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Angular distribution of hadrons in two-jet events
Volume 214, number 1
PHYSICS LETTERS B
10 November 1988
ANGULAR DISTRIBUTION OF H A D R O N S IN T W O - J E T E V E N T S Brian R. H I L L
Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 4 August 1988
Well-known simple arguments suggest that the distribution ofhadrons in an e+e - two-jet event should be approximately invariant under boosts along the jet axis. One consequence of this is the rapidity plateau. In this paper, stronger predictions based on this invariance are made for the angular distribution of hadrons, and its relationship to fan invariance is discussed. Comparison of these predictions with experiment is fair.
Simple arguments suggest that the distribution o f hadrons in an e+e - two-jet event should be approximately invariant u n d e r boosts along the j e t axis. F o r m o t i v a t i o n only, here is one argument based on confinement. W h e n a q u a r k - a n t i q u a r k pair is created at a large center o f mass energy, W, the quarks separate rapidly and their color electric field, confined to a tube, stretches between them. The configuration away from the ends o f this string looks the same to all observers boosted along its axis. So if most o f the hadrons in the resulting jet arise from processes not involving the ends o f the string, the d i s t r i b u t i o n will be approximately invariant under boosts along the jet axis. Only the conclusion o f this argument, which can be reached in a n u m b e r o f other ways, will be used in what follows. One consequence o f the conclusion is the plateau in D ( y ) , the distribution o f h a d r o n s in rapidity, y. However, stronger predictions about the distribution o f hadrons can be m a d e solely on the basis o f the invariance. In this paper, these predictions are d e r i v e d a n d c o m p a r e d with data. A nice p a r a m e t r i z a t i o n o f the f o u r - m o m e n t a o f the particles in a jet is
F o r simplicity, I assume that all the particles in the j e t are pions a n d take the jet axis to be the z-axis. Under a boost along the jet axis, with r a p i d i t y Yo,
Y--*Y+Yo, while ~0a n d p± are unchanged. This simple transform a t i o n p r o p e r t y is what makes the p a r a m e t r i z a t i o n nice. It implies that for a distribution o f hadrons, D(y, P-I, ~o), to be invariant u n d e r boosts along the jet axis, it must be i n d e p e n d e n t ofy. I f I also d e m a n d invariance u n d e r rotations about the jet axis, the distribution must be i n d e p e n d e n t o f ~0. N o w let me describe a second parametrization: (x/P~ + p 2 secZo/+m2, Pii tan a cos ~0, p~ tan a sin ~0,PH )"
p . sin ~o, x/P~- + m 2 s i n h y ) .
This p a r a m e t r i z a t i o n was used in a p a p e r by the TASSO collaboration [ 1 ] because it exhibits clearly a r e m a r k a b l e p r o p e r t y o f hadronic distributions they discovered a n d called fan invariance. Later I will discuss how the a p p r o x i m a t e Lorentz invariance ~ c o m p l e m e n t s fan invariance. F o r now just accept it as a slightly obscure p a r a m e t r i z a t i o n . The implications o f Lorentz invariance for D(p~, a, ~o) are not i m m e d i a t e l y clear. To get them, note that
Current address: Theory Department, Fermi National Accelerator Laboratory, MS 106, P.O. Box 500, Batavia, IL 60510, USA. Bitnet BHILL@FNAL.
~ From here on, I will say "Lorentz invariance", when to be more accurate I should say "invariance of the inclusive distributions under Lorentz boosts along the jet axis".
(X/P~- + m 2 cosh y, p i cos tp,
0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o U a n d Physics Publishing D i v i s i o n )
157
Volume 214, number 1
PHYSICS LETTERS B
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Fig. 1. TASSO data (squares) for the distribution o f hadrons in o~ at 34 GeV for seven values o f p and the predictions o f Lorentz invariance (solid line) based on the distribution for the second value o f p in graph (b). (Thus in the case o f graph ( b ) , the agreement with the theoretical curve is automatic. ) 158
Volume 214, number 1 0.30 l a
'
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PHYSICS LETTERS B I'
'''
I''''
l0 November 1988
D(pli , t~' ) = g(P~l , a' ) D(pll , a ), g(PIJ , ol )
0.25
(3)
where 0.20
or' = t a n - ~ (pP-~tan a ) .
(4)
0.15
The ratio o f the two jacobians can be expressed fairly simply as
0,10
12
0,00
2
2
g(Pll , a' ) PIt P II + P JItan a g(Pil, a ) - P~IP~ +P] tan2a
0,05
/p2 + m 2 +p~ tan2o~ 0
20
40
60
X N/p]2 + r n 2 + p ~ tan2 a •
80
4.0
Fig. I. Continued.
