Improved determination of d ¯ (x)− u ¯ (x) flavor asymmetry in the proton by BONuS experiment at JLAB and using an approach by Brodsky, Hoyer, Peterson, and Sakai
Maral Salajegheh, Hamzeh Khanpour, S. Mohammad Moosavi Nejad
aa r X i v : . [ h e p - ph ] D ec Improved determination of ¯ d ( x ) − ¯ u ( x ) flavor asymmetry in the proton by BONuSexperiment at JLAB and using an approach by Brodsky, Hoyer, Peterson, and Sakai Maral Salajegheh a , ∗ Hamzeh Khanpour b,c , † and S. Mohammad Moosavi Nejad a,c ‡ a Physics Department, Yazd University, P.O.Box 89195-741, Yazd, Iran b Department of Physics, University of Science and Technology of Mazandaran, P.O.Box 48518-78195, Behshahr, Iran c School of Particles and Accelerators, Institute for Research inFundamental Sciences (IPM), P.O.Box 19395-5531, Tehran, Iran (Dated: August 3, 2018)The experimental data taken from both Drell-Yan and deep-inelastic scattering (DIS) experimentssuggest a sign-change in ¯ d ( x ) − ¯ u ( x ) flavor asymmetry in the proton at large values of momentumfraction x . In this work, we present a phenomenological study of ¯ d ( x ) − ¯ u ( x ) flavor asymmetry. First,we extract the ¯ d ( x ) − ¯ u ( x ) distribution using the more recent data from the BONuS experimentat Jefferson Lab on the ratio of neutron to proton structure functions, F n /F p , and show that itundergoes a sing-change and becomes negative at large values of momentum fraction x , as expected.The stability and reliability of our obtained results have been examined by including target masscorrections (TMCs) as well as higher twist (HT) terms which are particularly important at thelarge- x region at low Q . Then, we calculate the ¯ d ( x ) − ¯ u ( x ) distribution using the Brodsky, Hoyer,Peterson, and Sakai (BHPS) model and show that if one chooses a mass for the down quark smallerthan the one for the up quark it leads to a better description for the Fermilab E866 data. In order toprove this claim, we determine the masses of down and up sea quarks by fitting to the available andup-to-date experimental data for the ¯ d ( x ) − ¯ u ( x ) distribution. In this respect, unlike the previousperformed theoretical studies, we have shown that this distribution has a sign-change at x > . PACS numbers: 11.30.Hv, 14.65.Bt, 12.38.Lg
Contents
I. Introduction II. ¯ d ( x ) − ¯ u ( x ) from recent CLAS data III. ¯ d ( x ) − ¯ u ( x ) from BHPS model IV. Summary and Conclusion Acknowledgments Appendix: The ¯ d and ¯ u intrinsicdistributions References I. INTRODUCTION
The parton distribution functions (PDFs) content fornucleon is usually determined from global fits to ex-perimental data at the large momentum transfer Q .Over the past decade, our knowledge of the quark andgluon substructure of the nucleon has been extensivelyimproved due to the high-energy scattering data from ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] the fixed target experiments, the data from ep colliderHERA [1–3] and also from high energy p ¯ p scattering atthe Tevatron [4, 5]. More recently, the data taken fromvarious channels in pp collisions at the CERN LHC play amain role to constrain the sea quarks and gluon distribu-tions at the proton [6]. In recent years, various up-to-dateefforts are being made to extract more complete infor-mation about the nucleon’s quark and gluon structure inthe form of parton distribution functions for the unpolar-ized PDFs [7–14] and the polarized PDFs [15–21] cases.These analysis are mainly focused on the extraction ofthe parton distribution functions at small and large val-ues of x up to next-to-next-to-leading order (NNLO) ac-curacy. Similar efforts have also been performed for thecase of fragmentation functions (FFs) [19, 22–28], nu-clear PDFs [29–33] and generalized parton distributions(GPDs) [34–37].Since the Gottfried sum rule [38] has been proposedin 1967, many experimental and theoretical researcheshave been widely performed so far to check the validityor violation of it and also to study the antiquark flavorasymmetry ¯ d − ¯ u in the nucleon sea (see Ref. [39] andreferences therein). If we adopt that the ¯ u and ¯ d dis-tributions in the nucleon are the same and the isospininvariance is also valid, then the Gottfried sum rule isobtained by integrating the difference between the F structure functions of the proton and neutron over x as I G ≡ R [ F p ( x ) − F n ( x )] /x dx = 1 /
3, where x is theBjorken scaling variable. However, assuming the flavorasymmetry of the nucleon sea, the Gottfried sum rule isviolated by an extra term as 2 / R [¯ u ( x ) − ¯ d ( x )] dx . Inthis way, if there is a ¯ d excess over ¯ u in the nucleon, weexpect a smaller value for the Gottfried sum than 1 / I G = 0 . ± .
