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Measuring velocity ratios 3 9 with correlation functions 9 1 n a J David Seibert∗ 8 Department of Physics 1 v 4 Kent State University, Kent, OH 44242 USA 2 2 1 Kevin Haglin† and Charles Gale‡ 0 3 Department of Physics 9 / h p McGill University, Montréal, P.Q., H3A 2T8, Canada - p e February 1, 2008 h : v i X r a Abstract We show how to determine the ratio of the transverse velocity of a source to the velocity of emitted particles, using split–bin correlation functions. The technique is to measure S and Sφ, 2 2 ∗On leave until 12 October 1993 at: Theory Division, CERN, CH–1211 Geneva 23, Switzerland; Internet: [email protected]. †Internet: [email protected]. ‡Internet: [email protected]. 1 subtract the contributions from the single–particle distribution, and take the ratio as the bin size goes to zero. We demonstrate the technique for two cases: each source decays into two particles, and each source emits a large number of particles. PACS Indices: 25.70.Np, 12.38.Mh, 12.40.Ee, 24.60.Ky. 2 1 Introduction There has been considerable study of correlation functions in high energy and nuclear physics. However, with the exception of source size measurements using identical particle (Hanbury–Brown/Twiss) interferometry [1], the connection between measurements of correlation functions and the physics of particle collisions is tenuous. For example, it is possible to measure the size of correlation sources (the number of pions produced) in high energy collisions [2], but no other source characteristics are currently identifiable. This is unfortunate, as the source size is anomalously large in ultra–relativistic nuclear collisions [2, 3] (the sources decay into at least fifteen pions each [4]), and the nature of these sources has not yet been determined. In this article, we propose to measure the velocities of correlation sources, the objects that make up the intermediate structure of collisions, by using the rapidity, y, and azimuthal angle, φ, correlation functions [5] ρ(2) δy S (δy;∆Y) = , (1) 2 D E ρ(2) ∆Y D E and φ ρ(2) Sφ(δy;∆Y) = δy , (2) 2 D Eφ ρ(2) ∆Y D E where measurements are made over a window from Y to Y +∆Y. Here 0 0 Y0+(i−1/2)x Y0+ix dy dy ρ(2) ∆Y/x 1 2 ρ(2) = ZY0+(i−1)x ZY0+(i−1/2)x , (3) x x∆Y/4 D E Xi=1 3 and Y0+ix π Y0+ix 2π dy dφlab dy dφlabρ(2) ∆Y/x 1 1 2 2 ρ(2) φ = ZY0+(i−1)x Z0 ZY0+(i−1)x Zπ , (4) x x∆Y/4 D E Xi=1 where ∆Y is an integer multiple of x. Split–bin correlation functions are useful for measuring correlations with maximum statistics (without re–use of data). S is closely related to the standard two–particle correlation 2 function: ρ(2)(0,δy/2) S (δy;∆Y) ≃ . (5) 2 ρ(2)(0,∆Y/2) For large ∆Y, the correlation functions are almost independent of ∆Y, but in practice it is impossible to make correlation measurements in particle collisions without using a large bin for reference, as the amount of produced matter fluctuates from event to event. The proposed technique is to measure the ratio Sφ(0;∆Y) − Σφ(∆Y) R(∆Y) = 2 2 . (6) S (0;∆Y) − Σ (∆Y) 2 2 Here Y0+∆Y ∆Y dyρ2(y) Σ (∆Y) = ZY0 (7) 2 Y0+∆Y/2 Y0+∆Y 4 dy dy ρ(y )ρ(y ) 1 2 1 2 ZY0 ZY0+∆Y/2 and Y0+∆Y π 2π ∆Y dy dφlab dφlabρ(y,φlab)ρ(y,φlab) 1 2 1 2 Σφ(∆Y) = ZY0 Z0 Zπ , 2 Y0+∆Y π Y0+∆Y 2π dy dφlab dy dφlabρ(y ,φlab)ρ(y ,φlab) 1 1 2 2 1 1 2 2 ZY0 Z0 ZY0 Zπ (8) are the values of S and Sφ from the single–particle distributions, ρ(y) and 2 2 ρ(y,φlab). In the case of a flat distribution [ρ(y,φlab) = const.], 4 Σ = Σφ = 1. It is also possible to measure emission velocities using 2 2 standard two–particle interferometry techniques; however, these measurements are somewhat more difficult than the technique proposed here, as they require particle identification. The technique proposed here has the advantage of being useable even in the absence of particle identification. We assume isotropic emission and show that, in the non–relativistic limit, R depends only on the velocity ratio, v /v , where v is the source s π s transverse velocity, and v is the velocity of emitted particles in the source π rest frame. In section 2, we calculate R for the case of a resonance that decays isotropically into two particles. In section 3, we calculate R for isotropic decay of a thermal source, finding that the shape is not very different from the first case. Finally, we summarize our results and discuss applications and future work in section 4. 