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FERMILAB - PUB - 96 - 242 - T CBPF - NF - 046/96 hep - ph / 9608???? August 26, 1996

? in Proton - Antiproton Di?ractive Production of bb Collision at the Tevatron

arXiv:hep-ph/9608443v1 26 Aug 1996

G I L V A N A L V E Sa ) ? , E U G E N E L E V I N b ) c ) and ? A L B E R T O S A N T O R Oa) c)

a)

?

LAFEX, Centro Brasileiro de Pesquisas F? ?sicas (CNPq) Rua Dr. Xavier Sigaud 150, 22290 - 180 Rio de Janeiro, RJ, BRAZIL

b)

Theory Department, Petersburg Nuclear Physics Institute 188350, Gatchina, St. Petersburg, RUSSIA Fermi National Accelerator Laboratory P.O. 500 Batavia Illinois -60510 U.S.A.

c)

Abstract: We show that the cross section of the di?ractive production of b? b can be

described as the sum of two contributions: the ?rst is proportional to the probability of ?nding a small size b? b color dipole in the fast hadron wave function before the interaction with a target, whilst the second is the b? b-production after or during the interaction with the target. The formulae are presented as well as the discussion of the interrelation between these two contributions and the Ingelman -Schlein and coherent di?raction mechanisms. The main prediction is that the coherent di?raction mechanism dominates, at least at the Tevatron energies, giving the unique possibility to study it experimentally.

? ? ?

E-mail: gilvan@lafex.cbpf.br E-mail: levin@fnal.gov; leving@ccsg.tau.ac.il E-mail: santoro@fnal.gov;santoro@lafex.cbpf.br

1

Introduction.

The main goal of this letter is to consider the possibility of measureing the inelastic cross section in the di?ractive kinematic region and to discuss the di?ractive production of b? b - pair as a way to extract the value of gluon structure function (xBj G(xBj , Q2 )) in the region of small xBj . New HERA data [1] shows a rapid increase of F2 (xBj , Q2 ) in the region of small xBj ( xBj < 10?2 ), which could be interpreted as a manifestation of the growth of the gluon structure function at small xBj . However, the data on F2 does not allow the extraction of the value of the gluon structure function within good accuracy. At the present we know the gluon structure function with accuracy up to factor two ( see Fig. 1, that shows the gluon structure in three parametrizations GRV94 [2], MRS(A) [3] and CTEQ [4] at di?erent values of Q2 as function of xBj ). Data on photoproduction of J/Ψ seems to favor the MRS(A) parametrization [5]. This question, however, is still open. We will argue that the large rapidity coverage collider detectors at the Tevatron o?er 2 an unique oportunity to measure the gluon structure function at 2 GeV 2 ≤ Q2 ≤ m2 b + pt , ?4 ?2 where mb is the b - quark mass and pt is its transverse momentum, at 10 < x < 10 using the process of the di?ractive dissociation of proton into b? b - pair. This process lego plot and amplitude are pictured on Fig. 2 and Fig.3 respectively. It is clear from these ?gures that this process is a typical large rapidity gap (LRG) process, suggested by Bjorken [6]. As pointed out by Bjorken, and as we demonstrate below, such a process can be described as the exchange of a “hard” Pomeron, which could be rewritten through the gluon structure function due to the intimate relation between inelastic and elastic process given by the optical theorem (Fig.4), ( see ref. [7] [8]for more details). We will show that the cross section of the di?raction dissociation ( DD) can be described as the sum of two di?erent contributions ? : 1. the ?rst is proportional to the probability of ?nding a ? bb color dipole with a small 1 2 size, of the order of rt ∝ m2 +p2 , in the fast hadron wave function before the interaction t b with the target. This dipole scatters with the target and produces the measured ?nal state of the DD process. This mechanism has a normal partonic interpretation and, in the Bjorken frame for the projectile ( in other words in the frame where the ? bb color dipole is at rest), it looks as a measurement of the partonic content of the Pomeron and corresponds to the Ingelman-Schlein (IS) hypothesis of the Pomeron structure function [9].

In what follows we use at large the parton picture of interaction. It is easier to discuss the di?ractive processes in this picture in the frame where the antiproton is at rest ( the ?xed target frame). Of course, all results will be given in relativistic invariant way.

?

