Effects of nonuniform acceptance in anisotropic flow measurement
Abstract
Applicability of anisotropic flow measurement techniques and their extension for detectors with nonuniform azimuthal acceptance are discussed. Considering anisotropic flow measurement with two and three (mixed harmonic) azimuthal correlations we introduce a set of observables based on the and components of the event flow vector. These observables provide independent measures of anisotropic flow, and can be used to test selfconsistency of the analysis. Based on these observables we propose a technique that explicitly takes into account the effects of nonuniform detector acceptance. Within this approach the acceptance corrections, as well as parameters which define the method applicability, can be determined directly from experimental data. For practical purposes a brief summary of the method is provided at the end.
pacs:
25.75.Ld, 25.75.q, 25.75.GzI Introduction
One of the most important observable in the study of ultrarelativistic nucleusnucleus collisions is the anisotropic transverse flow Back et al. (2005a); Adams et al. (2005a); Adcox et al. (2005); Voloshin (2003). It is quantified by coefficients in Fourier decomposition of particle azimuthal distributions relative to the collision reaction plane Voloshin and Zhang (1996); Poskanzer and Voloshin (1998a), the latter is defined by the beam direction and the impact parameter. The second harmonic in such a decomposition is called the elliptic flow. Observation of strong inplane elliptic flow increasing with collision energy from top AGS (Alternating Gradient Synchrotron) energies Barrette et al. (1994, 1997), then at CERN SPS (Super Proton Synchrotron) Appelshauser et al. (1998); Alt et al. (2003), and recently at RHIC (Relativistic Heavy Ion Collider) Ackermann et al. (2001); Adler et al. (2001, 2003); Adcox et al. (2002); Back et al. (2002, 2005b, 2005c) shows that the bulk matter created in the high energy heavy ion collision strongly interacts and behaves collectively. Taken together with a key feature of anisotropic flow to be sensitive to the early stage of the collision, this indicates rapid attainment of local thermal equilibrium in the created system. For central collisions, RHIC results are in agreement with ideal (zero viscosity) hydrodynamic predictions Ollitrault (1992); Teaney et al. (2002, 2001). These observations have played an important role in the discovery of the strongly interacting Quark Gluon Plasma (sQGP), the new form of matter formed in heavy ion collisions.
There exist a set of different techniques for anisotropic transverse flow measurement, which have been successfully applied at a variety of experimental setups worldwide Barrette et al. (1994, 1997); Appelshauser et al. (1998); Alt et al. (2003); Ackermann et al. (2001); Adler et al. (2001, 2003); Adcox et al. (2002); Back et al. (2002, 2005b, 2005c). As the reaction plane is not known experimentally, various methods exploit azimuthal correlations between particles as all of the particles are correlated to the same reaction plane. The most widely used are the event plane Barrette et al. (1997); Poskanzer and Voloshin (1998a), the scalar product Poskanzer and Voloshin (1998b); Adler et al. (2002), and mixed harmonic Poskanzer and Voloshin (1998a); Borghini et al. (2002); Adams et al. (2004, 2005b) methods. Analysis of anisotropic flow with azimuthal correlations requires to examine contribution from effects not related to orientation of the reaction plane, such as resonance decays, jets, as well as effects of conservation laws (charge, momentum, etc). Most of these so called nonflow correlations are due to few particle interactions, and their relative contribution scales inversely proportional to particle multiplicity in the event. For that reason in multiparticle correlations these effects are suppressed compared to collective effects such as anisotropic flow. The complete consideration of methods to study nonflow effects is beyond the scope of this paper (for more discussions see, for example, Adler et al. (2002) and references therein). In this paper we assume that the azimuthal distributions of particles produced in the collision depends only on the orientation relative to the reaction plane.
High statistics experimental data collected in recent years at RHIC allow to perform very precise measurements Barrette et al. (1994, 1997); Appelshauser et al. (1998); Alt et al. (2003); Ackermann et al. (2001); Adler et al. (2001, 2003); Adcox et al. (2002); Back et al. (2002, 2005b, 2005c). With the new data, the systematic uncertainty in the analysis becomes dominant compared to the statistical errors. It is vital to carefully investigate the systematic errors, in particular due to nonperfect azimuthal acceptance, as well as to review the applicability of different methods in this respect. In addition, the availability of different experimental setups with complicated azimuthal acceptance, such as central arms of PHENIX (A Physics Experiment at RHIC) detector Aizawa et al. (2003); Adcox et al. (2003), NA49 TPC (Time Projection Chamber) Afanasev et al. (1999), and PHENIX/STAR (Solenoidal Tracker At RHIC) ZDC SMD (Zero Degree Calorimeter Shower Maximum Detector) STAR ZDCSMD proposal (2003), requires further development of new and/or generalization of known methods for use with detectors with significant acceptance nonuniformity. Such generalization would allow to enrich already available data with anisotropic flow measurement results from a wider range of experimental setups. This further provides an opportunity for a comprehensive comparison of available theoretical predictions against the experimental data.
