Forward - Backward Multiplicity in High Energy Collisions

Speaker: Lai Weichang

National University of Singapore

Introduction

Introduction In our work, we attempt to determine the forward-backward

multiplicity correlation in high-energy hadron-hadron collisions.

Colliding proton proton and proton anti-proton.

Done by choosing a probability distribution to predict the number of forward and backward particles formed.

Contents Review

Chow-Yang Model Negative Binomial Distribution (NBD) Cluster Model

Generalized Multiplicity Distribution (GMD) Results

Discussions on cluster size r for GMD Comparing plots of GMD and NBD Correlation Strength b

Review

Review:Chou-Yang Model In 1984 T.T. Chou and C.N. Yang suggested that for high

energy collisions, the distribution with respect to the charge asymmetry is a binomial. (for a given number of particles produced, n)

[ at fixed ] =

Relation observed in 1984 by Chou and Yang in experiment.

€

Z 2

€

n

€

2n

€

z= n f − nb

- T.T. Chou, C.N. Yang: Phys Lett. B135 (1984)

€

n= n f + nb

Review:Chou-Yang Model

used to satisfy the simple formula that T.T. Chou and C.N. Yang observed of collisions at 540 Gev

Forward-backward multiplicity distribution separate into two components

- T.T. Chou, C.N. Yang: Phys Lett. B135 (1984)

€

P(n,z) = (Function of n)C(n +z ) / 4n / 2

€

C(n f ) / 2n / 2

Review:Chou-Yang Model More explicitly,

€

P(n,z) =ψ (n / n )C(n +z ) / 4n / 2 [B(n)]−1

€

ψ(n / n )

€

[B(n)]−1

= KNO scaling function

= Normalization Constant

€

n = Mean charges multiplicity

Review:Negative Binomial Distribution

NBD gives better parameterization of multiplicity distribution, rewrite as

€

P(n,z) = PNB (n)C(n f ) / 2n / 2 [B(n /2)]−1

€

PNB (n) =Γ(n + k)

Γ(n +1)Γ(k)

k

n + k

⎛

⎝ ⎜

⎞

⎠ ⎟

kn

n + k

⎛

⎝ ⎜

⎞

⎠ ⎟

n

- S.L. Lim, C.H. Oh et al.: Z. Phys. C 43 (1989) 621€

Γ−Gamma Function

Review:Negative Binomial Distribution Average backward multiplicity at fixed forward

multiplicity:

Experimentally, a linear correlation of the type:

Plot

€

n f

nb =

nbP(n f ,nb )nb

∑

P(n f ,nb )nb

∑

€

nb n f= a + bn f

- S. Uhlig et la.: Nucl. Phys B132 (1978) 15- UA5 Coll. K. Alpgard et. al.: Phys. Lett. 123B (1983) 361- UA5 Coll. R.E. Ansorge et. al.: Z. Phys. C 27 (1988)191

Review:Negative Binomial Distribution

€

nfnb =

nbP(n f ,nb )nb

∑

P(n f ,nb )nb

∑

€

nb n f= a + bn f

Observed for various energy

Collider energy fits well, disagreements exist in ISR energies

- S. Uhlig et la.: Nucl. Phys B132 (1978) 15- UA5 Coll. K. Alpgard et. al.: Phys. Lett. 123B (1983) 361- UA5 Coll. R.E. Ansorge et. al.: Z. Phys. C 27 (1988)191

Review:Cluster Model Each cluster is assumed to fragment into 2 charged particles

Since this is only observed experimentally at 540 GeV, no reason for other energies to be the same.

Each cluster is assumed to fragment into exactly r charged particles besides neutrals.