1

2. the second is the production of the ? bb - pair after or during the interaction with a target. We will show that this mechanism corresponds to the so called coherent di?raction ( CD ) (see ref.[10]) and we will demonstrate that the measurement of the ? bb di?raction will allow, thanks to the the di?erent dependance on the transverse momenta of produced quarks for both mechanisms, the separation of the CD contribution from the (IS) one. We would like to stress that the above two contributions are closely related to the classic di?ractive dissociation picture suggested by Good and Walker [11] 25 years ago. Indeed, there are two di?erent possibilities for the dissociation of a hadron h into a pair of hadrons ( h1 and h2 ): ?rst, the beam particle (h) interacts with the target and dissociates into the pair of hadrons ( h1 and h2 ); second, the beam particle dissociates ?rst and one of the produced particle interacts with the target ( see ref. [12] for details). However, we will argue that the di?ractive production of the heavy quark system is originated from the small distances where we can develop a theoretical approach based on perturbative QCD (pQCD). The pQCD approach allows us to calculate the di?ractive dissociation process of ? bb in such details which are beyond our reach in “soft” high energy phenomenology.

2

Notations and kinematics.

+ pL 1 1. y = 2 ln E is the rapidity of a particle with energy E and longitudinal momentum E ? pL (along the beam direction ) pL . For the rapidity in the center of mass frame we use the notation y ? .

2. P1 and P2 are the momenta of colliding proton and antiproton (in the c.m. frame P1 = P2 ): √ √ s 2 m2 s P1 = { (1 + ), , 0, 0 } ; 2 s 2 √ √ 2 m2 s s (1 + ), ? , 0, 0 } . P2 = { 2 s 2 (1)

3. y1 and y2 are the rapidities of produced b and ? b quarks, p1t and p2t are their transverse momenta and mb is the b mass. 4. M 2 is the mass of the produced b ? b -pair. m is the mass of the proton or antiproton. s is the squared energy of the reaction in the c.m. frame and it is equal to s = (P1 + P2 )2 2

5. ?y = y1 ? y2 is the di?erence of rapidities between the produced b and ? b. y1 + y2 6. Y = 2 is the mean rapitity of the ? bb system. 2 2 2 7. mit = mb + pit where i = 1, 2. 8. For the purpose of obtaining the kinematic relation in the simplest way we use the Sudakov decomposition [13] of the momenta of all particles, namely pi? = αi P1? + βi P2? + pit? , where vector pit is orthogonal to P1? and P2? . 2 At high energy p2 i? = αi βi s ? pit and the rapidity of particle “i” is equal to

? yi =

(2)

1 αi ln . 2 βi

(3)

9. Using Eqs.(1),(2) and (3) we obtain, for produced b quarks: m1t ? m1t m2t ? m2t ? ? α1 = √ ey1 ; β1 = √ e? y1 ; α2 = √ ey2 ; β2 = √ e? y2 . s s s s and 2 M 2 = 2 m1t m2t cosh( ?y ) + m2 1t + m2t

(4) (5)

10. Let us introduce x1 - the energy fraction of hadron “1” carried by gluon k in Fig.3 and - the energy fraction of hadron “2” carried by the Pomeron with momentum q ( gluon “ladder” in Fig.3). We show below that x1 ,x2 will be the arguments of the gluon structure functions in the cross section expression. Directly from Fig.3 one can see that x1 = α1 + α2 + αq ; x2 = β1 + β2 + βk , (6) where ( x1 , βk ) and ( αq , x2 ) are the longitudinal components of the four momenta of gluon 1 and the Pomeron, respectively. The main property of high energy scattering is the fact that αq ? α1 and or α2 and βk ? β1 and or β2 (see, for example, ref. [14] ). Therefore, we can easily derive from eq. (6), assuming m1t = m2t : 2 m1t 2 m1t ? ?y ?y ? ) ; x2 = √ e? Y cosh( ). (7) x1 = √ eY cosh( s 2 s 2 11. Throughout the paper we will choose a frame where antiproton (see Fig.2 and Fig.3) is essentially at rest and where all momenta (li ) of fast particles look as follows: li = (li + , li ? , lit ) = li + ,

2 m2 + lt , lt li + ,

,

(8)

where li+ = li0 + li3 and li? = li0 ? li3 . 12. xG(x, Q2 ) everywhere in the paper is the gluon structure function. 3

3

The value of the cross section in the generalized parton model.

p (P 1 ) + p ?(P2 ) → b(y1 , p1t ) + ? b(y2 , p2t ) + X + [LRG(Y )] + p ?(P2 ? q )

From Figures 3 and 4 we can see that the value of the cross section of our process (9)

is equal to dσ dσ G 2 2 2 =0 , | = ( x G ( x , ? ) ) |q t 1 1 qt =0 2 2 dY dqt d?ydp2 dY dqt d?ydp2 t t where σ G is the reaction cross section.

