The EMC ratios of 4 He , 3 He and 3 H nuclei in the k t factorization framework using the Kimber-Martin-Ryskin unintegrated parton distribution functions
TThe EMC ratios of H e , H e and H nuclei in the k t factorizationframework using the Kimber-Martin-Ryskin unintegrated partondistribution functions M. Modarres ∗ and A. Hadian † Physics Department, University of Tehran, 1439955961, Tehran, Iran.
Abstract
The unintegrated parton distribution functions (UPDFs) of H , He and He nuclei are gen-erated to calculate their structure functions (SFs) in the k t -factorization approach. The Kimber-Martin-Ryskin (KMR) formalisn is applied to evaluate the double-scale UPDFs of these nuclei fromtheir single-scale parton distribution functions (PDFs), which can be obtained from the constituentquark exchange model (CQEM). Afterwards, these SFs are used to calculate the European MuonCollaboration (EMC) ratios of these nuclei. The resulting EMC ratios are then compared with theavailable experimental data and good agreement with data is achieved. In comparison with ourprevious EMC ratios, in which the conventional PDFs were used in the calculations, the accord ofthe present outcomes with experiment at the small x region becomes impressive. Therefore, it canbe concluded that the k t dependence of partons can reproduce the general form of the shadowingeffect at the small x values in above nuclei. PACS numbers: 13.60.Hb, 21.45.+v, 14.20.Dh, 24.85.+P, 12.39.KiKeywords: Unintegrated parton distribution function, KMR and MRW frameworks, Constituent quarkexchange model, Structure function, EMC ratio. ∗ Corresponding author, Email: [email protected], Tel: +98-21-61118645, Fax: +98-21-88004781 † [email protected] a r X i v : . [ h e p - ph ] J a n . INTRODUCTION The traditional parton distribution functions (PDFs), a ( x, µ ) ( a = xq and xg ), dependon the Bjorken variable x (the longitudinal momentum fraction of the parent hadron) andthe squared scattering factorization scale µ . Conventionally, they are called the integratedPDFs, since the integration over transverse momentum k t up to the scale k t = µ is per-formed on them. Therefore, they are not explicitly depend on the scale k t . Additionally,these functions are obtained from the global analysis of deep inelastic and related hard scat-tering data, and satisfy the standard Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP)evolution equations [1–4].However, recently, it is observed that unintegrated parton distribution functions(UPDFs), f a ( x, k t , µ ), are necessary to consider for less inclusive processes, which are sen-sitive to the values of transverse momentum of partons. These distributions depend notonly on the factorization scale µ , but also on the transverse momentum k t . Therefore, theyare dependent on two hard scales, k t and µ . Application of the UPDFs to the nuclei, whichwas investigated by Martin group [5], have demonstrated that it can significantly affect thenucleus structure function (SF) at the small x region. In addition, very recently, we illus-trate that especially at the small Bjorken values ( x (cid:28) . Li nucleus [7] whichis known as shadowing effect [8, 9]. Due to dependency of the UPDFs on the extra hardscale k t , compared with the usual PDFs, we potentially have to deal with the much morecomplicated Ciafaloni-Catani-Fiorani-Marchesini (CCFM) evolution equations [10–14].Working with the CCFM equations, of course, confront two major problems. First, prac-tically, these equations are used only in the Monte Carlo event generators [15–19], and sosolving them is a mathematically complicated task. Second, these kind of equations areincapable to generate a complete quark version and can be exclusively used for the gluoncontributions [10–14]. Therefore, to overcome these obstacles, Kimber, Martin and Ryskin(KMR) introduced the more efficient k t -factorization framework [20–22]. The KMR ap-proach was constructed around the standard LO DGLAP evolution equations, along witha modification due to the angular ordering condition (AOC), which is the essential dynam-ical property of the CCFM formalism. This prescription was successfully applied by us toinvestigate different hard scattering processes in the various studies; e.g. see the references223–34]. In the section 3, we briefly introduce this approach as a method to generate thedouble-scale UPDFs from the conventional single-scale PDFs.To generate the UPDFs by using the KMR procedure, the integrated PDFs are requiredas inputs. So, we use the constituent quark exchange model (CQEM) to obtain the PDFsof He , He and H nuclei at the hadronic scale µ = 0.34 GeV [35–37]. These resultingPDFs at the initial scale µ , are then evolved to any required higher energy scale Q byusing the standard DGLAP evolution equations [38]. We will discuss about this process inthe section 2.So, in what follows, first in the section 2, based on the CQEM, the PDFs of He , He and H will be calculated. The sections 3 contains a brief introduction to the KMR formalismand the formulation of SF ( F ( x, Q )) in the k t factorization framework. Finally, results,discussions and conclusion are presented in the section 4. II. THE PDF s OF THE He , He AND H NUCLEI IN THE CQEM
