‘Effective’ pomeron model at large b

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Nuclear Physics A 736 (2004) 351–368

www.elsevier.com/locate/np

‘Effective’ pomeron model at largeb

Sergey Bondarenko

HEP Department, School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel

Received 5 January 2004; received in revised form 24 February 2004; accepted 27 February 200

Abstract

In this paper we consider the influence of non-perturbative corrections on the largeb (impactparameter) behavior of the BFKL amplitude. This is done in the framework of a model where“soft” corrections are taken into account in the BFKL kernel. We show, that these correctionsa power-like decreasing behavior of amplitude, which differs from the BFKL case. 2004 Published by Elsevier B.V.

1. Introduction

In this paper we consider the influence of the non-perturbative corrections tbehavior of the perturbative amplitude at largeb (impact parameter). We do not haa consistent theoretical approach for taking into account non-perturbative correctiQCD, so we examine our problem in the framework of an ‘effective’ pomeron mintroduced in [1,2]. In this model we take for the perturbative amplitude the solutiothe BFKL equation in leading order of QCD with fixed coupling constant [3]. Here‘effective’ pomeron appears as the solution of the Bethe–Salpeter equation, resultinthe ‘ladder’, where a ‘soft’ kernel is included, together with the BFKL kernel. Therethis pomeron represents a particular example of including non-perturbative correctthe QCD high energy scattering amplitude.

It is well known, that the BFKL amplitude has a power-like decreasing behavilarge b, see [4–6]. Such behavior contradicts the general postulates of analyticitcrossing symmetry. From these postulates we know, that the absence in thespectrum of particles with zero mass, dictates exponentiale−2bmπ decrease of the scatterinamplitude, see [10]. At the same time, the ‘soft’ pomeron, as a Regge trajeprovides sharp exponentiale−b2B decreasing behavior of the amplitude. There are diffe

E-mail address: [email protected] (S. Bondarenko).

0375-9474/$ – see front matter 2004 Published by Elsevier B.V.doi:10.1016/j.nuclphysa.2004.02.024

ee [6–9].degt suchin theple ofaddingt

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352 S. Bondarenko / Nuclear Physics A 736 (2004) 351–368

approaches which were suggested so as to achieve such an exponential decrease, sIn papers [6,7] it was suggested, that to obtain the necessary result it is sufficient to incluthe non-perturbative corrections only in the Born term of the BFKL amplitude, leavinthe kernel unchanged. On the other hand, there is a point of view, which claims thaa decrease may be achieved by only including non-perturbative QCD correctionsBFKL kernel, see [8]. The model, considered in this paper, gives a particular examsuch an approach, but will not lead to the unitarization of the cross section. Indeed,short range correction we do not modify the largeb behavior of the BFKL kernel, thanecessary for the unitarization of the amplitude. But exploring the largeb behavior ofthe amplitude, which is the admixture of the non-perturbative and the BFKL kernelinvestigate the differences between this amplitude and the BFKL one at largeb, that clarifythe importance of the non-perturbative corrections even in the usual high-energy proof pQCD.

In the next section we consider the BFKL amplitude forq �= 0. In the following sectionsthis amplitude will be used to construct the ‘effective’ pomeron, and only the diffuapproximation for this ‘hard’ amplitude will be considered. In Section 3, we conside‘effective’ pomeron in theq representation. In Section 4 we discuss theb representation othe amplitude which was found in Section 3. In Section 5 we compare the largeb behaviorof the BFKL and ‘effective’ pomeron. In Section 6 we summarize our results.

2. Dipole amplitude in the diffusion approach

We start this section considering the dipole–dipole BFKL amplitude, which for larb

has the form

NBFKL(y, r1,t , r2,t ;b) =∫

dν

2πφine

ω(ν)y24−8iν

(r21,t r

22,t

b4

) 12+iν

, (1)

whereb > r2,t > r1,t , see [4,5]. The factor 24−8iν is the scale factor which is determinby the conformal invariance of the solution of the BFKL equation, see also [11,12]Eq. (1) is divergent at small values ofb due the fact that we use the approximate soluat largeb for the BFKL Green function. The correct solution of the BFKL equationcourse, gives finite result for all range ofb, but has not been used in the treatment ofproblem due the mathematical difficulties of such consideration. To avoid this diverarising in integration overb in our calculations, we will be use the following expressinstead of Eq. (1)

N(y, r1,t , r2,t ;b) =∫

dν

2πφine

ω(ν)y24−8iν

(r21,t r

22,t

(b2 + r22,t /4)2

) 12+iν

. (2)

We see that the BFKL amplitude Eq. (1) has approximately a 1/b4 behavior at largeb,and in integration overb such behavior leads to the divergence of the integral in the re

2 2
of small b. Therefore, we introduce a cut overb in Eq. (2), writing(b + r2/4) insteadb2 in denominator of expression. This substitution has no affect on the results obtained at

e

that

S. Bondarenko / Nuclear Physics A 736 (2004) 351–368 353

largeb, see [6]. The form ofφin(r1,t ;b) is defined by the largeb behavior of the Born termof the BFKL amplitude