O ( p . , y) D(p;,,o~)= ~ D(p±,y), where I have suppressed ~0, and where O(pi, y)
= t a n h y sec2o~
p~ sec2t~ = x/p2 sec2o~+m 2 =g(Pll, oQ.
(1)
I have already argued from approximate Lorentz invariance that D ( p . , y ) = f ( p ± ) and since p± =PH × tan o~ we have the result that
D(p, , a ) =g(PH, a)f(p~, tan a ) .
(2)
(5)
In fig. 1 I have taken the highest energy (34 GeV) data of ref. [ 1 ], used the second o f the curves as input, and graphed the predictions obtained from eqs. (3), (4) and (5) for the other six curves. The lines are the predicted distributions. The squares are the data. In the second of the graphs since the line is taken from the data itself, the nearly perfect agreement is meaningless. (The data points do not lie on the line even in this case because the curve has been slightly smoothed.) Just as the prediction of a rapidity plateau is good for a few units o f rapidity and then breaks down, the predictions for the angular distribution are good and then break down at higher energy. O f course, kinematics alone told one this breakdown must occur. A Lorentz invariant distribution o f particles would have an infinite amount o f energy. A possible way of breaking Lorentz invariance is to cut the distribution offat some value ofy. The energy flow in the resulting jet was discussed in ref. [ 2 ] with
f ( P i )oc6(p± - p ) ~2. f i s an unknown function o f a single variable. This is the necessary and sufficient condition for invariance o f the distribution under boosts along the jet axis. To see how good the prediction is I will turn eq. (2) into a statement that directly relates to the data presented in fig. 50 ofref. [ 1 ]. The distribution in PI~ and c~ is displayed by taking a fixedpl I (actually small ranges) and graphing D(p[J, c~) as a function o f a. This is done for seven different values ofp~. The content ofeqs. ( 1 ) and (2) is that I can use one of these curves for a given value ofpH as input to determine the function f, and as output predict the curves for all other values, PI~. The equation is
N o w I will discuss how Lorentz invariance is complementary to fan invariance. I will add a subscript 1,2
Since completion of this work, a preprint with a way of implementing Lorentz invariance which avoids infinite energy without using a cutoffhas appeared [ 3 ]. Briefly, instead of puning Lorentz invariance into the inclusive distributions themselves, they build it into an assumed form for the matrix element for the production of N hadrons. Then they use the usual field theoretic framework for turning matrix elements into inclusive distributions. Legendre transformations of their equations in N and W help them obtain many results analytically. Although they have compared their data with e+e - data in several ways, they have not used the distributions in two variables as a possible stringent test of their assumptions. 159
Volume 214, number 1
PHYSICS LETTERS B
W to the inclusive distributions to denote the dependence on the C O M energy. F a n invariance is the observation that D,.(pll , o~) = h ( W', Pl~ )Dw, (P11, a ) . That is, up to a n o r m a l i z a t i o n d e p e n d i n g on the total C O M energy a n d PI~, the curves in a are the same. The equality o f this agreement is remarkable. A partial explanation based on Lorentz invariance goes as follows. Lorentz invariance d e t e r m i n e s the shape o f the distribution in a for fixed W at one p~ from any other and it does so in a W-independent way. Thus, if a way o f explaining the fan invariance for just a single value ofp~ were given, Lorentz invariance would extend it to all values ofpjj. The weaknesses o f this observation are that it reduces the p r o b l e m to one previously unsolved, a n d in any case the quality o f the predictions o f fan invariance are much better than the predictions o f Lorentz invariance. The fair quality o f the predictions o f Lorentz in-
160
10 November 1988
variance partially explains the success o f so m a n y different jet models in fitting two-jet data: any m o d e l that embodies this a p p r o x i m a t e Lorentz invariance a n d fits the d a t a at one value ofp~ is guaranteed to fit the data for quite a range ofp~h. Hopefully the results here are a step toward a realistic, simple, model-ind e p e n d e n t description o f the shape o f jets. I thank H o w a r d Georgi for m a n y helpful discussions. This work was s u p p o r t e d in part by N S F G r a n t PHY87-14654. The m a n u s c r i p t and figures were c o m p l e t e d at Fermilab.
References [ 1] TASSO Collab., M. Althoffet al., Z. Phys. C 22 (1984) 307. [3] H. Georgi, Phys. Lett. B 195 (1987) 581. [3] Y. Kurihara, J. Hfifner and J. Aichelin, Hadron production in e÷e - annihilation and ~t-p collisions under the assumption of longitudinal phase space dominance, Max Planck Institute preprint MPI H-1988-V22.
