Accurate impact parameter determination in a heavy-ion collision is crucial for almost all further analysis. We investigate the capabilities of an artificial neural network in that respect. First results show that the neural network is capable of improving the accuracy of the impact parameter determination based on observables such as the flow angle, the average directed inplane transverse momentum and the difference between transverse and longitudinal momenta. However, further investigations are necessary to discover the full potential of the neural network approach

Bass, Steffen A.

Bischoff, Arnd

Greiner, Walter

Hartnack, Christoph

Maruhn, Joachim

Reinhardt, Joachim

Stöcker, Horst

English

Hochschulschriftenserver - Universität Frankfurt am Main

I. Php. G: Nucl. Pm.  Phys. 20 (1994) L21-U6.  %nted  in the UK LETTER TO THE EDITOR Neural networks for impact parameter determination* S  A Basstf, A Bischofft, C Hartnacktg, J A Maruhni, J Reinhardtt, H Stiickert and W Greinerf 7  lnstitut fir  Theoretische Physik der Universioit, Poslfach 1  I1932  60054 Frankfurf am Main, Federal Republic of Germany t GSI  Dannstadf  Poslfach I10552.64220  DarmstadS Federal Republic of Germany $ Laboratoire de Physique Nucl&=,  Nantes, France Received 8 November 1993 Abstract.  Accme  impacl parameter determination in a heavy-ion collision is crucial for almost all further analysis. We  investigate the capabilities of an attificial neural network in lhst respeck First resulls show  lhal the neural network is capable of improving the accuracy of the impact parameter determination based  on observables such as the  Row angle, the average directed in- plane uansvene momentum and  the difference between lnnwerse and longitudinal momenla. However. further investigations are  necessary U)  discover the full potential of  lhe mural network approach. The  physics  of heavy-ion  collisions  is motivated  by  the  idea to  leam  something about the  properties  of  hot  and  dense  nuclear  matter [MI.  In  order to  investigate  highly compressed nuclear matter in heavy-ion collisions  it is important to select only  the most central collisions.  On the other hand, recently discovered new phenomena such as pionic bounce-off and squeeze-out are only observed in  semiperipheral collisions.  Therefore the proper determination of the impact parameter in a heavy-ion collision is crucial to almost all fuither analysis.  There have been various approaches towards the determination of  the impact parameter, most of which are either based on the  mean  particle multiplicity or on a transverse momentum analysis (directivity-cut). However, all of these methods have one thing in common:  they tend to break down for impact parameters smaller than 2 fenni. In general the accuracy is about dz1 to f1.5 fermi. In this contribution, an artificial neural network has been  used to determine the impact parameter.  For a proper  analysis of the  network's  performance, the  analysed heavy-ion collisions have to be supplied by a theoretical event-generator rather than by  experiment. Otherwise it would be impossible to compare the network output with a target value for the impact parameter of the heavy-ion collision. For our investigations, we applied an extension of  the quantum  molecular dynamics  model  (QMD)  [9-l1],  which explicitly  incorporates isospin and pion production via the delta resonance (IQMD)  [  12-14].  In the QMD model the nucleons are represented by  Gaussian-shaped density distributions.  They are initialized in a sphere of  a radius R = I .14A'I3 fm,  according to the liquid-drop model. Each nucleon is supposed to occupy a volume of h3,  so that the phase space is uniformly filled. The  initial momenta are randomly chosen between 0 and the local Thomas-Fermi  momentum.  The Ap and  AT  nucleons interact via two- and three-body Skyrme forces, a Yukawa potentid. '  Supponed by  GSI. BMFT and DFG 0954-38YY/94fl10021+06$0750  Q 1994 IOP Publishing Lld  L21 L22  Lefter to the Edifor momentum-dependent interactions, a symmetry potential (to achieve a correct distribution of  protons and neutrons in  the nucleus) and explicit Coulomb foxes between  the Zp  and ZT protons. They are propagated according to Hamilton’s equations of motion.  