φin = πα2s

N2c − 1

N2c

1

1/2− iν. (3)

In Eq. (2)ω(ν) is the eigenvalue of the full BFKL kernel. In the following, to simplify thcalculations, we will use the diffusion approximation forω(ν), (whereω(ν) = ω0 −Dν2),which is valid at small values ofν. Together Eqs. (2) and (3) give

N(y, r1,t , r2,t ;b) = πα2s

N2c − 1

N2c

∫dν

eω(ν)y24−8iν

2π(1/2− iν)

(r21,t r

22,t

(b2 + r22,t /4)2

) 12+iν

. (4)

The Born term of Eq. (4) with theφin of Eq. (3) is

N(y = 0, r1,t , r2,t ;b)B = πα2s

N2c − 1

N2c

r21,t r

22,t

(b2 + r22,t /4)2

, (5)

see [6].The ‘effective’ pomeron ‘ladder’, see Fig. 1, was studied in Ref. [2] atq = 0, the

generalization toq �= 0, which we need here, is very simple. Indeed, the equationsums ‘ladder’ diagrams is of Bethe–Salpeter type, and the momentum transferredq , is a

Fig. 1. The diagrams and the graphic form of the‘admixture’ of ‘soft’ and ‘hard’ kernels.

gthe:

ion

s. Theter


parameter which is preserved along a ‘ladder’. Therefore, the equation forq �= 0 can bewritten in the same way as forq = 0, by introducing the kernel withq dependence. Doinso, the form of solution will be the same as forq = 0, see next section. Consequently, infurther calculations we need to know the BFKL amplitude written in theq representation

N(y, r1,t , r2,t ;q)

= πα2s

N2c − 1

N2c

∫dν

eω(ν)y(r21,t r

22,t )

12+iν24−8iν

2π(1/2− iν)

∫db

2πJ0

(|q||b|) b

(b2 + r22,t /4)1+2iν

= α2s

N2c − 1

N2c

(r1,t r2,t )

∫dν

eω(ν)y(r21,t q

2)iν24−8iν

2π(1/2− iν)

K2iν(|q||r2,t |/2)

�(1+ 2iν), (6)

whereK2iν is the McDonald function. In the limit of the Born term aty = 0, performingcontour integration and returning to theb representation, we again obtain the expressgiven by Eq. (5). Finally we have

N(y, r1,t , r2,t ;q) =∫

dω

2πieωyNω(r1,t , r2,t ;q)

= α2s

N2c − 1

N2c

(r1,t r2,t )

∫dω

2πi

×∫

dν

2π

eωy(r21,t q

2)iν24−8iν

(ω − ω(ν))(1/2− iν)

K2iν(|q||r2,t |/2)

�(1+ 2iν). (7)

In the further calculations we will useNω(r1,t , r2,t ;q) given by Eq. (7).

3. ‘Effective’ pomeron

In this section we consider the approachfor the ‘effective’ pomeron, which waintroduced in the Refs. [1,2]. The main idea behind this pomeron is shown in Fig. 1resulting solution for the ‘ladder’ of Fig. 1,can be found as a solution of a Bethe–Salpetype equation, and this solution is

Nωfull (r1,t , r2,t ;q)

= Nω(r1,t , r2,t ;q) + Nω(r1,t , r2,t ;q)

1− A∫

d2k1∫

d2k2 φ(k1, q)k21N

ω0 (k1, k2;q)k2

2φ(k2, q),

(8)

where isNω(r1,t , r2,t ;q) is defined in Appendix C, and whereNω0 (k1, k2;q) in Eq. (10) is

the Fourier transform of

Nω0 (r1,t , r2,t ;q)

= (r1,t r2,t )

∫dν

2π

(r21,t q

2)iν24−8iν

(ω − ω(ν))(1/2− iν)

K2iν(|q||r2,t |/2)

�(1+ 2iν). (9)

The second term of Eq. (8) is the ‘effective’ pomeron, which is an admixture of ‘soft’ and‘hard’ kernels written in theq representation

r theirstly,

ndourotherefore,nalrefore,l

, buthave a

sformf

in


NωS–H (r1,t , r2,t ;q)

= Nω(r1,t , r2,t ;q)

1− A∫

d2k1∫

d2k2 φ(k1, q)k21N

ω0 (k1, k2;q)k2

2φ(k2, q). (10)

Throughout the rest of the paper we will concentrate on this part of the full solution fo‘effective ladder’. Eq. (10) has a different form than the expression obtained in [2]. Fas mentioned in the previous section, the function of Eq. (10) depends onq , whereas in [2]it was obtained as a solution for the caseq = 0. Another difference between Eq. (10) athe solution in paper [2], is in the form of the ‘hard’ Green’s function. The rank of‘effective ladder’ in Fig. 1 contains two propagators, one from the soft kernel and anfrom the hard, see Fig. 1. In [2] the fully truncated Green’s function was used. Therin Ref. [2], the additional propagator 1/k2 was included in integrations over each intermomenta. In our case, the function of Eq. (2) has four external propagators, and, thewe have to truncate two of them. We do so, includingk2 in the integration over this internamomenta.

