- Email: [email protected]
c
Physics Letters B 558 (2003) 29–33 www.elsevier.com/locate/npe
Erraticity of rapidity gaps in π + p and K+p collisions at 250 GeV/c EHS/NA22 Collaboration M.R. Atayan , Bai Yuting b , E.A. De Wolf c , A.M.F. Endler d , Fu Jinghua b , H. Gulkanyan a , R. Hakobyan a , W. Kittel e,∗ , Liu Lianshou b , Liao Hongbo b , Z.V. Metreveli f,1 , L.A. Tikhonova g , A.G. Tomaradze f,1 , F. Verbeure a , Wu Yuanfang b , S.A. Zotkin g,2 a
a Institute of Physics, AM-375036 Yerevan, Armenia b Institute of Particle Physics, Hua-Zhong Normal University, Wuhan 430070, China c Department of Physics, Universiteit Antwerpen, B-2610 Wilrijk, Belgium d Centro Brasileiro de Pesquisas Fisicas, BR-22290 Rio de Janeiro, Brazil e High Energy Physics Institute (HEFIN), University of Nijmegen/NIKHEF, NL-6525 ED Nijmegen, The Netherlands f Institute for High Energy Physics of Tbilisi State University, GE-380086 Tbilisi, Georgia g Scobeltsyn Institute of Nuclear Physics, Lomonosow Moscow State University, RU-119992 Moscow, Russia
Received 28 January 2003; received in revised form 6 February 2003; accepted 13 February 2003 Editor: L. Montanet
Abstract Two suggested moments of rapidity gaps are studied on π + p and K+ p collisions at 250 GeV/c. The experimental results are compared with those obtained from P YTHIA simulation. The efficiency of the method is discussed. 2003 Elsevier Science B.V. All rights reserved.
1. Introduction In the recent decade, the study of event-to-event fluctuations has become an important direction in the field of multiparticle production [1]. How to effectively measure these fluctuations is a current prob* Corresponding author.
E-mail address: [email protected] (W. Kittel). 1 Now at Northwestern University, Evanston, USA. 2 Now at DESY, Hamburg, Germany.
lem in the field. In the previous Letter [2], we have demonstrated that a method [3] based on event factorial moments is dominated by statistical noise at multiplicities as encountered in our experiment. In this Letter, we present an analysis in terms of the recently suggested moments of rapidity gaps [4]. This still contains important information on the momentum distribution of produced particles and is considered to be more effective than the former method, in particular when the average multiplicity is low.
0370-2693/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0370-2693(03)00214-4
30
EHS/NA22 Collaboration / Physics Letters B 558 (2003) 29–33
The rapidity gap is defined by the difference of rapidities of two neighboring particles in an event, xi = Xi+1 − Xi ,
i = 0, . . . , n,
(1)
where X0 = 0 and Xn+1 = 1 are the boundaries, Xi (i = 0, . . . , n) is the cumulative variable [5] b −1 yi Xi (yi ) =
ρ(y) dy a
ρ(y) dy
,
(2)
a
where a and b are the extreme points of the singleparticle distribution ρ(y) (ordered from small to large) of rapidity yi of the ith particle, which is free from the influence of energy conservation on the rapidity distribution and is uniformly distributed from 0 to 1, n is the total number of particles in the event. The set Se of n + 1 numbers: Se = {xi | i = 0, . . . , n} provides information on the rapidity distribution of all particles in the event. A quantitative characteristic of a single event is, therefore, given by the moments of its rapidity gaps: 1 q xi , n+1 n
Gq =
(3)
i=0
or: 1 Hq = (1 − xi )−q , n+1 n
n q + 1,
(4)
i=0
where the moment order q is an integer. Gq and Hq vary from event to event. Since the region of xi is between 0 and 1, the Gq are less than unity and the Hq amplify the Gq and are larger than unity. Hwa and Zhang [4] suggest the event sample averages: sq = −Gq ln Gq ,
(5)
σq = Hq ln Hq
(6)
as descriptions of event-to-event fluctuations of the sample. However, according to this description, the statistical fluctuations are included. In order to reduce the statistical fluctuations, the authors suggest to determine sq and σq for a purely statistical sample as st sqst = − Gst (7) q ln Gq , σqst = Hqst ln Hqst
(8)
and to define sq Sq = st , sq Σq =
σq , σqst
(9)
(10)
as measures of event-to-event fluctuations giving the so-called erraticity in terms of rapidity gaps. The measure Sq has recently been tested successfully in heavy ion collision [6]. In this Letter, these two moments are presented for π + p and K+ p collisions at 250 GeV/c. They are compared with the corresponding results from P YTHIA 5.720 [7].
