The origin of the breaking of conventional linear k_\perp-factorization for
hard processes in a nuclear environment is by now well established. The
realization of the nonlinear nuclear k_\perp-factorization which emerges
instead was found to change from one jet observable to another. Here we
demonstrate how the pattern of nonlinear k_\perp-factorization, and especially
the role of diffractive interactions, in the production of dijets off nuclei
depends on the color properties of the underlying pQCD subprocess.Comment: 4 pages, 1 eps-fi

Nikolaev, N. N.

Schäfer, W.

Zakharov, B. G.

English

arXiv

N. N. Nikolaev

W. Schäfer

B. G. Zakharov

Crossref

Nonuniversality Aspects of Nonlineark⊥Factorization for Hard Dijets

The origin of breaking of conventional linear k(perpendicular to) factorization for hard processes in a nuclear environment is by now well established. The realization of the nonlinear nuclear k(perpendicular to) factorization which emerges instead was found to change from one jet observable to another. Here we demonstrate how the pattern of nonlinear k(perpendicular to) factorization, and especially the role of diffractive interactions, in the production of dijets off nuclei depends on the color properties of the underlying pQCD subprocess

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PRL 95, 221803 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending25 NOVEMBER 2005Nonuniversality Aspects of Nonlinear k? Factorization for Hard DijetsN. N. Nikolaev,1,2,* W. Scha¨fer,1,† and B. G. Zakharov2,1,‡1Institut fu¨r Kernphysik, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany2L. D. Landau Institute for Theoretical Physics, Moscow 117940, Russia(Received 2 February 2005; published 23 November 2005)0031-9007=The origin of breaking of conventional linear k? factorization for hard processes in a nuclearenvironment is by now well established. The realization of the nonlinear nuclear k? factorization whichemerges instead was found to change from one jet observable to another. Here we demonstrate how thepattern of nonlinear k? factorization, and especially the role of diffractive interactions, in the productionof dijets off nuclei depends on the color properties of the underlying pQCD subprocess.DOI: 10.1103/PhysRevLett.95.221803 PACS numbers: 13.87.a, 11.80.La, 12.38.Bx, 13.85.tFactorization theorems are part and parcel of the theoryof hard processes. In the perturbative quantum chromodynamics (pQCD) description of high energy hard reac-tions one uses the so-called k? factorization [1] in terms ofthe unintegrated gluon densities [2] in the target and beamhadrons [for a review see [3] ]. A concise discussion of theinadequacy of the collinear approximation and the neces-sity of fully unintegrated parton densities with explicitallowance for transverse momenta is found in [4].Fundamentally, the familiar pQCD factorization for free-nucleon interactions implies that hard cross sections arelinear functionals of parton densities in the projectile andtarget.An extension of factorization theorems to nuclear targetsis of burning urgency—the notion of nuclear gluon den-sities can be made meaningful only if they furnish a unifieddescription of various nuclear hard processes. Analyzingforward dijets in deep inelastic scattering (DIS) off nucleiwe found a striking breaking of k? factorization [5], con-firmed in a related analysis of single-jet spectra in hadron-nucleus (hA) collisions [6]. Based on the pQCD treatmentof diffractive dijets [7] one can define the collective nuclearunintegrated gluon density such that in terms of it the lineark? factorization would hold for the nuclear structure func-tion F2Ax;Q2 and the forward single jets in DIS off nucleibecause of their special Abelian features. Dijets in DIS,and dijet and single-jet spectra in hA collisions, however,prove to be highly nonlinear functionals of the collectivenuclear glue. For a criticism of unwarranted applications oflinear k? factorization to nuclear targets, see, e.g., [6]; ourconclusions on the breaking of linear k? factorization forhard scattering off nuclei have recently been corroboratedin [8,9].In this Letter, we relate the diverse k? factorizationproperties of dijet production to the color structure of thepQCD subprocess: g ! Q Q for leading quark-antiquarkdijets in DIS, qg ! qg for the dominant dijets in theproton (deuteron) hemisphere of hA collisions (h  p; d)at the Relativistic Heavy Ion Collider (RHIC), and gg !Q Q for open charm production. The nonlinear k? factori-zation derives from the interaction cross section for