In this section, we tend to obtain the point-like valence quark, sea quark and gluondistributions of He , He and H nuclei. To reach our purpose, the CQEM, which indeedconsists of two more basic schemes, is applied. These two primary approaches are thequark exchange framework (QEF) [39, 40] and the constituent quark model (CQM) [41–43]. The QEF was first suggested by Hoodbhoy and Jaffe to calculate the valence quarkmomentum distributions of A = 3 iso-scalar system [39, 40], and afterwards, was successfullyreformulated by us for the Li and He nuclei [35, 44, 45]. However, this approach is unableto generate the other partonic degrees of freedom, i.e., the sea quarks and the gluons. Toconsider these extra distributions, the CQM, which was first introduced by Feynman [41–43], is incorporated in the QEF. This combination, like our previous works (e.g. references[7, 35, 36, 45]), is denominate the CQEM (=QEF ⊕ CQM).The up and down constituent quark momentum distributions of He and H nuclei, whichwere calculated by using the QEF in the reference [37], can be written as follows: ρ He U ( k ) = ρ H D ( k ) = (cid:104) A ( k ) + 29 B ( k ) − C ( k ) + 2827 D ( k ) (cid:105)(cid:104) I (cid:105) − , (1) ρ He D ( k ) = ρ H U ( k ) = (cid:104) A ( k ) + 19 B ( k ) − C ( k ) + 2627 D ( k ) (cid:105)(cid:104) I (cid:105) − , (2)3here ρ U and ρ D represent the up and down constituent quark momentum distributions,respectively. For the He iso-scalar nucleus, the up and down momentum distributions areequal, and these distributions, which were computed in the reference [35], can be presentedas follows: ρ He U ( k ) = ρ He D ( k ) = (cid:104) A ( k ) + 2 B ( k ) + 43 C ( k ) + 23 D ( k ) (cid:105)(cid:104) I (cid:105) − . (3)In the above equations, the coefficients A, B, C, D, and the overlap integral I are defined asfollows: A ( k ) = (cid:16) b π (cid:17) exp (cid:104) − b k (cid:105) , (4) B ( k ) = (cid:16) b π (cid:17) exp (cid:104) − b k (cid:105) I , (5) C ( k ) = (cid:16) b π (cid:17) exp (cid:104) − b k (cid:105) I , (6) D ( k ) = (cid:16) b π (cid:17) exp (cid:104) − b k (cid:105) I . (7) I = 8 π (cid:90) ∞ x dx (cid:90) ∞ y dy (cid:90) − d ( cosθ ) exp (cid:104) − x b (cid:105) | χ ( x, y, cosθ ) | , (8)where χ is the nuclear wave function and parameter b is the nucleon’s radius. Note that thebasic expressions in this section are based on the naive harmonic oscillator model for theconstituent quarks. In the present study, we intend to concentrate only on the pure quark-exchange effect, dynamically. Therefore, to reduce the number of variables, we suppose thesame nucleons radius, b = 0.8 f m , for the He , He and H nuclei, with correspondingoverlap integral I . The thorough discussions about calculating the above momentum distri-butions for the He and He nuclei in the QEF, were given in the references [35] and [37],respectively. Now, the constituent quark distributions in the nucleons of the nucleus A i , ateach Q , can be related to the above momentum distributions, as follows ( j = p , n ( a = U , D ) for the proton (up quark) and neutron (down quark), respectively) [39]: f ja ( x, Q ; A i ) = (cid:90) ρ ja ( (cid:126)k ; A i ) δ (cid:16) x − k + M (cid:17) d(cid:126)k, (9)the reason for the Q dependence of the right hand side of the equation (9) will be explainedbelow. The light-cone momentum of the constituent quark in the target rest frame is usedand k is considered as a function of | (cid:126)k | ( k = [( (cid:126)k + m a ) − (cid:15) a ). The two free parameters,i.e., m a and (cid:15) a , are the quark masses and their binding energies, respectively. We can4etermine these free parameters such that the best fit to the valence quark distributionfunctions of Martin et al. , i.e., MSTW 2008 [46–48], is achieved, at Q = 0 . GeV . Bydoing so, for the He , the pair of ( m a , (cid:15) a ) is chosen as (320, 120 MeV) ( a = U , D ), andfor the He , the pairs of ( m U , (cid:15) U ) and ( m D , (cid:15) D ) are taken as (300, 130 MeV) and (325, 115MeV) (they will be interchanged for H ), respectively. After doing the angular integration,the equation (9) leads to the following constituent quark distributions: f aj ( x, Q ; A i ) = 2 πM (cid:90) ∞ k amin ρ aj ( (cid:126)k ; A i ) kdk, (10)with, k amin ( x ) = ( xM + (cid:15) a ) − m a xM + (cid:15) a ) , (11)where M indicates the nucleon mass. Because of the above fitting the right hand side of theequations (9) and (10) become Q dependent.By determination of the constituent distributions of He , He and H nuclei via the QEF,it’s the time to present a brief description of the CQM to complete our discussion about theCQEM. In the CQM, it is supposed that the constituent quarks are not fundamental objects,but instead consist of point-like partons [41–43]. Therefore, their structure functions canbe expressed by a set of functions, φ ab ( x ), which define the number of partons of type b inside the constituent of type a with the fraction x of its total momentum. The varioustypes and functional forms of the constituent quarks structure functions are extracted fromthree natural assumptions, namely: (i) the determination of the point-like partons by QCD,(ii) the Regge behavior for x → φ P q v (cid:16) xz , µ (cid:17) = Γ( A + )Γ( )Γ( A ) (cid:16) − xz (cid:17) A − (cid:112) xz , (12)for the sea quarks, φ P q s (cid:16) xz , µ (cid:17) = C xz (cid:16) − xz (cid:17) D − , (13)and finally, for the gluons, φ P g (cid:16) xz , µ (cid:17) = G xz (cid:16) − xz (cid:17) B − . (14)5he momentum carried by the second moments of the parton distributions are known ex-perimentally at high Q . Their values at the low scale Q could be obtained by performinga next-to-leading-order evolution downward. These procedure is used to extract the valueof the constants A, B, G and the ratio C / D . For example, at the hadronic scale Q = 0.34 GeV , 53.5 of the nucleon momentum is carried by the valence quarks, 35.7 by the gluonsand the remaining momentum are belong to the sea quarks. So, in this scale, the mentionedparameters take the following values: A = 0 . B = 0 . C = 0 . D = 2 .
