Table Of ContentMeasuring velocity ratios
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with correlation functions
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J David Seibert∗
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Department of Physics
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4 Kent State University, Kent, OH 44242 USA
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1 Kevin Haglin† and Charles Gale‡
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Department of Physics
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p McGill University, Montr´eal, P.Q., H3A 2T8, Canada
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e February 1, 2008
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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].
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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
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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.],
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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
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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)
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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)
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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
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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 !
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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,
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