2 G(x1 , kt ) + p ?(P2 ) → b(y1 , p1t ) + ? b(y2 , p2t ) + [LRG(Y )] + p ?(P2 ? q )

(10)

(11)

The physical meaning of eq. (10) is very simple: x1 G(x1 , ?2) is the probability of ?nding a gluon with the fraction of energy x1 inside of the proton and σ G is the cross section of its interaction with the antiproton. In the spirit of the factorization theorem [15] we 2 introduce the factorization scale ?2 , the maximal value of kt at which we still can neglect G 2 the dependence of σ on kt . To simplify the color algebra we adopt throughout the paper the colorless probe approach, replacing the gluon with the transverse momentum kt and the fraction of energy x1 by a colorless probe with the same kinematics. The physical motivation is clear and based on the factorization equation (eq. (10)). Indeed, we can measure the gluon structure function using a colorless probe like the graviton or heavy Higgs boson. The properties of such a probe have been studied in details in ref. [16]. The cross section for the reaction of eq. (11) can be easily calculated. It is clear that we have two mechanisms for ? bb-production by the colorless probe which we will discuss in the rest target rest frame ( antiproton in Fig.2). 1. The ?rst mechanism is the following: there is a ? bb component in the wave function of the fast probe before its interaction with the target. This ? bb pair is a color dipole with 1 2 su?ciently small transverse size of the order of rt ∝ m2 +p2 which scatters with the target t b producing the measured ?nal state. 2. The second mechanism is the production of the ? bb - pair after or during the interaction with the target. These two mechanisms correspond to the two set of the Feyman diagrams pictured in ?guress 5a and 5b, respectively. 4

Let us start from the ?rst one which looks normal from partonic point of view in the sense that, in the Bjorken frame for the probe, it looks like the measurement of the partonic content of the Pomeron and corresponds to the Ingelman - Schlein hypothesis of the Pomeron structure function [9]. For the set of the Feyman diagrams of Fig. 5a the amplitude of ? bb production can be written as a product of two factors: (i) the wave ? function of ? bb pair in a virtual gluon ΨG λ1 λ2 and (ii) the rescattering amplitude of the quark - antiquark pair on the target Tλ1 λ2 , where λi is the quark polarization. Following the conventions of ref.[17], we have: Mf = Nc d2 p′t 16π 3

1 0 ′ ′ ′ ′ dz ′ ΨG λ1 λ2 (pt , z ) Tλ1 λ2 (pt , z ; pt , z ) ,

?

(12)

where pt is the transverse momentum of the produced quark and z is the fraction of energy carried by b-quark with respect to energy of the gluon. It is easyly found from eq. (4) and eq. (6) that ?y α1 α1 e2 z = (13) = = ?y ?y , x1 α1 + α2 + αq e 2 + e? 2 where αq we can be found from the equation: ( P2 ? q )2 = m2 and it is equal to αq = q2 ? α1 + α2 at large s. In deriving eq. (13) we have also assumed that p1t = (1?x2 )s The virtual gluon breaks into a quark - antiquark pair with a large lifetime which is equal to τG? . In leading log(1/x) approximation of pQCD, which we will use here, the time of interaction is much smaller than τG? and during this time the exchange of gluons does not change the fraction of energy carried by quark or/and antiquark. It is instructive to recall the argument of why this is so. According to the uncertainty principle the lifetime of the ? bb ?uctuation (τG? ) is τG? ? 1 x1 z (1 ? z ) P1 + 1 | = 2 . = | 2 ?E k ? ? p1 ? ? p2 ? mt + z (1 ? z )kt (14)

? p2t + kt + qt → ? p2t .