026 in measuring the pro-ton and deuteron F structure functions [40] from deep-inelastic muon scattering on hydrogen and deuterium tar-gets, which is approximately 28% smaller than the Got-tfried sum. This measurement provided the first clearevidence for the breaking of this sum rule. In addition tothe deep-inelastic scattering (DIS) experiments, the vio-lation of the Gottfried sum rule can be investigated fromsemi-inclusive DIS (SIDIS) and Drell-Yan cross sectionmeasurements. The related study has been performedby the HERMES collaboration [41] in the case of SIDISexperiment. In this study a measurement of ¯ d ( x ) − ¯ u ( x )was reported over the range of 0 . < x < .
3, but witha rather large experimental uncertainty. On the otherhand, the NA51 [42] and FNAL E866/NuSea [43] collab-orations studied this violation by measuring pp and pd Drell-Yan processes and established again that there is a¯ d excess over ¯ u in the nucleon sea. Although, the ratio of¯ d/ ¯ u was only measured at the mean x -value of h x i =0.18in the NA51 experiment. The x -dependence of this ra-tio and the ¯ d ( x ) − ¯ u ( x ) flavor asymmetry have also beenmeasured over the kinematic region 0 . < x < .
35 inthe Fermilab E866 experiment.In addition to the violation of the Gottfried sum ruleas well as the existence of the ¯ d − ¯ u flavor asymmetryin the nucleon sea, one could take another important re-sult from the Fermilab E866 data. In fact, the last datapoint suggested a sign-change for the ¯ d ( x ) − ¯ u ( x ) dis-tribution at x ∼ .
3, despite of their large uncertainty.To be more precise, it indicates that this distributionmust be negative at the x -values approximately largerthan 0 .
3. This can be very important issue because theperturbative regime of quantum chromodynamics (QCD)can not lead to a remarkable flavor asymmetry in the nu-cleon sea. Furthermore, according to the studies whichhave yet been performed (for a review see Refs. [39, 44–46]) the current theoretical models, regardless of theirability to describe an enhancement of ¯ d over ¯ u , can notpredict a negative value for the ¯ d ( x ) − ¯ u ( x ) distributionat any value of x . These theoretical studies are basedon the various models such as Pauli-blocking [47–50],meson-cloud [51–54], chiral-quark [55–57], chiral-quarksoliton [58–61], intrinsic sea [62–64] and statistical [65–67] models. Except the Pauli-blocking model which con-siders a perturbative mechanism to describe the enhance-ment of ¯ d over ¯ u , other models consider a nonpertur-bative origin for this effect and are almost successful.However, the Pauli-blocking model is not successful toproduce the distribution of the ¯ d ( x ) − ¯ u ( x ) when it iscompared with the experimental data.Recently, Peng et al. [68] have presented an indepen-dent evidence for the ¯ d ( x ) − ¯ u ( x ) sign-change at x ∼ . F p − F n [40] and F d /F p [69] can also lead to a negative value for the ¯ d ( x ) − ¯ u ( x ) at x & .
