The major achievement for our group in the past year has been the highly successful first NA49 160 GeV/u lead beam run at CERN this past November and December. In preparation for this run the UW group has accepted a leading role in the production of Main TPC tracking software (Sections 7.5 and 7.6) and slow-controls system coordination software (Sections 7.4 and 9.4). The CERN and heavy ion communities have recognized that NA49 has carried out a major achievement by bringing into operation a highly complex experimental system with almost perfect functioning during its first running period. This experiment is considered to be the flagship experiment for the CERN heavy ion program through the year 2000.
The full NA49 system will consist of two 1.5T vertex magnets, with one vertex TPC installed in each magnet gap, and two large main TPCs on either side of the beam and 10 m downstream from the main target. Of these the second vertex TPC and the right Main TPC have been installed for this first lead beam run. The full TPC complement will be installed in time for the 1995 run next fall.
The Main TPCs are each 3.5 m square by 1.2 m high in active volume. Each TPC has about 64,000 electronics channels with 512 ADC samples per channel per event. This produces a data rate for one main TPC of about 3-4 Mb/s. The total data volume after five days of NA49 operation was about 1.5 Tbyte.
Successful Main TPC tracking was accomplished minutes after the first data were recorded. Track and charge distributions were displayed with a UW-produced display package (Section 7.7) and were used to check out and optimize the functioning of the TPC on line and to analyze the performance of the tracking software offline.
The UW group also played a lead role in preliminary analysis of Main TPC data (Section 7.3) in preparation for the Quark Matter 95 meeting at Monterey, CA in January. Charged-particle momentum spectra, collision system temperature distributions and net-charge momentum distributions were extracted during the week following the end of the run. These UW efforts, in combination with other VTPC and MTPC analysis efforts by NA49 collaborating institutions such as MPI-Munich, Lawrence Berkeley Laboratory, and IKF-Frankfurt provided a good preliminary description of this new lead-lead collision system only weeks after the first availability of beam.
In addition to our NA49 activities we have continued an active role in the STAR collaboration, which will mount a solenoidal detector at the RHIC accelerator in time for turn on in 1999. Some of these activities included development of STAR trigger algorithms based on correlation measures (Sections 7.11, 7.12 and 7.13), development of a servo-controlled TPC high voltage control system, and simulations in support of the STAR silicon vertex tracker (Section 7.8).
In addition to collaboration-specific activities we continue to pursue a
strong interest in development of HBT or Bose-Einstein correlation
determinations of collision system space-time geometry (Sections 7.9 and 7.10).
This past November and December NA49 carried out at CERN the first data acquisition with 33 TeV lead projectiles incident on a lead target. Operational detector elements included an in-magnetic-field vertex TPC, an out-of-field main TPC, a ring calorimeter covering the pseudorapidity region 2.1 < {EMBED EQUATION |} < 3.4, and a veto calorimeter covering the very forward pseudorapidity region which accepts spectator projectile particles, i.e., those which have not participated in the collision.
A minimum-bias correlation plot between {EMBED EQUATION |} and {EMBED EQUATION |} for the veto and ring (transverse energy) calorimeters respectively shows an expected anticorrelation of these observables extending from {EMBED EQUATION |} = 33 TeV at {EMBED EQUATION |} = 0 (no collision) to {EMBED EQUATION |} = 6 TeV at {EMBED EQUATION |} = 0.5 TeV (central collision). Position of an event along this correlation is determined mainly by the particular collision geometry (impact parameter) for the event. The observed minimum {EMBED EQUATION |} energy is equivalent to about 38 noninteracting projectile particles (spectators) incident on this calorimeter, whereas simple model calculations predict about 13 spectator particles or 2 TeV. Therefore, about 4 TeV equivalent energy in produced particles must fall within the veto calorimeter acceptance.
Using these calorimeter data the degree of stopping of projectile partic les while passing through the target can be compared for these new Pb-Pb results with that for S-Au obtained by NA35. The degree of stopping is found to be sim ilar, for a corresponding mean number of interactions of projectile particles, i ndicating that collective effects in the stopping process do not seem to dominat e.
More detailed analysis of calorimeter data indicate an energy density in the collision system of about 3 GeV/{EMBED EQUATION |} in a volume (from HBT me asurements) roughly 3.5 times larger than that for 200 GeV/c sulfur-sulfur.
