Parton distributions from nonlocal chiral SU(3) effective theory: Flavor asymmetries

Y. Salamu , Chueng-Ryong Ji , W. Melnitchouk , A.W. Thomas , P. Wang , and X. G. Wang Institute of High Energy Physics, CAS, Beijing 100049, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA Jefferson Lab, Newport News, Virginia 23606, USA CoEPP and CSSM, Department of Physics, University of Adelaide, Adelaide SA 5005, Australia Theoretical Physics Center for Science Facilities, CAS, Beijing 100049, China


I. INTRODUCTION
It is well known that a complete characterization of nucleon substructure must go beyond three valence quarks. One of the great challenges of modern hadron physics is to unravel the precise role of hidden flavors in the structure of the nucleon. The observation of thed −ū flavor asymmetry in the light quark sea of the proton [1][2][3][4], following its prediction by Thomas a decade earlier [5] on the basis of chiral symmetry breaking [6,7], has been one of the seminal results in hadronic physics over the past two decades. It has led to a major reevaluation of our understanding of the role of the nonvalence components of the nucleon and their origin in QCD [8][9][10].
The role that strange quarks, in particular, play in the nucleon has also been the focus of attention in hadronic physics for many years. Early polarized deep-inelastic scattering (DIS) experiments suggested that a surprisingly large fraction of the proton's spin might be carried by strange quarks [11], in contrast to the naive quark model expectations [12]. One of the guiding principles for understanding the nonperturbative features of strange quarks and antiquarks in the nucleon sea has been chiral symmetry breaking in QCD. While the generation of ss pairs through perturbative gluon radiation typically produces symmetric s ands distributions (at least up to two loop corrections [13]), any significant difference between the momentum dependence of the s ands distributions would be a clear signal of nonperturbative effects [14][15][16][17][18][19].
In the previous paper [20], we presented the proton → pseudoscalar meson (ϕ) þ baryon splitting functions for the intermediate octet (B) and decuplet (T) baryon configurations in nonlocal chiral effective theory [21,22]. From the calculated splitting functions, the parton distribution functions (PDFs) of the nucleon are obtained as convolutions of these with PDFs of the intermediate state mesons and baryons [23][24][25]. Here we apply the results from [20] to compute, for the first time within the nonlocal theory, sea quark PDF asymmetries in the proton, including the light antiquark flavor asymmetryd −ū and the strange quark asymmetry s −s. Using SU(3) relations for the intermediate state hadron PDFs, the only free parameters in the calculation of the asymmetries are the mass parameters appearing in the ultraviolet regulator functions. These will be determined by fitting cross section data from inclusive baryon production in high energy pp scattering, using the same splitting functions that appear in the PDF asymmetries. All other parameters, including coupling constants and intermediate state baryon and meson PDFs, are fixed.
We begin in Sec. II by summarizing the convolution formulas for the quark and antiquark PDFs in terms of the fluctuations of the nucleon into its meson-baryon lightcone components. The calculation of the PDFs of the intermediate state baryons and mesons in the chiral theory is discussed in detail in Sec. III. Numerical results for the sea quark asymmetries are presented in Sec. IV, where we compare the results ford −ū with those extracted from Drell-Yan and semi-inclusive DIS measurements, and compare predictions for s −s asymmetries with some recent PDF parametrizations. Finally, in Sec. V we summarize our results and discuss future measurements which could further constrain the PDF asymmetries experimentally.

II. CONVOLUTION FORMULAS
Using the crossing symmetry properties of the spinaveraged PDFs, qð−xÞ ¼ −qðxÞ, the nth Mellin moment (n ≥ 1) of the distribution for a given flavor q (q ¼ u; d; s; …) is defined by In the operator product expansion, the moments Q ðn−1Þ are related to matrix elements of local twist-two, spin-n operators O μ 1 ÁÁÁμ n q between nucleon states with momentum p, where the operators are given by with D ↔ ¼ 1 2 ð ⃗ D − ⃖ DÞ, and the braces fÁ Á Ág denote symmetrization of Lorentz indices. The effective theory allows the quark operators O q to be matched to hadronic operators O j with the same quantum numbers [23], where the coefficients c ðnÞ q=j are the nth moments of the PDF q j ðxÞ in the hadronic configuration j, with y the light-cone momentum fraction of the nucleon carried by the hadronic state j. The operator relation in Eq. (4) then gives rise to the convolution formula for the PDFs [23,24], where q v j ≡ q j −q j is the valence distribution for the quark flavor q in the hadron j. The complete set of splitting functions f j ðyÞ for octet and decuplet baryons is given in Ref. [20].
In the present analysis we work under the basic assumption that the bare baryon states are composed of three valence quarks plus quark-antiquark pairs that are generated perturbatively through gluon radiation. Such contributions will effectively cancel in any differences of PDFs, such asd −ū or s −s. We therefore focus only on the nonperturbative contributions to sea quark PDFs which arise from pseudoscalar meson loops. In this approximation antiquark distributions arise only from diagrams involving direct coupling to mesons, as in the meson rainbow and bubble diagrams in  Contributions to quark PDFs can in principle come from both meson coupling and baryon coupling diagrams. The latter are illustrated in Fig. 2  where q ð0Þ is the quark PDF in the bare nucleon, and the wave function renormalization Z 2 arises from the summation over the diagrams in Figs. 2(a)-2(c) [26]. Following Ref. [24], we will work in terms of the same momentum fraction y for all meson and baryon coupling diagrams in Figs [20]. The corresponding quark PDFs for the intermediate state octet and decuplet baryons are discussed in the next section.

III. BARE BARYON AND MESON PDFS
To calculate the contributions to the quark and antiquark distributions in the proton in the convolution formulas (9) and (10) requires the proton → baryon þ meson splitting functions and the PDFs of the baryons and mesons to which the current couples. The full set of splitting functions was presented in our previous paper, Ref. [20]. In this section we derive the (valence) PDFs of the bare baryon and meson intermediate states using the same chiral SU(3) effective field theory framework that was used to compute the splitting functions.

