Evaluation of the Gottfried sum with use of the truncated moments method
aa r X i v : . [ h e p - ph ] J a n Evaluation of the Gottfried sum with use of the truncated moments method
A. Kotlorz, ∗ D. Kotlorz,
1, 2, † and O. V. Teryaev ‡ Opole University of Technology, 45-758 Opole, Proszkowska 76, Poland Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia (Dated: January 26, 2021)We reanalyze the experimental NMC data on the nonsinglet structure function F p − F n andE866 data on the nucleon sea asymmetry ¯ d/ ¯ u using the truncated moments approach elaborated inour previous papers. With help of the special truncated sum one can overcome the problem of theunavoidable experimental restrictions on the Bjorken x and effectively study the fundamental sumrules for the parton distributions and structure functions. Using only the data from the measuredregion of x , we obtain the Gottfried sum R F ns /x dx and the integrated nucleon sea asymmetry R ( ¯ d − ¯ u ) dx . We compare our results with the reported experimental values and with the predic-tions obtained for different global parametrizations for the parton distributions. We also discussthe discrepancy between the NMC and E866 results on R ( ¯ d − ¯ u ) dx . We demonstrate that thisdiscrepancy can be resolved by taking into account the higher-twist effects. PACS numbers: 11.55.Hx, 12.38.-t, 12.38.Bx
I. INTRODUCTION
The deep inelastic scattering (DIS) of leptons on hadrons and hadron-hadron collisions are a gold mine to studythe hadron structure and fundamental particle interactions at high energies. Especially, so-called DIS sum rules canprovide important information on partonic structure of the nucleon and a good test for the quantum chromodynam-ics (QCD). Nowadays, there are known a number of polarized and unpolarized sum rules for structure functions.Some of them are rigorous theoretical predictions and other are based on model assumptions which can be verifiedexperimentally. An example of the latter is the Gottfried sum rule (GSR) [1]. Thus, the GSR violation in a seriesof experiments [2–8] revealed that, unlikely to the assumed simple partonic model of the nucleon with the symmetriclight sea, the light sea of the proton was flavor asymmetric, i.e., ¯ u ( x ) = ¯ d ( x ). This unexpected result has prompteda large interest for many further studies, for review, see, e.g., [9, 10], related to theoretical explanations of the flavorasymmetry of the nucleon sea.In our paper, we present a phenomenological analysis of the experimental NMC data on the nonsinglet structurefunction F p − F n [3] and E866 data on the nucleon sea asymmetry ¯ d/ ¯ u [7], utilizing a very effective method fordetermination of the DIS sum rules in a restricted region of Bjorken x – the so-called truncated Mellin moments(TMM) approach [11].In the next section, we give a brief recapitulation of the Gottfried sum rule violation problem and discuss someeffects modifying the GSR like the perturbative QCD corrections, higher-twist terms, small- x behavior and nuclearshadowing. The method of the evaluation of the DIS rules from the experimental data with help of the truncatedMellin moments approach is shortly summarized in Section III. In Section IV, we present our numerical resultson the GSR value and compare them to those provided by the NMC and E866, and also to other determinations ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] based on the global parton distribution functions (PDFs) fits. Furthermore, we discuss the higher-twist effects as apossible explanation of the discrepancy between the NMC and E866 results on the integrated nucleon sea asymmetry R ( ¯ d − ¯ u ) dx . Finally, we discuss shortly our prediction for the iso-vector quark momentum fraction h x i u − d .In Section V, we give conclusions for this study. II. VIOLATION OF THE GOTTFRIED SUM RULE
The Gottfried sum rule [1] states that the integral over Bjorken variable 0 < x < /
3) under flavor symmetry in the nucleon sea (¯ u ( x ) = ¯ d ( x )),which is independent of the transferred four-momentum q [1]: S G ( Q ) = Z (cid:2) F p ( x, Q ) − F n ( x, Q ) (cid:3) dxx = 13 . (1)Here, x = Q / (2 P q ), where Q = − q , P = m , and m is the nucleon mass. This form of the GSR originates from asimple partonic model of the nucleon structure functions in which the isospin symmetry of the nucleon (the u-quarkdistribution in the proton is equal to the d-quark distribution in the neutron), u pv ( x ) = d nv ( x ) ≡ u v ( x ) , (2)and, similarly, d p = u n , ¯ u p = ¯ d n , ¯ d p = ¯ u n , etc., and the flavor symmetry of the light sea in the nucleon,¯ u ( x ) = ¯ d ( x ) , (3)are assumed. Then, the difference between the proton and neutron structure functions incorporating implicit pertur-bative QCD Q corrections to the parton model is given by F p ( x, Q ) − F n ( x, Q ) = 13 x (cid:2) u v ( x, Q ) − d v ( x, Q ) (cid:3) + 23 x (cid:2) ¯ u ( x, Q ) − ¯ d ( x, Q ) (cid:3) , (4)where the valence-quark distribution q v , ( q = u, d ), is defined by q v ≡ q − ¯ q , with ¯ q being the sea-quark distribution.Taking into account the charge conservation law for the nucleon, Z u v ( x, Q ) dx = 2 , Z d v ( x, Q ) dx = 1 , (5)we obtain Z (cid:2) F p ( x, Q ) − F n ( x, Q ) (cid:3) dxx = 13 + 23 Z (cid:2) ¯ u ( x, Q ) − ¯ d ( x, Q ) (cid:3) dx. (6)If the light sea is flavor symmetric, Eq. (3), the second term in Eq. (6) vanishes giving the Gottfried sum rule (1).Though the isospin symmetry, Eq. (2), is not exact, and can also contribute to the GSR violation, usually theexperimental results on the GSR breaking are interpreted as an evidence of the light flavor asymmetry of the nucleonsea, ¯ u ( x ) = ¯ d ( x ) . (7)The first clear indication of the GSR violation in DIS experiment was provided by the New Muon Collaboration(NMC) [2] and from the reanalyzed NMC data [3]. The obtained NMC measurement of S G ,NMC 1994 : S G ( Q = 4 GeV ) = 0 . ± .