778 and G = 0 . q ( x, µ ) = (cid:90) x dzz (cid:104) U ( z, µ ) φ U q (cid:16) xz , µ (cid:17) + D ( z, µ ) φ D q (cid:16) xz , µ (cid:17)(cid:105) , (15)where q denotes the various point-like partons, i.e., valence quarks ( u v , d v ), sea quarks ( u s , d s , s ), sea anti-quarks (¯ u s , ¯ d s , ¯ s ) and gluons ( g ). The U and D indicate the distributionsof up and down constituent quarks, respectively. Actually, these quantities are the same asthe functions f aj ( a = U , D ) of the equation (10), and for simplicity, since then, we replacethe f U j and f D j labels by U and D , respectively. The µ = 0.34 GeV is the initial hadronicscale at which the CQM is defined. In the CQM, the sea quark and anti-quark distributionsare independent of iso-spin flavor. Therefore, in the following, the label q s represents bothsea quark and anti-quark distributions. It should be noted that, the structure functions φ U d (cid:16) xz , µ (cid:17) and φ D u (cid:16) xz , µ (cid:17) in the equation (15) are zero, because in the constituent quarkof type U , there is no point-like valence quark of type d and vice versa (see the reference[49] about the origin of this assumption). In addition, for the He nucleus, the constituentup and down quark distributions are equal, because unlike the He and H cases, it is aniso-scalar system.Therefore, eventually, the single-scale PDFs of He , He and H nuclei at the hadronicscale µ can be specified in the CQEM as follows:(i) for the He nucleus, u Hev ( x, µ ) = d Hev ( x, µ ) = (cid:90) x dzz U He ( z, µ ) φ U q v (cid:16) xz , µ (cid:17) , (16) q Hes ( x, µ ) = 2 (cid:90) x dzz U He ( z, µ ) φ U q s (cid:16) xz , µ (cid:17) , (17)6 He ( x, µ ) = 2 (cid:90) x dzz U He ( z, µ ) φ U g (cid:16) xz , µ (cid:17) , (18)where U He ( z, µ ) = 2 πM (cid:90) ∞ k min ρ He U ( k ) kdk, (19)(ii) for the He and H nuclei, u Hev ( x, µ ) = d Hv ( x, µ ) = (cid:90) x dzz U He ( z, µ ) φ U u v (cid:16) xz , µ (cid:17) , (20) d Hev ( x, µ ) = u Hv ( x, µ ) = (cid:90) x dzz D He ( z, µ ) φ D d v (cid:16) xz , µ (cid:17) , (21) q Hes ( x, µ ) = q Hs ( x, µ ) = (cid:90) x dzz (cid:104) U He ( z, µ ) φ U q s (cid:16) xz , µ (cid:17) + D He ( z, µ ) φ D q s (cid:16) xz , µ (cid:17)(cid:105) , (22) g He ( x, µ ) = g H ( x, µ ) = (cid:90) x dzz (cid:104) U He ( z, µ ) φ U g (cid:16) xz , µ (cid:17) + D He ( z, µ ) φ D g (cid:16) xz , µ (cid:17)(cid:105) , (23)where U He ( z, µ ) = 2 πM (cid:90) ∞ k min ρ He U ( k ) kdk, (24)and D He ( z, µ ) = 2 πM (cid:90) ∞ k min ρ He D ( k ) kdk. (25)These resulted PDFs for the He and He nuclei, at the hadronic scale µ = 0.34 GeV , areshown in the panels (a) and (b) of figure 1, respectively.Now, by using the standard DGLAP equations, the above PDFs which are obtained fromthe CQEM at the initial scale µ , can be evolved to any higher energy scale Q [38]. However,these conventional PDFs are not k t -dependent distributions. So, to consider the transversemomentum explicitly, in the next section the KMR approach will be introduced to generatethe double-scale UPDFs from these single-scale PDFs. III. THE KMR FORMALISM AND THE UPDF s AND SF CALCULATIONS
It is well known that there are problems at small x region [54–57]. So one should usethe general formalism such as CCFM which the transverse momentum of partons play thecrucial role or the reggeon theory such as pameron model. However it was shown that the k t -factorization formalism is capable to consider the precise kinematics of the process and7n important part of the virtual loop corrections, via the survival probability factor T (seebelow). On the other hand, if we work with integrated partons, we have to include the NLO(and sometimes the NNLO) contributions to account for these effects. These differencesappear to cause a discrepancy between the integrated and unintegrated frameworks [20–22].A brief description of the KMR formalism as well as the SF formula in the k t -factorizationframework, is presented in the following subsections (A and B), respectively. A. The KMR formalism
In this subsection, we briefly discuss about the KMR scheme to extract the UPDFs fromthe resulted integrated PDFs of the previous section, as inputs. The KMR formalism wasfirst proposed by Kimber, Martin and Ryskin [20–22]. From the two scheme discussed inthe reference [21] we use the second approach which directly relates the UPDFs to theconventional PDFs. This formalism was also separately discussed in the reference [22].Based on this scheme, the LO DGLAP equations can be modified by separating the realand virtual contributions of the evolution, and the two-scale UPDFs, f a ( x, k t , µ ) ( a = q or g ), can be defined as follows: f a ( x, k t , µ ) = T a ( k t , µ ) (cid:88) b = q,g (cid:104) α s ( k t )2 π (cid:90) − ∆ x dzP ( LO ) ab ( z ) b (cid:16) xz , k t (cid:17)(cid:105) , (26)where P ( LO ) ab represent the LO splitting functions, which account for the probability of aparton of type a with momentum fraction x (cid:48)(cid:48) , a ( x (cid:48)(cid:48) , Q ), emerging from a parent partonof type b with a larger momentum fraction x (cid:48) , b ( x (cid:48) , Q ), through z = x (cid:48)(cid:48) /x (cid:48) . The survivalprobability factor, i.e., Sudakov form factor T a , which gives the probability that parton a with transverse momentum k t remains untouched in the evolution up to the factorizationscale µ , is defined via the following equation: T a ( k t , µ ) = exp (cid:16) − (cid:90) µ k t α s ( k )2 π dk k (cid:88) b = q,g (cid:90) − ∆0 dz (cid:48) P ( LO ) ab ( z (cid:48) ) (cid:17) , (27)The infrared cut-off, ∆ = 1 − z max = k t / ( µ + k t ), is determined by imposing the AOCon the last step of the evolution, and protects the 1 / (1 − z ) singularity in the splittingfunctions arising from the soft gluon emission. In the KMR formulation, the key idea is thatthe dependence on the second scale µ of the UPDFs appears only at the last step of the8volution. By completing the procedures of producing the UPDFs from the KMR scheme,the UPDFs of the He , He and H nuclei can be evaluated by using their conventionalPDFs (which were determined in the previous section), as inputs. B. The SF in the k t -factorization framework Here it is briefly described the different steps to calculate the SF ( F ( x, Q )) in the k t -factorization framework, by using the KMR UPDFs as inputs. We explicitly investigate theseparate contributions of gluons and (direct) quarks to the SF expression [20–22].The unintegrated gluons can contribute to F via an intermediate quark. As shown inthe figure 2, both the quark box and crossed-box diagrams must be regarded as the gluonportions. The variable z denotes the fraction of the gluon’s momentum that is transferred tothe exchanged struck quark. The parameters k t and κ t indicate the transverse momentumof the parent gluons and daughter quarks, respectively. In the k t -factorization framework,the unintegrated gluon contributions to F can be obtained via the following equation [20–22, 58–60]: F g → q ¯ q ( x, Q ) = (cid:88) q e q Q π (cid:90) dk t k t (cid:90) dβ (cid:90) d κ t α s ( µ ) f g (cid:16) xz , k t , µ (cid:17) Θ (cid:16) − xz (cid:17) × (cid:110) [ β + (1 − β ) ] (cid:16) κ t D − κ t − k t D (cid:17) + [ m q + 4 Q β (1 − β ) ] (cid:16) D − D (cid:17) (cid:111) . (28)The variable β is defined as the light-cone fraction of the photon’s momentum carried bythe internal quark line. In addition, the denominator factors are defined as follows: D = κ t + β (1 − β ) Q + m q D = ( κ t − k t ) + β (1 − β ) Q + m q . (29)In the equation (28), the summation goes over various quark flavors q with different masses m q which can appear in the box. In the present study, we consider three lightest flavor ofquarks ( n f = 3), i.e., u , d and s , whose masses are neglected with a good approximation.So, n f = 3 throughout of our calculations. Additionally, the variable z is defined as follows:1 z = 1 + κ t + m q (1 − β ) Q + k t + κ t − κ t . k t + m q βQ . (30)9hich is the ratio of the Bjorken variable x and the fraction of the proton momentum carriedby the gluon. As in the reference [58], the scale µ , which controls the unintegrated gluondistribution and the QCD coupling constant α s , is chosen as follows: µ = k t + κ t + m q . (31)The equation (28) gives the contributions of unintegrated gluons to F in the perturbativeregion, k t > k , where the UPDFs are defined. The smallest cutoff, k , we can choose, isthe initial scale of order 1 GeV , at which the k t -factorization scheme is defined [59]. For thecontribution from the nonpertubative region, k t < k , it can be approximated (cid:90) k dk t k t f g ( x, k t , µ ) (cid:104) remainder of equation (28) k t (cid:105) (cid:39) xg ( x, k ) T g ( k , µ ) (cid:104) (cid:105) k t = a , (32)where a is belong to the interval (0, k ). The dependence on the choice of a is numericallyunimportant to the nonperturbative contribution [20–22].Now, the contributions of unintegrated quarks must be added to F . If an initial quarkwith Bjorken scale x / z and perturbative transverse momentum k t > k , splits to a radiatedgluon and a quark with smaller Bjorken scale x and transverse momentum κ t , this finalquark can then couple to the photon and contributes to F , as follows: F q ( perturbative )2 ( x, Q ) = (cid:88) q = u,d,s e q (cid:90) Q k dκ t κ t α s ( κ t )2 π (cid:90) κ t k dk t k t (cid:90) Q/ ( Q + k t ) x dz × (cid:104) f q (cid:16) xz , k t , Q (cid:17) + f ¯ q (cid:16) xz , k t , Q (cid:17)(cid:105) P qq ( z ) , (33)where during the quark evolution, AOC is imposed on the upper limit of the z integration.Again, one must consider the nonperturbative contributions for the k t < k , F q ( nonperturbative )2 ( x, Q ) = (cid:88) q e q (cid:16) xq ( x, k ) + x ¯ q ( x, k ) (cid:17) T q ( k , Q ) , (34)which physically can be assumed as a quark or anti-quark, which does not experience realsplitting in the perturbative region, and interacts unchanged, with the photon at the scale Q . So, a Sudakov-like factor, T q ( k , Q ), is written to indicate the probability of evolutionfrom k to Q without radiation.Finally, by summing both gluon and quark contributions, one can obtain the overall SFin the k t -factorization framework. Subsequently, the EMC ratio, which is defined as the10atio of the SF of the bound nucleon to that of the free nucleon, can be evaluated as follows[39]: R EMC = F T ( x ) F T (cid:63) ( x ) , (35)where T stands for the target, averaged over nuclear spin and iso-spin and T (cid:63) is a hypothet-ical target with exactly the same quantum numbers but with no parton exchange [39]. So,if the overlap integral I is omitted in the momentum distribution formula, i.e., the equa-tions (1)-(3), we can compute the SF of free nucleons. The effects of nuclear Fermi motionare neglected from both T and T (cid:63) . We utilize the KMR UPDFs to calculate the SFs, andthe EMC ratios of He , He and H nuclei in the k t factorization approach, which will bepresented in the next section. IV. RESULTS, DISCUSSIONS, AND CONCLUSIONS
The overall SF, F , of He nucleus in the k t -factorization framework, using the KMRUPDFs, at the energy scales Q = 4.5 and 27 GeV are plotted in the panels (a) and (b)of Figure 3, respectively (the full curves). As expected, by increasing the scale Q from 4.5to 27 GeV , a considerable rise in F at the smaller values of x occurs. The dash curvesin each panel, are the SFs of the free proton in the k t -factorization framework, in which togenerate the KMR UPDFs, the MSTW 2008 PDF sets are used as inputs. The SF of ahypothetical He target, without any quark exchange between its nucleons (by ignoring theoverlap integral I in the momentum density equation), i.e., the hypothetical free nucleon,in the k t -factorization framework using the KMR UPDFs, are also exhibited in this figurefor comparison (the dotted curves). The three lightest flavors of quarks, i.e., u , d and s ,are considered in calculation of these SFs. According to the equation (35), the EMC ratioin the k t -factorization formalism at each energy scale, can be evaluated by regarding theratio of the full curve ( He SF) to the dotted curve (the hypothetical free nucleon SF). It isobserved that the SFs of our hypothetical free nucleon are in overall good agreement withthe SF of the free proton. Especially at the small x region, as one should expect, the SFs offree nucleon (the dotted curves) and free proton (the dash curves) are approximately equal,since in this area, u = d = ¯ u = ¯ d and the proton and neutron SFs must be the same. Thesimilar conclusions have been made for the Li nucleus in our recent work, i.e., the reference[7]. 11igures 4 and 5 are the same as figure 3, but for the He and H nuclei, respectively.Similar to the figure 3, the SFs of free nucleon (the dotted curves) and free proton (the dashcurves) are again approximately equal at the small x . Also, to obtain the EMC ratio in the k t -factorization framework for these nuclei, one should again consider the ratio of the fullcurves to the dotted curves in each panel. In addition, as we increase the Q value to 27 GeV , again, the overall SFs become greater.The resulting EMC ratios of He , He and H nuclei in the k t -factorization framework,using the KMR UPDFs, are plotted in the figures 6, 7 and 8, respectively. For each ofthese nuclei, the ratio is calculated at the energy scales Q = 4.5 and 27 GeV . Due toneglecting the Fermi motion, the EMC ratios monotonically decrease and the growth in theEMC ratios at the large x values do not occur. Therefore, the EMC ratios are illustratedfor the x ≤ . He EMC ratios more clearly at the small x , the experimental NMC dataare illustrated in the distinct diagrams with logarithmic scale, i.e., panels (b) and (d) of thefigure 6. The dash curves in the panels (a) and (b) of the figure 6, are given from our priorwork [35], in which the k t dependence of parton distribution functions were neglected in the He EMC calculations. Obviously, at the small x region, the present He EMC results areextremely improved with respect to our previous outcomes [35]. However, when the Bjorkenscale x is increased, the differences between the full and dash curves decrease, which showthat the k t -factorization scheme has an important effect on the EMC calculations at thesmall x values [7–9], i.e., shadowing region. Therefore, the inclusion of k t -dependent PDFsin the EMC calculation, can reproduce the general form of shadowing effect [7–9]. Thesimilar behavior is seen in the EMC curves of He and H nuclei (see the figures 7 and 8,respectively) as well as the EMC ratio of Li nucleus (see the figure 10 of reference [7]). Inaddition, for all three nuclei which discussed here, the EMC curves at the energy scales 4.5and 27 GeV have approximately the same behavior (see also the EMC ratio of Li nucleusin the figure 10 of reference [7]). This similarity is expected, because the EMC ratio are not Q dependent, significantly (e.g. see the reference [64]).The comparisons of EMC ratios of Li (the dash-dotted curves), He (the dash curves),12 He (the dotted curves) and H (the full curves) nuclei in the KMR approach at the energyscales 4.5 and 27 GeV are displayed in the left and right panels of figure 9, respectively. The Li EMC ratios are plotted from the reference [7]. As expected, the EMC curves of He and H mirror nuclei are very close together, because of iso-spin symmetry assumption. However,by increasing the number of nucleons in the nucleus, the probabilities of quark exchangesamong the nucleons are increased, which make the R EMC to have greater deviation fromunity.In conclusion, the CQEM and the KMR UPDFs were used to obtain the EMC ratiosof He , He and H nuclei in the k t -factorization framework. To calculate the double-scale UPDFs, we needed the conventional single-scale PDFs for each nucleon as inputs.Therefore, the CQEM was employed to elicit the PDFs of these nuclei at the hadronic scale0.34 GeV . Then, the resulted PDFs were evolved by the DGLAP evolution equations tothe higher energy scales. Subsequently, by using the KMR UPDFs, the SFs of these nucleiin the k t -factorization scheme were calculated at the energy scales Q = 4.5 GeV and 27 GeV . Subsequently, we compared the resulted SFs with the corresponding SF of free proton.Eventually, after computing the EMC ratios of He , He and H nuclei, they were comparedwith the experimental data. It was seen that the outcome EMC ratios astonishingly wereconsistent with the various experimental data. Especially, the k t -factorization approachextremely improved the EMC ratios of mentioned nuclei at the shadowing region. Therefore,similar to our previous work [7], the reduction of EMC effect at the small x region, whichtraditionally is known as the ”shadowing phenomena” [8, 9], can be successfully explainedin the k t -factorization framework by using the KMR UPDFs. Acknowledgements
M M would like to acknowledge the Research Council of University of Tehran for thegrants provided for him. [1] V. N. Gribov and L. N. Lipatov, Yad. Fiz. 15 (1972) 781.[2] L. N. Lipatov, Sov. J. Nucl. Phys. 20 (1975) 94.[3] G. Altarelli and G. Parisi, Nucl. Phys. B 126 (1977) 298.