Hard  N-N collisions are included by employing the collision term of the well known vuu/Buu equation [5, 15-18].  The collisions are done stochastically, in a similar way as in the cascade models [ 19,201. In addition, the Pauli blocking (for the final state) is taken into account by regarding the phase-space densities in the final states of  a two-body collision. As  input  for  the  neural  net  we  use  the  flow  angle  Sflow, the  average directed  in- plane transverse momentum Pr,dit, and an observable QZZ  which compares transverse with longitudinal momentum. The flow angle can be  extracted from the flow-tensor “=I using a sphericity analysis [ZI]. The average directed in-plane transverse momentum px.ar  is defined as and  Qzz  can be  calculated as A+Av n=I Qzz=~P:-P~=~P:-P:-P;  P,=  pi(n)  i=x,y,z. The impact parameter dependence of these obsewables has been  well established [ 141. We  now sketch the neural network algorithm used, a standard feedforward two-layer perceptron trained  by  error-backpropagation [22,23]. The network consists of  a  single ‘hidden’  layer  of  nonlinear  units  receiving  inputs  from  the  applied  data  vector  (e.g. (Snow.  p=.d,,,  Qzz)) and transfemng their signals to the output unit (we use a single linear output unit, whose continuous-valued output represents the impact parameter). In our feedforward network, every unit (of the hidden or  output layer) is  connected to each  unit of  the preceding layer, performing a weighted sum  over all  input signals, and finally calculating its own  signal by  applying a ‘squashing’ function to the result: for hidden units, and for  output  units.  (As  squashing functions we  use  O(Z) = tanh(Z),  and  e(Z) = Z. respectively.)  The connection weight, between  units  j  to  unit  i, is  given by  wi,.  yk  is a component of the data vector. For each unit, we include a connection to a constant signal, So = I, which  provides an activity threshold. Letter to the Editor  L23 First,  the network's  weights  are initialized with  small random values.  During training, for each learning pattem an output is  produced and rated by  the error function where SYget is the desired output.  Successively for each pattem, the weights  are updated according to a gradient descent in the weight space with  respect to the error function, with g as a learning parameter.  This leads to  the learning rules using the definition and where For a complete learning session, typically several hundred cycles through the entire training data are necessary.  Training is  stopped when  the performance  on a set of test  data does not improve any further.  In contrast to the extensive training phase, very little calculation time is needed  for the application  of  a trained network.  A  trained network  may even be transformed  into electronic hardware, which would be faster than our computer simulation by orders of magnitude. Two sets, each containing 2600 Au(1 A GeV)Au events, generated by the IQMD model [ 141,  form the simulation data. The first data set contains a minimum bias calculation, thus giving a sample from the full impact parameter range, whereas the second one is limited to impact parameters smaller than three fermi.  Both data sets have been subjected to an angular cut according to the acceptance of the FOP1 spectrometer in its phase I1 setup [24].  For each impact parameter range, the network is provided with  10% of the data as learning samples and then it has to estimate the impact parameter for the remaining 90% of events.  To avoid overfitting, we controlled the network's performance after each learning cycle by applying the  full test  data-set.  The network  used  for the full  input  vector  with  three components consisted  of five hidden  units  and one output  neuron.  In order to  determine the amount of hidden units needed for the best network performance,  the number of hidden units was increased  from one to  eight  in single increments  and the respective  network  performance was tested.  No  significant performance  increase was  achieved for more than  five hidden units. L24  Letter to fhe Editor Au(  1AGeV)  Au Ofm(bI12fm  . ~.,  . .  ..  .  0 fm 5 b 53  fm .. I impact parameter  impact parameter 0.0  0,s  L.0  1.5  2.0  1.E  8.0  1  4  D  0  LO Figure 1.  Dependence between impact parameter b and  VflRoxr.  