Preliminary analysis of vertex and main TPC data indicate that the negat ive charged particle (mainly pion) distribution in pion rapidity has a peak valu e at midrapidity of about 230 per rapidity unit. This distribution is significa ntly broader than would be expected for particle emission isotropic in the CM. A preliminary study of Bose-Einstein correlations of negative particles indicate that the source size is about 7.4 fm, as compared to a source size of about 4.7 fm for sulfur beam.
These preliminary results indicate the achievement of energy densities
and produced particle multiplicities consistent with predicted conditions
for color deconfinement. The similarity of the degree of stopping for sulfur
and lead on heavy targets indicates the absence of rescattering or other
collective effect.
When the 160 GeV/u lead beam became available at the CERN SPS last fall, we carried out a preliminary spectrum analysis for experiment NA49's main TPC in order to provide an initial look at the data. This analysis also served as a check on the status of the hardware and analysis software.
The NA49 right main TPC is located 11 m downstream from the target and d ouble magnet system with a lateral displacement of 2 m to the right of the beam. Straight tracks are deduced from correlation of ionization deposited by charge d particles traversing the TPC active volume. For each reconstructed track, the corresponding particle's physical parameters, such as transverse momentum {EMBE D EQUATION |}, rapidity assuming pion and proton mass ({EMBED EQUATION |}, {EMBE D EQUATION |}), and transverse mass ({EMBED EQUATION |}) are determined. Due to the condition of the software environment in these early stages, only 100 event s of negative particles ({EMBED EQUATION |}) and 70 events of positive particles ({EMBED EQUATION |}) were used for this analysis. On average, there are 320 re constructed tracks per event for {EMBED EQUATION |} and 390 for {EMBED EQUATION |}.
The only correction applied to the raw spectrum was the TPC geometrical acceptance. Two acceptance correction factors were determined, one for {EMBED E QUATION |} and one for {EMBED EQUATION |} spectra. The {EMBED EQUATION |} accep tance correction was determined by comparing the simulated output of GEANT with flat phase space input. On the other hand, a VENUS-generated phase space distri bution was used to determined the acceptance correction factor for the {EMBED EQ UATION |} distribution. Due to the lack of statistical power over certain regio ns, the acceptance correction was most reliable for 4 <{EMBED EQUATION |} < 5.
For a pion mass hypothesis, {EMBED EQUATION |} acceptance corrected data
are then used to produce event averaged {EMBED EQUATION |} < dN/d{EMBED EQUATIO
N |}> and rapidity
A "net baryon'' distribution can be deduced from the charge excess betwe
en the {EMBED EQUATION |} and {EMBED EQUATION |} spectra. The preliminary analy
sis shows a mean {EMBED EQUATION |} of 580 MeV/c for this distribution. The rap
idity distribution peaks near mid rapidity.
This set of results from the preliminary data analysis provides us with
a first substantive glance at the data obtained from a new generation of ultrare
lativistic heavy ion experiments. It also makes clear the challenge to our soft
ware analysis system in analyzing the bulk of data to be generated by NA49 in co
ming years.
SControl was designed (see Section 9.4) for CERN Experiment NA49 and was
used extensively in the initial Autumn 1994 run of NA49. The program operated
on a dedicated HP-712 workstation in the NA49 counting room. In this applicatio
n its function as a "control'' system for actively changing experimental paramet
ers was minimal. The principal tasks of SControl were: (a) to collect experim
ent-status information from 6 satellite processors, (b) to maintain an archive o
f this information by updating an archive file, (c) to maintain a data structure
(BOS bank) that was read at irregular intervals by the data acquisition system,
(d) to provide user-controlled displays of experiment parameters of interest, a
nd (e) to generate and manage alarms.
The experiment was represented by pages organized in a hierarchical stru
cture, with a "home'' page at the top level showing an orthographic representati
on of the whole experiment. On this diagram the major subsystems were outlined
in white, and each of these outlined regions provided a hyper-link to the top-le
vel page of the subsystem. The subsystem top-level pages typically showed a blo
ck diagram of the subsystem, with blocks outlined in red providing hyper-links t
o the appropriate sub-sub-systems. This pattern of diagrams with hyper-links wa
s repeated at one or more levels until a page was reached that was designed to m
onitor several related functions of a particular subsystem. For example, a page
displayed as a strip chart the measured drift velocity of gas used in a TPC, th
e pressure and temperature at which the measurement was made, a "normalized'' dr
ift velocity that had been corrected for variations in temperature, pressure, an
d electric field, and a scatter plot correlation of measured drift velocity with
measured pressure.