4] Y. L. Dokshitzer, Sov. Phys. JETP 46 (1977) 641.[5] E.G. de Oliveira, A.D. Martin, F.S. Navarra, M.G. Ryskin, J. High Energy Phys. 09 (2013)158.[6] J.J. Aubert, et al., Phys. Lett. B 105 (1983) 403.[7] M. Modarres, A. Hadian, Phys. Rev. D 98 (2018) 076001.[8] L. L. Frankfurt and M. I. Strikman, Phys. Rep., 160 (1988) 235.[9] L. L. Frankfurt, V. Guzey,and M. I. Strikman, Phys. Rep., 512 (2012) 255.[10] M. Ciafaloni, Nucl. Phys. B 296 (1988) 49.[11] S. Catani, F. Fiorani, and G. Marchesini, Phys. Lett. B 234 (1990) 339.[12] S. Catani, F. Fiorani, and G. Marchesini, Nucl. Phys. B 336 (1990) 18.[13] G. Marchesini, Proceedings of the Workshop QCD at 200 TeV, Erice, Italy, edited by L.Cifarelli and Yu. L. Dokshitzer (Plenum, New York, 1992), p. 183.[14] G. Marchesini, Nucl. Phys. B 445 (1995) 49.[15] G. Marchesini, B. Webber, Nucl. Phys. B 349 (1991) 617.[16] G. Marchesini, B. Webber, Nucl. Phys. B 386 (1992) 215.[17] H. Jung, Nucl. Phys. B 79 (1999) 429.[18] H. Jung, G.P. Salam, Eur. Phys. J. C 19 (2001) 351.[19] H. Jung, J. Phys. G: Nucl. Part. Phys. 28 (2002) 971.[20] M. A. Kimber, Ph.D. thesis, University of Durham, 2001.[21] M. A. Kimber, A. D. Martin, and M. G. Ryskin, Phys. Rev. D 63, (2001) 114027.[22] M.A. Kimber , A.D. Martin, and M.G. Ryskin, Eur. Phys. J. C 12 (2000) 655.[23] M. Modarres and H. Hosseinkhani, Nucl. Phys. A 815, (2009) 40.[24] M. Modarres and H. Hosseinkhani, Few-Body Syst. 47, (2010) 237.[25] H. Hosseinkhani and M. Modarres, Phys. Lett. B 694, (2011) 355.[26] H. Hosseinkhani and M. Modarres, Phys. Lett. B 708, (2012) 75.[27] M. Modarres, H. Hosseinkhani, and N. Olanj, Nucl. Phys. A 902, (2013) 21.[28] M. Modarres, H. Hosseinkhani, N. Olanj, and M. R. Masouminia, Eur. Phys. J. C 75, (2015)556.[29] M. Modarres, et al., Phys. Rev. D 94, (2016) 074035.[30] M. Modarres, et al., Nucl. Phys. B 926 (2018) 406.[31] M. Modarres, et al., Phys. Lett. B 772 (2017) 534.
32] M. Modarres, et al., Nucl. Phys. B 922 (2017) 94.[33] M. Modarres, H. Hosseinkhani, and N. Olanj, Phys. Rev. D 89, (2014) 034015.[34] M. Modarres, M. R. Masouminia, H. Hosseinkhani, and N. Olanj, Nucl. Phys. A 945, (2016)168.[35] M. Modarres, A. Hadian, Eur. Phys. J. A (2018) to be published.[36] M. Modarres, M. Rasti, M. M. Yazdanpanah, Few-Body Syst. 55 (2014) 85.[37] M. Modarres, F. Zolfagharpour, Nucl. Phys. A 765 (2006) 112.[38] M. Botje, Comput. Phys. Commun. 182 (2011) 490, arXiv:1005.1481.[39] P. Hoodbhoy, R.L. Jaffe, Phys. Rev. D 35 (1987) 113.[40] P. Hoodbhoy, Nucl. Phys. A 465 (1987) 113.[41] R.P. Feynman, Photon Hadron Interactions, Benjamin, New York (1972).[42] F.E. Close, An Introduction to Quarks and Partons, Academic Press, London (1989).[43] R.G. Roberts, The Structure of the Proton, Cambridge University Press, New York (1993).[44] M. Modarres, A. Hadian, Nucl. Phys. A 966 (2017) 342.[45] M. Modarres, A. Hadian, Int. J. Mod. Phys. E 24 (2015) 1550037.[46] A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt, Eur. Phys. J. C 63, (2009) 189.[47] A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt, Eur. Phys. J. C 64, (2009) 653.[48] A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt, Eur. Phys. J. C 70, (2010) 51.[49] G. Altarelli, N. Cabibbo, L. Maiani, R. Petronzio, Nucl. Phys. B 69 (1974) 531.[50] A. Manohar, H. Georgi, Nucl. Phys. B 234 (1984) 189.[51] S. Scopetta, V. Vento, M. Traini, Phys. Lett. B 421 (1998) 64.[52] M. Traini, V. Vento, A. Mair, A. Zambarda, Nucl. Phys. A 614 (1997) 472.[53] M. M. Yazdanpanah, M. Modarres, M. Rasti, Few-Body Syst. 48 (2010) 19.[54] S. Catani and F. Hautmann, Nucl.Phys.B, 427 (1994) 475.[55] A. Donnachie, H.G. Dosch, P.V. Landshoff and O. Nachtman, Pomeron physics and QCD,Cambridge University press, Cambridge (2002).[56] A. Donnachie and P.V. Landshoff, Eur.Phys.J., 77 (2017) 524.[57] Andersson et al, Eur.Phys.J.C, 25 (2002) 77.[58] J. Kwiecinski, A. D. Martin, and A. M. Stasto, Phys. Rev. D, 56 (1997) 3991.[59] A. J. Askew, J. Kwiecinski, A. D. Martin, and P. J. Sutton, Phys. Rev. D 47, (1993) 3775.[60] J. Kwiecinski, A. D. Martin, and A. M. Stasto, Acta Phys. Pol. B 28, (1997) 2577.
