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Mean, median and mode of hadron spectrum in QCD jets
Physics Lette(s B 273 ( 1991 ) 319-322 North-Holland
PHYSICS LETTERS B
Mean, median and mode of hadron spectrum in QCD jets Yu.L. Dokshitzer a,b, V.A. Khoze c,a C.P. Fong d and B.R. Webber d a b ¢ d
LeningradNuclearPhysicslnstitute, Gatchina, SU-188350Leningrad, USSR Department of Theoretical Physics, University ofLund, S6lvegatan 14A, S-223 62 Lund, Sweden CERN, INFNEloisatron Project, CH-1211 Geneva 23, Switzerland Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB30HE, UK
Received 11 September 1991
We present predictions of the three common statistical measures of the "typical" value of the quantity ~= In ( 1/x) where x is the momentum fraction of a hadron in a QCD jet. Although the distribution of ~ is asymptotically gaussian, its mean, median and mode (or peak) should differ by constants plus corrections of order x/~. An experimental study of these quantities would provide a detailed test of the underlying QCD dynamics of jet fragmentation.
I. Introduction During the past year, the LEP experiments have provided a wealth of new information on the properties of Q C D jets. One o f the most striking features to emerge so far has been the close similarity between the observed distribution [ 1 ] o f the single-hadron m o m e n t u m fraction, x, and the theoretical predictions [ 2 - 5 ] based on perturbative Q C D supplemented by the hypothesis of local p a r t o n - h a d r o n duality ( L P H D ) [6]. Q C D predicts that the m o m e n t u m fraction distribution has the form of a "hump-backed plateau" [ 7 ] which is asymptotically gaussian in the variable ~ = l n ( 1 / x ) . Taking proper account of leading and next-to-leading infrared logarithms to all orders, the shape of the distribution has calculable next-to-leading distortions that are substantial at present energies [ 2-4 ]. According to the L P H D hypothesis this shape should not be strongly modified by non-perturbative hadronization corrections, since these involve only small m o m e n t u m transfers. The observed distribution is indeed in very good agreement with the perturbative predictions, including the next-to-leading corrections, over a wide range o f energies.
Prompted by this success, we consider in this paper some more refined tests that could be performed making full use of the very precise data available from LEP in the near future. One natural feature to investigate more fully is the "typical" m o m e n t u m of a single hadron in a jet, as represented by the typical value of ~. For this purpose statisticians commonly employ three seemingly unrelated measures of the "centre" of the probability distribution P(~), namely: ( 1 ) the mean value, ~= f ~ t , ( ~ ) d~; (2) the median value ~m, which is such that the probabilities of lying above or below this value are equal, i.e., f o ~ P ( ~ ) d~= ~; (3) the modal or peak value ~p, at which P(~) reaches its maximum. We shall show that although the distribution P(~) is asymptotically gaussian, nevertheless these three measures do not coincide. They should differ by prescribed constant amounts asymptotically, and should have next-to-leading corrections proportional to where Q is of the order o f the centre-ofmass energy. These predictions are not inconsistent with the gaussian asymptotic form, since the width of the distribution grows like "t"3/4 where z = l n ( Q / A ) , and therefore a constant difference represents a vanishing fraction of the total width.
Research supported in part by the UK Science and Engineering Research Council. 0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
319
Volume 273, number 3
PHYSICS LETTERS B
2. Calculation
po = l + ~ k ,
Each of the quantities ¢i=~ cm or cp should have an energy dependence of the form ~i = a i z + b , v / r + c i
+ d J x / % + 0 (r - l ) ,
(1)
where the leading and next-to-leading coefficients ai and bi have been computed for the mean and the peak and are identical in these cases: ap=½,
bp=pl
19 December 1991
where p = 11 + 2 N f / N 3 and fl= 1 1 - 2 N J N c for Nc colours and Nf flavours. This is in good agreement with the observed energy dependence of the peak position [ 1 ]. For the median ~m, no previous results are available, so we consider first the calculation of this quantity. In the course of the calculation we shall see that information on the subleading coefficients ci and d~ can also be obtained for certain combinations of the mean, median and mode. Since the distribution P(~) is asymptotically gaussian, it may be expanded around the mean in the quantity d= ( ~ - ~)/0. where a is the width: 1
l P3 = ~1 s - -~k5
+
~ssk
' Ps = i~6ks - ~sk 1
P7 = T ~ s k .
