HBT shape analysis with q -cumulants

HBT shape analysis with q -cumulants

H. C. Eggers^I; P. Lipa^II

^IDepartment of Physics, University of Stellenbosch, 7602 Stellenbosch, South Africa

^IIArizona Research Laboratories, Division of Neural Systems, Memory and Ageing, University of Arizona, Tucson AZ 85724, USA

ABSTRACT

Taking up and extending earlier suggestions, we show how two- and three-dimensional shapes of second-order HBT correlations can be described in a multivariate Edgeworth expansion around Gaussian ellipsoids, with expansion coefficients, identified as the cumulants of pair momentum difference q , acting as shape parameters. Off-diagonal terms dominate both the character and magnitude of shapes. Cumulants can be measured directly and so the shape analysis has no need for fitting.

Keywords: Particle correlations; Intensity interferometry

I. INTRODUCTION

Early measurements of the Hanbury-Brown Twiss (HBT) effect made use of momentum differences in one dimension, for example the four-momentum difference q_inv [1]. The huge experimental statistics now available permits measurement of the effect in the three-dimensional space of vector momentum differences q = p₁ - p₂ and, in many cases, also its dependence on the average pair momentum K = ( p₁ + p₂)/2. Increasing attention has therefore been paid in the last decade to the second-order correlation function in its full six-dimensional form,

with r the density of like-sign pairs in sibling events and r^ref the event-mixing reference. While C₂ data can be visualized and quantified reasonably well in two dimensions [2, 3], it is harder to quantify three- or higher-dimensional correlations. Projections onto marginal distributions are inadequate [4], while sets of conditional distributions ("slices") require many plots and miss cross-slice features.

Under these circumstances, efforts to quantify the shape of the multidimensional correlation function with Edgeworth expansions [5, 6] or spherical harmonics [4] represent welcome progress. More ambitious programmes seek to extend connections between Gaussian source functions and the "radius parameters" of the correlation function to sets of higher-order coefficients using imaging techniques [7-9] and cartesian harmonics [10].

In this contribution, we extend the Edgeworth expansion solution proposed in [5, 6] to a fully multivariate form, including cross terms. Generically, the intention is to expand a measured normalized probability density f(q) in terms of a reference density f₀(q) and its derivatives,

so as to characterize f(q) by its expansion coefficients. While we have previously made use of a discrete multivariate Edgeworth form with Poissonian reference f₀ to describe multiplicity distributions [11], the shape analysis of R₂ requires the more traditional continuous version [12] with a Gaussian reference f₀. For the purpose of analysing the shape of the experimental correlation function in HBT, we hence define the measured non-Gaussian probability density as

where R₂(q) is itself a normalized cumulant of pair counts [13]. For the reference distribution (null case), we take the multivariate Gaussian, which in its most general form is

where D is the dimensionality of q , Einstein summation convention is used (here and throughout this paper), the l_i are the first " q -cumulants" of f₀ [14, 15],

l_ij(K) is the covariance matrix (the set of second-order q -cumulants) in the components of q ,

and the inverse matrix. While we suppress K in our notation from now on, all results are valid for K-dependent first moments l_i and covariance matrix elements l_ij.

II. REFERENCE DISTRIBUTION

The vector difference is normally decomposed into components q = (q₁,q₂,q₃) = (q_o,q_s,q_l) in the usual (out, side, long) coordinate system; for illustrative purposes we will also make use of a two-dimensional vector q = (q₁,q₂). (This is not the two-dimensional decomposition into (q_t,q_l) used in some experimental HBT analyses [2, 3], because q_t = is always positive, while (q₁,q₂) in our two-dimensional example can be positive or negative.) For the two-dimensional decomposition, the covariance matrix

has the inverse

where we have introduced standard deviations s_i = as well as the Pearson correlation coefficient r = /(s₁s₂) = l₁₂ / and c = 1 – r² [12, 16]. Similarly, in three dimensions the covariance matrix

has the general inverse

where r_ij = /(s_is_j) and the determinant is given by

For azimuthally symmetric sources [17], r_os = r_sl = 0, so that the inverse simplifies to

Identifying in the second part of Eq. (13) the inverse cumulant matrix with the usual radii of the parametrization f₀ ~ exp[- å_ij

q_i q_j], we note that the notation is misleading in that a positive covariance between the out and long directions, r_ol > 0, results in a negative .