2. The data sample [8] In the CERN experiment NA22, the European Hybrid Spectrometer (EHS) was equipped with the Rapid Cycling Bubble Chamber (RCBC) as an active target and exposed to a 250 GeV/c tagged, positive, meson enriched beam. In data taking, a minimum bias interaction trigger was used. Charged-particle tracks are reconstructed from hits in the wire- and drift-chambers of the two-leverarm magnetic spectrometer and from measurements in the bubble chamber. The momentum resolution varies from 1–2% for tracks reconstructed in RCBC, to 1–2.5% for tracks reconstructed in the first lever arm and is 1.5% for tracks reconstructed in the full spectrometer. Events are accepted for the analysis when the measured and reconstructed charged-particle multiplicity are the same, charge balance is satisfied, no electron is detected among the secondary tracks and the number of badly reconstructed (and therefore rejected) tracks is 0. The loss of events during measurement and reconstruction is corrected for by applying a multiplicitydependent event weight normalized to the topological cross sections. Elastic events are excluded. Furthermore, an event is called single-diffractive and excluded from the sample if the total charged-particle multiplicity is smaller than 8 and at least one of the positive tracks has a Feynman variable |xF | > 0.88. For laboratory-momenta pLAB < 0.7 GeV/c, the range in the bubble chamber and/or the change of track
EHS/NA22 Collaboration / Physics Letters B 558 (2003) 29–33
curvature is used for proton identification. In addition, a visual ionization scan is used for pLAB < 1.2 GeV/c. Positive particles with pLAB > 150 GeV/c are given the identity of the beam particle. Other particles with momenta pLAB > 1.2 GeV/c are not identified and are treated as pions. After all cuts, the inelastic, non-single-diffractive sample consists of 44 524 π + p and K+ p events. In spite of the electron rejection mentioned above, residual Dalitz decay and γ conversion near the vertex still contribute to the two-particle correlations. Their influence on our results has been investigated in detail in [9]. In this Letter, the analysis is performed in the rapidity range −2 y 2.
31
Fig. 1. ln Sq vs. ln q. Solid points are NA22 data and open triangles and circles are P YTHIA with and without BEC, respectively. The solid line is the fit to the experimental data, and the dashed line the fit to P YTHIA with BEC, according to Eq. (11).
3. Statistical sample The statistical sample is constructed in the following way: for an event of n charged particles, its cumulative rapidity variables are produced by n random numbers uniformly distributed in [0, 1]. Its multiplicity distribution is taken from the experimental data.
4. Results and discussions The two rapidity-gap moments, Eqs. (9) and (10), are presented in Figs. 1 and 2, respectively. The solid points are the results from the data and the open triangles and circles are those from P YTHIA MC with and without Bose–Einstein correlations (BEC), respectively. It is interesting to note that both the Sq and Σq values significantly deviate from unity for both data and P YTHIA. This implies that these measures of erraticity are indeed statistically significant [4]. The differences between the results of P YTHIA MC with and without Bose–Einstein correlations (BEC) are very small in ln Sq . They are relatively larger in Σq , but they do not account for those with the data and the tendency of variation with q is similar. The lines in Fig. 1 correspond to a fit from q = 3 to 9, indicating power-law behavior, ln Sq = A + α ln q,
(11)
in both the data and P YTHIA with BEC. Here, α = 0.319 ± 0.015 for the data and α = 0.188 ± 0.005
Fig. 2. ln Σq vs. q. Solid points are NA22 data and open triangles and circles are P YTHIA with and without BEC, respectively. The solid line is the fit to the experimental data, the dashed line the fit to P YTHIA with BEC, according to Eq. (12).
for P YTHIA, while the value obtained from E COMB (Eikonal Color Mutation Branching) model in [10] was α = 0.156. We see that the result of Sq from P YTHIA is closer to those from the NA22 data than E COMB, although P YTHIA fails to reproduce the power-law behavior of the factorial moments (intermittency) observed in the data [11], while E COMB is a new model specially constructed to account for the intermittency effect. In Fig. 2, the fit corresponds to ln Σq = A + βq,
(12)
32
EHS/NA22 Collaboration / Physics Letters B 558 (2003) 29–33
Fig. 3. ln sq and ln sqst vs. q. Solid points and triangles correspond to ln sq of the NA22 data and P YTHIA, respectively. Open circles and triangles correspond to ln sqst of purely statistical samples for NA22 and P YTHIA, respectively.
