Critical Exponent Of Gmd Chan Lim Oh

Critical Exponents for Multiplicity Distributions C.W. Lim, A.H. Chan and C.H. Oh Department of Physics, Faculty of Science, National University of Singapore 10 Kent Ridge Road, Singapore 117546

Abstract A Bose-Einstein model of particle multiplicity distribution developed by Mekjian, Cs¨org¨o and Hegyi has two important results. The combinants of the distribution is related to the power law critical exponent τ and allows various distribution models to be unified in the mould of Gauss hypergeometric series. Here we try to to use their first result to develop the critical exponent τ from the Generalized Multiplicity Distribution(GMD) developed by Chan and Chew[1], which is a convolution of both the negative binomial distribution and Furry-Yule distribution. By using the same approach, the combinants Cj of GMD yield a critical exponent that is similar to the negative binomial distribution. 1. Introduction In high-energies inelastic collisions, the number of hadrons produced varies with the events. Hence, multiplicity distributions give us the method to interpret and understand the underlying dynamics of the particles’ interactions. In addition, due to the statistical nature of the method used in studying the collisions, we can investigate the associated fluctuations and correlations. For example, some distributions can predict large fluctuations such as the disoriented chiral condensate and fluctuations arising from first order phase transition. The key to understanding the critical phenomena is the scaling properties that are related to the multiplicity distributions. Examples of such properties are the Koba-Nielson-Olesen(KNO) scaling features, void scaling, and the hierarchical structure relations. Interesting, some of these features are associated with the Levy distribution from the probability theory. From the review of Ref.[2,3,4], we try to find the critical exponent τ from the Generalized Multiplicity Distribution proposed by Chan and Chew[1, 5,6]. In this report, we want to introduce the generating function and its connections with the properties of the distribution, then proceed to study negative binomial distribution(NBD) and the Furry-Yule Distribution(FYD) which are both special cases of the generalized multiplicity distribution(GMD). The reason why we pay attention to both NBD and FYD is because the GMD is a convolution of both the NBD and FYD. Hence, by studying both the distributions will give us a more systematic approach towards the development of the critcal exponent of GMD. In the result and discussion section, we will also briefly discuss about second order phase transition and correlated length then conclude with the analysis of τ = 1 that is derived from the GMD.

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2. Generating function and its relations Our starting point will be a set of probabilities that is characterised to observe n particles in an event within a designated region of phase space. However, dealing with Pn can be inconvenient, thus representing it in the generating function form will be more simpler. The generating function of a probability distribution Pn is G(z) =

∞ X

Pn z n

(1)

n=0

The G(z) is related to the combinants Cj by G(z) = Exp

∞ X

! Cj (z j − 1)

(2)

j=1

The unnormalized factorial moments Fq of Pn is determined by G(z) as ∞ q X d G(z) n(n − 1) · · · (n − q + 1)Pn = Fq = dz q n=0

(3)

z=1

and the unnormalized cumulants Kq is determined by LnG(z) as ∞ X

dq LnG(z) Kq = j(j − 1) · · · (j − q + 1)Cj = dz q j=1



(4) z=1

Hence, the mean multiplicity n is related to the combinants by n = F 1 = K1 =

∞ X

jCj

(5)

j=1

The fluctuation of Pn is given as n2 − n2 = (n − n)2 is the second moment of the Cj distribution, the third moment will give us the skewness and the fourth moment will give us the kurtosis. (n −

n)2

=

(n − n)3 = (n − n)4 =

∞ X j=1 ∞ X j=1 ∞ X

j 2 Cj = F 1 + F 2

(6)

j 3 Cj = F1 + F3 + 3F2

(7)

j 4 Cj = F4 + 6F3 + 7F2 + F1

(8)

j=1

From the above connections between combinants, generating function and probability distribution, we can now investigate the asymptotic behaviour of the particles as n → ∞. This is because in multiparticle distribution, the cluster size k and z parameter determine 2

