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The gap survival probability and diffractive dissociation
Physics Letters B 320 (1994) 373-376 North-Holland
PHYSICS LETTERS B
The gap survival probability and diffractive dissociation R.S. F l e t c h e r Bartol Research Institute, University of Delaware, Newark. DE 19716, USA
Received 11 .hdy 1993 Editor: }1. Georgi
We study the survival probability for rapidity gaps in a model which includes diffractive dissociation. A GoodWalker model implementation of the minijet model, which consistently includes inelastic diffraction, predicts a much larger survival probability than previous calculations.
1. Introduction There has recently been much interest in the study of "rapidity gaps", regions o f rapidity in individual hadronic interactions which contain no hadrons [ 1 3]. H a r d interactions between partons mediated by electroweak bosons, or by some combination o f gluons forming a color singlet (the hard pomeron) lead to these gaps. When no color is exchanged between the scattered partons, there is a suppression o f Q C D radiation between the particles, and because there is no color separation between the beam jets, there are no central panicles from the fragmentation o f the underlying event. Estimating the rate for these event topologies depends on two calculations. First one must calculate the cross sections for weak boson or pomeron exchange [2,4]. Second one must estimate the survival probability [1,5,6], the probability that there are no secondary soft interactions which cause the underlying event to look like a typical m i n i m u m bias event, and destroy the gap. This paper is concerned with the second part o f this calculation. Calculations o f the survival probability are closely tied to the a model for the rise o f the total hadronic cross section. Most calculations of the survival probability in the literature are based on eikonal models for the total cross section, with an eikonal described in the mini-jet model, or another model for the rise o f the total cross section. These calculations share one
Elsevier Science B.V. SSDI 0370-2693 (93)E1518-3
drawback. These eikonal models give good descriptions o f the total and inelastic cross section, but do not consider single or double inelastic diffraction. Naively, it is not unreasonable that the existence o f inelastic diffraction would affect the survival probability. Typical estimates o f the survival probability at Tevatron energies, are around 10%. But between 15% and 20% o f m i n i m u m bias events have a rapidity gap [7]. These are the diffractive events [8,9]. One or both beam panicles breaks up, producing a low mass cluster o f hadrons, and leaving the central region empty o f particles. To estimate the effect o f diffractive dissociation on the survival probability, we use a G o o d - W a l k e r model for diffraction in the minijet model [10] to calculate the survival probability.
2. The Good-Walker expansion The G o o d - W a l k e r picture [10-13] is an optical modset o f el for diffraction. It supposes that the incoming proton (anti-proton) beam is a superposition o f 2 states: IP) = c111) + czl2),
(1)
c ~ + c 2 z = 1.
(2)
Then the initial 2-particle state for a colliding beam system is given by 373
Volume 320, number 3,4
IPP) = ~
c~cjli, j).
(3)
id = 1:2
In the G o o d - W a l k e r model we choose the states i, j , to be the states which diagonalize the elastic scattering amplitude
(i jlTIk l ) = d,kajt tij.
(4)
Thus, the p r o t o n - p r o t o n elastic scattering amplitude is given by
hJ
and
f~! ~
Ci Cj tij
- (tij) 2 ,
(6)
where Ci - c~. To calculate the the cross section for elastic and inelastic diffractive scattering (aa¢), we sum the cross sections for scattering into any of the complete set o f states [i j )
where ftot is given by the optical theorem and eq. (5). Ref. [ 10] describes how to implement the ideas o f the minijet model in the G o o d - W a l k e r framework. What is needed is a model for the four scattering amplitudes tgj. Each o f these amplitudes is constructed in impact parameter space, as is done for the pp amplitude in the conventional mini-jet model [14-16]. Each amplitude is writtcn in eikonal form, and is proportional to the overlap function, A (b), which is the convolution of two form factors, multiplied by a sum of two terms. A constant term describes the low energy cross sections, and a term proportional to the minijet cross section fQCD controls the rise, o f the cross sections.
tij(b) = 1 - e -z'j(b) ,
(13)
~ij = (giejfs
(14)
S-" [(ijlVlpp) ] ^ 2 = ~CgCjt~j =_ (t2j). i,j
g
2 1 dZb Z C i C j (1 - e -z~a) , d
(7)
i,j
Finally, for single diffractive and elastic scattering, (single elastic) we sum over the states o f one particle, and require the other to be a proton
+ ~i~j fQCD) Ao(b).
