- Email: [email protected]
Color evaporation induced rapidity gaps
.
1 P&EDIN& SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 99 (2001) 257-264
ELSEVIER
Color Evaporation 0. J. P. Ebolia*,
Induced Rapidity
www.elsevier.nl/locate/npe
Gaps
E. M. Gregores at, and F. Halzenb$
“Institute de Fisica Teorica, Universidade Estadual Paulista Rua Pamplona 145, 01405-900, S&o Paul0 - SP, Brazil bDepartment of Physics, University Madison, WI 53706, USA
of Wisconsin
We show that soft color rearrangement of final states can account for the appearance of rapidity gaps between jets. In the color evaporation model the probability to form a gap is simply determined by the color multiplicity of the final state. This model has no free parameters and reproduces all data obtained by the ZEUS, Hl, DO, and CDF collaborations.
1. Introduction We show that the appearance of rapidity gaps between jets, observed at the HERA and Tevatron colliders, can be explained by supplementing the string model with the idea of color evaporation, or soft color. The inclusion of soft color interactions between the dynamical partons, which rearranges the string structure of the interaction, leads to a parameter-free calculation of the formation rate of rapidity gaps. The idea is extremely simple. Like in the string model, the dynamical partons are those producing the hard interactions, and the left-over spectators. A rapidity gap occurs whenever final state partons form color singlet clusters separated in rapidity. As the par-tons propagate within the hadr&ic medium, they exchange soft gluons which modify the string configuration. These large-distance fluctuations are probably complex enough for the occupation of different color states to approximately respect statistical counting. The probability to form a rapidity gap is then determined by the color multiplicity of the final states formed by the dynamical partons, and nothing else. All data obtained by ZEUS [l], Hl [2], DO [3], and CDF [4] collaborations are reproduced when this color structure *Research supported by CNPq, FAPESP and PRONEX. ‘Research supported by FAPESP. TResearch supported by Wisconsin Alumni Research Foundation and DOE grant DE-FG02-95ER40896.
of the interactions is superimposed on the usual perturbative QCD calculation for the production of the hard jets. Rapidity gaps refer to intervals in pseudorapidity devoid of hadronic activity. The most simple example is the region between the final state protons, or its excited states, in pp elastic scattering and diffractive dissociation. Such processes were first observed in the late 50’s in cosmic rays experiments [5] and have been extensively studied at accelerators [6]. Attempts to describe the formation of rapidity gaps have concentrated on Regge theory and the pomeron [7,8], and on its possible QCD incarnation in the form of a colorless 2-gluon state [9,10]. After the observation of rapidity gaps in deep inelastic scattering (DIS), it was suggested [ll] that events with and without rapidity gaps are identical from a partonic point of view, except for soft color interactions that, occasionally, lead to a region devoid of color between final state partons. We pointed out [12] that this soft color mechanism is identical to the color evaporation mechanism [13] for computing the production rates of heavy quark pairs produced in color singlet onium states, like J/$. Moreover, we also suggested that the soft color model could provide a description for the production of rapidity gaps in hadronic collisions [12]. Color evaporation assumes that quarkonium formation is a two-step process: the
0920-5632/01/$ - see frontmatter0 2001 ElsevierScience B.V: All rights reserved. PII SO920-5632(01)01344-5
OJP kboli et al. /Nuclear Physics B (Proc. Suppl.) 99 (2001) 257-264
258
Table 1 Color multiplicities and gap probabilities FN for the reaction p y + jl jz X Y. Subprocess
Q”Q + QQ Q’Q + QQ
5 3 3
j2 3 3
X 383 3838383 3@3@3@3
Y 3 3 3
l/9 V 771 l/9 V 71 l/9 V VI
FN
3
I
Q”& -+ Qo 0”s -+ So &“Q + QQ
3
B
3
l/9 for 81 >
82
3
3
3@3@3@3
3
8 8
3838383 3@3 3@3@3@3
3 3 3
~2
Q”& + GG Qs& + GG Q”Q + GG
3 8 8
l/9 for ~1 > l/9 for 711>
8 3 3 z 8 8 3 8
8 8 8 8 3 3 I 8
3@[email protected]@3 3@3 3@3@3@3 3@3@3@3 38383 38383 38383 3@3@3
&“&
+
@J
QvG + QG QsG + QG QSG + QG
GQ+GQ
GQ+GQ GG-+QQ GG+GG
B
383
pair of heavy quarks is formed at the perturbative level with scale kfQ, and bound into quarkonium at scale AQco. Heavy quark pairs of any color below open flavor threshold can form a colorless asymptotic quarkonium state provided they end up in a color singlet configuration after the inevitable exchange of soft gluons with the final state spectator hadronic system. The final color state of the quark pairs is not dictated by the hard QCD process, but by the fate of their color between the time of formation and emergence as an asymptotic state. The success of the color evaporation model to explain the data on quarkonium production is unquestionable [14]. We show here that the straightforward application of the color evaporation approach to the string picture of QCD readily explains the formation of rapidity gaps between jets at the Tevatron and HERA colliders. 2. Color
Counting
Rules
In the color evaporation scheme for calculating quarkonium production, it is assumed that all color configurations of the quark pair occur with equal probability. For instance, the probability that a QQ pair ends up in a color singlet state is l/ (1 + 8) because all states in 3 6~)3 = 8 @ 1 are
~2
0 0
0 1127 for q71> 772 1172 for 91 > 712 1172 for ~1 > 772 l/9 for 171> 172 l/9 for 171> 772 0 1172 V ~1
3 383 3633 3@3 3 3 383 3@3
equally probable. We propose that the same color counting applies to the final state partons in high ET jet production. In complete analogy with quarkonium, the production of high energy jets is a two-step process where a pair of high ET partons is perturbatively produced at a scale ET, and hadronize into jets at a scale of order AQcD by stretching color strings between the partons and spectators. The strings subsequently hadronize. Rapidity gaps appear when a cluster of dynamical partons, i.e. interacting partons or spectators, form a color singlet. As before, the probability for forming a color singlet cluster is inversely proportional to its color multiplicity. In order to better understand the soft color idea let us consider the formation of rapidity gaps between two jets in opposite hemispheres, which happens when the interacting parton forming the jet and the accompanying remnant system form a color singlet. This may occur for more than one subprocess N and, therefore, the gap fraction is F gap
=
-&c
Fdm
,
(1)
N where FN is the probability for gap formation in the Nth subprocess, duN is the corresponding differential par-ton-parton cross section, and
O.A? kboli et al. /Nuclear
10
__..___.__...... ---.-... :
i+
‘i
I
-
Total
---..