(7)
Substituting these expressions we find
(2)
4V~,
P(~=~+a6) = ~ e x p ( - -
P2 = - l k ,
~-~=-~s
20 s
Cp-(=-½os ( 1- ~l sk5 + ~ k ) .
(8)
Comparing with the known results [ 3 ] 0.= V / ~ (~sfl,t.) 1/4 (1 -~a/~ l
/4~~)
+o(,-,/4),
~3 ( 4 8 ~ 1/4
s=-~p ~7\~7. s +o(~-~/'), 27(
k=-
½d2) ~pn ~n .
1
/~T __ 2_~) .at.0 (T_ 3/2 )
(3) 9 p ( 3 ) 3/2
To the precision required, one easily finds by Taylor expansion about the mean that the median is at dm = - (P3 + 7p5 + 57p7 )/Po,
(4)
while the peak is at dp = - 3(p3 + 5ps + 35p7 ) / ( p o - 2p2) .
,
(9)
we see that the differences given in eq. (8) are asymptotically constant at the values ~i
~m--4~lp=O.12, ~ p - & f i p = 0 . 3 5 .
/~1 ~ )
x~ =a2= ( ( ¢ - ~ ) ~) , x~ - 0.3s= < (~_~)3) , /~4 = 0.4k= ( (~__ ~)4) __30.4,
(6)
where s is the skewness and k the kurtosis of the distribution. The cumulants are the derivatives of the logarithm of the Laplace transform of P({); they vanish for n > 2 in the case ofa gaussian distribution. We find that to the accuracy needed in eqs. (4), (5) 320
(AflZ)I/4+O(72--7/4)
(lO)
(5)
Now the coefficients p, are related to the cumulants x, that were introduced in ref. [ 3 ]:
Ks = 0.Sk5 = ( ( { _ ~')5) _ 100.2(({_~)3) ,
k5 = ~
Thus the median, like the peak, has the same leading and next-to-leading behaviour as the mean, given by eqs. ( 1 ), (2), but lies three times closer to the mean than does the peak asymptotically. Notice that both the peak and the median lie above the mean, i.e. at smaller momentum, corresponding to about 90% of the momentum at the mean value of ~ in the case of the median and 70% for the peak. The subasymptotic behaviour of the skewness s cannot be calculated in a next-to-leading approach since it vanishes in leading order. Thus we cannot predict the first non-leading corrections to the results (10) separately, but we can predict the ratio, for ~ Numerical values are given for Nf= 3 active flavours, but would be almost identical for Nr= 5.
Volume 273, number 3
PHYSICS LETTERSB
which the correction is given by the leading behaviour of s, k and ks: 'P-~
3(1
0.50
19 December 1991 ....
l ....
I ....
I ....
1 k5 + ~ k ) lO s 0.10
=3 (I+ 9
~4B8z).
(II>
f
i
/
tin-
/
0.05
/ /
3. Discussion
(a)
The fact that the ratio ( m o d e - mean) / (med i a n - mean) is asymptotically equal to 3 for a number of"atmost gaussian" probability distributions was discovered empirically long ago by Pearson [ 8 ]. The conditions under which Pearson's "rule of 3" applies have been investigated by Haldane [ 9 ]. A sufficient condition is that the reduced cumulants k, = K,,/a" with n > 2 should vanish more and more rapidly asymptotically with increasing n. It follows from the results of ref. [ 3 ] that in general tcn ~ O ( T (2-n)/4)
0.01
'
l
'
....
I
5
0
5.0
'
~
....
I
. . . .
I , ,
i
l
15
i0
"r =
20
In(Q/A)
I
. . . .
I
. . . .
(b) 4.5
\ \ \
(12)
and therefore Haldane's conditions are satisfied for the distribution of ~. The leading subasymptotic correction to Pearson's rule in this case is proportional to r-t/z, i.e. of order ~ ) , and is already numerically small at LEP energies ( 14% for as = 0.12). The mean, median and modal values of ~ can also be computed from the explicit formulae for the soft parton spectrum in modified leading logarithmic approximation (MLLA), given in refs. [2,5 ]. Here one can also study the dependence of the spectrum on the cutoff parameter of the QCD parton cascade, Qo [ 4 ]. According to the LPHD hypothesis, one would expect the spectrum of hadrons of mass mh to be approximated by the patton spectrum with Qo related to mh. The observed ~ distribution for pions is in fact well described by the limiting spectrum with Qo = A ~ m~. Fig. 1 shows the differences ~p - ~ ~m - ( a n d their ratio for the limiting (Qo =A) spectrum, as functions of r = l n ( Q / A ) . The differences tend to the asymptotic values (10) from below while the ratio approaches the form ( 11 ) from above. The energy dependence of all these MLLA predictions is weak for the limiting spectrum. For larger values of Qo the in-
(~
3.5
--
~
Eq.(tt) 3.0
....