A. Operators and moments
In the effective theory the quark level operators are matched to a sum of hadronic level operators whose matrix elements [see Eq. (4)] are given by the moments of the splitting functions, as in Eq. (6). Identifying all possible contributions from octet and decuplet baryon intermediate states that transform as vectors, the most general expression for the quark vector operator O μ 1 ÁÁÁμ n q is given by [24,27] where "Tr" denotes traces over Lorentz indices. In the first term of Eq. (11), the operator U represents pseudoscalar meson fields ϕ, where f ϕ is the pseudoscalar decay constant and the coefficients a ðnÞ are related to moments of quark and antiquark PDFs in the pseudoscalar mesons. The flavor operators λ q AE are defined by where λ q ¼ diagðδ qu ; δ qd ; δ qs Þ are diagonal 3 × 3 quark flavor matrices.
In the remaining terms of Eq. (11), the operators B and T α represent octet and decuplet baryon fields, respectively, and we define the Dirac tensors γ αβρ ¼ 1 2 fγ μν ; γ α g and γ μν ¼ 1 2 ½γ μ ; γ ν . The coefficients fα ðnÞ ; β ðnÞ ; σ ðnÞ g and fᾱ ðnÞ ;β ðnÞ ;σ ðnÞ g are related to moments of the spin-averaged and spin-dependent PDFs in octet baryons, while fθ ðnÞ ; ρ ðnÞ g and fθ ðnÞ ;ρ ðnÞ g are related to moments of spin-averaged and spin-dependent PDFs in decuplet baryons, respectively. The coefficientsω ðnÞ are given in terms of moments of spindependent octet-decuplet transition PDFs, where the octetdecuplet transition tensor operator Θ μν is defined as Here Z is the decuplet off-shell parameter, and since physical quantities do not depend on Z, it is convenient to choose Z ¼ 1=2 to simplify the form of the spin-3=2 propagator [28,29]. For the Kroll-Ruderman diagrams in Figs. 2(e), 2(f), 2(h), and 2(i), the presence of the pseudoscalar field at the vertex introduces hadronic axial vector operators, whose contribution to the quark axial vector operator can in general be written as From the transformation properties of the operators O μ 1 ÁÁÁμ n q and O μ 1 ÁÁÁμ n Δq under parity [30], the sets of coefficients fα ðnÞ ;β ðnÞ ;σ ðnÞ ;θ ðnÞ ;ρ ðnÞ g and fᾱ ðnÞ ;β ðnÞ ;σ ðnÞ ;θ ðnÞ ;ρ ðnÞ g in (15) are the same as those in the spin-averaged operators in (11).
The operators B and T α appearing in Eqs. (11) and (15) can be written in terms of the SU(3) baryon octet fields B ij (which include N, Λ, Σ, and Ξ fields) and decuplet baryon fields T ijk α (which include Δ, Σ Ã , Ξ Ã , and Ω fields) using the relations [24,31] ðBBÞ ¼ Tr½BB; ð16aÞ and where ε klm is the antisymmetric tensor. Applying the relations (16) and (17), the vector operator O μ 1 ÁÁÁμ n q in Eq. (11) can then be more intuitively rearranged in the form The individual vector hadronic operators in (18) are given by where the PDFs correspond to those appearing in the convolution expressions in Eqs. (9) and (10). Each of the moments Q ðn−1Þ j can be expressed in terms of the coefficients fa ðnÞ ;α ðnÞ ;β ðnÞ ;σ ðnÞ ;θ ðnÞ ;ρ ðnÞ g appearing in Eq. (11), as discussed below.
In particular, for the meson PDFs, the contributions to the U  Table I for the ϕ ¼ π þ ; K þ , and K 0 mesons. Conservation of the valence quark number fixes the normalization of the n ¼ 1 moment of the meson distribution, such that Note that in the SU(3) symmetric limit, the u-quark moments in π þ and K þ are equivalent, as are the s-quark moments in K þ and K 0 , while the d-quark moments in π þ and K 0 have equal magnitude but opposite sign, The results for other charge states (π − , π 0 , K − , andK 0 ) are obtained from those in Table I using charge symmetry. of the quark distributions qð¼u; d; sÞ in the pseudoscalar mesons π þ , K þ , and K 0 . The moments are normalized such that a ð1Þ ¼ 2.  Table II for baryons B ¼ p; n; Σ AE;0 ; Λ; Ξ −;0 as well as for the Λ − Σ 0 interference. Solving for the coefficients, one can write these as linear combinations of the individual u, d, and s quark moments in the proton, Assuming the strangeness in the intermediate state nucleon to be zero (or equivalently, that the u content of Σ − , for example, vanishes), one finds for the lowest (n ¼ 1) moments, For the quark PDFs in the decuplet baryon intermediate states T, the moments Q ðn−1Þ T for the individual u, d, and s flavors are given in terms of combinations of fθ ðnÞ ; ρ ðnÞ g, and are listed in Table III for T ¼ Δ; Σ Ã ; Ξ Ã , and Ω − . Solving for the coefficients θ ðnÞ and ρ ðnÞ in terms of the moments in the Δ þ baryon, one has Again, assuming zero strangeness in the Δ þ , the n ¼ 1 moments are given by For the moments of the distributions generated by the tadpole diagrams in Fig. 2(j), in Table IV we list the contributions Q ðn−1Þ ðtadÞBϕϕ for the u, d, and s flavors in each octet baryon B. Note that the combinations involving K 0K0 do not contribute to the u-quark moments, those involving K þ K − do not contribute to the d-quark moments, and the contributions from π þ π − to the s-quark moments are also zero.