An estimate of the interaction time can be obtained from the typical time for the emission of a gluon with momentum l, from the quark p1 , say. Then τi ? | p′1 ? x1 P1 + 1 | = | m2 m2 t ? p1 ? ? l? ? zt ? z′

2 lt αl

|,

(15)

+ and z ′ = z + αl . In the leading log (1/x) approximation of pQCD we where αl = x1 lP 1+ have αl ? z and hence αl x1 P1 + (16) ? τG? . τi ≈ 2 lt

5

Therefore, the interaction only changes the transverse momenta of quarks (see Fig.3). The vertices also do not depend on the type of the diagram since the exchange of gluons preserves helicity at high energy. Finally the amplitude T can be reduced to the form [17]: Tλ1 λ2 = (17) d2 lt dl+ , 4 16π 3 lt where the function φ corresponds to the “ladder” diagram (see Fig.3) and only weakly (logarithmically ) depends on lt . l+ is the large component of vector l? which we have introduced in the previous section. The di?erence in signs between the terms in Eq.(17) re?ects the di?erent color charge of quark and antiquark. = 16π 3 { 2δ (k ′ t ? kt ) ? δ (k ′ t ? kt ? lt ) ? δ (k ′ t ? kt + lt ) } · δ (z ? z ′ ) φ(lt , x) Substituting eq. (17) in eq. (12) we obtain: Mf = where ?ΨG (pt , lt , z ) = 2 ΨG (pt , z ) ? ΨG (pt ? lt , z ) ? ΨG (pt + lt , z ) . Function Ψ has been found to be (see for example ref. [17] ):

? ΨG ± (p t , z ) ? ? ? ?

Nc

?ΨG λ1 λ2 (pt , lt , z ) φ(lt , x)

?

d2 lt dl+ 4 16π 3 lt

(18)

(19)

= ?g

z ( 1 ? z ) ( k2 ?

u ?λ1 (p1 )γ · ?G vλ2 (p2 )

?

m2 +p 2 t b z (1?z )

)

=

(20)

= ?g ·

2 ?

a2

1 ? {δλ1 ?λ2 [ λ1 (1 ? 2z ) ± 1 ] ?G ± · pt ± mb δλ1 λ2 ] } 2 + pt

?

g 1 G √ ( 0, 1, ± 1, 0 )) ; ?G where αS = 4 ± is the circular polarization vector of the gluon ( ?± = π 2 2 2 2 and a = mb + k z (1 ? z ). We have used formulae from refs.[18] and [16] in the above calculations. 2 Considering lt ? m2 bt we obtain : 2 ?ΨG ± (pt , lt , z ) = ?2 g · lt ·

?

(21)

·{ 4

a2 a2 ? p2 t G? δ [ λ (1 ? 2 z ) ± 1 ] ? ± δλ1 ?λ2 } . · p + m 1 t b ± 3 λ1 λ2 2 + p 2 )3 ( a2 + p2 ) ( a t t

In the leading log approximation in ln(1/x) and ln(a2 /Λ2) [19, 17] φ(lt , x) d2 lt dl+ 4π 2 TR αS = i (s + k 2 )xG(x, a2 + p2 t) , 2 16π 3 lt Nc 6 (22)

where TR /Nc arises from averaging over colors (TR = 1/2). Collecting all previous equations we can calculate cross section: dσ G 2 =0 = |q t 2 dY dqt d?ydp2 t · { [ (z 2 + (1 ? z )2 ) p2 t ](

2 2 Mf dz dz 2 16π = αα · S 16πs2 d?y d?y 9 λ1 λ2

(23)

a2 1 2 a2 ? p2 t 2 2 ) + m ( )2 } · ( xG(x, a2 + p2 b t )) 2 3 2 3 2 2 ( a + pt ) 4 ( a + pt )

Finally, we can rewrite eq. (23) in the form ( Nc = 3 ): dσ G 2 =0 = |q t 2 dY dqt d?ydp2 t = (24)

3 16 π 2αS p2 a2 1 cosh(?y ) 2 m2 t 2 b 2 ( 1 ? ) ] · { }2 (xG(x, a2 +p2 [ p + t )) . t ?y ?y 2 2 + p 2 )3 2 2 9 4 a ( a 4 cosh ( 2 ) 2cosh ( 2 ) t

For the cross section of the di?ractive dissociation we have (after sum over gluon polarization and correct averaging over color (Nc = 3)): dY ·

2 =0 |q t 2 dqt d?ydp2 t

dσ

=

x1 G(x1 , ?2)

·

(25)

3 16 π 2αS 1 p2 a2 cosh(?y ) 2 m2 t 2 b 2 ( 1 ? ) ] · { }2 (x2 G(x2 , a2 +p2 [ p + t )) y 2 2 + p 2 )3 2 ( ?y ) t 9 4 a ( a ) 4 cosh2 ( ? 2 cosh t 2 2

From the expression for a we can set the factorization scale, since our cross section 2 2 2 2 ceases to depend on kt if kt ≤ 4m2 bt . Therefore, the reasonable choice is ? = 4mbt . We 2 2 can neglect the scale dependance in our cross section and put a = mb . All ingredients of Eq.(25) are clearly seen in Fig. 6.