I
....
5
I 10
"r =
....
I 15
.... 20
In(Q/A)
Fig. I. (a) Limiting M L L A prcdictions of the diffcrcnces between m c a n (~), mcdian (~m) and modal (~p) values of ~ = In(!./x) as functions of r=In(Q/A). (b) Limiting M L L A prediction for Pearson's ratio.
dividual differences are expected to have stronger energy dependence, but their ratio should be less sensitive to Qo [4 ]. To investigate the behaviour of these quantities in a more specific model that obeys the LPHD hypothesis, we used the Monte Carlo simulation program HERWIG [ 10], which also describes the observed distribution of ~ quite well at present energies. Fig. 2 shows the HERWIG predictions for the differences and ratio as functions of the e+e - CM energy W. The "data" points and errors are based on Monte Carlo runs of approximately 10 6 events. The curves through the difference points, drawn simply to guide the eye, approach the values (10) asymptotically. They show 321
Volume 273, number 3 0.50
. . . .
I . . . .
PHYSICS LETTERS B
I . . . .
I . . . .
I . . . .
19 December 1991
l i m i t i n g M L L A p r e d i c t i o n . T h u s w i t h i n the f r a m e w o r k o f the M L L A - L P H D a p p r o a c h , illustrated specifically by the H E R W I G s i m u l a t i o n , we find that the p r e d i c t i o n ( 11 ) is not strongly m o d i f i e d by h a d r o n ization or n o n - l e a d i n g p e r t u r b a t i v e c o r r e c t i o n s and r e p r e s e n t s a clean test o f Q C D .
I . . . .
0.10
0,05 I /
iI1" /
/
Acknowledgement
t
/
(1v
....
0.01
I .... 100
(a) I .... 200
I .... 300
W
I .... 400
I .... 500
600
V.A.K. and B.R.W. w o u l d like to t h a n k t h e Dep a r t m e n t o f T h e o r e t i c a l Physics, U n i v e r s i t y o f Lund, for h o s p i t a l i t y while part o f this w o r k was done.
(CeV)
References
-"
MLLA
Eq.(11)
I
v
0
o
a00
200
aoo w (Gev)
400
500
800
Fig. 2. (a) H ERWIG Monte Carlo predictions of the differences between mean (~), median (~m) and modal (~p) values of~ for final-state hadrons in e+e - annihilation, as functions of the centreof-mass energy IV. (b) Analytical and HERWIG predictions for Pearson's ratio. a stronger energy d e p e n d e n c e than the limiting M L L A p r e d i c t i o n s in fig. 1, i n d i c a t i n g the p r e s e n c e o f further s u b l e a d i n g t e r m s in the M o n t e C a r l o s i m u l a t i o n , but r e m a i n c o n s i s t e n t with an a p p r o a c h t o w a r d s the e x p e c t e d a s y m p t o t i c v a l u e s f r o m below. In the M o n t e C a r l o results on the ratio (fig. 2 b ) , the a d d i t i o n a l s u b l e a d i n g t e r m s seen in the i n d i v i d ual differences t e n d to cancel, yielding v a l u e s close to the a s y m p t o t i c c u r v e a n d in g o o d a g r e e m e n t w i t h the
322
[ 1] OPAL Collab., M.Z. Akrawy et al., Phys. Lett. B 247 (1990) 617; L3 Collab., B. Adeva et al., L3 preprint ~025 ( 1991 ); T. Hebbeker, Plenary talk Intern. Lepton-photon Symp. and Europhysics Conf. on High energy physics (Geneva, JulyAugust, 1991 ). [2]Yu.L. Dokshitzer, V.A. Khoze and S.I. Troyan, in: Perturbative quantum chromodynamics, ed. A.H. Mueller (World Scientific, Singapore, 1989). [ 3 ] C.P. Fong and B.R. Webber, Phys. Lett. B 229 (1989 ) 289; Nucl. Phys. B 355 ( 1991 ) 54. [4]Yu.L. Dokshitzer, V.A. Khoze and S.I. Troyan, Lund preprint LU TP 91-12 ( 1991 ). [5]Yu.L. Dokshitzer, V.A. Khoze, A.H. Mueller and S.I. Troyan, Basics of perturbative QCD (Editions Fronti~res, Gif-Sur-Yvette, 1991 ). [6] D. Amati and G. Veneziano, Phys. Lett. B 83 (1979) 87; Ya.A. Azimov, Yu.L. Dokshitzer, V.A. Khoze and S.I. Troyan, Z. Phys. C 27 (1985) 65. [7] A.H. Mueller, in: Proc. 1981 Intern. Symp. on Lepton and photon interactions at high energies, ed. W. Pfeil (Bonn, 1981) p. 689; Yu.L. Dokshitzer, V.S. Fadin and V.A. Khoze, Phys. Lett. B 115 (1982)242. [ 8 ] K. Pearson, Phil. Trans. 186 ( 1895 ) 343. [9] J.B.S. Haldane, Biometrika 32 (1942) 294. [ 10] G. Marchesini and B.R. Webber, Nucl. Phys. B 310 (1988) 461; G. Marchesini, B.R. Webber, G. Abbiendi, I.G. Knowles, M.H. Seymour and L. Stanco, Cambridge preprint Cavendish-HEP-90/26 (1990), Comput. Phys. Commun., to be published.
