Analyses of dn ch /dη and dn ch /dy distributions of BRAHMS Collaboration by means of the Ornstein-Uhlenbeck process

Transcription

1 Analyses of dn /d and dn /dy distributions of BRAHMS Collaboration by means of the Ornstein-Uhlenbeck process M. Ide 1, M. Biyajima 1, and T. Mizogui 3 1 Department of Physics, Faculty of Science, Shinshu University, Matsumoto , Japan The Niels Bohr Institute, DK-1, Copenhagen, Denmark 3 Toba National College of Maritime Tenology, Toba , Japan arxiv:nucl-th/33v1 3 Feb 3 December 3, 17 Abstract Interesting data on dn /d in Au-Au collisions ( = lntan(θ/)) with the centrality cuts have been reported by BRAHMS Collaboration. Using the total multiplicity N = (dn /d)d, we find that there are scaling phenomena among (N ) 1 dn /d = /d with different centrality cuts at s NN = 13 GeV and GeV, respectively. To explain these scaling behaviors of /d, we consider the stoastic approa named the Ornstein- Uhlenbeck process with two sources. The following Fokker-Planck equation is adopted for the present analyses, P(x,t) t = γ [ x x+ 1 σ γ ] P(x, t) x where x means the rapidity(y) or pseudo-rapidity(). t, γ and σ are the evolution parameter, the frictional coefficient and the variance, respectively. Introducing a variable of z r = / rms ( rms = ) we explain the /dz r distributions in the present approa. Moreover, to explain the rapidity (y) distributions from distributions at GeV, we have derived the formula as dy = J 1 d, where J 1 = M(1+sinh y)/ 1+M sinh y with M = 1+(m/p t). Their data of pion and all hadrons are fairly well explained by the O-U process. To compare our approa with another one, a phenomenological formula by Eskola et al. is also used in calculations of /d. 1 Introduction RecentlyinterestingdataondN /d ( = lntan(θ/))and(.5 N part ) 1 dn /d = inau+au collision at s NN = 13 GeV and GeV have been reported by BRAHMS Collaboration [1, ]. ( N part and N mean the numbers of participants (nuclei) and arged particles produced in collisions, respectively.) Very recently the BRAHMS Collaboration has reported preliminary data on rapidity (y) distribution at GeV in Ref. [3]. We are interested in theoretical analyses of these data. Ontheotherhand,inRefs.[4,5]wehaveinvestigatedthepropertyof scalingof(n ) 1 dn /d = /d by PHOBOS Collaboration and found that the scaling holds. As a possible theoretical approa, we have adopted the stoastic theory named the Ornstein-Uhlenbeck (O-U) process with two sources at ±y max = ln( s NN /m N ). In this paper, we would like to analyses data [1,, 3] by the stoastic approa in terms of the pseudo-rapidity and/or rapidity variables. The approa named the O-U process is described by the following Fokker-Planck equation, P(y,t) t = γ [ y y + 1 σ γ y ] P(y, t), (1) 1

2 (a) d t = small (b) d t = finite T = max B = max d t (c) Figure 1: (a) Initial distribution of Eq. (). (b) Final distribution at t = finite. (c) Evolution of Eq. (). where t, γ and σ are the evolution parameter, the frictional coefficient and the variance, respectively 1. Assuming two sources at ±y max = ln( s NN /m N ) at t = and P(y, ) =.5[δ(y + y max ) + δ(y y max )], we obtain the following distribution function for /d (assuming y ) using the probability density P(y, t)[6, 7, 8, 9] { 1 P(y, y max, t) = exp [ (y +y maxe γt ) ] 8πV (t) V (t) +exp [ (y y maxe γt ) ]} V, () (t) where V (t) = (σ /γ)p with p = 1 e γt. The physical picture of Eq. () with the assumption of y are shown in Fig. 1. In our approa, it is assumed that N / particles are created at ±y max at t =. Then these N = (N / + N /) particles are evolved according to Eq. (). It is worthwhile to mention that a similar approa for the proton spectra has been given in Ref. [11]. The contents of the present paper are organized as follows. In Sec. II scaling of BRAHMS Collaboration is investigated. In Sec. III Analyses of distribution by means of Eq. () are performed. The physical meaning of evolution parameter γt with the frictional coefficient is also considered. In Sec. IV z r = / rms ( rms = ) scaling is considered. In Sec. V Analysis of y distribution derived /d distribution is presented. In the final section concluding remarks are given. 1 The equivalent Langevin stoastic equation with the white noise f w(t) is given as dy = γy +fw(t). dt