An alarm system was created which generated appropriate levels of alarms
when (a) certain temperatures went out of range and (b) when one of the "heartb
eat'' signals provided by the satellite processors failed to recur at the expect
ed interval.
Archives were updated every 10 minutes and organized into daily files, e
ach initialized at midnight. About 200 Mb of archived parameter files were accu
mulated during the NA49 run. Measured gas temperatures, pressures, and other da
ta from these files have already proved very valuable in providing comprehensive
time-dependent estimates of the drift velocities in the NA49 TPCs.
Experience during the 1994 run of NA49 provided a number of lessons that
bear on the application of SControl to future experiments and NA49 runs: (1) I
t is important to establish and enforce a naming convention for the slow control
parameters archived. Names supplied by satellites were used to identify parame
ters in the archives. Parameters with time-dependent names proved troublesome.
(2) Whenever an alarm link is created, a message describing the action to be ta
ken when the alarm occurs should also be provided. No alarm is "obvious''. (3)
In the 1994 NA49 run, all slow control information was written each day to a ma
ster archive file. Because of the large volume of array information (up to 12 M
b/day) stored in the archive file by the two time-of-flight systems, this arrang
ement proved unwieldy. In future runs, when there will be more detectors on lin
e, we will maintain separate archive file structures for several subsystems. (4
) A better way of reviewing archived information is needed. We are considering
a facility for converting slow control archives to a PAW ntuple for analysis. (
5) One of the great advantages of SControl is the ease with which the user inter
face can be modified and expanded while running. However, this can lead to conf
usion and duplication unless the application developer exercises restraint.
7.5 NA49 main TPC tracking software
S.J. Bailey, P. Chan, D.J. Prindle, S. Schönfelder,* T.A. Trainor, P. Ve
nable and X. Zhu
The original NA49 tracking software was an extension of NA35's TRAC prog
ram, modified to match the geometry and tracking needs of NA49. All steps of th
e analysis from reading the raw data through the momentum calculation were incor
porated into a single large stand-alone program (TNT).
To allow a more modular design and interactive access to the data, the N
A49 Server Environment (NASE) was created at IKF, Frankfurt, for use by all NA49
analysis programs. This server provided a central memory manager that separate
client programs could access, thus sharing data. After a client had finished,
its output data was stored in the central server where users could access the da
ta to immediately view the results from a PAW-like environment.
The stand-alone tracking software was modified and split up into several
separate clients for NASE. This collection of clients, STIRN, became the basis
for NA49 MTPC tracking. STIRN includes modules for reading in init files with
setup parameters, raw-data reading and space-point finding, tracking, dE/dx calc
ulation, and momentum determination. It also comes with a collection of utility
clients for matching simulated tracks to reconstructed tracks, calculating two-
track resolution, and other programs for debugging and testing the primary clien
ts.
Although improvements continue to be made to STIRN, the clients were rea
dy and tested on simulated data before the first data acquisition run took place
at CERN this last Fall. Thus, we were able to view reconstructed tracks litera
lly within minutes of the initial raw data acquisition. The modular nature of S
TIRN allowed numerous improvements and suggestions to be rapidly implemented dur
ing the run.
Although useful, NASE suffers from being slow, using memory inefficientl
y, and instability. Because of these problems STIRN was recently converted to r
un under DSPACK, an alternate server system. Because DSPACK uses shared memory
instead of copying data through Unix sockets, the performance of STIRN under DSP
ACK is significantly enhanced. DSPACK is also a more stable system than NASE an
d initial results have been promising.
Until both systems have been fully tested, STIRN is being simultaneously
developed for both servers with the choice of which server to use being made at
compile time. This allows both systems to be tested and compared while maintai
ning the same source code for the core of the analysis to ensure accountability
between tests of the different systems.