III. MULTIVARIATE EDGEWORTH EXPANSION

A. Derivation

In order to derive the Edgeworth expansion, we need to distinguish between the moments and cumulants of f₀(q) and f(q) respectively. The cumulants of the reference f₀ have been fully specified already: the order-1 and 2 cumulants are the set of (initially free) parameters l_i and l_ij respectively, while all cumulants of order 3 or higher vanish identically [12] for the Gaussian reference (4). For the measured non-Gaussian f(q), we denote the first- and second-order moments as µ_i = ò dqf(q) q_i, and µ_ij = ò dqf(q) q_i q_j, and in general

Cumulants k_ijk··· of f(q) are found from these moments by inverting the generic moment-cumulant relations [12]

where we have introduced the notation [3] to indicate the number of index partitions, and therefore terms, of a given combination of k's. The relations of order 4, 5 and 6, which we will need in a moment, are

For identical particles, all moments and cumulants are fully symmetric under index permutation.

The derivation of the Edgeworth expansion starts with the generic Gram-Charlier series [12, 18, 19], which is expressed in terms of differences between the measured and reference cumulants

and moment-like entities z_ijk··· which are related to the h_ijk··· by the same moment-cumulant relations (15)-(20), i.e.,

and so on. The Gram-Charlier series

is an expansion in terms of the zs and partial derivatives

which for Gaussian f₀ are called Hermite tensors; they will be discussed below.

The generic Gram-Charlier series is reduced to a simpler HBT Edgeworth series in three steps. First, the freedom of choice for the parameters l_i and l_ij of the reference distribution (4) allows us to set these to the values obtained from the measured distribution, i.e., we are free to set l_iº k_i and l_ijº k_ij, so that h_i = h_ij = 0 and hence z_i = z_ij = 0.

Second, we make use of the fact that all cumulants of order 3 or higher are identically zero for the Gaussian distribution, l_ijk = l_ijkl = ··· = 0, so that z_ijk = k_ijk, z_ijkl = k_ijkl, z_ijklm = k_ijklm and in sixth order z_ijklmn = k_ijklmn + k_ijk k_lmn[10].

Finally, the contribution to the correlation function C₂(q) of the momentum difference q_{a b} = p_{a = 1} - p_{b = 2} of a given pair of identical particles (a,b) is always balanced by an identical but opposite contribution q_{b a} = p_{b = 1} - p_{a = 2} = - q_{a b} by the same pair, so that C₂(q) must be exactly symmetric under " q -parity",

This implies that all moments and cumulants of odd order of the measured f(q) must be identically zero, k_ijk = k_ijklm º 0, so that terms of third and fifth order and the k_ijkk_lmn contribution to sixth order are also eliminated.

The end result of these simplifications is a multivariate Edgeworth series in which only terms of fourth and sixth order survive,

For three-dimensional q , there are 81 terms in the fourth-order sum and 729 in sixth order, but due to the symmetry of both the k and h, many of these are the same. Defining n = n₁ + n₂+ n₃, we introduce the "occupation number" notation

and correspondingly define cumulants as with n₁ occurrences of the index 1, n₂ occurrences of 2, and n₃ occurrences of 3 in (i₁ ...i_n), e.g. C₁₂₁ = k₁₂₂₃ = k₃₁₂₂ = ···. Similar definitions hold for and for the two-dimensional case. Combining terms in (31), we obtain for the two- and three-dimensional cases respectively,

where the square brackets here indicate the number of distinct cumulants related by index permutation to those shown. In two dimensions, we therefore have 5 distinct cumulants of fourth order and 7 of sixth order, while in three dimensions there are 15 distinct fourth-order and 28 sixth-order cumulants respectively. We note that these cumulants can be nonzero even when the reference Gaussian is uncorrelated, i.e., even if the Pearson coefficients are zero.

