BNL-113189-2016-JA Procedure for measuring photon and vector meson circular polarization variation with respect to the reaction plane in relativistic heavy-ion collisions A.H. Tang, G.Wang Submitted to Physical Review C August 2016 Physics Department Brookhaven National Laboratory U.S. Department of Energy USDOE Office of Science (SC), Nuclear Physics (NP) (SC-26) Notice: This manuscript has been co-authored by employees of Brookhaven Science Associates, LLC under Contract No. DE-SC0012704 with the U.S. Department of Energy. The publisher by accepting the manuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. 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Procedure for measuring photon and vector meson circular polarization variation with respect to the reaction plane in relativistic heavy-ion collisions A.H. Tang1, G.Wang2 1Brookhaven National Laboratory, Upton, New York 11973 2Department of Physics and Astronomy, University of California, Los Angeles, California 90095 The electromagnetic (EM) field pattern created by spectators in relativistic heavy-ion collisions plants a seed of positive (negative) magnetic helicity in the hemisphere above (below) the reaction plane. Owing to the chiral anomaly, the magnetic helicity interacts with the fermionic helicity of the collision system, and causes photons emitted in upper- and lower-hemispheres to have different preferences in the circular polarization. Similar helicity separation for massive particles, due to the global vorticity, is also possible. In this paper, we lay down a procedure to measure the variation of the circular polarization w.r.t the reaction plane in relativistic heavy-ion collisions for massless photons, as well as similar polarization patterns for vector mesons decaying into two daughters. We propose to study the yield differentially and compare the yield between upper- and lower-hemispheres in order to identify and quantify such effects. PACS numbers: 25.75.Ld I. INTRODUCTION II. PHOTONS Photon circular polarization can be measured with dedicated polarimeters [10], by studying the count rate asymmetries with a local magnet. Such a setup is infea- sible for large-scale heavy-ion experiments like the Rela- tivistic Heavy Ion Collider or the Large Hadron Collider, In a non-central relativistic heavy-ion collision, an as the local magnet will complicate the magnetic field ultra-strong magnetic field will be produced inside the that is used for the track detection of charged particles. participant zone by energetic spectator protons [1]. On Nevertheless, one can still probe photon circular polar- average, the magnetic field is perpendicular to the so- ization by tracking e+e− pairs from photon conversion. called \reaction plane" (RP) that contains the impact A photon will not decay by itself. However, when a parameter and the beam momenta. This strong mag- photon is near an atomic nucleus, its energy can be con- netic field, coupled with quantum anomalies, will cause a verted into an electron-position pair. Panels a) and b) series of novel non-dissipative transport effects that could in Fig. 1 show the typical conversion process for left and survive the expansion of the fireball and be detected in right circularly polarized photons, respectively. Here ζ is experiments. For a review on these subjects, see Ref. [2]. the angle of the electron relative to the positron, when Furthermore, the initial magnetic helicity of the collision arXiv:1607.04489v2 [nucl-th] 3 Sep 2016 both are projected onto the plane perpendicular to the system can be quite large, and bears opposite signs in motion direction of the photon. Owing to the opposite the upper- and lower-hemispheres. It was recently real- circular polarizations, the preferred direction orderings ized [3{6] that owing to the chiral anomaly, the helic- between the emitted electron and positron are opposite ity can be transferred back and forth between the mag- in the two cases. netic flux and fermions as the collision system evolves, so that the magnetic helicity could last long enough to a) Left circularly polarized b) Right circularly polarized yield photons with opposite circular polarizations in the hemispheres above and below the RP [3, 7, 8]. A similar - ζ + + ζ - asymmetry in photon polarization can also result from e e e e the initial global quark polarization [9] which could ef- fectively lead to a polarization of photons [7]. This local γ into ✕ γ into ✕ imbalance of photon circular polarization could be ob- the paper the paper served in experiments, e.g., by studying the polarization recoil recoil preference w.r.t the RP for photons that convert into e+e− pairs. In this paper, we discuss a few practical FIG. 1: A left (a) or right (b) circularly polarized photon is thoughts of carrying out such a measurement, first for converted into an electron-positron pair with the help of the photons and then for vector mesons. recoil of an atomic nucleus. 2 The cross-section for the pair production by completely hemisphere, circularly polarized high-energy photons is given as [11, Z π Z 2π 12] γ ∆Yup−down = dσdφ − dσdφ 0 π dσ / 1 ± a · sinζ; (1) / 4Hsinζ (5) where the + and − signs are for right and left circularly and the difference between the left hemisphere and right polarized photons, respectively. a (2 [0; 1]) is a positive hemisphere (separated by φ = π=2) would be zero owing quantity dependent on the photon energy and the ener- to the symmetry of the system: gies and momenta of its two daughters, as well as the 3π π atomic number Z of the recoiling nucleus. Z 2 Z 2 γ In the presence of the magnetic helicity effect, the ∆Yleft−right = dσdφ − dσdφ π − π yields of left and right circularly polarized photons can 2 2 each be parametrized as a sine wave modulation w.r.t. = 0; (6) the RP, which is similar to the directed flow (v1) [13] which can be regarded as a consistency check in experi- that is modulated by a cosine wave. Furthermore, the γ sine coefficients have the same magnitude and opposite ments. Note that when there is no signal, ∆Yup−down = γ signs for left and right circularly polarized photons. With ∆Yleft−right = 0, and a finite efficiency (ζ) will not cause γ this parametrization scheme, the effect of the magnetic an artificial, finite ∆Yup−down. helicity aforementioned is strongest in the direction per- In heavy-ion collider experiments, photons are usually pendicular to the RP (along the direction of the magnetic reconstructed at secondary vertices where they strike de- field), and weakest in the RP, and so is the difference in tector materials and emit e+e− pairs. The photon sample yield between the two groups of photons. The cross- is inevitably contaminated by combinatorial background sections for right and left circularly polarized photons formed by random e+e− pairs. However, the background can be written separately as contribution should be symmetric w.r.t both φ = 0 and φ = π=2 plane, and be canceled in this procedure by tak- dσR / (1 + a · sinζ)(1 + b · sinφ) (2) ing the difference in yield between up and down (or left dσL / (1 − a · sinζ)(1 − b · sinφ); (3) and right) hemispheres. The same argument also holds for the study of vector mesons discussed below. where φ is the photon's azimuthal angle relative to the RP, and b (2 [0; 1]) represents the magnitude of the mod- ulation. The total cross-section is the sum of dσR and III. VECTOR MESONS dσL: dσ / 1 + Hsinζsinφ, (4) Unlike photons that can be only transversely polarized (with the spin direction either parallel or anti-parallel to where H ≡ ab, and H 2 [0; 1]. the momentum direction), vector mesons can be both ζ, φ are both experimentally measurable quantities, longitudinally and transversely polarized. Below, we de- so in principle H can be obtained from the cross-section scribe a procedure to measure a potential difference, if a measurement. A finite H implies a finite b or a finite similar effect exists, between left and right circular polar- effect of modulation w.r.t the RP due to the external izations for vector mesons, as well as a way to study the magnetic field. Note that only the modulation w.r.t RP variation of such a difference w.r.t the RP. The procedure carries the information of the asymmetry between left is applicable to vector mesons decaying into two daugh- and right circularly polarized photons, so other sources of ters, such as !, φ, J= etc, and applicable to virtual photons (mostly from hadron decays) do not contribute photons decaying into two leptons by firstly fluctuating to H as long as they are produced symmetrically w.r.t RP into vector mesons.