026 (8)implies the integrated antiquark flavor asymmetry, Eq. (7), Z (cid:2) ¯ d ( x, Q ) − ¯ u ( x, Q ) (cid:3) dx = 0 . ± .
039 (9)which means that in the proton, d -sea is larger than u -sea.Later, the Gottfried sum rule was tested at the Fermilab in E866 Drell-Yan (DY) experiments which measured¯ d/ ¯ u as a function of x over the kinematic range of 0 . < x < .
35 at Q = 54 GeV [7]. Again, the datasuggested a significant deficit in the sum rule consistent with the DIS results and also with semi-inclusive DIS (SIDIS)measurements of the HERMES collaboration for 0 . < x < .
30 and 1 < Q <
20 GeV [8]:HERMES 1998 : Z (cid:2) ¯ d ( x, Q ) − ¯ u ( x, Q ) (cid:3) dx = 0 . ± .
03 (10a)E866 2001 : Z (cid:2) ¯ d ( x, Q ) − ¯ u ( x, Q ) (cid:3) dx = 0 . ± .
012 (10b)The surprisingly large difference between light sea in the nucleon, Eq. (7), observed in different experiments likeDIS, DY and SIDIS, has triggered many theoretical efforts to understand and accurately describe the experimentalresults (for review, see, e.g., [9, 10]). While the perturbative QCD fails in description of the sea asymmetry, thenonperturbative mechanisms as Pauli-blocking, meson cloud, chiral-quark, intrinsic sea, soliton seem to be morepromising in explanation of the GSR breaking. Recently, the statistical parton distributions approach was developedto study the flavor structure of the light quark sea [12]. The authors obtained a remarkable agreement of the statisticalmodel prediction for the ratio ¯ d/ ¯ u with the E866 data [10, 13] up to x = 0 .
2. Unfortunately, none of the studiesmentioned above predicts correctly the ¯ d/ ¯ u behavior in the whole x region, i.e. none of them predicts a sign-changefor ¯ d ( x ) − ¯ u ( x ) at x ≈ . x behavior and nuclear shadowing effects. A. pQCD corrections to the GSR
Here, we show that the perturbative QCD corrections to the GSR are too small to explain the light sea asymmetry[14]. The corrections of order α s to the GSR were obtained in [15] basing on numerical calculation of the order α s contribution to the coefficient function.From the renormalization group equation analysis for S G ( Q ) Kataev and Parente [15] obtained for the number ofactive flavors n f =4 the following QCD corrections to the GSR: S G ( Q ) = 13 (cid:20) . (cid:16) α s π (cid:17) − . (cid:16) α s π (cid:17) (cid:21) . (11)Using the above formula we find S G ( Q = 4 GeV ) = 13 − . . S G ( Q = 54 GeV ) = 13 − . . . (12b)This means that the magnitude of order α s perturbative QCD effects turn out to be about − .
4% at Q = 4 GeV ( α s ≈ . − .
08% at Q = 54 GeV ( α s ≈ .
20) of the original constant value of the GSR, S G = 1 / − .