PHYSICS LETTERS B
Mean, median and mode of hadron spectrum in QCD jets Yu.L. Dokshitzer a,b, V.A. Khoze c,a C.P. Fong d and B.R. Webber d a b ¢ d
LeningradNuclearPhysicslnstitute, Gatchina, SU-188350Leningrad, USSR Department of Theoretical Physics, University ofLund, S6lvegatan 14A, S-223 62 Lund, Sweden CERN, INFNEloisatron Project, CH-1211 Geneva 23, Switzerland Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB30HE, UK
Received 11 September 1991
We present predictions of the three common statistical measures of the "typical" value of the quantity ~= In ( 1/x) where x is the momentum fraction of a hadron in a QCD jet. Although the distribution of ~ is asymptotically gaussian, its mean, median and mode (or peak) should differ by constants plus corrections of order x/~. An experimental study of these quantities would provide a detailed test of the underlying QCD dynamics of jet fragmentation.
I. Introduction During the past year, the LEP experiments have provided a wealth of new information on the properties of Q C D jets. One o f the most striking features to emerge so far has been the close similarity between the observed distribution [ 1 ] o f the single-hadron m o m e n t u m fraction, x, and the theoretical predictions [ 2 - 5 ] based on perturbative Q C D supplemented by the hypothesis of local p a r t o n - h a d r o n duality ( L P H D ) [6]. Q C D predicts that the m o m e n t u m fraction distribution has the form of a "hump-backed plateau" [ 7 ] which is asymptotically gaussian in the variable ~ = l n ( 1 / x ) . Taking proper account of leading and next-to-leading infrared logarithms to all orders, the shape of the distribution has calculable next-to-leading distortions that are substantial at present energies [ 2-4 ]. According to the L P H D hypothesis this shape should not be strongly modified by non-perturbative hadronization corrections, since these involve only small m o m e n t u m transfers. The observed distribution is indeed in very good agreement with the perturbative predictions, including the next-to-leading corrections, over a wide range o f energies.
Prompted by this success, we consider in this paper some more refined tests that could be performed making full use of the very precise data available from LEP in the near future. One natural feature to investigate more fully is the "typical" m o m e n t u m of a single hadron in a jet, as represented by the typical value of ~. For this purpose statisticians commonly employ three seemingly unrelated measures of the "centre" of the probability distribution P(~), namely: ( 1 ) the mean value, ~= f ~ t , ( ~ ) d~; (2) the median value ~m, which is such that the probabilities of lying above or below this value are equal, i.e., f o ~ P ( ~ ) d~= ~; (3) the modal or peak value ~p, at which P(~) reaches its maximum. We shall show that although the distribution P(~) is asymptotically gaussian, nevertheless these three measures do not coincide. They should differ by prescribed constant amounts asymptotically, and should have next-to-leading corrections proportional to where Q is of the order o f the centre-ofmass energy. These predictions are not inconsistent with the gaussian asymptotic form, since the width of the distribution grows like "t"3/4 where z = l n ( Q / A ) , and therefore a constant difference represents a vanishing fraction of the total width.