3 .15 d.1.5 BRAHMS 13 Gev 4-5% -3% 5-1% -5% (a) /d BRAHMS GeV.6-5% 5-1%.4 1-% -3%. 3-4% 4-5% Figure : (a) A set of /d s with different centrality cuts at s NN = 13 GeV. Ea symbols have error-bars of about 8 1% of the magnitude. (b) /d with different centrality cuts at snn = GeV. About the error bars, the situation is the same as (a). (b) Analysis of scaling of /d by BRAHMS Collaboration First of all, we consider the problem on scaling in Fig., plotting the data of /d at 13 GeV and GeV. The scaling seems to be held. These distributions show d c (constant). (3) = Moreover, we examine the intercept at =. Authors of Ref. [11], WA98 Collaboration, noticed that the intercepts divided by(.5 N part ) should be described by the power-like law, as (.5 N part ) 1dN d = A N part α, (4) = provided that the participants (nuclei) have lost memory and every participant contribute a similar amount of energy to particle production in collisions. Actually it can be said that the power-like law holds, as is seen in Fig 3. See Tables 1 and. This physical picture with Eq. (4) indirectly supports the availability of the stoastic approa. Combining Eqs. (3) and (4), we have the following relations c Ex = 1 dn N d, (5) = c Sp =.5 N part N A N part α, (6) where the suffix Sp means the semi-phenomenological formula. Comparisons between Eqs. (5) and (6) with A and α in Fig. 3 are shown in Tables 1 and. As is seen in Tables 1 and, the intercept at = is fairly well explained by the semiphenomenological expression, Eq. (6). This implies that the stoastic approa may be available, because the participants lost their memory in collision. 3 Analyses of data by Eq. () Using the O-U process with two sources, Eq. (), we have analyzed the data. The results at snn = 13 GeV and GeV are shown in Figs. 4 and 5, and Tables 3 and 4. In our analyses we use Eq. () the pseudo-rapidity () instead of the rapidity (y). As is seen in Tables 3 and 4, 3

4 (a) 3.5 (b) (dn /d) =/ ( N part /) A=1.64 α=.19 c.c.=.97 (dn /d) =/ ( N part /) A=.9 α=.8 c.c.= N part N part Figure 3: (a) Estimation of parameters A and α at s NN = 13 GeV. The method of linearregression is used. A= 1.64, α =.19, and the correlation coefficient (c.c.) is.97. A power-like law is seen. (b) s NN = GeV. A=.9, α =.8, and (c.c.)=.74. Table 1: Empirical examination of Eqs. (5) and (6) at s NN = 13 GeV. δc e = and δc s = centrality (%) N part N 75±6 116±9 17±13 47±19 318±5 386±43 c Ex.131±δc e.134±δc e.138±δc e.141±δc e.143±δc e.137±δc e c Sp.131±δc s.135±δc s.137±δc s.141±δc s.144±δc s.141±δc s Table : The same as Table 1 but s NN = GeV, δc e = and δc s = centrality (%) N part 73±8 114±9 168±9 39±1 36±11 357±8 N 89±7 138±11 ±16 9±3 381±3 463±37 c Ex.14±δc e.16±δc e.17±δc e.131±δc e.135±δc e.19±δc e c Sp.11±δc s.16±δc s.131±δc s.133±δc s.133±δc s.19±δc s 4