B. Hermite tensors

In the Edgeworth series (33) and (34), the cumulants C are coefficients fixed by direct measurement, while the Hermite tensors H are, through eqs. (27)-(29) and explicit derivatives of (4), known functions of q . Defining dimensionless variables

which can also be written in terms of the usual radii as z_i = q_i R_ii, the lowest-order Hermite tensors are, for the azimuthally symmetric out-side-long system,

Fourth-order derivatives (32) of the Gaussian (4) yield,

where the inverse cumulant elements are functions of the parameters l_i and l_ij that can be read off from Eq. (13). The differences between various permutations of (n₁n₂n₃) above arise from the fact that = 0 for azimuthal symmetry.

Note that only one of the above Hermite tensors can be written in terms of Hermite polynomials at this level of generality, namely H₀₄₀ = H₄(z₂)/. Generally, the Hermite tensors factorize into products of Hermite polynomials H_n(z_i) only if all Pearson coefficients r_ij in the Gaussian reference are zero,

In sixth order, the tensors are generically

where again the square brackets indicate the number of distinct index partitions. Sixth-order tensors can then be constructed from these as usual, for example

which closely resembles the Hermite polynomial H₆(z) = z⁶ - 15z⁴ +45 z² - 15 but reduces to the latter only when r_ol = 0 and hence .

C. A gallery of shapes

In Fig. 1, we show surface plots for individual fourth-order Hermite tensors times the two-dimensional reference Gaussian, f₀, with r set to zero. As these are plotted in terms of z_i = q_i /s_i¹, the axes are scaled by the standard deviations, meaning that all Gaussians with r = 0 will be circular in (z₁,z₂) plots. The individual Hermite tensors clearly reflect the symmetry of their respective occupation number indices n_i and probe different parts of the (z₁,z₂) phase space as shown. Comparing the fourth-order terms of Fig. 1 with the sixth-order ones of Fig. 2, we note that the latter probe regions up to several s_i.

In order to exhibit the influence of combinatoric factors, we show in Fig. 3 individual terms of the two-dimensional Edgeworth series (33) in the form f₀(q) (1 + (q)), where the combinatoric factors are fixed in (33). All fourth- and sixth-order cumulants have been set to 1.0 and 2.0 respectively. (This is obviously for illustrative purposes only; in real data, smaller values are expected and shapes will be more Gaussian than those shown here.) The plots for H₃₁ and H₁₃ illustrate the correspondence between index permutation and symmetry about the z₁ = z₂ axis. We note that the diagonal terms H_n0 have little influence on the overall shape, while the off-diagonal ones have a larger effect, not least because of the combinatoric pre-factors.

Testing the influence of fourth- versus sixth-order terms, we show in Fig. 4 some "partial" two-dimensional Edgeworth series including only fourth-order terms, only sixth-order terms, and both orders; again, cumulants were set to the arbitrary values of 1 and 2 respectively.

In Fig. 5, a selection of shapes for individual terms f₀(q)(1 +) of the three-dimensional Edgeworth expansion (34) is shown. While in the two-dimensional case full contour plots could be shown, the surfaces shown here in each case represent only a single contour. In Fig. 6, we show two examples with two selections of fourth-order cumulants nonzero; the shape obviously depends strongly on their selection and magnitude. Clearly, effects of the different cumulants on the overall shape often cancel out. We emphasize again that the shapes shown are for illustrative purposes only and do not represent real data.

IV. DISCUSSION

The multivariate Edgeworth expansions (33)-(34) appear to be a promising tool for quantitative shape analysis in HBT. While the real test will be to gauge their performance in actual data analysis, they do seem to have the right features and behaviour. A number of issues deserve further comment:

1. It has been noted previously [14, 15] that the traditional radii of a Gaussian-shaped R₂(q) could be found by direct measurement rather than from fits. In the present formulation, this amounts to the direct measurement of the second-order cumulants l_ij, which can be directly converted to "radius parameter" form via Eq. (13). Going beyond Refs. [14, 15], we suggest that higher-order cumulants can be measured directly also.

2. Many people have rightly expressed concern that these radii do not adequately represent the true shapes and behaviour of HBT correlations. Our Edgeworth expansion confirms that such radii are clearly not the whole story, but that they do represent the appropriate lowest-order approximation (for Gaussian reference) with respect to which non-Gaussian shapes should be measured.