a05=95(22)=221803(4)$23.00 22180color-singlet system of n partons being a (matrix) super-position of elementary dipole cross sections, and theirnuclear S matrices being a (matrix) product of nuclear Smatrices for elementary dipoles. We establish universalityclasses for nonlinear k? factorization. Our results shedcertain light onto which extent hard processes in a nuclearenvironment can be described entirely in terms of theclassical gluon field of a whole nucleus [10].We require coherency over the whole nucleus, x Q2  M2JJ=2mpEa & xA  2RAmp1 	 0:1A1=3,where RA is the radius of the target nucleus of mass numberA, Q2 and Ea are the virtuality and energy of the beamparton a in the target rest frame, MJJ is the dijet mass, andmp is the proton mass [11]. At RHIC this correspondsprecisely to qg dijets in the largest rapidity bins in theproton hemisphere of pA collisions.To the lowest order in pQCD, all the above processes, inthe laboratory frame, can be viewed as an excitation of theperturbative jbci Fock state of the physical projectile jaiby one-gluon exchange with the target nucleon. For nucleartargets one has to deal with the non-Abelian intranuclearevolution due to multiple gluon exchanges which are en-hanced by a large target thickness. The kinematical con-straints in pA collisions at RHIC justify the focus on lowestFock states only; understanding the rich pattern of non-linear k? factorization found in this regime is a must forthe further studies of small-x evolution.The master formula for the dijet spectrum is derived in[6] based on [5,12,13]; here we only reproduce the mainresult:da ! bcdzbd2pbd2pc 124Zd2bbd2bcd2bbd2b0c expipb  bbb0b ipc  bcb0czb;bbbczb;b0bb0c fS4b ccbb0b;b0c;bb;bc S2aab0;bS3b cab;b0b;b0cS3abcb0;bb;bcg: (1)If ba  b is the projectile’s impact parameter, then bb 3-1 © 2005 The American Physical Societya bc+--(3)S(3)SS(4)bc(2)Sa a+ - - bcaaacbbbc cFIG. 1 (color online). The S-matrix structure of the two-bodydensity matrix for the excitation a ! bc.PRL 95, 221803 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending25 NOVEMBER 2005b zcr, bc  b zbr, where zb;c stand for the fraction ofthe light cone momentum of the projectile a carried bypartons b and c. For all reactions, z; r and its momen-tum space version z;p stands for the light cone wavefunction of the jbci Fock state of the projectile; its con-nection to the parton-splitting functions is found in [6]. AllSn describe a scattering of color-singlet systems of npartons, as indicated in Fig. 1. S2 and S3 are readilycalculated in terms of the 2-parton and 3-parton dipolecross sections [13,14]; general rules for the multiple scat-tering theory calculation of the coupled-channel S4 arefound in [5].The unintegrated gluon density in the target nucleon,F x; 2  @Gx; 2=@ log2, furnishes a universal de-scription of the proton structure function F2px;Q2 andof the final states. For instance, the linear k? factorizationfor forward dijets reads [for applications, see [15] andreferences therein]222dN!Q Qdzd2pd2fx;jz;pz;pj2;(2)where   p p Q is the jet-jet decorrelation momentum,and p  p Q, z  z Q refer to the Q jet.We turn to nuclear targets. We define the collectivenuclear gluon density, b; x;, per unit area in theimpact parameter plane, in terms of the q q nuclear profilefunction [5,7]:221802Ab; x; r  1 exp 12x; rTbZd2b; x;1 expi  r: (3)Here Tb is the optical thickness of a nucleus,x; r Zd2fx;1 expi  r (4)is the q q dipole cross section [14], andfx;  4SrNc14F x; 2: (5)The nuclear glue b; x; is a highly nonlinear functionalof the free-nucleon glue, its expansion in terms of thecollective glue for j overlapping nucleons in the Lorentz-contracted nucleus [11], its saturation properties at small ,and the saturation scale QAb; x are found in [5,7]. Theinclusive spectrum of leading quarks from the excitation ! Q Q off nuclei is linear k? factorizable in terms ofb; x; as its counterpart for the free-nucleon target interms of fx;, see Eq. (2)—such an Abelianization onlyholds for the color-singlet projectile. We shall also makeuse of b; x;  Sabsb2 b; x;, whereSabsb  exp 120xTb and 0x Rd2fx;is the dipole cross section for large dipoles.The t-channel pQCD gluon exchange leaves the targetnucleon debris in a color excited state. In the case ofnuclear targets one must distinguish truly inelastic pro-cesses and coherent diffraction aA ! bcA with retentionof the target nucleus in the ground state.Here we report closed-form analytic results for nucleardijet spectra to the leading order in 1=Nc; higher orders canbe derived following Ref. [5]. In DIS off nuclei, dipolespropagate