Strange quark helicity in the proton from chiral effective theory

We compute the helicity-dependent strange quark distribution in the proton in the framework of chiral effective theory. Starting from the most general chiral SU(3) Lagrangian that respects Lorentz and gauge invariance, we derive the complete set of hadronic splitting functions at the one meson loop level, including the octet and decuplet rainbow, tadpole, Kroll-Ruderman and octet-decuplet transition configurations. By matching hadronic and quark level operators, we obtain generalized convolution formulas for the quark distributions in the proton in terms of hadronic splitting functions and quark distributions in the hadronic configurations, and from these derive model-independent relations for the leading nonanalytic behavior of their moments. Within the limits of parameters of the Pauli-Villars regulators derived from inclusive hyperon production, we find that the polarized strange quark distribution is rather small and mostly negative.

Data from these and other polarized high-energy scattering processes, such as jet and W boson production in polarized pp collisions at RHIC [28][29][30], have been utilized in global QCD analyses of spin-dependent parton distribution functions (PDFs) by a number of groups [31][32][33][34][35][36][37][38][39][40][41][42]. The latest results from the JAM Collaboration's simultaneous analysis [42] of helicity PDFs and fragmentation functions give a fraction ∆Σ = 0.36±0.09 of the proton's spin carried by quarks and antiquarks at a scale of Q 2 = 1 GeV 2 . Parallel efforts from lattice QCD have also been made on calculations of moments of PDFs through the matrix elements of appropriate quark and gluon local operators within nucleon states [43][44][45][46][47], and more recently first studies have been explored of the feasibility of extracting information on the dependence of PDFs on the parton momentum fraction x from quasi-PDF and pseudo-PDF lattice calculations [48,49].
Among the three light quark flavors, the contribution to the proton spin from the strange quark is the least well determined, and phenomenological studies often rely on assumptions such as SU(3) flavor symmetry and equivalence of the strange and antistrange polarizations, ∆s = ∆s, to simplify the analyses. In many of the studies which have made these assumptions the strange quark polarization has typically been found to be in the vicinity of ∆s + ≡ ∆s + ∆s ≈ −0.1. Recent direct lattice simulations of disconnected loop contributions have yielded slightly smaller magnitudes for the strange quark polarization, ∆s + latt = −0.046(8) [47], while an analysis of the spin problem taking into account the angular momentum carried by the meson cloud [50][51][52], suggests a value of order −0.01 [53,54].
The recent JAM global QCD analysis, which used inclusive and semi-inclusive DIS data in order to relax the SU(3) symmetry constraint, also supports a smaller magnitude for the strange polarization, ∆s + JAM = −0.03 (10) [42] at a scale of Q 2 = 1 GeV 2 , but with a larger uncertainty. A review of the status and results from the global QCD analysis and lattice QCD communities can be found in Ref. [55].
It was shown recently by de Florian and Vogelsang [56] that a nonzero integrated asymmetry between ∆s and ∆s can arise from perturbative QCD evolution at three-loop order.
The effect was found to be small, however, with the difference ∆s − ∆s predicted to be negative and around 1% of the sum ∆s + ∆s. This is in contrast to the unpolarized case, where the total number of strange and antistrange quarks must be equal, even though the shape of their momentum fraction distributions in x need not be the same at three loops [57].
On the other hand, meson cloud models, in which the proton's strangeness content is generated by fluctuations to kaon-hyperon states such as p → ΛK + , naturally predict zero polarization for antistrange quarks. In the limit in which the kaon mass is much smaller than the baryon masses, the P -wave nature of the kaon emission would require the Λ to be polarized in the opposite direction to the proton. Since in a nonrelativistic quark model picture the strange quark carries all of the spin of the Λ, the expectation would be for the strange quark polarization to be negative. On the other hand, inclusion of relativistic effects [58,59], as well as Fock states with higher-mass hyperons and K * mesons [60][61][62], can significantly affect the shape and even the sign of the ∆s distribution.
A more systematic approach to computing the effects of pseudoscalar meson loops lies in the framework of chiral effective field theory, which establishes a more direct connection between the meson cloud of the nucleon and the underlying QCD theory. This methodology has been applied recently in studies of the unpolarized light quark asymmetryd −ū and the strange-antistrange asymmetry s−s in the proton, using both local [63][64][65] and nonlocal [66,67] formulations. Here, we extend our previous analysis [65] of the chiral loop contributions to the nonperturbative strange quark PDF to the polarized sector. We work within the local formulation of the chiral effective theory, using Pauli-Villars to regularize the integrals and consider both the SU(3) octet and decuplet hadronic states.
In Sec. II, we begin with presenting the lowest order meson-baryon chiral effective Lagrangian, consistent with Lorentz and gauge invariance. The convolution formalism for the nucleon PDFs in the framework of chiral effective theory is discussed in Sec. III, including the effective twist-2 operators relevant for the spin-dependent distributions. Hadronic split-ting functions are derived in Sec. IV, including for the octet and decuplet rainbow diagrams, Kroll-Ruderman, tadpole, and octet-decuplet transition contributions, and from these the model-independent leading nonanalytic (LNA) behavior of the loop contributions to the moments of the PDFs is deduced in Sec. V. The regularization procedures dealing with the divergent loop integrals are discussed in Sec. VI A, and the detailed numerical results for the polarized strange quark distributions in the proton are shown in Sec. VI B. Finally, we summarize our analysis and discuss future possible extensions of this work in Sec. VII. In Appendix A, we present some details about the derivation of the decuplet rainbow splitting function and the octet-decuplet splitting function.