7.6 NA49 InitEdit database editor
S. Bailey, P. Chan, J.G. Cramer, D.J. Prindle and T.A. Trainor
The NA49 main TPC tracking code (see previous section of this Annual Rep
ort) has had all parameters relating to track finding excised from the code and
placed in initialization files. This makes it easy to check the effect of each
parameter on track finding. The drawback is that we need an accountable way to
check and modify what is in the initialization file. To do this we have written
a program specifically to edit these files.
InitEdit can be thought of as a user interface to the tracking code para
meter files. InitEdit uses Motif widgets as its interface elements. The tracki
ng parameters are typically read from a file and after editing can be written ba
ck to a file or sent directly to the NASE server. This ability to send the para
meters directly to NASE makes it very easy to test a variety of parameter values
before committing them to a file.
The initialization data are divided into sections, with a given program module t
ypically using only one section of the initialization data. InitEdit edits each
section independently, making it easier for the user to worry only about the re
levant parameters. InitEdit can check that the parameters are within an accepta
ble range. Typically it checks that the user has entered a valid number and mak
es no restriction on the range.
Fig. 7.6-1. Example of an InitEdit screen. This screen allows one to modify t
he track-finding parameters.
7.7 Na49 display
S. Bailey, P. Chan, J.G. Cramer, D.J. Prindle and T.A. Trainor
To help check our track finding and fitting algorithms we have developed
an event display program. This program uses the Motif widget set for its user
interface elements and draws to the screen using X primitives. This allows it to
be used from any X terminal. Currently the program knows how to draw Monte-Car
lo and reconstructed space points in all NA49 TPCs as well as Monte-Carlo and re
constructed tracks in the Main TPCs.
Navigation through the data can be done by holding one of the mouse butt
ons down while moving the mouse. Holding the left mouse button down while movin
g the mouse simply moves the display objects. Holding the middle mouse button d
own causes a zoom toward or away from the point under the cursor. Holding the r
ight mouse button down while moving the mouse causes the objects in the display
to rotate about the origin. One of the main features of the program is that it
draws quite quickly, allowing one to view the wire frame TPC outlines and all tr
acks during a rotation. This makes it very easy to orient the display to look a
long a precise direction.
There are numerous other features, including the ability to select point
s or tracks, attach scales to points and hide categories of objects in addition
to an extensive help selection.
Fig. 7.7-1. An example of the display screen.
7.8 Energy deposition and particle identification in a TPC
H. Bichsel
Because the individual detector cells in a TPC encompass a limited volum
e (of the order of 5x10x40 mm), the ionization J observed (which corresponds to
the energy deposited in that volume) differs from the energy {SYMBOL 68 \f "Symb
ol"} lost by a charged particle traversing it. It is quite easy to calculate th
e energy lost.1 It will include large energy losses which produce secondary ele
ctrons, "{EMBED EQUATION |} rays". If the path of a {EMBED EQUATION |} ray carr
ies it outside of the measurement volume, the energy deposited will be reduced.
It is also possible that {EMBED EQUATION |} rays from neighboring volumes will e
nter the volume under consideration ("crossers"), and the energy deposited will
be increased.
Since J is related to particle speed, it is possible to determine from i
t the particle mass if the momentum of the particle is determined in another mea
surement. The process of mass determination is called "particle identification"
(PID). The ionization is a stochastic quantity, thus a large number of values
of J (of order 100) are measured in successive layers of a detector. Then, a su
itable average value Ja must be selected from all the values of J to get the spe
ed. Monte Carlo calculations simulating the process have been made, and the ran
ge of particle momenta for which PID is possible has been determined.
The energy loss spectrum for particles with charge z = 1 and {EMBED EQUA
TION |} = 5.2 is shown by the solid line in the Fig. 7.8-1. The use of 60% of a
ll {SYMBOL 68 \f "Symbol"} to obtain {EMBED EQUATION |} excludes values above ab
out {EMBED EQUATION |} = 7 keV. Thus the difference between {SYMBOL 68 \f "Symb
ol"} and J needs to be known only for {SYMBOL 68 \f "Symbol"} < 7 keV. Monte Ca
rlo calculations for the effect have been made. The result is shown by the dash
ed line: the difference between {SYMBOL 68 \f "Symbol"} and J is large2 only for
large {SYMBOL 68 \f "Symbol"}. The largest contribution to the difference is
the presence of crossers: their energy deposition is at least 10 keV, thus any J
which includes a crosser will be well above {EMBED EQUATION |} and will not be
relevant to PID. The total number of crossers is of order of 2%, depending on t
he geometry, and the net effect is simply that the energy deposition spectrum is
reduced by 2% compared to the energy deposition spectrum. Electrons escaping f
rom the counting volume occur for only 0.6% of all events.