Color Evaporation
...‘.....’Background ’
2
I , I
2.25
2.5
I
2.75
‘,.
Yh,i
: .___.._:)4-......‘_ .._._......_...... - ._.... %... ._._..__.___._......- --.-._-.-_ ‘.:,. .1. Total .‘\. ..-.. Color Evoporatkm “\i\< ...‘...
hi,,
I . I
3
3.25
x;,
IL.
3.5
259
Physics B (Proc. Suppl,) 99 (2001) 257-264
I
3.75
4
’
2
I.
2.25
I.
2.5
Background
I.
2.75
I.
3
“‘..
I.
3.25
I.
3.5
‘\:,.
I “Q.: .I_
3.75
4
b
A?
Figure 1. Fraction of rapidity gap events as a function of the gap size. da (= EN daN) is the total cross section. In our model, the probabilities FN are determined by the color multiplicity of the state and spatial distribution of partons while duN is evaluated using perturbative &CD. The soft color procedure is obvious in a specific example: let us calculate the gap formation probability for the subprocesses pg
+QVQV+QQXY,
where Q” stands for u or d valence quark, and X (Y) is the diquark remnant of the proton (antiproton). The final state is composed of the X (3 @ 3) color spectator system with rapidity 77~ = +oo, the Y (3 @ 3) color spectator system with ny = -co, one 3 parton jr, and one 3 parton js. It is the basic assumption of the soft color scheme that by the time these systems hadronize, any color state is equally likely. One can form a color singlet final state between X and ji since 3 @ 3 @ 3 = 10 @ 8 @ 8 g, 1, with probability l/27. Because of overall color conservation, once the system X @jr is in a color singlet, so is the system Y @ j2. On the other hand, it is not possible to form a color singlet system with ji and Y. Moreover, to form a rapidity gap these systems (ji @ X and jz @ Y) must not overlap in rapidity space. Since the experimental data consists of events where the two jets are in opposite hemispheres, the only additional requirements are ji to be in the same hemisphere as X, i.e. ~1 > 0, and jz to be in the opposite hemisphere (vi +772 < 0). In this configuration, the color strings linking the remnant and the parton
in the same hemisphere will not hadronize in the region between the two jets. We have thus produced two jets separated by a rapidity gap using the color counting rules which form the basis of the color evaporation scheme for calculating quarkonium production. 3. Rapidity
Gaps
at HERA
The partonic subprocess for dijet photoproduction is related to the ep cross section by ~‘ep--tj~jzx~
(~1
=
Irma=
LTa’
F:(Y,
lmin mln x ~py-+jljzx~ W>
&
Q2) dQ2(2)
where W is the center-of-mass energy of the m system, y = W2/s is the fraction of the electron momentum carried by the photon, and Q2 is the photon virtuality. Q2 ranges from Qkin = M:y”/(l - y) to Qs,, which depends on the kinematic coverage of the experimental apparatus. The distribution function of photons in the electron is F,7(y,Q2)
= $&
[I + (I - 2/)2- y]
~(3)
where Me is the electron mass and o is the finestructure constant. The m cross section is related to the partonparton cross section by
0.M
260
Table 2 Color multiplicities Subprocess
Q”Q” + QQ QsQs + QQ
kboli
et al. /Nuclear
and gap probabilities jl
j2
3
3
3
3
PhysicsB (Proc. Suppl.)
FN for the reaction
99 (2001) 257-264
p~3 +
X Y.
jlj2 Y
X 3@3 3@3@3@99
FN
8@3@3@3
1127 v
8@1@3@3
l/81
Q”Q”
-+ &&
I
3
3@3@3@3
B@L
~“~”
+
QQ
3
I
3@3@3@3
S@f@3@3
Q”&” Q”@ QsQv 9’9” @QS
+ -+ -+ + +
Ofi? Q& 00 Q& OQ
3 3
I 3
Q”Q”
1127 V Vl l/81 V 711 l/27 for 71 >
f@l
3@3 383
B@SL%J@3
l/27
for for for for
3
z
3@3@3@3
B@Z
l/27
3
3@3@3L%$
8@3@8@3
l/81
8
3 3
3@3@3@3
S@S@B@J
l/81
-+ GG
8
8
383
S@S
0
Q”a”
-+ GG
8
8
3@3
8@3@3@3
0
Q”Q”
-+ GG
8
8
3@3@3@3
S@$
0
Qsgs
+
GG
8
8
3@3@3@3
Z@S@$@3
@Q”
+
GG
8
8
3@3@3@3
2@8@8@8
QG
3
8
QG QG GQ GQ GQ
3
8
3@J3@3@3
B@I3C_aS
l/108
z
8
3@3@3@3
8@3@3
l/108
8
I
3@3@3
L@I
8
f
8@,fCsS@3
l/108
8
3
3@3@3 38383
S@IzI@pf@Z
l/108
3
z
3@3@3
8@3@8
8
8
3@3@3
8@8@8
QvG
+
QsG -+ QSG + GQV -+ GQs -i GQs + GG+QQ GG-+GG
where F~(z~) (F;(q)) is the distribution function for parton a (b) in the proton (photon) and fi = m W is the parton-parton center-ofFor direct m reactions (b q y), mass energy. F;(Q) = 6(1 - 26). The hadronic system X (Y) is the proton (photon) remnant, and ji(s) is the jet which is initiated by the parton ~~(2). The proton is assumed to travel in the positive rapidity direction. 3.1. ZEUS Results The ZEUS collaboration [l] has measured the formation of rapidity gaps between jets produced in ep collisions with 0.2 < y < 0.85 and photon virtuality Q2 < 4 GeV2. Jets were defined by a cone radius of 1.0 in the (~,4) plane, where v is the pseudorapidity and 4 is the azimuthal angle. In the event selection, jets were required to have ET > 6 GeV, to not overlap in rapidity (A7 = 1771- 7721 > 2), to have a mean position ]q] < 0.75, and to be in the region n < 2.5. The cross sections were measured in A77 bins in the range 2 2 A17 5 4.