II. EFFECTIVE LAGRANGIAN
In this section we review the basic effective chiral SU(3) Lagrangian describing the relativistic interactions of pseudoscalar mesons (φ) and SU(3) octet (B) and decuplet (T ) baryons [68][69][70]. To lowest order, this can be written as where D and F are the meson-octet baryon coupling constants, and C and H are the mesonoctet-decuplet and meson-decuplet-decuplet baryon couplings, respectively. In the meson sector the operator u µ is defined as with u given in terms of the pseudoscalar fields φ, and f φ is the pseudoscalar meson decay constant. The pseudoscalar pion, kaon and η meson fields can be collected in the matrix φ, The octet-decuplet transition tensor operator Θ µν is defined as where Z is the decuplet off-shell parameter. To simplify the calculations, in this analysis we will choose Z = 1/2 [71], although the physical results should be independent of the value of Z chosen. The octet-decuplet-meson interaction term in Eq. (1) can be written explicitly in component form as [72] T Expanding the effective Lagrangian (1) up to O (φ/f φ ) 2 , we can write this in more explicit fashion as a sum of specific meson-baryon interactions, where the first two terms, representing the meson-octet baryon interaction and the Weinberg-Tomozawa term, are given in Ref. [65]. The third term involves the meson-octet-decuplet vertex and is given by The final term in Eq. (11) involving the meson-decuplet-decuplet baryon vertices is not shown as it is not relevant to the matrix elements at the one-loop level when the initial and final states are both nucleons.

III. PARTON DISTRIBUTIONS IN THE NUCLEON
In this section, we derive the polarized PDFs in the nucleon within the convolution formalism by matching the spin-dependent twist-2 quark operators to hadronic operators with the same quantum numbers. We identify the complete set of hadronic operators contributing to the polarized quark distributions, and relate the matching coefficients to the moments of PDFs in the hadronic configurations.

A. Convolution formalism
The n-th Mellin moment of the spin-dependent quark distribution ∆q(x) is defined as where we have used the crossing symmetry relation ∆q(−x) = +∆q(x) between the quark and antiquark distributions. (Note that spin-averaged PDFs, in contrast, have the opposite crossing symmetry property [65].) From the operator product expansion these moments can be related to the matrix elements of local twist-2 operators O µ 1 ···µn ∆q between nucleon states, where p µ is the four-momentum of the nucleon and s µ its polarization vector, with s 2 = −1, and the braces {· · · } represent total symmetrization of Lorentz indices. The spin-dependent twist-two operators are defined as with In an effective field theory, these quark operators are matched to hadronic operators with the same quantum numbers (but not necessarily with the same twist) [73], where the subscript h labels different types of hadronic operators. The c-number coefficients ∆q/h can be defined through the n-th moments of the spin-dependent PDFs ∆q h (x) in the hadronic configuration h, Matrix elements of the hadronic operators O µ 1 ···µn h are used to define the moments of the hadronic splitting functions ∆f h by taking the "+" components of the Lorentz indices, In analogy with the unpolarized case [65], the operator relation in Eq. (16) then gives rise to a convolution form for the spin-dependent PDFs in the nucleon, where ∆q + h = ∆q h +∆q h is the spin-dependent valence quark distribution for quark flavor q in the hadronic configuration h. The convolution expression (19) is the basis for the calculation of the contributions to the quark helicity distributions from the chiral loop corrections generated from the Lagrangian (1).

B. Twist-2 operators
The spin-dependent quark operators in Eq. (15) can be matched to hadronic operators derived from the lowest order Lagrangian in Eq. (11) [67,74], where the trace "Tr" here is over the Lorentz indices. The a priori unknown coefficients {ᾱ (n) ,β (n) ,σ (n) } and {α (n) , β (n) , σ (n) } correspond to the octet baryonic pseudovector and vector operators, respectively, whileγ (n) andω (n) correspond to decuplet-decuplet and octetdecuplet transition operators, respectively. Note that only those operators that contribute to matrix elements with initial and final nucleon states are listed in Eq. (20).
Writing the spin-1/2 octet baryon operator B in a three-index tensor representation, one can relate this to the octet baryon field matrix B by with the corresponding conjugate representation giving where ijk is the antisymmetric tensor. In Eq. (20) the flavor operator λ q ± is defined as withλ q = diag(δ qu , δ qd , δ qs ) being diagonal 3 × 3 matrices. Expanding λ q ± up to O(φ 2 ), one has Finally, the combinations of operators B · · · B , T µ AT ν and T µ AB in Eq. (20) involving the three-index tensors are given by [72] (BB) = Tr B B , and With these relations we can write the hadronic operators explicitly for each of the spindependent u, d and s quark distributions as The hadronic operators appearing in Eqs. (27)-(29) are given by for octet baryon operators, and for operators involving decuplet baryon fields.
In the present work we will focus on the polarized strange quark distributions in the proton, ∆s(x). Correspondingly, the matrix elements of the hadronic operators give rise to the octet rainbow, tadpole, Kroll-Ruderman, decuplet rainbow, and octet-decuplet transition splitting functions, as illustrated by the diagrams in Fig. 1. The convolution representation (19) then gives the strange quark PDF in terms of the explicit hadronic configurations as where for notational convenience we define the splitting functionsf j (y) ≡ f j (ȳ), with y ≡ 1 − y the baryon momentum fraction when the meson carries momentum fraction y. For that while the convolution result in Eq. (32) involves the ∆s + j distribution in the hadronic configuration, in our calculations we shall assume that all of the antiquarks reside in the pseudoscalar meson loops, so that the antiquark polarization is zero, ∆s j = 0. In the next section we discuss the calculation of these PDFs in more detail.

C. PDFs in hadronic configurations
The spin-dependent strange quark distributions in the hadronic configurations as appear in Eq. (32) can be computed by relating their moments to the coefficients of the various terms in the twist-2 operator for the strange quark in Eq. (29). Starting with the PDFs in the bare octet baryons, ∆s B [ Fig. 1(a)], the moments can be expressed in terms of the For the kaon tadpole distributions ∆s , the moments are given by are given in terms of the coefficients α (n) , β (n) and (in principle) σ (n) , Using SU(3) flavor symmetry, the axial vector and vector coefficients can also be written in terms of the spin-dependent and spin-averaged PDFs in the proton [65], and ∆s (tad) and the spin-dependent strange Kroll-Ruderman PDFs ∆s in terms of the unpolarized nonstrange PDFs in the proton, For the PDFs involving decuplet baryons, the moments of the spin-dependent distribu- while for the octet-decuplet transitions [ Fig. 1(e)] the moments of ∆s T B are expressed in terms of the coefficientω (n) , From SU(6) symmetry the coefficientγ (1) can be related to the meson-baryon coupling from which the decuplet spin-dependent strange PDFs can be expressed as For the coefficient of the octet-decuplet transition operators in Eq. (29), SU(3) symmetry gives the relationω which allows the spin-dependent strange transition PDFs to be written as With these relations, we have expressed all of the necessary strange quark distributions in the hadronic configurations in Fig. 1 in terms of PDFs in the bare proton, which, together with the hadronic splitting functions, constitute the input to the convolution formula in Eq. (19). In the next section we will derive the complete set of the hadronic splitting functions necessary to complete the evaluation of the PDFs.

IV. HADRONIC SPLITTING FUNCTIONS
The spin-dependent hadronic splitting functions ∆f j defined in Eq. (18)

A. Octet baryon rainbow
For the meson-octet baryon rainbow diagram of Fig. 1(a), the splitting function is given by ∆f (rbw) where D φ and D B are the meson and octet baryon virtualities, with m φ and M B the kaon and octet baryon masses, respectively. The spinor u(p) is normalized such thatū(p) u(p) = 2M , and s + is the "+" component of the external proton spin vector s µ . The coefficients C 2 Bφ can be obtained from the effective Lagrangian (1), and for the ΛK and ΣK configurations are explicitly given in terms of the D and F couplings as Using the Dirac equation, the integrand in Eq. (47) can be decomposed into several terms with different combinations of meson and octet baryon propagators, ∆f (rbw) where with In a frame of reference in which p ⊥ = 0, the two combinations (k · p s + − k · s p + ) and (k 2 s + − 2 k·s k + ) appearing in Eqs. (51) become independent of k − . After integration over k + , these two terms take the forms yM 2 s + and (y 2 M 2 −k 2 ⊥ ) s + , respectively. It is convenient, therefore, to write the total octet baryon rainbow function ∆f (rbw) Bφ as a sum of three splitting functions associated with the on-shell, off-shell and δ-function contributions, ∆f (rbw) Integrating over the k − component in Eq. (50) and using the residue theorem, one can write the individual functions in (53) in terms of integrals over k 2 ⊥ . In particular, for the on-shell function one has ∆f (on) where and F (on) The result in Eq. (54) for the on-shell splitting function is in agreement with that in Refs. [58,60]. On the other hand, the new off-shell splitting function in Eq. (53) is given by where here F is the corresponding regulating function for the k 2 ⊥ integration (which can in practice be different from the on-shell regulating function F (on) B in Eq. (54)). For the δ-function term, ∆f (δ) φ , which arises from meson loops with zero light-cone momentum (k + = 0), one has ∆f (δ) where is the corresponding regulating function. Compared with the splitting functions for the spin-averaged case derived in Ref. [65], the spin-dependent on-shell function ∆f

B. Tadpole
The distribution functions associated with the meson tadpole diagram in Fig. 1(b), with an operator insertion at the two nucleon-two meson vertex, can be written as The tadpole splitting functions for the charged and neutral kaon loop contributions are then given by where the generic tadpole function ∆f C. Kroll-Ruderman The light-cone momentum distribution associated with the Kroll-Ruderman diagrams in Straightforward calculation gives ∆f (KR) The Kroll-Ruderman splitting function can then be written in terms of the off-shell and δ-function contributions as ∆f (KR) with the off-shell function ∆f   Fig. 1(d) can be written where the usual spin-3/2 Rarita-Schwinger energy projector is This expression for the decuplet propagator corresponds to the particular choice Z = 1/2 in Eq. (9), for which the octet-decuplet transiton tensor operator Θ µν takes the simple form g µν − γ µ γ ν . The coefficients C 2 T φ can be derived from the effective Lagrangian (12), and for the Σ * 0 K and Σ * + K configurations are explicitly given by In our analysis, we will take C = −2D from SU (6) as a sum of 3 terms involving different numbers of decuplet baryon propagators, D T . In analogy with the octet baryon splitting function in (50) and (51), the numerators N T i in Eq. (67) can be written as linear combinations of the structures 2M s + , (p·k s + − k ·s p + ) and (k 2 s + − 2k·s k + ), where we define the difference and sum of the masses for the decuplet baryons as in Eq. (52), This structure then allows the decuplet rainbow splitting function to be decomposition into decuplet on-shell, off-shell and δ-function terms, Details of the derivations of the individual functions in Eq. (70) are given in Appendix A.
After the k − integration we therefore obtain and ∆f (off) T (y) = 1 for the decuplet on-shell and off-shell functions, respectively, with F For the δ-function contribution, we have ∆f (δ) where the two functions proportional to δ(y) are given by ∆f (δ) with regulating functions F (δ1) T (y, k 2 ⊥ ) and F (δ2) T (y, k 2 ⊥ ), respectively. Explicit expressions for each of the regulating functions are given in Sec. VI A for Pauli-Villars regularization.