3 Taking into account the running QCD coupling constant we have to replace αS in 2 2 2 2 Eq.(25) by αS (? )αS (a + pt ).

Now, let us consider the diagrams of Fig. 5b. They correspond to the possibility of producing a ? bb pair inside of the Pomeron. Indeed, the Pomeron is not a point-like particle; gluons inside it live su?ciently long time and during this time they can create a ? bb-pair which rescatters with the proton by exchange only one gluon. Indeed, the lifetime x1 s of Gl Gl - pair in the diagram of Fig. 5b is equal to τl = , where zl is the l2

2+ kt t zl (1?zl )

7

energy fraction of gluon l. This time is much bigger than the time τb that ? bb - pair lives x1 s (τb = M 2 ? τl ). As has been discussed many times (see, for example refs. [16] [20] [21]) we can safely calculate the diagram of ?g. 5b by closing the contour of integration over βl on the propagator marked by cross in Fig. 5b. We anticipate that lt < kt and that the vertex of emission of the gluon 1′ is proportional to l?t (see ref.[14]). The interaction of the gluon with transverse momentum kt + lt with the target is calculated using eq. (18) with ?Ψ(pt , lt , z ) = Ψ(pt + lt , z ) ? Ψ(pt ? lt , z ) . (26)

Substituting eq. (20) in eq. (26) one obtains after integration over the azimuthal angle of vector lt : 2 ?Ψ(pt , lt , z ) = ?2 g lt pt · (27) ·{ δλ1 ?λ2 [ (1 ? 2z )λ1 ± 1 ] a2 1 ? δ m λ }. λ λ b 1 1 2 2 2 ( a2 + pt )2 ( a2 + p2 t )

Using eq. (22) and eq. (23) we obtain (Nc = 3 ): dσ G [CD] 2 =0 = |q t 2 dY dqt d?ydp2 t =

?

(28)

3 1 1 4 π 2 αS cosh(?y ) 4 1 2 2 2 2 }2 (x2 G(x2 , kt )) . ?y [ ?y a + mb pt ] · 2 · { 2 2 2 2 9 4 cosh ( 2 ) 2 cosh ( 2 ) kt (a + p 2 ) t

2 Notice that extra factor 1/kt in eq. (28) comes from the fact that ?Ψ of eq. (27) does not depend on the polarization of the gluon with transverse momentum kt which is proportional to kt and cancels one of the gluon propagators in eq. (23). We would like to recall that in the previous calculation we assumed that lt < kt and this inequality establish the scale in the gluon structure function in eq. (28). The answer for the cross section of the coherent di?raction has the form:

dσ [CD] 2 =0 = |q t 2 dY dqt d?ydp2 t

m2 bt

2 2 dkt ?x1 G(x1 , kt ) 4 2 kt ? ln kt

2 x2 G(x2 , kt )

2

·

(29)

3 2 1 cosh(?y ) 4 1 4 π 2αS (k t ) 2 2 . ·[ · ?y ?y a + mb pt ] · 2 4 2 2 9 ( a + p2 4 cosh ( 2 ) 2 cosh ( 2 ) t )

This equation gives the contribution for so called coherent di?raction (CD) [10]. The most contribution to the integral comes from the region of su?ciently small kt due to the factor 1 4 kt in the dominator and for the proton kt ∝ R where Rp is the proton radius. It means p 8

that we cannot trust our perturbative calculation for the CD contribution. However,if we calculate the integral

m2 bt 2 2 dkt ?x1 G(x1 , kt ) 4 2 kt ? ln kt 2 x2 G(x2 , kt ) 2

,

(30)

using the current parametrization of the gluon structure function, we can ?nd out that 2 the typical kt , which is essential in the integral, is not very small but about 1 - 2 GeV 2 . To understand this fact we have to remember that the gluon structure function behaves 2 <γ> as (kt ) (at least in semiclassical approach) and the value of < γ > calculated in the current parametrization for the gluon structure function turns out to be rather big in the region of k 2 ≈ 1 ? 2GeV 2 (see Fig. 9 ). One can see that if < γ > > 0.5 the integral converges on the upper limit or, in other words, the small distances start to be important.