We also use the soft kernel of the modelKsoft(k1, k2, q) = ∆Sφ(k1, q)φ(k2, q), where

φ(k, q) = exp

[− k2

2q2s

− (�q − �k)2

2q2s

]. (11)

In the following calculations, for the sake of simplicity, we take the kernel in this formthis form of the kernel is also dictated by the instanton approach, see [1], and couldmore general basis. We will find later the numerical factorA in Eq. (10), which providesthe correct soft pole position in denominator of Eq. (10). Performing the Fourier tranfor the transverse momentak1, k2 to the size of dipolesr1,t , r2,t in the denominator oEq. (10), we obtain

NωS–H (y, r1,t , r2,t ;q)

= Nω(y, r1,t , r2,t ;q)

×[1− A

∫d2k1 k2

1

∫d2k2 k2

2φ(k1, q)φ(k2, q)

∫d2r1,t

(2π)2e−i�k1�r1

×∫

d2r2,t

(2π)2e−i�k2�r2Nω0 (r1,t , r2,t ;q)

]−1

. (12)

We find the constantA of Eq. (10) considering the Born term for the BFKL amplitudeq representation

Nω0 (k1, k2;q) = δ2(�k2 − �q + �k1)

k21k2

2

.

We obtain

A

ω

∫d2k1

∫d2k2φ(k1, q)δ2(�k2 − �q + �k1)φ(k2, q)

∫ [ ]
= A
ωd2k1 exp − k2

1

q2s

− (�q − �k1)2

q2s

= A

2ωπq2

s e−q2/2q2s . (13)

jectory


From Ref. [2] we know, that in this case, the ‘effective’ amplitude has the form:

1

ω

KS/q2s

ω − ∆S + α′q2 . (14)

We consider here the case of smallq , q2/q2s < 1. The comparison with Eq. (13) gives

A

2πq2

s e−q2/2q2s = ∆S − α′q2, (15)

and

A = 2∆S

πq2s

, α′ = Aπ

4= ∆S

2q2s

. (16)

These expressions show the relationships of the parameters of the ‘soft’ pomeron trawith the ‘soft’ kernel given by Eq. (11).

The ‘effective’ pole of Eq. (10) is the solution of the following equation:

1− 2∆S

πq2s

∫d2k1 k2

1

∫d2k2k2

2φ(k1, q)φ(k2, q)

∫d2r1,t

(2π)2e−i�k1�r1

×∫

d2r2,t

(2π)2e−i�k2�r2Nω

0 (r1,t , r2,t ;q) = 0. (17)

This equation is solved in Appendix A, and the solution obtained is.

(1) In the region where 2ν0 < 1 and 2ν0 ln(q2s /q2) < 1 with

ν0 = 4∆S

Dln

(q2s

q2

), (18)

the solution of Eq. (17) is given by

ω = ωS–H = ω0 + (∆S)2 ln2(q2s /q2)

216ν0−4D. (19)

(2) In the region 2ν0 < 1 and 2ν0 ln(q2s /q2) > 1, where

ν0 =√

2∆S

D, (20)

the solution for ‘effective’ pole reads:

ω = ωS–H = ω0 + ∆S

28ν0−1 − 2∆S

(q2

16q2s

)2ν0

. (21)

Below we denote both solutions byωS–H , unless mentioned otherwise.

.

fl

e

s,


Returning to the Eq. (10), and integrating this equation overω we obtain

NS–H (y, r1,t , r2,t ;q)

=∫

dω

2πieyωNω

S–H (r1,t , r2,t ;q)

=∫

dω

2πi

f (ωS–H ,ω)

ω − ωS–HeωyNω(r1,t , r2,t ;q), (22)

with theNω(r1,t , r2,t ;q) defined by Eq. (C.4) andf (ωS–H ,ω) = 2√

ω − ω0√

ωS–H − ω0in the case of solution Eq. (19), orf (ωS–H ,ω) = ω − ω0 in the case of solution Eq. (21)

For high energies we have thatωS–H − ω0 > Dν2SP, whereνSP = ln(q2/q2

s )

2Dyor νSP =

ln(r1,t qs)

2Dy, are the values of saddle points in integration overν1 or ν2 in Eq. (C.4) by the

method of steepest descent. Therefore, after the integration overω, we have

NS–H (y, r1,t , r2,t ;q) = f (ωS–H ,ωS–H)eωS–H yN(ω=ωS–H )(r1,t , r2,t ;q). (23)

We can calculateN(ω=ωS–H )(r1,t , r2,t ;q), see Appendix D. In approximation of smallν

and qqs

< 1 we obtain

NS–H (y, r1,t , r2,t ;q)

= Cπ∆SeωS–H y (r1,t r2,t )

164ν0−1Dν0

(q

qs

)2ν0

(r1,t qs)2ν0K2ν0

(|q||r2,t |/2), (24)

whereν0 =√

ωS–H −ω0D

andC = 2 in the case of solution Eq. (19), orC = 1 in the case osolution Eq. (21). We see, that instead of the saddle point value ofν in Eq. (6), the smalq behavior of Eq. (24) is governed by non-perturbative contributions inν, which are givenby Eq. (18) or by Eq. (20).