In this paper we describe a procedure to broaden the applicability of known methods to measure anisotropic flow for a range of detectors with nonuniform azimuthal acceptance. Distinctive feature of the suggested approach is that the acceptance corrections can be determined directly from experimental data. This eliminates the need to perform timeconsuming and model dependent MonteCarlo simulations of the detector acceptance and efficiency. The main idea is demonstrated on an example of twoparticle correlations, but for completeness we also provide formulae for the case of three particle correlations (mixed harmonic) technique, which, as discussed above are less susceptible to nonflow effects. We keep the discussion of more complicated three particle correlation case in separate subsections, such that they can be easily skipped if the reader is interested only in the main idea.
The paper is organized as follows. In section II we review notations and definitions of basic quantities used in anisotropic flow analysis. We formulate them in a way that later helps us to introduce new observables based on different components of the event flow vector. These observables provide independent measures of anisotropic flow, and can be used to test selfconsistency of the results. In section III we discuss the effects of nonuniform detector acceptance, and describe the procedure of correcting the single particle and event flow vector such that observables derived for the perfect detector can be used. We provide the smallness parameters that can be used to quantify the range of applicability of the proposed method. These parameters can be estimated directly from experimental data. Finally, we summarize the method.
A similar problem of flow analysis with nonuniform acceptance detector was discussed in Borghini et al. (2001); Bhalerao et al. (2003) in the context of cumulant and LeeYang zeros analyses based on the use of generating functions. Though in some cases these techniques might yield to similar results, our independent approach clarifies the nature of the acceptance effects in flow studies, and further provides the required tools for the most often used analyses based on correlations with the event flow vector.
Ii Methods
ii.1 Definition and notations
Anisotropic transverse flow of particles produced in heavy ion collision is quantified by coefficients in Fourier decomposition of particle azimuthal distribution Voloshin and Zhang (1996); Poskanzer and Voloshin (1998a). In this paper we use particle azimuthal spectra normalized to unity (particle production probability density):
(1) 
Here is the particle azimuthal angle, and is the th harmonic anisotropic flow. The reaction plane angle in equation (1) is an azimuthal angle of the impact parameter. In general, coefficients are functions of particle transverse momentum and pseudorapidity . Such dependences are hereafter assumed implicitly and not indicated in the notation for the sake of brevity and simplicity.
We introduce a unit vector (), defined for each particle:
(2) 
In this equation is given as a complex number with angle and absolute value of unity. Throughout this paper we use complex number representation, but all equations can be rewritten in terms of commonly used algebra of 2dimensional vectors. An estimate of the reaction plane orientation is usually obtained with the so called harmonic event flow vector , which is defined as a sum of vectors over a specific subset ”” of particles produced in the collision:
(3)  
Here and are the event vector components, and is th harmonic event plane angle. For each the event flow vector (or ) provides an independent estimate of the reaction plane orientation. Note that itself can serve as a vector. However this is not very practical, since the more particles used to define the event plane the closer its orientation will be to that of the reaction plane.
To derive the main formulae of proposed technique we find it very useful to introduce, though experimentally unobservable, averaged over events with fixed orientation of the reaction plane:
(4)  
In this section we consider the case of a detector with perfect azimuthal acceptance. This implies that an integration over azimuthal angle goes over without any weight. We will relax this assumption in section III when considering detector acceptance effects. Similarly:
(5)  
Here is an average harmonic anisotropic flow convoluted with multiplicity of particles from a subset “EP” used to calculate the event flow vector .
From equation (5) it follows that is defined as a vector in transverse plane, which on average has an orientation of that of the reaction plane. This feature can be used to define the event flow vector with detectors without tracking that are only sensitive to the shape of the particle distribution in the transverse plane (for example, calorimeters). The only, but very important, requirement to be fulfilled is that the components, and , should be on average proportional to and , respectively.
ii.2 Anisotropic flow from different components
ii.2.1 Two particle correlations
Anisotropic flow via two particle correlations can be obtained with the socalled scalar product method Adler et al. (2002). According to the scalar product technique, one considers the average of the product of and vectors over all events. With the help of Eqs. (4) and (5) this average can be written as an average over all events with fixed reaction plane with further average over all reaction plane orientations:
Here, angle brackets with subscripts , , denote the average over events with fixed ; angle brackets without subscripts, , correspond to the average over entire ensemble of events with all possible orientations of the reaction plane.