5% and − . B. Higher-twist effects
In light of the results obtained in [16], also the higher-twist terms seem not to be much helpful in description ofthe large discrepancy between the theoretical prediction of the GSR, Eq. (1), and the experimental value of Eq. (8).The authors of [16], basing on the DIS world data, fitted in the NNLO analysis the twist-4 coefficient H τ =42 ( x ) forthe nonsinglet function F p − n ( x ) and found the HT corrections marginal in comparison with the leading twist (LT)terms, F p − n ( x, Q ) = F p ( x, Q ) − F n ( x, Q ) = (cid:2) F p − n ( x, Q ) (cid:3) LT + H τ =42 ( x ) Q . (13)In Fig. 1, we plot the coefficient of the twist-4 term H τ =42 ( x ), Eq. (13), for the nonsinglet structure function F p − n ( x )obtained in [16] and compare the corresponding HT corrections with the results of NMC for F p − n ( x ) at Q = 4 GeV .However, when applied to the Gottfried sum rule, these HT corrections though too small to be responsible for the −0.08−0.06−0.04−0.02 0 0.02 0.04 0 0.2 0.4 0.6 0.8 1x H −0.05 0 0.05 0.1 0.15 0.01 0.1 1x F NMCHT
FIG. 1:
Left: the central values (solid line) and the error band for the coefficient of the twist-4 term H τ =42 ( x ), Eq. (13), obtained fromthe NNLO fit for the nonsinglet structure function F p − n ( x ) [16]. Right: comparison of the NMC data for F p − n ( x ) at Q = 4 GeV withthe corresponding HT corrections. observed flavor asymmetry, are not marginal and can accurately explain the relatively large discrepancy between thetwo central values of the experimental results: NMC, Eq. (9), and E866, Eq. (10b).Namely, the HT effects modify the original GSR giving the contribution on the level of − .
4% at Q = 4 GeV (NMC)and − .
4% at Q = 54 GeV (E866) of the sum 1 /
3. Hence, the corresponding difference between the NMC and E866results for the flavor asymmetry of the light sea∆( Q ) ≡ Z (cid:2) ¯ d ( x, Q ) − ¯ u ( x, Q ) (cid:3) dx , (14)implied by the HT effects at different scales of Q , is∆ HT ( Q = 4 GeV ) − ∆ HT ( Q = 54 GeV ) ≈ . ± . . (15)This is in a very good agreement with the experimental data:∆ NMC − ∆ E ≈ . . (16)Taking into account also the perturbative QCD radiative corrections of Eq. (11), we arrive at the value even closer tothe data: ∆ Rad + HT ( Q = 4 GeV ) − ∆ Rad + HT ( Q = 54 GeV ) ≈ . . (17)It is seen that the Q -dependence of the GSR can resolve a discrepancy between the flavor asymmetry of the lightsea in the nucleon measured in different experiments. Similar suggestion was made by the authors of Ref. [17].We have found that the QCD-improved parton model including the NNLO radiative corrections and also the twist-4contributions predicts for the Gottfried sum rule at Q = 4 GeV S G ( Q = 4 GeV ) = 13 − . | {z } pQCD − . | {z } HT ≈ − . . (18)This means that the large deficit of the GSR observed in the experiments ( S G ≈ / − .
1) comes from another sourcesthan perturbative mechanisms and HT effects.
C. Low- x contribution Experimental verification of the most sum rules faces the difficulty that in any realistic experiment one cannotreach arbitrarily small values of the Bjorken x . This is a serious obstacle also in the determination of the Gottfriedsum rule which involves the first Mellin moment, i.e. integral of the nonsinglet structure function F ns over the wholerange of x : 0 x
1. The lack of low- x data with good accuracy makes reasonable the idea that a significantcontribution to the integral of the GSR can come just from the small- x region. We illustrate this in Fig. 2 where weshow different low- x behaviors of F ns /x ∼ x a and the corresponding truncated GSR, R x F ns ( x ) dx/x , together withthe NMC data [3]. We use three values for a : − . − . − .
6. It is seen that the experimental uncertainties inthe small- x region are too large to favor any of them. ( F - F ) / x x NMC (cid:242) x ( F - F ) d x / x x NMC
FIG. 2:
Left: possible parametrizations of F ns /x reflecting different small- x behavior ∼ x a : a = − . − . − . R x F ns ( x ) dx/x , respectively. The different ∼ x a behaviors predict significant different very low- x contributions to the GSR, R . F ns ( x ) dx/x ,namely 0.006 for a = − .
2, 0.011 for a = − . a = − .