5 Table 3: Estimated parameters at s NN = 13 GeV in our analyses by Eq. () with two sources. Evolution of Eq. () is stopped at minimum χ s. max = 4.8. R = N (Th) /N. rms =. centrality (%) N (Th) 789± ±37 373±68 395±83 R rms.3±.1.7±.1.4±.1.1±.1 p.841±.7.858±.7.865±.7.871±.7 V (t).79±.3.8±.3.64±.1.56±. c (Th).14±.7.133±.7.136±.8.139±.8 χ /n.d.f..877/13.434/13.57/13.758/13 d % -3% d % 4-5% Figure 4: Analyses of /d at s NN = 13 GeV by Eq. (). See Table 3. R = N (Th) /N is always larger than 1. In the measurements of BRAHMS Collaboration, as the observable region is restricted with 4.7, we can conjecture the number of N (Th) is always 3% 7% larger than N. The different values of χ in Tables 3 and 4 are attributed to the magnitude of the error bars at 13 GeV and GeV. The intercepts of /d at = is explained by the following expression in the O-U process, { [ c (Th) 1 = exp (± ]} max 1 p) πv (t) V. (7) (t) Since our theory is based on the O-U process, the intercept c (Th) is relating to y max, the width of /d and the evolution parameter. Next we consider physical meaning of the evolution parameter γt. When we assign the meaning of second [sec] to t, γ has the dimension of [sec 1 ]. For the magnitude of the interaction region of Au-Au collisions, we assume to be 1 fm. See discussions in Ref. [1]. See also Tables 5 and 6. The averaged γ [fm 1 ] are almost the same as estimated values from PHOBOS Collaboration [13, 14] and ones estimated from the proton spectra at SPS energies in Ref. [1]. 5

6 Table 4: Estimated parameters at s NN = GeV in our analyses by Eq. () with two sources. Evolution of Eq. () is stopped at minimum χ s. max = 5.4. R = N (Th) /N, rms =, δp.5 and δc t =.4.5. centrality (%) N (Th) 955± ±4 158±34 311±49 434± ±76 R rms.41±.8.4±.6.39±.9.37±.8.35±.8.3±.8 p.854±δp.859±δp.86±δp.866±δp.871±δp.878±δp V (t) 3.169±. 3.17± ± ± ± ±.19 c (Th).115±δc t.117±δc t.118±δc t.11±δc t.13±δc t.18±δc t χ /n.d.f. 7./33 5./33 4.3/33 5.4/33 4.9/33 5.1/33.1 /d % 1-%.1 /d % 5-1%.1 /d.8.4-3% -5% Figure 5: The same as Fig. 4, but GeV. See Table 4. Table 5: Values of γ and σ at s NN = 13 GeV provided that t sec. centrality (%) average γ [fm 1 ] σ [fm 1 ] σ /γ

7 Table 6: Values of γ and σ at s NN = GeV provided that t sec. centrality (%) average γ [fm 1 ] σ [fm 1 ] σ /γ The z r = / rms scaling To investigate the z r = / rms scaling whi has been proposed in Ref. [4], we use rms = = /d at s NN = 13 GeV and GeV. We can consider the following formula with z r : rms d = dz r = f(z r = / rms ). (8) The right hand side with multiplying rms is obtained from Eq (), as { 1 = exp [ (z r +z max e γt ) ] dz r 8πV r (t) Vr [ (t) (z r z max e γt ) ]} +exp V r (t), (9) where z max = max / rms and Vr (t) = V (t)/rms. rms is the averaged quantity in the set of data. In concrete analyses of data, Vr (t) and p are treated as the free parameters. The z r scaling at 13 GeV are compared with that of the hemisphere ( 6) at GeV in Fig. 6 (b). It is difficult to distinguish them without the labels of incident energies. The behavior of full space is given in Fig. 6 (c). This situation is also observed in analyses of data at 13 GeV and GeV by PHOBOS Collaboration [13, 14]. 5 Rapidity (y) distribution derived from distribution It is well known that one can usually calculate the distribution from the y distribution. In this present study, on the contrary, we consider an inverse problem as follows. First we regard Eq. () as the correct description of the data, because of small χ values. Using the following formula we can obtain the y distribution as dy = where M = 1+m /p t. The right hand side, /d, is given as y = 1 d ln E +pz E p z = 1 ln = [ M(1+sinh y) 1+M sinh y d, (1) { 1 exp 8πV (t) [ ((y)+y maxe γt ) ] V (t) +exp [ ((y) y maxe γt ) ]} V, (11) (t) 1+m /p t +sinh +sinh 1+m /p t +sinh sinh ] ( ) = tanh 1 pz lntan(θ/). E = 1 p+pz ln and p p z dy = d d dy, where (y) = arcsinh( M sinhy). For /d = (p/e)/dy, we have p/e = cosh/ 1+m /p t +sinh. Moreover, we have confirmed that (/dy)dy = 1 and (/d)d = 1. 7