Finally, to complete the set of the contributions to the unpolarized PDFs, in Table V   can be obtained from the entries here by the replacements fα ðnÞ →ᾱ ðnÞ ; β ðnÞ →β ðnÞ ; σ ðnÞ →σ ðnÞ g. ðKRÞT are given in terms of the coefficientsᾱ ðnÞ ,β ðnÞ , andω ðnÞ , which are related to moments of the spin-dependent parton distributions.
For the latter, recall that spin-dependent PDFs are related to matrix elements of the axial vector operators O μ 1 ÁÁÁμ n Δq in Eq. (15), which, using the relations (16) and (17), can be expanded in terms of hadronic axial vector operators with coefficients given by moments ΔQ ðn−1Þ j of the spin-dependent distributions. In analogy to the expansion in Eq. (18), we therefore expand the axial vector operators as where only the operators relevant for the calculation of unpolarized PDFs are listed. [The remaining terms not listed in Eq. (27) will be relevant for the calculation of spindependent PDFs in the proton [32].] More explicitly, the axial vector hadronic operators in (27) are given bỹ with the corresponding moments ΔQ ðn−1Þ j of the spindependent PDFs defined by For simplicity, in Eq. (29) we restrict ourselves to the diagonal octet (B ¼ B 0 ) and diagonal decuplet (T ¼ T 0 ) cases, with respective spin-dependent PDFs Δq B ðxÞ and Δq T ðxÞ, and the octet-decuplet transition distribution, Δq BT ðxÞ. In particular, the moments ΔQ of the spindependent PDFs in octet baryons can be obtained from the entries in Table II  BT , are given in terms of the coefficientω in Eq. (15) and are listed in Table VI. Solving for the octet coefficients in terms of the moments of the spin-dependent proton PDFs in the proton, one has, in analogy with Eq. (23), the relations Similarly, for the decuplet case, the coefficientsθ ðnÞ and ρ ðnÞ can be written in terms of the moments of the spindependent PDFs of quarks in the Δ þ bayron, The moments of the octet-decuplet transition operators can be related to the moments of the octet baryon operators via the SU(3) relation for all n. For the n ¼ 1 octet baryon moments, in particular, the coefficients are given in terms of axial vector charges F and D,ᾱ In terms of moments of the spin-dependent proton PDFs, for the octet-decuplet transition vertex,ω ð1Þ is given by the SU(3) symmetry relation [27], which also reproduces the relation C ¼ −2D between the meson-octet-decuplet baryon coupling C and the mesonoctet coupling D [33]. Note that through Eq. (32)  For the valence distributions in the pion and kaon, from Eq. (22) and Table I one has q π ðxÞ ≡ u π þ ðxÞ ¼d π þ ðxÞ ¼ d π − ðxÞ ¼ū π − ðxÞ for all values of x. For the PDFs in the baryons, to simplify notations we shall label the bare distributions in the proton without an explicit baryon subscript, qðxÞ ≡ q p ðxÞ, and those in the Δ þ baryon by q Δ ðxÞ ≡ q Δ þ ðxÞ. Starting with the quark distributions in the SU(3) octet baryons, from   For the quark distributions in the SU(3) decuplet baryons, from Table III the u-, d-, and s-quark PDFs can be written in terms of the PDFs in the Δ þ as In our actual numerical calculations for the strange asymmetry, for simplicity we approximate d Δ ðxÞ ≈ dðxÞ and assume valence quark dominance for the bare states, so that sðxÞ ≈ s Δ ðxÞ ≈ 0. Numerically we find that the octet baryon contributions to the s −s asymmetry are in fact dominant, and the approximations for the input PDFs of the decuplet states do not affect the conclusions of our analysis.
For the PDFs arising from the tadpole diagrams in Fig. 2 The distributions associated with the tadpole gauge link diagrams in Fig. 2(g) turn out to be the same as those for the regular tadpole diagrams, Turning now to the Kroll-Ruderman diagrams in Figs. 2(e) and 2(h), for a proton initial state the corresponding PDFs are expressed in terms of spin-dependent PDFs in the proton, ΔqðxÞ ≡ Δq p ðxÞ. From

ðxÞ: ð42bÞ
With this set of distributions in the SU(3) octet and decuplet baryons and mesons, and the proton → meson þ baryon splitting functions from Ref. [20], we can finally proceed with the computation of the meson loop contributions to the quark and antiquark PDFs in the proton, as in Eqs. (9) and (10). In the following section we focus on the calculation of specific PDF asymmetries in the proton numerically.

IV. SEA QUARK ASYMMETRIES IN THE PROTON
To illustrate the calculation of the contributions to PDFs from pseudoscalar meson loops within the nonlocal chiral effective theory framework, we consider the examples of the flavor asymmetry in the light antiquark sea in the proton,d −ū, and the strange-antistrange asymmetry in the nucleon, s −s. In both quantities perturbatively generated contributions from gluon radiation effectively cancel, at least up to next-to-next-to-leading order corrections in α s [13], so that observation of large asymmetries may be indicative of nonperturbative effects [14][15][16].
For the numerical calculation of the meson-bayron splitting functions, earlier work used various regularization prescriptions, including sharp transverse momentum cutoffs, Pauli-Villars regularization, as well as phenomenological vertex form factors [24,25,[34][35][36][37]. At times the prescriptions have been imposed in rather ad hoc ways, without necessarily ensuring that the relevant symmetries, such as Lorentz, chiral, and local gauge symmetries, are necessarily respected. In the present work we for the first time perform the calculation within nonlocal regularization, which is consistent with all of the above symmetry requirements. An advantage of the nonlocal method is that only a single parameter, Λ, is needed to regulate all of the onshell, off-shell, and δ functions associated with each of the diagrams in Figs. 1 and 2.
Following Ref. [20], in the present analysis we adopt a dipole shape in the meson virtuality k 2 for the regulator functions for the one-loop contributions, parametrized by a cutoff parameter Λ, where The cutoff Λ can be determined by fitting the calculated meson-exchange cross section to differential cross sections data for inclusive baryon production in high-energy pp scattering, pp → BX, for different species of baryon B. Summing over the particles X in the final state, the differential inclusive baryon production cross sections can be written as where E is the incident proton energy andȳ ≡ 1 − y is the longitudinal momentum fraction of the incident proton carried by the produced baryon B. In Eq. (44) we have used the fact that for spin-averaged scattering the differential cross section is independent of the azimuthal angle. Available data exist on inclusive neutron and Δ þþ production [38][39][40], as well as on Λ and Σ Ãþ production [39,41,42] in the hyperon sector. In principle, the cutoffs may depend on the baryon B, although within the SU(3) symmetry framework we do not expect large variations among the different Λ values.
Once the cutoffs are determined and the one-loop splitting functions are fixed, these can then be convoluted with the various meson and baryon PDFs in Eqs. (9) and (10) to compute the contributions to the PDFs in the proton. In the numerical calculations the input PDFs of the pion and kaon are taken from Aicher et al. [43]. The spinaveraged PDFs of the proton are from Ref. [44], while the spin-dependent PDFs are taken from Ref. [45]. Since the valence pion and proton PDFs are reasonably well determined, at least compared with the sea quark distributions, using other pion [46][47][48][49][50] or proton [51,52] parametrizations will not lead to significant differences.

A.d −ū asymmetry
Turning to the light antiquark asymmetry in the proton sea, within the chiral effective theory framework the primary source of the asymmetry is the meson rainbow and bubble diagrams in Fig. 1. In this approximation thē d −ū difference does not depend directly on the structure of the baryon coupling diagrams in Fig. 2, but only on the splitting functions and the substructure of the pion. More specifically, from Eq. (9) one can write the contribution to thed −ū difference in the proton as the convolution where the first (octet rainbow) term in the brackets is from Fig. 1(a), the second and third (decuplet rainbow) terms correspond to Fig. 1(b), and the fourth (bubble) term is from Fig. 1(c). Using the notations of Ref. [20], the splitting functions in Eq. (45) for the rainbow and bubble diagrams can be expressed in terms of octet and decuplet basis functions. In particular, for the πN configuration the function f ðrbwÞ π þ n is given by a sum of nucleon on-shell and δfunction contributions, where D and F are the SU(3) flavor coefficients, and f ¼ 93 MeV is the pseudoscalar meson decay constant. Explicit forms for the basis functions are given in Ref. [20] for the dipole regulatorFðkÞ in Eq. (43). The on-shell function f ðonÞ N is nonzero for y > 0, while the local f ðδÞ π and nonlocal δf ðδÞ π functions are proportional to δðyÞ [20], and therefore contribute to thed −ū asymmetry only at x ¼ 0 [34,35]. In the pointlike limit, in which the form factor cutoff Λ → ∞, the nonlocal function δf ðδÞ π vanishes; however, at finite Λ values it remains nonzero.
In the standard formulation of the chiral effective theory, the δ functions at y ¼ 0 represent the contributions from the so-called zero modes on the light front. These are essential in order to ensure that fundamental symmetries, such as Lorentz invariance, chiral symmetry, and local gauge symmetry, are respected in the calculation. Experimentally, the physical appearance of this contribution at x ¼ 0 is still an open question. One may imagine that in more elaborate treatments that better capture the short-range structure of the intermediate state baryon in QCD, the δ function contribution may soften into a peak near x ¼ 0 [34]. However, in the absence of a solution to QCD in the strong coupling regime this remains speculative, and in the present formulation of the effective theory the δ-function terms are a natural consequence.
For the πΔ contributions to the asymmetry in Eq. (45), the splitting function for the rainbow diagram in Fig. 1(b) includes several regular and δ-function terms, where M ¼ M þ M Δ and C is the meson-octet-decuplet baryon coupling, which is related to the πNΔ coupling constant g πNΔ by C ¼ ffiffi ffi 2 p fg πNΔ [20,35]. As for the πN case, the on-shell function for the Δ intermediate state, f ðonÞ Δ , is nonzero for y > 0, with a shape that is qualitatively similar to f ðonÞ N , but peaking at smaller y because of the positive Δ-nucleon mass difference [20,36,37]. The onshell end-point function, f ðon endÞ Δ , also has a similar shape for finite Λ, but in the Λ → ∞ limit is associated with an end-point singularity that gives a δ function at y ¼ 1. Although this term gives a nonzero PDF only at x ¼ 0, since it contributes to the integral ofd −ū, it will indirectly affect the normalization for x > 0. On the other hand, experimental cross sections are in practice available only for x > 0, so that the δ-function pieces are generally difficult to constrain directly, especially in regularization schemes that use different regulator parameters for the δfunction and y > 0 contributions [35]. The advantage of the nonlocal approach employed here is that by consistently introducing a vertex form factor in coordinate space in the nonlocal Lagrangian [20], the same regulator function then appears in all splitting functions derived from the fundamental interaction, which in our case is parametrized through the single cutoff Λ. Even if experimental data constrain only contributions at x > 0, such as from the onshell functions, once determined these can then be used to compute other contributions, including those at x ¼ 0. Following Refs. [24,25,36], we can constrain the parameter Λ for the octet intermediate states by comparing the one-pion exchange contribution with the differential cross section for the inclusive charge-exchange process pp → nX at y > 0, The cross section σ π þ p tot ðysÞ in Eq. (49) is the total π þ p scattering cross section evaluated at the center of mass energy ys. In the numerical calculations, we use the (approximately energy independent) empirical value σ π þ p tot ¼ 23.8ð1Þ mb [53]. For the SU(3) couplings we take D ¼ 0.85 and F ¼ 0.41, which gives a triplet axial charge g A ¼ ð2ᾱ ð1Þ −β ð1Þ Þ=3 ¼ D þ F ¼ 1.26 and an octet axial charge g 8 ¼ᾱ ð1Þ þβ ð1Þ ¼ 3F − D ¼ 0.38.