2 To check this statement we calculate the integrand of eq. (30) as a function of ln(kt /Q2 0) 2 ?x1 G(x1 ,kt ) 2 2 with ? ln k2 = 1. Q0 = 0.34 GeV is the initial virtuality in the GRV parametrization. t The result of the calculation is plotted in Fig. 8a for the GRV, in Fig. 8b for the MRS(A’) 2 and in Fig. 8c for the CTEQ parametrizations. We see a de?nite maximum in ln kt /Q2 0 2 2 dependence around kt ≈ 1 ? 1.5GeV which becomes more pronounced at smaller values of x2 . It means that we can safely use the perturbative QCD approach to calculate the CD contribution.

2) ?x1 G(x1 ,kt , 2 ? ln kt

In numerical esstimates of eq. (29) we use the GLAP equation [24] to calculate namely :

2 2 ?x1 G(x1 , kt ) αS (kt ) 4 = { 2 ? ln kt 2π 3 1 1 x

dz 2 [z + 2(1 ? z ) ] z

i

x x 2 qi ( , kt ) + z z

(31)

1 zdz dz 2 x x 2 x x 2 2 [ z (1 ? z ) + 1 ? z ] G( , kt ) + 6 [ G( , k t ) ? xG(xkt )] + z z z z x x 1?z z Nf 11 2 ? ] xG(x, kt )}, + 6[ 12 18 where Nf is the number of ?avours and NC = 3 is the number of colors. The running 4π 2 coupling QCD constant αS (kt ) = . k2

+6

( 11 ?

2 3

Nf ) ln

t Λ2

It is worthwhile mentioning that in the case of the di?ractive dissociation in the deep 2 inelastic scattering the smallest value of kt is kt = Q2 and eq. (29) gives the contribution of the order of 1/Q2 . In other words, the coherent di?raction in this case is a high twist contribution while the (IS) di?raction (see eq. (25)) occurs in the leading twist.This result has been obtained in ref. [22]. 9

It should be stressed that there is no interference between the Ingelman-Schlein and the coherent di?raction contributions. Indeed, the interference term vanishes due to integration over the azimuthal angle of pt and summation over gluon polarizations, as one can see comparing Eq.(21) and Eq.(27)

4

Numerical estimates.

Setting x1 = 0.1 we can estimate x2 ≈ 0.6 10?3. As far as the value of the cross section is concerned, we get at ?y = 0 and pt = 0 the value ( at αS = 0.25) dY

2 =0 |q t 2 dqt d?ydp2 t

dσ

≈ 0.1 10?3

mbarn GeV 4

Here, we took x2 G(x2 , m2 bt ) = 20, which is the value in the GRV parametrization. The result of a detailed calculation is presented in Fig. 10. To test the sensitivity of our result to high order QCD corrections we plotted the cross section for the coherent di?raction for two cases: ?xed and running QCD coupling constant. The di?erence is rather big but it does not change the main conclusion: the coherent di?raction gives much bigger cross section than the Ingelman-Schlein contribution of Eq.(25). (IS in Fig. 9a). Terefore, the measurement of the di?ractive dissociation in b? b system gives the unique opportunity to study the CD unlike the deep inelastic processes where the CD is suppressed. One can see from Fig. 9b that the value of the di?erential cross section crucialy depends on the parametrization of the gluon structure function with the di?erence about factor 2. This di?erence encourages us to claim that the measurement of the b? b di?ractive production could provide the selection of the parameterization and give an important contribution to the extraction of the value of the gluon structure function from experiment. We also calculate the integrated cross section de?ned as dσ = dY dp2 t min

∞ +∞ ?∞

pt

d?y

∞ 0

2 dqt

dσ , 2 dY d?ydqt dp2 t

(32)

