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The Running BFKL Resolution of Caldwell's Puzzle

a r X i

v

:h e p -p h /9812446v 1 19 D e c 1998

ITEP-PH-6/98FZ-IKP(TH)-1998-32

The running BFKL:resolution of Caldwell’s puzzle

N.N.Nikolaev αand V.R.Zoller β

α

Institut f¨u r Kernphysik,Forschungszentrum J¨u lich,

D-52425J¨u lich,Germany

E-mail:kph154@ikp301.ikp.kfa-juelich.de β

Institute for Theoretical and Experimental Physics,

Moscow 117218,Russia E-mail:zoller@heron.itep.ru

Abstract

The HERA data on the proton structure function,F 2(x,Q 2),at very small x and Q 2show the dramatic departure of the logarithmic slope,?F 2/?log Q 2,from theoretical predictions based on the DGLAP evolution.We show that the running BFKL approach provides the quantitative explanation for the observed x and/or Q 2-dependence of ?F 2/?log Q 2.

Caldwell’s presentation of the HERA data in terms of the logarithmic derivative ?F 2/?log Q 2for the proton structure function (SF)F 2(x,Q 2)exhibits the turn-over of the slope towards small x and/or Q 2up to currently attainable x ~10?6and Q 2~0.1GeV 2[1,2].The DGLAP-evolution [3]with GRV input [4]predicts a steady increase of the derivative

?F DGLAP T

1

The preliminary results have been reported at the DIS’98Workshop [6]

component of the BFKL pomeron.

The s-channel approach to the BFKL equation[7]was developed in terms of the color dipole cross sectionσ(x,r)[8,9](hereafter r is the color dipole moment).The positive feature of the color dipole picture,to be referred to as the running BFKL approach,is consistent incorporating the two crucial properties of QCD:i)asymptotic freedom(AF), i.e.,the running QCD couplingαS(r)and,ii)the?nite propagation radius R c of perturbative gluons.

The BFKL equation for the interaction cross sectionσ(x,r)of the color dipole r with the target reads

?σ(x,r)

8π3 d2 ρ1| E( ρ1)? E( ρ2)|2[σ(x,ρ1)+σ(x,ρ2)?σ(x,r)].(2) Here the kernel K is related to the wave function squared of the color-singlet qˉq g state with the Weizs¨a cker-Williams(WW)soft gluon.The quantity

E( ρ)=?g

(ρ) ?ρK0(μGρ)=g S(ρ)μG K1(μGρ) ρ/ρ,(3)

S

where R c=1/μG and Kν(x)is the modi?ed Bessel function,describes a Yukawa screened transverse chromoelectric?eld of the relativistic quark and| E( ρ1)? E( ρ2)|2describes the ?ux(the modulus of the Poynting vector)of WW gluons in the qˉq g state in which r is theˉq-q separation and ρ1,2are the q-g andˉq-g separations in the two-dimensional impact parameter plane.Our numerical results are for the Yukawa screening radius R c=0.27fm. The recent?ts to the lattice QCD data on the?eld strength correlators suggest similar R c [10].

The asymptotic freedom of QCD uniquely prescribes the chromoelectric?eld be com-puted with the running QCD charge g S(r)=

The color dipole factorization[14]in conjunction with the explicit form of the qˉq light-cone wave function,Ψqq(z,r),relates the dipole cross sectionsσn(r)with the eigen-SF, f n(Q2),

f n(Q2)=

Q2

1+R20Q2 1+c0log(1+r20Q2) γ0,(9) which has the large-Q2asymptotics[16,9]

f0(Q2)∝[αs(Q2)]?γ0,γ0=

4

1+R2n Q2n max i=1 1?z

the accessible range of Q2[11,12].The?rst nodes of sub-leading f n(Q2)are located at Q2~20?60GeV2,the second nodes of f2(Q2)and f3(Q2)are at Q2?5·103GeV2and Q2?2·104GeV2,respectively.The parameterization tuned to reproduce the numerical results for f n(Q2)at Q2~<105GeV2is given by eq.(11).For n=3we take a simpli?ed form with only two?rst nodes,because the third node of f3(Q2)is at~2·107GeV2,way beyond the reach of accelerator experiments at small x.The found parameters are listed in the Table.

n c n R2n,GeV?2z(2)n?n

0.0232 1.1204

10.1113 3.46480.220

0.195 1.5682 1.7706 1.2450

30.0653 2.7756 6.91600.111

Asymptotically,at1/x→∞,the expansion(8)is dominated by the term f0(Q2)(x0/x)?0. At moderately small x the sub-leading terms are equally important since?n~1/n.How-ever,as it has been pointed out in[11,12],for Q2~<104GeV2all f n(Q2)with n≥3are very close in shape to each other.Then we arrive at the truncated expansion

F2(x,Q2)=

3

n=0f n(Q2)(x0/x)?n+F soft2(Q2)+F val2(x,Q2),(13)

where the term f3(Q2)(x0/x)?3with the properly adjusted weight factor,a3,stands for all terms with n≥3.The addition of this“background”term in eq.(13)improves signi?cantly the agreement with data for large Q2thus expanding the applicability region of eq.(13)over the whole small-x kinematical domain of HERA.