as color singlets until at depth  from the frontface of a nucleus they excite into the octet state; the non-Abelian evolution in the slice ; 1 consists of colorrotations within the octet state [16].Combining the truly inelastic and diffractive compo-nents of DIS,22dA ! Q Qd2bdzd2pd2 12TbZ 10dZd21d2fx;1 ; b; x; 1  1 ; b; x;1 j; z;p 1 ; z;p 1  j2  2j1; z;p z;pj2; (6)where ; z;p  Rd2;b; x;z;p  is thewave function of the incident dipole distorted by the co-herent initial state interaction (ISI) in the slice 0;  of anucleus, and the collective nuclear glue for the slice  of anucleus, ;b; x;, is defined by [5]exp 12x; rTbZd2; b; x; expi  r:(7)The diffractive component of Eq. (6), / 2, givesexactly back-to-back dijets [for the  dependence forfinite-size nuclei, see Ref. [7] ]. It is a quadratic functionalof the collective nuclear glue and makes 50% of the totalDIS off heavy absorbing nuclei [17]. The first componentin (6)—truly inelastic DIS—is of fifth order in gluondensities: a quartic functional of the collective nuclearglue for the two slices of a nucleus which describe thecoherent ISI’s of color-singlet dipoles in the slice 0; 3-2PRL 95, 221803 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending25 NOVEMBER 2005and incoherent FSI’s of color-octet dipoles in the slice; 1, and a linear one of the free-nucleon glue fx;which describes the hard singlet-to-octet transition. Thestriking distinction between ISI and FSI requires the col-lective nuclear glue (7) for slices of the nucleus and not theclassical gluon field of the whole nucleus. The nonlineark? factorization result (6) must be contrasted to the free-nucleon spectrum (2); it entails a nuclear enhancementof the decorrelation of dijets from truly inelastic DIS,while semihard dijets, jpQ Qj2 & Q2Ab; x, are completelydecorrelated.The k? factorizable free-nucleon q ! qg dijet crosssection is the differential form of the single-jet spectrum,22180Eq. (82) of Ref. [6] (here p  pg, z  zgg):222dNq!gqdzd2pd2fx;jz;pz;pj2jz;pz;pzj2:(8)We proceed to nuclear targets. The set of color-singlet 4-parton states qg q0g0 of interest consists of j33i, j66i, andj1515i states. The nuclear dijet spectrum takes the form22dAq ! qgd2bdzd2pd2 12TbZ 10dZd22d21d2;b; x;21 ; b; x; 1  2  CACF1 ;b; x;1fx;j; z;p 1  2 ; z;p 1  2  j2b; x;j1; z;p  z;p zj2  2Sabsbj1; z;p z;pj2: (9)Coherent diffractive excitation of color-triplet qg dipoles,/ 2, is suppressed by the nuclear attenuation becausethe initial parton q is colored. Note that 1; z;p iscoherently distorted over the whole thickness of the nu-cleus; consequently, the second term in (9)—the excitationof the color-triplet qg states—is a cubic functional of thenuclear glue. The first component of the nuclear spectrum(9)—excitation of the color sextet and 15-plet qg states—resembles strongly the truly inelastic dijet spectrum (6) forDIS: the free-nucleon glue fx; describes the hard ex-citation of the qg color dipole from the lower (triplet) tohigher (sextet and 15-plet) states; the coherent ISI distor-tions in the slice 0;  are similar, too. The incoherentdistortions are somewhat different, however. In DIS wehave an equal distortion of both outgoing parton waves bypure FSI. In the qg case, the uncorrelated FSI distortion ofthe outgoing quark and gluon waves in the slice ; 1 arecomplemented by incoherent ISI distortions of the incidentquark wave in the slice 0; . The ratio of Casimir opera-tors CA=CF ’ 2 in 1  CACF ;b; x; 1 for FSI of glu-ons is a reminder that the collective nuclear glue derivesfrom a density matrix in color space [5,6]. As such, exci-tation of color sextet and 15-plet final states exhibits a sixthorder nonlinearity: it is linear in the free-nucleon glue andfifth order in the collective nuclear glue.Heavy flavor excitation, g ! Q Q, off free nucleons isthe differential form of the single-jet spectrum derived in[6] (here p  p, z  z Q):222dNg!Q Qdzd2pd2fx;jz;pz;pzj2jz;pz;pzj2:(10)Here one starts with the color-octet Q Q dipole; intranu-clear interactions are color rotations in the space of theoctet states. Transitions to the color-singlet Q Q dipoles andcoherent diffraction are 1=Nc suppressed [5]. The non-Abelian evolution of the Q QQ0 Q0 system becomes asingle-channel problem and the resulting nuclear dijetcross section equals22dAg ! Q Qdzd2pd2bd2 Sabsbb; x;fjz;p z;p zj2  jz;p  z;p zj2gZd2b; x;b; x;   jz;p  z;p zj2: (11)Interestingly, this is precisely the differential version of thesingle-quark spectrum, Eq. (31) of Ref. [6], if in the non-linear term one makes a proper identification of momenta.It satisfies the quadratic-nonlinear