6. This means that the very low- x contributionsto the GSR can vary from 1 − / x region confirm very well the expectations of the theoretical studies on F based on theRegge theory. In the Regge approach, the small- x behavior of F ns ( x ) is controlled by the reggeon A exchange [18]: F p − n ( x ) = F p ( x ) − F n ( x ) ∼ x − α A , (19)where α A ≈ . A reggeon intercept. Taking into account the Regge predictions in the NMC data analysis,we can estimate the small- x contribution to the Gottfried sum R . F ns ( x ) dx/x as 4 − / D. Nuclear shadowing
Since there is no fixed target for the neutron, the deuteron is usually used for measuring the neutron structurefunction F n . The same method was used by the NMC for determination of the Gottfried sum rule. In order to obtainthe difference of the structure functions F p − F n of free nucleons which enters into the GSR, the extracted F n fromthe deuteron data has to be corrected by the shadowing effects: F d = 12 ( F p + F n ) − δF d , (20)where δF d > S G ( Q ) than that determinedexperimentally assuming no shadowing, and the GSR violation is even magnified. The nuclear shadowing which isdominated by the vector-meson-dominance (VMD) mechanism is non-negligible in the region of x . Q relevant for the NMC measurements and has to be taken into account in the data analysis [19].This leads to the following expression for the difference between the proton and neutron structure functions in theintegrand of the Gottfried sum, Eq. (1): F p ( x, Q ) − F n ( x, Q ) = (cid:0) F p ( x, Q ) − F n ( x, Q ) (cid:1) NMC − δF d ( x, Q ) , (21)where ( F p ( x, Q ) − F n ( x, Q )) NMC obtained by NMC is related to the measured F d and F n /F p via (cid:0) F p ( x, Q ) − F n ( x, Q ) (cid:1) NMC = 2 F d − (cid:16) F n F p (cid:17) NMC (cid:16) F n F p (cid:17) NMC (22)Using the results of [19] for δF d , in Fig. 3, we compare the NMC data with the corrected one by the nuclearshadowing effect. We find that the negative shadowing correction to the experimental result for the Gottfried sum, S G (0 . , . , Q ) = 0 . . ≈ −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.01 0.1 1xNMCNMC corrected2 d F (x) FIG. 3:
The nuclear shadowing corrected NMC data for F ns (open circles) calculated in [19]. Solid: fit to the shadowing contribution tothe deuteron structure function, 2 δF d . III. TMM METHOD FOR DETERMINATION OF SUM RULES
Here, we briefly present an effective method which allows one to determine any sum rule value from the experimentaldata in the available restricted kinematic range of the Bjorken variable x . The method was elaborated in [11, 20]for the Bjorken sum rule, and successfully applied to the experimental data at COMPASS, SLAC and JLab [11, 20–22].The main philosophy of the method presented in [11] is construction of a special truncated sum Γ which approachesthe limit of the sum rule value more quickly, i.e. for larger x , than the ordinary sum. In other words, the use of Γ“mimics” the extension of the experimental kinematic region of x to the lower values.Below, we give useful formulas for determination of the sum rule value in the TMM approach which in the nextsection we shall apply to the GSR. The details on theoretical aspects of the Γ construction and the description ofdifferent approximations of the TMM method can be found in [11].Determination of the sum rules involves the integrals of the parton density or structure function f ( x, Q ) over thewhole range (0 ,
1) of x : S (0 ,
1) = Z f ( x ) dx , (23)where for clarity we omit the Q dependence. The experimental measurements provide data on f ( x ) only in thelimited range of x : 0 < x min ≡ x < x < · · · < x max ≡ x N <
1, where x min ≡ Q / (2( P q ) max > S ( x min , x max ) = Z x max x min f ( x ) dx . (24)The truncation at the upper limit x max is less important in comparison to the low- x limit x min because of the rapiddecrease of the parton densities and structure functions as x → n th truncated moment of the structure function f ( x, Q ) as M n ( x min , x max , Q ) = Z x max x min x n − f ( x, Q ) dx , (25)the Gottfried sum rule S G ( Q ) is the first moment M (0 , , Q ) of the nonsinglet function F p ( x, Q ) − F n ( x, Q ).The special sum Γ is constructed based on the ordinary sum S in the following way:Γ( x , r ) = S ( x ,
1) + A Z x /rx f ( x ) dx , (26)(27)where x is the smallest value of x accessible in the experiment and A , and r are parameters calculated from the data.In the limit x →
0, Γ( x , r ) is equal to S ( x ,
1) providing the sum rule value S (0 , x > x , r ) approaches S (0 ,
1) much earlier than S ( x ,
1) itself. This is illustrated in Fig. 4 where we compare S ( x , x , r ), Eq. (26), plotted for different values of A . We use smooth fits to the NMC and E866 datasetting the ratio of two experimental points r = x /x equal to 0 . . A I while the restof calculations will be performed with use of the pure data f ( x ) from the measured x -region.In our approach, as described in [11], we utilize the quasi-linear regime of Γ( x , r ) which starts already for x significantly larger than the smallest experimental value of x . This ensures the applicability of the first (or evenzero as in the NMC case) order approximation for estimation of the value of S (0 ,
1) with help of Γ( x , r ). Thus,requiring the second derivative to vanished, Γ ′′ ( x , r ) = 0, we obtain A I and the sum rule value can be determinedvery effectively in the first order of Taylor expansion: S (0 ,
1) = Γ(0 , r ) ≈ Γ( x , r ) − x Γ ′ ( x , r ) = Γ( x , r ) + ( A I + 1) x f ( x ) − A I x r f ( x /r ) , (28) A I = (cid:20) r f ′ ( x /r ) f ′ ( x ) − (cid:21) − , (29) NMC x , 1), G (x ,r) E866 x , 1), G (x ,r) FIG. 4: S ( x , x , r ), Eq. (26), for different values of A . Upper (blue) solid line corresponds to A = A (left panel) and A = A I (right panel), see description in the text. where Γ( x , r ) is given by Eq. (26) and f ′ ( x ) denotes the first order derivative with respect of x .In a special case, where the small- x experimental data can be well described by a simple form f ( x ) = N x a , we have r n f ( n − ( x ) f ( n − ( x /r ) = r f ( x ) f ( x /r ) (30)and all derivatives Γ ( n ) ( x , r ) vanish for the same A = A , A = (cid:20) r f ( x /r ) f ( x ) − (cid:21) − . (31)Hence, we arrive at the zero order approximation for the sum rule value which reads S (0 ,
1) = Γ( x , r ) = S ( x ,
1) + A Z x /rx f ( x ) dx . (32)The method of estimation of the sum rule value based on the special truncated sum Γ is effective for differentsmall- x behavior of the function f ∼ x a , also for a <
0, as in the case of the Gottfried sum rule.