2 Two–particle decay Suppose that we have an event with a single resonance that decays into two particles, and a flat background of N −2 particles in a window of rapidity ∆Y. The background contribution to the two–particle density is then constant: ρ(2)(y ,φlab,y ,φlab) = (N −2)(N −3)/(4π2∆Y2) . (9) b 1 1 2 2 5 Adding the contribution from combining one background particle with one particle from the resonance, we obtain the total uncorrelated signal, ρ(2)(y ,φlab,y ,φlab) = [N (N −1) − 2] /(4π2∆Y2) . (10) u 1 1 2 2 Given the resonance azimuthal angle, φ , and the resonance rapidity, y , r r we calculate the laboratory coordinates, y and φlab, as functions of the center-of-momentum coordinates, θ and φ. (1−v2)1/2v cosθ y = tanh−1 s π + y , (11) r 1 + vsvπ sinθ sinφ! v + v sinθ sinφ φlab = tan−1 s π + πH(−cosφ) + φ , (12) (1−vs2)1/2vπ sinθ cosφ! r where 0 x ≤ 0, H(x) = (13)   1 x > 0. [We use standard high–energy units, with c = 1. In deriving eq. (12), we assume that −π/2 ≤ tan−1x < π/2.] We assume that y and φ are r r distributed randomly, with flat distributions. The resonance contribution to ρ(2) is simplest in the resonance rest frame: sinθ sinθ δ(cosθ + cosθ ) δ(|φ −φ | − π) ρ(2)(θ ,φ ,θ ,φ ) = 1 2 1 2 1 2 , (14) r 1 1 2 2 2π where δ is the Dirac δ–function. The resonance contribution to S (δy;∆Y) is 2 ∆Y δy/2 δy 2π 2π ∆Y dy dy dy dφ dφ ρ(2)(y ,φ ,y ,φ ) r 1 2 1 2 r 1 1 2 2 S (δy;∆Y) = Z0 Z0 Zδy/2 Z0 Z0 , 2,r δy2π2∆Y2ρ(2)(y ,y ) u 1 2 (15) 6 where ∂θ ∂θ ρ(2)(y ,φ ,y ,φ ) = 1 2 ρ(2)(θ ,φ ,θ ,φ ) . (16) r 1 1 2 2 (cid:12)∂y1(cid:12) (cid:12)∂y2(cid:12) r 1 1 2 2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) As we assume flat distributions, th(cid:12)e co(cid:12)n(cid:12)tribu(cid:12)tion from all bins is the same, (cid:12) (cid:12) (cid:12) (cid:12) so we use only the bin from y = 0 to δy, and take Y = 0 The contribution 0 at δy = 0 is δy/2 2yr−δy/2 3δy/4 δy/2 4∆Y dy dy + dy dy r 1 r 1 S (0;∆Y) = lim "Zδy/4 Z0 Zδy/2 Z2yr−δy #, (17) 2,r δy→0 δy2 [N (N −1) − 2] (1−v2)1/2v s π h ∆Y i = .(18) 2 [N (N −1) − 2] (1−v2)1/2v s π Evaluating Sφ in the same manner, we obtain 2,r 2π π 2π ∆Y dφ dφ dφ δ(|φ −φ |−π) r 1 2 1 2 Sφ (0;∆Y) = Z0 Zφl1ab=0 Zφl2ab=π . (19) 2,r 2π2[N(N −1)−2](1−v2)1/2v s π Note that φlab is now independent of θ, as in the limit δy → 0 the only contribution is from cosθ = 0. We then perform the integral over φ . r 2π ∆Y dφg φlab(φ) − φlab(φ+π) S2φ,r(0;∆Y) = 2πZ02[N(N −(cid:16)(cid:12)(cid:12)1)−2](1−v2)1/2v (cid:12)(cid:12)(cid:17) , (20) (cid:12) s π (cid:12) where x 0 ≤ x < π , g(x) = (21)   2π − x π ≤ x < 2π . In the limit ∆Y → ∞,  S = Σ + S , Sφ = Σφ + Sφ . (22) 2 2 2,r 2 2 2,r Our ratio is then 1 2π R = dφg φlab(φ) − φlab(φ+π) , (23) π2 Zφ=0 (cid:16)(cid:12) (cid:12)(cid:17) (cid:12) (cid:12) (cid:12) (cid:12) 7 Note that this result is independent of the size of the background, N! In fact, if we allow the presence of more than one resonance, we find that S 2,r and Sφ are both proportional to the number of sources, n , so R is also 2,r s independent of the number of resonances. For the case of non–relativistic resonance decay, eq. (12) reduces to (v /v )+sinφ φlab = tan−1 s π + πH(−cosφ) + φ , (24) r cosφ ! as we can take cosθ = 0. It is clear from eqs. (23) and (24) that, in the non–relativistic limit, R depends only on the ratio v /v . In Fig. 1, we s π show R for v = c/2 and v = c, along with the value in the non–relativistic π π limit. In both the relativistic and non–relativistic cases, R(0) = 2; this is an identity, due to the different topologies of φ and y (φ = 2π is equivalent to φ = 0, while y = δy is not equivalent to y = 0). The behavior as v → 1 is π analytically intractable, but from Fig. 1 it is clearly not too different from the non–relativistic case. The limit v → 1 can be taken analytically, s yielding 2(1−v2)1/2 v + v R = s ln s π . (25) π2 v − v (cid:20) s π(cid:21) The limit of large v differs only by a factor γ = (1−v2)1/2 in the s s s relativistic and non–relativistic cases. 