The formulas for the mini-jet model, with a G o o d Walker treatment o f diffraction are as follows. O'tot
O'de ,-~
13 January 1994
PHYSICS LETTERS B
eel
f__ d2b d
(15)
i,j
rz
CiCj (1 - e -z~J)
I.. i,j
1
,
(16)
f d2b Z C , Cj (1 - e-Yt~) 2 ,
aide
d
(17)
i,)'
f~ .~ ~ _ C , [(i plTIp p )] 2 i
= Y~
fse= f d 2 b ~ C i . G t~j
- ((tijl~lj.
(8)
C j ( 1 - e-X~J)
o'na = atot - fae,
'
(18) (19)
i"
These expressions for diffractive cross sections sum over all the states o f the scattered proton, so these cross sections include elastic scattering as well as the total or partial diffractive cross section. We can express the usual partial cross sections as follows f~ = f~ -fcl,
(9)
add = f de - - f c l - - 204
(10)
=fae +f el- 2f~,
(11)
fad = f t o t - - f d e , 374
(12)
[ d2bZ c cj (l,I
(20)
i,j
The important fact here is that the simple probablistic form for the non-diffractive cross section, which is the basis for the probabilistic interpretation of the mini-jet model [15], is preserved in this formalism. The factors Ci are probabilities for the incoming particles to be in the state i. The factor e x p ( - 2 X , / ) is the probability that, if they are in the state [i, j ], the incoming particles do not break up through jet production o f soft gluon exchanges, so 1 - e x p ( - 2 X i j ) is the probability that the protons do scatter and exchange
Volume 320, number 3,4
PHYSICS LETTERS B
color, causing the protons to break up. It is this probablistic picture of non-diffractive scattering that lets us calculate the survival probability. We can write the total cross section for photon exchange, or some other color singlet exchange cross section in the form e~glet = /
d2b Z CiCjAi,j ( b ) (70 = O0, i,j
d
(21 )
where we have used the fact that Aij is normalized to one, and eq. (2). Then the cross section for such a hard exchange, and that there are no soft exchanges to destroy the gap, is given by aGap
tro
100
'
.......
I
50 _
........
/
'
I
~
~(pp)
ZO *d ~a0
5
[ d:b Z C,CjA,,(b)(l,1
_
13 January 1994
2
i,j
(22) The ration of these two integrals is the survival probability. In fig. 1 we plot the total and diffractive cross section, and the survival probability according to the above model. Also plotted is a selection of data, described in ref. [9]. All model parameters arc given in table 1, column 1, of ref. [10]. The Good-Walker minijet model gives a larger value of the survival probability than suggested by most previous models. At the Tevatron this model predicts a survival probability of 44%, compared to the prediction of 8% given in ref. [5], and a value of 33% at 40 TeV, compared to the 5% predicted in ref. [ 1 ].
3. C o n c l u s i o n s
Gap production offers the opportunity to study new features of hadronic interactions, and may provide important new signals for the production of new particles at the SSC. For the production of new particles such as the Higgs boson, the low production rates expected make it difficult to see the expected signal. These rates are directly proportional to the survival probability. If the Good-Walker model's prediction is correct, it will be significantly easier to see gap signals. It is however difficult to judge the accuracy of this model. Though formally simple, a two-component model is certainly two simple to describe all the features of hadronic interactions. The real lesson is that if we expect to understand rapidity gaps in
i
|
i
" " "' .... I
' " .... I
.......
.......