The conclusion is that the difference between energy loss and energy dep
osition is negligibly small for purposes of PID.
{EMBED EQUATION |}
{EMBED EQUATION |}
Fig. 7.8-1. Energy loss spectra for z=1 and {EMBED EQUATION |} = 5.2.
7.9 STAR SVT efficiency for D meson detection
S. Bailey, P. Chan, J.G. Cramer, D.J. Prindle and T.A. Trainor
One very interesting observable in ultra relativistic heavy ion collisio
ns is D meson production. Previous attempts1 to demonstrate that STAR is capabl
e of observing D meson production have used the flight path of the D as a signat
ure. The D+ and D- mesons have {EMBED EQUATION |} (in the rest frame) while the
{EMBED EQUATION |} and {EMBED EQUATION |} have {EMBED EQUATION |}. An energeti
c D meson may travel a few millimeters before decaying, and a suitable vertex de
tector should be able to distinguish its decay point from the primary vertex.
However, STAR's acceptance is centered at mid-rapidity and the D mesons
produced there are expected to be thermal and hence low velocity. This implies
a very short flight path for the D mesons. Multiple scattering of the rather lo
w energy decay products in the beam pipe and vertex detector material significan
tly modifies the track directions. This has led to the conclusion that the curr
ent STAR vertex detector design is not suitable for detecting D mesons produced
in central Au-Au collisions via the detection of a secondary vertex.
We have therefore examined the possibility of using the special kinemati
cs of the D*+ {EMBED EQUATION |} decay mode to detect {EMBED EQUATION |} mesons.
2 Briefly, the branching ratio of D*+ into {EMBED EQUATION |} is 55%. Since th
e mass difference is so small ({EMBED EQUATION |}) the {EMBED EQUATION |} is ess
entially emitted with the D*+ meson velocity. Typically one looks for a {EMBED
EQUATION |} candidate, then given this candidate one looks for a matching {EMBED
EQUATION |} to form a D*+ candidate and makes a tight cut on that mass. In som
e cases this cut can dramatically reduce the background under the {EMBED EQUATIO
N |} invariant mass spectrum.3
Our findings are not highly encouraging. Briefly, we found that the sma
ll mass difference is still large enough that for a thermal D*+ meson the labora
tory {EMBED EQUATION |} - {EMBED EQUATION |} opening angle spans a large range.
We can make some kinematic cuts to accept less than one {EMBED EQUATION |} per
{EMBED EQUATION |} candidate in an average central Au-Au event. This increases
the signal-to-background ratio, but the net loss of signal decreases its overall
statistical significance. It is possible that selecting the high energy tail o
f the D distribution will make possible both the secondary vertex selection and
the use of special D*+ decay kinematics, but this method is sensitive to details
of the transverse momentum distribution of the D mesons.
7.10 Scale-distorted Gaussians and HBT "Wiggles''
J.G. Cramer
The Hanbury-Brown-Twiss interferometry technique is widely used in ultra
-relativistic heavy ion measurements with pions and other Bose-Einstein particle
s to extract information on source geometry and duration. The technique is prim
arily sensitive to the 2nd moment, i.e., width, of the source distribution. Howe
ver, deviations from a Gaussian source shape will produce "wiggles'', i.e., osci
llations of a few percent around a value 1, in the momentum space correlation at
large relative momentum. These, in principle, can be used to gain more source-
shape information.
We have demonstrated this phenomenon in Monte Carlo simulations using no
n-Gaussian source distributions and find that the wiggle amplitudes are stronger
for the correlation of 3 or more particles than for the more standard 2-particl
e correlations. This work was described in an Annual Report contribution last y
ear.1 Unfortunately, the HBT analyses of experimental data that have been perfo
rmed up to now are based on at most a few thousand events, so the statistics are
not sufficient to support investigation of this phenomenon. However, HBT analy
sis of CERN data with 104 to 106 high-multiplicity events should soon provide th
e needed statistics.