The value of pmin can be taken from the experimental values obtained by the Tevatron t Collider experiments. In reference Ref. [25] analysis techniques are used to separate muons coming from di?erent sources and, in particular, from b → ? + ν + c process for 2 pmin ≥ 5 GeV. We assumed, in the integration over qt , an exponential behaviour of dσ t 2 with respect to qt 10

dσ dσ 2 ?bqt 2 =0 e = 2 |q t 2 dqt dqt

(33)

We take the slope b = 4.9GeV ?2 as it has been measured at HERA [23]. In our estimates we took αS = 0.25 which corresponds to the value of the running QCD coupling constant (αS (k 2 ) ) at k 2 ? m2 b. Fig. 10 shows that the value of the integrated cross section is not very small and can be meausered by the Tevatron detectors in the next run. One can also see that the di?erences between the estimates in di?erent parameterizations is rather big. It is about a factor 2-3 between the highest values of the cross section in the GRV parameterization and the lowest one in the MRS(A’) parameterization. Finally, we would like to stress that the Tevatron provides an unique possibility to look inside the microscopic mechanism of di?ractive dissociation by measuring the coherent di?raction which gives a small contribution to the deep inelastic proccesses. The formulae written in this paper give us the basis for the Monte Carlo simulation of the di?ractive events in 3-dimension phase space (η, φ, pt ) which we are going to present in further publications. This Monte Carlo will provide a more detailed estimates of the experimental possibilies at the Tevatron and, we hope, will encourage future experiments on large rapidity gap physics. We ?rmly believe that the di?ractive dissociation opens a new window to study such di?cult questions as the Pomeron structure, the matching between hard and soft processes and the search of new collective phenomena in QCD related to the high density parton system. We thank the Fermilab Theory, Computing and Research Divisions for the hospitality. E.L. is very grateful to LAFEX-CBPF/CNPq for the support and warm atmosphere during his stay in Brazil. We would like to thank H.Montgomery for fruitfull discussions in Rio, and to Helio da Motta for reading the manuscript.

References

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[4] H.L. Lai, et. al., CTEQ collaboration: Phys. Rev. D51 (1995) 4763 [5] M.G.Ryskin et.al: Durham University preprint, DTP/95/96, November 1995. [6] Yu.L. Dokshitzer, V.A. Khoze and T.Sjostrand: Phys. Lett. B274 (1992) 116; J.D. Bjorken: Phys. Rev. D45 (1992) 4077; Phys. Rev. D47 (1992) 101;Nucl. Phys. B ( Proc. Suppl.) 23C (1992) 250; Acta Phys. Pol. B23 (1992) 637. [7] M.Albrow et.al.:“Future Experimental Studies of QCD at Fermilab: report of the QCD section: option for a Fermilab Strategic Plan” FERMILAB-FN 622,Aug.1994. [8] A.H. Mueller, B. M¨ uller,C. Rebbi and W.H. Smith: “Report of the DPF Long Range Planning Group on QCD”. [9] G. Ingelman and P.Schlein: Phys. Lett. B152 (1985) 256. [10] J. Collins, L. Frankfurt and M. Strikman: Phys. Lett. B307 (1993) 161. [11] M.L.Good and W.D.Walker: Phys. Rev. 120 (1960) 1857 [12] G. Cohen Tannoudji, A. Santoro and M. Souza: Nucl. Phys. B125 (1977) 445; E.L.Berger and P.Pirila: Phys. Rev. D12 (1975) 3448; Phys. Lett. B59 (1975) 361; E.L.Berger and R.Cutler: Phys. Rev. D15 (1977) 1903; G.Alberi and G.Goggi: Phys. Rep. 74 (1981) 1 and references therein. [13] V.V. Sudakov: ZhETF 30(1956) 187. [14] L. V. Gribov, E. M. Levin and M. G. Ryskin: Phys.Rep. 100 (1983) 1. [15] J. Collins, D.E. Soper and G. Sterman: Nucl. Phys. B308 (1988) 833;In Perturbative Quantum Chromodynamics,ed. A.H. Mueller. Singapore: World Scienti?c (1989) and reference therein. [16] A.H. Mueller: Nucl. Phys. B335 (1990) 115; [17] S.J. Brodsky et al: Phys. Rev. D50 (1994) 3134. [18] S.J.Brodsky and G.P.Lepage: Phys. Rev. D22 (1980) 2157 [19] E.M. Levin and M.G.Ryskin: Sov. J. Nucl. Phys. 45 (1987) 150. [20] E.A. Kuraev, L.N. Lipatov and V.S. Fadin: Sov. Phys. JETP 45 (1977) 199 ; Ya.Ya. Balitskii and L.V. Lipatov:Sov. J. Nucl. Phys. 28 (1978) 822; L.N. Lipatov: Sov. Phys. JETP 63 (1986) 904; J.Bartels: Nucl. Phys. B175 (1980) 365; J. Kwiecinski: Phys. Lett. B94 (1980) 413. 12