The left hand side of Eq. (II.2.1) can be measured from experimental data. To obtain one needs to evaluate . This can be done by using random subevents, i.e. randomly assigning particles used to construct the event flow vector into two subsets and Poskanzer and Voloshin (1998a):
(7) 
The factor of here takes into account the multiplicity difference between the full event and that of subevents and .
From Eqs. (II.2.1) and (7) we obtain:
(8) 
We further note that the two terms in Eqs. (II.2.1) and (7) are statistically independent, which allows to consider them separately:
(9)  
(10) 
thus providing two independent measures of anisotropic flow:
(11) 
Independent observables (11) can be used to check the selfconsistency of the results.
Note, that with normalization of the vector to unity, , the average in (II.2.1) reduces to , and Eq. (8) leads to the main observable of the event plane method Poskanzer and Voloshin (1998a):
(12) 
Similarly, the second equality in formula (11) gives an observable used by the NA49 Collaboration Alt et al. (2003):
(13) 
Here and in Eq. (12) we use the event plane angle defined for the subevents, which resulted in an extra factor of two compared to Eqs. (8, 11).
ii.2.2 Three particle correlations
In the case of three particle correlations one considers:
(14)  
where:
(15) 
Then, the anisotropic flow is given by:
(16) 
If one normalizes vector to unity, formula (16) leads to an observable of the mixed harmonic method Adams et al. (2004, 2005b):
(17) 
All terms in formula (II.2.2) are statistically independent, which leads to a set of equalities:
(18) 
Thus one obtains four independent observables to measure anisotropic flow from three particle correlations:
(19)  
As in the case of the two particle correlations, each of the four terms in Equation (19) provides an independent measure of anisotropic flow, and can be used to check the selfconsistency of the results.
Note, that one can construct three particle correlation function from and vector components other than that defined by formula (II.2.2). Some examples are or . Derivation of observables based on these correlators are similar, but in this paper we only consider combination (II.2.2), which in the case of leads to the known observable for directed flow Adams et al. (2004, 2005b).
Iii Effects of nonuniform acceptance
In order to generalize our consideration for the case of imperfect acceptance we introduce the acceptance function which we normalize to unity (similar to Borghini et al. (2001); Bhalerao et al. (2003)):
(20) 
Then the average of some function , which depends on particle azimuthal angle , at fixed reaction plane orientation is given by the integral:
Here for brevity we introduce notation and , and denote by , the average over the detector acceptance:
(22) 
One might note that and represent th harmonic coefficients in the Fourier expansion of the acceptance function . An important observation is that the acceptance average of , , coincides with the event average, :
(23) 
We assume here that the distribution of the reaction plane angle, , is uniform within a given centrality event sample. Experimentally, this can be achieved by using for the collision centrality determination the independent detector with a good azimuthal coverage. Consequently, all acceptance average quantities can be extracted directly from the experimental data by the corresponding average over all particles in the event sample.
Formula (III) allows to rewrite the expressions for and taking into account the effects of the nonuniform detector acceptance. For clarity of comparison with Eq. (4) we separate the term with , which is the only nonvanishing term in case of perfect acceptance:
(24)  
(25)  
Here we have introduced the acceptance coefficient :
(26) 
and the following smallness parameters:
(27) 
These parameters define the relative contribution of different terms in Eq. (24) and (25). For a particular case of , values of are defined only by detector acceptance, while in general they also depend on the ratio of anisotropic flow and . For a perfect detector (and consequently all parameters ), and Eqs. (24) and (25) are reduced to Eq. (4).
Eqs. (24) and (25) show that acceptance effects result in coupling of equations for flow of different harmonics, and in general a simultaneous analysis of all harmonics is required. However we proceed below neglecting terms. The relative contribution of terms is of the order of , and this case can be understood as either when the th harmonic flow is dominant: (such an assumption, for example, is made in Bhalerao et al. (2003) when discussing acceptance effects), or the acceptance effects for higher harmonics are small: , or both.
In the following, we distinguish three types of acceptance effects:

Twist of the vector that results in appearance of () terms in () components of vector. Determined by nonzero values of in Eqs. (24) and (25), this effect can be corrected for by the diagonalization procedure (after recentering has been applied):
(29) Twist effect is zero if , e.g., in case for detectors symmetric in x or y (such as of rectangular shape).

Rescaling of the vector, which is defined by the coefficients in Eqs. (24) and (25). This effect is the most important one next to the shift of the vector, and it can be corrected for by rescaling the vector components with acceptance coefficients (after the recentering and twist corrections have been applied):
(30)
The acceptance corrected vector has the same average as in the case of a detector with perfect acceptance (compare with Eq. (4)):
(31) 
Similar corrections can be applied for the vector components, which we write in the following form (contributions from terms have been neglected):
The symmetry requires . Applying corrections (2830) for the vector (, have to be replaced with , ), one gets:
(32) 
From Eqs. (31) and (32) it follows that all equations given in section II.2 can be applied to and vectors, and the same observables (11), (II.2.2), and (19) can be used for acceptance corrected and vector components.