Our Monte Carlo simulations up to now have used simple but unphysical sh
apes, e.g., sharp-edge spheres and hemispheres. The next step in this investiga
tion is to devise more physically reasonable non-Gaussian source shapes having s
ignificant 3rd moments (skewness) and 4th moments (kurtosis). One suggested tec
hnique for achieving this2 is to add higher Gaussian derivatives as Laguerre po
lynomials in an Edgeworth expansion. This technique offers the advantages that
distributions have analytic integrals and that distribution moments are coeffici
ents of the expansion. Unfortunately, the resulting distributions are not posit
ive-definite and are therefore unphysical.
We have devised an alternative prescription for producing a physically r
easonable non-Gaussian source shape, the scale-distorted Gaussian. The procedur
e is to replace the Gaussian function G(x) with G(f(x)), where f(x) is a scale d
istortion function. To produce an unsymmetric distribution function which has a
significant 3rd moment and skewness, we have used an arc-tangent scale-distorti
on function afn(x,a3) = 2 arctan(x/a3)/{EMBED EQUATION |} and f3(x,d3,a3) = x[1-
d3afn(x,a3)]. The Type-3 scale-distorted Gaussian function is therefore G3(x,{E
MBED EQUATION |}) = {EMBED EQUATION |}. Typically, a3 is set to 0.1 and d3 is v
aried between 0.0 (a standard Gaussian) and 1.0 (a limiting case in which the ri
ght side of the distribution is constant out to infinity).
To produce a Gaussian-like function that has a significant 4th moment an
d kurtosis, i.e., is either more flat-topped or sharper that a normal Gaussian,
we have used a modified Gaussian scale-distortion function {EMBED EQUATION |} an
d {EMBED EQUATION |}. The Type-4 scale-distorted Gaussian function is therefore
{EMBED EQUATION |}. Typically, a4 is set to 2.5 and d4 is varied from -0.5 (a
"sharp'' Gaussian) to 1.0 (a "flat-top'' Gaussian).
The disadvantage of using functions like these is that their integrals c
annot be calculated analytically, and thus distribution moments must be calculat
ed numerically. On the other hand, the strong advantages are (1) the functions
produced are guaranteed to be positive-definite, (2) they are fairly simple to c
alculate, and (3) they are fairly simple to incorporate in Monte Carlo calculati
ons. We are now modifying our HBT Monte Carlo program to include these new dist
ribution shapes, which will permit investigation of this phenomenon with more re
alistic source distributions.
7.11 "No-Background'' maximum likelihood HBT analysis
J.C. Cramer
The probability density function for the "signal'' in HBT interferometry
i.e., the probability density of finding a correlated particle pair with relati
ve momentum Q from a source of geometry R, is S(Q,R) = [1+C(Q,R))]G(Q,R))B(Q)/N(
R), where C(Q,R) is the reduced correlation function, G(Q) is the Coulomb correc
tion, and B(Q) is the uncorrelated background distribution. The background dist
ribution B(Q) in the standard approach to HBT analysis is constructed by correla
ting the momenta of particles from separate and therefore uncorrelated collision
s.
The probability normalization factor is N(R) = N0 + N1 (R), where N0 =
{SYMBOL 242 \f "Symbol"}{SYMBOL 87 \f "Symbol"}G(Q)B(Q)dQ, a constant independen
t of R, and N1(R) = {SYMBOL 242 \f "Symbol"}{SYMBOL 87 \f "Symbol"}C(Q,R)G(Q)B(Q
)dQ, a function of R. Here {SYMBOL 87 \f "Symbol"} is the complete space of al
l allowed Q values. We define the normalization correction factor M(R) = N1(R)/
N0, so that N(R) = N0(1+M(R)] and note that for expected R-values M(R) {SYMBOL 2
25 \f "Symbol"}{SYMBOL 225 \f "Symbol"} 1.