[21] E. Levin and M. W¨ ustho?: Phys. Rev. D50 (1994) 4306. [22] E.Levin: “Deep Inelastic Scattering and Related Subjects” Eilat, Israel, 6-11 February 1994, ed. Aharan Levy, WS, 1994, p.83; A. Berrera and D. Soper: Phys. Rev. D50 (1994) 4328. [23] H1 collaboration. A. De Roeck et al.: DESY 95 - 52, August 1995. ZEUS collaboration. M.Derrick et al.: DESY 95 - 133, July 1995. [24] V.N.Gribov and L.N. Lipatov:Sov.J.Nucl.Phys. 15 (1972) 438: L.N. Lipatov: Yad. Fiz. 20 (1974) 181; G. Altarelli and G. Parisi: Nucl. Phys. B126 (1977) 298; Yu. L. Dokshitser: Sov. Phys. JETP46 (1977) 641. [25] D0 Collaboration, S.Abachi et al., Phys. Rev. Lett. 74 (1995) 3548, and CDF Collaboration, F.Abe et al., Phys. Rev. Lett. 75 (1995) 1451

13

Fig.1a

Fig.1b

Fig.1c Figure 1: Gluon structure function xG(x, Q2 ) in di?erent parameterizations: GRV [2] ( Fig.1a ), MRS(A’)[3] (Fig.1b ) and CTEQ [4] ( Fig.1c).

14

φ b p b y LRG (Y) y1 bb

– – –

0

y2

X

Figure 2: Lego - plot of b? b di?ractive production in pp ? collision.

15

p P1 ? x1, kt

? ? 2 ?M ? ?

→

? ? ?X ? x G(x , k2) ? ? 1 1 ? ? ? ?

→ y1, p1t

b

–

? ? 2 ? x2 G ( x2, l ) ? ? – ? p ? P2 ?

l

2

b

→ y2, p2t

P2? - q?

Figure 3: Amplitude of b? b - di?ractive production.

16

2

p

x1 G(x1, m2 bt)

x1

b y1 b y2

–

x1

=

x2

b b

–

x2

x2 G(x2, mbt)

2

p

–

Figure 4: Optical Theorem.

17

→ x1, kt

ΨG*

→ lt

2

→ → x1,kt kt → + lt

(a)

T

x

? ?φ ?

→ lt

x

(b)

Figure 5: Feyman diagrams for b? b di?ractive production by colorless gluon probe.

18

p

p

x1GGLAP( x1,k )

k2 k2

2

l2

l2

x2G(x2,mbt)

p

–

2

p p

–

–

p

–

Figure 6: The cross section for b? b di?ractive production in the Ingelman - Schlein approach [9].

19

3.0

<γ>

2.5

GRV

Q =0.5 Gev 2 =1.0 Gev 2 =5.0 Gev 2 =10. Gev 2 =15 Gev

2 2

2.0

1.5

1.0

0.5

0.0 0.0

5.0

10.0

15.0

20.0

ln(1/x)

Figure 7: The behaviour of average < γ > for the GRV parameterization of the gluon structure function. 20

Fig. 8a

Fig.8b

Fig.8c

2 2 t 2 versus ln(kt /Q2 Figure 8: 2 0 ) in di?erent parameterizations: GRV [2] ( Fig.1a kt ), MRS(A’) [3] ( Fig. 1b ) and CTEQ [4] (Fig. 1c).

(x G(x ,k 2 ))2

21

Fig. 9a

Fig.9b

Figure 9: (a) The cross section for b? b di?ractive production for the coherent di?raction (CD) (Eq. (29)) and the Ingelman - Schlein (IS) di?raction ( Eq. (25)); (b) The cross section for b? b for the coherent di?raction (CD) in di?erent parameterizations for the gluon structure function.

22

Figure 10: The integrated cross section for the coherent di?raction ( Eq.(32)) versus pmin in di?erent parameterizations of the gluon structure function. t

23

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