Noting that the number of gluons in the hadron wave function is discrete, and
their formation in the chain of small-x evolution occurs over discrete rapidity
intervals of Delta y~1/alpha_s, we formulate the discrete version of the
Balitsky--Kovchegov evolution equation and show that its solution behaves
chaotically in the phenomenologically interesting kinematic region.Comment: 4 pages, 4 figures. Contributed to the proceedings of DIS 2005
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Tuchin, Kirill

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arXiv

Kirill Tuchin

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Development of Chaos in the Color Glass Condensate

arXiv:hep-ph/0507077v1  6 Jul 2005Development of Chaosin the Color Glass CondensateKirill TuchinNuclear Theory Group,Physics Department,Brookhaven National Laboratory,Upton, NY 11973-5000, USAAbstract. Noting that the number of gluons in the hadron wave function is discrete, and theirformation in the chain of smallx evolution occurs over discrete rapidity intervals of∆y ≃ 1/αs,we formulate the discrete version of the Balitsky–Kovchegov evolution equation and show that itssolution behaves chaotically in the phenomenologically interesting kinematic region.Keywords: small-x, evolution equation, chaosPACS: 24.60.Lz,12.38.-tEVOLUTION AS A DISCRETE PROCESSThe color field of an ultra–relativistic hadron is a quasi-classical non-AbelianWeizsacker-Williams field [1, 2]. It emerges when the occupation number of thebremsstrahlung gluons emitted at a given impact parameter exc eds unity and eventu-ally saturates at∼ 1/αs. It has been argued that in a big nucleus, such thatαsA1/3 ≫ 1,and not very high energies the mean-field treatment is a reasonable approximation tothe evolution equations.As a result of broken scale symmetry of QCD there exist the dimensional scaleΛwhich is the infrared cutoff on the gluon’s momenta. An introduction of an infraredcutoff Λ on the momentum of the emitted gluons amounts to imposing theboundarycondition. This is equivalent to the quantization of the gluon modes in a box of sizeL ∼Λ−1, in which case the spectrum of the emitted gluons and their number become discrete.The formation of a gluon occurs over a rapidity interval of∆y ≃ 1/αs. Therefore,the evolution in rapidity can be considered as a discrete quantum process, where eachsubsequent step occurs when∆yαs ≃ 1 [3].Assuming that∆yαs is a certain number for all steps in evolution process neglectsa stochastic nature of quantum evolution. Full treatment ofthe discrete BK equationrequires taking these effects into account. However, unfortunately BK equation is knownto resist all attempts of analytical solution, and our hope at present is to develop ameaningful approximation. Thus, we suggest an approximation in which the gluons areemitted over a fixed “time" defined bȳαs∆y =C with C = 1. To justify this assumption,let us note that BFKL takes into account only fast gluons, i.e. those withC ∼ 1. It isbeyond the leading logarithmic (LL) approximation to take into account slow gluons.Moreover, it is known that an account of NLL corrections effectively leads to imposinga rapidity veto [4] on the emission of gluons with close rapidities, which restricts2 4 6 8 10 12 14 kT0.20.40.60.81ΦFIGURE 1. Discrete BK equation atω = 2.8. Different lines from left to right correspond ton =1,4,7,10 (black, red, green, blue).production of gluons with smallC (this is due to an effective repulsion between theemitted gluons induced at the NLL level). ThereforeC is bounded from below by anumber close to one. On the other hand the probability that noglu n is emitted whenCbecomes larger than one is very small if we choose the high density i itial condition,such as the one given by the McLerran-Venugopalan model[1].Therefore,C takesrandom values around 1, but the effective dispersion can be exp cted quite small.DISCRETE EVOLUTION EQUATIONEquation which describes the gluon evolution in the high parton density regime of QCDis the Balitsky-Kovchegov equation [5, 6]. This equation isformulated for the forwardscattering amplitude of a color dipole of transverse momentumk and rapidityy to scatterof a big nucleus at impact parameterb. Assumption that the dispersion of the dipoletransverse momenta withy is a slow process one can develop a diffusion approximationto the BK equation. This approximation is meaningful in the saturation region. Thediscrete version of BK equation takes the form (see also [7])Ñn+1(k,y) = (1+ χ(γ0)) Ñn(k,y) − Ñ2n (k,y) , (1)where we introduced the discrete variablen to enumerate emitted gluons andχ(γ0) =4ln2≡ ω −1. It is convenient to re-scale the scattering amplitudeφn = χ(γ0)Ñn so