Exploring origin of small x saturation in collinear approach
aa r X i v : . [ h e p - ph ] O c t Exploring origin of small x saturation in collinear approach
A.M. Snigirev , and G.M. Zinovjev Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 119991, Moscow, Russia Bogoliubov Laboratory of Theoretical Physics, JINR, 141980, Dubna, Russia and Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev 03143, Ukraine (Dated: October 3, 2019)A modification of the collinear evolution equations as an appropriate approach to improve thebehavior of parton distribution functions in the region of small longitudinal momentum fractions,and to find more theoretical arguments to clarify the possible appearence of saturation regime issuggested. It is argued that parton diffusion in the rapidity space at large parton densities along thespace-time evolution could result in the emergence of a natural saturation scale on which freezingactually occurs.
PACS numbers: 12.38.-t, 12.38.Bx, 13.85.-t, 11.80.La
I. INTRODUCTION
Nowadays it is widely recognized the hadron interac-tions at very high energies are driven by the states withvery high densities of partons (quarks and gluons), inparticular, with small longitudinal momentum fractions x . The routine theoretical framework for analyzing suchsystems is essentially grounded on the QCD collinearfactorization where the calculated cross sections are de-composed in the perturbative coefficient functions andnonperturbative parton densities of which evolvement istreated according the DGLAP equations [1–4]. Alreadythese linear equations capture qualitatively the traits as-sociated with an increase in the gluon densities at small x with extremely large Q values. The latter turn out alsoquite instrumental, for example, to justify by neglectingany type of higher twist corrections and some pertur-bative resummation contributions. An idea to follow theevolution within the perturbative paradigm and to evalu-ate the leading contributions at small x for not very large Q led to the development of the BFKL approach associ-ated to so-called high energy factorization. However, re-solving the corresponding BFKL equations [5–7] exhibitsa very strong raise (power-like) of the gluon density atsmall x that is stronger than the experimental data anal-ysis demonstrates and leads to apparent violation of uni-tarity at very small x . It signals some theoretical prob-lems generated by appearance of an infrared instabilityrelated to a diffusion with the rapidity evolution and theconsistent description of QCD coupling constant α s be-havior that should reflect a very sophisticated interplayof perturbative and nonperturbative QCD physics. Ap-parently, both look like an ensuing result of taking intoaccount the linear evolution only with resummation inthese approaches. From phenomenological point of viewan observation of a scaling law in wide range of small x and Q was done [8] thereby demonstrating an onset ofsaturation scale. This fact is quite interesting because itmay provide a perturbative scale in high density regionof small x where linear evolution approximtion works andprovides, in a sense, a boundary condition to the linearevolution equations. In fairness, remember it has been long time ago [9, 10] argued that eventually the systemunder consideration should enter a new regime, where therate of growing gluon density slows down and saturatespossibly curing, thus, a potential conflict with unitarityof the underlying scattering. Actually, the restorationof the unitarity in high energy limit of QCD remains achallenging problem, although several approaches, draw-ing a scenario with nonlinear behavior, are being ex-plored in past years (see, for example, [11–18] and ref-erences therein) but those allow us to conclude only thatwe are still no essentially closer to knowing where theproblem solution lies besides of very general claim aboutthe nonperturbative finite density effects which are leftout entirely from the BFKL evolution. The interest inphysics of high density regime of small x QCD is greatlyincreasinng and dictated by an avalanche of experimentaldata on collisions of relativistic heavy ions overwhelmingthis area of research in the last decades.Meanwhile, there is another opportunity to address theproblem in the framework of well-known DGLAP ap-proach that we would like to draw attention to in thisletter. It concerns one possible modificaion of collineartime-like evolution equations that has been also discussedlong time ago [19, 20] as well in the context of increasingparton multiplicity in electron-positron annihilation intohadrons. We adapt this modification for the space-likeevolution of parton distribution functions and demon-strate it develops a saturated regime of color glass con-densate [21]. This is a regime of strong color fields inwhich nonlinear dynamics come to the perceptible playand signals thereby an appearance of the natural satura-tion scale on which the evolution is, in fact, frozen, thusindicating also the universality of both phenomena. Ac-tually, such an approach is treated as an effective theoryof high energy scattering successefuly describing the datameasured in experiments.The paper is organized as follows. In Sec. II we re-view briefly the principal features of the DGLAP evolu-tion necessary in further. The particular modified QCDevolution is discussed in Sec. III. In Sec. IV the exten-sion to the double parton distribution functions is consid-ered. The possible phenomenological issues are discussedin Sec. V, together with some conclusions.
