Nuclear longitudinal structure function in eA processes at the LHeC
aa r X i v : . [ h e p - ph ] M a r Nuclear longitudinal structure function in eA processes at the LHeC
G.R.Boroun, ∗ B.Rezaei, † and S.Heidari Physics Department, Razi University, Kermanshah 67149, Iran (Dated: March 25, 2018)The nucleon and nuclear longitudinal structure functions are determined by the Kharzeev-Levin-Nardin (KLN) model of the low x gluon distribution. The behavior of the gluon distributionratio R g = G A /AG p and the ratio R totalL = F A − totalL /AF p − totalL in this processes are found. Theheavy longitudinal structure function ratios in eA processes at the LHeC region are discussed.Heavy contributions to the ratio of the total longitudinal structure function are considerable andshould not be neglected especially at smaller x of the LHeC project. In the KLN model the newgeometrical scaling for transition from the linear to nonlinear regions in accordance with the LHeCprocesses is used, whose results intensively depended on the heavy quarks mass effect. I. Introduction
Our knowledge of the gluon distribution function offree nucleons comes from the deep inelastic scattering(DIS ) measurements in lepton-nucleon collisions, as atlow- x the gluon distributions are predominant at allvalues of Q . There is a transition from the linear tononlinear regions as it can be tamed by screening effects.These nonlinear terms reduce the growth of the gluondistribution at low x values. Therefore DIS processesin LHeC provide very important tools for probing thegluon distribution in the nucleons and nuclei. Thenuclear gluon distribution xg A ( x, Q ) can be determinedfrom the gluon distribution of nucleons which arebounded in a nucleus. Also, the nuclear distributionfunctions can be extracted from the measurements ofdeep inelastic lepton-nucleus scattering (eA processes).In the electron-proton/ion collider LHeC, we intendedto demonstrate how the low x data are possible fornuclear targets and could constrain the nuclear gluondistribution function.The LHeC shows an increase in the kinematic rangeof the deep inelastic scattering (DIS) because theDIS kinematics are 2 < Q < , GeV and0 . < x < . √ s ep > T eV . Clearly this increase in theprecision of parton distribution functions (PDF , s) atlow- x kinematic region is expected to cause by thenon-linear effects in the so-called saturation region [1-4].The nuclear parton distribution functions (nPDF , s) canbe determined based on the DGLAP [5-6] evolution,analogous to the parton distributions of the free proton.At low x , the data show a reduction of the nucleardistribution functions with respect to the free distri-bution functions. This phenomenon is caused by the ∗ Electronic address: [email protected]; [email protected] † [email protected] nuclear shadowing effects as xg A ( x, Q ) < Axg N ( x, Q ).These shadowing corrections give rise to nonlinear termsin the evolution equation of the gluon distributionfunction. Indeed, these behaviors are tamed by thesaturation effects. In the gluon saturation approach, animportant point is the x -dependence saturation scale Q s ( x ) where it is the critical line between the linearand nonlinear regions. It is expected that the nonlineareffects are small in Q > Q s ( x ) and it should be strongin Q < Q s ( x ) which generate geometrical scaling inthis region. Therefore the nuclear reduced cross sectionis dependent upon the single variable τ = Q Q s ( x ) , as σ γ ∗ A ( x, Q ) = σ γ ∗ A ( τ ) and the saturation scale is givenby Q s ( x ) = Q ( xx ) − λ [7-9]. Here Q = 0 . GeV , x = 3 . × − and exponent λ is a dynamical quantityat the order of λ ≃ . II. KLN model and saturation scale
We focused on the nuclear longitudinal structure func-tion based on the geometrical scaling at low x . The mainpurpose of this study was to analyze possible compatibil-ity of the new geometrical scaling with the KLN modelfor transition from the linear to nonlinear nuclear be-havior. In heavy production, the geometrical scaling isexpected to be violated by heavy quarks mass, since thetraditional geometrical scaling ( τ ) can be modified totake into account heavy quarks mass [9]: τ H = (1 + 4 m H Q ) λ Q Q ( xx ) λ . (1)In the KLN model [10], a simple relation for the uninte-grated gluon distribution was observed as it is related tothe gluon distribution by the following form G ( x, Q ) = Z Q dk t ϕ ( x, k t ) , (2)where ϕ is the unintergrated gluon distribution of a nu-cleon or nucleus. Authors in Ref.[10] used a simplifiedassumption about the form of G ( x, Q ) by two regions ofintegration over Q defined in accordance with the criti-cal line Q s . For a nucleon we use the KLN Ansatz as G ( x, Q ) = (cid:26) K Sαs ( Q s ) Q (1 − x ) , Q Q s ( τ > , (3)where the numerical coefficient K can be determinedfrom the gluon density, which is usually taken from theparameterization groups and S is the area correspondsto the target. Here the factor (1 − x ) is to describe thefact that the gluon density is small at high x values.The gluon distribution function, for a nucleus with themass number A , can be exploited to G A ( x, Q ) with re-placements S → S A = A S and Q s → Q As = A Q s [11].The gluon distribution for a nucleus with respect to Eq.3can be written as G A ( x, Q ) = (cid:26) K SAαs ( Q As ) Q (1 − x ) , Q
Q As ( τ A > . (4)The transition point between the linear and nonlinearregions in accordance with the critical line for the proton( A = 1) and a nucleus ( A ) is x c = x ( Q A / Q ) /λ . (5)Table.1 shows that the critical point is dependent on themass number A and Q values. At low x values, the tran-sition point between the linear and nonlinear behavior isobservable for light nuclei at low Q and for heavy nucleiat low and moderate Q values in eA processes. Thesecritical points refer to zero quark mass so that the geo-metrical scaling defined by τ A (= Q Q As ( x ) ). Since masse ofheavy quarks in the LHeC region is not negligible, there-fore the geometrical scaling is sensitive to the mass of theheavy quarks. This is consistent with a new geometricalscaling τ AH into the one as follows τ AH = (1+ m H Q ) λ Q Q As .We expect that the transition points shift to lower x val-ues as transition between the linear and nonlinear behav-iors tamed at low x values. As to the mass correction, the critical point is given by the following form x c = x ( Q A / Q (1 + m H Q ) ) /λ . (6)In Tables 2-4, we observe the heavy quark mass effectson the critical points. Indeed, the difference betweenthe (traditional) geometrical scaling variable ( τ ) andthe ”new” geometrical scaling variable ( τ H ), particularlyfor eA ( τ A and τ AH ) is strong dependence on the heavyquark mass as ∆ ττ or ∆ τ A τ A = (1 + m H Q ) λ − R g = G A ( x,Q ) AG p ( x,Q ) forlight and heavy nuclei A = 12 and A = 208 respectivelyat Q values of 2 ,
10 and 100
GeV in order to determinethe gluon densities in nuclei. The magnitude of shad-owing effects are considered by the geometrical scalingbehavior at low x values. We observe the saturationeffects for the ratio R g at x < − and for small valuesof Q at light and heavy nuclei by using the traditionalscaling. In Figs.2-3, we observe that the critical pointsrelated to the new geometrical scaling are noticeable ina wide region of x ( x < − upto x < − ). Theseobservations are essential in determining of these ratioswhen we take into account the heavy quarks mass effects. III. Nuclear longitudinal structure function
Now, we consider the nuclear longitudinal structurefunction at eA processes with respect to the nuclear gluondensity behavior. The nuclear longitudinal structurefunction is interested because it is directly sensitive to thenuclear gluon density through the transition g A → q A q A ineA-DIS. Indeed a measurement of F AL ( x, Q ) can be usedto extract the nuclear gluon structure function. There-fore the measurement of F AL provides a sensitive test ofperturbative QCD (pQCD). Since the longitudinal struc-ture function F L contains rather large heavy flavor con-tributions at small-x region, so the measurements of theseobservables in the eA processes have told us about theratio of the heavy quarks contribution to the nuclear lon-gitudinal structure function and also the dependence ofnuclear parton distribution functions (nPDFs) on heavyquarks mass. In perturbative QCD, the nuclear longitu-dinal structure function can be written as x − F AL = C L,ns ⊗ q Ans + < e > ( C L,q ⊗ q As + C L,g ⊗ g A )+ x − F Heavy − AL , (7)where the symbol ⊗ indicates convolution over the vari-able x as: A ( x ) ⊗ B ( x ) = R x dyy A ( y ) B ( xy ). Here q Ans , q Ai and g A represent the distributions of quarks and gluonsin nuclei, respectively. < e > is the average squaredcharge (= for light quarks) and C L,a is the perturba-tive expansion of the coefficient functions as it follows C L,a ( α s , x ) = X n =1 ( α s π ) n c ( n ) L,a ( x ) . (8)At low x , the gluon contribution to the total nuclear lon-gitudinal structure function dominates over the singletand nonsinglet contributions as F AL | x → ≃ F g − AL + F Heavy − AL . (9)The gluonic nuclear longitudinal structure function isgiven by F g − AL ( x, Q ) = X n =1 ( α s π ) n < e > c ( n ) L,g ( x ) ⊗ G A ( x, Q ) . (10)The nuclear heavy quark-longitudinal structure function,at low- x , is depends on the nuclear gluon distributionwhen neglecting the contributions due to incoming lightquarks and anti-quarks in boson gluon fusion processes.Therefore F Heavy − AL ( x, Q ) = C HeavyL,g ( x, Q ) ⊗ G A ( x, Q ) ≡ X n =1 F ( n ) ,Heavy − AL ( x, Q ) , (11)where n indicates the order of α s . At low x values, Eqs.10and 11 are explicitly dependent on the strong couplingconstant and nuclear gluon density.Similarly, in the electron-proton collision, the gluonic lon-gitudinal structure function is directly dependent on thegluon distribution function. Some analytical solutionsof the Altarelli- Martinelli equations [13] using the ex-panding method and hard pomeron behavior initializedby Cooper-Sarkar et al. , have been reported in last years [14-15] with considerable phenomenological success.At leading order (LO) analysis, the gluonic nuclear longe-tudinal structure function is given by F g − AL ( x, Q ) = α s π [ N f X i =1 e i ] Z x dyy [8( x/y ) (1 − x/y )] × G A ( y, Q ) , (12)and F Heavy − AL ( x, Q ) = F c − AL ( x, Q ) + F b − AL ( x, Q )+ F t − AL ( x, Q ) , (13)where F Heavy − AL ( x, Q , m H ) = 2 e H α s ( µ H )2 π Z xa H xdyy × C HL,g ( xy , m H Q ) G A ( y, µ H ) . (14)Here a H = 1 + 4 m H Q , C HL,g is the hevay coefficientfunction related to the heavy quarks mass. The scale µ H (= q Q + 4 m H ) is the mass factorization and therenormalization scale, and α s ( µ H ) is the running cou-pling constant. The heavy longitudinal coefficient func-tion can be expressed as C HL,g ( z, ζ ) = − z ζ H ln β H − β H + 2 β H z (1 − z ) , (15)where β H = 1 − zζ H − z ( ζ H ≡ m H Q ).The low- x behavior of the nuclear gluon distributionfunction in accordance with the KLN model can be ex-ploited to the nuclear longitudinal structure function.Therefore F A − totalL at low x can be found as F A − totalL = α s π [ N f X i =1 e i ] Z x dyy [8( x/y ) (1 − x/y )] G A ( y, Q ) + 2 e c α s ( µ c )2 π Z a c x xdyy C cL,g ( xy , ζ c ) G A ( y, µ c )+2 e b α s ( µ b )2 π Z a b x xdyy C bL,g ( xy , ζ b ) G A ( y, µ b )+2 e t α s ( µ t )2 π Z a t x xdyy C tL,g ( xy , ζ t ) G A ( y, µ t ) . (16)The behavior of the ratio R L = F g − AL AF g − pL as a functionof x for Q = 2 ,
10 and 100
GeV and nuclei A = 12and A = 208 is presented in Fig.1. The transition pointbetween the linear and nonlinear regions is shown at low Q values in this figure. These results are comparablewith EKS [16] and EPS [17] analysis in comparison withDS [18] and HKN [19] parameterizations at low x andalso with nuclear PDFs from the LHeC perspective [20].The magnitude of the shadowing effect is dependent on Q values and mass number ( A ). In fact, this modelpredict a large value for the shadowing effects at lowand high Q values. The shadowing effects for heavynuclei are larger than light nuclei at a wide range of x and Q . Figures 2-3 show the transition point betweenlinear and nonlinear regions with respect to the newgeometrical scaling shifted towards very low x values,and this relates to the nuclear mass in eA processes.In Figs.4-5, we present the small- x behavior of the ratio R HL in accordance with the traditional transition point(Eq.5) as a function of x for Q = 2 ,
10 and 100
GeV and nuclei A = 12 and A = 208. In these figures,we observed antishadowing and shadowing behaviorsat low x and low Q values along the traditional geo-metrical scaling (Eq.5). In the all cases, the depletionand enhancement in these ratios reflect the linear(at x > x c )/linear(at x > x c ), nonlinear(at x < x c )/linear(at x > x c ) and nonlinear(at x < x c )/nonlinear(at x < x c )behavior for nuclei/nuclon related to Eq.5. The en-hancement in the ratio nuclei/nuclon (i.e. antishadowingbehavior) is related to the nonlinear behavior of nuclei.This is dependence to the coherent multiple scatteringwhere introduces the medium size enhanced (in powersof A / )nuclear effects [22]. Indeed nuclear shadowingis controlled by the interplay of photon fluctuationslifetime and coherent time for transition between noshadowing and saturated shadowing at very small x .The gluon shadowing is negligibly at x > .
01 whichcovers the whole range on the NMC data [23]. Indeedtransition of nuclei from linear to nonlinear regions isfaster than transition of nucleon.The new transition points for these ratios are shownin figures 6-7. This is consistent with Eq.6 for charm,bottom and top quarks. Indeed the nonlinear effectsare predominant for light and heavy nuclei at low- Q values. The shadowing effects for A = 12 are observableat Q = 2 GeV and x < − and for A = 208 areobservable at Q = 2 and 10 GeV at x < − by thecharm content of the nuclei and nucleon. For bottomand top contribution to the longitudinal structurefunctions, the shadowing effects will be noticeable at x < − and low Q values, which may be expected tobe predict at the LHeC energies. A comparison between R L (Fig.1) and R HL (Figs.6-7) shows that transition pointfor going to the shadowing region is at larger valuesof x . This is consistent with light and heavy quarks mass. IV. Total longitudinal structure function
Let us now discuss about the ratio of the totallongitudinal structure functions. It is well known thatthe inclusive observable F totalL is strongly dependenton the gluon distribution and heavy contributions tothe structure function. Fig.8 shows the results of the total longitudinal structure function ratio for A = 12and A = 208 at Q value of 2 GeV , where saturationeffect is more observable than other Q values. In thiscase, the significant nonlinear effect is observable as thiseffect start to appear at x ≤ − for heavy nuclei anddecrease to lower x values for light nuclei. Thereforethe shadowing effect For light and heavy nuclei areobservable at low x values. An enhancement at behaviorof the ratio ( R totalL ) for light nuclei (in Fig.8 on the leftpanel) is due to the KLN gluon model and caused bythe anti-shadowing effects. Finally the ratio of the totallongitudinal structure function decreases as A increases. V. Summary
In conclusion, we have observed that the KLN modelfor the total longitudinal structure function ratio R totalL gives the saturation effect of the heavy quarks effect tothe light flavors at small x . This ratio shows shadowingeffects for heavy nuclei at low x . But for light nucleian enhancement in addition to depletion is shown atthis region. The results are close to EPS nuclear distri-bution. Lastly, one important conclusion is that heavycontribution to the total longitudinal structure functionratio R totalL = F A − totalL /AF p − totalL is considerable andcannot be neglected especially at smaller x of the LHeCproject. Acknowledgments
We thank F.O.Dur e a es for useful discussions, com-ments and reading the manuscript. REFERENCES
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Fig.1. R g and R L evaluated as a function of x at Q = 2 ,
10 and 100
GeV for nuclei A = 12 and A = 208 with the KLN model.Fig.2. R g and R L evaluated as a function of thegeometrical scalings at Q = 2 GeV for nuclear A = 12.Fig.3. The same Fig.2 for A = 208.Fig.4. The ratio R HL = F H ( A ) L AF H ( p ) L for A = 12 at Q = 2 ,
10 and 100
GeV .Fig.5. The same Fig.4 for A = 208.Fig.6. The nonlinear and shadowing behavior of R HL for A = 12 in accordance with new geometricaltransition point.Fig.7. The same Fig.6 for A = 208.Fig.8. R totalL for A = 12 and A = 208 at Q = 2 GeV . R i Q =2 GeV ,A=12 R g R L Q =10 GeV ,A=12 Q =100 GeV ,A=12 R i xQ =2 GeV ,A=208 R g R L 1E-6 1E-5 1E-4 1E-3 0.01 0.1 1 xQ =10 GeV ,A=208 xQ =100 GeV ,A=208 Fig.1 R i Q =2 GeV R g ( ) R L ( ) c Q =2 GeV R g ( c12 ) R L ( c12 )
10 100 10000.00.20.40.60.81.01.21.4 R i b Q =2 GeV R g ( b12 ) R L ( b12 ) t Q =2 GeV R g ( t12 ) R L ( t12 ) Fig.2 R i Q =2 GeV R g ( ) R L ( ) c Q =2 GeV R g ( c208 ) R L ( c208 )
10 100 10000.00.20.40.60.81.01.21.4 R i b Q =2 GeV R g ( b208 ) R L ( b208 ) t Q =2 GeV R g ( t208 ) R L ( t208 ) Fig.3 R L c A=12Q A=12Q =10 GeV A=12Q =100 GeV R Lb A=12Q =2 GeV A=12Q =10 GeV A=12Q =100 GeV R L t x A=12Q =2 GeV x A=12Q =10 GeV x A=12Q =100 GeV Fig.4 R L c A=208Q =2 GeV A=208Q =10 GeV A=208Q =100 GeV R Lb A=208Q =2 GeV A=208Q =10 GeV A=208Q =100 GeV R L t xA=208Q =2 GeV x A=208Q =10 GeV x A=208Q =100 GeV Fig.5 R L c A=12Q =2 GeV A=12Q =10 GeV R Lb A=12Q =2 GeV A=12Q =10 GeV R L t x A=12Q =2 GeV x A=12Q =10 GeV Fig.6 R L c A=208Q =2 GeV A=208Q =10 GeV R Lb A=208Q =2 GeV A=208Q =10 GeV R L t x A=208Q =2 GeV x A=208Q =10 GeV Fig.7 R L t o t a l xA=12Q =2 GeV xA=208Q =2 GeV2