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Wave Function Collapse During ρ⁰ Photoproduction in STAR

By Spencer Klein, Lawrence Berkeley National Laboratory

The phrase "relativistic heavy-ion collisions" usually brings to mind violent interactions and complex, 1000-particle final states. However, some RHIC collisions are the exact opposite - the colliding ions act as point sources of fields; the fields interact to produce a few-particle final state, leaving the relativistic nuclei intact. This happens at large impact parameters, where the nuclei do not interact hadronically; these are ultra-peripheral collisions (UPCs) [1]. The ions interact electromagnetically, by photon exchange. Many different types of UPCs are possible, including e⁺e^- pair production from two (or more) photons, or vector meson photoproduction. In vector meson photoproduction, one nucleus emits a photon, which scatters elastically from the other target, emerging as a vector meson [2, 3].

In photoproduction, the two ions act as a two-source interferometer. Interference occurs because ρ⁰ photoproduction at low transverse momentum (p_T) leaves the target intact, so it is impossible to tell which ion emitted the photon, and which ion was the target. The two possibilities (ion #1 emitting a photon which interacts with ion #2, or vice-versa) are indistinguishable, and are related by a parity transformation (mirror inversion through the center of mass of the system switches which ion is the photon emitter). Quantum mechanics requires that the amplitudes be combined. Since ρ⁰ are negative parity, the two amplitudes enter with opposite signs, and, at impact parameter b, the cross-section to produce a ρ⁰ is at mid-rapidity, y=0, is [4,5]:

where A₁ and A₂ are the amplitudes for ρ⁰ production. At y=0 A₁ = A₂.

The exponential (cosine term in the right-hand expression) is a propagator from one nucleus to the other. At small p_T (p_T < hbar/<b>), the two amplitudes interfere destructively and ρ⁰ production is suppressed.

In ρ⁰ photoproduction, the median impact parameter is 46 fm [5]. This distance is large compared to the size of the ions; so the two production points are nearly point sources. Further, the ρ⁰ decay almost immediately (the ρ⁰ lifetime is t < 10^-23s, and ct < 1 fm), long before the amplitudes from the two sources can overlap.

This leads to an interesting paradox [6], which is more clearly illustrated by considering J/ψ decay, as in Figure 1, below. The J/ψ can decay to a many different final states: e⁺e^-, μ⁺μ^-, π⁺π^-π⁰, etc.

Figure 1. A transverse view (the ion trajectories are perpendicular to the paper) of J/ψ photoproduction relativistic heavy ion collisions. The two orange dots are the ions; b is the impact parameter vector. The two J/ψ amplitudes decay almost immediately. Since production is simultaneous and the two ions are well separated in space, there can be no communication before the J/ψ decay, and no reason to expect them to decay to the same final state. However, since the two amplitudes to produce J/ψ at each ion have opposite signs, they should interfere with each other.

Since production is simultaneous, the amplitudes at the two sources share no common history or possibility of communication, so the two amplitudes must decay independently. One might expect that, most of the time, they will decay differently - one to e⁺e^-, and the other to three pions, for example. When the final states are different, then interference is impossible. Yet, one expects interference.

The solution to this apparent paradox is found in quantum mechanics: wave functions don't collapse until one actually observes the final state. So, even though the J/ψ decay shortly after production, the post-decay wave function includes amplitudes for all possible final states - e⁺e^-, π⁺π^- π⁰, etc. All of these amplitudes remain until the final state interacts with our detector. Then wave function collapses to a single final state. By the time of this collapse, the amplitudes from the two sources are fully overlapped.

The final state that STAR observed was simpler, since the ρ⁰ almost always decays to π⁺π ^-. However, the pions can go in different directions; the different final state angles play the same role as the different J/ψ final state particles.

STAR observed the interference in the p_T spectrum of photoproduced ρ⁰. This is more conveniently analyzed in terms of t ~ t_^= p_T². In the absence of interference, the t spectrum should be roughly exponential dN/dt ~ exp(-kt). Interference suppresses the production of ρ⁰ with p_T < hbar/<b>.

Figure 2. The t_⊥ = p_T² spectrum for coherently photoproduced ρ⁰ with rapidity 0.05< |y| < 0.5. The symbols are the STAR data points. The two histograms are calculations with (dashed) and without (solid) interference.

Unfortunately, there is no simple expression for the t spectrum when interference is included; a numerical calculation is required to find the t spectrum for a finite rapidity range. Figure 2 shows the t-spectrum measured by STAR, along with calculations with and without interference for 0.05 <|y|<0.5 when the photoproduction is accompanied by mutual Coulomb excitation. The pronounced drop in the data for t<0.001 GeV²/c² matches the interference calculation. STAR fit the t spectrum to the form

where A is an arbitrary normalization and Int(t) and Noint(t) are the predicted spectra, with and without interference, respectively. c determines the degree of interference; c=0 corresponds to no interference, and c=1 for full interference.

STAR studied four separate datasets: ρ⁰ photoproduction with and without mutual Coulomb excitation (this selects different impact parameter ranges), in two different rapidity ranges. All four datasets showed the expected interference, and the combined result was that the interference was 87±5 (stat.) ± 8 (syst.)% of the expected value.

This measurement shows that waveforms do not collapse when particles decay - the post-decay wave function includes amplitudes for all possible decays; these amplitudes do not collapse until later, after the system has spread out and the wave functions from the two amplitudes overlap. This extended system is another example of the Einstein-Podolsky-Rosen paradox.

References

[1] C. Bertulani, S. Klein and J. Nystrand, Ann. Rev. Nucl. Part. Sci. 55, 271 (2005); G. Baur et al., Phys. Rept. 364, 459 (2002).
[2] B. I. Abelev et al., (STAR Collaboration), Phys. Rev. C77, 034910 (2008).
[3] S. Afanasiev et al. (PHENIX Collaboration), arXiv:0903.2041.
[4] B. I. Abelev et al. (STAR Collaboration), Phys. Rev. Lett. 102, 112301 (2009).
[5] S. Klein and J. Nystrand, Phys. Rev. Lett. 84, 2330 (2000); A. Baltz, S. Klein and J. Nystrand, Phys. Rev. Lett. 89, 012301 (2002); K. Hencken, G. Baur and D. Trautmann, Phys. Rev. Lett. 96, 012303 (2006).
[6] S. Klein and J. Nystrand, Phys. Lett. A308, 323 (2003).

The Relativistic Heavy Ion Collider at Brookhaven National Laboratory is a world-class scientific research facility primarily funded by the U.S. Department of Energy Office of Science. Hundreds of physicists from around the world use RHIC to study what the universe may have looked like in the first few moments after its creation. What physicists learn from these collisions may help us understand more about why the physical world works the way it does, from the smallest subatomic particles, to the largest stars.