E. Octet-decuplet baryon transition
For the octet-decuplet rainbow transition diagrams in Fig. 1(e), the splitting function can be written as for the T Bφ = Σ * 0 Σ 0 K + and Σ * + Σ + K 0 configurations, with C Bφ and C T φ given by Eqs. (49) and (66), respectively. The two terms in the brackets of Eq. (76) correspond to the two orderings of BT and T B in Fig. 1 (50) and (67), we write the octet-baryon transition rainbow splitting function as a sum of 3 terms with different number of baryon propagators, where the numerators of the terms in the brackets are given by and we define T Bφ (y) = Following the steps given in Appendix A, the on-shell octet-decuplet transition function in (80) can be written as where the regulator functions F T B are given in Sec. VI A below. The off-shell transition function is given by in terms of the same regulators F T B as in the on-shell function (81). Finally, for the δ-function contribution to the octet-decuplet transition, we find ∆f (δ) where the function ∆f To begin with, we define the n-th moment of the spin-dependent splitting function ∆ f for the i = {on, off, δ} contribution. From the convolution expression for the ∆s PDF in the nucleon in Eqs. (19) and (32), and the definition of the nucleon PDF moment in Eq. (13), we can write the n-th moment of the strange PDF in the nucleon as where  Fig. 1 explicitly, we can compute the LNA behavior of the strange PDF moments as In the following we focus specifically on the n = 1 moment of the strange quark PDF, x 0 LNA ∆s ≡ ∆S where The spin-dependent off-shell and δ-function terms are equivalent to the corresponding unpolarized splitting functions, and for the n = 1 moments have the NA behavior [66], respectively.
For the decuplet rainbow splitting functions, the NA behavior of the n = 1 moments of the on-shell and off-shell functions is given by and Note that the results for the individual on-shell and off-shell contributions in (91) and (92) depend on the choice of the decomposition into the two pieces, the sum of the on-shell and off-shell contributions of the separation, and gives rise to The LNA contribution arising from the δ-function term is given by For the octet-decuplet transition splitting functions, the NA behavior is slightly more involved because of the presence of two baryon mass differences, ∆ B and ∆ T . For the on-shell and off-shell splitting functions, the first moments are given by for ∆ B < m φ and ∆ T < m φ . There is strong cancelation between the on-shell and off-shell pieces, resulting in a sum that is given by In the chiral limit, one has ∆ B < m φ while ∆ T > m φ , and the corresponding NA behavior is given by Finally, for the δ-function contribution the LNA behavior is In the chiral limit, m φ → 0, the mass difference ∆ B ∼ O(m 2 φ ) approaches zero first, while ∆ T remains a constant. Further expanding R T = ∆ T −m 2 φ /2∆ T +O(m 4 φ ), the LNA behavior in Eqs. (93) and (98) can be evaluated as for the T and T B contributions, respectively.
Finally, combining the derived LNA behaviors for the splitting function moments with Eq. (87), the LNA contribution to the n = 1 moment of the spin-dependent strange-quark PDF in the nucleon is given by Summing over all the relevant octet B and decuplet T states, and using the expressions for the couplings in Eqs. (49) and (66) and the moments ∆S We stress that any calculation of the strange quark PDFs in the nucleon or its moments must obtain this behavior, if it is to be consistent with the chiral symmetry properties of QCD, which provides an important, model-independent constraint on nonperturbative models of the nucleon.