The need for a soft pomeron contribution F soft

2in addition to the perturbative gBFKL

SF’s described previously is brought about by phenomenological considerations.A viable gBFKL phenomenology of the rising component of the proton structure function over the whole range of Q2studied at HERA(real photo-absorption included)is obtained if one starts with the Born dipole cross sectionσB(r)as a boundary condition for the gBFKL evolution at x0=0.03[17,12].However,such a purely perturbative input,σB(r),with R c=0.27fm strongly underestimates the cross sections of soft processes and the proton SF at moderate Q2~1GeV2.Therefore,at r~>R c,the above described perturbative gBFKL dipole cross sectionσpt(x,r),must be complemented by the contribution from the non-perturbative soft pomeron,σnpt(x,r).In terms of the relationship[17]betweenσ(x,r)and the gluon structure function of the proton,G(x,Q2),the non-perturbative dipole cross sectionσnpt(r)at r~>R c must be associated with soft non-perturbative gluons in the conventional G(x,Q2).The contribution to G(x,Q2)from the non-perturbative transverse momenta k2~

Because the BFKL rise ofσ(x,r)is due to production of s-channel perturbative gluons, which does not contribute toσnpt(r)in[17,12]we argued that to a?rst approximation one must consider the energy independentσnpt(r)and additivity of scattering amplitudes from both the hard BFKL and soft non-perturbative mechanisms.For recent suggestions to identify ourσnpt(r)with the soft pomeron of the two-pomeron picture see[18,19].In the models of soft scattering via polarization of the non-perturbative QCD vacuum[20,21],σnpt(r)is interpreted in terms of the non-perturbative gluon distributions.

To our opinion,the recently encountered troubles with the small-Q2extrapolations of DGLAP evolution[5]and the failure of DGLAP?ts in the Caldwell plot[1],[2]are due to illegitimate enforcing the DGLAP evolution upon the non-perturbative glue.

The non-perturbative term F soft

2(Q2)in eq.(13)calculated from eq.(6)withσ=σnpt(r)

from[22]can be parameterized as follows

F soft 2(Q2)=b

R2Q2

Λ2n x

[2]ZEUS Collaboration,J.Breitweg et al.,ZEUS results on the measurement and phe-

nomenology of F2at low x and low Q2,Report No.DESY-98-162

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new global analysis,Report No.DTP/98/10;RAL-TR-98-029

[6]V.R.Zoller,in Proc.6th Int.Workshop on DIS and QCD(DIS98),Brussels,1998.

Report No.FZ-IKP(TH)-1998-5

[7]E.A.Kuraev,L.N.Lipatov and V.S.Fadin,Sov.Phys.JETP44(1976)443;45(1977)

199;Ya.Ya.Balitskii and L.N.Lipatov,Sov.J.Nucl.Phys.28(1978)822.

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No.HD-THEP-98-34

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[13]L.N.Lipatov,Sov.Phys.JETP63(1986)904.

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[18]A.Donnachie and https://www.wendangku.net/doc/bc1846757.html,ndsho?,Phys.Lett.B437(1998)408.

[19]K.Golec-Biernat and M.W¨u stho?,DTP-98-50,Jul1998.

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Figure Captions

Fig.1Caldwell’s plot of?F2/?log Q2for the ZEUS data[2](Fig.1a)and?xed target data

[1](Fig.1b).Our predictions(BFKL-Regge)are shown by the solid lines.Shown by

the dashed lines is the leading BFKL pole approximation(LPA).

Fig.2Description of the H1,ZEUS and E665F2(x,Q2)data by the BFKL-Regge expansion

(8):the large-Q2data(Q2=3.5,12,25,65,120and200GeV2)are shown in Fig.2a,

the small-Q2data(Q2=0.11,0.20,0.40,0.65,0.85and1.2GeV2)are in Fig.2b.For display purposes we have multiplied F2by the numbers shown in brackets.

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