:--
0.8 ~.6
-
D'40. 4 _ 0.2
-
0.1001
....... '
I
toz
I
aoa
I ao 4
i
(oev)
Fig. 1. The total, single diffractive, and double diffractive, cross sections (above), and the survival probability (below), calculated as described in the text.
high pt processes, we will also have to improve our understanding of rapidity gaps in soft interactions.
Acknowledgement
I would like to thank Tim Stelzer, and Francis Halzen for useful conversations and comments.
References
[1] J.D. Bjorken, Phys. Rev. D 47 (1993) 101. [2] H. Chehime et al., Phys. Lett. B 286 (1992) 397. [3] Y.L. Dokshitzer, V. Khoze, and T. Sj6strand, Phys. Lett. B 116 (1992). [4] V. Del Duca and W.-K. Tang, SLAC-6095. 375
Volume 320, number 3,4
PHYSICS LETTERS B
[5] R.S. Fletcher and T. Stelzer, MAD/PH/, BA-93-31, submitted to Phys. Rev. D. [6] E. Gotsman, E.M. Levin and U. Maor, Fermilab-Pub93/018-T, Taup 2030-93. [7] E-710 Collab., Amos et al., FERMILAB-Pub-92/377E. [81K. Goulianos, Phys. Rep. 110 (1983) 170. [91K. Goulianos, Nucl. Phys. B (Proc. Supl.) 12 (1989) 110. [10l R.S. Fletcher, Phys. Rev. 46 (1992) 187.
376
13 January 1994
[ 11 ] M.M. Good and W.D. Walker, Phys. Rev. 120 (1960) 1857. [12] G. Alberi et al., Phys. Lett. B 109 (1982) 481. [ 13] H. Miettinen and J. Pumplin, Phys. Rev. D 18 (1978) 1696. [14] P. L'Heureux, B. Margolis and P. Valin, Phys. Rev. D 32 (1985) 1681. [15] L. Durand and H. Pi, Phys. Rev. D 38 (1988) 78. [16] M. Block, R.S. Fletcher, F. Halzen, B. Margolis and P. Valin, Phys. Rev. D 41 (1990) 978.
PHYSICS LETTERS B
The gap survival probability and diffractive dissociation R.S. F l e t c h e r Bartol Research Institute, University of Delaware, Newark. DE 19716, USA
Received 11 .hdy 1993 Editor: }1. Georgi
We study the survival probability for rapidity gaps in a model which includes diffractive dissociation. A GoodWalker model implementation of the minijet model, which consistently includes inelastic diffraction, predicts a much larger survival probability than previous calculations.
1. Introduction There has recently been much interest in the study of "rapidity gaps", regions o f rapidity in individual hadronic interactions which contain no hadrons [ 1 3]. H a r d interactions between partons mediated by electroweak bosons, or by some combination o f gluons forming a color singlet (the hard pomeron) lead to these gaps. When no color is exchanged between the scattered partons, there is a suppression o f Q C D radiation between the particles, and because there is no color separation between the beam jets, there are no central panicles from the fragmentation o f the underlying event. Estimating the rate for these event topologies depends on two calculations. First one must calculate the cross sections for weak boson or pomeron exchange [2,4]. Second one must estimate the survival probability [1,5,6], the probability that there are no secondary soft interactions which cause the underlying event to look like a typical m i n i m u m bias event, and destroy the gap. This paper is concerned with the second part o f this calculation. Calculations o f the survival probability are closely tied to the a model for the rise o f the total hadronic cross section. Most calculations of the survival probability in the literature are based on eikonal models for the total cross section, with an eikonal described in the mini-jet model, or another model for the rise o f the total cross section. These calculations share one
Elsevier Science B.V. SSDI 0370-2693 (93)E1518-3
drawback. These eikonal models give good descriptions o f the total and inelastic cross section, but do not consider single or double inelastic diffraction. Naively, it is not unreasonable that the existence o f inelastic diffraction would affect the survival probability. Typical estimates o f the survival probability at Tevatron energies, are around 10%. But between 15% and 20% o f m i n i m u m bias events have a rapidity gap [7]. These are the diffractive events [8,9]. One or both beam panicles breaks up, producing a low mass cluster o f hadrons, and leaving the central region empty o f particles. To estimate the effect o f diffractive dissociation on the survival probability, we use a G o o d - W a l k e r model for diffraction in the minijet model [10] to calculate the survival probability.