Research supported in part by the UK Science and Engineering Research Council. 0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
319
Volume 273, number 3
PHYSICS LETTERS B
2. Calculation
po = l + ~ k ,
Each of the quantities ¢i=~ cm or cp should have an energy dependence of the form ~i = a i z + b , v / r + c i
+ d J x / % + 0 (r - l ) ,
(1)
where the leading and next-to-leading coefficients ai and bi have been computed for the mean and the peak and are identical in these cases: ap=½,
bp=pl
19 December 1991
where p = 11 + 2 N f / N 3 and fl= 1 1 - 2 N J N c for Nc colours and Nf flavours. This is in good agreement with the observed energy dependence of the peak position [ 1 ]. For the median ~m, no previous results are available, so we consider first the calculation of this quantity. In the course of the calculation we shall see that information on the subleading coefficients ci and d~ can also be obtained for certain combinations of the mean, median and mode. Since the distribution P(~) is asymptotically gaussian, it may be expanded around the mean in the quantity d= ( ~ - ~)/0. where a is the width: 1
l P3 = ~1 s - -~k5
+
~ssk
' Ps = i~6ks - ~sk 1
P7 = T ~ s k .
(7)
Substituting these expressions we find
(2)
4V~,
P(~=~+a6) = ~ e x p ( - -
P2 = - l k ,
~-~=-~s
20 s
Cp-(=-½os ( 1- ~l sk5 + ~ k ) .
(8)
Comparing with the known results [ 3 ] 0.= V / ~ (~sfl,t.) 1/4 (1 -~a/~ l
/4~~)
+o(,-,/4),
~3 ( 4 8 ~ 1/4
s=-~p ~7\~7. s +o(~-~/'), 27(
k=-
½d2) ~pn ~n .
1
/~T __ 2_~) .at.0 (T_ 3/2 )
(3) 9 p ( 3 ) 3/2
To the precision required, one easily finds by Taylor expansion about the mean that the median is at dm = - (P3 + 7p5 + 57p7 )/Po,
(4)
while the peak is at dp = - 3(p3 + 5ps + 35p7 ) / ( p o - 2p2) .
,
(9)
we see that the differences given in eq. (8) are asymptotically constant at the values ~i
~m--4~lp=O.12, ~ p - & f i p = 0 . 3 5 .
/~1 ~ )
x~ =a2= ( ( ¢ - ~ ) ~) , x~ - 0.3s= < (~_~)3) , /~4 = 0.4k= ( (~__ ~)4) __30.4,
(6)
where s is the skewness and k the kurtosis of the distribution. The cumulants are the derivatives of the logarithm of the Laplace transform of P({); they vanish for n > 2 in the case ofa gaussian distribution. We find that to the accuracy needed in eqs. (4), (5) 320
(AflZ)I/4+O(72--7/4)
(lO)
(5)
Now the coefficients p, are related to the cumulants x, that were introduced in ref. [ 3 ]:
Ks = 0.Sk5 = ( ( { _ ~')5) _ 100.2(({_~)3) ,
k5 = ~
Thus the median, like the peak, has the same leading and next-to-leading behaviour as the mean, given by eqs. ( 1 ), (2), but lies three times closer to the mean than does the peak asymptotically. Notice that both the peak and the median lie above the mean, i.e. at smaller momentum, corresponding to about 90% of the momentum at the mean value of ~ in the case of the median and 70% for the peak. The subasymptotic behaviour of the skewness s cannot be calculated in a next-to-leading approach since it vanishes in leading order. Thus we cannot predict the first non-leading corrections to the results (10) separately, but we can predict the ratio, for ~ Numerical values are given for Nf= 3 active flavours, but would be almost identical for Nr= 5.
Volume 273, number 3
PHYSICS LETTERSB
which the correction is given by the leading behaviour of s, k and ks: 'P-~
3(1
0.50
19 December 1991 ....
l ....
I ....
I ....
1 k5 + ~ k ) lO s 0.10
=3 (I+ 9
~4B8z).
(II>
f
i
/
tin-
/
0.05
/ /
3. Discussion
(a)
The fact that the ratio ( m o d e - mean) / (med i a n - mean) is asymptotically equal to 3 for a number of"atmost gaussian" probability distributions was discovered empirically long ago by Pearson [ 8 ]. The conditions under which Pearson's "rule of 3" applies have been investigated by Haldane [ 9 ]. A sufficient condition is that the reduced cumulants k, = K,,/a" with n > 2 should vanish more and more rapidly asymptotically with increasing n. It follows from the results of ref. [ 3 ] that in general tcn ~ O ( T (2-n)/4)
0.01
'
l
'
....
I
5
0
5.0
'
~
....
I
. . . .
I , ,
i
l
15
i0
"r =
20
In(Q/A)
I
. . . .