PHYSICS LETTERS B
10 November 1988
ANGULAR DISTRIBUTION OF H A D R O N S IN T W O - J E T E V E N T S Brian R. H I L L
Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 4 August 1988
Well-known simple arguments suggest that the distribution ofhadrons in an e+e - two-jet event should be approximately invariant under boosts along the jet axis. One consequence of this is the rapidity plateau. In this paper, stronger predictions based on this invariance are made for the angular distribution of hadrons, and its relationship to fan invariance is discussed. Comparison of these predictions with experiment is fair.
Simple arguments suggest that the distribution o f hadrons in an e+e - two-jet event should be approximately invariant u n d e r boosts along the j e t axis. F o r m o t i v a t i o n only, here is one argument based on confinement. W h e n a q u a r k - a n t i q u a r k pair is created at a large center o f mass energy, W, the quarks separate rapidly and their color electric field, confined to a tube, stretches between them. The configuration away from the ends o f this string looks the same to all observers boosted along its axis. So if most o f the hadrons in the resulting jet arise from processes not involving the ends o f the string, the d i s t r i b u t i o n will be approximately invariant under boosts along the jet axis. Only the conclusion o f this argument, which can be reached in a n u m b e r o f other ways, will be used in what follows. One consequence o f the conclusion is the plateau in D ( y ) , the distribution o f h a d r o n s in rapidity, y. However, stronger predictions about the distribution o f hadrons can be m a d e solely on the basis o f the invariance. In this paper, these predictions are d e r i v e d a n d c o m p a r e d with data. A nice p a r a m e t r i z a t i o n o f the f o u r - m o m e n t a o f the particles in a jet is
F o r simplicity, I assume that all the particles in the j e t are pions a n d take the jet axis to be the z-axis. Under a boost along the jet axis, with r a p i d i t y Yo,
Y--*Y+Yo, while ~0a n d p± are unchanged. This simple transform a t i o n p r o p e r t y is what makes the p a r a m e t r i z a t i o n nice. It implies that for a distribution o f hadrons, D(y, P-I, ~o), to be invariant u n d e r boosts along the jet axis, it must be i n d e p e n d e n t ofy. I f I also d e m a n d invariance u n d e r rotations about the jet axis, the distribution must be i n d e p e n d e n t o f ~0. N o w let me describe a second parametrization: (x/P~ + p 2 secZo/+m2, Pii tan a cos ~0, p~ tan a sin ~0,PH )"
p . sin ~o, x/P~- + m 2 s i n h y ) .
This p a r a m e t r i z a t i o n was used in a p a p e r by the TASSO collaboration [ 1 ] because it exhibits clearly a r e m a r k a b l e p r o p e r t y o f hadronic distributions they discovered a n d called fan invariance. Later I will discuss how the a p p r o x i m a t e Lorentz invariance ~ c o m p l e m e n t s fan invariance. F o r now just accept it as a slightly obscure p a r a m e t r i z a t i o n . The implications o f Lorentz invariance for D(p~, a, ~o) are not i m m e d i a t e l y clear. To get them, note that
Current address: Theory Department, Fermi National Accelerator Laboratory, MS 106, P.O. Box 500, Batavia, IL 60510, USA. Bitnet BHILL@FNAL.
~ From here on, I will say "Lorentz invariance", when to be more accurate I should say "invariance of the inclusive distributions under Lorentz boosts along the jet axis".
(X/P~- + m 2 cosh y, p i cos tp,
0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o U a n d Physics Publishing D i v i s i o n )
157
Volume 214, number 1
PHYSICS LETTERS B
0.3C
0.80 . . . .
a
I
. . . .
I
. . . .
I
. . . .
I
'
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10 N o v e m b e r 1988
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I ' ~ e ~ r ~ - - r - ~ , ~0 40
60
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Fig. 1. TASSO data (squares) for the distribution o f hadrons in o~ at 34 GeV for seven values o f p and the predictions o f Lorentz invariance (solid line) based on the distribution for the second value o f p in graph (b). (Thus in the case o f graph ( b ) , the agreement with the theoretical curve is automatic. ) 158
Volume 214, number 1 0.30 l a
'
'
'
I
PHYSICS LETTERS B I'
'''
I''''
l0 November 1988
D(pli , t~' ) = g(P~l , a' ) D(pll , a ), g(PIJ , ol )
0.25
(3)
where 0.20
or' = t a n - ~ (pP-~tan a ) .
(4)
0.15
The ratio o f the two jacobians can be expressed fairly simply as
0,10
12
0,00
2
2
g(Pll , a' ) PIt P II + P JItan a g(Pil, a ) - P~IP~ +P] tan2a
0,05
/p2 + m 2 +p~ tan2o~ 0
20
40
60
X N/p]2 + r n 2 + p ~ tan2 a •
80
4.0
Fig. I. Continued.