711> 171> ~1 > 91 >
0 0 0 0 0
0 0 l/27
8@$@zi
3@3
q1
V 71
l/27
for qi > 0 for ~1 > 0 for 71 > 0 for qi > 0 for ~1 > 0 for ~1 > 0 0
l/l08
V ~1
For the above event selection, we evaluated the dijet differential cross section dajet”/dAq, which is the sum of the direct (dud+) and the resolved photon (doyes) cross sections. We used the GRVLO [15] distribution function for the proton, and the GRV [16] for the photon. We fixed the renormalization and factorization scales at PR = PF = ET/~, and calculated the strong coupling constant for four active flavors with AQ~D = 350 MeV. Now we turn to dijet events showing a rapidity gap. We evaluated the differential cross section da@p/dAv which has two sources of gap events: color evaporation gaps (dc$z$) and background gaps (da:?). In our model, the gap cross section is the weighted sum over resolved events d,m,WP cem
-c _
FN
do:, >
(5)
N
with the gap probability FN for the different processes given in Table 1. Background gaps are formed when the region of rapidity between the jets is devoid of hadrons because of statistical
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et al. /Nuclear
Physics B (Proc.
Suppl.)
99 (2001)
261
257-264
I.‘,I...(.“,‘.,(..‘I”.I...I.
4s = 1600 GeV
‘20
25
30
40
35
45
50
CDF
60 55 ET (GeV)
4s = 630 GeV
ai
1
i ,,..,....,....,....,,,,,
0 10
15
20
2.5
z
25
I _ I i z I 0 ~.:,,,..,.,.,...,,..,.t.,.f.,.,.,,1.8 2 2.2 2.4 2.6 2.6
30E, (Gel’;
Figure 2. Dependence of the gap frequency the jet transverse energy.
9
on
(6)
where b is a constant, fitted as b = 2.9. This value of b agrees with b = 2.7 f 0.3 found by ZEUS collaboration, when they approximated the nonbackground gap fraction by a constant. The background gap cross section is then written as dc$y
= Fbg(Aq) (dc+
- da::;)
.
3
3.2
3.4
0%-)7W2
fluctuation of ordinary soft particle production. Their rate should fall exponentially as the rapidity separation An between the jets increases [l]. We parametrized the background gap probability as Fbg(Al;l) = e b(s-Ag)
-
(7)
The gap frequency Fgap(Aq) = dogaf’/dojets is shown in Fig. la, where we show the contributions of the color evaporation mechanism and the background. 3.2. Hl Results We also performed an identical analysis for the data obtained by the Hl collaboration [2]. They used the same cone size for the jet definition (AR = l), and collected events produced in proton-photon reactions with center-of-mass energy in the range 158 < W < 247 GeV and with photon virtuality Q2 < 0.01 GeV2. They also imposed cuts on the jets: -2.82 < q < 2.35 and ET > 4.5 GeV. Our results are compared with the preliminary experimental data in Fig. lb
Figure 3. The same as Fig. 2 for half the gap size.
where we used b = 2.3 to describe the background in the Hl kinematic range. As before, color evaporation induces gap formation with a rate compatible with observation. 4. Rapidity Gaps at Tevatron For dijet production in pp collisions, we denoted by X (Y) the proton (antiproton) remnant, and by jrt2) the parton giving rise to a jet. The proton is assumed to travel in the positive rapidity direction. The dijet production cross section is related to the parton-parton one via ~P‘PP-+A~~XY (3)
=
c //- F;(G)
F;(Q)
a,b X flab+plpz
where s (S = &&,S) of-mass
(a> dx,
is the (subprocess)
energy squared
and F$$
dxb
,
(8)
center-
is the distribu-
tion function for the parton a (b). We evaluated the dijet cross sections using MRS-J distribution functions [17] with renormalization and factorization scales PR = ,LF = A. The color evaporation model prediction for the gap production rates in pp collisions is analogous to the one in m interactions, with the obvious replacement of the photon by the antiproton, represented as a 3 @ 3 @ 3 system. The color subprocesses and their respective gap formation probabilities are listed in Table 2. Both experimental collaborations presented
OJP kboli et al. /Nuclear Physics B (Proc. Suppl.) 99 (2001) 257-264
262
‘.
t4F
v’s = ” 18OOGeV ‘.
”
0: I,
I.
0.2
0.3
I.
I
0.4
0.5
“4 COF
*
2
I.
0.6
0
15GeV<
E,<25GeV
T
0.7 &,X2
10 4s = 630 GeV
e
5*
0-l 0.2
I T
+ I
0.3
1
+ .
+
I
‘1
I
I
I
0.4
0.5
0.6
‘I.
