Nonperturbative strange-quark sea from lattice QCD, light-front holography, and meson-baryon fluctuation models

We demonstrate that a nonzero strangeness contribution to the spacelike electromagnetic form factor of the nucleon is evidence for a strange-antistrange asymmetry in the nucleon's light-front wave function, thus implying different nonperturbative contributions to the strange and antistrange quark distribution functions. A recent lattice QCD calculation of the nucleon strange quark form factor predicts that the strange quark distribution is more centralized in coordinate space than the antistrange quark distribution, and thus the strange quark distribution is more spread out in light-front momentum space. We show that the lattice prediction implies that the difference between the strange and antistrange parton distribution functions, $s(x)-\bar{s}(x)$, is negative at small-$x$ and positive at large-$x$. We also evaluate the strange quark form factor and $s(x)-\bar{s}(x)$ using a baryon-meson fluctuation model and a novel nonperturbative model based on light-front holographic QCD. This procedure leads to a Veneziano-like expression of the form factor, which depends exclusively on the twist of the hadron and the properties of the Regge trajectory of the vector meson which couples to the quark current in the hadron. The holographic structure of the model allows us to introduce unambiguously quark masses in the form factors and quark distributions preserving the hard scattering counting rule at large-$Q^2$ and the inclusive counting rule at large-$x$. Quark masses modify the Regge intercept which governs the small-$x$ behavior of quark distributions, therefore modifying their small-$x$ singular behavior. Both nonperturbative approaches provide descriptions of the strange-antistrange asymmetry and intrinsic strangeness in the nucleon consistent with the lattice QCD result.

I.

INTRODUCTION
The unveiling of nucleon structure in terms of fundamental quark and gluonic degrees of freedom is a main goal of nuclear and particle physics. The strangeness distribution of the nucleon is of particular interest since it is a purely sea-quark distribution. The nonperturbative dynamics of the strange-antistrange quark asymmetry s(x) −s(x) poses a challenging theoretical problem.
It has become of major interest in both experimental and phenomenological studies, not only because of its important role in understanding strong-interaction dynamics but also because the s(x)−s(x) asymmetry is an important input for testing electroweak theory and new physics models.
For example, a precise test of electroweak physics in neutrino and antineutrino-induced dimuon production depends in detail on the intrinsic strange and antistrange distributions in the nucleon [1].
The intrinsic nonperturbative strangeness distributions and asymmetry also give insight, via the operator product expansion, into the nonperturbative physics of the intrinsic charm and bottom contributions to the nucleon structure functions [2][3][4].
Lattice QCD calculations [5][6][7], at the physical pion mass and extrapolated to the continuum and infinite volume limits, have provided estimates of the strangeness contribution to the electromagnetic (EM) form factors of the nucleon with better accuracy than that available from the global analyses [8][9][10] of the experimental data. A direct lattice calculation of s(x) −s(x) has not as yet been achieved [11]. However, we shall show that one can constrain the s(x) −s(x) asymmetry by comparing the lattice QCD results for the strange form factor with predictions based on a baryon-meson fluctuation model [12]. We will also introduce a new model based on the structural behavior of the light-front holographic approach to hadron structure [13], form factors and parton distribution functions [14]. We shall show that the s(x) −s(x) asymmetry in the nucleon can be predicted up to a normalization factor constrained by lattice results.
Parton distribution functions (PDFs) are interpreted, at leading twist, as distributions of quarks and gluons carrying the light-front momentum fraction x of the nucleon's momentum at fixed lightfront time τ = t + z/c. The global QCD analysis of PDFs is based on factorization theorems of physical observables, such as the cross section of deep inelastic lepton-nucleon scattering [15].
Although equal numbers of s ands are required by their nonvalence nature in the nucleon, no fundamental principles prohibit different s(x) ands(x) distributions. A nonzero s(x) −s(x) has also been allowed for in global analyses of PDFs [16][17][18]. Furthermore, the first moment of the difference of PDFs, can also be used to quantify the s(x) −s(x) asymmetry.
The strange-quark sea in the nucleon has both "extrinsic" and "intrinsic" components [2][3][4]. The extrinsic one is produced by gluon splitting g → ss triggered by a hard probe, e.g., the virtual photon exchanged between the lepton and the nucleon in a deep inelastic scattering process. Since the QCD coupling α s is small at high momentum scale, the extrinsic strange-sea derived from the splitting function can be calculated perturbatively. The nonperturbative intrinsic strange-sea encoded in the nucleon's nonvalence light-front (LF) Fock state wave function can in principle be obtained by solving the LF Hamiltonian eigenvalue problem [19]; e.g., by matrix diagonalization. However, to capture the nonperturbative dynamics in the bound state equations, one should integrate out all higher Fock states, corresponding to an infinite number of degrees of freedom, a formidable problem.