3. Theyhave also discussed the significance of this sign-changeand the fact that none of the current theoretical modelscan predict this asymmetry. Future Drell-Yan experi-ment at J-PARC P04 [70] and also Fermilab E906 [71]experiments will give us more accurate information onthe ¯ d − ¯ u flavor asymmetry, especially, at the larger val-ues of x . This motivates us to study on this topic.In the present paper, following the studies performedby Peng et al. for the extraction of ¯ d ( x ) − ¯ u ( x ), wefirst investigate whether such behavior can be seen inthe analysis of data from other experiments. If it is, weshall study the approximate position of the ¯ d ( x ) − ¯ u ( x )sign-change in x and also estimate the magnitude of itsnegative area. In addition, since our study is in the lowQ at high value of x in which the target mass correc-tions (TMCs) and higher twist (HT) effects are signifi-cant, then we develop our analysis by considering thesenonperturbative contributions. Therefore, we calculatethe ¯ d ( x ) − ¯ u ( x ) distribution using the Brodsky, Hoyer,Peterson, and Sakai (BHPS) model [72] and show thatthe available experimental data for this quantity suggesta smaller value for the down quark mass than the upquark one in the BHPS formalism. Note that, this isin contrast to the previous studies in this context [62–64] where was assumed equal masses for the down andup quarks in the proton. This difference between massesleads to a sign-change for ¯ d ( x ) − ¯ u ( x ) when we evolve thisquantity to the scale of experimental data [43].The content of the present paper goes as follows: wecompare the Fermilab E866 [43] data with the predictionof the latest parton distribution functions from variousgroups and also extract the ¯ d ( x ) − ¯ u ( x ) using the updatedCLAS collaboration data for the F n /F p ratio in Sec. II.This section also includes detailed discussions on the nu-clear corrections as well as the effects arising from thenonperturbative TMCs and HT terms. In Sec. III, webriefly introduce the BHPS model and explain the ideafor choosing a smaller mass for the down quark than upquark in the BHPS formalism. Then, we prove our claimand determine the masses of down and up sea quarks byfitting the available experimental data for the ¯ d ( x ) − ¯ u ( x ).Finally, we summarize our results and present our conclu-sions in Sec. IV. Appendix presents our FORTRAN packagecontaining the ¯ d and ¯ u intrinsic distributions using theBHPS model. II. ¯ d ( x ) − ¯ u ( x ) FROM RECENT CLAS DATA
In recent years, our knowledge of the nucleon struc-ture have been developed to a large extent, but it is notstill enough. In this respect, an updated global analy-sis of PDFs including a broad range of the experimentaldata from the various observables and also theoreticalimprovements can play an important role. In the the-oretical studies, generally, an independent parametriza-tion form is chosen for the ¯ d ( x ) − ¯ u ( x ) distribution in the x d - u E866HERMESJR14NNPDF3.0MMHT14CT14
FIG. 1: A comparison between HERMES [41] and Fermi-lab E866 [43] collaborations data for the ¯ d ( x ) − ¯ u ( x ) andthe NNLO theoretical predictions of JR14 [7], NNPDF3.0 [8],MMHT14 [9] and CT14 [10] PDFs at Q = 54 GeV . global analysis of PDFs at the initial scale Q . Fig. 1shows the ¯ d ( x ) − ¯ u ( x ) data from the HERMES and theFermilab E866 at Q = 2 . , respectively,which have been compared with the NNLO theoreticalpredictions of JR14 [7], NNPDF3.0 [8], MMHT14 [9] andCT14 [10] PDFs for Q = 54 GeV . Although, all pre-dictions are in good agreement with these data, but theyhave major differences with each other. For example,there is no possibility to change the ¯ d ( x ) − ¯ u ( x ) sign atlarge- x in JR14 parametrization unlike other PDF setsor the CT14 parametrization predicts ¯ d ( x ) − ¯ u ( x ) < x -region. There is also another important con-clusion which can be taken from the E866 data. As itis clear from Fig. 1 the last data point, despite of itslarge uncertainty, indicates that the ¯ d ( x ) − ¯ u ( x ) must benegative at x -values approximately larger than 0 . et al. [68] showed that in addition tothe Drell-Yan data, there is an independent evidence forthe ¯ d ( x ) − ¯ u ( x ) sign-change at x ∼ .