2. The Good-Walker expansion The G o o d - W a l k e r picture [10-13] is an optical modset o f el for diffraction. It supposes that the incoming proton (anti-proton) beam is a superposition o f 2 states: IP) = c111) + czl2),
(1)
c ~ + c 2 z = 1.
(2)
Then the initial 2-particle state for a colliding beam system is given by 373
Volume 320, number 3,4
IPP) = ~
c~cjli, j).
(3)
id = 1:2
In the G o o d - W a l k e r model we choose the states i, j , to be the states which diagonalize the elastic scattering amplitude
(i jlTIk l ) = d,kajt tij.
(4)
Thus, the p r o t o n - p r o t o n elastic scattering amplitude is given by
hJ
and
f~! ~
Ci Cj tij
- (tij) 2 ,
(6)
where Ci - c~. To calculate the the cross section for elastic and inelastic diffractive scattering (aa¢), we sum the cross sections for scattering into any of the complete set o f states [i j )
where ftot is given by the optical theorem and eq. (5). Ref. [ 10] describes how to implement the ideas o f the minijet model in the G o o d - W a l k e r framework. What is needed is a model for the four scattering amplitudes tgj. Each o f these amplitudes is constructed in impact parameter space, as is done for the pp amplitude in the conventional mini-jet model [14-16]. Each amplitude is writtcn in eikonal form, and is proportional to the overlap function, A (b), which is the convolution of two form factors, multiplied by a sum of two terms. A constant term describes the low energy cross sections, and a term proportional to the minijet cross section fQCD controls the rise, o f the cross sections.
tij(b) = 1 - e -z'j(b) ,
(13)
~ij = (giejfs
(14)
S-" [(ijlVlpp) ] ^ 2 = ~CgCjt~j =_ (t2j). i,j
g
2 1 dZb Z C i C j (1 - e -z~a) , d
(7)
i,j
Finally, for single diffractive and elastic scattering, (single elastic) we sum over the states o f one particle, and require the other to be a proton
+ ~i~j fQCD) Ao(b).
The formulas for the mini-jet model, with a G o o d Walker treatment o f diffraction are as follows. O'tot
O'de ,-~
13 January 1994
PHYSICS LETTERS B
eel
f__ d2b d
(15)
i,j
rz
CiCj (1 - e -z~J)
I.. i,j
1
,
(16)
f d2b Z C , Cj (1 - e-Yt~) 2 ,
aide
d
(17)
i,)'
f~ .~ ~ _ C , [(i plTIp p )] 2 i
= Y~
fse= f d 2 b ~ C i . G t~j
- ((tijl~lj.
(8)
C j ( 1 - e-X~J)
o'na = atot - fae,
'
(18) (19)
i"
These expressions for diffractive cross sections sum over all the states o f the scattered proton, so these cross sections include elastic scattering as well as the total or partial diffractive cross section. We can express the usual partial cross sections as follows f~ = f~ -fcl,
(9)
add = f de - - f c l - - 204
(10)
=fae +f el- 2f~,
(11)
fad = f t o t - - f d e , 374
(12)
[ d2bZ c cj (l,I
(20)
i,j
The important fact here is that the simple probablistic form for the non-diffractive cross section, which is the basis for the probabilistic interpretation of the mini-jet model [15], is preserved in this formalism. The factors Ci are probabilities for the incoming particles to be in the state i. The factor e x p ( - 2 X , / ) is the probability that, if they are in the state [i, j ], the incoming particles do not break up through jet production o f soft gluon exchanges, so 1 - e x p ( - 2 X i j ) is the probability that the protons do scatter and exchange
Volume 320, number 3,4
PHYSICS LETTERS B
color, causing the protons to break up. It is this probablistic picture of non-diffractive scattering that lets us calculate the survival probability. We can write the total cross section for photon exchange, or some other color singlet exchange cross section in the form e~glet = /
d2b Z CiCjAi,j ( b ) (70 = O0, i,j
d
(21 )
where we have used the fact that Aij is normalized to one, and eq. (2). Then the cross section for such a hard exchange, and that there are no soft exchanges to destroy the gap, is given by aGap
tro
100
'
.......