the mean multiplicity n or the peak position of the Pn and the width of the distribution or fluctuation (n − n)2 . We must also remember that for multiparticle production, the n is the total number or multiplicity of the produced particles and this number is not fixed. So we study the multiplicity distribution of produced particles as a function of n in a grand canonical ensemble[2]. In general, the combinants decrease and converge in the fashion of power law 1/j τ as the number of particles tend to ∞ at a critical point. From this view, we can see that the critical P∞ exponent actually determines the grand canonical partition function Z(Cj ) = Exp( j=1 Cj ). In summary, if we can develop the combinants Cj from any multiplicity distribution, we can determine the critical exponent τ from the asymptotic behaviour of the j parameter. 3. Generalized Multiplicity Distribution and it’s critical exponent Chan and Chew proposed to use the generalized multiplicity distribution(GMD) to fit data at various pseudorapidity ranges and had seen better results in higher energies where the NBD fails to describe. In this section, we will discuss the overview of the GMD, NBD and it’s relation to the critical exponent τ . The idea is to decompose GMD into NBD and FYD, then analysis it’s multiplicity and combinants as n → ∞. After indentifying both NBD and FYD’s combinants, we will derive the critical exponent from the GMD. The probability distribution of GMD is Γ(n + k) PGM D (k, k , n) = Γ(k + k 0 )Γ(n − k 0 + 1) 0



n − k0 n+k

n−k0 

k + k0 n+k

k+k0 (9)

Where k is related to the inital number of quarks by k = mA/A, k 0 is the initial number of gluons and Γ is the usual gamma function. The eAt is a term that arises from the exact solution of PGM D and t is the QCD evolution parameter. This distribution is derived from the Markov branching process and the probability for producing m quarks and n gluons are casted in 4 branching processes: g → gg q → qg g → qq g → ggg

− − − −

A A B C

The above branching processes are the average probabilities A, A, B and C respectively. From the review of Ref.[5], using Eq.(1), we obtain the generating function of GMD.  −k  −k0 1 − z At At GGM D (z) = z + (1 − z)e 1+ e z

(10)

Next,we look at the negative binomial distribution and determine its generating function, mean multiplicity and it’s combinants.

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Negative Binomial Distribution The probability distribution is given as Γ(n + k) PN BD (k, n) = n!Γ(k)

k n

!k

!−(n+k)

k 1+ n

(11)

Using the same approach, the generating function is ∞ X

−k  n GN BD (z) = PN BD z = z + (1 − z) k n=1 n

(12)

The mean mulitiplicity n is given by Eq.(5) n=

kz 1−z

(13)

And we know that the infinite series of z n from n=1 to ∞ is ∞ X

zn =

n=1

z 1−z

(14)

So now, we can determine the combinant distribution that is given by Eq.(5) n=

∞ X

jCj =

j=1

kz 1−z

(15)

The combinant of negative binomial distribution is Cj = kz j /j and the exponent τ is the power in the 1/j τ of the combinant for large j, thus the critical exponent for the NBD is τ = 1. Furry-Yule Distribution The FYD probability distribution is given as: Γ(n) PF Y D (k 0 , n) = Γ(n − k 0 + 1)Γ(k 0 )

k0 n

!k 0

k0 1− n

!n−k0 (16)

The generating function is  −k0 (1 − z) n GF Y D (z) = PF Y D z = 1 + z k0 n=1 ∞ X

The mean mulitiplicity n is n=

n

(17)

k0 1−z

(18)

4

given that

∞ X

1 1−z

z n−1 =

n=1

(19)

Hence, the combinants of FYD are ∞ X

∞ X k 0 z n−1 n= jCj = j j j=1 k=1

(20)

From Eq.(10,12,17), we can see that the GMD’s generating function is a convolution of both the NBD and FYD. We also want to note that the combinants of the FYD are similar to those of the NBD because the FYD is an alternative form of NBD[7]. With all these results in hands, we will then develop the combinants of the GMD and compare them with NBD and FYD. Critical exponent of GMD The convolution of two distributions with different and independent parameters is given as ∞ ∞ X X P2 δn1 +n2 , n P1 (21) Pn = n1 =0

n2 =0

As we can see, it will be extremely uneasy to deal with summations. However if we are dealing in terms of generating functions, the convolution of two distributions will be nothing more than the product of each distribution’s generating function. G(z) = G(1) (z)G(2) (z)