The results for the differential neutron production cross section are shown in Fig. 3 vsȳ. The experimental data are typically presented as a function of the ratio 2p L = ffiffi ffi s p , where p L is the longitudinal momentum of the produced baryon in the center of mass frame; at high energies, however, this is equivalent toȳ. In Fig. 3(a) we compare our results with the neutron production data from the ISR at CERN at energies ffiffi ffi s p between ≈31 and 63 GeV for 0°n eutron production angles, or k 2 ⊥ ¼ 0 [38]. Data from the hydrogen bubble chamber experiment at the CERN proton synchrotron at ffiffi ffi s p ≈ 5 and 7 GeV [39] are shown in Fig. 3(b) for the k ⊥ -integrated neutron cross section. Because the pion-exchange process is dominant only at largeȳ [54], with contributions from background processes such as the exchange of heavier mesons [36,55,56] becoming more important at lowerȳ, we include data only in the regionȳ > 0.7. Corrections from rescattering and absorption are also known to play a role in inclusive hadron production, and are estimated to be around 20% at high values ofȳ [56][57][58]. A good description of the single and double differential neutron data can be achieved with a cutoff parameter Λ πN ¼ 1.0ð1Þ GeV. A marginally larger value is found if fitting only the double differential data, and slightly smaller value for just the k ⊥ -integrated cross section, but consistent within the uncertainties. For the inclusive production of decuplet baryons, the invariant differential cross section for an inclusive Δ þþ in the final state can be written for y > 0 as where σ pπ − tot is the total π − p scattering cross section. In our numerical calculations we assume this to be charge independent, so that σ π − p tot ≈ σ π þ p tot , and for the coupling constant C we take the SU (6)  respectively. The k 2 ⊥ -integrated Δ þþ cross section is shown in Fig. 3(c) compared with hydrogen bubble chamber data taken at Fermilab for ffiffi ffi s p ≈ 20 GeV [40]. A good fit to the data is obtained with a value of the decuplet cutoff of Λ πΔ ¼ 0.9ð1Þ GeV, which is slightly smaller than that for the neutron production cross sections. To examine the model dependence of the analysis, for comparison we also fitted the hadron production cross sections in Figs. 3(a)-3(c) using instead the Pauli-Villars regularization for the local effective theory [24,25]. The explicit forms of the Pauli-Villars regularized octet on-shell splitting functions can be found in [24,25]. The result for the sum of the decuplet on-shell and on-shell end-point functions is as in Eq. (96) of [20], with the integral regularized by a factor of ð1 þ 4D πΔ =D ΛΔ Þ, where D πΔ and D ΛΔ are momentum dependent functions given in Eq. (86) of [20]. The results for the best fit Pauli-Villars mass parameters Λ are similar to those for the nonlocal calculation, even though the numerical values for the cutoffs are rather different. While there is some difference in the shape of the calculated k ⊥ -integrated neutron production cross section in Fig. 3(b) at smaller values ofȳ, in the region where the data provide constraints the Pauli-Villars results lie within the uncertainty bands of the nonlocal calculation using with the dipole regulator.
Using the values of Λ πN and Λ πΔ for our nonlocal calculation constrained by the pp cross sections in Fig. 3, we next evaluate the flavor asymmetryd −ū from the convolution of the splitting functions and the pion PDF in Eq. (45). The results for xðd −ūÞ are shown in Fig. 4 and compared with the asymmetry extracted from the E866 Drell-Yan lepton-pair production data from Fermilab [4]. At nonzero x values only the on-shell nucleon and Δ and end-point Δ terms contribute to the asymmetry, each of which is indicated in Fig. 4. The positive nucleon on-shell term makes the largest contribution, which is partially canceled by the negative Δ contributions. For the values of the cutoffs used here, the end-point term is relatively small compared with the on-shell Δ component.
Although the δ-function contributions to the flavor asymmetry are not directly visible in Fig. 4, their effect can be seen in the lowest moment of the asymmetry, The contributions from the individual on-shell, end-point, and δ-function components of the πN and πΔ rainbow and the π bubble diagrams to the moment are shown in Fig. 5 vs the dipole cutoff parameter Λ (¼ Λ πN or Λ πΔ ), for the approximate ranges of values found in the fits in Fig. 3. For the best fit values Λ πN ¼ 1.0ð1Þ GeV and Λ πΔ ¼ 0.9ð1Þ GeV, the contributions from the individual terms in Eqs. (45)- (48) are listed in Table VII, along with the combined contributions from the x > 0 and x ¼ 0 terms, and the local and nonlocal terms, to the total integrated result. The nucleon on-shell term is the most important component, with a contribution that is within ≈20% of the total integrated value hd −ūi ¼ 0.127 þ0.044 −0.042 , where the errors reflect the uncertainties on the cutoff parameters. The on-shell and end-point πΔ terms yield overall negative contributions, with magnitude ≈30% of the on-shell πN. Furthermore, the breakdown into the local and nonlocal pieces shows that the latter is negative, with magnitude ≈20% of the local.  The various δ-function terms from all three diagrams in Fig. 1 cancel to a considerable degree, with the x ¼ 0 contribution making up ≈20% of the total. This contribution is smaller than the total uncertainty due to the variation of the cutoff parameter Λ, making it difficult to draw unambiguous conclusions about the phenomenological role of the δ-function terms. With more precise experimental data on the inclusive pp baryon production cross sections and better theoretical constraints on our model, the range of Λ could be further reduced. Eventually, if the total uncertainty on Λ is smaller than the δ-function contributions, one could hope to make firmer conclusions about the x ¼ 0 terms. Experimentally, the asymmetry at x ¼ 0 is of course not directly measurable, and typically extrapolations are made to estimate contributions from outside of the measured region. The New Muon Collaboration, for instance, found hd −ūi ðexpÞ NMC ¼ 0.169ð32Þ from their analysis of F p 2 − F n 2 in the experimentally accessible region 0.004 ≤ x ≤ 0.8, and hd −ūi ðtotÞ NMC ¼ 0.148ð39Þ when including x → 0 and x → 1 extrapolations [1]. The E866 Collaboration, on the other hand, extracted hd −ūi ðexpÞ E866 ¼ 0.080ð11Þ in the experimentally measured interval 0.015 ≤ x ≤ 0.035, and hd −ūi ðtotÞ E866 ¼ 0.118ð12Þ for the entire x range after extrapolation. Note that the extrapolations by different analyses are often based on different assumptions for the asymptotic x → 0 and x → 1 behavior, so that a direct comparison of extrapolated results is problematic. Nevertheless, the general magnitude of the asymmetry is comparable with that found in our calculation, even with the uncertainties about the x ¼ 0 and extrapolated contributions.