IV. DATA ANALYSIS
Below we present our numerical results for the Gottfried sum rule value S (0 ,
1) based on the experimental NMC [3]and E866 [7] data following the approach described in the previous section.
A. NMC
The violation of the GSR was first observed by the New Muon Collaboration at CERN in 1991 [2]. NMC measuredthe cross section ratio for deep inelastic scattering of muons from hydrogen and deuterium targets in the kinematicrange extended to the low- x region, 0 . x .
8. The difference of the structure functions was calculatedby Eq. (22) and the ratio F n /F p = 2 F d /F p − F d was taken from a fit to various experimental data. The results were obtained by interpolationor extrapolation to Q = 4 GeV . In the reanalyzed data [3], which are under study in this section, NMC used theirown data for F d and revised F n /F p ratios.The NMC data for F p − n which form the GSR, S G (0 ,
1) = Z (cid:2) F p ( x, Q ) − F n ( x, Q ) (cid:3) dxx , (33)can be for x . ∼ . x . [3], which agrees with theoretical prediction of theRegge-like behavior, Eq. (19), [18]. The corresponding truncated function Γ( x , r ) saturates to the constant S G (0 , x (see upper solid line in the left panel of Fig. 4) and we can estimate S G (0 ,
1) using the zero orderformulas, Eqs. (31) and (32), which take the form S G (0 ,
1) = S G ( x , NMC + A Z x /rx F p − n ( x, Q ) dxx , (34) A = " F p − n ( x /r, Q ) F p − n ( x , Q ) − − . (35)All quantities in Eqs. (34) and (35) are directly provided by the data or can be calculated from the data withoutnecessity to use of any fit function. Namely, S G ( x , NMC is the contribution to the GSR from the measured region of x together with the correction for x > . x denotes the smallest h x i in the analysis, and r = x /x k is a ratio of twoexperimental points where x k > x . The integral in Eq. (34) can be calculated as a sum of the partial experimentalcontributions, respectively: Z x i +1 x i F p − n ( x ) dxx = x i +1 − x i h x i i F p − n ( h x i i ) . (36)In Table I we show our estimations for S G (0 ,
1) obtained for two values of x : 0 .
007 and 0 .
015 and corresponding r and A , up to x k ≈ .
4. The experimental value of S G (0 , x contribution from the region x < . TABLE I: Estimations of the Gottfried sum rule value S G (0 ,
1) obtained in the zero order approximation, Eqs. (34) and (35),for two values of x : 0 .
007 and 0 .
015 based on the NMC data at Q = 4 GeV [3]. The ratio r = x /x i , where i = 1 , , ... S G (0 ,
1) is displayed in the last row. x = 0 . x = 0 . r A S G (0 , r A S G (0 , .
47 2 . .
236 0 .
50 1 .
07 0 . .
23 0 .
53 0 .
230 0 .
30 0 .
94 0 . .
14 0 .
48 0 .
236 0 .
19 0 .
52 0 . .
09 0 .
29 0 .
234 0 .
12 0 .
34 0 . .
06 0 .
20 0 .
233 0 .
09 0 .
31 0 . .
04 0 .
19 0 .
236 0 .
06 0 .
22 0 . .
03 0 .
14 0 .
235 0 .
04 0 .
19 0 . .
02 0 .
12 0 .
235 0 .
03 0 .