(22)

Thus, if we multiply both Eq.(12) and Eq.(17), we get Eq.(10) which the left term is NBD and right term is FYD. Now that we have proven that the GMD is actually a convolution of both NBD and FYD, we can proceed to find the mean multiplicity of GMD and determine it’s critical exponent. The mean multiplicity n is computed as
At −k

s + (1 − s) e n=−

At

k 1−e




1+

(1−s)eAt s

−k0

s + (1 − s) eAt
At −k

s + (1 − s) e −



1+

(1−s)eAt s

1+

−k0

k

0



At − es

−

(1−s)eAt s2

(1−s)eAt

 (23)

s

By setting z = 1, we will obtain the mean multiplicity in its standard form.
n = k eAt − 1 + k 0 eAt

(24)

We will want to take note that the GMD has two special cases. If we set k 0 = 0 we will obtain NBD and k = 0 we get FYD. The mean multiplicity of these two cases are 5

equivalent to Eq.(12) and Eq.(17). Hence, Eq.(24) can be rewritten as  j  X ∞ ∞ X kz k0 z + z j−1 0 n= + = f (k, k ) j = jCj 1−z 1−z j j=1 j=1

(25)

The combinants for GMD are given as Cj = f (k, k 0 )(z j + z j−1 )/j which gives an critical exponent of τ = 1. We also factor out k and k 0 as f (k, k 0 ) function to simplify the expression as these variables will not determine the value of τ . This is the same as for negative binomial distribution or Furry-Yule distribution. We can now summarize the results of Generalized Multiplicity distribution, negative binomial distribution and Furry-Yule distribution in Table 1. Table 1. Results of multiparticle distributions Model Generalized Multiplicity Distribution

Pn Γ(n+k) Γ(k+k0 )Γ(n−k0 +1)

Γ(n+k) n!Γ(k)

Negative Binomial Distribution

Furry-Yule Distribution



n−k0 n+k

 k  k n

Γ(n) Γ(n−k0 +1)Γ(k0 )

Ck n−k0 

1+

k n

 0 k 0  k n

k+k0 n+k

k+k0

−(n+k)

1−

k0 n

n−k0

f (k, k 0 ) (z

n j +z j−1 )

j

kz 1−z

+

τ k0 1−z

kz j j

kz 1−z

1

k0 z j−1 j

k0 1−z

1

We have see that the combinants of GMD, NBD and FYD have the same power law 1/j τ as j → ∞ and this result agrees with the fact that GMD is a convolution of both NBD and FYD(FYD is an alternative form of NBD). 4. Results and Discussion Order parameter and Phase transition In a first order phase transition(e.g. boiling of water or melting of ice), the transition will almost results in the absorption or release of heat into the surroundings. In addition, the entropy is not continuous across the phase boundary and the heat capacity diverges as well. But we are more interested in the second kind of phase transition which is the second order phase transition whereby the entropy is continuous across the phase boundary. This second kind of phase transition is of great interest due to behaviour of the particles whereby they show simplicity and follow the same direction, thus breaking the symmetry. In general, at high temperature the system is dis-ordered and is symmetrical about any point while at low temperature the system is orderd but loses symmetry as the particles will tend to follow a prefered direction. This is where the order parameter, 6

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η is required to fully describe the low temperature ordered phase and the point where the temperature T reaches Tc is the critical point. The order parameter can be wriiten as the volume integral over an order parameter density function ψ(~r) with the space vector as ~r = (r1 , r2 , r3 ). Z (26) η = d3 rψ(~r) The integral is over the volume with the order parameter density function and is taken as the statistical average. Now we will like to introduce the correlation function, which shows the difference in the order parameter of one point to another point within the system. ζ(~r) = ψ(~r)ψ(0) − ψ(0) ∗ ψ(~r)

(27)