3. Their results havebeen achieved by analyzing the NMC DIS data for the F p − F n [40] and the F d /F p [69]. In this section, we aregoing to investigate if such behavior can be seen in theanalysis of data from other experiments such as BarelyOff-shell Nucleon Structure (BONuS) experiment at Jef-ferson Lab. In this way, we can compute the position ofthe ¯ d ( x ) − ¯ u ( x ) sign-change in x and it is also possible toestimate the magnitude of its negative area.From the parton model, one knows that the F p,n struc-ture function of the nucleon at the leading-order (LO) ofstrong coupling constant α s is expressed as an expansionof parton distributions f i ( x ), F p,n ( x ) = P i e i xf i ( x ),where i denotes the flavor of the quarks and e i is thecharge of i ’th quark. It should be noted that, in gen-eral, the parton distributions and in conclusion the struc-ture functions depend on the four-momentum transfersquared Q . Now, if we adopt the charge symmetry of parton distributions in proton and neutron and also as-sume that the perturbatively generated s, c, b quark dis-tributions are equal in different nucleons, the followingrelation is obtained for the F p − F n at LO F p ( x ) − F n ( x ) = 13 x [ u ( x ) + ¯ u ( x ) − d ( x ) − ¯ d ( x )] . (1)In consequence, using the definition of valence quark, q v = q − ¯ q , the above relation can be used to extractthe ¯ d ( x ) − ¯ u ( x ) as follows¯ d ( x ) − ¯ u ( x ) = 12 [ u v ( x ) − d v ( x )] − x [ F p ( x ) − F n ( x )] . (2)According to Eq. (2), having two quantities u v ( x ) − d v ( x ) and F p ( x ) − F n ( x ) for a given value of x , one canextract the ¯ d ( x ) − ¯ u ( x ) flavor asymmetry. For the firstterm in Eq. (2), we can use the related parameteriza-tions from the various PDFs [7–10] and the last term( F p ( x ) − F n ( x )) in the second bracket can be calculated,for example, from the new CLAS Collaboration data re-ported for the F n /F p [73]. Since we are looking for a pos-sible sign-change in the ¯ d ( x ) − ¯ u ( x ) at a large value of x ,in this work we use the NNLO JR14 parametrization [7]for the u v − d v that its prediction for the ¯ d ( x ) − ¯ u ( x ) isclearly positive in all x , as seen in Fig. 1. In this way,if this sign-change occurs, we ensure that it is not re-sulted due to the selected PDFs. On the other hand,the CLAS Collaboration [73] has recently published thedata for the neutron structure function F n , and its ratioto the inclusive deuteron structure function ( F n /F d ) aswell as an updated extraction of Ref. [74] for the ratio R ( x ) = F n /F p from the BONuS experiment at Jeffer-son Lab. The data covers both the resonance and deep-inelastic regions including a wide range of x for Q be-tween 0 . and invariant mass W between1 and 2 . F p ( x ) − F n ( x ) inEq. (2) can be calculated from the data for the ratio R ( x )and by using the parametrization of F d ( x ) from Ref. [75],according to the following relation F p − F n = 2 F d (1 − F n /F p ) / (1 + F n /F p ) . (3)Fig. 2 shows our final results for the ¯ d ( x ) − ¯ u ( x ) dis-tribution, related to three lower cuts on the range offinal-state invariant mass; W ∗ > . W ∗ > . W ∗ > . Q -dependent and not related to a fixed value of Q ,we have allowed all quantities in Eqs. (2) and (3) to bealso Q -dependent. Therefore, the extracted ¯ d ( x ) − ¯ u ( x )data points in x are related to the different Q valuesapproximately between 1 and 4 . . For example,for the case in which W ∗ > .
6, the first and last datapoints are related to Q = 1 .
086 and 4 .
