02 06 06 0 v 2 8 J an 2 00 3 Does particle decay cause wave function collapse : An experimental test

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. Typeset using REVTEX 1 In 1935, Einstein, Podolsky and Rosen (EPR) showed that quantum mechanics required that wave functions can be non-local [1]. When a system is observed, the wave function collapses from one which contains amplitudes for a host of possible outcomes to smaller set of possibilities, in accord with the measurement. This collapse is instantaneous; much has been written about its superluminous nature. Most studies of the EPR paradox have tested 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 angle between them. Bell found that models with non-local wave functions and models with hidden variables produced different angular correlation spectra. Previous tests of non-locality used discrete variables like ’pseudo-spin’ for CP violation, as with studies using the reaction Φ → KK [4]. We describe a very different system that, in contrast to the KLKS system, is sensitive to 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 though the 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 source separation, so the mesons decay before their wave functions can spatially overlap. Any interference between the two sources must involve the decay products. Interference is 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 decay channels and angular distributions long after the decay takes place [7]. We have previously calculated the interference pattern [8]. This letter will focus on the effects of the wave function collapse and Bells inequality-like tests, and sketch an alternate derivation of the interference, to emphasize the symmetries of the system. Figure 1 shows electromagnetic VM production in relativistic heavy ion collisions at large impact parameters, ~b. A photon from the electromagnetic field of one nucleus fluctuates to a virtual quark-anti-quark pair which elastically scatters from the other nucleus, emerging as a real vector meson [9]. Either nucleus can emit the vector meson. The momentum 2 transfers from the nuclei are similar, and they remain in the ground state, so it is impossible to determine which nucleus emitted the photon and which is the target. The electromagnetic interaction (photon) has a long range, while the elastic scattering has a short range, around 0.6 fm [10], far smaller than the < 7 fm radius of a heavy nucleus or the typical impact parameter. So, VM production takes place essentially ‘on top of’ the emitting 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 up to 200 GeV per nucleon. Starting in 2006, the Large Hadron Collider (LHC) at CERN will collide 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] with the photon spectrum given by the Weizsäcker-Williams virtual photon method [12]. The calculated photonuclear cross sections agree with data to within 20%. The cross sections are large, about 10% of the hadronic cross section at RHIC, rising to 50% at the LHC. The corresponding production rates, more than 100 ρ/sec at RHIC, rising to 230,000 ρ/sec at the LHC, are large enough that it will be easy to collect adequate statistics to study wave function collapse. Already, the STAR collaboration [13] has observed more than 10,000 ρ, 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 fm at the LHC. All are much larger than RA ≈ 7 fm for heavy ions. It is possible to select events with smaller 〈b〉, but still with b > 2RA, by choosing events where VM production is accompanied by nuclear breakup [14]. The 〈b〉 are larger than the distance travelled by most VMs before they decay. The VM lifetimes τ range from 4 × 10s for the ρ up to 7.5 × 10s for the J/ψ. The VM are produced with typical transverse momentum pT ≈ 2h̄/RA ≈ 60MeV/c; at mid-rapidity, the longitudinal momentum is zero, so VM have a median decay distance d = 2h̄cτ/RAMV . 3 Except 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 schematically ψ(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 a conventional two-source interferometer. The interference can be seen by examining the symmetries of the system. The total amplitude AT for observing the VM with momentum ~p at position ~r, and time, t, depends on the production amplitude A(~ p, ~x, t) and a propagator P (~ p, ~x, t, ~r, t) which transports the meson 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), nuclear density ρ(~x, t) and the amplitude f(~ p,~k) for a photon with momentum ~k to fluctuate to a qq 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 with a width b/γ where γ is the Lorentz boost. When γ ≫ 1, the electric and magnetic fields are perpendicular 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 scattering amplitude is obtained from data; only its symmetries are important here. Absorption of the nascent ρ is neglected, but could be included with an additional position-dependent variable, effectively modifying ρ(x, t). At large distances, the propagator for a VM with energy ω = √ M V + |~ p| 2 may be modelled with a plane wave. Neglecting, for now, VM decays, P (~ p, ~x, t′, ~r, t) = e p·(~r−~x)−ω(t−t )) (4) 4 The nuclear density is symmetric around the center of mass (origin), giving it positive parity, while the antisymmetric electric field has negative parity): ρ(~x, t) = ρ(−~x, t) and E(~x, t) = −E(−~x, t). With this, the range of integration in Eq. (2) can be restricted to a single nucleus: