Gluon density and F 2 functions from BK equation with impact parameter dependence
aa r X i v : . [ h e p - ph ] F e b Gluon density and F functions from BK equation with impactparameter dependence S. Bondarenko a ) ∗ a ) University Santiago De Compostela, Spain
December 10, 2018
Abstract
In this note we fix the preliminary results obtained in the study of gluon density function of the paper[33]. The LO BK equation for unintegrated gluon density with impact parameter dependence is considered inorder to fix the parameters of the proposed model. In particular the form of initial condition for the equationsof proton-proton scattering from [33] is determined, which is similar to the form of fenomenological GBWansatz. The gluon density function and F function are also calculated and compared with the results forthe gluon density and F functions from the GRV parameterization for different values of Q . It is shown,that the results for F structure function of the considered model are in the good accordance with the resultsobtained from the GRV parameterization of parton densities. The attempts to understand the aspects of high energy scattering of nuclei and hadrons in terms of QCD BFKLpomerons, [1], led in the last time to a number of papers concerning the phenomenological applications of thehigh energy scattering as well as the pure theoretical properties of the theory, [2, 3, 4, 5, 6, 7, 8, 9]. In presentpaper we fix the first result obtained in the framework of the model proposed in [2, 3, 5], introducing the impactparameter dependence into the initial conditions and solving the equations of the model for each point in impactparameter space. The found parameters of the model, tuned with the help of DIS, are the first step toward thesolution of the problem of proton-proton scattering considered in [33].The DIS process is well described in the frameworks related with the BK equation, [10, 11], fenomenologicalmodels with the saturation properties and CGC models, see [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].Nevertheless, in rapidity evolution the impact parameter dependence of the amplitude was treated only approx-imately, see for example [17, 18], whereas the fenomenological models, such as GBW model [12, 13], neglect theevolution of the amplitude with rapidity. In papers [14, 21, 23] the approximate treatment of impact parameterdependence of gluon structure function was accounted together with DGLAP evolution of the function, and ∗ Email: [email protected] n papers [24, 25, 26] the factorized from momentum function impact parameter dependence in modified BKequation was considered. But still, the treatment of impact parameter dependence of the simple BK equationcompatable with the fenomenological models was missed.In present calculations of solution of BK equation we include the impact parameter dependence of theamplitude at initial values of rapidity and find the amplitude in each point of impact parameter space, solvingthe evolution BK equation with the help of the methods developed in [5], see also the papers [24, 25, 26] for similartechnics of calculations. In order to simplify the calculations, the solution is obtained in LO approximationand we discuss a possible generalization of the solution till NLO order in the conclusion. Another importantquestion, which we tried to answer on, it is a problem of the form of initial condition function for the BKequation with impact parameter dependence. The form of this function is general in the given framework ofinteracting BFKL pomerons and, as we mentioned above, the same function could be used in the proton-protonscattering, see [5, 33].Certainly, the results of calculations must be clarified with the help of well established results for gluondensity and/or with the help of DIS data. We perform the check of our calculations comparing calculated gluondensity function (integrated gluon density) and F function with the results given by the LO and NLO GRVparameterizations for DIS data, [27]. This comparison shows, that in the present framework we achieved thesatisfactory description of DIS data. The model based on BK equation with impact parameter dependenceshows a good coincidence with GRV parameterization and could be used as a independent parameterization ofunintegrated gluon density, [33].The paper is organizes as follows. In the next section we shortly describe a formalism of calculations. InSec.3 and Sec.4 we present the results of calculations for F structure function and gluon density