Fig. 4. ln σq and ln σqst vs. q. Solid points and triangles correspond to ln sq of the NA22 data and P YTHIA, respectively. Open circles and triangles correspond to ln sqst of purely statistical samples for NA22 and P YTHIA, respectively.
where β = 0.335 ± 0.038 in the data and β = 0.064 ± 0.022 in P YTHIA with BEC. Again, P YTHIA underestimates the results both in absolute value and slope of the q dependence. The corresponding slope of E COMB is β = 0.28 [10], much closer to the data than that of P YTHIA. To fully understand the results, it should be noted that ln Σq depends strongly on the number of events [12] and that the number of the present experiment is marginal in this respect. The very different behavior of ln Sq and ln Σq vs. q shown by P YTHIA demonstrates that Hq is not simply an amplification of Gq . For comparison, ln sq and ln σq for the data, Eqs. (5) and (6), and for the purely statistical sample, Eqs. (7) and (8), are presented in Figs. 3 and 4, respectively. Solid points and triangles correspond to NA22 data and P YTHIA, respectively, open circles and triangles to the purely statistical sample using the multiplicity distributions of NA22 and P YTHIA, respectively. It can be seen from Fig. 3 that ln sqst of NA22 and P YTHIA are very close to each other. The small difference is due to the difference in average multiplicity of these two samples, n NA22 = 6.97 and n P YTHIA = 6.47. This difference becomes more obvious when the moment order is higher. An important observation is that the measure for purely statistical fluctuations, ln sqst , is indeed smaller than that of the corresponding data samples, ln sq . The results from P YTHIA underestimate those from the data. The difference increases with increasing moment order.
The corresponding measures ln σq and ln σqst given in Fig. 4 support the conclusions drawn from Fig. 3.
5. Summary Two kinds of rapidity gap moments are studied on π + p and K+ p collisions at 250 GeV/c. Significant non-statistical event-to-event fluctuations (erraticity) are indeed observed. Current models, such as P YTHIA and E COMB, no matter whether they can reproduce the power-law behavior of factorial moments (intermittency) or not, underestimate these erraticity measures for the NA22 data.
Acknowledgements This work is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)”, which is financially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)”. We further thank NWO for support of this project within the program for subsistence to the former Soviet Union (07-13-038). The Yerevan group activity is financially supported, in the framework of the theme No. 0248, by the Government of the Republic of Armenia. This work is also supported in part by the National Natural Science Foundation of China and the Ministry of Education of China and by the Royal
EHS/NA22 Collaboration / Physics Letters B 558 (2003) 29–33
Dutch Academy of Sciences under the Project numbers 01CDP017 and 02CDP011, and by the US Department of Energy. References [1] A. Białas, in: N.G. Antoniou, et al. (Eds.), Proc. XXVII Int. Symp. on Multiparticle Dynamics, World Scientific, Singapore, 1999, p. 145; H. Heiselberg, Phys. Rep. 351 (2001) 161. [2] M.R. Atayan, et al., EHS/NA22 Collaboration, Phys. Lett. B 558 (2003) 22. [3] Z. Cao, R.C. Hwa, Phys. Rev. Lett. 75 (1995) 1268; Z. Cao, R.C. Hwa, Phys. Rev. D 53 (1996) 6608; Z. Cao, R.C. Hwa, Phys. Rev. D 54 (1996) 6674.
33
[4] R.C. Hwa, Q.-H. Zhang, Phys. Rev. D 62 (2000) 014003. [5] W. Ochs, Z. Phys. C 50 (1991) 339; A. Białas, M. Ga´zdzicki, Phys. Lett. B 252 (1990) 483. [6] D. Ghosh, et al., Phys. Lett. B 540 (2002) 52. [7] T. Sjöstrand, Comput. Phys. Commun. 82 (1994) 74. [8] N.M. Agababyan, et al., EHS/NA22 Collaboration, Phys. Lett. B 382 (1996) 305. [9] F. Botterweck, Ph.D. Thesis, University of Nijmegen, 1992. [10] R.C. Hwa, Acta Phys. Pol. B 27 (1996) 1789; Z. Cao, R.C. Hwa, Phys. Rev. D 61 (2000) 074011. [11] I.V. Ajinenko, et al., EHS/NA22 Collaboration, Phys. Lett. B 222 (1989) 306; C. Albajar, et al., UA1 Collaboration, Nucl. Phys. B 345 (1990) 1. [12] H. Liao, Y. Wu, hep-ph/0206184.