8 (a) BRAHMS 13 GeV -5% 5-1% -3% 4-5% (b) BRAHMS GeV -5% 5-1% 1-% -3% 3-4% 4-5% /dz r. /dz r z r = / rms z r = / rms.4.35 (c).3.5 /dz r BRAHMS GeV -5% 5-1% 1-% -3% 3-4% 4-5% z r = / rms Figure 6: Normalized distribution of /dz r with z r = / rms scaling and estimated parameters using Eq. (9). (a) s NN = 13 GeV, p = 1 e γt =.889 ±.3, Vr (t) =.57 ±.1 and χ /n.d.f. = 5.4/61. (b) and (c) s NN = GeV, p = 1 e γt =.865±., Vr (t) =.559 ±.15 and χ /n.d.f. = 3.1/189. (b) is taken from hemisphere data ( 6) of Fig. 6(c). (c) The full space of /dz r. The dotted lines represent the magnitude of error-bars in the centrality cut -5%. 8

9 where (y) = arcsinh( M sinhy). From Eq. () with the averaged parameters p and V (t), we obtain y distributions at GeV for π meson and all hadrons (π ±, K ±, p and p). They are compared with the data in Ref. [3] in Fig. 7. The small peak is due to the inverse Jacobian factor. Indeed the data at GeV show these behaviors at y, even large error bars. To confirm these phenomena, measurements in wider region as well as y are necessary. A phenomenological approa proposed in Ref. [15] (whi is named as EKRT) is also shown in Fig 7. dy (EKRT) = 1 (1+e y/d ) c N (1+e ( y y)/d )(1+e (y y)/d ), (1) where c N is the normalization factor 3. y = 3.3 and d =.65 are parameters 4 given in Ref. [15]. Eq. (1) also reproduces the both data in Fig. 7. From Eq. (1) we can calculate /d (centrality cut -3%) at 13 GeV and GeV whi is presented in Fig. 8. The coincidences between data and theory are very well, when y and d are treated as free parameters. 6 Concluding Remarks 1) We have observed that the behaviors of scaling of /d by BRAHMS Collaboration hold fairly well among the various centrality cuts at s NN = 13 GeV and GeV. ) To explain those scaling behaviors, we have assumed that /d is governed by the O-U stoastic process with two sources at ±y max ( = ln s NN /m N ). The intercept of /d at = is expressed by Eq. (7). See Tables 3 and 4. The constant c s are reflecting the scaling property relating to the O-U process. 3) From the evolution parameter γt and the assumed size of the interaction region of Au+Au collision (1 fm), we have obtained the following value, γ.1 fm 1, whi is almost the same value as that estimated in Ref. [11]. 4) From Fig. 6, it can be said that the z r scaling holds at 13 GeV and GeV. It is difficult to distinguish them, as compared both data without the labels of incident energies. 5) Using Eq. (11) with distributions at GeV, we have calculated the y distributions whi explain the data of Ref. [3]. The comparison with different approa given in Ref. [15] is also shown. In a future both approaes can be distinguished by the existence of a projection (small peak) at y. Finally, it can be concluded that the O-U process is one of possible explanations for the scaling property of /d at s NN = 13 GeV and GeV by BRAHMS Collaboration [1, ] as well as distributions by PHOBOS Collaboration [14]. Acknowledgements One of authors (M. B.) would like to thank the Scandinavia-Japan Sasakawa Foundation for financial support, and H. Bøggild, J. P. Bondorf, H. Ito and K. Tuominen for their kind hospitality and useful conversations at the Niels Bohr Institute as well as PANIC. 3 We have estimated the normalization factor c N as follows (1+e y/d ) c N = (1 +e ( y y)/d )(1 +e (y y)/d dy = ) 4 Notice that a similar expression with its symmetrization can be seen in Ref. [16]. A different expression based on the fractional Fokker-Planck equation for /dy is found in Ref. [17]. Both are proposed for analyses of pp (or pp) collisions. 9