px,&  and  Qrz for central and minimum bias Aut1 A GeV)Au evenls as  calculated by the tQMD model. Each even1 is rcpresenled as  a dol. Both the leaming sample and the test sample are  plolled. Figure 1 shows scatter plots (each dot representing one event) describing the functional dependence of the impact parameter on either one of the three inputs, Onow, Px.dir and Qzz, as calculated by  the  lQMD model.  Using the minimum  bias data set, the neural  network output  essentially  produced  a nonlinear  fit to  the  impact  parameter  dependence  of  these inputs which is also intuitively  apparent from the figure.  In order to  minimize errors due to  statistical  fluctuations  in the events,  it is,  however,  desirable  to  simultaneously  tit  the impact parameter as a function of O&,  Px,d,,  and  QZZ. This gives a trajectory  in a three- dimensional phase space with the impact parameter as an evolution parameter.  For standard fitting techniques this poses a more difficult problem.  which  we  instead  left again to the neural  network.  Table  I  shows the average absolute error (in  fermi) between  the impact parameter  estimated by the neural network and the actual impact parameter (using the test data set) for the functions h(On,),  h(p&dir),  ~(Qzz)  and b(OB,,,  pr,dir, QZZ)  as leamed by the neural net. Letter to the Editor  L25 .. 0' 0  2  4  6  8  10  12 actual impact parameter Figure 2.  True  impact parameter  versus impact parameter predicted by the  neural network  for the minimum bias data-set using the full input vector. The best performance is achieved in the 3 to 9 fm  impact parameter range. Table 1.  Average absolute emr  (in fermi) between the impact paramew  h leamed by the neural network and the  actual impact  parameter as a function of either  ~9%~.  h.da  or  QZZ  and as a function of r9fiOw. pl,*  and Qzz simultaneously. Imoact Dammeter  OhnSb<3fm  Ofm<h<12fm For the minimum bias data-set the neural network is able to extract a better fit using all three inputs than using only one of them.  For central events this is not the case:  a simple mapping of ~(Qzz)  shows almost identical results.  This becomes understandable looking at figure  1, where b(Qzz) clearly shows the least fluctuations. Figure 2 shows a graphical representation of the network performance on the minimum bias data-set using the full input vector.  The hue impact parameter is plotted versus the impact parameter estimated by the neural network.  The best performance is achieved in the range between  3 fm and 9 fm.  For very central and very peripheral collisions the width of the output deviations increases. With  the selected preprocessed  event input,  classical  fitting  techniques suffice  for a good estimate of  the impact parameter.  Compared with  previously  used techniques [251, the suggested observables can improve the accuracy of the impact parameter determination. However,  we surely have not  used  the neural network approach up to its full potential:  a larger  network  will  most  likely  be  capable of  handling  raw  event-data and  recognizing patterns  which  are  indistinguishable  for  classical  types of  analysis.  In  addition,  more sophisticated network  models (e.g.  with  higher  order or optimized connectivity 126,271) could  be  used  instead  of  our  exploratory  straightforward  model  to  gain  calculational efficiency.  In  the  form  of a  VLSI  chip it might  even  serve as a  hardware event-trigger for the impact parameter. L26  Letter to the Editor References Scheid W.  Ligensa R and Greiner W 1968 Phys. RN. Len. 21 1479 Scheid W.  MUller  H and Greiner W  1974 Phys. Re,.  Len.  32 741 Csemai L P and Kapusta I I  1966 Phys. Rep, 131 7.25 Stock R  1986 Phys. Rep. 135 261 St6cker H and Greiner W 1966 Phys. Rep. 137 277 Ciare R B and Stroman D 1986 Phys. Rep. 141 179 Schurmann B.  Zwermann W and  Malfliet R  1986 Phys. Rep. 147 3 Cassing W,  Melag V. Mosel  U and Niita K  1990 Phys, Rep. 1%  365 Aichelin J and Stacker H 1986 Phys. Len. 1768 14 Aichelin J.  Roxnhauer A, Peilen G. Sucker H and Greiner W  1987 Phys. Rrr. Leff.  58 1926 Aichelin J  1991 Phys. 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Neural networks for impact parameter determination

Neural networks for impact parameter determination

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