Erraticity of rapidity gaps in π + p and K+p collisions at 250 GeV/c EHS/NA22 Collaboration M.R. Atayan , Bai Yuting b , E.A. De Wolf c , A.M.F. Endler d , Fu Jinghua b , H. Gulkanyan a , R. Hakobyan a , W. Kittel e,∗ , Liu Lianshou b , Liao Hongbo b , Z.V. Metreveli f,1 , L.A. Tikhonova g , A.G. Tomaradze f,1 , F. Verbeure a , Wu Yuanfang b , S.A. Zotkin g,2 a
a Institute of Physics, AM-375036 Yerevan, Armenia b Institute of Particle Physics, Hua-Zhong Normal University, Wuhan 430070, China c Department of Physics, Universiteit Antwerpen, B-2610 Wilrijk, Belgium d Centro Brasileiro de Pesquisas Fisicas, BR-22290 Rio de Janeiro, Brazil e High Energy Physics Institute (HEFIN), University of Nijmegen/NIKHEF, NL-6525 ED Nijmegen, The Netherlands f Institute for High Energy Physics of Tbilisi State University, GE-380086 Tbilisi, Georgia g Scobeltsyn Institute of Nuclear Physics, Lomonosow Moscow State University, RU-119992 Moscow, Russia
Received 28 January 2003; received in revised form 6 February 2003; accepted 13 February 2003 Editor: L. Montanet
Abstract Two suggested moments of rapidity gaps are studied on π + p and K+ p collisions at 250 GeV/c. The experimental results are compared with those obtained from P YTHIA simulation. The efficiency of the method is discussed. 2003 Elsevier Science B.V. All rights reserved.
1. Introduction In the recent decade, the study of event-to-event fluctuations has become an important direction in the field of multiparticle production [1]. How to effectively measure these fluctuations is a current prob* Corresponding author.
E-mail address: [email protected] (W. Kittel). 1 Now at Northwestern University, Evanston, USA. 2 Now at DESY, Hamburg, Germany.
lem in the field. In the previous Letter [2], we have demonstrated that a method [3] based on event factorial moments is dominated by statistical noise at multiplicities as encountered in our experiment. In this Letter, we present an analysis in terms of the recently suggested moments of rapidity gaps [4]. This still contains important information on the momentum distribution of produced particles and is considered to be more effective than the former method, in particular when the average multiplicity is low.
0370-2693/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0370-2693(03)00214-4
30
EHS/NA22 Collaboration / Physics Letters B 558 (2003) 29–33
The rapidity gap is defined by the difference of rapidities of two neighboring particles in an event, xi = Xi+1 − Xi ,
i = 0, . . . , n,
(1)
where X0 = 0 and Xn+1 = 1 are the boundaries, Xi (i = 0, . . . , n) is the cumulative variable [5] b −1 yi Xi (yi ) =
ρ(y) dy a
ρ(y) dy
,
(2)
a
where a and b are the extreme points of the singleparticle distribution ρ(y) (ordered from small to large) of rapidity yi of the ith particle, which is free from the influence of energy conservation on the rapidity distribution and is uniformly distributed from 0 to 1, n is the total number of particles in the event. The set Se of n + 1 numbers: Se = {xi | i = 0, . . . , n} provides information on the rapidity distribution of all particles in the event. A quantitative characteristic of a single event is, therefore, given by the moments of its rapidity gaps: 1 q xi , n+1 n
Gq =
(3)
i=0
or: 1 Hq = (1 − xi )−q , n+1 n
n q + 1,
(4)
i=0
where the moment order q is an integer. Gq and Hq vary from event to event. Since the region of xi is between 0 and 1, the Gq are less than unity and the Hq amplify the Gq and are larger than unity. Hwa and Zhang [4] suggest the event sample averages: sq = −Gq ln Gq ,
(5)
σq = Hq ln Hq
(6)
as descriptions of event-to-event fluctuations of the sample. However, according to this description, the statistical fluctuations are included. In order to reduce the statistical fluctuations, the authors suggest to determine sq and σq for a purely statistical sample as st sqst = − Gst (7) q ln Gq , σqst = Hqst ln Hqst
(8)
and to define sq Sq = st , sq Σq =
σq , σqst
(9)
(10)
as measures of event-to-event fluctuations giving the so-called erraticity in terms of rapidity gaps. The measure Sq has recently been tested successfully in heavy ion collision [6]. In this Letter, these two moments are presented for π + p and K+ p collisions at 250 GeV/c. They are compared with the corresponding results from P YTHIA 5.720 [7].