. 0.7
%,X2
Figure 4. The same as Fig. 2 for the Bjorken-z of each jet.
their data with the background subtracted. The CDF collaboration measured the appearance of rapidity gaps at two different p~? center-of-mass For the data taken at fi = 1800 energies. GeV, they required that both jets to have ET > 20 GeV, and to be produced in opposite sides (~1 . 772< 0) within the region 1.8 < 171 < 3.5. For the lower energy data, fi = 630 GeV, they required both jets to have ET > 8 GeV, and to be produced in opposite sides within the region 171 > 1.8. Since the experimental distributions are normalized to unity, on average, we do not need to introduce an ad-hoc gap survival probability. Therefore, our predictions do not exhibit any free parameter to be adjusted, leading to a important test of the color evaporation mechanism. In Figs. 2, 3, and 4 we compare our predictions with the experimental observations of the gap fraction as a function of the jets transverse energy, their separation in rapidity, and the Bjorken-z of the colliding partons, respectively. As we can see, the overall performance of the color evaporation model is good since it describes correctly the shape of almost all distributions. This is an impressive result since the model has no free parameters to be adjusted. The DO collaboration has made similar observations at fi = 1800 GeV. They required that both jets to have ET > 15 GeV, to be produced in opposite sides (71 . 72 < 0) within
0~“““‘““‘“““““““‘“” 20 30 40
50
60
70 80 G (GeV)
Figure 5. Gap fraction as a function of the jet ET as measured by the DO collaboration at fi = 1800 GeV.
the region 1.9 < 1~1< 4.1, and to be separated by I&[ > 4.0. In Fig. 5 our results are compared with experimental observations of the dependence of the gap frequency on jet transverse energy, where we used a gap survival probability S, = 30% to reproduce the absolute normalization. This is consistent with qualitative theoretical estimates; see discussion below. As we can see, the fraction of gap events increases with the transverse energy of the jets. This is expected once. the dominant process for the rapidity gap formation is quark-quark fusion, which becomes more important at larger ET [18]. Apart from the lowest transverse energy bin, data and theory are in good agreement. In Fig. 6 we compare our prediction for the dependence of the gap frequency with the separation between the jets. Agreement is satisfactory although the absolute value of our predictions for low transverse energy is somewhat higher than data as shown in Fig. 5. Finally, in Fig. 7 we show our results for the mean value of the Bjorken-z of the events, where all correlations between the jet transverse energy and rapidity have been included. Again, the agreement between theory and data is satisfactory except for the low transverse energy bins. 4.1. Survival Probability at Tevatron We estimated the survival probability of rapidity gaps formed at pji collisions comparing our pre-
O.Jl? holi
rs E
3 :““,““,”
et al. /Nuclear
..(..I.,....,....,..,.,,...,...
.-
: 0 -
E,>JOGeV
0 _.._.
15GeV
263
Physics B (Proc. Suppl.) 99 (2001) 257-264
09
:
ij t 2.5
Qa 0 Aq bins (15GeV < E, C 25GeV) W Aq bins (E, > 3OGeV)
2
2:
1.5 :
T .
i....I....l....I....I....I....I....I....I.. ’ 4 4.25 4.5 4.75 5 5.25 5.5 5.75 Figure 6. Gap fraction rapidity separation.
as a function
6
ii5
of the jets
’
t.
1..
0.1
1..
0.15
. .I..
0.2
. . I..
0.25
Figure 7. Gap fraction age Bjorken-z.
. .I.,
0.3
. .fi..
0.35
,I.
0.4 h+xJ/2
as a function
*,
.
Ii
0,45
of the aver-
dictions with the values of gap fraction actually observed. Assuming that the survival probability varies only with the collision center-of-mass energy, and not with the jets transverse energy, we evaluated the average survival probability
tained R = 2.0 f 0.4 for the same arrangement.
F&Y sp=Fgap. cem
In summary, the occurrence of rapidity gaps between hard jets can be understood by simply applying the soft color, or color evaporation, scheme for calculating quarkonium production, to the conventional perturbative QCD calculation of the production of hard jets. The agreement between data and this model is impressive.
In order to extract 3, we combined the DO and CDF available data at each center-of-mass energy; 630 and 1800 GeV. We found ,.$(lSOO) = 34.4 f 3.3% and &(630) = 65.4 f 12% , a value compatible with the calculation of Ref. [19] based on the Regge model, which yields S,(1800) = 32.6%. For individual contributions and further Moreover, we have that details see Table 3. $,(630)/,!$(1800) = 1.9f0.4, which is compatible with the theoretical expectation 2.2f0.2 obtained in Ref. [20]. Using the extracted values of the survival probability, we contrasted the color evaporation model predictions for the gap fraction corrected by 3, (Fgap daCOT = F,E$ x 3,) with the experimental ta in Table 3. We can also compare the ratio R = F,g,“,p(63O)/F~~~(1800) with the experimental result. DO has measured this fraction for jets with ET > 12 GeV for both energies, and they found R = 3.4 f 1.2; we predict R = 2.5 f 0.5. On the other hand, CDF measured this ratio using different values for EFin at 630 GeV and 1800 GeV; they obtained R = 2.0 f 0.9 while we ob-
kinematical
5. Conclusion
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0.H
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Physics B (Proc. Suppl.)
99 (2001) 257-264
Table 3 Gap frequencies and survival probabilities. Theoretical uncertainties are not included. fi (GeV) 1800 1800 1800 630 630
5. 6. 7. 8. 9.
10.
11
12.
13.
Eman (GeV) 3: 20 12 12 8
F!:$ (%) 2.91 2.49 2.24 2.97 2.55
Fj!$’ (%) 0.94 f 0.13 (DS) 1.13 f 0.16 (CDF) 0.54 f 0.17 (DS) 1.85 f 0.38 (DS) 2.3 f 1.0 (CDF)
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14.
15. 16. 17. 18. 19. 20.
S, (%> 32.3 f 4.5 45.4 f 6.4 24.1 f 7.6 62.3 f 12.8 90.2 f 39.2
Fj;$ 1.00 0.85 0.77 1.94 1.67
f f f f f
x S, (%) 0.10 0.08 0.07 0.36 0.31
1344 (1978); M. Gluck, J. Owens, and E. Reya, Phys. Rev. D17, 2324 (1978). R. Gavai et al., Int. J. Mod. Phys. AlO, 3043 (1995); R. Vogt and G. Schuler, Phys. Lett,. B387, 181 (1996); J. Amundson, 0. J. P. Eboli, E. M. Gregores, and F. Halzen, Phys. Lett. B390, 323 (1997); A. Edin, G. Ingelman, and J. Raths,man, Phys. Rev. D56, 7317 (1997); 0. J. P. Eboli, E. M. Gregores, and F. Halzen, Phys. Lett. B395, 113 (1997); ibidem 451, 241 (1999); idem, Phys. Rev. D 60, 117501 (1999). M. Gluck, E. Reya, and A. Vogt, Z. Phys. C67, 433 (1995). M. Gluck, E. Reya, and A. Vogt, Phys. Rev. D46, 1973 (1992). A. Martin, R. Roberts, and W. Stirling, Phys. Lett. B387, 419 (1996). 0. J. P. Eboli, E. M. Gregores, and F. Halzen, Phys. Rev. D58, 114005 (1998). E. Gotsman, E. Levin, and U. Maor, Phys. Lett. B309, 199 (1993). E. Gotsman, E. Levin, and U. Maor, Phys. Lett. B438, 229 (1998).