3 Thermal sources A typical example of a source that emits a large number of particles is a thermal source, such as a large droplet. Imagine that we have a single 8 source, as before, but that the source emits n particles, where n ≫ 1. π π The two–particle density from this source is n (n −1) sinθ sinθ ρ(2)(θ ,φ ,θ ,φ ) = π π 1 2. (26) r 1 1 2 2 16π2 The δ–function constraints are absent, as for large n we can essentially π ignore conservation of energy and momentum. Again moving to the resonance rest frame, and taking ∆Y → ∞, we obtain ∞ 2π 2π ∂θ ∂θ 1 2 S (0) ∝ n (n −1) dy dφ dφ sinθ sinθ , (27) 2,r π π 1 2 1 2 Z−∞ Z0 Z0 (cid:12)(cid:12) ∂y (cid:12)(cid:12) (cid:12)(cid:12) ∂y (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) and (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ∞ π 2π ∂θ ∂θ Sφ (0) ∝ 4n (n −1) dy dφ dφ 1 2 sinθ sinθ . (28) 2,r π π Z−∞ Zφl1ab=0 1Zφl2ab=π 2(cid:12)(cid:12) ∂y (cid:12)(cid:12)(cid:12)(cid:12) ∂y (cid:12)(cid:12) 1 2 (cid:12) (cid:12)(cid:12) (cid:12) Taking the ratio as before, we find (cid:12) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12) ∞ 2π 2π ∂θ ∂θ 2 dy dφ dφ 1 2 sinθ sinθ g φlab − φlab R = Z−∞ Z0 1Z0 2 (cid:12)(cid:12) ∂y (cid:12)(cid:12) (cid:12)(cid:12) ∂y (cid:12)(cid:12) 1 2 (cid:16)(cid:12) 1 2 (cid:12)(cid:17) . (29) ∞(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)2π (cid:12)(cid:12) ∂θ 2 (cid:12)(cid:12) (cid:12)(cid:12) π (cid:12)dy (cid:12) (cid:12) dφ(cid:12) sinθ Z−∞ "Z0 (cid:12)(cid:12)∂y(cid:12)(cid:12) # (cid:12) (cid:12) Note that R is independent of n as well(cid:12)as(cid:12)being independent of N! In π (cid:12) (cid:12) fact, R is even independent of the number of sources, as before. We cannot evaluate the relativistic denominator for the thermal source, but we can take the non–relativistic limit as before. We obtain 1 π 2π 2π R = dθ sinθ dφ dφ g φlab − φlab , (30) 4π3 1 2 1 2 Z0 Z0 Z0 (cid:16)(cid:12) (cid:12)(cid:17) (cid:12) (cid:12) where (cid:12) (cid:12) (v /v )+sinθsinφ φlab = tan−1 s π +πH(−cosφ)+φ . (31) r sinθcosφ ! 9 It is again clear that, in the non–relativistic limit, R depends only on v /v . s π In Fig. 2, we display R as in the case of two–particle decay. We find now that R(0) = 1 and R(∞) = 0, so the normalization is clearly different from that obtained for two–particle decay. Because the curves are different, there will be some dependence of the measured velocity on the source size. If the sources are known to emit many pions, as is the case in ultra–relativistic nuclear collisions, this dependence is weak. However, any velocity measurement using the proposed technique depends somewhat on knowledge of the source size. 4 Conclusions We have demonstrated a new procedure for measuring the ratio of a source transverse velocity to the velocity of emitted particles, using the split–bin correlation functions S and Sφ. We showed the technique for two simple 2 2 cases: two–body decay, and unconstrained decay. In both cases, the technique gives a measurement that is independent of the number of sources, and the background size. In addition, we find that the measurement is independent of the source size for large sources. We have considered somewhat simple cases for pedagogical purposes, in order to illustrate our technique in a simple, model–independent framework. The specific issue of contamination by dynamical effects should be addressed separately. Similar techniques could be used to measure many other quantities, so the potential usefulness of this line of enquiry is enormous. For example, 10