2. The data sample [8] In the CERN experiment NA22, the European Hybrid Spectrometer (EHS) was equipped with the Rapid Cycling Bubble Chamber (RCBC) as an active target and exposed to a 250 GeV/c tagged, positive, meson enriched beam. In data taking, a minimum bias interaction trigger was used. Charged-particle tracks are reconstructed from hits in the wire- and drift-chambers of the two-leverarm magnetic spectrometer and from measurements in the bubble chamber. The momentum resolution varies from 1–2% for tracks reconstructed in RCBC, to 1–2.5% for tracks reconstructed in the first lever arm and is 1.5% for tracks reconstructed in the full spectrometer. Events are accepted for the analysis when the measured and reconstructed charged-particle multiplicity are the same, charge balance is satisfied, no electron is detected among the secondary tracks and the number of badly reconstructed (and therefore rejected) tracks is 0. The loss of events during measurement and reconstruction is corrected for by applying a multiplicitydependent event weight normalized to the topological cross sections. Elastic events are excluded. Furthermore, an event is called single-diffractive and excluded from the sample if the total charged-particle multiplicity is smaller than 8 and at least one of the positive tracks has a Feynman variable |xF | > 0.88. For laboratory-momenta pLAB < 0.7 GeV/c, the range in the bubble chamber and/or the change of track
EHS/NA22 Collaboration / Physics Letters B 558 (2003) 29–33
curvature is used for proton identification. In addition, a visual ionization scan is used for pLAB < 1.2 GeV/c. Positive particles with pLAB > 150 GeV/c are given the identity of the beam particle. Other particles with momenta pLAB > 1.2 GeV/c are not identified and are treated as pions. After all cuts, the inelastic, non-single-diffractive sample consists of 44 524 π + p and K+ p events. In spite of the electron rejection mentioned above, residual Dalitz decay and γ conversion near the vertex still contribute to the two-particle correlations. Their influence on our results has been investigated in detail in [9]. In this Letter, the analysis is performed in the rapidity range −2 y 2.
31
Fig. 1. ln Sq vs. ln q. Solid points are NA22 data and open triangles and circles are P YTHIA with and without BEC, respectively. The solid line is the fit to the experimental data, and the dashed line the fit to P YTHIA with BEC, according to Eq. (11).
3. Statistical sample The statistical sample is constructed in the following way: for an event of n charged particles, its cumulative rapidity variables are produced by n random numbers uniformly distributed in [0, 1]. Its multiplicity distribution is taken from the experimental data.
4. Results and discussions The two rapidity-gap moments, Eqs. (9) and (10), are presented in Figs. 1 and 2, respectively. The solid points are the results from the data and the open triangles and circles are those from P YTHIA MC with and without Bose–Einstein correlations (BEC), respectively. It is interesting to note that both the Sq and Σq values significantly deviate from unity for both data and P YTHIA. This implies that these measures of erraticity are indeed statistically significant [4]. The differences between the results of P YTHIA MC with and without Bose–Einstein correlations (BEC) are very small in ln Sq . They are relatively larger in Σq , but they do not account for those with the data and the tendency of variation with q is similar. The lines in Fig. 1 correspond to a fit from q = 3 to 9, indicating power-law behavior, ln Sq = A + α ln q,
(11)
in both the data and P YTHIA with BEC. Here, α = 0.319 ± 0.015 for the data and α = 0.188 ± 0.005
Fig. 2. ln Σq vs. q. Solid points are NA22 data and open triangles and circles are P YTHIA with and without BEC, respectively. The solid line is the fit to the experimental data, the dashed line the fit to P YTHIA with BEC, according to Eq. (12).
for P YTHIA, while the value obtained from E COMB (Eikonal Color Mutation Branching) model in [10] was α = 0.156. We see that the result of Sq from P YTHIA is closer to those from the NA22 data than E COMB, although P YTHIA fails to reproduce the power-law behavior of the factorial moments (intermittency) observed in the data [11], while E COMB is a new model specially constructed to account for the intermittency effect. In Fig. 2, the fit corresponds to ln Σq = A + βq,
(12)
32
EHS/NA22 Collaboration / Physics Letters B 558 (2003) 29–33
Fig. 3. ln sq and ln sqst vs. q. Solid points and triangles correspond to ln sq of the NA22 data and P YTHIA, respectively. Open circles and triangles correspond to ln sqst of purely statistical samples for NA22 and P YTHIA, respectively.