VI. NUMERICAL RESULTS
Combining the results derived in Secs. III and IV for the splitting functions and the PDFs in the hadronic configurations, in this section we present the results for the numerical computation of the spin-dependent strange quark distributions in the proton. We begin by discussing the regularization procedure for the splitting functions, and then compare the computed PDFs with some recent phenomenological parametrizations from global QCD analyses.

A. Regularization of splitting functions
The hadronic splitting functions computed in Sec. IV in the framework of chiral effective theory generally involve loop integrals that are ultraviolet divergent. A regularization prescription is therefore required to regulate the high-energy behavior and render the loop integrals finite. Various prescriptions have been utilized in previous analyses, including dimensional regularization [78], finite momentum cutoffs, Pauli-Villars [64,65], as well as finite-range regularization within local [79][80][81] and nonlocal [82,83] formulations. Following our earlier analysis of spin-averaged strange-antistrange quark asymmetries [64,65], we adopt here the Pauli-Villars regularization scheme, which has the advantages of preserving the Lorentz invariance, gauge invariance, and chiral symmetry of the effective theory. It allows us to use the same phenomenological parameters as those determined in the unpolarized strange analysis [65].
As discussed in Refs. [64,65], the Pauli-Villars method regularizes divergent integrals by subtracting from the pointlike results expressions in which the propagator masses are replaced by finite cutoff masses, such that in the high-energy limit the difference between them vanishes. For the on-shell baryon octet splitting function, ∆f (on) B , we employ the subtraction which corresponds to using a regulating function in Eq. (54) given by where µ 1 is the subtraction mass parameter, and D Bφ is given in Eq.
For the δ-function term, ∆f (δ) B , in Eq. (57), two subtractions are necessary to take into account the divergences in both the k − and k 2 ⊥ integrations, where µ 1 and µ 2 are the mass parameters for the subtraction terms, whose coefficients a 1 and a 2 must satisfy the relation This leads to an effective regulating function in Eq. (57) given by with Ω µ i = k 2 ⊥ + µ 2 i . In the decuplet sector, the loop integrals associated with the on-shell and off-shell functions are more divergent than those of the octet contributions due to the presence of derivative couplings. To regularize the integrals for the decuplet splitting functions, therefore, requires several subtractions, which we take to have the form where the coefficients b i satisfy To reduce the number of free parameters, in our numerical analysis we take µ 1 = µ 2 = µ 3 = µ 4 ≡ µ for the decuplet baryon contributions, in which case we have the replacement For the on-shell and off-shell decuplet splitting functions in Eqs. (71) and (72), the regulating functions can be written as, respectively. For the decuplet δ-function contributions, Eq. (74), Pauli-Villars regularization gives the regulating functions for the two functions in Eqs. (75a) and (75b), respectively. Finally, for the octet-decuplet transition splitting functions, the regulators in the on-shell and off-shell functions in Eqs. (81) and (82) are given by In our previous analysis of meson loop contributions to the spin-averaged strange quark PDFs in the proton [64,65], the cut-off parameter µ 1 was fixed by fitting the pp → ΛX differential cross section data, and an upper limit was set on µ 2 by requiring that the T B are multiplied by the couplings C T φ C Bφ in Eq. (80), which for the Σ * 0 Σ 0 K + case is negative [Eqs. (66) and (49)], the sign of the overall contribution of these terms can be opposite to that shown in Fig. 2.  [39,42,85] and unpolarized [86,87] cross section data. For the spin-averaged u(x) and d(x) quark distributions in the proton, for convenience we use the parametrization from Ref. [88], while the polarized PDFs, ∆u and ∆d, are taken from Ref. [37]. For our applications, the dependence on the choice of input parametrization is relatively mild, however.
