aa r X i v : . [ h e p - ph ] F e b Physical limits in the Color Dipole Model Bounds

G.R.Boroun ∗ Physics Department, Razi University, Kermanshah 67149, Iran (Dated: February 19, 2021)The ratio of the cross sections for the transversely and longitudinally virtual photon polarizations, σ γ ∗ pL/T , at high photon-hadron energy scattering is studied. I investigate the relationship betweenthe gluon distribution obtained using the color dipole model and standard gluons obtained fromthe Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution and the Altarelli-Martinelliequations. It is shown that the color dipole bounds are dependent on the gluon distributionbehavior. This behavior is considered by the expansion and Laplace transform methods. Numericalcalculations and comparison with the color dipole model (CDM) bounds can indicate the range ofvalidity of this method at small dipole sizes, r ∼ /Q ≪ /Q s .

1. Introduction

Dipole model provides a convenient description of deepinelastic scattering (DIS) at small x . In this region, par-tons in the proton form a dense system. Processes suchas mutual interaction and recombination leads to the sat-uration of the total cross section. We know that the DIScross section is factorized into a light-cone wave function.As the virtual photon splits up into a quark-antiquarkpair (dipole). Usually, this contribution ( γ ∗ → qq ) is de-ﬁned as a convolution of the inﬁnite momentum framewave function with the pQCD calculable coeﬃcient func-tions. These coeﬃcient functions describing the shortdistance propagation of the particles between two vir-tual photon vertices. Many of experimental provided atHERA have been analysis in terms of this model whichthe dipole picture acts like a quark-antiquark dipole [1-4]. We need to know that the lifetime of the ﬂuctuationof the photon into the color dipole is much larger thanthe typical timescale of the dipole-proton interaction [5].In this reference frame, the photon splits into qq -dipoleand then interacts with proton as qq lifetime is about 1 /x times longer than time of interaction with proton.The saturation region is approached when the reactionis mediated by multi-gluon exchange. This aspect of sat-uration is closely linked to unitarity. Indeed the growthof the gluon density is slowed down at very small x bygluon -gluon recombination ( g + g → g ). In this regionthe gluon density in hadrons can become nonperturba-tively large. It means that it is the regime of gluon sat-uration. This is the origin of the shadowing correctionin pQCD interactions [1]. One of the implementation ofmultiple scattering in colour dipole model is based onthe Glauber-Mueller (GM) eikonal approach [2], whichis used in Golec-Biernat-W¨usthoﬀ (GBW) model [3]. Inthis approach the multiple colour dipole scatters are in- ∗ Electronic address: [email protected]; [email protected] dependent of each other which describing the multi-gluondensity in the proton by the b-Sat model [4]. As the largesaturation eﬀects are required to transition from the hardPomeron behavior at small dipole sizes to soft Pomeronbehavior at large dipole sizes [6].An eﬀective ﬁeld theory describing the small- x regime ofQCD is the color glass condensate (CGC) [7,8,9]. Thismodel is based on the Balitsky-Kovchegov (BK) [10]non-linear evolution equation and improves the Iancu-Itakura-Munier (IIM) dipole model [11]. The non-lineareﬀects appear in CGC formalism when the gluon densitybecomes large. Indeed in high energy collisions which x decreases, the number of gluons increase. When gluonsare highly coherent at infrared scale, the gluon satura-tion leads to the Glasma [8,9,10,11]. Indeed the Glasmais matter produced from the CGC after a collision. Afterthe collisions, Glasma is formed in the region betweenthe two sheets of colored glass. At high energy scatter-ing with evolution and recombination of gluon density,one can probes the number of gluons with a given x andtransverse momenta k ⊥ ≤ Q as the gluon number is de-ﬁned by x dN g dx ( Q ) = α s C R π Z Q Λ QCD dk ⊥ k ⊥ = α s C R π ln( Q Λ QCD ) (1)where Λ

QCD is the QCD cutoﬀ and C R is the SU ( N c )Casimir operator [9]. The HERA data collected on theinclusive γ ∗ p cross section for x ≤ .

01 indicate a scalingas a function of the ratio Q /Q s ( x ). Which Q s ( x ) = Q ( x /x ) λ is the saturation scale with dimensions givenby a ﬁxed reference scale Q [12]. Indeed saturation scaleis a border between dense and dilute gluonic system. For Q < Q s the linear evolution is strongly perturbed bynonlinear eﬀects, while for Q > Q s the linear evolutionis dominated and evolution of parton densities is gov-erned by the DGLAP equations.The proton structure function F and the longitudinalstructure function F L can be can be written in terms of γ ∗ p cross section as follows F ( x, Q ) = Q π α em [ σ γ ∗ pL ( x, Q ) + σ γ ∗ pT ( x, Q )] (2) F L ( x, Q ) = Q π α em σ γ ∗ pL ( x, Q ) (3)where α EM is the electromagnetic ﬁne structure con-stant. Here the subscripts L and T denote the longitu-dinal and transverse polarizations of the virtual photon.The reduced cross section σ r is expressed in terms of theinclusive proton structure function F and F L as σ r ( x, y, Q ) = F ( x, Q ) − y − y ) F L ( x, Q ) (4)where Q is the virtuality of exchanged photon, √ s denotes the center of mass energy in ep collisions and y = Q / ( sx ) is the inelasticity variable.The total cross section behavior in some literatures [3,13]is based on a logarithmic behavior in x which do notviolate unitarity. This behavior has been supplementedby unitarity correction. In accordance with the Froissartpredictions [14], authors in Ref.[15] have suggested aparameterization of the structure function F ≤ ln (1 /x )at large s . This parameterization implies that its growthis limited by the Froissart bound as Bjorken x → x providevaluable information about the unitarity limits of QCDand parton saturation eﬀects at future experiments suchas an Electron-Ion Collider (EIC) [16], and also thelarge Hadron Electron collider (LHeC) [17,18] and theFuture Circular Collider program (FCC-eh)[19]. Alsosome theoretical analysis at low x have considered thelongitudinal structure function F L ( x, Q ) to describeprocess [20,21].The paper is organized as follows. In sect.2, we give asummary about the color dipole model. This sectionreviews the relevant formulae of the dipole picture. Wewill study the color dipole model bounds with respectto the proton structure function F at low values of x insection 3. It is the purpose of this paper to predict thebehavior of the color dipole bounds at large Q values atleading order and next-to-leading order approximations.I will attempt to preserve analysis of Altarelli-Martinelliequation and its solution in DGLAP framework for verysmall dipoles. In the following the Laplace transformmethod of the gluon distribution function with respectto the transversal and longitudinal structure functions,which both obey the Froissart boundary conditions,in the LO and NLO approximations at low values of x are discussed. Explicit expressions for the Wilsoncoeﬃcients at the LO are given as well and at theNLO are given for each Q values for simplicity. Theseparameters at LO and NLO are compared with bounds in the color dipole model. Section 4 contains the resultsand discussions. Numerical results for the extractedthese parameters in the expansion method and Laplacetransform method, together with comparisons withthe color dipole bounds are presented in this section.Also the W dependence of the extracted parameters isdiscussed (where W is the energy in the photon-protoncenter of mass). Conclusions and summary are summa-rized on Sec.5. Three Appendices contain results usedin the main text. In Appendix A the gluon density isdiscussed with respect to expansion at distinct points ofexpansion. Appendix B explains the steps for obtainingthe gluon density by Laplace transform method. Themost cumbersome expressions for the parameterizationof the longitudinal structure function at LO and NLOapproximations are relegated to Appendix C.