4. Large b behavior of ‘effective’ pomeron

In this section we will consider the largeb representation of the amplitude Eq. (24). WdefineNS–H (y, r1,t , r2,t ;b) as

NS–H (y, r1,t , r2,t ;b) =qs∫

0

d2q ei �q �bNS–H (y, r1,t , r2,t ;q). (25)

The calculation ofNS–H(y, r1,t , r2,t ;b) is performed in Appendix B. The resultobtained there, are the following.

(1) In the region ofb where

4bqs

(1

2∆Sy

)1/4ν0

< 1, (26)

or

b < bmax= 1

4qs

(2∆Sy)1/4ν0, (27)

n

of1)

romyone,


the amplitude is

NS–H (y, r1,t , r2,t ;b)

= 24ν0π2

164ν0−1

√2∆S

D

(8

∆Sy

)1/2ν0(r1,t r2,t q

2s

)( r1,t

r2,t

)2ν0

× exp

[ω0y + y

∆S

28ν0−1

]�

(1+ 1

2ν0

), (28)

with

ν0 =√

2∆S

D. (29)

We see, that for suchb, the amplitude does not depend onb.(2) For impact parameters which are such that

b > bmax= 1

4qs

(2∆Sy)1/4ν0, (30)

the amplitude reads as

NS–H (y, r1,t , r2,t ;b)

= 2π2

164ν0−1

√8∆S

D

(r1,t r2,t )1+2ν0

(b2 + r22,t /4)1+2ν0

exp

[ω0y + y

∆S

28ν0−1

]�(1+ 2ν0), (31)

with the sameν0 as in Eq. (29).

5. Large b behavior of BFKL amplitude

We return to Eq. (4) and consider the largeb behavior of this amplitude. The integratiooverν, may be performed by the method of steepest descent, and for largeb the answer is

NBFKL(y, r1,t , r2,t ;b) ∝ 1√y

r1,t r2,t

b2 + r22,t /4

eω0y. (32)

The full expression for the ‘effective ladder’ of Fig. 1, is given by the sumthe usual BFKL amplitude and ‘effective’pomeron, which are Eqs. (4), (28) and (3correspondingly, see Eq. (8). From Eq. (8) it is also clear, that when∆S = 0 we stay onlywith the BFKL amplitude, which is the lower bound for the ‘effective’ pomeron. So, fthe value ofb larger than someb0, the largeb behavior of the ‘ladder’ is governed bthe BFKL amplitude. The ‘effective’ pomeron has a larger intercept than the BFKLbut the BFKL amplitude decreases more slowly at largeb. We find the value ofb0 bycomparison Eqs. (31) and (32). It is approximately

( )2ν [ ]
1√y
� r1,t r2,t

b20

0

exp y∆S

28ν0−1 , (33)

vein thehis

ativens are

meterIn the

ty this

in thisn

g thell

, this.


which gives

b0 = y1/8ν0√

r1,t r2,t exp

[y

∆S

28ν0+1ν0

]. (34)

For smallν0 and largey, b0 is a large number, from which the largeb behavior of the fullsolution for the ‘effective ladder’ is governed by the BFKL amplitude.

6. Conclusion

In this paper we studied the influence of non-perturbative corrections on perturbatiamplitude at large values of impact parameter. We investigated the influenceframework of the model of the ‘effective’ pomeron, which is given by Eq. (10). Tpomeron is the ‘admixture’ of the pQCD BFKL amplitude and the soft, non-perturbpomeron, and gives an example of the model where non-perturbative correctioincluded in the perturbative QCD kernel.

The main result of this paper is given by Eqs. (28) and (31). The impact paradependence of the amplitude, at large values of impact parameter is the following.region ofb where

b < bmax= 1

4qs(2∆Sy)1/4ν0,

theb dependence of amplitude is defined by Eq. (28). For a constant value of rapidiamplitude is constant, and does not depend onb, but only on values of∆S andy. In theregion ofb, from bmax to theb0 of Eq. (34),

b0 > b > bmax,

the amplitude is given by Eq. (31). There is a power-like decrease of the amplituderegion ofb. From the pointb = b0 and to the some value ofb, which defines the regioof applicability of the diffusion approach, and which isbdiff = √

r1,t r2,t eDy/2, the larger

impact parameter behavior is governed by usual BFKL amplitude Eq. (32). Insertinvalueb0 of Eq. (34) in Eq. (32) we find, that the at this point the intercept is very sma

NBFKL(y, r1,t , r2,t ;b) ∼ Const×exp

[y(ω0 − ∆S

28ν0ν0

)]y1/4ν0

.

For small values ofν0, the amplitude decreases withy. It is also possible, thatb0 > bdiff ,and only the ‘effective’ pomeron will define the largeb behavior. This picture of thebdependence is presented in the Fig. 2.