II. COLLINEAR EVOLUTION
One may take the value of hard scale as the evolutionvariable in the DGLAP approach. The most popularchoice is the transfer momentum squared Q , or its log-arithm ξ = ln( Q /Q ). The double logarithm that takesinto account explicitly the behavior of the effective cou-pling constant in the leading logarithm approximationproves very instrumental as well t = 2 β ln " ln( Q Λ )ln( Q Λ ) , (1)where β = (11 N c − n f ) / Q is the some char-acteristic scale above which the perturbative theory isapplicable, n f is the number of active flavors, Λ is theQCD dimensional parameter and N c = 3 is the colornumber. In Eq. (1) the one loop running QCD coupling α s ( Q ) = 4 πβ ln( Q / Λ ) (2)was used.The DGLAP evolution equations [1–4] assume the sim-plest form if we use the natural dimensionless evolutionvariable t ; that is, dD jh ( x, t ) dt = X j ′ Z x dx ′ x ′ D j ′ h ( x ′ , t ) P j ′ → j xx ′ ! . (3)These equations describe the evolution of single distri-butions D jh ( x, t ) of bare quarks, antiquarks and gluons( j = q, ¯ q, g ) within a hadron h in response to the changeof evolution variable t . The kernels, P , of these equationsinclude a regularization at x → x ′ and are known in theirappropriate forms.Equations (3) are explicitly solved by introducing theMellin transforms M jh ( n, t ) = Z dxx n D jh ( x, t ) , (4)which reduce those to a system of ordinary linear-differential equations at the first order: dM jh ( n, t ) /dt = X j ′ M j ′ h ( n, t ) P j ′ → j ( n ) , (5)where P j ′ → j ( n ) = Z x n P j ′ → j ( x ) dx. (6) In order to obtain the distributions in x representationthe inverse Mellin transformation should be performed xD jh ( x, t ) = Z dn πi x − n M jh ( n, t ) , (7)where the integration runs along the imaginary axis tothe right from all n singularities. It can be done in ageneral form numerically only. However, the asymptoticbehavior can be estimated in some interesting and simpleenough limits with the technique under consideration.The solutions of the DGLAP equations with the giveninitial conditions D jh ( x,
0) at the reference scale Q ( t =0) can be expressed by the Green functions D ji ( z, t ) inthe following way: D jh ( x, t ) = X i ′ Z x dzz D ih ( z, D ji ( xz , t ) . (8)These Green functions (gluon distributions at the partonlevel) D ji ( z, t ) are the solutions of Eqs. (3) at the partonlevel with the singular initial conditions D ji ( z, t = 0) = δ ( x − δ ij and in the double logarithm approximation(see, for instance, [3, 9]) look like xD gg ( x, t ) = 4 N c t exp [ − at ] I ( v ) /v ≃ N c tv − / exp [ v − at ] / √ π, (9)where v = p N c t ln (1 /x ) , a = 116 N c + 13 n f /N c , (10)and I is the standard modified Bessel function. This re-sult just illustrates the unitarity violation at very small x .In addition, one should also note that the mean numberof partons of type j in a parton of type i< n j > i = M ji (0 , t ) = [exp P (0) t ] ji (11)can not be correctly determined in the collinear approachbecause the kernels P g → g (0) and P q → g (0) are divergentand some improvements are necessary to be done at verysmall x . III. COLLINEAR EVOLUTION WITHDISSIPATION
The modification of collinear time-like evolution equa-tions was discussed in Refs. [19, 20] to take into ac-count the formation (so-called pionization) of soft quark-antiquark pairs at a hard quark (gluon) propagating. Inanalogy with the electron-photon showers the energy out-flow was phenomenologically simulated by the dissipativeterms in the evolution equations with rather interestingincome. Such a modification for the space-like evolutionhas, of course, another physical motivation in our casedue to the parton diffusion in the rapidity space at largeparton densities, and the following evolution equationsare suggested: ∂D ji ( x, t ) ∂t = X j ′ Z x dx ′ x ′ D j ′ i ( x ′ , t ) P j ′ → j xx ′ ! + γ j ∂D ji ( x, t ) ∂x (12)with γ j as some parameters characterizing the process ofthe energy outflow.In the situation of small dissipation, γ j ≪