2. A Short Theoretical Input

In this section we brieﬂy present the theoretical part ofthe color dipole model. For small values of the Bjorkenvariable x , the correct degrees of freedom in the highenergy γ ∗ p scattering ( γ ∗ emitted by the incident elec-tron) are given by qq colorless dipoles [1]. In the dipolepicture, the γ ∗ p interaction process can be described asfollows that ﬁrst the virtual photon splits into a qq col-orless dipole. Then quark and antiquark interact withproton through radiant gluons. In the leading order, thegeneric structure of the ( qq ) p is well described by two-gluon exchange [4]. In deep inelastic scattering (DIS), thesmall- x saturation means that the partons in the protonform a dense system. This system with mutual inter-action and recombination leads to the saturation of thecross section [22]. In the saturation region, the single-gluon exchange changes into multi-gluon exchange. Thisprocess also can be extended to the next-to-leading or-der (NLO) corrections as described in Refs.[23,24]. InRef.[24] authors discussed the corresponding correctionsin the γ ∗ Fock state by adding a new qqg component tothe qq -state.Within the dipole framework of the γ ∗ p scattering σ γ ∗ pL,T ( x, Q ) = Z d r Z dzψ ∗ ( Q, r, z )ˆ σ ( x, r ) ψ ( Q, r, z ) . (5)Indeed the scattering of the virtual photon on the protoncan be conceived as a virtual photon ﬂuctuation into aquark-antiquark pair, then the produced quark-antiquarkdipole interacts with the proton via gluon exchanges [7].The integrands are given by the squared of the light conewave functions of the virtual photon and the scatteringamplitudes for the dipole cross section. The ﬁrst integra-tion is so-called dipole representation where transversemomentum k T is replaced by its Fourier conjugate vari-able r . Here r ( ≡ | r | ) is the ﬁxed transverse separation ofthe quarks in the qq pair. Here the quark (or antiquark)carries a fraction z of the incoming photon light-cone en-ergy (0 < z < σ ( x, r ) isusually assumed to be independent of z and it is a so-lution of the generalized BFKL equation [25]. Also itis universal for all ﬂavors and the x dependence of itscomes from the QCD evolution eﬀects described by thegeneralized BFKL equation. The squared wave functionof the qq Fock states of the virtual photon is given by thefollowing equations | Ψ T ( z, r ) | = 6 α em π n f X e f { [ z + (1 − z ) ] ǫ K ( ǫr )+ m f K ( ǫr ) } , and | Ψ L ( z, r ) | = 6 α em π n f X e f { Q z (1 − z ) K ( ǫr ) } , (6)where n f is the number of active quark ﬂavor. In theabove equations ǫ = z (1 − z ) Q + m f and m f is the quarkmass. e f is the quark charge and the functions K , arethe Bessel-McDonald functions. The mass of qq dipole isrealized by M qq = −→ k ⊥ z (1 − z ) . Which the transverse momen-tum −→ k ⊥ is introduced into four momenta of the quark andantiquark. If the three momenta −→ q = −→ k + −→ k ′ is deﬁnedin the direction of the z -axis of a coordinate system, thenthe quark and antiquark momenta represented by −→ k = z −→ q + −→ k ⊥ , −→ k ′ = (1 − z ) −→ q − −→ k ⊥ (7)where −→ k ⊥ . −→ q = 0. With respect to the center-of-massenergy W , the restriction on masses of the qq states isdeﬁned by M qq W ≪ . , (8)where the Bjorken variable x ∼ = Q W ≪ .

1. In this approachthe photoabsorption cross section can be factorized in thefollowing form σ γ ∗ pL,T ( x, Q ) = Z dzd r ⊥ | Ψ L,Tγ ( r ⊥ , z (1 − z ) , Q ) | × b σ qq ( r ⊥ , W ) , (9)where σ qq ( r ⊥ , W ) is the color-dipole cross-section b σ ( qq ) p ( r ⊥ , W ) = Z d −→ ℓ ⊥ e σ ( qq ) p ( −→ ℓ ⊥ , W ) × (1 − e − i −→ ℓ ⊥ −→ r ⊥ ) , (10)which the variable r ⊥ determines the transverse qq -separation variable and −→ ℓ ⊥ stands for the transverse momentum of the absorbed gluon. In the above inte-gral (i.e., Eq.(10)), the ﬁrst term is associated with thegluon transverse momentum distribution and the secondterm is the QCD gauge theory structure [26,27].On the other hand, the dipole cross section was proposedto have the following form [3,20] b σ ( qq ) p ( x, r ) = σ { − exp( − π r α s ( µ ) xg ( x, µ )3 σ ) } , (11)where σ is a parameter of the model and determinedfrom a ﬁt to small- x data. This form of the dipole crosssection imposes the unitarity condition at large dipolesizes r as b σ ( qq ) p ≤ σ . For small dipole sizes r , the dipolecross section is in agreement with the phenomenon ofcolor transparency resulting from pQCD. The right-handside of Eq.(10) is proportional to Eq.(11) in the small- r region as [26] α s ( Q ) xg ( x, Q ) = 34 π Z d −→ ℓ ⊥ −→ ℓ ⊥ e σ ( qq ) p ( −→ ℓ ⊥ , W ) . (12)Plotting the experimental data for σ γ ∗ p (where σ γ ∗ p = σ γ ∗ pT + σ γ ∗ pL = 4 π α em F /Q ) as a function of the scalingvariable η ( W , Q ) = Q + m Λ sat ( W ) shows a unique behavioras σ γ ∗ p ∼ σ ( ∞ ) ( η ( W ,Q ) , for η ≫ η ( W ,Q ) , for η ≪ . (13)Here the quantity σ ( ∞ ) is independent of the photon en-ergy, and Λ sat ( W ) is the saturation scale. The colortransparency or saturation of the dipole cross section de-pend on that Q ≫ Λ sat ( W ) or Q ≪ Λ sat ( W ) respec-tively. Refs.[26] and [27] show that the saturation scaleis deﬁned byΛ sat ( W ) = πσ ( ∞ ) Z d −→ ℓ ′ ⊥ −→ ℓ ′ ⊥ e σ ( qq ) J =1 L ( −→ ℓ ′ ⊥ , W ) , (14)which is ﬁxed spin J = 1 and ℓ ′ is deﬁned into the gluontransverse momentum. Also the light-cone variable z reads as −→ ℓ ′ ⊥ = −→ ℓ ⊥ z (1 − z ) . Therefore, the relationship between gluon distributionand saturation scale is expressed in the form α s ( Q ) xg ( x, Q ) = 18 π σ ( ∞ ) Λ sat ( W ) . (15)We know that the leading contribution to F ( x, Q ) atsuﬃciently large Q in terms of the J = 1 projectionbecomes F ( x, Q ) = Q π α ( σ γ ∗ T p ( W , Q ) + σ γ ∗ L p ( W , Q ))= R e + e − π ( Z d −→ ℓ ′ ⊥ −→ ℓ ′ ⊥ e σ ( qq ) J =1 T ( −→ ℓ ′ ⊥ , W )+ 12 Z d −→ ℓ ′ ⊥ −→ ℓ ′ ⊥ e σ ( qq ) J =1 L ( −→ ℓ ′ ⊥ , W )) , (16)where R e + e − = 3 P f e f . As shown in Ref.[28], the lon-gitudinal and transverse terms on the right-hand side in(16) becomes Z d −→ ℓ ′ ⊥ −→ ℓ ′ ⊥ e σ ( qq ) J =1 T ( −→ ℓ ′ ⊥ , W )= ρ Z d −→ ℓ ′ ⊥ −→ ℓ ′ ⊥ e σ ( qq ) J =1 L ( −→ ℓ ′ ⊥ , W ) , (17)The parameter ρ is associated with the enhanced trans-verse size of qq ﬂuctuations in the CDM originating fromtransverse, γ ∗ T → qq , and longitudinal, γ ∗ L → qq , photons.Indeed the ρ parameter describes the ratio of the averagetransverse momenta ρ = < −→ k ⊥ > L < −→ k ⊥ > T . It can also be relatedto the ratio of the eﬀective transverse sizes of the ( qq ) J =1 L,T states as < −→ r ⊥ > L < −→ r ⊥ > T = ρ . The ratio of the longitudinal tothe transversal photoabsorption cross sections is given by R = σ γ ∗ pL σ γ ∗ pT = 12 ρ , (18)where factor 2 originates from the diﬀerence in the pho-ton wave functions. In terms of the proton structurefunctions, F ( x, Q ) and F L ( x, Q ), the ratio becomes F L/ ≡ F L F = 11 + 2 ρ . (19)The quantity of ρ in previous analysis [26,27,28,29] is con-sidered equal to 1(i.e., ρ = 1) and for Q ≫ Λ sat ( W )was used the value ρ = 4 /

3. The deviation from ρ = 1quantiﬁes the deviation between the scatterings of longi-tudinally polarized versus transversally polarized qq ﬂuc-tuations of the photon. In the large- Q limit, the struc-ture function (according to Eq.(14)) takes the form F ( x, Q ) = R e + e − π σ ( ∞ ) Λ sat ( W )( ρ + 12 ) . (20)The structure function F ( x, Q ) in the small x limit inthe DIS scheme is proportional to the ﬂavor-singlet quarkdistribution, Σ( x, Q ), as F ( x, Q ) = R e + e − x Σ( x, Q ) , (21)which x Σ( x, Q ) = n f ( xq ( x, Q ) + xq ( x, Q )). Due tothat the sea-quark and gluon distributions have identicaldependence on the kinematic variables, therefore they areassumed [26,27,28,29] to be proportional to each otherwith respect to the ρ parameter as x Σ( x, Q ) = 83 π α s ( Q ) xg ( x, Q )( ρ + 12 ) . (22)Consequently the ρ parameter into the singlet structurefunction and gluon distribution function reads as ρ = 3 π α s ( Q ) F s ( x, Q ) G ( x, Q ) − , (23) where F s ( x, Q ) = x Σ( x, Q ) and G ( x, Q ) = xg ( x, Q ).Indeed Eq.(23) is valid for very small dipoles which it isin agreement with the phenomenon of color transparencyresulting from perturbative QCD. Therefore I will obtainthe relation between gluon distribution function anddipole cross-section in order to rewrite the ratio ofstructure functions for the color dipole framework.

3. Model Description

In order to describe the ratio of structure functions itis necessary to make speciﬁc methods about the gluonbehavior in the color dipole model upon applying theDGLAP evolution. First, I give a simple method forthe gluon distribution due to the expansion methodat the appropriate points of expansion and contrastthis method with other methods that have appearedpreviously. Then I deﬁne the gluon distribution andthe proton structure functions according to the param-eterization method. Then I analyse the consistencybetween the Altarelli-Martinelli relation of the gluondistribution and the color dipole model bounds. Inthe following I deﬁne and discuss the ρ parameter inthe color dipole picture with respect to the Laplacetransform method at LO and NLO approximations.Finally after the parameterization of structure functionsare speciﬁed, I determined the parameters by rely on theFroissart-bounded parameterization of the DIS structurefunctions. The authors in Refs.[26-29] have used the same corre-lation between gluon distribution and derivative of struc-ture function mentioned in Ref.[30] at suﬃciently lowvalues of the Bjorken variable x ≃ Q /W ≪ .

1. Theevolution of the structure function with respect to ln Q is determined by ∂F ( x, Q ) ∂ ln Q ≃ α s ( Q )3 π X q e q G (2 x, Q ) . (24)Also another similar relation for the longitudinal struc-ture function into the gluon density is given as follows[31] F L ( x, Q ) ≃ α s ( Q )5 . π X q e q G (2 . , Q ) . (25)Eqs.(24) and (25) actually indicates that the structurefunctions are dependent to the gluon density at low x approximation of the pQCD. That means for x ≃ Q /W ≪

1, the photon-proton interaction is dominatedby the gluon fusion process via γ ∗ gluon → qq . In thelast years these methods [30,31] were proposed to iso-late the gluon density by its expansion around z = .One method was proposed in Ref.[32] by the expansionof the gluon density at an arbitrary z = a (For futurediscussion please see Appendix A), as ∂F ( x, Q ) ∂ ln Q ≃ α s ( Q )9 π G ( x − a ( 32 − a ) , Q ) . (26)A similar relationship for the behavior of the longitudinalstructure function at low x is also described in Ref.[33] . Therefore the ratio ρ for a point of expansion a < ρ = 12 [ F ( x, Q ) / ∂F ( X, Q ) ∂ ln Q − , (27)where X = x − a − a . When the point a = is used, we getthe result expressed in the literatures [26-29]. Authors inRef.[33] showed that according to the expansion method,a general relationship for the longitudinal structure func-tion into the gluon distribution at low values of x and atLO approximation can be obtained by the following form F L ( x, Q ) = 10 α s ( Q )27 π G ( − a − a x, Q ) . Therefore, we can express Eq.(23) in terms of the longi-tudinal structure function as ρ = 12 [ F ( x, Q ) /F L ( X, Q ) − , (28)which the longitudinal structure function in connectionwith the Froissart bound at LO and NLO approxima-tions is investigated at Ref.[20]. In another method,the expansion of the longitudinal structure function isdescribed into the high order corrections in Ref.[34]. Using a parameterization suggested by authors inRef.[15] on the proton structure function in a full accor-dance with the Froissart predictions [14]. The explicitexpression for the F parameterization, which obtainedfrom a combined ﬁt of the H1 and ZEUS collaborationsdata [35] in a range of the kinematical variables x and These articles [32,33] use the fact that quark densities can beneglected at low x , and the nonsinglet contribution F NS can beignored safely at this limit. Q ( x < .

01 and 0 . < Q < ), is given bythe following form F ( x, Q ) = D ( Q )(1 − x ) n X m =0 A m ( Q ) L m , (29)where A ( Q ) = a + a ln(1 + Q µ ) ,A ( Q ) = a + a ln(1 + Q µ ) + a ln (1 + Q µ ) ,A ( Q ) = a + a ln(1 + Q µ ) + a ln (1 + Q µ ) ,D ( Q ) = Q ( Q + λM )( Q + M ) ,L m = ln m ( 1 x Q Q + µ ) . (30)Here M and µ are the eﬀective mass a scale factor re-spectively. The additional parameters with their statis-tical errors are given in Table I.The point to be considered in Eqs.(24), (25) and (26)is that both F and F L are related at small x mainlythrough the gluon density. For this purpose, we thor-oughly examine the equations of evolution. According tothe LO DGLAP [37] evolution equation for 4 masslessquarks the formalism introduced in Refs.[38], the evolu-tion of the proton structure function is given by F ( x, Q ) = x Z x G ( z, Q ) K qg ( xz ) dzz , (31)where K qg is the gluon → quark splitting in leading or-der of QCD (For more on other quantities, please seeAppendix B). With respect to the Laplace transformmethod, the analytical equation for the gluon distribu-tion G ( x, Q ) for massless quarks is given by G ( x, Q ) = 3 FF ( x, Q ) − ∂ FF ( x, Q ) ∂ ln x − Z x FF ( z, Q )( xz ) / (cid:8) √ √

72 ln zx ]+2 cos[ √

72 ln zx ] dzz (cid:9) , (32)where FF ( x, Q ) = ( α s π X q e q ) − F ( x, Q ) , (33)and F ( x, Q ) ≡ ∂F ( x, Q ) ∂ ln Q − α s π x Z x F ( z, Q ) K qq ( xz ) dzz . (34)Therefore the ratio ρ is ρ = 27 π α s ( Q ) F ( x, Q )(i . e ., Eq . (29)) G ( x, Q )(i . e ., Eq . (32)) − . (35)Indeed the above equation (i.e., Eq.(35)) expressed basedon the Froissart-bounded parameterization of F ( x, Q )while Eq.(27) is expressed based on the condition ofgluon dominant at low x as gluon carries the z = a fraction from the proton momentum. In the following I developed model due to the Laplacetransform method at LO and NLO approximation. Theparameterization of the structure functions should sat-isfy the CDM bounds. In Ref.[20] the longitudinal struc-ture function F L ( x, Q ) extracted as it follows the Frois-sart boundary conditions (For more discussion please seeAppendix C). Now I want to express a new interpreta-tion for Eq.(23), based on which the ratio ρ will be de-termined in terms of F ( x, Q ) and F L ( x, Q ) structurefunctions, both of which follow the Froissart boundarycondition. The standard collinear factorization formulafor F L ( x, Q ) at low values of x reads [43] F L ( x, Q ) = a s ( Q )[( c (0) L,q ( x ) + a s ( Q ) c (1) L,q ( x ) + ... ) ⊗ F ( x, Q )+ < e > ( c (0) L,g ( x )+ a s ( Q ) c (1) L,g ( x ) + ... ) ⊗ G ( x, Q )] , (36)where < e > is the average squared charge and thesinglet-quark coeﬃcient function is deﬁned by c ( n ) L,q = c ( n ) L,ns + c ( n ) L,ps which decomposed into the non-singlet andpure singlet contribution. Some analytical solutions ofthe Altarelli-Martinelli [44] equation have been reportedin recent years [45] with considerable phenomenologicalsuccess.Now I use the coordinate transformation in υ -space. Thelongitudinal structure function reads as b F L ( υ, Q ) = a s ( Q ) Z υ [( b c ( n ) L,q ( υ − w ) + ... ) b F ( w, Q )+ < e > ( b c ( n ) L,g ( υ − w ) + ... ) b G ( w, Q )] dw, (37)where the functions b f (i . e ., b F , b c and b G ) are deﬁned by b f ( υ, Q ) ≡ b f ( e − υ , Q ) . Which the non-singlet quark distribution become negligibly smallin comparison with the singlet distributions.

With respect to the Laplace transforms (Please see Ap-pendix B) we have F L ( s, Q ) = a s ( Q )[ k ( s ) F ( s, Q ) + h ( s ) g ( s, Q )] , (38)where at LO approximation h ( s ) = < e > c (0) L,g ( s ) and k ( s ) = c (0) L,q ( s ). Solving Eq.(38) for g , we ﬁnd that g ( s, Q ) = 1 a s ( Q ) F L ( s, Q ) h ( s ) − k ( s ) h ( s ) F ( s, Q ) . (39)Then we take the inverse laplace transform as the aboveequation (i.e., Eq.(39)) can be written as b G ( υ, Q ) = 1 a s ( Q ) L − [ F L ( s, Q ) h ( s ) − ] −L − [ F ( s, Q ) k ( s ) h ( s ) ] , (40)where L − [ g ( s, Q ); υ ] = b G ( υ, Q ). Therefore we ﬁndthat b G ( υ, Q ) = 1 a s ( Q ) Z υ b F L ( w, Q ) b J ( υ − w ) dw − Z υ b F ( w, Q ) b L ( υ − w ) dw, (41)where b J ( υ ) and b L ( υ ) are new auxiliary functions, deﬁnedby b J ( υ ) ≡ L − [ h − ( s ); υ ] , b L ( υ ) ≡ L − [ k ( s ) h ( s ) − ; υ ] . (42)The calculations of b J ( υ ) and b L ( υ ), using the inverseLaplace transform, are straightforward and are given interms of the Dirac delta function and its derivatives, aswe ﬁnd that b J ( υ ) = 34 n f δ ( υ ) + 58 n f δ ′ ( υ ) + 18 n f δ ′′ ( υ ) , b L ( υ ) = 2 C F ( 34 n f δ ( υ ) + 14 n f δ ′ ( υ )) . (43)Using the properties of Dirac delta function, we there-fore obtain an explicit solution for the gluon distributionin terms of the parameterization of F ( x, Q ) [15] and F L ( x, Q ) [20] by G LO ( x, Q ) = 1 a s ( Q ) < e > [ 18 n f x ∂ ∂x F LOL ( x, Q ) − n f x ∂∂x F LOL ( x, Q ) + 34 n f F LOL ( x, Q )] − C F < e > [ − n f x ∂∂x F ( x, Q )+ 34 n f F ( x, Q )] , (44)where the parameterization of F is given by (29). Theexplicit expression for the parameterization of F L at theLO approximation is obtained by the following form [20] F LO L ( x, Q ) = (1 − x ) n X m =0 C m ( Q ) L m . (45)The coeﬃcient functions and future discussion about theabove relation can be found in Appendix C. In this for-malism both F and F L obey the Froissart boundary con-dition. Therefore we ﬁnd the ratio ρ with respect to theLaplace transform method at LO approximation by thefollowing form ρ = 27 π α s ( Q ) Eq . (29)Eq . (44) − . (46) Finally I discuss how the higher-order components ofthe coeﬃcient functions may aﬀect these bounds. Thelongitudinal structure function within the NLO approxi-mation in υ -space reads as b F L ( υ, Q ) = a s ( Q ) Z υ [( b c (0) L,q ( υ − w ) + a s ( Q ) × b c (1) L,q ( υ − w )) b F ( w, Q )+ < e > ( b c (0) L,g ( υ − w )+ a s ( Q ) b c (1) L,g ( υ − w )) b G ( w, Q )] dw. (47)Then transform the NLO longitudinal structure functioninto s -space is F L ( s, Q ) = a s ( Q )[ K ( s ) F ( s, Q ) + H ( s ) g ( s, Q )] , (48)where the coeﬃcient functions at NLO approximation areextended by H ( s ) = < e > [ c (0) L,g ( s ) + a s ( Q ) c (1) L,g ( s )] and K ( s ) = c (0) L,q ( s ) + a s ( Q ) c (1) L,q ( s ). Now the inverse Laplacetransforms of the above equation (i.e., Eq.(48)) can beperformed in the following form b G ( υ, Q ) = 1 a s ( Q ) L − [ F L ( s, Q ) < e > ( c (0) L,g ( s )+ a s ( Q ) c (1) L,g ( s )) − ] − L − [ F ( s, Q ) < e > × c (0) L,q ( s ) + a s ( Q ) c (1) L,q ( s ) c (0) L,g ( s ) + a s ( Q ) c (1) L,g ( s ) ] . (49) Indeed the gluon distribution at NLO approximation canbe represented as b G ( υ, Q ) = 1 a s ( Q ) Z υ b F L ( w, Q ) b T ( υ − w ) dw − Z υ b F ( w, Q ) b U ( υ − w ) dw, (50)where b T ( υ ) ≡ L − [ H − ( s ); υ ] , b U ( υ ) ≡ L − [ K ( s ) H ( s ) − ; υ ] . (51)The inverse Laplace transform of the terms b T ( υ ) and b U ( υ ) at NLO approximation are straightforward but theyare too lengthy. In the limit case for these terms, thesimplest form can be expressed by the following form for Q = 100 GeV as b T ( υ ) = 0 . δ ( υ ) + 0 . δ ′ ( υ ) + 0 . δ ′′ ( υ )+0 . e − . υ − . e − . υ +0 . e − . υ + 0 . e . υ , b U ( υ ) = 0 . δ ( υ ) + 0 . δ ′ ( υ ) + 0 . e − . υ +0 . e − . υ + 0 . e . υ − . e − υ − . e − . υ − . e − υ . (52)Therefore an analytical solution for the NLO gluon dis-tribution in terms of the parameterization of F ( x, Q )[15] and F L ( x, Q ) [20] at NLO approximation at Q =100 GeV is obtained by G NLO ( x, Q ) = 268 . (cid:20) . x ∂ ∂x F NLO L ( x, Q ) − . x ∂∂x F NLO L ( x, Q ) + 0 . F NLO L ( x, Q )+ Z x dzz F NLO L ( z, Q ) (cid:26) . xz ) . − . xz ) . + 0 . xz ) . + 0 . zx ) . (cid:27)(cid:21) − (cid:20) . F ( x, Q ) − . x ∂∂x F ( x, Q ) + Z x dzz F ( z, Q ) (cid:26) . xz ) . + 0 . xz ) . − . xz ) . + 0 . zx ) . − . xz ) − . zz ) (cid:27)(cid:21) , (53)Therefore, at NLO approximation, the ratio ρ is obtainedas follows ρ = 27 π α s ( Q ) Eq . (29)Eq . (53) − , (54)and F NLOL/ = 11 + 2 ρ (Eq . . The NLO corrections for the dipole factorization of DISstructure functions at low x values have been consideredin Ref.[46]. In Ref.[46] the LO approximation for thelongitudinal wave-function of photon is essentially thesame as described in the literature but with an eﬀectivevertex. But at NLO approximation, the colored sector ofthe virtual photon wave-functions contains both qq and qqg components. Expansion of the structure functions, F and F L , in Fock state in the CDM are given by F ,L ( x, Q ) = F qq ,L ( x, Q ) + F qqg ,L ( x, Q ) + ... (55)The bound F LOL/ = or is valid only for the ﬁrstcomponent in the Fock states. Authors in Ref.[47] showedthat at higher Fock states one can be derived the modiﬁedCDM bound for the ratio F L/ as F NLOL/ = F LOL/ δǫ ( x, Q )1 + ǫ ( x, Q ) , (56)where ǫ ( x, Q ) = F qqg ( x,Q ) F qq ( x,Q ) and 0 ≤ δ ≤ . ǫ value is not more than %20.Finally Eq.(54) is my ﬁnal expression for the ratio ρ within the NLO approximation for low values of x . Indeed I shown that due to the analysis of theAltarelli-Martinelli equation in DGLAP approach, theCDM bounds which described for the longitudinal totransverse ratio at high order correction is valid onlyfor very small dipoles. The new feature of the modelis a parameter dependent dipole cross section whichrelying on the Froissart-bounded parameterization ofthe structure functions.

4. Results and Discussions

QCD evolution eﬀects are taken into account byevolving the gluon structure function due to the DGLAPevolution and the Altarelli-Martinelli equation. In thecolor dipole model the gluon distribution is modiﬁedwith respect to the parameterization of structure func-tions at LO and NLO approximations. The ratio of thelongitudinal to the transverse photoabsorption cross sec-tions, σ γ ∗ pL /σ γ ∗ pT , is extracted. This result correspondsto the explicit form of the ratio F ( x, Q ) /G ( x, Q ) bythe expansion and Laplace transform methods. Theparameters ρ , R and F L/ are obtained with respect tothe the expansion method at LO approximation andextended to the NLO approximation due to the Laplacetransform method. The results obtained in the Laplacetransform method at LO and NLO approximations aredependent on the parameterization of the structurefunctions (i.e., F and F L ). These parameterizations[15,20] are valid at low x in a wide range of the momen-tum transfer 1 < Q < . The active ﬂavoris selected in these calculations to be equal to n f = 4and the QCD parameter Λ has been extracted from therunning coupling constant α s ( Q ) normalized at theZ-boson mass, as it is chosen [48] to be α s ( M Z ) = 0 . ρ , R and F L/ are shown interms of the invariant mass W ( ≃ Q x ) in the interval10 GeV < W < GeV for three expansion points a = 0 . , .

50 and 0 .

75. In traditional literature, theexpansion point of the gluon distribution is chosen by a = 0 .

50. Indeed this is the proton momentum fractioncarried by gluon in DIS process. In these calculationsI used the expansion of the gluon distribution at somearbitrary points and compared with the CDM bounds.Parameters are almost dependent on the invariant massin the small expansion points. At high expansion points,the behavior of these parameters is almost independentof the invariant mass, that corresponds to the expansionat a = 0 .

75. Also a detailed comparison with the CDMbounds has been shown in this ﬁgure (i.e., Fig.1). Ascan be seen, the values of these parameters are in goodagreement with the CDM bounds in a wide range ofthe invariant mass at ﬁxed value of Q . The errorbares are in accordance with the statistical errors of theparameterization of F as presented in Table I.An explicit expression for the gluon density into theproton structure function at LO approximation by usingthe Laplace transform method is derived in Refs.[37].The result is comparable to the CTEQ5L [50] andMRST2001LO [51], though there are some diﬀerenceswith CTEQ5L for large x values. With respect to thismethod, a comparison of the parameterization methodwith expansion method can be seen for the parameters(i.e., ρ , R and F L/ ) in Figs.2-4. In the parameterizationmethod, the gluon obtained by authors in Ref.[38] isused directly. In the expansion method we use the same a = 0 . Q values is much lower than in low Q values. Accordingto the range of errors, it can be seen in these ﬁguresthat the results are in the CDM bounds range. Also,the results of parameterization method are better thanexpansion method.Now I proceed with an analysis of the gluon distributioninto the parameterization of the structure functionsas this is of interests in connection with theoreticalinvestigations of ultra-high energy processes with cosmicneutrinos. Also this method is in the context of theFroissart restrictions at low values of x . The longitudi-nal structure function at low and mediate x values iswritten in ﬂavour-singlet quark and gluon distributions.Therefore in a new method using the Laplace transformmethod, the gluon distribution function is expressed interms of the structure functions at LO and NLO approx-imations. The method relies on the Altarelli-Martinelliequation and on the Froissart-bounded parameterizationof the structure functions.In Fig.(5) we present the parameters ρ , R = 1 / (2 ρ )and F L/ = 1 / (1 + 2 ρ ) at LO approximation related toEq.(46) in comparison with the CDM bounds using theparameterizations of F and F LOL [15,20]. As can beseen in this ﬁgure, one can conclude that the behaviorof these parameters are almost constant and comparablewith the CDM bounds at high Q values for x ≤ . ≤ Q ≤ . These results showthat the parameters are almost x independent for low x values. But they are dependent on Q values. Howeverthis behavior is consistent with the experimental data.However, I need to emphasize that such results arepossible only in a limited kinematics, when virtuality Q is very large and signiﬁcantly exceeds the saturationscale Q s . These results indicate that the relationshipbetween gluon PDFs and the dipole cross-section isconsistent with the CDM bound at high Q values in order to rewrite the well-known Altarelli-Martinellirelationship for the color dipole framework.A very important point that can be seen in all the ﬁgures(i.e., Figs.1-5) is that the comparability of CDM boundsand results for large Q values in these calculations takesplace in the color transparency region where η ≫

1. Weobserve that for Q = 5 and 10 GeV we are practicallyin the saturation region where η ≪

1, which is why theresults are inconsistent with CDM bounds. Figure 6can be seen to express the two concepts of saturationand color transparency in the results. In this ﬁgure(i.e., Fig.6) a comparison between the ratio of structurefunctions at LO approximation with the H1 data [52]and the CDM bounds is shown. For W & GeV , thecomparison between the ratio F L/ and the H1 data withthe bounds is good. This depends on Q ≫ Λ sat ( W ).For Q ≪ Λ sat ( W ) in the saturation region that compat-ibility is not appropriate at all. The results in the colortransparency depend on strong interference between allpossible diagrams. While in the saturation region somediagrams are closed which causes the photoabsorptioncross section to be lnΛ sat ( W ) dependent, actually turnsinto the soft energy dependence. These behaviors arethe result of a general and direct consequence of thecolor dipole nature of the interaction of the hadronicﬂuctuations of the photon with the color ﬁeld in thenucleon. Indeed the parameters obtained in the region η ≫ η ≫ η ≪ r is signiﬁcantly larger, r ∼ /Q s .In other words, the formulas mentioned in this articlecannot be used in the saturation regime. For this reasonall discussions and ﬁnal conclusions apply to the colortransparency region.The validity of the results obtained in the LO approx-imation is established in the η > F L/ ( W ) = F L ( W ,Q ) F ( W ,Q ) = ρ and comparison withthe CDM bounds at LO and NLO approximations arepresented in Fig.7. Calculations have been performed atﬁxed value Q = 100 GeV at low values of x , allowingthe invariant mass variable W to vary in the interval10 GeV < W < GeV . Figure 7 clearly demon-strates that the ratio extracted is comparable with theNLO CDM bound. In fact, I have shown that Eq.(22),which represents the relation between the gluon densityand CDM bounds in the color transparency region, canalso be described at NLO approximation. In conclu-0sion this method deﬁned consistency between the CDMbounds and gluon PDFs in the color transparency region.

5. Conclusion

In this paper the CDM parameters (i.e., ρ , R and F L/ ) based on one general expression for approx-imative determination of the gluon distribution forvery small dipoles presented. The gluon behavior atarbitrary point of expansion of G ( x − z ) is considered.Comparing the obtained results with the CDM bounds,it can be concluded that the more suitable points ofexpansion are in the range a ≥ .

75. Then a methodbased on the Altarelli-Martinelli equation and on theFroissart-bounded parameterization of F and F L withrespect to the Laplace transform method at LO andNLO approximations when virtuality Q is large isproposed. The parameters obtained on the kinematicregion of low and ultra-low values of x in a large intervalof the momentum transfer at LO approximation. AtNLO approximation we focus our attention on value Q = 100 GeV . The obtained explicit expressions forthe parameters are entirely determined by the eﬀectiveparameters of the F and F L parameterizations. The re-sults obtained for the ratio F L/ at NLO approximationin the color transparency region are in good agreementwith the NLO CDM bound which have considered thecontribution of the qqg component in the the coloredsector of the virtual photon wave-function.According to the relationships obtained in the colortransparency region at large Q values, we observethat the results obtained are comparable to the H1data and the CDM bounds. These results indicatethat for Q ≫ Λ sat ( W ) the relationships obtained onthe basis of the parameterization of F and F L arecomparable to the proposed constraints. The predictionsare most reliable for 20GeV ≤ Q ≤ at x ≤ .

01. The behavior of the parameters is independentof x for large Q values and are almost dependent on Q in a wide range of Q values. They become lessreliable, when Q decreases to Q < , since inthis case the transition to the saturation region hasto be reﬁned by the nonlinear eﬀects. Indeed in thesaturation region the dipole size r is signiﬁcantly largewhich causes neither DGLAP evolution nor the formu-las listed in this paper to be used in the saturated regime. ACKNOWLEDGMENTS

Author is grateful the Razi University for ﬁnancialsupport of this project. The author is especially gratefulto N.Nikolaev and D.Schildknecht for carefully reading the manuscript and fruitful discussions.

Appendix A. Details on the Derivation of (26)

In this Appendix, I provide a brief exposition of thederivation Eq.(26) in Ref.[32]. The evolution equationsfor the parton distributions are deﬁned by ∂∂ ln Q f i ( x, Q ) = P ij ( x, Q ) ⊗ f j ( x, Q ) , (57)where f i ( x, Q ) stands for the number distributions ofpartons in a hadron and ⊗ stands for the Mellin convolu-tion. This equation (i.e., Eq.(57)) represents a system of2 n f + 1 coupled integro-diﬀerential equations. The split-ting functions P ij ( x, Q ) for N m LO approximation aredeﬁned by P N m LOij ( x, Q ) = P mk =0 a k +1 s ( Q ) P kij ( x ) with a s ( Q ) = α s ( Q ) / π . The ﬂavor singlet quark density isdeﬁned f s = P n f i =1 [ f i + f i ]. The LO evolution equationfor F at low x for four ﬂavors is deﬁned by ∂F ( x, Q ) ∂ ln Q = 10 α s π Z − x P qg ( z ) G ( x − z , Q ) dz. (58)Here the fact is used that at low values of x quark densitycan be neglected and the nonsinglet contribution can beignored. The authors [32] used the expansion of the gluondistribution at an arbitrary point z = a as at the limit x →

0, the equation obtained is ∂F ( x, Q ) ∂ ln Q ≃ α s ( Q )9 π G ( x − a ( 32 − a ) , Q ) . (59)Therefore the gluon distribution can be expressed by G ( x, Q ) = 9 π α s ( Q ) 32 ∂F ( x − a − a , Q ) ∂ ln Q . (60)The result of comparing them with GRV94(LO) [39]showed that the better choices have been in the range0 . ≤ a ≤ . et al. [40] corresponds to a = 0 . Appendix B: Details on the Derivation of (32)

The gluon density used in this analysis obeys the fol-lowing Laplace-transform method [41,42], as the coordi-nate transformation introduced by υ ≡ ln(1 /x ). Further,Eq.(31) rewritten by the following form b F ( υ, Q ) = Z υ b G ( w, Q ) b H ( υ − w ) dw, (61)1where b H ( υ ) ≡ e − υ b K qg ( υ ) and b f ( υ, Q ) ≡ f ( e − υ , Q ) as b K qg ( υ ) = 1 − e − υ + 2 e − υ . The Laplace transform ofthe right-hand of Eq.(61) is deﬁned by L (cid:20) Z υ b G ( w, Q ) b H ( υ − w ) dw ; s (cid:21) = g ( s ) × h ( s ) (62)where h ( s ) ≡L [ b H ( υ ); s ] = R ∞ b H ( υ ) e − sυ dυ with the con-dition b H ( υ ) = 0 for υ < υ -space is obtained interms of the inverse transform of a product to the con-volution of the original functions as b G ( υ, Q ) = L − [ f ( s, Q × h − ( s ); υ ]= Z υ b F ( w, Q ) b J ( υ − w ) dw, (63)where b J ( υ ) ≡ L − [ h − ( s ); υ ]= 3 δ ( υ ) + δ ′ ( υ ) − e − υ/ (cid:8) √ √ υ ]+2 cos[ √ υ ] (cid:9) . (64)Therefore the explicit solution for the gluon distributionin υ -space is deﬁned in terms of the integral b G ( υ, Q ) = 3 b F ( υ, Q ) + ∂ b F ( υ, Q ) ∂υ − Z υ b F ( w, Q ) × e − υ − w ) / (cid:8) √ √

72 ( υ − w )]+2 cos[ √

72 ( υ − w )] (cid:9) . (65) Appendix C: Details about the parameterization of F L ( x, Q ) The authors in Ref.[20] obtained two analytical rela-tions for the longitudinal structure function at LO andNLO approximations in terms of the eﬀective parametersof the parameterization of the proton structure function.The results show that the obtained method provides re-liable longitudinal structure function at HERA domainand also the structure functions F L ( x, Q ) manifestlyobey the Froissart boundary conditions. The structurefunctions, F ( x, Q ) and F L ( x, Q ), and their derivativesinto ln Q are deﬁned with respect to the singlet andgluon distribution functions xf a ( x, Q ) as F k { =2 ,L } ( x, Q ) = < e > X a = s,g [ B k,a ( x ) ⊗ xf a ( x, Q )]and ∂∂ ln Q xf a ( x, Q ) = − X a,b = s,g P ab ( x ) ⊗ xf b ( x, Q ) . (66) The quantities B k,a ( x ) and P ab ( x ) are the Wilson coef-ﬁcient and splitting functions respectively. The high or-der corrections to the coeﬃcient functions can be seen inRef.[20]. With respect to the Mellin transform method,the leading order longitudinal structure function is ob-tained at low x by the following form F LO L ( x, Q ) = (1 − x ) n X m =0 C m ( Q ) L m , (67)where C = b A + 83 a s ( Q ) DA C = b A + 12 b A + 83 a s ( Q ) D [ A + (4 ζ −

72 ) A ] C = b A + 14 b A − b A + 83 a s ( Q ) D [ A + (2 ζ −

74 ) A +( ζ − ζ −

178 ) A ] , (68)and b A = e A b A = e A + 2 DA µ µ + Q b A = e A + DA µ µ + Q e A i = e DA i + DA i Q Q + µ e D = M Q [(2 − λ ) Q + λM ][ Q + M ] A m = a m + 2 a m L , a = 0 . (69)The standard representation for QCD running couplingconstant in the LO and NLO approximations have beendescribed a LO s ( Q ) = 1 β ln( Q / Λ ) , (70) a NLO s ( Q ) = 1 β ln( Q / Λ ) − β ln ln( Q / Λ ) β [ β ln( Q / Λ )] , which the QCD parameter Λ at LO and NLO approxima-tions has been extracted with Λ LO ( n f = 4) = 136 . NLO ( n f = 4) = 284 . β -functions are β = 13 (11 C A − n f ) , β = 13 (34 C A − n f (5 C A + 3 C F ))where C F and C A are the Casimir operators in the SU ( N c ) color group.The NLO longitudinal structure function at small x is2deﬁned by the following form F NLO L ( x, Q ) = 1[1 + a s ( Q ) L C ( b δ (1) sg − b R (1) L,g )] (cid:26) [1 − a s ( Q )( δ (1) sg − R (1) L,g )] F LO L ( x, Q ) − a s ( Q )[ 13 b B (1) L,s L A + B (1) L,s ] F ( x, Q ) (cid:27) (71)where the coeﬃcient functions read as b B (1) L,s = 8 C F [ 259 n f − C F + (2 C F − C A )( ζ + 2 ζ − B (1) L,s = 203 C F (3 C A − n f ) b δ (1) sg = 263 C A δ (1) sg = 3 C F − C A b R (1) L,g = − C A R (1) L,g = − C F − C A L A = L + A A L C = L + C C L = ln(1 /x ) + L L = ln Q Q + µ . (72) TABLE I: The eﬀective Parameters at low x for 0 .

15 GeV

068 GeV and µ = 2 . ± .

290 GeV .parameters value a . × − ± . × − a . × − ± . × − a . × − ± . × − a − . × − ± . × − a − . × − ± . × − a . × − ± . × − a . × − ± . × − a . × − ± . × − n . ± . λ . ± . χ (goodness of ﬁt) 0 . REFERENCES

1. N.N.Nikolaev and B.G.Zakharov, Z.Phys.C ,607(1991); Z.Phys.C , 331(1992); A.H.Mueller,Nucl.Phys.B , 373(1994); Nucl.Phys.B ,107(1995); K.Golec-Biernat, Acta Phys.Pol.B ,3103(2004).2. R.J.Glauber, Phys.Rev. , 1515(1955); A.H.Mueller,Nucl.Phys.B , 115(1990).3. K.Golec-Biernat and M.W¨usthoﬀ, Phys.Rev.D ,014017(1998); K.Golec-Biernat and M.W¨usthoﬀ,Phys.Rev.D , 114023(1999).4. K.Golec-Biernat, Acta Phys.Pol.B , 2771(2002);J.Phys.G , 1057(2002); H.Kowalski and D.Teaney,Phys.Rev.D , 114005(2003); H.Kowalski, L.Motykaand G.Watt, Phys.Rev.D , 074016(2006).5. C.Ewerz, A.von Manteuﬀel and O.Nachtmann,JHEP , 102(2010); D.Britzger et al., Phys.Rev.D ,114007 (2019); C.Ewerz, A. von Manteuﬀel andO.Nachtmann, Phys.Rev.D , 074022(2008).6. H.Kowalski and D.Teaney, Phys.Rev.D ,114005(2003).7. G.Watt and H.Kowalski, Phys.Rev.D ,014016(2008); A.H.Rezaeian et al., Phys.Rev.D87,034002(2013).8. L.McLerran, arXiv:0804.1736 [hep-ph] (2008);Acta.Phys.Polon.B , 2307 (2014).9. F. Gelis et al., Annu. Rev. Nucl. Part. Sci. , 463(2010); G.M.Peccini et al., Phys. Rev. D , 074042(2020); M.Genovese, N.N.Nikolaev and B.G.Zakharov,J.Exp.Theor.Phys. , 633(1995); J.Exp.Theor.Phys. ,3625(1995); Amir H.Rezaeian and I.Schmidt, Phys.Rev.D , 074016 (2013).10. I.Balitsky, Nucl.Phys.B , 99(1996);Phys.Rev.D , 014001(2007); Y.V.Kovchegov,Phys.Rev.D , 034008(1999); Phys.Rev.D ,074018(2000).11. E.Iancu,K.Itakura and S.Munier, Phys.Lett.B ,199(2004).12. A.M.Stasto, K.Golec-Biernat, J.Kwiecinski,Phys.Rev.Lett. , 596(2001); F.Gelis et al.,Phys.Lett.B , 376(2007); J.Kwiecinski andA.M.Stasto, Phys.Rev.D , 014013(2002).13. D.Schildknecht and H.Spiesberger, arXiv [hep-ph]:9707447 (1997); W.Buchmuller and D.Haidt, arXiv[hep-ph]: 9605428 (1996) .14. M.Froissart, Phys.Rev. , 1053(1961).15. M. M. Block, L. Durand and P. Ha, Phys. Rev.D ,no. 9, 094027 (2014); M.M.Block et al., Phys. Rev.D ,no. 1, 014006 (2013).16. D.Boer et al., arXiv: [nucl-th]1108.1713.17. M.Klein, arXiv [hep-ph]:1802.04317; M.Klein,Ann.Phys. , 138(2016); N.Armesto et al.,Phys.Rev.D , 074022(2019).18. J.Abelleira Fernandez et al., [LHeC Study GroupCollaboration], J.Phys.G , 075001(2012); P.Agostiniet al. [LHeC Collaboration and FCC-he Study Group ],CERN-ACC-Note-2020-0002, arXiv:2007.14491 [hep-ex](2020).19. A. Abada et al., [FCC Study Group Collaboration],Eur.Phys.J.C , 474(2019).20. L.P.Kaptari et al., Phys.Rev.D , 096019 (2019).21. B.Rezaei and G.R.Boroun, Phys.Rev.C , 045202(2020); G.R.Boroun and B.Rezaei, Nucl.Phys.A ,244(2019).22 K.Golec-Biernat and S.Sapeta, JHEP , 102(2018); J.Bartels, K.Golec-Biernat and H.Kowalski,Phys.Rev.D , 014001(2002).23. G.Beuf, Phys.Rev.D , 054016(2016);Phys.Rev.D , 074033 (2017); G.Beuf et al.,arXiv:2008.05233[hep-ph] (2020).24. B.Duclou´e et al., Phys.Rev.D , 094017(2017);J.Bartels et al., Phys.Rev.D , 054017(2010).25. N.N.Nikolaev and B.G.Zakharov, Phys.Lett.B ,250(1994); L.Frankfurt, A.Radyushkin and M.Strikman,Phys.Rev.D , 98(1997); M. Genovese, N.N. Nikolaevand B.G. Zakharov, JETP , 633(1995); I.P. Ivanovand N.N. Nikolaev, Phys. Rev. D , 054004 (2002).26. M.Kuroda and D.Schildknecht, Phys.Lett.B , 84(2005); M.Kuroda and D.Schildknecht,Acta Phys.Polon. B , 835(2006); M.Kuroda andD.Schildknecht, Phys.Lett. B , 129(2008); M.Kurodaand D.Schildknecht, Phys.Rev. D , 094013(2017).27. D.Schildknecht and M.Tentyukov, arXiv[hep-ph]:0203028; M.Kuroda and D.Schildknecht,Phys.Rev. D , 094001(2012); D.Schildknecht,Mod.Phys.Lett.A , 1430028(2014); M.Kuroda and D.Schildknecht, Int. J. Mod. Phys. A , 1650157(2016).28. M.Kuroda and D.Schildknecht, Phys.Rev.D ,094001(2014).29. D.Schildknecht, B.Surrow and M.Tentynkov,Phys.Lett.B499, 116(2001); G.Cvetic , D.Schildknecht,B.Surrow and M.Tentynkov, Eur.Phys.J.C ,77(2001); D.Schildknecht, B.Surrow and M.Tentynkov,Mod.Phys.Lett.A16, 1829(2001); M.Kuroda andD.Schildknecht, Eur.Phys.J.C37, 205(2004).30. K.Prytz, Phys.lett.B , 286(1993).31. A.M.Cooper-Sarkar et al., Z.Phys.C , 281(1988);A.M.Cooper-Sarkar et al., Acta Phys.Pol.B ,2911(2003).32. M.B.Gay Ducati and Victor P.B.Goncalves,Phys.Lett.B , 401(1997).33. G.R.Boroun and B.Rezaei, Eur.Phys.J.C72,2221(2012).34. B.Rezaei and G.R.Boroun, Eur.Phys.J.A ,262 (2020); G.R.Boroun and B.Rezaei, arXiv [hep-ph]:2010.15357 (2020).35. F.D.Aaron et al., [H1 and ZEUS Collaborations],JHEP , 109(2010).36. M.M.Block and F.Halzen, Phys.Rev.Lett. ,212002 (2011); Phys.Rev.D , 091901 (2004).37. Yu.L.Dokshitzer, Sov.Phys.JETP , 641(1977);G.Altarelli and G.Parisi, Nucl.Phys.B , 298(1977);V.N.Gribov and L.N.Lipatov, Sov.J.Nucl.Phys. ,438(1972).38. Martin M.Block, Eur.Phys.J.C , 1 (2010); M.M. Block, L. Durand and Douglas W.McKay, Phys.Rev.D , 0140131 (2009); Phys. Rev.D , 094003(2008).39. M.Gluk, E.Reya and A.Vogt, Z.Phys.C , 433(1995).40. M.G.Ryskin, Yu. M.Shabelski and A.G.Shuvaev,Z.Phys.C , 111(1996).41. M.M.Block, L.Durand and D.W.McKay,Phys.Rev.D , 014031(2009); M.M.Block,Eur.Phys.J.C , 1(2010).42. F.Taghavi-Shahri, A.Mirjalili andM.M.Yazdanpanah, Eur.Phys.J.C , 1590(2011);S.M.Moosavi Nejad et al., Phys.Rev.C , 045201(2016);H.Khanpour, A.Mirjalili and S.Atashbar Tehrani,Phys.Rev.C , 035201(2017); G.R. Boroun, S. Zarrinand F. Teimoury, Eur.Phys.J.Plus , 214(2015);G.R. Boroun and S. Zarrin, Phys.Atom.Nucl. ,1034(2015); G.R. Boroun, S. Zarrin and S. Dadfar,Nucl.Phys.A , 21(2016); G.R. Boroun, S. Zarrinand S. Dadfar, Phys.Atom.Nucl. , 236(2016); F.Teimoury Azadbakht, G.R. Boroun and B. Rezaei,Int.J.Mod.Phys.E , 1850071(2018); S.Dadfar andS.Zarrin, Eur.Phys.J.C , 319(2020).43. S.Moch, J.A.M.Vermaseren and A.Vogt,Phys.Lett.B , 123(2005); D.I.Kazakov et al.,Phys.Rev.Lett. , 1535(1990).44. G.Altarelli and G.Martinelli, Phys.Lett.B ,4 -6-4-2024 10 -1.0-0.50.00.51.0 10 -1.0-0.50.00.51.0 Q =100 GeV a=0.25 a=0.5 a=0.75 =1 =4/3 (W ) Q =100 GeV a=0.25 a=0.5 a=0.75 R=1/2 R=3/8 R(W ) W [GeV ] W [GeV ] Q =100 GeV a=0.25 a=0.5 a=0.75 F L/2 =1/3 F