1 P&EDIN& SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 99 (2001) 257-264
ELSEVIER
Color Evaporation 0. J. P. Ebolia*,
Induced Rapidity
www.elsevier.nl/locate/npe
Gaps
E. M. Gregores at, and F. Halzenb$
“Institute de Fisica Teorica, Universidade Estadual Paulista Rua Pamplona 145, 01405-900, S&o Paul0 - SP, Brazil bDepartment of Physics, University Madison, WI 53706, USA
of Wisconsin
We show that soft color rearrangement of final states can account for the appearance of rapidity gaps between jets. In the color evaporation model the probability to form a gap is simply determined by the color multiplicity of the final state. This model has no free parameters and reproduces all data obtained by the ZEUS, Hl, DO, and CDF collaborations.
1. Introduction We show that the appearance of rapidity gaps between jets, observed at the HERA and Tevatron colliders, can be explained by supplementing the string model with the idea of color evaporation, or soft color. The inclusion of soft color interactions between the dynamical partons, which rearranges the string structure of the interaction, leads to a parameter-free calculation of the formation rate of rapidity gaps. The idea is extremely simple. Like in the string model, the dynamical partons are those producing the hard interactions, and the left-over spectators. A rapidity gap occurs whenever final state partons form color singlet clusters separated in rapidity. As the par-tons propagate within the hadr&ic medium, they exchange soft gluons which modify the string configuration. These large-distance fluctuations are probably complex enough for the occupation of different color states to approximately respect statistical counting. The probability to form a rapidity gap is then determined by the color multiplicity of the final states formed by the dynamical partons, and nothing else. All data obtained by ZEUS [l], Hl [2], DO [3], and CDF [4] collaborations are reproduced when this color structure *Research supported by CNPq, FAPESP and PRONEX. ‘Research supported by FAPESP. TResearch supported by Wisconsin Alumni Research Foundation and DOE grant DE-FG02-95ER40896.
of the interactions is superimposed on the usual perturbative QCD calculation for the production of the hard jets. Rapidity gaps refer to intervals in pseudorapidity devoid of hadronic activity. The most simple example is the region between the final state protons, or its excited states, in pp elastic scattering and diffractive dissociation. Such processes were first observed in the late 50’s in cosmic rays experiments [5] and have been extensively studied at accelerators [6]. Attempts to describe the formation of rapidity gaps have concentrated on Regge theory and the pomeron [7,8], and on its possible QCD incarnation in the form of a colorless 2-gluon state [9,10]. After the observation of rapidity gaps in deep inelastic scattering (DIS), it was suggested [ll] that events with and without rapidity gaps are identical from a partonic point of view, except for soft color interactions that, occasionally, lead to a region devoid of color between final state partons. We pointed out [12] that this soft color mechanism is identical to the color evaporation mechanism [13] for computing the production rates of heavy quark pairs produced in color singlet onium states, like J/$. Moreover, we also suggested that the soft color model could provide a description for the production of rapidity gaps in hadronic collisions [12]. Color evaporation assumes that quarkonium formation is a two-step process: the
0920-5632/01/$ - see frontmatter0 2001 ElsevierScience B.V: All rights reserved. PII SO920-5632(01)01344-5
OJP kboli et al. /Nuclear Physics B (Proc. Suppl.) 99 (2001) 257-264
258
Table 1 Color multiplicities and gap probabilities FN for the reaction p y + jl jz X Y. Subprocess
Q”Q + QQ Q’Q + QQ
5 3 3
j2 3 3
X 383 3838383 3@3@3@3
Y 3 3 3
l/9 V 771 l/9 V 71 l/9 V VI
FN
3
I
Q”& -+ Qo 0”s -+ So &“Q + QQ
3
B
3
l/9 for 81 >
82
3
3
3@3@3@3
3
8 8
3838383 3@3 3@3@3@3
3 3 3
~2
Q”& + GG Qs& + GG Q”Q + GG
3 8 8
l/9 for ~1 > l/9 for 711>
8 3 3 z 8 8 3 8
8 8 8 8 3 3 I 8
3@[email protected]@3 3@3 3@3@3@3 3@3@3@3 38383 38383 38383 3@3@3
&“&
+
@J
QvG + QG QsG + QG QSG + QG
GQ+GQ
GQ+GQ GG-+QQ GG+GG
B
383
pair of heavy quarks is formed at the perturbative level with scale kfQ, and bound into quarkonium at scale AQco. Heavy quark pairs of any color below open flavor threshold can form a colorless asymptotic quarkonium state provided they end up in a color singlet configuration after the inevitable exchange of soft gluons with the final state spectator hadronic system. The final color state of the quark pairs is not dictated by the hard QCD process, but by the fate of their color between the time of formation and emergence as an asymptotic state. The success of the color evaporation model to explain the data on quarkonium production is unquestionable [14]. We show here that the straightforward application of the color evaporation approach to the string picture of QCD readily explains the formation of rapidity gaps between jets at the Tevatron and HERA colliders. 2. Color
Counting
Rules
In the color evaporation scheme for calculating quarkonium production, it is assumed that all color configurations of the quark pair occur with equal probability. For instance, the probability that a QQ pair ends up in a color singlet state is l/ (1 + 8) because all states in 3 6~)3 = 8 @ 1 are
~2
0 0
0 1127 for q71> 772 1172 for 91 > 712 1172 for ~1 > 772 l/9 for 171> 172 l/9 for 171> 772 0 1172 V ~1
3 383 3633 3@3 3 3 383 3@3
equally probable. We propose that the same color counting applies to the final state partons in high ET jet production. In complete analogy with quarkonium, the production of high energy jets is a two-step process where a pair of high ET partons is perturbatively produced at a scale ET, and hadronize into jets at a scale of order AQcD by stretching color strings between the partons and spectators. The strings subsequently hadronize. Rapidity gaps appear when a cluster of dynamical partons, i.e. interacting partons or spectators, form a color singlet. As before, the probability for forming a color singlet cluster is inversely proportional to its color multiplicity. In order to better understand the soft color idea let us consider the formation of rapidity gaps between two jets in opposite hemispheres, which happens when the interacting parton forming the jet and the accompanying remnant system form a color singlet. This may occur for more than one subprocess N and, therefore, the gap fraction is F gap
=
-&c
Fdm
,
(1)
N where FN is the probability for gap formation in the Nth subprocess, duN is the corresponding differential par-ton-parton cross section, and
O.A? kboli et al. /Nuclear
10
__..___.__...... ---.-... :
i+
‘i
I
-
Total
---..