For representing the contributions to the polarized strange PDF from the various terms in Eq. (32), it is convenient to express the total distribution in terms of the diagrams in Fig. 1. Decomposing each diagram into on-shell, off-shell and δ-function contributions, in analogy with the unpolarized case in Ref. [65], one can write the total ∆s PDF as ∆s(x) = ∆s (on) + ∆s (off) + ∆s (δ) B rbw + ∆s (off) + ∆s (δ) KR + ∆s (δ) tad + ∆s (on) + ∆s (off) + ∆s (δ) T rbw + ∆s (on) + ∆s (off) + ∆s (δ)  octet-decuplet transition contributions, whose overall magnitude is smaller than those from the octet states. Furthermore, in contrast to the octet case, the on-shell contributions are positive, but cancelled somewhat by the negative off-shell and δ-function terms, which turn out to have a very similar shape and magnitude. The net result is a total positive effect, with about 1/5 of the magnitude of the octet contribution.
Comparing the calculated polarized strange distribution with phenomenological PDFs obtained from global QCD analyses, in Fig. 4 we show the total x∆s from the chiral theory together with parametrizations from the NNPDF [39] and JAM [42] analyses at Q 2 = 1 GeV 2 . The most striking observation is the small magnitude of the calculated strange polarization compared with the uncertainty bands of the global parametrizations, which reflects the relatively weak constraints on ∆s that exist from current experiments. The JAM study [42], in particular, performed a dedicated analysis of the strange quark PDF using data from inclusive and semi-inclusive DIS, without imposing the commonly used assumption about SU(3) flavor symmetry for the axial charges extracted from hyperon decays [89]. This leads to a significantly larger uncertainty on ∆s than that obtained in analyses that do impose SU(3) symmetry on the axial charges [31][32][33][34][35][36][37][38][39][40].
Furthermore, since existing data cannot discriminate between the strange quark and antiquark polarizations, in all of the global QCD analyses the assumption is made that ∆s = ∆s, so that in practice ∆s + ≡ ∆s + ∆s → 2∆s. In contrast, in the chiral theory calculation, Since all strange antiquarks reside in the spin-0 kaon, in this framework the antistrange polarization ∆s is identically zero. One may therefore expect the determinations of the strange polarization in the global QCD analyses to overestimate the ∆s contribution from the chiral calculation.
Integrating the calculated distribution over all x, in Table I we list the contributions of the various terms in Eq. (118b) to the lowest (n = 1) moment of ∆s(x), which from Eq. (13) we denote by x 0 ∆s ≡ ∆s . Numerically, a large degree of cancellation is seen between the various on-shell and off-shell terms, with the δ-function terms somewhat smaller. Within This can be compared with the value determined from the JAM global QCD analysis [42] of ∆s + JAM = −0.03 (10). While our central values are about an order of magnitude smaller than the phenomenological results, they are in good agreement within the relatively large uncertainty. Future data on semi-inclusive DIS and parity-violating inclusive DIS from the planned Electron-Ion Collider [90] should reduce the uncertainty on the extracted ∆s + and allow a better discrimination between the ∆s and ∆s distributions. The result is that the octet contributions are mostly responsible for the polarized strange PDF ∆s(x) to be negative at small x, with the lowest moment, ∆s , lying in the range (−5.3, −2.8) × 10 −3 . In comparison with the recent JAM global QCD analysis, ∆s + JAM = −0.03(10) [42], or the latest lattice QCD calculation from the ETM Collaboration, ∆s + latt = −0.046(8) [47], the chiral contribution is relatively small, although consistent with the phenomenological values within the uncertainties.
In the future it will be important to compare the current work with calculations within a nonlocal chiral theory, such as that used for the unpolarized sea quark asymmetries in Refs. [66,67]. Furthermore, extending the analysis to the nonstrange (valence quark) distributions ∆u(x) and ∆d(x) using the relativistic formalism presented here should provide robust estimates of the effect of the chiral effects on the axial charges g A and g 8 and total helicity ∆Σ carried by quarks.
For the octet-decuplet transition splitting function ∆f (rbw) T Bφ , following the same procedure we have for the first term in Eq. (77), For the second term in Eq. (77), we can write