1, the meannumber of partons can be calculated [20] by using theMellin technique. For credibility we bring here the re-sult for gluon multiplicity at the early evolution stage( t ≪
1) only referring to the transparent, but laborious,calculations performed in [20]: < n g > g = I ( V ) e − at + s N c t ln (1 /γ g ) ln s ln (1 /γ g )2 N c t I ( V ) e − at , (13)where V = p N c t ln (1 /γ g ) , (14)and I is another modified Bessel function. This result(13) reproduces exactly the mean number of gluons withthe longitudinal momentum fractions larger than x = γ g as calculated in the DGLAP unmodified approach. Theexercise above makes transparent the physical meaningof the dissipative term. It establishes the scale of en-ergy drift because gluons (partons) with the longitudinalmomentum fractions less than γ g are simply withdrawnfrom consideration. Moreover, the evolution is, in fact,frozen at the scale [19] Q fr = Λ ( Q / Λ ) γ g . (15)The origin of this freezing scale is similar to the sat-uration scale in the color glass condensate (CGC) ap-proach [13, 15–18, 21]. IV. GENERALIZING TO DOUBLE PARTONDISTRIBUTIONS
The extension of basic equations to double parton dis-tribution functions is straightforward: ∂D j j h ( x , x , t ) ∂t (16)= X j ′ − x Z x dx ′ x ′ D j ′ j h ( x ′ , x , t ) P j ′ → j x x ′ ! + γ j ∂D j j h ( x , x , t ) ∂x + X j ′ − x Z x dx ′ x ′ D j j ′ h ( x , x ′ , t ) P j ′ → j x x ′ ! + γ j ∂D j j h ( x , x , t ) ∂x + X j ′ D j ′ h ( x + x , t ) 1 x + x P j ′ → j j x x + x ! . Here, the splitting kernels,1 x + x P j ′ → j j ( x x + x ) , (17)which appear in the nonhomogeneous part of theequations, are the nonregularized one-loop well-knownDGLAP kernels without the “+” prescription. The un-modified equations were derived first in Refs. [22, 23]in framework of the DGLAP approach. The functions D j j h ( x , x , t ) in question have a specific interpretationin the leading logarithm approximation of perturbativeQCD. They are the inclusive probabilities which allowone to find two bare partons of types j and j with thegiven longitudinal momentum fractions x and x in ahadron h .The dissipative terms provide the energy outflow andestablish the scale of energy drift as well. Gluons (par-tons) with the longitudinal momentum fractions lessthan γ j are simply removed again from consideration foreach of two parton cascade branches practically indepen-dently. In the small x region we can restrict ourselves tohomogeneous evolution equations because the solutionsof nonhomogeneous unmodified equation are substantialat not parametrically small longitudinal momentum frac-tions only [24]. Moreover, the homogeneous evolutionequations (independent evolution of two branches) admitthe factorization of double parton distribution functions: D j j h ( x , x , t ) ≃ D j h ( x , t ) D j h ( x , t ) (18)as a good approximate solution, if such a factorizationwas assumed at the reference scale Q ( t = 0).Further we hold the leading exponential terms only ifthose have the same structure [25] both at the partonlevel and the hadron level under smooth enough initialconditions at the reference scale. Indeed, Eq. (8) in thedouble logarithm approximation reads xD gh ( x ; t ) ≃ Y Z dy [ zD gh ( z, | /z =exp y × exp [ p N c p t ( Y − y )] ∼ exp [ p N c √ tY ] (19)with Y = ln(1 /x ). The y integration is not as a saddle-point type, and, therefore, one of the edges, just y → z → z decreas-ing. Actually, one needs zD gh ( z, ∼ (1 /z ) a at z → a < A , where A = p N c t/Y >