To clarify better how the above described corrections work we consider below the correlations between uncorrected and vector components, and discuss what kind of effects are removed by each of the corrections (2830).
iii.1 Two particle correlations
Acceptance effects in conjunction with anisotropic flow may lead to various spurious correlations, such as correlations in multiplicity and/or transverse momentum. In particular, multiplicity correlations in two kinematic regions and are given by the following equation:
(33)  
where , and denotes the average over kinematic regions and . According to Eqs. (III, 33), the (event) average of the product of two functions and defined in regions and can be written as:
(34) 
Deviation of the denominator from unity (which is the value for no multiplicity correlation) is defined by nonzero terms in the sum in Eq. (33) over harmonics. Taking into account that the measured anisotropic flow at RHIC is , in the most pessimistic estimate, using , we obtain . In practice, , which reduces the acceptance effects on multiplicity correlations to the level of a tenth of a percent. In principle, such effects can be consistently taken into account, but for the sake of simplicity, below we proceed neglecting these multiplicity correlations.
In analogy to (II.2.1) we consider the correlations between uncorrected and vector components:
(35) 
The first terms, and , can be removed by the recentering procedure (28). Note, that and enter as a product, what allows to recenter only the event vector components. Similar, to remove the second order terms, , it is sufficient to apply the twist correction only for the vector components. Parameters , and can be obtained with the random subevent technique. In that case, they are defined by a set of coupled equations:
(36)  
After the recentering and twist corrections have been applied Eqs. (III.1) leads to the following observable for the anisotropic flow :
(37) 
Rescaling of the vector reduces this to Eq. (11), which should be written for rescaled and shifted and twisted vectors. Note, that correction factors and correspond to rescaling of the vector.
Eq. (37) shows that in case of two particle correlations recentering and twist corrections of the vector are not required. This equation can be also used for the vector, if the second order corrections defined by the terms are small, and can be neglected (twist correction is not required).
iii.2 Three particle correlations
In the case of three particle correlations we consider:
(38)  
Similar expressions can be written for other terms in Eq. (II.2.2). In contrast to the case of two particle correlations, recentering procedure (28) is required for all three vectors , , and . Acceptance coefficients and are given by a set of coupled equations (36) written for the vector. Twist corrections applied to and vectors removes the terms and , and Eq. (38) leads to observable (19) written for and vector components.
Iv Method summary and conclusion
In this paper we discuss new observables for anisotropic flow measurement based on correlations of and components of the flow vectors. Providing independent measures of anisotropic flow they can be used to check selfconsistency of the analysis. Moreover, these observables allow direct accounting for acceptance effects, which we discuss in detail for two particular cases of anisotropic flow measurement with two and three (mixed harmonic) particle correlations. Importantly, acceptance corrections and parameters, which define applicability of these observables, can be determined directly from experimental data.
Nonuniformity of the detector acceptance is quantified with coefficients and , which further define coefficients and given by Eqs. (26) and (27). Though accounting for the acceptance effects in general might be difficult as it requires a solution of a set of coupled equations with different harmonics involved, in the case when the contribution of harmonics can be neglected, the problem significantly simplifies. It becomes possible to correct the single particle and event flow vectors such that the conventional observables (derived for the perfect detector) can be used. Note, that both, the acceptance coefficients and , and the correlators between and vectors can be obtained during a single pass over the data. This can significantly reduce the amount of time needed for the calculation. At the same time, due to variation of detector acceptance in time, with collision centrality, vertex position, etc., it may be important to apply the acceptance corrections separately runbyrun, for different vertex position, etc. In that case it might be more convenient to split the procedure into a few steps with two passes over the data. During the first pass the acceptance coefficients and are extracted as a function of different centrality, time, etc. and all coefficients needed for acceptance corrections presented in Eqs. (2830), both, for and vectors, are calculated. During the second pass over the data the correlators of the standard procedure given by Eq. (910) (Eq. (II.2.2) in the case of three particle correlations) are calculated. Finally, the flow values are extracted as given in Eqs. (11), (19).
Acknowledgments
We thank the members of the STAR flow physics discussion group for fruitful discussions, and especially A.M. Poskanzer and C. Pruneau for reading the manuscript and fruitful suggestions, and J.Y. Ollitrault for critical comments. Financial support provided in part by US Department of Energy Grant No. DEFG0292ER40713.
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