B. s −s asymmetry
While thed −ū asymmetry is perhaps the best known consequence of pion loops on PDFs in the nucleon, an equally intriguing ramification of SU(3) chiral symmetry breaking is the s −s asymmetry generated by kaon loops. In analogy to the light antiquark PDFs in Eq. (45), the contribution to the antistrange PDF in the proton arising from kaon loops in Fig. 1 can be written as where here the sums are over the states ϕB ¼ fK þ Λ; K þ Σ 0 ; K 0 Σ þ g for the kaon-octet baryon rainbow diagram [ Fig. 1(a)], ϕT ¼ fK þ Σ Ã0 ; K 0 Σ Ãþ g for the kaondecuplet baryon rainbow diagram [ Fig. 1(b)], and for the ϕ ¼ K þ ðK − Þ and K 0 ðK 0 Þ loop in the bubble diagram [ Fig. 1(c)]. In terms of the on-shell and δ-function basis functions, the kaon-octet baryon rainbow function can be written is given by a form similar to that in Eq. (46),  Contributions to the integral hd−ūi≡ R 1 0 dxðd−ūÞ from the πN rainbow, πΔ rainbow, and π bubble diagrams in Fig. 1, for the best fit parameters Λ πN ¼ 1.0ð1Þ GeV and Λ πΔ ¼ 0.9ð1Þ GeV. The contributions from the various terms in Eqs. (45)- (48) are listed individually, as are the combined contributions from x > 0 and x ¼ 0, and the local and nonlocal terms, to the total. Note that some numbers do not sum to the totals because of rounding.
for the KΣ Ã states, where the coupling C is given in the previous section. For the bubble diagram, the splitting function for charged or neutral kaon loops is given by a form similar to that in Eq. (48), Explicit expressions for all the basis functions are given in Ref. [20]. where the sums are over the octet bayon-meson states Bϕ ¼ fΛK þ ; Σ 0 K þ ; Σ þ K 0 g, decuplet baryon-meson states Tϕ ¼ fΣ Ã0 K þ ; Σ Ãþ K 0 g, and mesons ϕ ¼ K þ ðK − Þ and K 0 ðK 0 Þ for the tadpole contributions. As in Eq. (10), the splitting functions for all the hyperon coupling diagrams in Eq. (58) use the shorthand notationf j ðyÞ ≡ f j ð1 − yÞ.
For the octet hyperon rainbow diagrams, Fig. 2(d), the individual splitting functions can be written in terms of the on-shell, off-shell, and δ-function basis functions as where the functions f respectively. For the decuplet hyperon contributions, the respective splitting functions are given by

K ðyÞ ð62Þ
for the decuplet rainbow diagram in Fig. 2(g), for the Kroll-Ruderman diagram in Fig. 2(h), and To determine the regulator mass parameter for the kaonhyperon-nucleon vertices in Figs. 1 and 2, we consider inclusive hyperon production cross sections in pp collisions, in analogy with the neutron and Δ production above. Data on inclusive Λ production are available from the 2 m hydrogen bubble chamber at the CERN proton synchrotron [39] and the 12 foot hydrogen bubble chamber at ANL [41], and on inclusive Σ Ã production from CERN bubble chamber experiments [42]. The corresponding differential cross sections for inclusive Λ and Σ Ã production (for y > 0) are given by In Eqs. (67) and (68) σ K þ p tot and σ K 0 p tot are the total kaon-proton scattering cross sections, evaluated at invariant mass ys. For the numerical calculations we take the empirical value for σ K þ p tot ¼ 19.9ð1Þ mb from Ref. [53], independent of energy. As there are no data for the K 0 p total cross section, we assume charge symmetry and relate this to the measured K þ n cross section, σ K 0 p tot ≈ σ K þ n tot ¼ 19.7ð1Þ mb [59].
In a similar vein to the pion exchange analysis of neutron and Δ production discussed above, in Fig. 6 we compare the inclusive pp → ΛX and Σ Ãþ X cross sections forȳ > 0.7 with the kaon exchange contributions calculated from Eqs. (67) and (68). It has been established in several analyses [53,54,56,58] that rescattering and absorption effects in inclusive baryon production result in ≲20% corrections at highȳ. The previous studies found that the t dependence at large values ofȳ is consistent with that produced by meson exchange, and in our numerical analysis we follow the earlier work [54,56,57] by focusing on the regionȳ > 0.7.
The best fit to the CERN bubble chamber Λ production data from Ref. [39] at k ⊥ ¼ 0.075 GeV [ Fig. 6(a)] and the k ⊥ -integrated data from Ref. [41] [ Fig. 6(b)] yields a dipole regulator mass Λ KΛ ¼ 1.1ð1Þ GeV, similar to the value found for the πN cutoff parameter from the inclusive neutron production data in Fig. 3. Comparison of the singly differential decuplet Σ Ãþ production data at largē y [ Fig. 6(c)] with the kaon exchange cross section in Eq. (68) gives a best fit for the decuplet regulator mass of Λ KΣ Ã ¼ 0.8ð1Þ GeV. The cutoff parameter for the decuplet baryon is again slightly smaller than that for the octet baryon, as was found for the pion exchange contributions to the neutron and Δ cross sections in Fig. 3. Of course, the cutoff parameters could also in principle be constrained from data on kaon production, where the kaon is produced in conjunction with scattering from an exchanged virtual baryon, as in Figs. 2(d) and 2(g). However, the backgrounds from kaons produced through quark fragmentation will be significant, making it considerably more difficult to identify mesons produced through u-channel baryon exchange and fragmentation.