Returning to equation Eq. (31) we see, that in the wide region of largeb, the ‘effective’pomeron preserves a power-like decreasing structure and has the interceptωS–H largerthan the interceptω0 of the BFKL amplitude, see Section 5 of the paper. Neverthelessdecrease is quite different from the structure of the pure BFKL largeb asymptotic behaviorIndeed, in the diffusion approach, the BFKL amplitude leads to the 1/b2 behavior, see
Eq. (32) while theb dependence of Eq. (31) is very different from this. This differenceis in the sharper decrease, than the BFKL amplitude has, but it is even more important,

eed, itctions,

rgeusual

n theh

willfullwork

aper

Israeli


Fig. 2. Amplitude behavior in impact parameter space.

that the power ofb is governed byν0 function which depends on the value of∆S , i.e.,by the soft, non-perturbative physics. Only from very largeb, defined by Eq. (34), thasymptotic behavior of amplitude is again governed by the BFKL amplitude. Indewas also mentioned in Introduction, that due the short range of the considered correthe behavior of ‘effective ladder’ from very largeb will be again defined by the BFKLamplitude. But still, the main result of this consideration is that there is a region of lab

where the admixture of the ‘soft’ and ‘hard’ pomerons has the very different from theBFKL amplitude behavior.

Another observation is that the results of Eqs. (28) and (31) are obtained iapproximation of smallν0, where 2ν0 < 1 in the framework of the diffusion approacfor the BFKL kernel. It easy to see, that in the case when 2ν0 > 1 we will obtain a verydifferent largeb behavior from those of Eq. (28) or Eq. (31), where the final answeralso depend on the value ofν0. In this case we need to include in the calculations theBFKL kernel, instead of the diffusion approximation, and this task is beyond the frameof the paper.

Acknowledgements

I wish to thank E. Levin and E. Gotsman, without whose help and advice this pwould not be written.

This research was supported by Israel Science Foundation, founded by theAcademy of Science and Humanities.

Appendix A

In this appendix we consider the expression for the denominator of Eq. (10), and willsolve Eq. (17). In the denominator of Eq. (10) we must perform the integration overk1, k2,

se the


r1,t andr2,t variables. The integration overk1, r1,t is simple

∫k2

1 d2k1 exp

[− k2

1

q2s

+ �k1�qq2s

]∫d2r1,t

(2π)2e−i�k1�r1r1+2iν

1

= (qs)1−2iν21+4iν

�(32 + iν)�(1

2 − iν)

�(−12 − iν)

1F1

(1

2− iν,1,

q2

4q2s

). (A.1)

From the integrals overk2, r2,t , we see, that the main contribution comes fromr2,t ∼1/qs . Assumingq/qs < 1 we replace theK2iν function by the first three terms of it’expansion overr2,t q . We will see later, that three terms are enough in order to achievdesired precision(q2/q2

s )2ν . We have

K2iν

(qr2,t

2

)= 2−1+4iν(qr2,t )

−2iν�(2iν) + 2−1−4iν(qr2,t )2iν�(−2iν)

+ 2−5+4iν(qr2,t )2−2iν �(2iν)

1− 2iν. (A.2)

Inserting expression Eq. (A.2) back into the integral, and integrating overk2, r2,t , weobtain∫

k22 d2k2 exp

[− k2

1

q2s

+ �k1�qq2s

]∫dr2,t

(2π)r22J0(k1r1,t )

×(

2−1+4iν(qr2,t )−2iν�(2iν) + 2−1−4iν(qr2,t )

2iν�(−2iν)

+ 2−5+4iν(qr2,t )2−2iν �(2iν)

1− 2iν

)

= 22iν �(2iν)�(3/2− iν)�(1/2+ iν)

�(−1/2+ iν)q1+2iνs q−2iν

1F1

(1

2+ iν,1,

q2

4q2s

)

+ 2−2iν �(−2iν)�(3/2+ iν)�(1/2− iν)

�(−1/2− iν)q1−2iνs q2iν

1F1

(1

2− iν,1,

q2

4q2s

)

+ 2−6+2iν �(2iν)�(5/2− iν)�(−1/2+ iν)

(1− 2iν)�(−3/2+ iν)q−1+2iνs q2−2iν

× 1F1

(1

2+ iν,1,

q2

4q2s

). (A.3)

So, now we are left with the integral overν:

∆S

π2q2s

e−q2/q2s

∫dν q1−2iν

s q2iν25−4iν

(ω − ω(ν))(1/2− iν)

�(32 + iν)�

( 12 − iν

)�(1+ 2iν)�

(−12 − iν

)1F1

(1

2− iν,1,

q2

4q2s

)

×(

22iν �(2iν)�(3/2− iν)�(1/2+ iν)

�(−1/2+ iν)q1+2iνs q−2iν

1F1

(1

2+ iν,1,

q2

4q2s

)

( )
+ 2−2iν �(−2iν)�(3/2+ iν)�(1/2− iν)
�(−1/2− iν)q1−2iνs q2iν

1F11

2− iν,1,

q2

4q2s

acket

rome

d inlf thefore,f

only

after


+ 2−6+2iν �(2iν)�(5/2− iν)�(−1/2+ iν)