If ζ(~r) decreases very fast with distance, then the system is localized and far neighbouring sub-systems are uncorrelated and we can say that the system is dominated by short-range forces and it’s microscopic structure. But if ζ(~r) decreases slowly with distance, we can say that the sub-systems are correlated and they influence each other as the system reaches critical point. In short, if ζ(~r) decreases slowly, the system becomes simplified at macroscopic level. Now, we can represent this distance and correlation relationship in the from of ξ, the correlation length. By doing so, we can rewrite Eq.(27) as   r −p (28) ζ(~r) → r Exp − ξ From the above expression, we can see that if distance r between two points in the system is larger than the correlation length ξ, the exponent will decreases at a faster rate compared to ξ > r. In experiments, ξ diverges as T → Tc . Thus, in the vicinity of the critical point the correlation function ζ(~r) can be approximated as −p ζ(~r) ∼ r (29) T →Tc

The critical exponent ν is related to the r−p dimension of the system as p = d − 2 + ν. By the scaling hypothesis, the correlation length ξ is the only characteristic length scale c −α near the critical point. Thus, we can replace r with ξ ∼ | T −T | and obtain the fisher T critical exponent τ which is the one we are interested in. τ = α(2 − ν)

(30)

Analysis of τ = 1 j

j−1

The combinants f (k, k 0 ) (z +zj ) that we obtained from GMD give us an exponent τ = 1. This exponent can be casted in a hypergeometric model (HGa) proposed by Lee and Mekjian[2,3]. The hypergeometric function has an arbitary value of a, b = 1 and c = 2.

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∞ X [a]m [1]m z m F (a, 1; 2; z) = [2]m m! m=0

(31)

With the combinants given as zj Cj = f (k, k )[a]j−1 j! 0

(32)

For large j using stirling approximation, the critical exponent is given as τ =2−a

(33)

5. Conclusion In this research, we aim to use the method developed by Mekjian, Cs¨org¨o and Hegyi to determine the critical exponent τ of the generalized multiplicity distribution and to analysis the exponent’s connections with a more generalized model. Proceeding in this direction, we decomposed the GMD into two distributions NBD and FYD and study each of them independently, obtaining the mean multiplicities and combinants. In doing so, we verified that the GMD is a convolution of both NBD and FYD. In addition, we want to note that the FYD is an alternative form of NBD therefore we should obtain the same power law for the GMD’s combinants. This shows that the GMD is an enhanced version of negative binomial distribution and can extend to higher energies collisions. In further analysis, it is from the combinants that leads to a generalized hypergeometric function(HGa) with b = 1 and c = 2. As a → 1, the HGa reduces to NBD and the critical exponent is related to the a parameter by τ = 2 − a. We also briefly discuss about the meaning of critical exponent and order parameter in the result and discussion section. This is to give the critical exponent that we are developing a more physical meaning, thus the importance of comparing it with a set of experiment data. In summary, the development of critical exponent allows us to unify various distribution models in the mould of hypergeometric model. For the case of GMD, we managed to show that the combinants of GMD have a power scaling of 1/j, which is the same as for negative binomial distribution. This is a good reflection of the GMD as NBD has been the most popular distribution model in describing high energies collisions. References [1] Fuglesang, C. (1989), “UA5 multiplicity distributions and fits of various functions”, CERN, EP-89-135. [2] Lee, S.J. and Mekjian, A.Z. (2004), “Development of particle multiplicity distributions using a general form of the grand canonical partition function”, Nucl. Phys. A,730, 514547 [3] Mekjian, A.Z, Cs¨org¨o, T. and Hegyi, S. (2007), “A Bose-Einstein model of particle multiplicity distribution”, Nucl. Phys. A,784, 515-535 [4] Dremin, I.M. and Gary, W.J. (2001), “Hadron Multiplicities”, Phys Rep, 349, 301-393. [5] Chan, A.H. and Chew, C.K. (1989), “Analysis of Generalized Multiplicity Distribution at Various Pseudorapidity Ranges”, Nuovo Cimento, 101A, 409-422. 8

[6] Chan, A.H. and Chew, C.K. (1989), “e+ e− multiplicity distribution from branching process”, Z.Phys., C55, 503-508. [7] Hwa, R.C. and Lam, C.S. (1986), “An energy-independent parametrization of the non-scaling multiplicity distribution”, Phys. Lett B, 173, 346-348. [8] Landau, L.D. and Lifshitz, E.M. (1980), Statistical Physics Part 1 3rd Ed, Pergamon Press Ptd.

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