259 GeV , respec-tively. However, we could also choose an average valuefor all data, i.e. Q = 2 . . We examined this sim-plification and found it leads to an overall reduction in x -0.100.10.2 d - u W * >1.4 GeVW * >1.6 GeVW * >1.8 GeVW * >1.4 GeV (Q =2.1 GeV )NNLO JR14 FIG. 2: The ¯ d ( x ) − ¯ u ( x ) flavor asymmetry as a functionof x . The results obtained by the NNLO JR14 parametriza-tion [7] and the CLAS data [73] related to the three lower cutson the range of final-state invariant mass W ∗ . The detailedexplanation is given in the text. x -0.0500.050.10.15 d - u W * >1.4 GeVW * >1.6 GeVW * >1.8 GeVW * >1.4 GeV (Q = 2.1 GeV ) LO CT14
FIG. 3: As in Fig. 2 but obtained from the LO CT14parametrization [10] using the CLAS data [73]. The plot isrelated to the three lower cuts on the range of final-state in-variant mass W ∗ . the magnitude of ¯ d ( x ) − ¯ u ( x ), specifically, at small andlarge values of x . The related results have been shown inFig. 2 as black triangles. To estimate the uncertainties,we have included both the uncertainties of F n /F p and F d in our calculation for the F p − F n (3), and also theJR14 PDFs uncertainties in the extraction of ¯ d ( x ) − ¯ u ( x )by using Eq. (2). As can be seen from Fig. 2, the high-quality data from the BONuS experiment leads to rathersmaller uncertainties. It should be noted that Eq. (2) isextracted at the LO approximation but in our analysis,shown in Fig. 2, we used the NNLO PDF parametriza-tion for more accuracy. However, as we show in Fig. 3, ifone uses the LO PDF parameterizations from CT14 [10],the results show a sign-change as well. x -0.200.20.40.6 d - u W * >1.4 GeVW * >1.6 GeVW * >1.8 GeV NNLO JR14 TMC +HT FIG. 4: The ¯ d ( x ) − ¯ u ( x ) asymmetry considering the TMC andHT corrections. The last important issue which should be consideredin our analysis is the effect of the nonperturbative targetmass corrections (TMCs) and higher-twist (HT) terms.At the region of low Q , nucleon mass correction can-not be neglected. Therefore, the power-suppressed cor-rections to the structure functions can make an impor-tant contribution in some kinematical regions. In addi-tion to the pure kinematical origin TMCs, the structurefunctions also receive remarkable contributions from HTterms. In the range of large values of x , their contri-butions are increasingly important. In this respect, weexamine the stability and reliability of our obtained re-sults by including the TMCs as well as the HT termswhich are particularly important at the large- x regionand low Q . Actually, since the CLAS measurementsbelong to the kinematical regions of W ≈ . Q ≈ − , and the Eq. (1) might be too naive touse for the data points at such low W and Q regions,we should check the validity of our results by consideringboth the TMCs and HT term. In this regards, we followthe formalization presented in Refs. [76] and [77] in or-der to taking into account the TMC and HT correctionsin the structure functions of Eq. (1). It should be alsonoted that for calculating the HT effect, we use the re-sults presented in Table 3 of Ref. [78]. Our final resultshave been shown in Fig. 4, again for three lower cut val-ues on W ∗ . Comparing Figs. 2 and 4, one can concludethat the TMCs and HT effect overall cause the resultshave larger value than before for positive area and thedata points which were in the negative area have becomemore negative. Although, the TMCs and HT effect havepaused some negative points to the positive area, we stillhave some data points which undergo the sign change.As a last point, note that if one uses the results obtainedin Ref. [79] for calculating the HT term, similar resultswill be achieved.The most important conclusion of our analysis in thissection is to show that the sign-change of ¯ d ( x ) − ¯ u ( x ) oc-curs at large- x , as suggested by Peng et al. [68] in theiranalysis of the NMC DIS data for the F p − F n [40] and F d /F p [69], and also seen by the Drell-Yan experimentaldata measured at the Fermilab Experiment (E866) [43].Although, this sign-change has occurred at x ∼ .
5, thatis larger in comparison to the case of Drell-Yan data x ∼ . Q in comparison to the E866 data. As another considerablepoint, note that in the definition of Eq. (3) the nuclear ef-fects in the deuteron, defined as R d EMC = F d / ( F p + F n ),have been ignored. Actually, the nuclear corrections inthe deuteron structure function are small and usually areneglected in calculations. This fact is checked in the re-cent studies of the EMC effect in the deuteron by Grif-fioen et al. [80] through analyzing the recently publishedCLAS data at Jefferson Lab [73]. However, we recalcu-lated the ¯ d ( x ) − ¯ u ( x ) considering the nuclear correctionsin the deuteron but only for the last data point that its re-lated R d EMC (= 1 .
07) is comparatively large, see Ref. [80].We found that it changes the result by 10% so that thenegativity of data at large- x is still remaining. III. ¯ d ( x ) − ¯ u ( x ) FROM BHPS MODEL
In this section, we present the results of our study forthe ¯ d ( x ) − ¯ u ( x ) in the basis of the BHPS model. As wasalready mentioned in the Introduction, since the Got-tfried sum rule has been violated by the NMC measure-ment [40], many theoretical studies based on the variousmodels have yet been extended to explain the ¯ d ( x ) − ¯ u ( x )flavor asymmetry. Similar efforts have been also donein the case of strange-antistrange asymmetry of the nu-cleon sea (for instance see Refs. [81–83]). In recent years,Chang and Pang [62] have demonstrated that a good de-scription of Fermilab E866 data for the ¯ d ( x ) − ¯ u ( x ) canbe also achieved using the BHPS model [72] for the in-trinsic quark distributions in the nucleons. In the pastthree decades, the intrinsic quarks have been a subject ofinterest in many researches including both intrinsic lightand heavy quark components (see Refs. [82, 84] and ref-erences therein). According to the BHPS model that ispictured in the light-cone framework, the existence of thefive-quark Fock states | uudq ¯ q i in the proton wave func-tion is natural and the momentum distributions of theconstituent quarks are given by P ( x , · · · , x ) = N δ (cid:18) − P i =1 x i (cid:19)(cid:18) m p − P i =1 m i x i (cid:19) , (4)where m p and m i refer to the masses of the proton andquark i , and x i stands for the momentum fraction carriedby quark i . It should be noted that in Eq. (4) the effectof the transverse momentum in the five-quark transition amplitudes is neglected and the normalization factor N is also determined through the following condition Z dx · · · dx P ( x , · · · , x ) ≡ P q ¯ q , (5)where P q ¯ q is a probability to find the | uudq ¯ q i -Fock statein the proton. Considering Eq. (4), one can integrateover x , x , x and x to obtain the ¯ q -distribution in theproton. As was mentioned in Ref. [72], the probabil-ity of the five-quark Fock state is proportional to 1 /m q ,where m q is the mass of q (¯ q ) in the Fock state | uudq ¯ q i .Although, the BHPS model prediction for the P q ¯ q is suit-able when the quarks are heavy, we expect that the lightfive-quark states have a larger probability in comparisonto the heavy five-quark states.It is worth noting that the BHPS model was applied,at first, for calculating the intrinsic charm distribution[72]. However, Chang and Pang [62] generalized it to thelight five-quark states to calculate their intrinsic distribu-tions in the proton and also to extract their probabilities( P q ¯ q ) using available experimental data. It is interestingto note that they obtained different values for the P d ¯ d and P u ¯ u and therefore they extracted ¯ d ( x ) − ¯ u ( x ) dis-tribution. This may leads us to a new idea so that wecan chose different masses for down and up quarks in theBHPS formalism. To make this point more clear, notethat in one hand the P q ¯ q is proportional to 1 /m q andon the other hand, Eq. (4) completely depends on theconstituent quark masses, so these facts inevitably leadto the difference masses for the up and down quarks.Moreover, from [62], since P d ¯ d (= 0 . P u ¯ u (= 0 . m d, ¯ d should besmaller than m u, ¯ u . Considering this assumption, if oneevolve the ¯ d ( x ) − ¯ u ( x ) distributions to the experimentaldata scale [43], it will provide a sing-change at large valueof x , x > . d ( x ) − ¯ u ( x ). To this end,considering the definition of the χ -function as [85] χ = X i (∆ data i − ∆ theory i ) ( σ data i ) , (6)we must minimize it to obtain the optimum values forthe up and down quark masses. Here, ∆ data i is the ex-perimental data for the ¯ d ( x ) − ¯ u ( x ). In our analysis weuse the HERMES [41] and E866 [43] data which are theonly available data for this quantity. In (6), the theo-retical result for the ¯ d ( x ) − ¯ u ( x ) distribution (∆ theory i ) isobtained from the BHPS model and σ data i is the exper-imental error related to the systematical and statisticalerrors as: ( σ data i ) = ( σ stat i ) + ( σ syst i ) .In our calculation of the theoretical result ∆ theory i , therequired probabilities of | uudu ¯ u i and | uudd ¯ d i states (inthe proton) are taken from the recent analysis of Changand Pang [64] who have done their analysis by consid-ering the new measurements of HERMES Collaboration[86] for the x ( s + ¯ s ). The related values are P u ¯ u = 0 . P d ¯ d = 0 .
347 for µ = 0 . P u ¯ u = 0 . P d ¯ d = 0 .
296 for µ = 0 . µ is the ini-tial scale for the evolution of the non-singlet ¯ d ( x ) − ¯ u ( x )distribution to the scale of experimental data.In this analysis, we merely extract the value of m d, ¯ d by performing a fit to the experimental data. In fact, itis not necessary to extract m u, ¯ u from data analysis, be-cause one can determine this quantity using the followingequation m u, ¯ u = m p − m d, ¯ d . (7)The equation above is obtained by the fact that the pro-ton consists of two up quarks and one down quark in theground state.To minimize the χ -function (6), we employ the CERNprogram MINUIT [87] and perform our analysis at the LOand next-to-leading order (NLO) approximations. Forboth LO and NLO, our results are evolved from the ini-tial scales µ = 0 . µ = 0 . = 54 GeV for the E866 data andQ = 2 . for the HERMES data). In Table I, ourresults for m d, ¯ d along with the corresponding χ / d . o . fvalues are presented for four scenarios, depend on theorder of perturbative QCD and the initial scale applied. TABLE I: The optimum values for the d -quark mass alongwith the corresponding χ / d . o . f values.Approach χ / d . o . f m d, ¯ d LO ( µ = 0 .
3) 6.3145 0.2020 ± × − NLO ( µ = 0 .
3) 1.0682 0.2779 ± × − LO ( µ = 0 .
5) 11.2947 0.2020 ± × − NLO ( µ = 0 .
5) 4.4402 0.2020 ± × − According to Table I and Eq. (7), the possible valuesfor the m d, ¯ d are smaller than the m u, ¯ u in all scenariosapplied. As it can be seen from Table I, the value of χ / d . o . f for the NLO approach considering the initialscale µ = 0 . m d, ¯ d are the same when differentscenarios are applied, i.e. LO ( µ = 0 . µ =0 . µ = 0 . m u, ¯ u = 0 .
330 GeV using the second scenario where µ = 0 . m u, ¯ u =0 .
368 GeV considering other three scenarios.We have provided a code that gives the ¯ d and ¯ u in-trinsic quark distributions in the proton for any arbi-trary down quark mass and momentum fraction x (seeAppendix). Now, we can recalculate the BHPS modelfor the ¯ d ( x ) − ¯ u ( x ) distribution using the new massesextracted for the up and down sea quarks. Because,the minimum value of χ / d . o . f appears in the NLO sce-nario for µ = 0 .
3, we expect that this scenario leads to a x d - u E866HERMESBHPS ( µ =0.3 GeV) LOBHPS ( µ =0.3 GeV) NLOBHPS ( µ =0.5 GeV) LOBHPS ( µ =0.5 GeV) NLO FIG. 5: A comparison between the experimental data fromthe HERMES [41] and E866 [43] collaborations and the theo-retical results obtained for ¯ d ( x ) − ¯ u ( x ) in four situations, usingthe BHPS model with masses listed in Table I. more convenient consistency with the experimental data.Fig. 5 shows a comparison between the experimental dataand obtained results for the ¯ d ( x ) − ¯ u ( x ) in four scenarios,using the BHPS model with the masses listed in Table I.Actually, these results show that our assumption is cor-rect so choosing a smaller mass for the down quark islogical.Another interesting finding has been achieved from ouranalysis is that the evolved distributions have a singe-change at the large value of x . The observed differencebetween ¯ d ( x ) and ¯ u ( x ) in this study for large value of x isnot significant as presented in Fig. 5. In this regard, forshowing this sign-change, we have plotted the ¯ d ( x ) / ¯ u ( x )distribution as a function of x for four analyzed scenarios.According to Fig. 6, at x & .