thatthe corresponding continuous amplitude is normalized toÑn(k,y)≤ 1:φn+1 = ω φn − (ω − 1)φ2n . (2)Numerical solution to the discrete BK equation is shown in Figures 1–4.Let us now consider how does the evolution proceeds for various values ofω. In Fig. 1the case of 1< ω < 3 is shown. In that case, for anyk there is one stable fixed point atφn = 1 and one unstable fixed point atφn = 0. The fixed points are determined from theconditionφn+1 = φn.In Fig. 2 we consider the case 3< ω < 3.442. . .. The pointφn = 1 ceases to beunstable. Instead two new stable points appear. These can bedetermined from the2 4 6 8 10 12 14 kT0.20.40.60.81ΦFIGURE 2. Discrete BK equation atω = 3.1. Different lines from left to right correspond ton =1,4,7,10 (black, red, green, blue).2 4 6 8 10 12 14 kT0.20.40.60.811.2ΦFIGURE 3. Discrete BK equation atω = 3.495. Different lines from left to right correspond ton = 1,4,7,10 (black, red, green, blue).conditionφn+2 = φn. The described multiplication of stable points is called ingeneralbifurcation. In our particular case it is referred to as theperiod doubling scenario.When 3.442. . . < ω < 3.56. . . there are four new fixed points, see Fig. 3. It isimportant to emphasize that atn → ∞ the value ofφn settles to a given set of fixed points(specified by the value ofω) independently of the initial condition. In other words,at very high energies the scattering amplitude in the saturation regimekT < Qs(n) isindependent of the initial condition.ONSET OF CHAOSThe period doubling proceeds at increasingly smaller increments ofω until the accu-mulation point ωF = 3.569. . . known also as theFeingenbaum’s number. At this pointthere is no more universal limiting behavior at largen. Instead small change in the initialcondition leads to large change in the final state. The onset of this chaotic behavior canbe observed in Fig. 4. Note now that at first,ωBFKL = 1+ 4ln2= 3.77> ωF and atsecond,ωBFKL is the absolute minimum of the function 1+ χ(γ). Thus, we conclude2 4 6 8 10 12 14 kT0.20.40.60.811.2ΦFIGURE 4. Discrete BK equation atω = 3.77. Different lines from left to right correspond ton =1,4,7,10 (black, red, green, blue).that the evolution of the scattering amplitude at high energies in the saturation regionmay be chaotic. At the same time, it is evident from Figs. 1–4 that he highkT tail ofthe scattering amplitude which describes the perturbativeregime is not affected by thepeculiar behavior of the discrete equation.By averaging over all events one can define the mean value of thscattering amplitude.However, this procedure hides a lot of interesting physics.The most obvious exampleof this is diffraction, which measures the strength of fluctuations in the inelastic crosssection. Figs. 1–4 imply that diffraction is a significant part of the total inelastic crosssection at very high energies, and is universal (independent of the properties of thetarget).The model used in this letter is admittedly oversimplified: we neglected the diffusionin transverse momentum, stochasticity of gluon emission and the dynamical fluctuationsbeyond the mean field approximation. Nevertheless, we hope that a least some of thefeatures of discrete quantum evolution at smallx will survive a more realistic treatment.The chaotic features of smallx evolution open a new intriguing prospective on the studiesof hadron and nuclear interactions at high energies.ACKNOWLEDGMENTSThis research was done in the collaboration with Dima Kharzeev. I would like tothank Yuri Kovchegov, Alex Kovner, Misha Kozlov, Genya Levin, Larry McLerran, AlMueller and Anna Stasto for informative and helpful discussion and comments. Thisresearch was supported by the U.S. Department of Energy under Grant No. DE-AC02-98CH10886. BNL preprint number is BNL-NT-05/24.REFERENCES1. L. D. McLerran and R. Venugopalan, Phys. Rev. D49, 3352 (1994) [arXiv:hep-ph/9311205];Phys. Rev. D 49, 2233 (1994) [arXiv:hep-ph/9309289]; Phys. Rev. D50, 2225 (1994)[arXiv:hep-ph/9402335].2. Y. V. Kovchegov, Phys. Rev. D54, 5463 (1996) [arXiv:hep-ph/9605446]; Phys. Rev. D55, 5445(1997) [arXiv:hep-ph/9701229].3. D. Kharzeev and K. Tuchin, “Chaos in the color glass condensate,” arXiv:hep-ph/0501271.4. C. R. Schmidt, Phys. Rev. D60, 074003 (1999) [arXiv:hep-ph/9901397];J. R. Forshaw, D. A. Ross and A. Sabio Vera, Phys. Lett. B455, 273 (1999) [arXiv:hep-ph/9903390];G. Chachamis, M. Lublinsky and A. Sabio Vera, Nucl. Phys. A748, 649 (2005)[arXiv:hep-ph/0408333].5. I. Balitsky, Nucl. Phys. B463, 99 (1996) [arXiv:hep-ph/9509348].6. Y. V. Kovchegov, Phys. Rev. D60, 034008 (1999) [arXiv:hep-ph/9901281].7. A. Bialas and R. Peschanski, arXiv:hep-ph/0502187.

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