functioncorrespondingly. Section 5 is a conclusion of the paper. In this section we shortly write the main formulae used in our calculations. The F structure function of DISwith impact parameter dependence we define as follows F ( x, Q ) = Q π α Z d b Z d kk f ( x, k , b )4 π (cid:0) Φ T ( k, m q ) + Φ L ( k, m q ) (cid:1) (1)The unintegrated gluon density function f ( x, k , b ) we find solving the BK equation for each point in impactparameter space: ∂ y f ( y, k , b ) = N c α s π k Z da a (cid:20) f ( a , b ) − f ( k , b ) | a − k | + f ( k , b )[4 a + k ] (cid:21) − πα s (cid:20) k Z k da a f ( a , b ) Z k dc c f ( c , b ) + f ( k , b ) Z k da a log (cid:18) a k (cid:19) f ( a , b ) (cid:21) (2)where we introduced the rapidity variable y = log(1 /x ). In Eq.2 we assumed that the evolution is local in thetransverse plane, i.e. impact parameter dependence of f ( y, k , b ) appear only throw the initial condition for f ( y, k , b ) f ( y = y , k , b ) = f in ( k , b ) (3)In order to exclude part of ambiguities in the solution of BK equation arising due the non included NLOcorrections, we perform the following substitute in the equation f ( y, k , b ) → f ( y, k , b ) α s ( k ) α s = ˜ f ( y, k , b ) α s , (4)2btaining ∂ ˜ y ˜ f (˜ y, k , b ) = N c π k Z da a " ˜ f ( a , b ) − ˜ f ( k , b ) | a − k | + ˜ f ( k , b )[4 a + k ] − π (cid:20) k Z k da a ˜ f ( a , b ) Z k dc c ˜ f ( c , b ) + ˜ f ( k , b ) Z k da a log (cid:18) a k (cid:19) ˜ f ( a , b ) (cid:21) (5)where ˜ y = α s y . The value of α s is a constant in the LO approximation and we consider α s as the parameterof the model which must be determined from the fit of DIS data.The impact factors in Eq.1 are usual impact factors of the problem with three light quarks flavors of equalmass included. They are the following (see [28] for example):Φ L ( k, m q ) = 32 π α X q =1 e q Z dρdη k η (1 − η ) ρ (1 − ρ ) Q (cid:0) Q ρ (1 − ρ ) + k η (1 − η ) + m q (cid:1) (cid:0) Q ρ (1 − ρ ) + m q (cid:1) (6)and Φ T ( k, m q ) = 4 π α X q =1 e q Z dρdη ·· k Q (cid:0) ρ + (1 − ρ ) (cid:1) ρ (1 − ρ ) (cid:0) η + (1 − η ) (cid:1) + k (cid:0) ρ + (1 − ρ ) (cid:1) m q + 4 ρ (1 − ρ ) η (1 − η ) m q (cid:0) Q ρ (1 − ρ ) + k η (1 − η ) + m q (cid:1) (cid:0) Q ρ (1 − ρ ) + m q (cid:1) (7)We exclude α s from the definition of the impact factors rewriting the F structure function in the following way F ( x, Q ) = Q π α Z d b Z d kk ˜ f ( x, k , b )4 π (cid:0) Φ T ( k, m q ) + Φ L ( k, m q ) (cid:1) = Q π α Z d b S ( x, b, Q ) (8)Due the including quark masses in the calculations, the rapidity variable y (Bjorken x) in BK equation is alsomodified, see details in [12, 13]. For each fixed rapidity y of the process, the value rapidity taken in BK equationis changed y → y − ln(1 + 4 m q Q ) (9)The form of the function ˜ f ( y, k , b ) at initial rapidity, i.e. initial condition for the BK equation Eq.5, hasbeen borrowed from the form of GBW ansatz, [12, 13], with introduced impact parameter dependence˜ f ( y = y , k , b ) = 34 π k R e b /R p exp ( − k R e b /R p ) (10)Additionally to the α s from the BK equation Eq.5 there are three more parameters, which are initial rapidityof evolution y , radius of the proton R p and ”saturation” radius R . These parameters must be found from thefitting of DIS data and they are presented in the next section. The plots of the functions S ( x, b, Q ) from Eq.8are given in Fig 1. F function and the parameters of the model The parameters of the model we determine fitting the DIS data for the total cross section and F structurefunction, [29]. In this note we present the parameters of the model and results of the calculations of F for onlyfew values of Q with only three light quarks flavors included, more results, including the application of themodel to the ”soft” proton-proton scattering, will be presented in the mentioned above paper [33]. The Table 1shows found values of the parameters of the model and plots Fig 2 present the results of calculations for the F structure function. It must be mentioned, that instead the the λ , λ GBW parameters which determines the3 = 0.25 GeV -1 ] 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009S(x,b,Q ) Q = 0.65 GeV -1 ] 0 0.001 0.002 0.003 0.004 0.005 0.006S(x,b,Q )Q = 2.5 GeV -1 ] 0 0.0005 0.001 0.0015 0.002 0.0025 0.003S(x,b,Q ) Q = 12 GeV -1 ] 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001S(x,b,Q )Q = 60 GeV -1 ] 0 5e-05 0.0001 0.00015 0.0002 0.00025S(x,b,Q ) Q = 120 GeV -1 ] 0 2e-05 4e-05 6e-05 8e-05 0.0001 0.00012 0.00014S(x,b,Q ) Figure 1:
The impact parameter profile of S ( x, b, Q ) as a function of x . y ( x ) R p ( GeV ) R ( GeV ) α s m q ( GeV )3 . . The parameters of the model. α s is the parameter which determines the energy behavior of the gluon density functions. The smallnessof obtained value of α s is explained by the LO precision of the calculations and by the variable change Eq.4.In present scheme the value of α s determines the evolution length in rapidity space independently on values of Q , i.e. this is some ”averaged” value for α s found from the data fitting. The found value of quark mass is alsodifferent from the numbers of [12, 13] for example, being nevertheless in the range of possible quark masses oflight quarks flavors used in [21, 23]. The radius of the proton from Table 1 is close to the experimental value ofthe proton shape found from the t-distribution of J/ψ meson of [29], in fact the results of this measurementsrestrict the possible numericall values of this parameter.The main difference of our calculations from the results of other models are the results for F structurefunction at very small values of Q . The two top plots of Fig 2, calculated at small values of Q , show thatour model fails to describe the F data at Q ∼ . , GeV . It means, that saturation effects which providesthe description of the low Q data in ”canonical” saturation model, such as [12] for example , in our frameworkare not so strong as there. The possible reasons for such a deviations from the ”normal” results obtained atlow Q and small x we discuss in conclusion. Nevertheless, the comparison of the obtained results with theGRV parameterization results for F function for Q > GeV shows a good coincidence, they stay in thelimits of differences between the results of GRV parameterization with the results given by other parton densityparameterizations, such as [30]for example . In order to estimate the possible effects of the variables change Eq.4 it is instructive to calculate the valuesof integrated gluon density. Indeed, as it seems from Fig 2 the LO and NLO GRV parameterization give veryclose results for the F structure function. In the same time, the integrated gluon density functions are verydifferent for LO and NLO GRV curves at the same values of Q . We present the obtained plots in Fig 3 for theintegrated gluon density functions at different values of Q .There are two possible resulting plots of the model in Fig 3, which we denotes as BK and BK curves.The reason for a existing of two curves is a following. Let’s consider the definition of integrated gluon densityfunction xG ( x, Q ) = Z d b Z Q d k k f ( x, k , b ) (11)Due the variable change Eq.4 we obtain a integrated gluon density function in terms of the new function ˜ fxG ( x, Q ) = Z d b Z Q d k k ˜ f ( x, k , b ) α s ( k ) → α s Z d b Z Q d k k ˜ f ( x, k , b ) (12)From the Eq.12 it is clear, that the definition of the xG ( x, Q ) in terms of ˜ f has a ambiguities in LO schemecalculations due the running coupling α s ( k ) under the integral over k . In our LO calculations we need tochoose the fixed LO value of the α s ( k ) and to extract it from the integration over k in Eq.12. Therefore, weconsidered two possibilities for the fixed α s value. As the first one we took the value of α s from the Table 1,obtained in the fitting of the data. This choice results are denoted as BK curves in the Fig 3. Another choicefor the α s is the values of α s from the NLO GRV parameterization taken separately for coresponding Q . Theresults for this value of α s is denoted as BK in the Fig 3.5 (x,Q ) Q = 0.25 GeV -5 -4 -3 F (x,Q ) Q = 0.65 GeV -4 -3 F (x,Q ) Q = 2.5 GeV -4 -3 -2 F (x,Q ) Q = 12 GeV -4 -3 -2 F (x,Q ) Q = 60 GeV -3 -2 F (x,Q ) Q = 120 GeV -3 -2 Figure 2:
The F data for different values of Q : the present model results (solid lines), LO GRV parame-terization (dashed lines) and NLO GRV (doted lines) as functions of x . The GRV results are restricted by Q > . GeV . G(x,Q ) Q = 2.5 GeV GRV (LO) BK GRV (NLO)BK -4 -3 -2 xG(x,Q ) Q = 12 GeV GRV (LO) BK GRV (NLO)BK -4 -3 -2 xG(x,Q ) Q = 60 GeV GRV (LO)BK GRV (NLO) BK -3 -2 xG(x,Q ) Q = 120 GeV GRV (LO)BK GRV (NLO) BK -3 -2 Figure 3:
The integrated gluon density function: the present model results (solid lines), LO GRV parameteriza-tion (dashed lines) and NLO GRV (doted lines) as functions of x . α s for each Q in Eq.12 gives more correct value of xG ( x, Q ) comparing withthe common α s value from the data fit. This fact related with the use of the LO ˜ f function in our scheme ofcalculations and, therefore, each calculation of the xG ( x, Q ) needs the redefinition of the present α s value. Thesecond conclusion concerns the shape of the found curves. It is easy to see, that both BK and BK curveshave a shapes similar to the NLO GRV curve and pretty different from the LO GRV curve for integrated gluondensity. This is a sign, that the simple redefinition of the variables Eq.4 in BK equation allows to include a somepart of NLO corrections to the integrated gluon density and F functions. It is important to underline again,that obtained integrated gluon density function is similar to the integrated gluon density function obtained withthe use of GRV parameterization. We demonstrated, that based on QCD BFKL pomerons BK evolution equation for unintegrated gluon densitywith impact parameter dependence could be used as a calculation tool for the F structure function andintegrated gluon density function. In the large range of energies and large range of values of Q we obtained agood description of DIS data for F structure function. We note, that the obtained results are in good agreementwith results obtained with the help of GRV parameterization of parton densities and therefore could be usedas independent parameterization of unintegrated gluon density. It is important, because for our frameworkit means that we not only reproduced the results for DIS using more complex theory than usual evolutionequations without impact parameter dependence, but also that we found the initial conditions for the proton-proton scattering in the framework of Braun equations [2, 3, 5]. Therefore, the obtained impact parameterdependent parameterization of proton shape Eq.10 with parameters of Table 1 allow to apply formalism of[7] to the important and more general case of proton-proton scattering, see [33]. Another interesting field ofthe application of the proposed model, is the description of the processes of exclusive particle production. Asit was shown in [4], the account of impact parameter dependence of the proton-proton scattering amplitudeis very important for the better understanding and better description of the low momentum region and NLLcorrections in the resulting amplitude of the process of exclusive Higgs production.The unexpected result, obtained in present calculations, it is a bad agreement between the calculations inour approach and results of similar approaches in description of F function at small Q = 0 . GeV . Usually,this region of the small values of Q at high energies is considered as a region where the saturation effects arelarge, see [12]. In our case, as it seems, the evolution over rapidity in impact parameter space does not leadto the saturation effects which will generate appropriate slope of F function at very small Q and small x .The reasons for such a distinction from usual saturation models behavior is not clear. The including of NLOcorrections into the calculation scheme could, in principal, to improve the situation. At small values of Q theeffect of NLO corrections must be large, it is clear if we will consider the averaged value of α s obtained in ourfit. This value is very small and in the theory with running coupling constant at small Q this value must bechanged a lot, giving a more appropriate result for F at small Q .Another approach to this problem, is that the DIS process at small Q physically is very similar to thehadron-hadron scattering, see [31] for example. From this point of view it is not clear why the simple ”fan”structure of BK equation must work well at small values of Q . More complicated ”net” diagrams of interactingpomerons became to be important in this case , see [2, 3, 7, 8, 32], and absence of these diagrams in BK equationcould lead to the wrong results for DIS at small Q . Interesting to note, that from the formal point of view8hese ”net” diagrams are also part of NLO correction to the unintegrated gluon density, which arise from thefield theory part of the process and not from the corrections to the BFKL kernel. We plan to investigate thisquestion in our future studies of the gluon density function in the framework with NLO corrections included. Acknowledgments
I am especially grateful to Y.Shabelsky for the discussion on the subject of the paper and to Leszek Motyka forthe help and useful comments. This work was done with the support of the Ministerio de Educacion y Cienciaof Spain under project FPA2005-01963 together with Xunta de Galicia (Conselleria de Educacion).
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