Color Evaporation
...‘.....’Background ’
2
I , I
2.25
2.5
I
2.75
‘,.
Yh,i
: .___.._:)4-......‘_ .._._......_...... - ._.... %... ._._..__.___._......- --.-._-.-_ ‘.:,. .1. Total .‘\. ..-.. Color Evoporatkm “\i\< ...‘...
hi,,
I . I
3
3.25
x;,
IL.
3.5
259
Physics B (Proc. Suppl,) 99 (2001) 257-264
I
3.75
4
’
2
I.
2.25
I.
2.5
Background
I.
2.75
I.
3
“‘..
I.
3.25
I.
3.5
‘\:,.
I “Q.: .I_
3.75
4
b
A?
Figure 1. Fraction of rapidity gap events as a function of the gap size. da (= EN daN) is the total cross section. In our model, the probabilities FN are determined by the color multiplicity of the state and spatial distribution of partons while duN is evaluated using perturbative &CD. The soft color procedure is obvious in a specific example: let us calculate the gap formation probability for the subprocesses pg
+QVQV+QQXY,
where Q” stands for u or d valence quark, and X (Y) is the diquark remnant of the proton (antiproton). The final state is composed of the X (3 @ 3) color spectator system with rapidity 77~ = +oo, the Y (3 @ 3) color spectator system with ny = -co, one 3 parton jr, and one 3 parton js. It is the basic assumption of the soft color scheme that by the time these systems hadronize, any color state is equally likely. One can form a color singlet final state between X and ji since 3 @ 3 @ 3 = 10 @ 8 @ 8 g, 1, with probability l/27. Because of overall color conservation, once the system X @jr is in a color singlet, so is the system Y @ j2. On the other hand, it is not possible to form a color singlet system with ji and Y. Moreover, to form a rapidity gap these systems (ji @ X and jz @ Y) must not overlap in rapidity space. Since the experimental data consists of events where the two jets are in opposite hemispheres, the only additional requirements are ji to be in the same hemisphere as X, i.e. ~1 > 0, and jz to be in the opposite hemisphere (vi +772 < 0). In this configuration, the color strings linking the remnant and the parton
in the same hemisphere will not hadronize in the region between the two jets. We have thus produced two jets separated by a rapidity gap using the color counting rules which form the basis of the color evaporation scheme for calculating quarkonium production. 3. Rapidity
Gaps
at HERA
The partonic subprocess for dijet photoproduction is related to the ep cross section by ~‘ep--tj~jzx~
(~1
=
Irma=
LTa’
F:(Y,
lmin mln x ~py-+jljzx~ W>
&
Q2) dQ2(2)
where W is the center-of-mass energy of the m system, y = W2/s is the fraction of the electron momentum carried by the photon, and Q2 is the photon virtuality. Q2 ranges from Qkin = M:y”/(l - y) to Qs,, which depends on the kinematic coverage of the experimental apparatus. The distribution function of photons in the electron is F,7(y,Q2)
= $&
[I + (I - 2/)2- y]
~(3)
where Me is the electron mass and o is the finestructure constant. The m cross section is related to the partonparton cross section by
0.M
260
Table 2 Color multiplicities Subprocess
Q”Q” + QQ QsQs + QQ
kboli
et al. /Nuclear
and gap probabilities jl
j2
3
3
3
3
PhysicsB (Proc. Suppl.)
FN for the reaction
99 (2001) 257-264
p~3 +
X Y.
jlj2 Y
X 3@3 3@3@3@99
FN
8@3@3@3
1127 v
8@1@3@3
l/81
Q”Q”
-+ &&
I
3
3@3@3@3
B@L
~“~”
+
3
I
3@3@3@3
S@f@3@3
Q”&” Q”@ QsQv 9’9” @QS
+ -+ -+ + +
Ofi? Q& 00 Q& OQ
3 3
I 3
Q”Q”
1127 V Vl l/81 V 711 l/27 for 71 >
f@l
3@3 383
B@SL%J@3
l/27
for for for for
3
z
3@3@3@3
B@Z
l/27
3
3@3@3L%$
8@3@8@3
l/81
8
3 3
3@3@3@3
S@S@B@J
l/81
-+ GG
8
8
383
S@S
0
Q”a”
-+ GG
8
8
3@3
8@3@3@3
0
Q”Q”
-+ GG
8
8
3@3@3@3
S@$
0
Qsgs
+
GG
8
8
3@3@3@3
Z@S@$@3
@Q”
+
GG
8
8
3@3@3@3
2@8@8@8
QG
3
8
QG QG GQ GQ GQ
3
8
3@J3@3@3
B@I3C_aS
l/108
z
8
3@3@3@3
8@3@3
l/108
8
I
3@3@3
L@I
8
f
8@,fCsS@3
l/108
8
3
3@3@3 38383
S@IzI@pf@Z
l/108
3
z
3@3@3
8@3@8
8
8
3@3@3
8@8@8
QvG
+
QsG -+ QSG + GQV -+ GQs -i GQs + GG+QQ GG-+GG
where F~(z~) (F;(q)) is the distribution function for parton a (b) in the proton (photon) and fi = m W is the parton-parton center-ofFor direct m reactions (b q y), mass energy. F;(Q) = 6(1 - 26). The hadronic system X (Y) is the proton (photon) remnant, and ji(s) is the jet which is initiated by the parton ~~(2). The proton is assumed to travel in the positive rapidity direction. 3.1. ZEUS Results The ZEUS collaboration [l] has measured the formation of rapidity gaps between jets produced in ep collisions with 0.2 < y < 0.85 and photon virtuality Q2 < 4 GeV2. Jets were defined by a cone radius of 1.0 in the (~,4) plane, where v is the pseudorapidity and 4 is the azimuthal angle. In the event selection, jets were required to have ET > 6 GeV, to not overlap in rapidity (A7 = 1771- 7721 > 2), to have a mean position ]q] < 0.75, and to be in the region n < 2.5. The cross sections were measured in A77 bins in the range 2 2 A17 5 4.
711> 171> ~1 > 91 >
0 0 0 0 0
0 0 l/27
8@$@zi
3@3
q1
V 71
l/27
for qi > 0 for ~1 > 0 for 71 > 0 for qi > 0 for ~1 > 0 for ~1 > 0 0
l/l08
V ~1
For the above event selection, we evaluated the dijet differential cross section dajet”/dAq, which is the sum of the direct (dud+) and the resolved photon (doyes) cross sections. We used the GRVLO [15] distribution function for the proton, and the GRV [16] for the photon. We fixed the renormalization and factorization scales at PR = PF = ET/~, and calculated the strong coupling constant for four active flavors with AQ~D = 350 MeV. Now we turn to dijet events showing a rapidity gap. We evaluated the differential cross section da@p/dAv which has two sources of gap events: color evaporation gaps (dc$z$) and background gaps (da:?). In our model, the gap cross section is the weighted sum over resolved events d,m,WP cem
-c _
FN
do:, >
(5)
N
with the gap probability FN for the different processes given in Table 1. Background gaps are formed when the region of rapidity between the jets is devoid of hadrons because of statistical
OJl?
&oli
et al. /Nuclear
Physics B (Proc.
Suppl.)
99 (2001)
261
257-264
I.‘,I...(.“,‘.,(..‘I”.I...I.
4s = 1600 GeV
‘20
25
30
40
35
45
50
CDF
60 55 ET (GeV)
4s = 630 GeV
ai
1
i ,,..,....,....,....,,,,,
0 10
15
20
2.5
z
25
I _ I i z I 0 ~.:,,,..,.,.,...,,..,.t.,.f.,.,.,,1.8 2 2.2 2.4 2.6 2.6
30E, (Gel’;
Figure 2. Dependence of the gap frequency the jet transverse energy.
9
on
(6)
where b is a constant, fitted as b = 2.9. This value of b agrees with b = 2.7 f 0.3 found by ZEUS collaboration, when they approximated the nonbackground gap fraction by a constant. The background gap cross section is then written as dc$y
= Fbg(Aq) (dc+
- da::;)
.
3
3.2
3.4
0%-)7W2
fluctuation of ordinary soft particle production. Their rate should fall exponentially as the rapidity separation An between the jets increases [l]. We parametrized the background gap probability as Fbg(Al;l) = e b(s-Ag)
-
(7)
The gap frequency Fgap(Aq) = dogaf’/dojets is shown in Fig. la, where we show the contributions of the color evaporation mechanism and the background. 3.2. Hl Results We also performed an identical analysis for the data obtained by the Hl collaboration [2]. They used the same cone size for the jet definition (AR = l), and collected events produced in proton-photon reactions with center-of-mass energy in the range 158 < W < 247 GeV and with photon virtuality Q2 < 0.01 GeV2. They also imposed cuts on the jets: -2.82 < q < 2.35 and ET > 4.5 GeV. Our results are compared with the preliminary experimental data in Fig. lb
Figure 3. The same as Fig. 2 for half the gap size.
where we used b = 2.3 to describe the background in the Hl kinematic range. As before, color evaporation induces gap formation with a rate compatible with observation. 4. Rapidity Gaps at Tevatron For dijet production in pp collisions, we denoted by X (Y) the proton (antiproton) remnant, and by jrt2) the parton giving rise to a jet. The proton is assumed to travel in the positive rapidity direction. The dijet production cross section is related to the parton-parton one via ~P‘PP-+A~~XY (3)
=
c //- F;(G)
F;(Q)
a,b X flab+plpz
where s (S = &&,S) of-mass
(a> dx,
is the (subprocess)
energy squared
and F$$
dxb
,
(8)
center-
is the distribu-
tion function for the parton a (b). We evaluated the dijet cross sections using MRS-J distribution functions [17] with renormalization and factorization scales PR = ,LF = A. The color evaporation model prediction for the gap production rates in pp collisions is analogous to the one in m interactions, with the obvious replacement of the photon by the antiproton, represented as a 3 @ 3 @ 3 system. The color subprocesses and their respective gap formation probabilities are listed in Table 2. Both experimental collaborations presented
OJP kboli et al. /Nuclear Physics B (Proc. Suppl.) 99 (2001) 257-264
262
‘.
t4F
v’s = ” 18OOGeV ‘.
”
0: I,
I.
0.2
0.3
I.
I
0.4
0.5
“4 COF
*
2
I.
0.6
0
15GeV<
E,<25GeV
T
0.7 &,X2
10 4s = 630 GeV
e
5*
0-l 0.2
I T
+ I
0.3
1
+ .
+
I
‘1
I
I
I
0.4
0.5
0.6
‘I.
