On the Momentum Dependence of the Flavor Structure of the Nucleon Sea

Difference between the $\bar u$ and $\bar d$ sea quark distributions in the proton was first observed in the violation of the Gottfried sum rule in deep-inelastic scattering (DIS) experiments. The parton momentum fraction $x$ dependence of this difference has been measured over the region $0.02<x<0.35$ from Drell-Yan and semi-inclusive DIS experiments. The Drell-Yan data suggested a possible sign-change for $\bar d(x)-\bar u(x)$ near $x \sim 0.3$, which has not yet been explained by existing theoretical models. We present an independent evidence for the $\bar d(x)-\bar u(x)$ sign-change at $x \sim 0.3$ from an analysis of the DIS data. We further discuss the $x$-dependence of $\bar d(x)-\bar u(x)$ in the context of meson cloud model and the lattice QCD formulation.

It is now a well established fact that theū andd distributions in the proton are strikingly different. The first evidence for this difference came from the observation of the violation of the Gottfried sum rule [1] in a deep-inelastic scattering (DIS) experiment by the NMC Collaboration [2]. The Gottfried sum rule, ]/x B dx B = 1/3, is obtained under the assumption of a symmetricū andd sea [1], where x B is the Bjorken variable and is effectively equal to parton momentum fraction x probed in DIS using the leading order QCD factorization formalism of the structure function F 2 (x B ). The NMC measurement of I G = 0.235 ± 0.026 implies that this assumption is invalid with an x-integrated difference of 1 0 [d(x) −ū(x)]dx = 0.148 ± 0.039. The NMC result was subsequently checked using two independent experimental techniques. From measurements of the Drell-Yan cross section ratios of [σ(p + d)]/[σ(p + p)], the NA51 [3] and the Fermilab E866 [4] experiments measuredd/ū as a function of x over the kinematic range of 0.015 < x < 0.35. As shown in Fig. 1, thed/ū ratios clearly differ from unity. From a semi-inclusive DIS measurement, the HERMES collaboration also reported the observation [5] ofd(x) −ū(x) = 0, consistent with the Drell-Yan results.
Thed(x)/ū(x) data obtained from the Drell-Yan experiments have provided stringent constraints for parametrizing the parton distribution functions (PDFs). Figure 1 compares the data measured at Q 2 = 54 GeV 2 from Fermilab E866 with parametrizations of several PDFs. The E866 data show the salient feature thatd/ū rises linearly with x for x < 0.15 and then drops as x further increases. At the largest value of x (x = 0.315), thed/ū ratio falls below unity, albeit with large experimental uncertainty. This intriguing x-dependence ofd/ū is reflected in recent PDFs including CTEQ6 [6], CT10 [7], and MSTW08 [8]. However, for the CTEQ4M [9] and GRV98 [10] PDFs, which predated the E866 data, thed/ū ratios at large x are not well described by the parametrizations. In particular,d(x)/ū(x) remains greater than unity, or equiv-FIG. 1: Ratio ofd(x) overū(x) versus Bjorken-x from experiments NA51 [3] and E866 [4]. Parametrizations from several parton distribution functions are also shown.
alently,d(x) −ū(x) > 0, at all x. The parametrizations of the more recent PDFs are sufficiently flexible to accommodate a sign-change ford(x) −ū(x) at x ∼ 0.3, as suggested by the E866 data.
Many theoretical models have been put forward to explain the surprisingly large difference betweend(x) andū(x). For reviews of various theoretical models, see references [11][12][13][14][15]. While these models can explain the enhancement ofd over u involving various mechanisms such as meson cloud, chiralquark, intrinsic sea, soliton, and Pauli-blocking, none of them predicts that thed/ū ratio falls below unity at any value of x [14]. In order to understand the origin of the sea-quark flavor structure, it is important to improve the accuracy and to extend the kinematic coverage of thed/ū measurement to the x > 0.3 region. This is the goal of an ongoing Fermilab Drell-Yan experiment, E906 [16], and a proposed experiment [17] at the J-PARC facility. The x-dependence ofd/ū (or the related quantityd −ū) at large x remains a topics of much interest both theoretically and experimentally.
In this paper we address the intriguing possibility thatd −ū changes sign at the x ∼ 0.3 region. We first show that an independent experimental evidence for this sign-change, other than the one shown in Fig. 1 from the Drell-Yan data, comes from an analysis of the NMC DIS data. We then discuss the significance of this sign-change and the stringent constraint it imposes on theoretical models. We also discuss the implications on the x-dependence ofd(x) −ū(x) using the lattice QCD formulation for the sea-quark parton distributions. Future measurements ofd(x)/ū(x) at x > 0.25 in Drell-Yan experiments could provide strong constraints and new insights on the origins of the flavor structure of the proton's sea.