A Maximum Likelihood fit to data using this function can be performed by
minimizing the function L(R) = - {EMBED EQUATION |}S(Qi,R), where i is the inde
x specifying all measured particle pairs and K is the total number of pairs. Th
is function can be rewritten as:
L(R) = - {EMBED EQUATION |} [1+C(QiR,)]G(Qi)B(Qi)/N0[1+M(R)
]
= - {EMBED EQUATION |}[1+C(Qi,R)] - {EMBED EQUATION |
}G(Qi) - {EMBED EQUATION |} B(Qi)
+ {EMBED EQUATION |}N0 + {EMBED EQUATION |}[1+M(R)].
(1)
Of the terms in Equation (1), only first and last depend on R. Thus, minimizing
L(R) by varying R is equivalent to minimizing L1(R), as defined by:
L1(R) = K ln[1+M(R)] - {EMBED EQUATION |}[1+C(Qi,R)]
{SYMBOL 64 \f "Symbol"} K M(R) _ {EMBED EQUATION |}[1+C(
Qi,R)] (2)
where the last line is uses the approximation ln[1+M(R)] {SYMBOL 64 \f "Symbol"}
M(R).
If we use the approximation M(R) {EMBED EQUATION |} 0 to eliminate the i
mplicit background dependence of the first M(R)-dependent term in Equation (2),
it will have two consequences. First, the probability density function for the
signal S(Q) will no longer be a true probability. This has no effect on the fit
. Second, the S(Q) function will tend to be slightly larger for large R-values
than for small R-values. Therefore, approximating M(R) {SYMBOL 187 \f "Symbol"}
0 will cause the maximum likelihood fit to be slightly biased toward smaller va
lues of R. For the case of single-event HBT analysis, where the fit to data is
only used as a guide for selecting an ensemble of events that are characterized
by unusually large radii, a small monotonic bias toward smaller radii should be
quite acceptable. Thus, Equation (2) with M(R) = 0 can be used without calculat
ing a background to perform HBT analysis on single high-multiplicity events in u
ltra-relativistic heavy ion collisions.
7.12 Scaled topological measures
J.G. Reid and T.A. Trainor
For this project we use the scaling behavior of topological properties o
f point sets to infer the nature of the parent or generating process. To carry
out such an analysis on a data set it must first be binned over a range of scale
. To do this we simply take the embedding space of the data set and divide it i
nto (rectangular) bins of characteristic size e. We then increase the size of t
he bins, and repeat the process until we have analyzed the entire scale range of
interest. However, the analysis is not quite so simple due to aliasing that re
sults from the density variations between the binnings. Our solution to this is
to 'dither' (phase shift) the binnings of the space at each scale value and for
m an ensemble average of the relevant quantities over the dithered binnings to r
educe the aliasing effects.
With the binning problem under control we examined the scale behavior of
the entropy, information and dimension of various point sets. There is a conti
nuum of defined entropies (each with its own information and dimension), of whic
h the standard thermodynamic (Gibbs) entropy is one, and we must in general cons
ider this continuum as a function of scale. To keep things as simple as possibl
e we usually consider only the 0th (Kolmogorov) and 1st (Gibbs) order entropies,
but it is important to note that this analysis can be extended to include whate
ver entropy measures are of interest. By examining a Poisson-filled embedding s
pace we find that the analysis behaves just as we expect for the Kolmogorov quan
tities; however we are still investigating details of the Gibbs quantities.
Since the main purpose of this analysis is to identify density variation
s in scale which would be missed by classical point set correlation analysis we
have also investigated the analysis of simple hierarchical point distributions.
These are point sets whose only significant features are scaled density variati
ons. We have been very pleased with these results. For these simple hierarchie
s the scaled dimension behaves exactly as we expect: the dimension reaches peak
s at each scale level of the hierarchy and falls off in between. The scaled dim
ension is very nearly the simple sum of the dimension fields of each hierarchy l
evel.
Another important characteristic of the data that can be determined by t
his analysis is the scaled set volume. We have only recently begun this part of
our analysis, but it is already very promising. At large scale the analysis se
es the perimeter of the data set, since at this level the analysis can only see
prominent features. As the scale decreases and approaches the point at which cl
assical volume determinations are made the calculated volume approaches the clas
sical value. Finally, our analysis continues beyond this point into the small-s
cale region. The 'volume' eventually corresponds to the length of a space-filli
ng curve through the data. Below this scale level the volume approaches asympto
tically the total number of points in the set.