I
. . . .
(b) 4.5
\ \ \
(12)
and therefore Haldane's conditions are satisfied for the distribution of ~. The leading subasymptotic correction to Pearson's rule in this case is proportional to r-t/z, i.e. of order ~ ) , and is already numerically small at LEP energies ( 14% for as = 0.12). The mean, median and modal values of ~ can also be computed from the explicit formulae for the soft parton spectrum in modified leading logarithmic approximation (MLLA), given in refs. [2,5 ]. Here one can also study the dependence of the spectrum on the cutoff parameter of the QCD parton cascade, Qo [ 4 ]. According to the LPHD hypothesis, one would expect the spectrum of hadrons of mass mh to be approximated by the patton spectrum with Qo related to mh. The observed ~ distribution for pions is in fact well described by the limiting spectrum with Qo = A ~ m~. Fig. 1 shows the differences ~p - ~ ~m - ( a n d their ratio for the limiting (Qo =A) spectrum, as functions of r = l n ( Q / A ) . The differences tend to the asymptotic values (10) from below while the ratio approaches the form ( 11 ) from above. The energy dependence of all these MLLA predictions is weak for the limiting spectrum. For larger values of Qo the in-
(~
3.5
--
~
Eq.(tt) 3.0
....
I
....
5
I 10
"r =
....
I 15
.... 20
In(Q/A)
Fig. I. (a) Limiting M L L A prcdictions of the diffcrcnces between m c a n (~), mcdian (~m) and modal (~p) values of ~ = In(!./x) as functions of r=In(Q/A). (b) Limiting M L L A prediction for Pearson's ratio.
dividual differences are expected to have stronger energy dependence, but their ratio should be less sensitive to Qo [4 ]. To investigate the behaviour of these quantities in a more specific model that obeys the LPHD hypothesis, we used the Monte Carlo simulation program HERWIG [ 10], which also describes the observed distribution of ~ quite well at present energies. Fig. 2 shows the HERWIG predictions for the differences and ratio as functions of the e+e - CM energy W. The "data" points and errors are based on Monte Carlo runs of approximately 10 6 events. The curves through the difference points, drawn simply to guide the eye, approach the values (10) asymptotically. They show 321
Volume 273, number 3 0.50
. . . .
I . . . .
PHYSICS LETTERS B
I . . . .
I . . . .
I . . . .
19 December 1991
l i m i t i n g M L L A p r e d i c t i o n . T h u s w i t h i n the f r a m e w o r k o f the M L L A - L P H D a p p r o a c h , illustrated specifically by the H E R W I G s i m u l a t i o n , we find that the p r e d i c t i o n ( 11 ) is not strongly m o d i f i e d by h a d r o n ization or n o n - l e a d i n g p e r t u r b a t i v e c o r r e c t i o n s and r e p r e s e n t s a clean test o f Q C D .
I . . . .
0.10
0,05 I /
iI1" /
/
Acknowledgement
t
/
(1v
....
0.01
I .... 100
(a) I .... 200
I .... 300
W
I .... 400
I .... 500
600
V.A.K. and B.R.W. w o u l d like to t h a n k t h e Dep a r t m e n t o f T h e o r e t i c a l Physics, U n i v e r s i t y o f Lund, for h o s p i t a l i t y while part o f this w o r k was done.
(CeV)
References
-"
MLLA
Eq.(11)
I
v
0
o
a00
200
aoo w (Gev)
400
500
800
Fig. 2. (a) H ERWIG Monte Carlo predictions of the differences between mean (~), median (~m) and modal (~p) values of~ for final-state hadrons in e+e - annihilation, as functions of the centreof-mass energy IV. (b) Analytical and HERWIG predictions for Pearson's ratio. a stronger energy d e p e n d e n c e than the limiting M L L A p r e d i c t i o n s in fig. 1, i n d i c a t i n g the p r e s e n c e o f further s u b l e a d i n g t e r m s in the M o n t e C a r l o s i m u l a t i o n , but r e m a i n c o n s i s t e n t with an a p p r o a c h t o w a r d s the e x p e c t e d a s y m p t o t i c v a l u e s f r o m below. In the M o n t e C a r l o results on the ratio (fig. 2 b ) , the a d d i t i o n a l s u b l e a d i n g t e r m s seen in the i n d i v i d ual differences t e n d to cancel, yielding v a l u e s close to the a s y m p t o t i c c u r v e a n d in g o o d a g r e e m e n t w i t h the
322
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