The NMC measurement of the Gottfried sum involves the F 2 structure functions on proton and neutron. In terms of QCD factorization, we have at the leading order in α s , where x = x B was used at this order. Equation (1) is obtained under the usual assumption of charge symmetry of parton distributions and the equality of heavy-quark (s, c, b) distributions in proton and neutron. Note that the (1) and the definition of valence quarks, , one readily obtains the following expression: Equation (2) shows that the x dependence ofd(x) −ū(x) can be extracted from the NMC measurement of F p 2 (x) − F n 2 (x) and the parametrization of To illustrate this, we show in Fig. 2 the values ofd(x) −ū(x) at Q 2 = 4 GeV 2 [2] using Eq. (2), where the first term of the right-hand side is from the GRV98 parametrization [10] and the second term is evaluated from the NMC data [2] at Q 2 = 4 GeV 2 . Also shown in Fig. 2 are the values ofd(x) − u(x) obtained from E866 at Q 2 = 54 GeV 2 . The indication from the E866 data thatd(x) −ū(x) changes sign at x ∼ 0.3 is consistent with the evaluation using Eq. (2) based on the NMC data and the GRV98 PDFs. It is important to note that GRV98 uses a parametrization ofd −ū which is positive at all x (see Fig. 1). We show that the NMC data together with the valence quark distributions of GRV98 lead to ad(x) −ū(x) distribution changing sign at x ∼ 0.3. Very similar results, all showing thatd(x) −ū(x) turns negative at x ∼ 0.3, were obtained if other recent PDFs were used instead of GRV98. This reflects the fact that the (2) is well constrained by the extensive DIS and hadronic scattering data.
Although Fig. 2 shows similar trends for the x-dependence of (d(x) −ū(x)) extracted from the E866 Drell-Yan and the NMC DIS data, these two data sets correspond to two different Q 2 scales. A more direct comparison can be obtained by analyzing the final results published by the NMC collaboration on the ratio )/2, and the parametrization of F d 2 (x) of Ref. [19]. These results included additional data which were not available for NMC's earlier evaluation of the Gottfried sum [2]. Moreover, the values of F p 2 (x) − F n 2 (x) were presented for different bins of Q 2 ranging from 1.12 to 90.68 GeV 2 . Since the E866 Drell-Yan data ond(x) −ū(x) correspond to Q 2 = 54 GeV 2 , a comparison could be made by using the NMC data at similar Q 2 . Figure 3 showsd(x) −ū(x) for four values of Q 2 using Eq. 2 with the GRV98 parametrization of the valence quark distributions and the NMC data [18] for F p 2 (x) − F n 2 (x). The agreement between the E866 and NMC results is now improved when compared with Fig. 2. In particular, both the NMC and the E866 experiments show evidence thatd(x) −ū(x) changes sign at x ∼ 0.3.
Since both the NMC data and the E866/NA51 Drell-Yan data are included in recent global fits for determining the parton distributions, it is conceivable that the NMC data have already played a role in constraining the behavior of d(x)−ū(x) at large x. Nevertheless, the possible sign-change ofd(x) −ū(x) for x ∼ 0.3 has only been attributed in the literature to the E866 data, which have large uncertainty at the highest x region. We show that an independent indication for this sign-change is already provided by the NMC DIS data, which were obtained prior to the E866 Drell-Yan data.
The significance of the sign-change ofd(x) −ū(x) for x > 0.3, if confirmed by future experiments, is that it would severely challenge existing theoretical models which can successfully explaind(x) −ū(x) at x < 0.25, but predict no sign-change at higher x. Take for example the meson-cloud model [20][21][22][23], which treats proton as a linear combination of a bare proton plus pion-nucleon and pion-delta Fock states: The subscript zeros denote bare baryons with flavor symmetric seas. Theū andd seas have contributions from the valence antiquarks of the pion cloud, i.e.,d in π + andū in π − . The pion-nucleon amplitude is larger than the pion-delta amplitude (a > b) due to the heavier mass for the ∆. The excess ofd overū arises because of the dominance of the n 0 π + configuration over the less probable ∆ ++ 0 π − configuration. This leads to an overall excess ofd overū. Moreover, the x distribution forū is softer than that ofd, since π − in the ∆ ++ 0 π − configuration carries a smaller fraction of the proton's momentum than π + in the n 0 π + configuration. As a consequence, d(x) −ū(x) remains positive and does not change sign at large x. The same conclusion can be obtained for the chiral quark model [24][25][26], in which the pions couple directly to the constituent quarks. Since there are two u quarks coupling to π + (u → π + + d) and only one d quark coupling to π − (d → π − + u), the larger probability for the π + meson cloud relative to the π − cloud would lead tod >ū for all x. Similar conclusions can be obtained in the intrinsic sea model [27], the chiral-quark soliton model [28], and the sta- tistical model [29].