With these values of the cutoffs, we can compute the kaon loop contributions to the strange and antistrange distributions in the proton, and estimate the shape and magnitude of the strange asymmetry s −s. In Fig. 7 the various octet and decuplet contributions to xs and xs are shown for the best fit parameters Λ KΛ ¼ Λ KΣ ¼ 1.1 GeV and Λ KΣ Ã ¼ 0.8 GeV. For the xs PDF in Fig. 7(a), the octet on-shell contribution from the rainbow diagram [ Fig. 1(a)] dominates over the decuplet on-shell and end-point terms from the decuplet rainbow [ Fig. 1(b)]. The resulting xs distribution peaks at x ≈ 0.1 and essentially vanishes beyond x ≈ 0.6. The δ-function terms from the rainbow diagrams as well as from the kaon bubble diagram [ Fig. 1(c)] contribute tos only at x ¼ 0 and so do not appear in Fig. 7(a).
In contrast, for the strange quark distribution, from the convolution in Eq. (58) one finds that all terms from each of the rainbow, Kroll-Ruderman, and tadpole diagrams in Figs. 2(d)-2(k) have nonzero contributions at x > 0. Since there are many individual terms, we display ones involving octet þ tadpole and decuplet baryons separately in Figs. 7(b) and 7(c), respectively. Unlike the on-shell term dominance of the antistrange PDF, for the strange distribution there are sizable contributions from many of the terms, with nontrivial cancellations between them. For the octet baryons, the off-shell terms change sign at around x ≈ 0.1, with significant cancellation occurring between the local and nonlocal (gauge link dependent) off-shell contributions. The (positive) local and (negative) nonlocal δfunction terms come with the largest magnitudes, but mostly cancel among themselves, leaving a total octet contribution that is positive and peaks around x ≈ 0.2, with a similar order of magnitude as the xs distribution.
A qualitatively similar scenario is evident in Fig. 7(c) for the decuplet intermediate state contributions to xs, where the individual on-shell, off-shell, δ-function, and gauge link terms are shown. (Note that the on-shell and off-shell terms include also the respective end-point pieces.) The predominantly positive on-shell, off-shell, and nonlocal δ-function contributions at x ≳ 0.2 largely cancel with the predominantly negative local δ-function and nonlocal off-shell terms, resulting in a very small overall decuplet contribution to xs, peaking at x ∼ 0.1, that is, an order of magnitude smaller than the octet.
Finally, the resulting asymmetry xðs −sÞ in Fig. 7(d) reflects the interplay between thes PDF, which peaks at lower x, and the s-quark PDF, which extends to larger values of x. A key feature of this result is the strong cancellations between positive local and negative nonlocal, gauge-link dependent contributions, in both the octet and the decuplet channels. The net effect is then a small positive xðs −sÞ asymmetry, peaking at x ≈ 0.2-0.3, and about an order of magnitude smaller than the asymmetry between thed andū PDFs resulting from pion loops.
In addition to the shape, it is instructive also to examine the contributions of the various terms to the lowest moments of the s ands PDFs, in particular, the average number of strange and antistrange quarks, and the average momentum carried by them, The conservation of strangeness of course requires equal numbers of s ands quarks in the nucleon, hsi ¼ hsi, as a direct consequence of local gauge invariance, although the shapes of the s ands distributions themselves are obviously rather different. The zero net strangeness can be verified by explicitly summing the contributions to hsi and hsi from the various diagrams in Figs. 1 and 2, as Table VIII indicates. Note also that the conservation of strangeness holds for the octet and decuplet contributions individually, as well as for the tadpole and bubble diagrams, Although the contribution to the total strange and antiquark quark number from the decuplet intermediate states is about an order of magnitude smaller than that from octet intermediate states, the role of the kaon bubble and tadpole terms is more significant, making up ≈60% of the total. For the antistrange moment, hsi, including the δfunction contributions from the rainbow diagrams, some 40% of the total moment comes from x ¼ 0. For the strange hsi moment, on the other hand, the structure of the convolution in Eq. (58) means that all of the contributions to sðxÞ are at x > 0, including ones involving δ-function splitting functions. Interestingly, significant cancellation occurs between the local terms and the gauge linkdependent nonlocal contributions, which turn out to be negative and about half as large in magnitude as the local.
While the lowest moments of the s ands are constrained to be equal, there is no such requirement for higher moments, including the x-weighted moment corresponding to the momentum carried by s ands quarks. Since the total s −s asymmetry is found to be mostly positive over the range of x relevant in this analysis, not surprisingly the total hxðs −sÞi moment is also positive. Including the uncertainties on the kaon-nucleon-hyperon vertex regulator parameters from Fig. 6, the combined asymmetry in our analysis is It is instructive, however, to observe the origin of the asymmetry in our chiral effective theory formulation. As mentioned above, there are no contributions to the momentum carried bys quarks from any of the δ-function terms from the rainbow or kaon bubble diagrams, so that only the on-shell and on-shell end-point terms are nonzero. In contrast, all terms, including the δ function, contribute to the momentum carried by s quarks. The result is a relatively small asymmetry that survives the cancellation of the (positive) on-shell s ands terms, with large contributions   Fig. 2(f)] diagrams. After the cancellations of various terms, the octet baryon contribution to the strange momentum asymmetry is actually negative, hxðs −sÞi oct ≈ −0.87 × 10 −3 . The terms involving decuplet hyperon states give relatively small absolute contributions, with significant cancellations arising that lead to a negligible overall strange decuplet asymmetry, hxðs −sÞi dec ≈ −0.05 × 10 −3 .