3. We have demonstrated that it is imperative to write Edgeworth expansions in a fully multivariate way: the combinatoric pre-factors in (34) are large for multivariate "off-diagonal" cumulants, while the influence of diagonal cumulants is strongly suppressed due to their small prefactors. The cumulant C₂₁₁, for example, has a weight 12 times larger than C₄₀₀, and indeed the entire expansion is dominated by the off-diagonal cumulants. Furthermore, even large diagonal cumulants do not change the shape much, as a glance at Fig. 3 will confirm.

4. Deviations from Gaussian shapes are consistently quantified by the sign and magnitude of higher-order cumulants, which are identically zero for a null-case pure Gaussian f( q ). The Edgeworth expansion using these cumulants, while recreating the shape of f( q ), therefore at the same time provides a quantitative framework for comparison of different shapes.

5. Operationally, we suggest a procedure of successive approximation, whereby in a first step all elements of the covariance matrix k_ij = l_ij are measured, thereby determining all the s's and the Pearson coefficient; this is equivalent to the usual determination of radii. This is followed by measurement of the set of fourth-order cumulants . The measured numbers for fourth-order cumulants then represent the basis for shape quantification and comparison. If statistics permit, sixth-order cumulants can be added as a further refinement.

6. The q -cumulants proposed here are numbers rather than functions of q . From the viewpoint of compactness of description, this will be an advantage compared to the shape decompositions in terms of spherical and cartesian harmonics [4, 10], in which each coefficient is a function of |q|. It may, however, in some cases be better to see the detail provided by such functions.

7. The procedure outlined above involves no fits. This represents a major advantage over fit-based quantification in two ways:

Firstly: In three-dimensional analysis, typical fits are dominated by phase space, i.e., by the fact that there are many more bins at intermediate and large |q| than at small |q|. This dominance suppresses the influence of the most interesting region on the c² for best-fit values of the parameters. In Ref. [3], for example, we found that the regions of intermediate |q| dominated the shape and quality of various fits.

Secondly, as shown in Fig. 2, cumulants are sensitive to the tails of distributions, and they will hence access the same information as these fits and parametrizations, but in a more direct and sensitive way. It is well known that a direct fit to a probability distribution that is close to Gaussian is an ineffective and inaccurate way to quantify non-Gaussian deviations, while cumulants do so in the most direct way possible.

8. It remains to be seen how the proposed procedure fares when the practical experimental difficulties of finding f(q) and the higher-order cumulants come into play. Much will also depend on the size and accuracy of statistical errors. Fortunately, current sample sizes are large enough to warrant some optimism in this respect.

9. The traditional chaoticity parameter l remains undetermined within the present Edgeworth framework, because it cancels already in the definition (3) of f(q). For a given level of approximation (Gaussian only, fourth-order cumulants, sixth-order), it and the overall normalization factor g may be recovered afterwards by using (34) in a two-parameter fit mode using parametrization

C₂(q) = g [1+l f₀(q) (Edgeworth expansion) ]

with the previously experimentally-determined radii and treated as constants, with g and l the fit parameters.

10. We note the importance of the parity argument C₂(-q) = C₂(q) in eliminating odd-order terms in the Edgeworth expansion. The parity argument falls away, however, in variables where this symmetry does not arise; for example, any one-dimensional Edgeworth expansion involving only positive differences (e.g. in q_inv) would have to include third- and fifth-order terms.

A corollary of the parity argument is that three-dimensional correlations may not be represented in terms of positive absolute values of the components (q_o,q_s,q_l) as this destroys the underlying symmetries. The best one can do to improve statistics is to combine bins that map onto each other under the transformation q ® - q and thereby eliminate four of the eight octants in the three-dimensional (q_o,q_s,q_l)-space.

11. In the present formulation, any dependence on average pair momentum K resides in the cumulants: all k_ijk···, including the traditional radii and the Pearson coefficient, are in principle functions of K .