I
50 _
........
/
'
I
~
~(pp)
ZO *d ~a0
5
[ d:b Z C,CjA,,(b)(l,1
_
13 January 1994
2
i,j
(22) The ration of these two integrals is the survival probability. In fig. 1 we plot the total and diffractive cross section, and the survival probability according to the above model. Also plotted is a selection of data, described in ref. [9]. All model parameters arc given in table 1, column 1, of ref. [10]. The Good-Walker minijet model gives a larger value of the survival probability than suggested by most previous models. At the Tevatron this model predicts a survival probability of 44%, compared to the prediction of 8% given in ref. [5], and a value of 33% at 40 TeV, compared to the 5% predicted in ref. [ 1 ].
3. C o n c l u s i o n s
Gap production offers the opportunity to study new features of hadronic interactions, and may provide important new signals for the production of new particles at the SSC. For the production of new particles such as the Higgs boson, the low production rates expected make it difficult to see the expected signal. These rates are directly proportional to the survival probability. If the Good-Walker model's prediction is correct, it will be significantly easier to see gap signals. It is however difficult to judge the accuracy of this model. Though formally simple, a two-component model is certainly two simple to describe all the features of hadronic interactions. The real lesson is that if we expect to understand rapidity gaps in
i
|
i
" " "' .... I
' " .... I
.......
.......
:--
0.8 ~.6
-
D'40. 4 _ 0.2
-
0.1001
....... '
I
toz
I
aoa
I ao 4
i
(oev)
Fig. 1. The total, single diffractive, and double diffractive, cross sections (above), and the survival probability (below), calculated as described in the text.
high pt processes, we will also have to improve our understanding of rapidity gaps in soft interactions.
Acknowledgement
I would like to thank Tim Stelzer, and Francis Halzen for useful conversations and comments.
References
[1] J.D. Bjorken, Phys. Rev. D 47 (1993) 101. [2] H. Chehime et al., Phys. Lett. B 286 (1992) 397. [3] Y.L. Dokshitzer, V. Khoze, and T. Sj6strand, Phys. Lett. B 116 (1992). [4] V. Del Duca and W.-K. Tang, SLAC-6095. 375
Volume 320, number 3,4
PHYSICS LETTERS B
[5] R.S. Fletcher and T. Stelzer, MAD/PH/, BA-93-31, submitted to Phys. Rev. D. [6] E. Gotsman, E.M. Levin and U. Maor, Fermilab-Pub93/018-T, Taup 2030-93. [7] E-710 Collab., Amos et al., FERMILAB-Pub-92/377E. [81K. Goulianos, Phys. Rep. 110 (1983) 170. [91K. Goulianos, Nucl. Phys. B (Proc. Supl.) 12 (1989) 110. [10l R.S. Fletcher, Phys. Rev. 46 (1992) 187.
376
13 January 1994
[ 11 ] M.M. Good and W.D. Walker, Phys. Rev. 120 (1960) 1857. [12] G. Alberi et al., Phys. Lett. B 109 (1982) 481. [ 13] H. Miettinen and J. Pumplin, Phys. Rev. D 18 (1978) 1696. [14] P. L'Heureux, B. Margolis and P. Valin, Phys. Rev. D 32 (1985) 1681. [15] L. Durand and H. Pi, Phys. Rev. D 38 (1988) 78. [16] M. Block, R.S. Fletcher, F. Halzen, B. Margolis and P. Valin, Phys. Rev. D 41 (1990) 978.