10 ..16 (a) π + meson /dy BRAHMS GeV y..16 (b) All hadrons /dy BRAHMS GeV y Figure 7: p and V (t) are adopted from Tables 3 and 4. The averagedparameters p =.865 (fixed) and V (t) = (fixed) are used. (a) /dy of all π meson. m/p t =.4 (fixed), N = 153±77. (b) /dy of all hadrons (π ±, K ±, p and p). m/p t =.5 (fixed), N = 3915±34. The dashed lines are obtained from Eq. (1) with 1/c N =

11 3 5 (a) (b) dn / d BRAHMS 13 GeV -3% (c) (d) BRAHMS GeV -3% dn / d BRAHMS 13 GeV -3% 4 BRAHMS GeV -3% Figure 8: Using Eqs. (11) and (1), we calculate dn /d (centrality cut -3%) at 13 GeV and GeV. (a) Dashed line is obtained by y = 3.3, d =.65, N (Th) = 1731±34, m/p t =.5 and χ /n.d.f. = 1.1/15. Solid line is obtained by O-U process [χ /n.d.f. =.43/13 from Table 3], (b) Dashedlineisobtainedbyy = 3.3,d =.65,N Th = 54±31,m/p t =.5andχ /n.d.f. = 134/35. Solid line is obtained by O-U process [χ /n.d.f. = 4.3/33 from Table 4]. When y and d are treated as free parameters, the following sets of parameters are obtained. (c) y = 3.3, d =.83, χ /n.d.f. =.4/13. (d) y = 3.7, d =.83, χ /n.d.f. = 4.4/33. References [1] I. G. Bearden et al. [BRAHMS Collaborations], Phys. Lett. B 53, 7 (1). [] I. G. Bearden et al. [BRAHMS Collaboration], Phys. Rev. Lett. 88, 31 (). [3] D. Ouerdane [BRAHMS Collaboration], nucl-ex/11. [4] M. Biyajima, M. Ide, T. Mizogui and N. Suzuki, Prog. Theor. Phys. 18, 559 () and Addenda-ibid. 19, 151 (3). See nucl-th/7 and also hep-ph/1135. [5] M. Biyajima, M. Ide, T. Mizogui and N. Suzuki, Analyses of (.5 N part ) 1 dn /d distributions of PHOBOS and BRAHMS Collaborations by means of a stoastic process in Proceedings of the 4th symposium on science of hadrons under extreme conditions (Mar 4-6,, JAERI, TOKAI, JAPAN), edited by S. Chiba and T. Maruyama, JAERI-Conf, -11, p [6] N. S. Goel and N. Riter-Dyn, Stoastic Models in Biology (Academic Press, New York, 1974). [7] N. G. van Kampen, Stoastic Processes in Physics and Chemistry (North-Holland Publ., Amsterdam, 1981). [8] K. Saitou, Probability and Stoastic Process for Engineers (in Japanese), (Saiensu-Sha, Tokyo, 198). [9] J. Hori, Langevin Equation (in Japanese), (Iwanami-Shoten, Tokyo, 198). 11

12 [1] G. Wolsin, Eur. Phys. J. A 5, 85(1999): His Fokker-Planck equation is given with a replacing R(Y,t) by P(y,t) as P(y,t) t = 1 τ y y [(y y eq)p(y,t)]+d y y P(y,t), where y eq is relating to the rapidity of the colliding energies. [11] M. M. Aggarwal et al. [WA98 Collaboration], Eur. Phys. J. C 18, 651 (1). [1] K. Morita, S. Muroya, C. Nonaka and T. Hirano, Phys. Rev. C 66, 5494 (). [13] M. Biyajima and T. Mizogui, nucl-th/94, to appear in Prog. Theor. Phys. 19 (3) No. 3. [14] R. Nouicer et al. [PHOBOS Collaboration], nucl-ex/83. See also, B. B. Back et al. [PHOBOS Collaboration], Phys. Rev. Lett. 87(1), 133, and B. B. Back et al.[phobos Collaboration], Phys. Rev. Lett. 88 (), 3. [15] K. J. Eskola, K. Kajantie, P. V. Ruuskanen and K. Tuominen, Phys. Lett. B 543, 8 (). [16] A. Ohsawa, Prog. Theor. Phys. 9, 15 (1994). [17] M. Rybczynski, Z. Wlodarczyk and G. Wilk, hep-ph/