To shed some light on the x-dependence ofd(x)−ū(x), we consider the origins of sea quarks in the lattice QCD approach. There are two sources for thed andū seas in the path-integral formalism of the hadronic tensor defining the structure function F 2 (x) [30], as shown in the two gauge-invariant and topologically distinct diagrams in Fig. 4. One is the connected sea (CS) from the connected insertion diagram (Fig. 4(a)) and the other is the disconnected sea (DS) from the disconnected insertion diagram (Fig. 4(b)). For the case with isospin symmetry, i.e. m u = m d , it is shown [31] that the DS does not distinguishū fromd. Hence, theū(x),d(x) difference must originate solely from the CS (Fig. 4(a)). It is well known that sea quark distribution generated by the disconnected diagram is a steeply falling function of momentum fraction x, because the gluon radiated from the initial quark line is dominantly soft due to the ∝ 1/x behavior of the splitting kernel. In contrast, sea quarks generated by the connected diagram have an x −1/2 behavior at small x and are most relevant in the medium and large x region. While lattice QCD so far could only generate the moments rather than the x-dependence of quark distributions, a first attempt to separate the CS and DS components of theū(x) +d(x) was reported recently [32]. The extracted CS and DS forū(x) +d(x) [32] have a distinct x dependence in qualitative agreement with expectation. The DS dominates the small x region (i.e. x < 0.05) while the CS dominates the x > 0.05 region.
It is instructive to considerd(x) −ū(x) in three different x regions. At small x, the DS with small x behavior of x −1 dominates. For Q 2 = 2.5 GeV 2 where the CS and DS are explicitly separated [32], this is the region where x < 0.05. Since the only difference betweenū DS andd DS is the u/d mass difference which is much smaller than the scale for the validity of the parton picture, we expect the difference between them due to isospin symmetry breaking to be very small. In the mid-x region (from x = 0.05 to x ∼ 1/3), dominated by the CS with a x-dependence of x −1/2 , the Fock space wavefunction of the quarks in the nucleon is important. It is in this region that the DIS and Drell-Yan experiments reveal thatū(x) <d(x). The dominance of the CS at this region of x suggests a greater chance for the CS partons to share the momentum with the valence quarks resulting in the meson-baryon configurations. Hence the pronounced feature ofū(x) <d(x) in this x region can be understood in terms of the pion cloud model. At even larger x (x > 1/3) the nature of the sea quarks is expected to be strongly influenced by the valence quarks. Intuitively, the connected sea diagram provides a natural mechanism for generating moreū(x) thand(x) at this region, since there are two u valence quarks capable of generatingū quarks as shown in Fig. 4. Figure 5 shows specific examples of diagrams responsible for generating antiquarks from one (top) or two (bottom) valence quarks. In Fig. 5, the antiquark mode of the QCD quantum fluctuation of a quark is probed by the currents J µ and J ν . The quantum fluctuation could be thought as a time sequence of four steps: 1) fluctuation of a valence quark into a quark and a highly virtual gluon, 2) a quick splitting of the gluon into a quark and antiquark pair, 3) annihilation or recombination of the quark and the newly produced antiquark into a highly virtual gluon, which is then, 4) absorbed by the quark. Since both valence u and d quarks can go through the same QCD quantum fluctuation to generateū andd quarks, the mechanism depicted in Fig. 5 could generate about twice ofū(x) overd(x) due to the 2-to-1 ratio of valence quarks. But, this fluctuation is the most probable only if partons involved have an excellent coalescence, and therefore, it should be very short-lived. That is, it is unlikely to generate enough imbalance betweenū andd to compete with what could be generated by the pion cloud or other mechanisms/models at small-x. However, this mechanism is not very sensitive to the parent quark's momentum fraction x, and would become relevant when the imbalance generated by other mechanisms/models dies away at large x. The data indicates that such transition takes place at x ∼ 1/3. A detailed calculation ofū(x) andd(x) in terms of connected (or recombination) diagrams, like that in Ref. [33], is beyond the scope of this letter, and will be presented later.
In summary, we have discussed the importance of the possible sign-change ford(x) −ū(x) at x ∼ 0.3 for understanding the flavor structure of the nucleon sea. We present an independent evidence for thed(x) −ū(x) sign-change at large x from an analysis of existing DIS data. This sign-change cannot be explained by any existing theoretical model on the nucleon sea. Nevertheless, a qualitative explanation for the sign-change at large x is provided in the context of lattice QCD formalism. We note that the x dependence ford(x) −ū(x) can be calculated on the lattice [30] from the structure function of the hadronic tensor or the recently proposed direct calculation via a Lorentz boost [34]. When reliable results are obtained, they would provide a direct check on the possible sign-change ford(x) −ū(x) at x ∼ 0.3. New experimental information on the x-dependence of thed(x) −ū(x) at large x, anticipated for future Drell-Yan experiments, would be critical for understanding the origins of the flavor structure of the nucleon sea.