The strange-antistrange asymmetry in the nucleon originates in QCD from the difference between quark-quark versus quark-antiquark interactions. Since the nucleon carries nonzero quark number-the number of quarks minus the number of antiquarks-the interaction of the strange quark with the spectators of the nonvalence Fock states is different from that of the antistrange quark with the remaining quarks, thus leading to different s ands distributions. The extrinsic strange-antistrange asymmetry in the nucleon PDF arises from perturbative QCD evolution at high orders due to the difference between quark-to-strange quark splitting function P qs and quarkto-antistrange quark splitting function P qs . Since the strange-antistrange pair is generated from a nonstrange quark at next-to-leading order, and the interaction between the strange/antistrange quark and the nonstrange quark is mediated by additional gluon exchange, this pQCD effect arises at the three-loop level. An explicit calculation has been performed in [20].
In addition to PDFs, one can also obtain information on nucleon structure from elastic form factors, which relate to the transverse coordinate space distributions at fixed LF time via a Fourier transform [21]. The nucleon spin-preserving amplitude is described by the Dirac form factor, which can be expressed as: where Q 2 is the momentum transfer squared, and the flavor form factor F q 1 (Q 2 ), with q = u, d, s, · · · , measures the q-flavor quark contribution minus theq-flavor antiquark contribution due to the opposite charge of the quark and antiquark. Therefore a nonvanishing F s 1 (Q 2 ) at Q 2 = 0 indicates a strange-antistrange asymmetry in LF coordinate space. The constraint F s 1 (0) = 0 is fixed by the sum rule (1).
Lattice QCD results for F s 1 (Q 2 ), obtained in the continuum limit [5][6][7], are shown in Fig. 2 with systematic and statistical uncertainties added in quadrature. The lattice QCD analyses are described in the Appendix A.
In the baryon-meson fluctuation model [12], the nonperturbative strange sea is generated from the fluctuation of the nucleon valence state to the lightest mass hadronic state with strangeness; i.e., a kaon and a hyperon (Λ or Σ). The different distributions of the strange quark in the hyperon and the antistrange quark in the kaon yield a nonvanishing s(x) −s(x) distribution. In this model, a meson-baryon configuration, e.g., the K + Λ 0 state, creates different radially separated distributions of the s and thes quarks from the center of mass. Since the kaon is lighter than the hyperon, one expects that the kaon-and thus thes quark-to be at a larger radial distance from the center of mass than the hyperon and its s quark. This picture leads to F s 1 (Q 2 ) > 0 at Q 2 > 0, consistent with the lattice QCD results [5][6][7].
As we will discuss below, a positive value of F s 1 (Q 2 ) at Q 2 > 0 indicates that the strange quark distribution is more centralized in coordinate space than the antistrange quark distribution, and results in an s(x)−s(x) asymmetry in momentum space. A narrower distribution in coordinate space corresponds to a wider one in momentum space, and therefore the lattice QCD result F s 1 (Q 2 ) > 0 implies a negative s(x) −s(x) distribution at small-x and a positive distribution at large-x.
We will also examine in this article the behavior of F s 1 (Q 2 ) and s(x) −s(x) using the nonperturbative structure of light-front holographic QCD (LFHQCD), a semiclassical approach to relativistic bound state equations which follows from the holographic embedding of light-front dynamics in a higher dimensional gravity theory, with the constraints imposed by the underlying superconformal algebraic structure [28][29][30][31][32][33]. This approach incorporates a nontrivial connection to the hadron spectrum and therefore to the Regge trajectories predicted by the model.
In Sec. II, we will describe the strange-antistrange asymmetries in coordinate and momentum spaces in the boost invariant light-front formalism, together with qualitative discussions. We will perform quantitative calculations of s(x) −s(x) and F s 1 (Q 2 ) in Sec. III using the baryon-meson fluctuation model, and in Sec. IV using the structural framework of LFHQCD. We will analyze the constraints imposed from lattice QCD for these two nonperturbative models. We will also use the lattice QCD data to quantitatively constrain each model in order to obtain more precise predictions.
The procedures discussed here can be applied to other approaches, e.g., by deriving constraints on the wave functions predicted by meson cloud and chiral quark models. Final discussions and conclusions are presented in Sec. V. Taking a complete basis of LF Fock states with quarks and gluons as the degrees of freedom, a nucleon state with four-momentum P µ = (P + , P − , P ⊥ ) and total spin S z can be expanded as where The index n = qqq, qqqg, qqqqq, · · · , represents the constituents of the Fock state, the internal LF variables x i , k i⊥ , and λ i are the longitudinal momentum fraction, the intrinsic transverse momentum, and the spin carried by the ith constituent, respectively, and ψ n/N is the light-front wave function (LFWF). It gives the probability of the n-particle LF Fock state and represents the transition amplitude of the on-shell nucleon eigenstate to the quark and gluon Fock states of the free LF Hamiltonian which are off-shell in invariant mass. All nucleon properties are encoded in the LFWFs, which in principle could be obtained by solving the LF Hamiltonian eigenvalue problem. Aiming at a first-principle calculation of the LFWFs, calculational methods based on matrix diagonalization, such as discretized LF quantization [35], the transverse lattice method [36] and the basis LF quantization [37], have been proposed.
In this paper, we will focus on the s ands quark contributions to the nucleon nonvalence LF Fock state wave functions, ψ s/N (x s , k s⊥ , λ s ) and ψs /N (xs, ks ⊥ , λs), where the sum over other degrees of freedom is implied. The s ands quark PDFs expressed in terms of the LFWFs are s(x) = λs d 2 ks ⊥ 16π 3 |ψs /N (xs, ks ⊥ , λs)| 2 .
The sum rule (1) requires the normalization λs dx s d 2 k s⊥ 16π 3 |ψ s/N (x s , k s⊥ , λ s )| 2 = λs dxsd 2 ks ⊥ 16π 3 |ψs /N (xs, ks ⊥ , λs)| 2 = I s , where I s gives the number of intrinsic strange/antistrange quarks in the nucleon. Perturbative QCD evolution needs to be performed to include contributions from the extrinsic sea and to compare with the PDFs extracted from high energy scattering experiments.
The EM form factors of the nucleon are defined as [38] where J µ = q e qψq γ µ ψ q is the current operator, M is the nucleon mass, F 1 (Q 2 ) and F 2 (Q 2 ) are the Dirac and Pauli form factors, respectively. Comparing with the decomposition (3), one observes that F s 1 (Q 2 ) and F s 2 (Q 2 ) are given by the matrix elements of the current operator J µ s =ψ s γ µ ψ s . In the LF formalism, F 1 (Q 2 ) and F 2 (Q 2 ) can be calculated from the overlap of spin-conserving and spin-flip matrix elements of the + component of the current, respectively, [39]: with q µ = (q + , q − , q ⊥ ) and transferred momentum squared q 2 = t = (P − P ) 2 = −Q 2 .
The Drell-Yan-West (DYW) frame [41,42] with q 2 = −q 2 ⊥ , can be used to avoid off-diagonal contributions n → n = n ± 2 from Fock states with different constituents. Here q ⊥ is the Fourier conjugate of the transverse LF coordinate a ⊥ .
From (10) and the Fock state expansion (4), the Dirac form factor, in terms of the LFWFs, is given by the DYW expression [41,42] where ρ s/s (q ⊥ ) represents the effective strange/antistrange density. The relative minus sign in (14) arises from the opposite strange and antistrange charges.
The density ρ s/s (q ⊥ ) is the inverse Fourier transform of the distributionρ(a ⊥ ), Following the normalization (8) or the sum rule (1), we require and thus F s based on the uniqueness of the Fourier transform. As illustrated in Fig. 1 for a simple Gaussian distribution, if the s (ors) quarks are more concentrated at small transverse separation than thes (or s) quarks, one obtains a positive (or negative) form factor F s 1 (Q 2 ) at Q 2 > 0. A similar concept based on the locality defined in the instant form was presented in [43].
The strange-antistrange asymmetries in LF coordinate space and LF momentum space are correlated. To show this, we express ρ s/s (q ⊥ ) in terms of the transverse impact variable b ⊥ using the Fourier transform of the k ⊥ -space LFWFs following Ref. [44], The coordinate space distribution is theñ Here, b ⊥ is not the usual LF transverse coordinate variable but related according to a ⊥ = (1 − x)b ⊥ .
As they are related by a Fourier transform, the strange-antistrange asymmetry in b ⊥ -space is equivalent to the asymmetry of the transverse momentum k ⊥ distribution. Since there is no privileged direction for an unpolarized nucleon, one should have a nonvanishing strange-antistrange asymmetry of the longitudinal momentum distribution if the asymmetry of the transverse momentum distribution is nonzero. A positive F s 1 (Q 2 ) implies that the s quarks in the nucleon sea are more centralized in coordinate space than thes quarks and are therefore more spread out in momentum space. This leads to a negative s(x) −s(x) distribution at small-x and a positive one at large-x.

III. THE BARYON-MESON FLUCTUATION MODEL
We first evaluate the s(x) −s(x) distribution in the nucleon using the baryon-meson fluctuation model of Ref. [12]. As in Ref. [45], we shall focus on the fluctuation of the proton to the K + Λ 0 state, the lightest kaon-hyperon configuration and thus the state with the minimum off-shellness in invariant mass. In this nonperturbative approach the momentum distribution of the constituents is maximal at minimum off-shellness; i.e., at equal rapidity: Thus the mean LF momentum fraction of each constituent is proportional to its transverse mass: Instead of expanding directly in terms of quarks and gluons as in Eq. (4), the expansion in the fluctuation model is performed using a two-level convolution approach [45] in which the proton state is expanded as the valence state plus the baryon-meson state |BM . The component baryon and the meson wave functions are then further expanded into their quark and gluon Fock states.
This LF cluster-decomposition procedure [46] for the baryon LFWF is similar to the expansion in the meson cloud model [47][48][49][50]. Considering only the fluctuation to the |BM = |ΛK state, the expansion yields where "· · · " represents states other than |ΛK in the expansion, x Λ/K is the longitudinal LF momentum fraction carried by the Λ/K, and k Λ/K⊥ is the intrinsic transverse momentum of the The wave function is normalized to the probability of the fluctuation: where I s is the intrinsic strange quark number in (8).
The intrinsic distribution s(x) is then expressed as a convolution of the strange distribution q s/Λ in the Λ and the Λ distribution f Λ/ΛK in the baryon-meson state, Likewise, the intrinsic distributions(x) is The Λ and K distributions in the baryon-meson state are where One can observe that which leads to the relation The equal numbers of strange and antistrange quarks in the nucleon, i.e., Eq. (1), is satisfied by the sum rules However, the distribution s(x) −s(x) remains nontrivial.
We now calculate F s 1 (Q 2 ). For definitive predictions we adopt the approach used in Ref. [51], in which the s quark wave function is evaluated from the strange quark-scalar diquark configuration |sD of the Λ, and thes quark is evaluated from the antistrange quark-spectator quark configuration |sq of the K. Similar to the expansion (19), the Λ and K states are expressed as Then F s 1 (Q 2 ) can be expressed in terms of the LFWFs as where For the phenomenological description of the LFWFs, we choose the Brodsky-Huang-Lepage prescription [52,53] as utilized in Ref. [45], with invariant mass where m 1 and m 2 are the masses of the two components. The s(x) −s(x) asymmetry has been calculated with this LFWF in Ref. [12] and reproduced in Ref. [51] with the parameters m q = 0.330 GeV, m s = 0.480 GeV, m D = 0.600 GeV, and the universal momentum scale β = 0.330 GeV.
A determination from the data of extended observables indicates 0.24 < β < 0.37 GeV [54]. For the masses of Λ and K, we use the values given in Ref. [55]. Alternative LFWFs have been assumed for the study of the s(x) −s(x) asymmetry using the same baryon-meson fluctuation picture in Ref. [51].
Taking the fluctuation probability I s = 1.27% from Ref. [24], we calculate F s 1 (Q 2 ). The results are shown in Fig. 2 along with the lattice QCD results [5]. This result is consistent with lattice QCD using the original parameters assumed in Ref. [12].  [5,7]. The predictions of the fluctuation model use the LFWFs from Refs. [52,53].
To further constrain the baryon-meson fluctuation model, we will match its predictions to the lattice QCD data by taking β and I s as free parameters. The result is shown in Fig. 3, with β = 0.31(11) GeV and I s = 1.06(51)%. These values are consistent with the original choice in Ref. [12] and the value determined in Ref. [54].
If we take the model parameters determined by the fits, we obtain a model-based phenomenological constraint on the s(x) −s(x) distribution based on the baryon-meson fluctuation approach.
A comparison with global PDF fits is shown in Fig. 4. The factorization scale is not specified in this nonperturbative model, so the comparison has been done assuming µ = 1 GeV. The PDF uncertainties are commonly represented in two ways: the Hessian matrix and the Monte Carlo samplings. In Fig. 4, the uncertainty bands of the global fits are standard deviations calculated from the Hessian matrix for MMHT2014 [17] and JR14 [18] and from Monte Carlo replicas for NNPDF3.0 [16]. The Monte Carlo replicas for MMHT2014 and JR14 are generated from the Hessian matrix following the procedure described in Ref. [56].