Fig. 4. ln σq and ln σqst vs. q. Solid points and triangles correspond to ln sq of the NA22 data and P YTHIA, respectively. Open circles and triangles correspond to ln sqst of purely statistical samples for NA22 and P YTHIA, respectively.
where β = 0.335 ± 0.038 in the data and β = 0.064 ± 0.022 in P YTHIA with BEC. Again, P YTHIA underestimates the results both in absolute value and slope of the q dependence. The corresponding slope of E COMB is β = 0.28 [10], much closer to the data than that of P YTHIA. To fully understand the results, it should be noted that ln Σq depends strongly on the number of events [12] and that the number of the present experiment is marginal in this respect. The very different behavior of ln Sq and ln Σq vs. q shown by P YTHIA demonstrates that Hq is not simply an amplification of Gq . For comparison, ln sq and ln σq for the data, Eqs. (5) and (6), and for the purely statistical sample, Eqs. (7) and (8), are presented in Figs. 3 and 4, respectively. Solid points and triangles correspond to NA22 data and P YTHIA, respectively, open circles and triangles to the purely statistical sample using the multiplicity distributions of NA22 and P YTHIA, respectively. It can be seen from Fig. 3 that ln sqst of NA22 and P YTHIA are very close to each other. The small difference is due to the difference in average multiplicity of these two samples, n NA22 = 6.97 and n P YTHIA = 6.47. This difference becomes more obvious when the moment order is higher. An important observation is that the measure for purely statistical fluctuations, ln sqst , is indeed smaller than that of the corresponding data samples, ln sq . The results from P YTHIA underestimate those from the data. The difference increases with increasing moment order.
The corresponding measures ln σq and ln σqst given in Fig. 4 support the conclusions drawn from Fig. 3.
5. Summary Two kinds of rapidity gap moments are studied on π + p and K+ p collisions at 250 GeV/c. Significant non-statistical event-to-event fluctuations (erraticity) are indeed observed. Current models, such as P YTHIA and E COMB, no matter whether they can reproduce the power-law behavior of factorial moments (intermittency) or not, underestimate these erraticity measures for the NA22 data.
Acknowledgements This work is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)”, which is financially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)”. We further thank NWO for support of this project within the program for subsistence to the former Soviet Union (07-13-038). The Yerevan group activity is financially supported, in the framework of the theme No. 0248, by the Government of the Republic of Armenia. This work is also supported in part by the National Natural Science Foundation of China and the Ministry of Education of China and by the Royal
EHS/NA22 Collaboration / Physics Letters B 558 (2003) 29–33
Dutch Academy of Sciences under the Project numbers 01CDP017 and 02CDP011, and by the US Department of Energy. References [1] A. Białas, in: N.G. Antoniou, et al. (Eds.), Proc. XXVII Int. Symp. on Multiparticle Dynamics, World Scientific, Singapore, 1999, p. 145; H. Heiselberg, Phys. Rep. 351 (2001) 161. [2] M.R. Atayan, et al., EHS/NA22 Collaboration, Phys. Lett. B 558 (2003) 22. [3] Z. Cao, R.C. Hwa, Phys. Rev. Lett. 75 (1995) 1268; Z. Cao, R.C. Hwa, Phys. Rev. D 53 (1996) 6608; Z. Cao, R.C. Hwa, Phys. Rev. D 54 (1996) 6674.
33
[4] R.C. Hwa, Q.-H. Zhang, Phys. Rev. D 62 (2000) 014003. [5] W. Ochs, Z. Phys. C 50 (1991) 339; A. Białas, M. Ga´zdzicki, Phys. Lett. B 252 (1990) 483. [6] D. Ghosh, et al., Phys. Lett. B 540 (2002) 52. [7] T. Sjöstrand, Comput. Phys. Commun. 82 (1994) 74. [8] N.M. Agababyan, et al., EHS/NA22 Collaboration, Phys. Lett. B 382 (1996) 305. [9] F. Botterweck, Ph.D. Thesis, University of Nijmegen, 1992. [10] R.C. Hwa, Acta Phys. Pol. B 27 (1996) 1789; Z. Cao, R.C. Hwa, Phys. Rev. D 61 (2000) 074011. [11] I.V. Ajinenko, et al., EHS/NA22 Collaboration, Phys. Lett. B 222 (1989) 306; C. Albajar, et al., UA1 Collaboration, Nucl. Phys. B 345 (1990) 1. [12] H. Liao, Y. Wu, hep-ph/0206184.