17 0 . S G (0 ,
1) = 0 . ± . We obtain S G (0 ,
1) = 0 . ± . SD . The low- x contribution S G (0 , . . ± . x contribution as an average deviation for the composed errors of S G (0 , . S G (0 ,
1) = 0 . ± . F ns /x and R . x F ns dx/x with the predictions of two parametrizationsbased on the global PDFs fit, CTEQ6 [23] and MSTW08 [24], and to our TMM estimation for the GSR. In the leftpanel we plot also the low- x fit function f NMCfit for illustration of the regular Regge behavior of the NMC data up to x ≈ .
4. We find1 x (cid:2) F p ( x, Q ) − F n ( x, Q ) (cid:3) NMC ≈ f NMCfit = ax b , a = 0 . ± . , b = − . ± . , (37)which is consistent with the form provided in [3]. F s / x x NMClow-x fitMSTW08CTEQ6
TMM (cid:242) x . F s d x / x x NMCMSTW08CTEQ6
FIG. 5: F ns ( x, Q ) /x (left) and R . x F ns ( x, Q ) dx/x (right) at Q = 4 GeV . A comparison of the NMC data to CTEQ6 [23] andMSTW08 [24] predictions and also to the TMM estimation for the Gottfried sum rule. The low- x fit function f NMCfit in the left panel hasthe form Eq. (37).
In Table II, we collect the contributions to the Gottfried sum rule, R F ns ( x, Q ) dx/x , obtained for different x ranges at Q = 4 GeV . A comparison of the NMC data to TMM, CTEQ6 and MSTW08 predictions is shown. TABLE II: The contributions to the Gottfried sum rule, R F ns ( x, Q ) dx/x , integrated over different x ranges at Q = 4 GeV .Compared are the NMC data to TMM, CTEQ6 [23] and MSTW08 [24] predictions. ( ∗ ) The result contains a fit to theunmeasured region of small- x . x range NMC TMM CTEQ6 MSTW080 < x < . ± . ∗ . ± .
022 0 .
255 0 . . < x < . . ± .
021 0 .
231 0 . < x < .
004 0 . ± . ∗ . ± .
004 0 .
023 0 . . < x < . ± .
001 0 .
001 0 . The small- x contribution to the GSR in the unmeasured region x < .
004 was determined by NMC with useof the fit to the data. Our method, which is totally based on the experimental data in the measured region of x ,provides almost the same result. The CTEQ6 prediction is slightly above the NMC estimation for S G (0 , . S G (0 , . S G (0 ,
1) are muchlarger than both the experimental and our estimations. A reason for this discrepancy is a small- x behavior of ¯ d − ¯ u assumed by MSTW08 which implies a decrease of R ( ¯ d − ¯ u ) dx and hence the increase of R F ns /x dx in this region incomparison to other global PDF fits. We shall discuss it also in the next subsection which is devoted to the E866experiment.1 B. E866
Fermilab experiment E866 [7] was a fixed target experiment that has measured the light sea quark asymmetryin the nucleon using Drell-Yan process of di-muon production in 800 GeV proton interactions with hydrogen anddeuterium targets. From the data, the ratio ¯ d/ ¯ u was determined over a wide range in Bjorken- x . The obtained resultsconfirmed previous measurements by E866/NuSea [5], which were the first demonstration of a strong x -dependenceof the ¯ d/ ¯ u ratio, and extended them to lower- x . To obtained the antiquark asymmetry ¯ d − ¯ u and also the integratedasymmetry R ( ¯ d − ¯ u ) dx , E866 used their data for ¯ d/ ¯ u and the PDF parametrization CTEQ5M [25] for ¯ d + ¯ u . In orderto estimate the contribution from the unmeasured region 0 < x < . x > .
35 was negligible.In our TMM analysis, presented below, we use the experimental data only from the measured region 0 . < x < . ,
1) = Z (cid:2) ¯ d ( x, Q ) − ¯ u ( x, Q ) (cid:3) dx , (38)we apply the universal first order approximation of the special truncated sum Γ method given by Eqs. (26)-(29). Inthe terms of the experimental data they read∆(0 ,
1) = ∆( x , E866 + A I Z x /rx [ ¯ d − ¯ u ]( x ) dx + ( A I + 1) x [ ¯ d − ¯ u ]( x ) − A I x r [ ¯ d − ¯ u ]( x /r ) , (39) A I = " r f ′ fit ( x /r ) f ′ fit ( x ) − − , (40)where the prime denotes a derivative with respect to x of the fit function f E866fit to the E866 data on ¯ d − ¯ u : (cid:2) ¯ d ( x, Q ) − ¯ u ( x, Q ) (cid:3) E866 ≈ f E866fit = ax b (1 − x ) c (1 + d x ) ,a = 0 . ± . , b = − . ± . , c = 2 . ± . , d = − . ± . . (41)The fit function, which we use only for calculation of A I in Eq. (40), is shown as a dotted line in the left panel ofFig. 6. All other quantities in Eqs. (39) and (40) are directly provided by the data. Again, as in the NMC analysis, x denotes the smallest h x i , and the ratio r = x /x k is determined from the kinematics. ∆( x , E866 denotes thecontribution to the light sea asymmetry from the measured region of x .In Table III we show our estimations for ∆(0 ,
1) obtained for two values of x : 0 .