. 0.7
%,X2
Figure 4. The same as Fig. 2 for the Bjorken-z of each jet.
their data with the background subtracted. The CDF collaboration measured the appearance of rapidity gaps at two different p~? center-of-mass For the data taken at fi = 1800 energies. GeV, they required that both jets to have ET > 20 GeV, and to be produced in opposite sides (~1 . 772< 0) within the region 1.8 < 171 < 3.5. For the lower energy data, fi = 630 GeV, they required both jets to have ET > 8 GeV, and to be produced in opposite sides within the region 171 > 1.8. Since the experimental distributions are normalized to unity, on average, we do not need to introduce an ad-hoc gap survival probability. Therefore, our predictions do not exhibit any free parameter to be adjusted, leading to a important test of the color evaporation mechanism. In Figs. 2, 3, and 4 we compare our predictions with the experimental observations of the gap fraction as a function of the jets transverse energy, their separation in rapidity, and the Bjorken-z of the colliding partons, respectively. As we can see, the overall performance of the color evaporation model is good since it describes correctly the shape of almost all distributions. This is an impressive result since the model has no free parameters to be adjusted. The DO collaboration has made similar observations at fi = 1800 GeV. They required that both jets to have ET > 15 GeV, to be produced in opposite sides (71 . 72 < 0) within
0~“““‘““‘“““““““‘“” 20 30 40
50
60
70 80 G (GeV)
Figure 5. Gap fraction as a function of the jet ET as measured by the DO collaboration at fi = 1800 GeV.
the region 1.9 < 1~1< 4.1, and to be separated by I&[ > 4.0. In Fig. 5 our results are compared with experimental observations of the dependence of the gap frequency on jet transverse energy, where we used a gap survival probability S, = 30% to reproduce the absolute normalization. This is consistent with qualitative theoretical estimates; see discussion below. As we can see, the fraction of gap events increases with the transverse energy of the jets. This is expected once. the dominant process for the rapidity gap formation is quark-quark fusion, which becomes more important at larger ET [18]. Apart from the lowest transverse energy bin, data and theory are in good agreement. In Fig. 6 we compare our prediction for the dependence of the gap frequency with the separation between the jets. Agreement is satisfactory although the absolute value of our predictions for low transverse energy is somewhat higher than data as shown in Fig. 5. Finally, in Fig. 7 we show our results for the mean value of the Bjorken-z of the events, where all correlations between the jet transverse energy and rapidity have been included. Again, the agreement between theory and data is satisfactory except for the low transverse energy bins. 4.1. Survival Probability at Tevatron We estimated the survival probability of rapidity gaps formed at pji collisions comparing our pre-
O.Jl? holi
rs E
3 :““,““,”
et al. /Nuclear
..(..I.,....,....,..,.,,...,...
.-
: 0 -
E,>JOGeV
0 _.._.
15GeV
263
Physics B (Proc. Suppl.) 99 (2001) 257-264
09
:
ij t 2.5
Qa 0 Aq bins (15GeV < E, C 25GeV) W Aq bins (E, > 3OGeV)
2
2:
1.5 :
T .
i....I....l....I....I....I....I....I....I.. ’ 4 4.25 4.5 4.75 5 5.25 5.5 5.75 Figure 6. Gap fraction rapidity separation.
as a function
6
ii5
of the jets
’
t.
1..
0.1
1..
0.15
. .I..
0.2
. . I..
0.25
Figure 7. Gap fraction age Bjorken-z.
. .I.,
0.3
. .fi..
0.35
,I.
0.4 h+xJ/2
as a function
*,
.
Ii
0,45
of the aver-
dictions with the values of gap fraction actually observed. Assuming that the survival probability varies only with the collision center-of-mass energy, and not with the jets transverse energy, we evaluated the average survival probability
tained R = 2.0 f 0.4 for the same arrangement.
F&Y sp=Fgap. cem
In summary, the occurrence of rapidity gaps between hard jets can be understood by simply applying the soft color, or color evaporation, scheme for calculating quarkonium production, to the conventional perturbative QCD calculation of the production of hard jets. The agreement between data and this model is impressive.
In order to extract 3, we combined the DO and CDF available data at each center-of-mass energy; 630 and 1800 GeV. We found ,.$(lSOO) = 34.4 f 3.3% and &(630) = 65.4 f 12% , a value compatible with the calculation of Ref. [19] based on the Regge model, which yields S,(1800) = 32.6%. For individual contributions and further Moreover, we have that details see Table 3. $,(630)/,!$(1800) = 1.9f0.4, which is compatible with the theoretical expectation 2.2f0.2 obtained in Ref. [20]. Using the extracted values of the survival probability, we contrasted the color evaporation model predictions for the gap fraction corrected by 3, (Fgap daCOT = F,E$ x 3,) with the experimental ta in Table 3. We can also compare the ratio R = F,g,“,p(63O)/F~~~(1800) with the experimental result. DO has measured this fraction for jets with ET > 12 GeV for both energies, and they found R = 3.4 f 1.2; we predict R = 2.5 f 0.5. On the other hand, CDF measured this ratio using different values for EFin at 630 GeV and 1800 GeV; they obtained R = 2.0 f 0.9 while we ob-
kinematical
5. Conclusion
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0.H
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I?boli el al. /Nuclear
Physics B (Proc. Suppl.)
99 (2001) 257-264
Table 3 Gap frequencies and survival probabilities. Theoretical uncertainties are not included. fi (GeV) 1800 1800 1800 630 630
5. 6. 7. 8. 9.
10.
11
12.
13.
Eman (GeV) 3: 20 12 12 8
F!:$ (%) 2.91 2.49 2.24 2.97 2.55
Fj!$’ (%) 0.94 f 0.13 (DS) 1.13 f 0.16 (CDF) 0.54 f 0.17 (DS) 1.85 f 0.38 (DS) 2.3 f 1.0 (CDF)
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14.
15. 16. 17. 18. 19. 20.
S, (%> 32.3 f 4.5 45.4 f 6.4 24.1 f 7.6 62.3 f 12.8 90.2 f 39.2
Fj;$ 1.00 0.85 0.77 1.94 1.67
f f f f f
x S, (%) 0.10 0.08 0.07 0.36 0.31
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