rapidity at RHIC [8]. The large dip for pT < h̄/〈b〉 isa distinctive experimental signature.Alternately, at least in a gedanken experiment, position sensitive detectors could be usedto localize the decay to a single nucleus, provided the ion trajectories are known. The decayJ/ψ → e+e− produces two relatistic electrons that are back-to-back in the transverse plane.For pT = 0, a line between the two measured electron positions will intersect one of the iontrajectories. When the electron ~pk are perpendicular to ~b then it is possible to determinewhich nucleus emitted the VM. The nonzero meson pT introduces some uncertainty, but notenough to encompass both ion trajectories. For detectors 500 fm from the collision point,the pointing uncertainty is 16 fm, less than the 〈b〉 ∼ 50 fm for J/ψ at the LHC.As with existing tests of Bell’s inequality, two detectors, on opposite sides of the collisionregion could randomly measure either position or momentum. All single-detector measure-ments are insensitive to what happens in the other detector. However, when both detectorsmeasured position, the production point could be determined, while when both measuredmomentum, a null at pT = 0 would be seen, showing the interference. These two possi-bilities are only compatible if the wave function collapses when a measurement is made,and not earlier, when the meson decays. For J/ψ → e+e−, the collapse would have to be8superluminal.One possible source of decoherence is the decay timing. The VM are produced nearly atrest, but the decay products may be relativistic. The maximum flight time difference fromthe two sources to a detector is b/c. If the detectors could resolve this time difference, thiscould partially localize the production, reducing the coherence.Since the probability of producing a VM in grazing collisions (b = 2RA) is high, about 1%at RHIC and 3% at the LHC [11], multiple VM production is also observable. Multi-mesonfinal states will exhibit more complicated entanglements, with possibly new behavior.We have described a 2-source interferometer for short lived particles, and showed that itsdescription requires a non-local wave function. The observation of interference will clearlydemonstrate that, after a decay, a systems’ wave function includes amplitudes for all possibledecay modes and angular distributions, and does not collapse to a specific decay modeuntil the wave function is externally observed. Measurements of this interference should beavailable soon. The STAR detector has observed exclusive ρ0 production in gold collisionsat RHIC [13]. Current data should provide higher statistics and an accurate ρ0 pT spectrum,probing the EPR paradox for continuous variables.This work was supported by the U.S. Department of Energy under Contract No. DE-AC-03076SF00098 and by the Swedish Research Council (VR).9REFERENCES[1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).[2] J. S. Bell, Physics 1, 195 (1965).[3] M. A. Horne, A. Shimony and A. Zeilinger, Phys. Rev. Lett. 62, 2209 (1989).[4] A. Apostolakis et al., Phys. Lett. B422, 339 (1998).[5] A. S. Parkins and H. J. Kimble, Phys. Rev. A61, 052104 (2000).[6] P. Grangier, Nature 419, 577 (2002)[7] T. Sudbery, in Quantum Concepts in Space and Time, ed. R. Penrose and C. J. Isham,(Oxford, 1986).[8] S. R. Klein and J. Nystrand, Phys. Rev. Lett. 84, 2330 (2000).[9] T. H. Bauer, R. D. Spital, D. R. Yennie and F. M. Pipkin, Rev. Mod. Phys. 50, 261(1978).[10] B. Müller and A. J. Schramm, Nucl. Phys. A523, 677 (1991).[11] S. R. Klein and J. Nystrand, Phys. Rev. C60, 014903 (1999).[12] J. D. Jackson, Electrodynamics, 2nd edition, Wiley, 1975.[13] C. Adler et al. (STAR Collaboration), nucl-ex/0206004, submitted to Phys. Rev. Lett.;F. Meissner, nucl-ex/0210028.[14] A. Baltz, S. Klein and J. Nystrand, Phys. Rev. Lett. 89, 012301 (2002).[15] M. Vidovic, M. Greiner, C. Best and G. Soff, Phys. Rev. C47, 2308 (1993).[16] J. Katzenstein, Phys. Rev. 78, 161 (1950).[17] R. C. Stabler, Nature 206, 922 (1965).[18] R. A. Bertlmann, W. Grimus and B. C. Hiesmayr, Phys. Rev. D60, 114032 (1999).10FIGURESγγAuAuxzyρ−> ππFIG. 1. Diagram showing ultra-peripheral ρ0 production and decay in heavy ion collisions.The nuclear momenta follow the z axis, and come closest at z = 0, when their separation (impactparameter), ~b follows the y axis.11p⊥ [GeV/c]dn/dp ⊥2Au+Auρ00.20.40.60.810 0.05 0.1 0.15 0.2 0.25FIG. 2. Perpendicular momentum spectra for ρ0 production at RHIC, at pz = 0, for gold ongold collisions at a center of mass energy of 200 GeV per nucleon. Plotted are dN/dpT , withand without interference. The curves are normalized to 1 for pT = 0 and no interference. Thecalculation assumes that the impact parameter is not measured, so the interference is washed out,except for pT < 25 MeV/c.12

We describe an experimental test of whether particle decay causes wave function collapse. The test uses interference between two well separated, but coherent, sources of vector mesons. The short-lived mesons decay before their wave functions can overlap, so any interference must involve identical final states. Unlike previous tests of non-locality, the interference involves continuous variables, momentum and position. Interference can only occur if the wave function retains amplitudes for all possible decays. The interference can be studied through the transverse momentum spectrum of the reconstructed mesons

Klein, SR

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Does particle decay cause wave function collapse: an experimental test

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Does particle decay cause wave function collapse: An experimental test

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