Proposed that energy has a correlation with cluster size:

[ at fixed n] = rn

€

Z 2

- S.L. Lim, C.H. Oh et. al.: Z. Phys. C 43 (1989) 621

€

C(n f ) / 2n / 2 → C(n f ) / r

n / r

Review:Cluster Model

Rewrite NBD with cluster size r

r is adjusted to reproduce the experimental forward-backward correlation strength b of

re-plotted again

€

P(n,z) = PNB (n)C(n f ) / rn / r [B(n /r)]−1

- S.L. Lim, C.H. Oh et al.: Z. Phys. C 43 (1989) 621

€

nb n f= a + bn f

€

nb n f

Review:Cluster Model

- S.L. Lim, C.H. Oh et. al.: Z. Phys. C 43 (1989) 621

€

nfnb =

nbP(n f ,nb )nb

∑

P(n f ,nb )nb

∑

€

nb n f= a + bn f

(r varied)

Review:Cluster Model

- S.L. Lim, C.H. Oh et. al.: Z. Phys. C 43 (1989) 621

- S. Uhlig et. la.: Nucl. Phys. B132 (1978) 15

Review:Cluster Model (Finding r analytically)

Since The slope b is a measure of correlation strength. Indeed, it can be shown that b is equivalent to the statistical

definition of the correlation coefficient.€

nb n f= a + bn f

€

b =cov(n f ,nb )

[Var( n f ) × Var( nb )]=

D2(n) − dn2 z( )

D2(n) + dn2 z( )

€

=( n /k) +1− r

( n /k) +1+ r

- UA5 Collaboration, K. Alpgard et. la.: Phys. Lett. B123 (1983)

- S.L. Lim, C.H. Oh et. al.: Z. Phys. C 43 (1989) 621

Review:Cluster Model (Finding r analytically)

- S.L. Lim, C.H. Oh et. al.: Z. Phys. C 43 (1989) 621

Generalized Multiplicity Distribution

Generalized Multiplicity Distribution: Interests in such studies has been revived since the data at

the TeV region became available. At high energy (900GeV), the NBD does not describe the

data very well. LHC will publish data this year.

The GMD is devised in NUS by Dr Chan and Prof Chew.

- L.K. Chen, C.K. Chew et. al.: Z. Phys. C 76 (1997) 263- T.Alexopoulos et. la.: Phys Lett. B 353 (1995) 155

€

PGMD (n) =Γ(n + k)

Γ(n − ′ k +1)Γ( ′ k + k)

n − ′ k

n + k

⎛

⎝ ⎜

⎞

⎠ ⎟

n− ′ k ′ k + k

n + k

⎛

⎝ ⎜

⎞

⎠ ⎟

′ k +k

Generalized Multiplicity Distribution: GMD is a convolution of NBD and FYD.

We use GMD for a better parameterization of the charged particle multiplicity distribution.

The physical meaning of k and can be explained

€

PGMD (n) =Γ(n + k)

Γ(n − ′ k +1)Γ( ′ k + k)

n − ′ k

n + k

⎛

⎝ ⎜

⎞

⎠ ⎟

n− ′ k ′ k + k

n + k

⎛

⎝ ⎜

⎞

⎠ ⎟

′ k +k

- A.H. Chan, C.K. Chew: Phys. Rev. 41 (1989) 851

€

′ k

Generalized Multiplicity Distribution:

In search of an even better parameterization of the multiplicity distribution, we rewrite as

Plot

€

P(n,z) = PGMD (n)C(n f ) / rn / r [B(n /r)]−1

- S.L. Lim, C.H. Oh et al.: Z. Phys. C 43 (1989) 621

€

nfnb =

nbP(n f ,nb )nb

∑

P(n f ,nb )nb

∑€

PGMD (n) =Γ(n + k)

Γ(n − ′ k +1)Γ( ′ k + k)

n − ′ k

n + k

⎛

⎝ ⎜

⎞

⎠ ⎟

n− ′ k ′ k + k

n + k

⎛

⎝ ⎜

⎞

⎠ ⎟

′ k +k

Results

Results:Discussions on cluster size r for GMD

Using the statistical definition of the correlation coefficient,

we calculate r for the GMD€

b =cov(n f ,nb )

[Var( n f ) × Var( nb )]=

D2(n) − dn2 z( )

D2(n) + dn2 z( )

€

r =n + k( ) n − ′ k ( )

k + ′ k

1− b

n 1+ b( )

⎛

⎝ ⎜

⎞

⎠ ⎟

Results:Discussions on cluster size r for GMD

Ranges of mean cluster size r which would give correlation strength b equal to experimental values within the quoted experimental errors for collisions at CERN ISR and SppS Collider energies.