(1− 2iν)�(−3/2+ iν)q−1+2iνs q2−2iν

× 1F1

(1

2+ iν,1,

q2

4q2s

)). (A.4)

The terms with the gamma functions can be simplified. For the first term in the brwe have

�(2iν)�(32 + iν)�(1

2 − iν)�(3/2− iν)�(1/2+ iν)

�(1+ 2iν)�(−1/2− iν)�(−1/2+ iν)= π(1/4+ ν2)2

2iν cosh(πν). (A.5)

According to Eq. (A.5), the main contribution in the integral of Eq. (A.4) comes fthe region of smallν, therefore, withω(ν) in diffusion approximation, the first term in thbracket gives the following function in the integral

π(1/4+ ν2)2

2iν cosh(πν)(ω − ω(ν))(1/2− iν)1F21

(1

2+ iν,1,

q2

4q2s

)

≈ π

16iν

eq2/4q2s I2

0 (q2/4q2s )

ω − ω0 + Dν2, (A.6)

where I0 is the Bessel function of the second kind. This answer is obtainethe approximation where 2ν < 1. The case, when 1< 2ν < 2, needs an additionaconsideration and must include the full BFKL kernel in the calculations, instead odiffusion approximation. This task is not in the framework of this paper, and, therein the following only the case of smallν with 2ν < 1 will be considered. In the limit osmall ν, the second term in the bracket yields the same answer as Eq. (A.6), withchangei → −i, and the third term gives

9π

(211)iν

eq2/4q2s I2

0 (q2/4q2s )

ω − ω0 + Dν2 . (A.7)

Now the equation Eq. (17) has the form

1− ∆S

πe−3q2/4q2

s

∫dν 24−8iν

ω − ω0 + Dν2

I20 (q2/4q2

s )

iν

×(

1

8− 1

8

(q2

q2s

)2iν

+ 9

211

(q2

q2s

))= 0. (A.8)

We close the contour of integration in Eq. (A.8) in the lower semi-plane, andintegration overν, we have the following equation for the ‘effective’ pole position

1− 2∆S

28ν0

e−3q2/4q2s

ω − ω0+ 2∆S

28ν0

e−3q2/4q2s

ω − ω0

(q2

q2s

)2ν0

+ 9∆S

27+8ν0

e−3q2/4q2s

ω − ω0

(q2

q2s

)= 0,

(A.9)√ω−ω0
whereν0 =
D. The solution of this equation depends on the value ofν0. There are

two possible cases, which we consider separately.

e

ththis

is


(1) The first solution we obtain assuming that 2ν0 < 1 and 2ν0 ln(q2s /q2) < 1. In this case

the equation may be rewritten in the form

1− 2∆S

28ν0

e−3q2/4q2s

ω − ω0

(1− e−2ν0 ln(q2

s /q2)) = 0, (A.10)

which has the following approximate solution:

ω = ωS–H = ω0 + (∆S)2 ln2(q2s /q2)

216ν0−4D, (A.11)

with

ν0 = 4∆S

Dln

(q2s

q2

). (A.12)

(2) The second solution is where 2ν0 < 1 and 2ν0 ln(q2s /q2) > 1. Here we consider th

following terms in our main equation Eq. (A.9)

1− 2∆S

28ν0

e−3q2/4q2s

ω − ω0+ 2∆S

28ν0

e−3q2/4q2s

ω − ω0

(q2

q2s

)2ν0

= 0. (A.13)

The solution, which we obtain from Eq. (A.13), is

ω = ωS–H = ω0 + ∆S

28ν0−1− 2∆S

(q2

16q2s

)2ν0

, (A.14)

where

ν0 =√

2∆S

D. (A.15)

Appendix B

The integral of Eq. (25) is

NS–H (y, r1,t , r2,t ;b)

= 2π2Cr1,t r2,t∆S

164ν0−1D(r1,t qs)

2ν0

×qs∫

0

q dq J0(|q||b|)eωS–H y K2ν0(|q||r2,t |/2)

ν0

(q

qs

)2ν0

. (B.1)

The value of this integral is defined by saddle points of exponent, together wiJ0function, or by initial point of integration. Initially, we consider the saddle points ofintegral, which depend on the form of solution for∆S–H , and theq region of applicabilityof this solution.

Let us consider the solution forωS–H given by Eq. (19). The contribution from thsolution comes from the region ofq defined by condition 2ν0 ln(q2

s /q2) < 1. Taking theasymptotic expression forJ0(qb) in Eq. (B.1), we have the saddle point equation:( )

d

dq∆2

Syln2(q2

s /q2)

216ν0−4D+ iqb = 0, (B.2)

ic

int


or

−4∆2S

y

q

ln(q2s /q2)

216ν0−4D+ ib = 0. (B.3)

The solution of this equation can be obtained by iteration and it is

q0SP= −4i∆2

S

y

216ν0−4bD, (B.4)

q1SP= −4i∆2

S

y

216ν0−4bDln

(b2q2

s D2216ν0−4

16∆4Sy2

). (B.5)

We see, that in this case for largey, we have|qSPb| ∝ y 1, therefore, the asymptotexpansion ofJ0 is, indeed, justified. This solution gives:

NS–H (y, r1,t , r2,t ;b) ∝ r1,t r1+2ν02,t

b2+2ν0exp

[ω0y + y

(∆S)2

216ν0−4Dln2

(216ν0−4q2

s b2D2

(∆S)4y2

)].