33 and for all approaches,the ratio of ¯ d ( x ) / ¯ u ( x ) is smaller than 1. From Fig. 6 one x d / u BHPS ( µ =0.3 GeV) LOBHPS ( µ =0.3 GeV) NLOBHPS ( µ =0.5 GeV) LOBHPS ( µ =0.5 GeV) NLO FIG. 6: ¯ d ( x ) / ¯ u ( x ) versus x which obtained in four situations,using the BHPS model with masses listed in Table I. can conclude that, for the NLO scenario and for µ = 0 . d ( x ) − ¯ u ( x ) usingthe new and up-to-date experimental set up are mostwelcome. IV. SUMMARY AND CONCLUSION
The experimental data taken from a Drell-Yan exper-iment by FNAL E866/NuSea collaboration [43] can berecognized as a cleanest evidence for the violation of theGottfried sum rule and the existence of the ¯ d ( x ) − ¯ u ( x )flavor asymmetry in the nucleon sea. Furthermore, thesedata suggest a sign-change for the ¯ d ( x ) − ¯ u ( x ) at x ∼ . et al. [68] haspresented an independent evidence for the ¯ d ( x ) − ¯ u ( x )sign-change at x ∼ .
3. They have showed that in addi-tion to the Drell-Yan data, the analysis of the NMC DISdata for the F p − F n [40] and F d /F p [69] can also lead toa negative value for the ¯ d ( x ) − ¯ u ( x ) at x & .
3. They havealso discussed the significance of this sign-change and thefact that none of the current theoretical models can pre-dict this effect. Following their studies, we have investi-gated this behavior in the DIS data analysis from otherexperiments. Then, we have tried to found the x -positionof the ¯ d ( x ) − ¯ u ( x ) in which the sign-change occurs. Inthe following, we estimated the magnitude of the nega-tive area of the ¯ d ( x ) − ¯ u ( x ) distribution. We have alsoenriched our formalism by considering the nonperturba-tive TMCs and HT terms. As a result, we fount thatusing the updated CLAS collaboration data for the struc-ture function ratio F n /F p [73] the extracted ¯ d ( x ) − ¯ u ( x )undergoes a sing-change and becomes negative at largevalues of x , as suggested by Drell-Yan E866 data.Then, we have used in the following the BHPSmodel [72] to calculate the ¯ d ( x ) − ¯ u ( x ) distribution. Ac-cording to the BHPS prediction, we assumed that theprobability of the Fock state | uudq ¯ q i in the proton wavefunction is proportional to 1 /m q , where m q is the massof q (¯ q ) in five quark Fock state. Under this assumption,the d ( ¯ d ) quark has a smaller mass than the u (¯ u ) quark inthe proton. To prove that, we obtained the real massesfor the down and up sea quarks by fitting the available experimental data. We considered the χ -function andminimized it to obtain the optimum down and up seaquarks masses. Our calculations have been done in fourscenarios: leading- and next-to-leading order approxima-tions considering two different initial scales µ = 0 . µ = 0 . d ( x ) − ¯ u ( x ) distribution with the newextracted masses, is in good agreement with the availabledn up-to-date experimental data. In addition, unlike thepreviously performed theoretical studies [44–46], our re-sults show a sign-change on the ¯ d ( x ) − ¯ u ( x ) distribution.The latter one is the more significant finding emerge fromthis study. Any further information both on theory andexprimental observables on ¯ d ( x ) − ¯ u ( x ) asymmetry wouldhelp us to establish a greater degree of accuracy on thismatter. Theses are important issues for future research,and hence, further studies with more focus on ¯ d ( x ) − ¯ u ( x )asymmetry are therefore suggested. Acknowledgments
Authors are thankful of School of Particles and Accel-erators, Institute for Research in Fundamental Sciences(IPM) for financially support of this project. HamzehKhanpour also gratefully acknowledge the University ofScience and Technology of Mazandaran for financial sup-port provided for this research.
Appendix: The ¯ d and ¯ u intrinsic distributions We have provided a
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