The r values derived from the NBD is compared to the GMD.

Results:Discussions on cluster size r for GMD In conclusion, the multiplicity correlations observed reveal the

following features for 30 - 900 GeV:

1. Mean cluster size r correlates to energy as reported by Lim et. la.

2. Obeys relation

3. No significance difference between the cluster size r of NBD and GMD

€

r = α log s + β

€

α =0.341± 0.028

€

β =0.042 ± 0.139

Results:Discussions on cluster size r for GMD In conclusion, the multiplicity correlations observed review the

following features for 1.8 - 14 TeV:

1. In 1995 E735 Collaboration produced some experimental results for r and b at 1.8 TeV

2. Using relation

3. We arrive at r = 2.60 0.35 for c.m.s energy of 1.8 TeV

4. This values compare favorably with experimental results from the E735 Collaboration for r = 2.62 0.12

€

r = α log s + β

- T.Alexopoulos et. la.: Phys Lett. B 353 (1995) 155

Results:Discussions on cluster size r for GMD In conclusion, the multiplicity correlations observed review the

following features for 1.8 - 14 TeV:

5. Using relation

6. We predict the value of r = 3.300.41 for c.m.s energy of 14 TeV if the cluster size is a function of only energy.

7. Extrapolation to these energies may not be meaningful since the validities of the parameterization of , and becomes in doubt.

8. Cluster size may level off at higher energies.

- T.Alexopoulos et. la.: Phys Lett. B 353 (1995) 155

€

r = α log s + β

€

n

€

′ k

€

k

Plot

Compare the NBD with the GMD.

Results:Comparing plots of GMD and NBD

€

P(n,z) = PGMD (n)C(n f ) / rn / r [B(n /r)]−1€

nfnb =

nbP(n f ,nb )nb

∑

P(n f ,nb )nb

∑

€

PGMD (n) =Γ(n + k)

Γ(n − ′ k +1)Γ( ′ k + k)

n − ′ k

n + k

⎛

⎝ ⎜

⎞

⎠ ⎟

n− ′ k ′ k + k

n + k

⎛

⎝ ⎜

⎞

⎠ ⎟

′ k +k

Results:Comparing plots of GMD and NBD

GMD shown here as black line. Experimental result is shown as red. Green and blue are NBD with different r values.

Notice that the line plotted by using the GMD follows the curve of the data points at low nf values.

Results:Comparing plots of GMD and NBD

GMD shown here as black line. Experimental result is shown as red. Green and blue are NBD with different r values.

Notice that the blue line plotted by using the NBD is almost indistinguishable from the distribution using the GMD

Results: Correlation Strength b

a and b are calculated from linear fits of GMD plots shown previously.

Comparing between the linear forward-backward correlation parameters, experimental and calculated by using the NBD and GMD.

- S.L. Lim, C.H. Oh et. al.: Z. Phys. C 43 (1989) 621

- S. Uhlig et. la.: Nucl. Phys. B132 (1978) 15

Results: Correlation Strength b

In conclusion, the multiplicity correlations observed review the following features for the correlation strength b:

1. Our calculated b agrees well with those proposed previously by Lim et. la.

2. Using our results, we are able to propose the relation:

3. This values fall into the experimental results proposed by Alexopoulos et. la.€

b = c log s + d

€

c = −0.174 ± 0.020

€

d = 0.120 ± 0.004

€

c = −0.181± 0.015

€

d = 0.120 ± 0.003

- T.Alexopoulos et. la.: Phys Lett. B 353 (1995) 155

Results:Correlation Strength b

In conclusion, the multiplicity correlations observed review the following features for the correlation strength b:

4. Our parameterization of b gives b = 0.980.19 at 14 TeV.

5. Agrees with the prediction of Chou and Yang that b saturates as energy approaches infinity.

- T.T. Chou, C.N. Yang: Phys Lett. B135 (1984)€

b =1−α

1+ α ⏐ → ⏐ 1 as energy ⏐ → ⏐ ∞

Thank you