026 and 0 .
038 and corresponding r and A I . To minimize a possible error implied by the decrease of r , [11], we proceed our analysis up to x k ≈ . d − ¯ u ]( x, Q ) and R . x [ ¯ d − ¯ u ]( x, Q ) dx with the predictions of twoparametrizations based on the global PDFs fits, CTEQ6 [23] and MSTW08 [24], and to the TMM results for R [ ¯ d − ¯ u ]( x, Q ) dx . In Table IV, we present a comparison of the light sea quark asymmetry ∆ integrated overdifferent x ranges for the E866 data, TMM approach and CTEQ6 and MSTW08 predictions. The TMM results areshown together with the average deviation of the composed errors calculated from Eqs. (39) and (40) for the set datafrom Table III.The low- x contribution ∆(0 , . . ± .
012 obtained in our analysis is essentially smaller than the E866estimation obtained with use of the combined fits MRST98 and CTEQ5M. It is also smaller than the CTEQ6prediction but larger than the more recent global fit prediction of MSTW08. Since the NMC and E866 data wereused in the global fit analysis, the CTEQ6 and MSTW08 predictions are in a good agreement with these experimentaldata from the measured x -region. The problem is in determination of the GSR and ∆ contributions coming from the2 TABLE III: Estimations of the integrated quark asymmetry ∆(0 ,
1) obtained in the first order approximation, Eqs. (39) and(40), for two values of x : 0 .
026 and 0 .
038 based on the E866 data at Q = 54 GeV , [7]. The ratio r = x /x i where i = 1 , , , ...
8. The experimental value of ∆(0 ,
1) is displayed in the last row. x = 0 . x = 0 . r A I ∆(0 , r A I ∆(0 , .
68 1 .
79 0 .
098 0 .
73 2 .
19 0 . .
50 0 .
79 0 .
098 0 .
57 1 .
03 0 . .
39 0 .
48 0 .
101 0 .
46 0 .
67 0 . .
32 0 .
35 0 .
099 0 .
39 0 .
50 0 . .
27 0 .
27 0 .
101 0 .
34 0 .
40 0 . .
23 0 .
22 0 .
101 0 .
30 0 .
33 0 . .
20 0 .
19 0 .
100 0 .
27 0 .
29 0 . .
18 0 .
17 0 .
101 0 .
24 0 .
26 0 . ,
1) = 0 . ± . dba r - uba r x E866fitMSTW08CTEQ6
TMM (cid:242) x . [ dba r - uba r ] d x x E866MSTW08CTEQ6
FIG. 6: [ ¯ d − ¯ u ]( x, Q ) (left) and R . x [ ¯ d − ¯ u ]( x, Q ) dx (right) at Q = 54 GeV . A comparison of the E866 data to CTEQ6 [23] andMSTW08 [24] predictions and also to the TMM estimation for the integrated asymmetry of the light sea quarks R [ ¯ d − ¯ u ]( x, Q ) dx . Thefit function f E866fit in the left panel has the form Eq. (41). unmeasured regions, especially from the small- x region. While all reasonable fits to the data assume ¯ u = ¯ d as x → d ( x ) − ¯ u ( x ) with CTEQ6 and MSTW08 NLO fits at Q = 54 GeV . Our result for the small- x contribution ∆(0 , . d ( x ) − ¯ u ( x ) at Q = 1 GeV goes to zero as x . at small- x and this behavior is not excluded by the E866 data.Let us finally comment the discrepancy between the NMC and E866 data. To this aim we shall use the TMMresults which provide even larger discrepancy than the results reported by NMC and E866. Namely, we compare∆(0 ,
1) calculated from the GSR value for the NMC data analysis with that obtained for the E866 data. We have∆(0 , , Q = 4 GeV ) = 0 . ± .
033 vs ∆(0 , , Q = 54 GeV ) = 0 . ± . H τ =42 ( x ) for the nonsinglet function F p − n ( x ) [16], the difference for ∆(0 ,
1) at Q = 4 and 54 GeV implied by the HT terms is 0 . ± . TABLE IV: Integrated light sea quark asymmetry ∆ over different x ranges at Q = 54 GeV obtained in the TMM approach.A comparison to the E866 data and CTEQ6 [23] and MSTW08 [24] predictions. ( ∗ ) The result contains a fit to the unmeasuredregion of small- x . x range E866 TMM CTEQ6 MSTW080 < x < . ± . ∗ . ± .