0. Let’s no-tice that the parametrization of the initial gluon distri-butions, usually used, satisfies this condition (e.g., theCTEQ parametrization of Ref. [26]). Thus, as a resultwe have for the double gluon distributions [25] in thisappproximation: x x D ggh ( x , x , t ) ∼ exp [ p N c ( p t ln (1 /x ) + p t ln (1 /x ))] (20)with the infinite mean number of such gluons. If thetwo branches evolves independently then by introducingthe dissipative terms slow down the rate of gluon densityincrease and one gets the finite mean gluon numbers as < n gg > h ∼ exp [ p N c ( p t ln (1 /γ g ) + p t ln (1 /γ g ))](21)since the gluons with the longitudinal momentum frac-tions less than γ g are simply excluded. V. DISCUSSION AND SUMMARY
Clearly, the dissipative parameters above can not bedetermined within the DGLAP approach. They aretreated as the phenomenological parameters in numericalsimulations and shoud be estimated in the other modelsfor further applications. The phenomena of saturationand slowing down an increase of gluon density take placealso in the CGC scenario [13, 15–18]. However, the sat-uration scale is energy dependent in that approach and,nevertheless, it comes about quite predictive. For exam-ple, in the Golec-Biernat-Wusthoff (GBW) model [27, 28]it is parametrized by three parameters : Q s = Q ( x /x ) λ , (22)with Q = 1 GeV, x ≃ . λ ≃ . x in Eq. (22) allows us to estimate the dis-sipative parameter γ g that has a physical meaning similar to x . In fact, it justifies the assumption of small dissi-pation used in the previous Sections to obtain the crucialestimates (13) and (21) which are pretty encouraging toinvestigate the properties of modified collinear equationsfurther as a new alternative insight into the saturationphysics extending the initial limits of linear approach.In summary, the modified collinear evolution equationsare suggested to extract information on the propertiesof hot and dense QCD medium created in the experi-ments on heavy ion collisions searching the quark-gluonplasma, a thermolized phase, that may exist in very spe-sific regimes for very short periods of time. Compre-hensive phenomenological analysis of proton-proton col-lisions based on the QCD factorization, as a key instru-ment, made it possible to extract the universal distribu-tion functions validating such an approximation and openup (quite often) transparent ways for introducing the ef-ficient corrections. Truly, these corrections at leadingpower of the large momentum transfer are fairly generaland easily traceable but the corrections within the fac-torized forms turn out very complicated and too muchsensitive to the process details, as it was shown againmany years ago, because of the QCD multiple scaterringswhich differ [30] hadronic and heavy ion collisions signif-icantly. The model presented in this letter shows a pos-sibility of forming perturbatively a dynamical regime inparticular kinematical configuration which could not beforeseen according to the theoretical dogmas. As argued,it concerns a regime of high parton densities and dynam-ical interactions described definitely by nonlinear equa-tions. The evolution of hadron scattering amplitudes, atleast, in the framework of color dipole picture in such aregime is quite similar [31] to the time evolution of a clas-sical particles system undergoing reaction-diffusion pro-cesses. Amazingly, by introducing the dissipative termsresults in an origin of natural saturation scale on whichthe evolution is frozen and the gluons with longitudinalmomentum less than γ g are simply excluded. In the phe-nomenological applications the direct numerical solutionsof suggested modified equations may occur simpler thanthe BFKL treatment of very small x region. Acknowledgments
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