L/2 =3/11 F L/2 (W ) W [GeV ] FIG. 1: Results of the parameters ρ, R and F L/ obtainedfrom the three expansion points a = 0 . ,

50 and 0 .

75 re-spectively. The parameters compared with the CDM bounds ρ = 1 and 4 / R = 1 / / F L/ = 1 / / Q value inthe interval 10 GeV < W < GeV at low values of x .The error bars are correspondent to the uncertainties of the F parameterization (i.e., Table I). , 1061(2014); N.Baruah, N.M.Nathand J.K.Sarma, Int.J.Theor.Phys. , 2464(2013);G.R.Boroun, Phys.Rev.C , 015206(2018); G.R.Boroun,Int.J.Mod.Phys.E , 131(2009); G.R.Boroun, B.Rezaeiand J.K.Sarma, Int.J.Mod.Phys.A , 1450189(2014);G.R.Boroun, Eur.Phys.J.Plus , 19(2014).46. G.Beuf, Phys. Rev. D , 034039(2012).47. M.Niedziela and M.Praszalowicz, Acta PhysicaPolonica B , 2018(2015).48. K.G.Chetyrkin, B.A.Kniehl and M.Steinhauser,Phys.Rev.Lett. , 2184(1997); Nucl.Phys.B ,61(1998).49. S.Chekanov et al.[ZEUS Collaboration],Eur.Phys.J.C ,443(2001).50. H.L.Lai et.al. [CTEQ Collaboration],Eur.Phys.J.C , 375(2000).51. A.D.Martin, R.G.Roberts, W.J.Stirling andR.S.Thorne, MRST2001, Eur.Phys.J.C , 73(2002).52. V.Andreev et al. [H1 Collaboration],Eur.Phys.J.C , 2814(2014).5 -10-50510 10 -10-50510 Q =5 GeV Q =10 GeV Q =100 GeV Q =1000 GeV =1 =4/3 (W ) W [GeV ] b) Expansion Method with a=0.5 W [GeV ] a) Parameterization Method FIG. 2: Comparison between the results obtained in param-eterization method and expansion method at a = 0 . ρ parameter. The obtained values compared with the CDMbounds ρ = 1 and 4 /

3. The results are presented at fourﬁxed values of Q ( Q = 5 , ,

100 and 1000 GeV ) in theinterval 10 GeV < W < GeV . The error bars are cor-respondent to the uncertainties of the F parameterization(i.e., Table I). -6-5-4-3-2-101234 10 -6-5-4-3-2-101234 Q =5 GeV Q =10 GeV Q =100 GeV Q =1000 GeV R=1/2 R=3/8

R(W ) W [GeV ] b) Expansion Method with a=0.5 W [GeV ] a) Parameterization Method FIG. 3: The same as Fig.2 for the ratio R which comparedwith the CDM bounds R = 1 / / -5-4-3-2-1012 10 -5-4-3-2-1012 Q =5 GeV Q =10 GeV Q =100 GeV Q =1000 GeV F L/2 =1/3 F

L/2 =3/11 F L/2 (W ) W [GeV ] b) Expansion Method with a=0.5 W [GeV ] a) Parameterization Method FIG. 4: The same as Fig.2 for the ratio F L/ which comparedwith the CDM bounds F L/ = 1 / / -5 -4 -3 -2 -1 ( x , Q ) x =1 =4/3 -5 -4 -3 -2 -1 R ( x , Q ) x R=1/2 R=3/8 -5 -4 -3 -2 -1 F L / ( x , Q ) x Q =5 GeV Q =10 GeV Q =100 GeV Q =1000 GeV F L/2 =1/3 F

L/2 =3/11

FIG. 5: Parameters ρ , R and F L/ at LO approximation as afunction of x for Q = 5 , ,

100 and 1000 GeV comparedwith the CDM bounds. LO Results for x=0.001 H1 data F

L/2 =1/3 F

L/2 =3/11

FIG. 6: The ratio of the longitudinal to transversal structurefunctions calculated within the LO approximation at ﬁxedvalue of the Bjorken variable x = 0 . . ≤ Q ≤

150 GeV . The obtained valuescompared with the CDM bounds F L/ = 1 / /

11. Theerror bars are correspondent to the uncertainties of the F and F LOL parameterization (i.e., Table I and Appendix C). F L / ( W , Q ) W [GeV ] Q =100 GeV NLO result F

L/2LO =0.333 F

L/2LO =0.273 F

L/2NLO =0.442 F

L/2NLO =0.392

FIG. 7: Ratios F L/ plotted as function of the invariant mass W . Straight lines correspond to the CDM bounds at LO andNLO approximations. The obtained values at NLO approxi-mation compared with the NLO CDM bounds F L/ = 0 . .