12. The Edgeworth analysis set out in this contribution is based on a Gaussian reference f₀. Shapes that differ significantly from Gaussian will not be described well in either the Edgeworth framework or the spherical or cartesian harmonics frameworks. One should not, for example, expect power laws such as a pure Coulomb wavefunction (whose square tails off like |q|^-2) to work in a Gaussian-based Edgeworth expansion. Indeed, it is known that large cumulants can lead to a situation where the truncated Edgeworth expansion of f(q) becomes negative in some regions. It is therefore suitable only for shapes that do not deviate strongly from Gaussians; for strong deviations, other expansions will become a necessity.

13. The Edgeworth framework is easily extended to the case of nonidentical particles. In that case, cumulants of all orders will have to be measured. It may well be that the fluctuations of lower-order quantities render the measurement of higher-order cumulants impossible, and great care will clearly have to be taken.

Acknowledgments

This work was funded in part by the South African National Research Foundation. HCE thanks the Tiger Eye Institute for hospitality and inspiration.

[1] G. Goldhaber, S. Goldhaber, W. Lee, and A. Pais, Phys. Rev. 120, 300 (1960).

[2] EHS/NA22 Collaboration, N. M. Agababyan et al., Z. Phys. C 71, 405 (1996).

[3] H.C. Eggers, B. Buschbeck, and F.J. October, Phys. Lett. B 635, 280 (2006) [hep-ex/0601039].

[4] Z. Chajecki, T.D. Gutierrez, M.A. Lisa, and M. Lopez-Noriega (STAR Collaboration), in: 21st Winter Workshop on Nuclear Dynamics, Breckenridge CO, February 2005 [nucl-ex/0505009].

[5] S. Hegyi and T. Csörgo, Proc. Budapest Workshop on Relativistic Heavy Ion Collisions, preprint KFKI-1993-11; T. Csörgo, in: Soft Physics and Fluctuations, Proc. Cracow Workshop on Multiparticle Production, Cracow, 1993, ed. A. Bia as, K. Fialkowski, K. Zalewski, and R.C. Hwa, World Scientific (1994), p. 175.

[6] T. Csörgo and S. Hegyi, Phys. Lett. B 489, 15 (2000).

[7] D.A. Brown and P. Danielewicz, Phys. Lett. B 398, 252 (1997) [nucl-th/9701010].

[8] D.A. Brown and P. Danielewicz, Phys. Rev. C 57, 2474 (1998) [nucl-th/9712066].

[9] D.A. Brown et al., Phys. Rev. C 72, 054902 (2005) [nucl-th/0507015].

[10] P. Danielewicz and S. Pratt, Phys. Lett. B 618, 60 (2005) [nucl-th/0501003].

[11] P. Lipa, H.C. Eggers, and B. Buschbeck, Phys. Rev. D 53, 4711 (1996) [hep-ph/9604373].

[12] A. Stuart and J.K. Ord Kendall's Advanced Theory of Statistics, 5th edition Vol. 1, Oxford University Press, New York (1987).

[13] P. Carruthers, H.C. Eggers, and I. Sarcevic, Phys. Lett. B 254, 258 (1991).

[14] U.A. Wiedemann and U. Heinz, Phys. Rev. C 56, 610 (1996) [nucl-th/9610043].

[15] U.A. Wiedemann and U. Heinz, Phys. Rev. C 56, 3265 (1996) [nucl-th/9611031].

[16] B.R. Schlei, D. Strottman, and N. Xu, Phys. Lett. B 420, 1 (1998) [nucl-th/9702011].

[17] S. Chapman, J.R. Nix, and U. Heinz, Phys. Rev. C 52, 2694 (1995) [nucl-th/9505032].

[18] J.M. Chambers, Biometrika 54, 367 (1967).

[19] O.E. Barndorff-Nielsen, Parametric Statistical Models and Likelihood, Lecture Notes in Statistics Vol. 50, Springer (1988).

Received on 25 November, 2006

1

In our figures and Eq. (54), we use for the Hermite polynomials the definition

H_n(

x) =

(

d/dx)

ⁿ

, which is related to the alternative definition

(

x) =

(

d/dx)

ⁿ

by

H_n(

x) = 2

^-n/2

(

x/

) . The extra

factors creep in because Mathematica uses the latter definition.

Publication Dates

• Publication in this collection
  24 Sept 2007
• Date of issue
  Sept 2007

History

• Received
  25 Nov 2006