016 0 .
119 0 . . < x < .
35 0 . ± .
011 0 .
083 0 . < x < .
015 0 . ± . ∗ . ± .
012 0 .
037 0 . . < x < − . − . able to reduce the difference ∆(0 , , Q = 4 GeV ) − ∆(0 , , Q = 54 GeV ) = 0 . ± .
049 by about 60% .
C. Second moment of F p − n The main aim of our paper is to study the Gottfried sum rule within the TMM approach, nevertheless, finally, wewould like also to discuss shortly our predictions for the second moment of the structure function F p − n , Z (cid:2) F p ( x, Q ) − F n ( x, Q ) (cid:3) dx = 13 (cid:0) h x i u − d + h x i ¯ u − ¯ d (cid:1) , (42)where h x i u − d ≡ Z x (cid:2) u ( x, Q ) − d ( x, Q ) (cid:3) dx . (43)The latter, h x i u − d , being the iso-vector quark momentum fraction, is recently of large interest for the analyses basedon the lattice QCD. This interest, which has triggered many theoretical and phenomenological investigations, ismainly motivated by a discrepancy of over 25% between the lattice predictions, h x i u − d > .
2, and the values obtainedfrom phenomenological fits to the experimental data, 0 . − .
17 [27].Below, we present our results for h x i u − d at Q = 4 GeV obtained within the TMM approach. Since the NMCdata provide knowledge only for the sum h x i u − d + h x i ¯ u − ¯ d , Eq. (42), we use combined results based on the NMC andE866 data. We take also into account the Q evolution effects for the E866 data provided for Q = 54 GeV . To thisaim, we correct the value of h x i ¯ u − ¯ d calculated from the E866 data by a mean difference h x i ¯ u − ¯ d obtained for the two Q values 4 and 54 GeV from the CTEQ6 and MSTW08 fits. Thus, finally, we obtain h x i u − d = 0 . ± .
007 and h x i ¯ u − ¯ d = 0 . ± . h x i u − d at Q = 4 GeV with the predictions of the world-wide fitsCTEQ6 and MSTW08, and also with the recent lattice result [28]. TABLE V: The iso-vector quark momentum fraction h x i u − d at Q = 4 GeV obtained in TMM approach from the combinedNMC and E866 data. A comparison to the global fit predictions CTEQ6 and MSTW08, and to the recent lattice result [28].TMM CTEQ6 MSTW08 LATTICE0 . ± .
007 0 .
158 0 .
161 0 . ± . For comparison, the recent analysis of the DIS data from fixed-target experiments on the structure function F performed in the valence-quark approximation at the NNLO approximation, and incorporating the NMC result onthe Gottfried sum rule, provides h x i ¯ u − ¯ d = 0 . ± .
021 [29].4
V. CONCLUSIONS
In this paper, based on the experimental NMC data on the nonsinglet structure function F p − F n at Q = 4 GeV [3], and E866 data on the ¯ d/ ¯ u asymmetry in the nucleon sea at Q = 54 GeV [7], we have reevaluated the Gottfriedsum rule [1]. In our analysis, we used the truncated moments approach in which, with help of the special truncatedsum, one can overcome in a study of the fundamental integral characteristics of the parton distributions the problemof the unavoidable kinematic restrictions on the Bjorken variable x [11].Using only the data from the measured region of x , we obtained for the Gottfried sum R F ns /x dx = 0 . ± . . ± .
016 for the integrated nucleon seaasymmetry R ( ¯ d − ¯ u ) dx . The latter, though still consistent with the E866 result 0 . ± . < x < . x behavior of the ( ¯ d − ¯ u ) ∼ x − . than the MRST and CTEQ5M parametrizations used by E866 for the determination of the R . ( ¯ d − ¯ u ) dx . For acomparison, the more recent global fit MSTW08, incorporating also the E866 data, assumes the small- x behavior ofthe ( ¯ d − ¯ u ) ∼ x . and provides R ( ¯ d − ¯ u ) dx = 0 .
09 which better agrees with our estimation.We have also discussed the well-known discrepancy between the NMC and E866 results on R ( ¯ d − ¯ u ) dx . Wedemonstrated that this discrepancy can be understood after taking into account the higher-twist effects which becomeimportant in the case of the NMC data with a relatively low Q = 4 GeV . Using the results obtained for the twist-4coefficient H τ =42 ( x ) for the nonsinglet function F p − n ( x ) [16], we found that the HT effects can be responsible for thedifference of 0 . ± .
022 between the two experimental results obtained at the different Q scales.In the last point of our paper, we obtained in the TMM analysis the iso-vector quark momentum fraction h x i u − d = 0 . ± . Acknowledgments
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