(B.6)

The condition of applicability of this solution, 2ν0 ln(q2s /q2) < 1 requires|qSP| > q0 =

qse−(D/2∆S)1/2

, therefore Eq. (B.6) is applicable in the limited range ofb:

b < bmax= 4∆2S

y

qsD216ν0−4e(D/2∆S)1/2, (B.7)

and reaches the maximum value at the pointb = bmax:

NS–H (y, r1,t , r2,t ;b) ∝ r1,t r1+2ν02,t

b2+2ν0max

exp

[ω0y + y

∆S

216ν0−3

]. (B.8)

For otherωS–H , given by Eq. (21), we have from Eq. (B.1) the following saddle poequation

d

dq

(−2∆Sy

(q2

16q2s

)2ν0

+ iqb

)= 0, (B.9)

or

−4∆Syν0

q

(q2

16q2s

)2ν0

+ ib = 0. (B.10)

The equation Eq. (B.10) has the approximate solution

qSP= −i4qs

(∆Sν0y

bqs

)1+4ν0

, (B.11)

where again we check that|qSPb| ∝ y > 1. This solution contributes in the region ofq ,where 2ν0 ln(q2

s /q2) < 1, this gives|qSP| < q0 = qse−(D/2∆S)1/2

. We obtain for Eq. (B.1)

NS–H (y, r1,t , r2,t ;b) ∝ 1

b2q2s

exp

[ω0y + y

∆S

28ν0−1− y2∆S

(∆Sν0y

bqs

)4ν0]. (B.12)

We now consider the contribution in our integral, from the region of smallq , which areclose to the initial point of integration. Initially, we suppose thatq is so small thatqb < 1.

tione

e

.14),


Later, we will define the region ofb where this condition is satisfied. For such smallq wehave, thatJ0(qb) ≈ 1, and can expandK2ν0 function aroundq = 0. We obtain

NS–H (y, r1,t , r2,t ;b) = 2−1+4ν0π2r1,t r2,t∆S

164ν0−1Dν0

(r1,t

r2,t

)2ν0

exp

[ω0y + y

∆S

8

]

×∞∫

0

dq2 exp

[−2∆Sy

(q2

16q2s

)2ν0]. (B.13)

Here, in the region of smallq , we used the solution given by Eq. (21), since our funcdecreases strongly withq , the integration overq in Eq. (B.13) goes to infinity. We sefrom Eq. (B.13), that the main contribution in the integral comes from the regionq whereq ∼ 4qs(

12∆Sy

)1/4ν0. Therefore, the conditionqb < 1 is satisfied for suchb which are

4bqs

(1

2∆Sy

)1/4ν0

< 1. (B.14)

In this region ofb, after integration in Eq. (B.13), we obtain

NS–H (y, r1,t , r2,t ;b)

= 24ν0π2

164ν0−1

√2∆S

D

(8

∆Sy

)1/2ν0(r1,t r2,t q

2s

)( r1,t

r2,t

)2ν0

exp

[ω0y + y

∆S

28ν0−1

]

× �

(1+ 1

2ν0

), (B.15)

where

ν0 =√

2∆S

D. (B.16)

In the region ofb, where

4bqs

(1

2∆Sy

)1/4ν0

> 1 (B.17)

in the integration we can neglect the exponent and keepJ0(qb) function. In this case thintegration overq is simple and gives

NS–H (y, r1,t , r2,t ;b)

= 2π2

164ν0−1

√8∆S

D

(r1,t r2,t )1+2ν0

(b2 + r22,t /4)1+2ν0

exp

[ω0y + y

∆S

28ν0−1

]�(1 + 2ν0). (B.18)

Comparing Eqs. (B.15) and (B.18) with Eqs. (B.6) and (B.12), and taking Eqs. (B(B.17) into account, we see, that Eq. (B.6) is suppressed at least by factor1

b4 , in comparison

to Eqs. (B.15), and (B.12) is suppressed at least by factor exp[−y2∆S(∆Sν0y

bqs)4ν0]/y1/2ν0,

in comparison to Eq. (B.18). Therefore, in the following we take the expression obtainedby the contribution around the initialpoint of integration as the solution.

btain


Appendix C

The functionNω(r1,t , r2,t ;q) is the Fourier transform of the functionNω(k1, k2;q).Correspondingly to [2] this function is defined as

Nω(r1,t , r2,t ;q)

=∫

d2k1

∫d2k2ei�k1�r1,t+i�k2�r2,t Nω(k1, k2;q)

= ∆S

q2s

∫d2k1

∫d2k2ei�k1�r1+i�k2�r2

∫d2k′ (k′)2φ(k′, q)Nω

0 (k′, k2;q)

