forward - backward multiplicity in high energy collisions speaker: lai weichang national university...
TRANSCRIPT
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Forward - Backward Multiplicity in High Energy Collisions
Speaker: Lai Weichang
National University of Singapore
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Introduction
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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.
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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
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Review
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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Review:Cluster Model (Finding r analytically)
- S.L. Lim, C.H. Oh et. al.: Z. Phys. C 43 (1989) 621
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Generalized Multiplicity Distribution
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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
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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
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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
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Results
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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( )
⎛
⎝ ⎜
⎞
⎠ ⎟
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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.
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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
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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
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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
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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
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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.
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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
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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
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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
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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 ⏐ → ⏐ ∞
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Thank you