k? factorization interms of the collective glue defined for the whole nucleus.We summarize: we established how the color propertiesof the relevant pQCD subprocess define the pattern ofnonlinearity: (a) excitation of dijets in higher color multip-lets from partons in a lower multiplet gives rise to theuniversality class with the fifth, or even sixth, order non-linearity of the dijet spectrum in gluon densities. Thisincludes the free-nucleon glue which describes hard exci-tation of higher color multiplets [see also the discussion ofthe 1=N2c  1 expansion in Ref. [5] ]; (b) in heavy flavor3-3PRL 95, 221803 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending25 NOVEMBER 2005excitation starting from already higher-multiplet partons(gluons) inelastic interactions can be viewed as color rota-tions within the same multiplet and the nonlinearity will bea quadratic one; (c) coherent diffraction is a universalityclass of its own, unsuppressed by large Nc and color sup-pressed for incident partons in lower and higher colormultiplets, respectively; this point, and the very existenceof coherent diffraction for incident colored partons, arealso new observations; (d) the contribution from dijets inthe same lower color multiplet as the incident parton formsa fourth universality class. Nonlinear k? factorization fur-nishes a unified pQCD description of dijet production in anuclear environment in terms of one and the same collec-tive nuclear glue of Eqs. (3) and (7). Such explicit quad-ratures for the dijet spectra are not contained in previousworks on the subject [8,9].Because of the important physics distinction of theinitial and final state interactions in class-A final statesthis collective glue must be evaluated for different slicesof a nucleus and nonlinear nuclear k? factorization cannotbe described in terms of the collective glue defined for thewhole nucleus. The latter remains useful for other univer-sality classes, though. By the coherency condition, x &xA 	 0:1A1=3, our formalism applies to the largest rapid-ity bins of the proton hemisphere of pA collisions at RHIC;it does not hold for the midrapidity dijets studied so far atRHIC [18].This work was partly supported by the Grant No. DFG436 RUS 17/101/04.*Electronic address: N.Nikolaev@fz-juelich.de†Electronic address: wo.schaefer@fz-juelich.de‡Electronic address: b.zakharov@fz-juelich.de[1] J. C. Collins and R. K. Ellis, Nucl. Phys. B360, 3 (1991);S. Catani, M. Ciafaloni, and F. Hautmann, Nucl. Phys.22180B366, 135 (1991); E. M. Levin et al., Sov. J. Nucl. Phys.53, 657 (1991).[2] E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, Sov. Phys.JETP 45, 199 (1977); I. I. Balitsky and L. N. Lipatov, Sov.J. Nucl. Phys. 28, 822 (1978).[3] J. R. Andersen et al. (Small x Collaboration), Eur. Phys. J.C 35, 67 (2004).[4] J. C. Collins and X. Zu, J. High Energy Phys. 03 (2005)059; J. Collins and H. Jung, hep-ph/0508280.[5] N. N. Nikolaev, W. Scha¨fer, B. G. Zakharov, and V. R.Zoller, J. Exp. Theor. Phys. 97, 441 (2003).[6] N. N. Nikolaev and W. Scha¨fer, Phys. Rev. D 71, 014023(2005).[7] N. N. Nikolaev, W. Scha¨fer, and G. Schwiete, Phys. Rev. D63, 014020 (2001); JETP Lett. 72, 405 (2000).[8] J. P. Blaizot, F. Gelis, and R. Venugopalan, Nucl. Phys.A743, 57 (2004).[9] J. Jalilian-Marian and Y. V. Kovchegov, Phys. Rev. D 70,114017 (2004).[10] L. McLerran and R. Venugopalan, Phys. Rev. D 49, 2233(1994); R. Venugopalan, hep-ph/0412396, and referencestherein.[11] N. N. Nikolaev and V. I. Zakharov, Sov. J. Nucl. Phys. 21,227 (1975); Phys. Lett. 55B, 397 (1975).[12] B. G. Zakharov, Sov. J. Nucl. Phys. 46, 92 (1987).[13] N. N. Nikolaev, G. Piller, and B. G. Zakharov, J. Exp.Theor. Phys. 81, 851 (1995); Z. Phys. A 354, 99 (1996).[14] N. N. Nikolaev and B. G. Zakharov, Z. Phys. C 49, 607(1991); 64, 631 (1994); J. Exp. Theor. Phys. 78, 598(1994).[15] A. Szczurek, N. N. Nikolaev, W. Scha¨fer, and J. Speth,Phys. Lett. B 500, 254 (2001).[16] Speaking of the adjoint and fundamental representations atarbitrary Nc as the octet and triplet states must not causeany confusion; the same refers to the triplet, sextet, and15-plet states after Eq. (8).[17] N. N. Nikolaev, B. G. Zakharov, and V. R. Zoller, Z. Phys.A 351, 435 (1995).[18] C. Adler et al. (STAR Collaboration), Phys. Rev. Lett.90, 082302 (2003).3-4

Nonuniversality aspects of nonlinear k(T)-factorization for hard dijets

Nonuniversality Aspects of Nonlinear k_\perp-factorization for Hard
  Dijets

Nonuniversality Aspects of Nonlinear k_\perp-factorization for Hard Dijets

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