×∫

d2k′′(k′′)2φ(k′′, q)Nω0 (k1, k

′′;q), (C.1)

where, as in second section,Nω0 (k1, k

′′;q) is the Fourier transform of

Nω0 (r1,t , r2,t ;q) = (r1,t r2,t )

∫dν

2π

(r21,t q

2)iν24−8iν

(ω − ω(ν))(1/2− iν)

K2iν(|q||r2,t |/2)

�(1+ 2iν), (C.2)

and the functionφ is defined by Eq. (11). From Eq. (C.1) we have

Nω(r1,t , r2,t ;q)

= ∆S

∫d2k1 k2

1φ(k1, q)Nω0 (k1, r2,t ;q)

∫d2k2 k2

2φ(k2, q)Nω0 (r1,t , k2;q)

= ∆S

q2s

∫d2k1 k2

1φ(k1, q)

∫d2r ′

(2π)2e−i�k1�r ′Nω

0 (r ′, r2,t ;q)

×∫

d2k2 k22φ(k2, q)

∫d2r ′′

(2π)2e−i�k2�r ′′Nω

0 (r1,t , r′′;q). (C.3)

Performing the same integration as in Appendix A with the function Eq. (C.2), we owith the precision of(q/qs)

2iν :

Nω(r1,t , r2,t ;q)

= ∆S(r1,t r2,t )e−q2/q2

s

∫dν1

2π

K2iν1(|q||r2,t |/2)�(3

2 + iν1)�

( 12 − iν1

)24−8iν1

(ω − ω(ν1))(1/2− iν1)�(1+ 2iν1)�(−1

2 − iν1)

×(

q

qs

)2iν1

21+4iν11F1

(1

2− iν1,1,

q2

4q2s

)

×(∫

dν2

2π

�(3/2− iν2)�(1/2+ iν2)24−8iν2

(ω − ω(ν2))(1/2− iν2)(2iν2)�(−1/2+ iν2)

× (r1,t qs)2iν222iν21F1

(1

2+ iν2,1,

q2

4q2s

)

+∫

dν2

2π

�(3/2+ iν2)�(1/2− iν2)24−8iν2

(ω − ω(ν2))(1/2− iν2)(−2iν2)�(−1/2− iν2)(r1,t q)2iν2

(q

)2iν2(

1 q2 ))
×
qs

2−2iν21F12

− iν2,1,4q2

s

. (C.4)

poles


Appendix D

We have to calculate the following expression


= ∆S(r1,t r2,t )e−q2/q2

s

×∫

dν1

2π

K2iν1(|q||r2,t |/2)�( 3

2 + iν1)�

( 12 − iν1

)24−8iν1

(ωS–H − ω(ν1))(1/2− iν1)�(1+ 2iν1)�(−1

2 − iν1)

×(

q

qs

)2iν1

21+4iν11F1

(1

2− iν1,1,

q2

4q2s

)

×(∫

dν2

2π

�(3/2− iν2)�(1/2+ iν2)24−8iν2

(ωS–H − ω(ν2))(1/2− iν2)(2iν2)�(−1/2+ iν2)

× (r1,t qs)2iν222iν21F1

(1

2+ iν2,1,

q2

4q2s

)

+∫

dν2

2π

�(3/2+ iν2)�(1/2− iν2)24−8iν2

(ωS–H − ω(ν2))(1/2− iν2)(−2iν2)�(−1/2− iν2)(r1,t q)2iν2

×(

q

qs

)2iν2

2−2iν21F1

(1

2− iν2,1,

q2

4q2s

)), (D.1)

whereω(ν) = ω0 − Dν2. We perform the integration over both variablesν1 andν2 byclosing the contours of integration in the lower half-plane, and taking into account the

ν1 = −iν0 = −i

√ωS–H −ω0

D, ν2 = −iν0 andν2 = 0. In the limit of smallν0 the integration

overν1 gives

25−8ν0eq2/8q2s

Dν0K2ν0

(|q||r2,t |/2)I0

(q2/4q2

s

)�(3/2)�(1/2)

�(−1/2)

(q

qs

)2ν0

. (D.2)

Integration overν2 gives

23−8ν0eq2/8q2s

Dν20

I0(q2/4q2

s

)�(3/2)�(1/2)

�(−1/2)(r1,t qs)

2ν0. (D.3)

Therefore, we obtain


= π∆S

164ν0−1Dν0

(r1,t r2,t )

ωS–H − ω0e−3q2/4q2

s I20

(q2/4q2

s

)( )2ν0 ( )
× q
qs

(r1,t qs)2ν0K2ν0 |q||r2,t |/2 . (D.4)

rons,

, hep-

nger-


References

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[2] S. Bondarenko, E. Levin, C.I. Tan, High energy amplitude as an admixture of ‘soft’ and ‘hard’ pomehep-ph/0306231, Nucl. Phys. A, in press.

[3] E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Sov. Phys. JETP 45 (1977) 199;Ia.Ia. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys. 28 (1978) 822.

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Documents

‘Effective’ pomeron model at large b