O ( p . , y) D(p;,,o~)= ~ D(p±,y), where I have suppressed ~0, and where O(pi, y)
= t a n h y sec2o~
p~ sec2t~ = x/p2 sec2o~+m 2 =g(Pll, oQ.
(1)
I have already argued from approximate Lorentz invariance that D ( p . , y ) = f ( p ± ) and since p± =PH × tan o~ we have the result that
D(p, , a ) =g(PH, a)f(p~, tan a ) .
(2)
(5)
In fig. 1 I have taken the highest energy (34 GeV) data of ref. [ 1 ], used the second o f the curves as input, and graphed the predictions obtained from eqs. (3), (4) and (5) for the other six curves. The lines are the predicted distributions. The squares are the data. In the second of the graphs since the line is taken from the data itself, the nearly perfect agreement is meaningless. (The data points do not lie on the line even in this case because the curve has been slightly smoothed.) Just as the prediction of a rapidity plateau is good for a few units o f rapidity and then breaks down, the predictions for the angular distribution are good and then break down at higher energy. O f course, kinematics alone told one this breakdown must occur. A Lorentz invariant distribution o f particles would have an infinite amount o f energy. A possible way of breaking Lorentz invariance is to cut the distribution offat some value ofy. The energy flow in the resulting jet was discussed in ref. [ 2 ] with
f ( P i )oc6(p± - p ) ~2. f i s an unknown function o f a single variable. This is the necessary and sufficient condition for invariance o f the distribution under boosts along the jet axis. To see how good the prediction is I will turn eq. (2) into a statement that directly relates to the data presented in fig. 50 ofref. [ 1 ]. The distribution in PI~ and c~ is displayed by taking a fixedpl I (actually small ranges) and graphing D(p[J, c~) as a function o f a. This is done for seven different values ofp~. The content ofeqs. ( 1 ) and (2) is that I can use one of these curves for a given value ofpH as input to determine the function f, and as output predict the curves for all other values, PI~. The equation is
N o w I will discuss how Lorentz invariance is complementary to fan invariance. I will add a subscript 1,2
Since completion of this work, a preprint with a way of implementing Lorentz invariance which avoids infinite energy without using a cutoffhas appeared [ 3 ]. Briefly, instead of puning Lorentz invariance into the inclusive distributions themselves, they build it into an assumed form for the matrix element for the production of N hadrons. Then they use the usual field theoretic framework for turning matrix elements into inclusive distributions. Legendre transformations of their equations in N and W help them obtain many results analytically. Although they have compared their data with e+e - data in several ways, they have not used the distributions in two variables as a possible stringent test of their assumptions. 159
Volume 214, number 1
PHYSICS LETTERS B
W to the inclusive distributions to denote the dependence on the C O M energy. F a n invariance is the observation that D,.(pll , o~) = h ( W', Pl~ )Dw, (P11, a ) . That is, up to a n o r m a l i z a t i o n d e p e n d i n g on the total C O M energy a n d PI~, the curves in a are the same. The equality o f this agreement is remarkable. A partial explanation based on Lorentz invariance goes as follows. Lorentz invariance d e t e r m i n e s the shape o f the distribution in a for fixed W at one p~ from any other and it does so in a W-independent way. Thus, if a way o f explaining the fan invariance for just a single value ofp~ were given, Lorentz invariance would extend it to all values ofpjj. The weaknesses o f this observation are that it reduces the p r o b l e m to one previously unsolved, a n d in any case the quality o f the predictions o f fan invariance are much better than the predictions o f Lorentz invariance. The fair quality o f the predictions o f Lorentz in-
160
10 November 1988
variance partially explains the success o f so m a n y different jet models in fitting two-jet data: any m o d e l that embodies this a p p r o x i m a t e Lorentz invariance a n d fits the d a t a at one value ofp~ is guaranteed to fit the data for quite a range ofp~h. Hopefully the results here are a step toward a realistic, simple, model-ind e p e n d e n t description o f the shape o f jets. I thank H o w a r d Georgi for m a n y helpful discussions. This work was s u p p o r t e d in part by N S F G r a n t PHY87-14654. The m a n u s c r i p t and figures were c o m p l e t e d at Fermilab.
References [ 1] TASSO Collab., M. Althoffet al., Z. Phys. C 22 (1984) 307. [3] H. Georgi, Phys. Lett. B 195 (1987) 581. [3] Y. Kurihara, J. Hfifner and J. Aichelin, Hadron production in e÷e - annihilation and ~t-p collisions under the assumption of longitudinal phase space dominance, Max Planck Institute preprint MPI H-1988-V22.