61] J. Seely, A. Daniel, D. Gaskell, J. Arrington et al., Phys. Rev. Lett. 103 (2009) 202301.[62] S. Malace, D. Gaskell, D.W. Higinbotham, I.C. Cloet, Int. J. Mod. Phys. E 23 (2014) 1430013.[63] New Muon Collaboration Collaboration (P. Amaudruz et al.), Nucl. Phys. B 441 (1995) 3.[64] J. Gomez, R. Arnold, P. E. Bosted, C. Chang et al., Phys. Rev. D 49 (1994) 4348.[65] A. Airapetisn et al., Phys. Lett. B 567 (2003) 339. .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 x q H e ( x , µ = . G e V ) (b) x x u Hev ( = x d Hv ) x d Hev ( = x u Hv ) x g He ( = x g H ) x q Hes ( = x q Hs ) (a) x x q H e ( x , µ = . G e V ) x u Hev ( =x d Hev ) x g He x q Hes
Fig. 1
FIG. 1: Panel (a): PDFs of He nucleus versus x , for ( m a , (cid:15) a ) pair of (320, 120 M eV ) ( a = U , D ) and b = 0.8 f m at the hadronic scale, µ = 0.34 GeV . The dash curve represents the gluondistribution, while the solid and dotted curves indicate the valence and sea quark distributions,respectively. Panel (b): PDFs of He nucleus versus x , for ( m U , (cid:15) U ) and ( m D , (cid:15) D ) pairs of (300,130 MeV) and (325, 115 MeV), respectively, and b = 0.8 f m at the hadronic scale, µ = 0.34 GeV .The solid and dotted-dash curves are the valence up (down) and down (up) quark distributions of He ( H ) nucleus, respectively, while the dotted and dash curves indicate the sea quark and gluondistributions, respectively. IG. 2: The quark box and crossed-box diagrams which demonstrate the contribution of theunintegrated gluon distributions, f g ( x / z , k t , µ ), to the structure function, F . -4 -3 -2 -1 free proton He nucleus free nucleon ( I overlap =0) Q = 27 GeV (b) F x -4 -3 -2 -1 free proton He nucleus free nucleon ( I overlap =0) Q = 4.5 GeV (a) x F Fig. 3
FIG. 3: The comparison of the SFs of the He nucleus in the KMR approach (the full curves) withthose of the free proton using the MSTW-2008 data sets as inputs (the dash curves), at the energyscales 4.5 GeV (panel (a)), and 27 GeV (panel (b)). the dotted curves indicate our hypotheticalfree nucleon (by setting the overlap integral I equal to zero in the momentum density formula of He nucleus) SFs in the KMR approach. All SFs are calculated with considering the three lightestquark flavors ( u , d , s ). -4 -3 -2 -1 = 27 GeV (b) x F free proton He nucleus free nucleon ( I overlap =0) -4 -3 -2 -1 free proton He nucleus free nucleon ( I overlap =0) Q = 4.5 GeV (a) x F Fig. 4
FIG. 4: The same as the figure 3, but for the He nucleus. -4 -3 -2 -1 free proton H nucleus free nucleon ( I overlap =0) Q = 27 GeV (b) F x -4 -3 -2 -1 free proton H nucleus free nucleon ( I overlap =0) Q = 4.5 GeV (a) x F Fig. 5
FIG. 5: The same as the figure 3, but for the H nucleus. ig. 6 -4 -3 -2 -1 (d) Q = 27 GeV x R H e E M C ( x ) KMR NMC x (c) R H e E M C ( x ) Q = 27 GeV KMR JLab SLAC -4 -3 -2 -1 x (b) Q = 4.5 GeV R H e E M C ( x ) KMR NMC Old ratio [33] x (a) R H e E M C ( x ) Q = 4.5 GeV KMR Old ratio [33] JLab SLAC
FIG. 6: The EMC ratio of He nucleus in the k t -factoization framework by using the KMR UPDFsas inputs (the full curves), at the energy scales 4.5 GeV (panels (a) and (b)), and 27 GeV (panels(c) and (d)). The circles, the triangles, and the squares are from JLab [61, 62], NMC [62, 63], andSLAC [62, 64] experimental data, respectively. the dotted-dash curves in the panels (a) and (b),are given from reference [35] at b = 0.8 f m and Q = 0.34 GeV , in which the contributions ofUPDFs are not accounted in the He EMC calculations. .0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.91.01.1 Q = 4.5 GeV R H e E M C ( x ) x KMR JLab HERMES Fig. 7 x R H e E M C ( x ) Q = 27 GeV KMR JLab HERMES
FIG. 7: The EMC ratio of He nucleus in the k t -factoization framework by using the KMR UPDFsas inputs (the full curves), at the energy scales 4.5 GeV (left panel), and 27 GeV (right panel).The filled circles and the filled squares are the experimental data from JLab [61, 62] and HERMES[62, 65], respectively. .0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.91.01.1 Q = 27 GeV R H E M C ( x ) x KMR JLab HERMES
Fig. 8 x R H E M C ( x ) Q = 4.5 GeV KMR JLab HERMES
FIG. 8: The same as the figure 7, but for the H nucleus. .0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.900.951.001.05 x KMR H KMR He KMR He KMR Li Q = 27 GeV R E M C ( x ) x Q = 4.5 GeV R E M C ( x ) KMR H KMR He KMR He KMR Li Fig. 9