We are pleased with the progress of this scaled topological analysis, an
d we expect to concentrate on applications in the near future, particularly for
STAR triggering.
7.13 Entropy analysis of STAR EM calorimeter energy distributions
J.G. Reid and T.A. Trainor
The STAR electromagnetic calorimeter can provide important information r
egarding both the early stages of a high-energy nucleus-nucleus collision and th
e subsequent evolution through hadronization. Because of its fast response to e
nergy deposition the EM calorimeter may be an especially valuable part of the ST
AR trigger system. We have developed an algorithmic approach to triggering base
d on information entropy analysis which may quickly be able to determine the deg
ree of correlation in a particular event energy distribution, and therefore serv
e as a model-independent trigger criterion.
Information entropy analysis is sensitive both to isolated features and
to slower variations in density. To test the analysis we obtained twenty simula
ted proton-proton events produced by the standard event generator HIJING and use
d a simpler event simulator to generate data for A-A events.
For the Kolmogorov (capacity) entropy, as the scale of the binning syste
m approaches the base width of an isolated jet this entropy is reduced with resp
ect to that of a uniform (Poisson-distributed) energy distribution. Once the sc
ale is comparable to the jet size there will be no further such reduction becaus
e there are no more empty bins to reject. Thus by identifying the minimum with
scale of the corresponding capacity dimension-change {EMBED EQUATION |} we can d
etermine the size and spacing of the prominent jets for p-p collisions. One suc
h event analysis is shown in the top panels of Fig. 7.13-1. By examining ensemb
les of such p-p jet events we can formulate trigger conditions based on informat
ion-related quantities such as dimension which will permit fast selection of spe
cial event classes. The bottom panels of Fig. 7.13-1 show an ensemble of p-p ev
ents which exhibit the degree of variation of the dimension-change distribution
for a given event generator.
The Gibbs (information) entropy may seem redundant for analyzing the pro
minent jets of a p-p system, but it is essential in analyzing heavier A-A collis
ion systems. With any A-A event there will be few or no void areas in the data
because we expect to see a background of so-called minijets. This background ma
kes the Kolmogorov entropy much less useful since it depends on identifying reje
cted bins to 'find' distribution features. The Gibbs entropy on the other hand
depends only on density variations. Thus, the minimum in the Gibbs dimension-ch
ange {EMBED EQUATION |} will be an indicator of the point in scale at which all
of the significant density features have been identified. This tells us at whic
h scale our smallest features occur for A-A events. The minijet background prob
lem is especially serious for standard jet-finding algorithms, and is an importa
nt motivation for this new approach.
Since there is a continuum of entropies which correspond to information
contained in higher-order correlations we can extend our analysis to include any
of these which may be found useful. At present we have concentrated our effort
s on developing the analysis system itself with jet distributions and triggering
in mind. In future work we hope to understand in more detail the relationship
of the jet-minijet distribution to changes in the scaled dimension for both the
Gibbs and Kolmogorov entropies.
Fig. 7.13-1. Dimensions lowering analysis of single event calorimeter jet distr
ibution (top) and similar analysis of an ensemble of events (bottom).
*Max-Planck Institut für Physik, Föhringer Ring 6, D-80805 München, Germany.
† Lawrence Berkeley Laboratory, Berkeley, CA.
*Max-Planck Institut für Physik, Föhringer Ring 6, D-80805 München, Germany.
†Lawrence Berkeley Laboratory, Berkeley, CA.
*Max-Planck Institut für Physik, Föhringer Ring 6, D-80805 München, Germany.
*Max-Planck Institute für Physik (MPI), Föhringer Ring 6, D-80805 München, Germa
ny.
1Nuclear Physics Laboratory Annual Report, University of Washington (1993) pp. 5
6-57.
2H. Bichsel, Radiat. Protection Dosimetry 13, 91 (1985).
1STAR Note 127.
2STAR Note 160.
3D. Cinabro et al., CLEO Collaboration, Phys. Rev. Lett. 72, 1410 (1994).
1Nuclear Physics Laboratory Annual Report, University of Washington (1994) p. 48
.
2T. Csörgö, private communication, (1993).
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7.4 Experience using SControl in CERN experiment NA49
J.G. Cramer, P.B. Cramer* and M.A. Howe