Interestingly, the most significant role played here is by the kaon tadpole terms [Figs. 2(j) and 2(k)]. With strong cancellations between the positive local f ðδÞ K and negative nonlocal δf ðδÞ K terms, the total asymmetry from the tadpole, hxðs −sÞi tad ≈ 2.57 × 10 −3 , is still about 3 times larger in magnitude than that from the rainbow and Kroll-Ruderman diagrams. The result is an overall asymmetry in Eq. (73) that is positive.
Experimentally, identifying an asymmetry of this size will be challenging, but not impossible. Traditionally, inclusive charm meson production in charged current neutrino and antineutrino DIS from nuclei has been used to provide information about the s ands PDFs in the nucleon, and analyses of data from neutrino experiments at BEBC [60], CDHS [61], CDHSW [62], CCFR [63], and NuTeV [64,65] have yielded values in the range hxðs −sÞi ∼ ð0 − 3Þ × 10 −3 [66][67][68]. Unfortunately, the neutrino-nucleus data are known to be affected by uncertainties in nuclear medium effects when relating nuclear structure functions to those of free nucleons [69], and in the nuclear dependence of charm quark energy loss and Dmeson interactions during hadronization in the nuclear medium [70,71].
Alternatively, the s ands distributions can be constrained by K AE meson production data from semiinclusive deep-inelastic scattering (SIDIS) off protons and deuterons, such as from the HERMES [72,73] or COMPASS [74] experiments. In a first of its kind global analysis, the JAM Collaboration recently fitted both the SIDIS and inclusive DIS data, along with other high energy scattering data, within a Bayesian likelihood analysis using Monte Carlo techniques to simultaneously determine both the spin-averaged PDFs and parton-tohadron fragmentation functions [75]. The analysis found a suppressed strange content in the nucleon at large x, and found no clear evidence for a nonzero s −s asymmetry within relatively large uncertainties. In the future, high-precision SIDIS data from the Jefferson Lab 12 GeV program or from the planned Electron-Ion Collider should provide better constraints on the s and s PDFs, as may Wþ charm production data from pp collisions at the LHC [76][77][78].

V. CONCLUSION
In this paper we have calculated the contributions to the sea quark distributions in the proton which are generated within a nonlocal chiral effective field theory. Both octet and decuplet intermediate states were included in the oneloop calculation using a four-dimensional dipole regulator to deal with the ultraviolet divergences. This regulator was introduced explicitly in the nonlocal Lagrangian density, with gauge invariance ensured through the presence of gauge links. A consequence of the introduction of the regulator are additional diagrams [Figs. 2(f), 2(i), and 2(k)] that arise from the expansion of the gauge links to lowest order in the electromagnetic coupling.
The free parameters entering the calculation, namely the dipole regulator masses, have been determined by fitting the available inclusive differential pp → nX, pp → ΔX, pp → ΛX, and pp → Σ Ã X cross section data. Using the fitted values of the dipole masses, Λ ¼ f1.0ð1Þ; 0.9ð1Þ; 1.1ð1Þ; 0.8ð1Þg GeV for the fπN; πΔ; KΛ; KΣ Ã g states, respectively, we computed the x dependence of the sea quark asymmetryd −ū, which is dominated at x > 0 by the on-shell contribution involving a nucleon intermediate state. The general shape and magnitude of the asymmetry extracted from the E866 Drell-Yan data [4] are described quite well, with the exception of the apparent change sign at higher x values, which is practically impossible to accommodate within the current theoretical framework. On the other hand, preliminary data from the SeaQuest experiment at Fermilab [83] suggest that the extractedd=ū ratio may flatten out at large x values and remain above unity. The integratedd −ū asymmetry was found to lie in the range between hd −ūi ≈ 0.09 and 0.17, which encompasses the values extracted by the New Muon [1] and E866 [4] Collaborations of ≈0.15 and 0.12, respectively. Remarkably, some 30% of our calculated value is associated with a δ-function contribution at x ¼ 0, which is not accessible experimentally at finite energy.
For the strange distributions in the proton, both the s and s PDFs were found to be positive at all values of x > 0. Interestingly in this case, while thes distribution receives δ-function contributions also at x ¼ 0 (around 2=3 of the total), the s PDF vanishes at x ¼ 0; both integrate to the same value, however, to ensure zero total strangeness, hsi ¼ hsi. Again, the contributions from the octet baryon intermediate states are dominant, with decuplet baryon contributions about an order of magnitude smaller. The gauge link dependent terms play a significant role in the nonlocal formulation of the chiral theory, contributing about half of the total hsi and hsi, but of opposite sign.
Large cancellations also appear in the x-weighted asymmetry xðs −sÞ, which remains small but positive across all x, with the integrated value lying in the range 0.9 × 10 −3 ≲ hxðs −sÞi ≲ 2.5 × 10 −3 . This is broadly consistent with previous determinations from neutrino scattering experiments [63,65], although the uncertainties on the empirical bounds are rather large. A nonzero moment hxðs −sÞi leads to a correction [68] to the NuTeV extraction of sin 2 θ W [79]. Our result supports the idea that the strange-antistrange quark asymmetry may indeed reduce the NuTeV anomaly by up to 1 standard deviation, which, along with other corrections such as charge symmetry breaking in the nucleon sea [84][85][86] and the isovector EMC effect [87], may account for the apparent anomaly entirely in terms of Standard Model physics.
Future progress on constraining the s −s asymmetry experimentally is expected to come on several fronts. Parity-violating inclusive DIS and semi-inclusive kaon electroproduction from hydrogen at Jefferson Lab and at a future Electron-Ion Collider will provide independent combinations of flavor PDFs, with the s ands distributions weighted by different electroweak charges and fragmentation functions, respectively. At higher energies, data on inclusive Wþ charm production in pp collisions at the LHC [76,77] can also provide sensitivity to differences between the s ands PDFs at small values of x, complementing the constraints at higher x values from fixed target experiments.