IV. LIGHT-FRONT HOLOGRAPHIC QCD
The EM form factors of nucleons were described in the nonperturbative holographic framework from the coupling of the ρ to a qq pair in the proton in the limit of massless quarks [57]. In this section we calculate F s 1 (Q 2 ) and s(x) −s(x) in the nucleon using the analytic structure of form factors and quark distribution functions in LFHQCD for bound states of arbitrary twist. Here, twist refers to the dimension minus spin of the interpolating operator for the hadron state; it is equal to the number of constituents in a given Fock component in the LF Fock expansion.
In LFHQCD [13], the EM form factors for a bound-state hadron with twist-τ can be expressed as [14,58] where the Euler Beta function is with ) a normalization factor, and α(t) is the Regge trajectory of the vector meson which couples to the EM current in the t-channel exchange.
The Beta function structure of the EM form factors (40), which follows from the gauge/gravity structure in LFHQCD, was obtained in the pre-QCD era by Ademollo   independently by Landshoff and Polkinghorne [60]. Their derivations were based on the Veneziano model [61], which is an incorporation of the concept of duality [62]  For linear Regge trajectories Eq. (40) incorporates the hard-scattering counting rules at large t [64,65]. Indeed, for fixed u and large v we have B(u, v) ∼ Γ(u)v −u , and therefore the first argument in the Euler Beta function determines the scaling behavior of (40) lim at large Q 2 = −t. The second argument in (40) determines the timelike pole structure of the form factor; the analytic structure of (40) thus leads to a nontrivial connection with the hadron spectrum. In fact, using the expansion of the Gamma function for integer twist N = τ , with N the number of constituents for a given Fock component, we find which is expressed as a product of τ − 1 poles located at The form factor (45) thus generates the radial excitation spectrum of the exchanged particles in the t-channel, while keeping the structural form found previously in the limit of zero quark masses [13].
For the lowest radial excitation the VM spectrum in LFHQCD is given by [13,33] (Appendix B) where the squared mass shift ∆M 2 incorporates the effect from finite light quark masses. The quantity λ = κ 2 is the emergent mass scale, the only dimensional quantity appearing in LFHQCD for massless quarks [13]. Its value determined from the best fit to all radial and orbital excitations of the light mesons and baryons is κ = √ λ = 0.523 ± 0.024 GeV [33].
There is no need to introduce additional procedures to include quark masses when using the structural form (40) to describe form factors, since the effect of quark masses only amounts to a shift of the Regge intercept. For example, for the ρ, a vector mesons we obtain from Eq. (47) the leading Regge trajectory with slope α = 1 4λ and intercept α ρ (0) = 1 2 − ∆M 2 ρ 4λ , which differs from the conformal limit 1 2 by the mass shift ∆M 2 ρ 4λ from quark masses. Likewise, the ω, f trajectory is with the same slope α = 1 4λ and similar intercept α ω (0) = 1 2 − ∆M 2 ω 4λ . We show in Fig. 5 the Chew-Frautschi plot for the leading ρ − a and ω − f trajectories.
The spectrum of the exchanged particles in the t-channel follows from (46) for the leading VM trajectory (48). We find which is precisely the spectrum of the ρ and its radial excitations [13] (Appendix B). In this case the shift in the intercept is rather small since ∆M 2 ρ = ∆M 2 ω = M 2 π ± and M 2 π 4λ 0.02.

A. Strange quark form factor
In contrast to the two-step convolution expansion of the fluctuation model, F s 1 (Q 2 ) and s(x) − s(x) from LFHQCD can be obtained directly from higher-twist terms in the Fock state expansion by matching to the quark degrees of freedom. To this end, let us recall that for the up and down quark form factors the ρ-trajectory is relevant because it dominantly couples to uū and dd quark currents in the proton [57]. Likewise, we compute F s 1 (Q 2 ) in the holographic framework by considering the Regge trajectory of the φ meson, which is nearly a pure ss state [66], and therefore couples dominantly to the ss sea current in the nucleon.
To determine the slope and intercept of the φ trajectory, we fix the ρ intercept from the pion mass and find the best value for the universal Regge slope from the simultaneous fit of the ρ and φ trajectories; this procedure determines the φ intercept and the universal slope α = 1 4λ . We obtain √ λ = 0.534 GeV and α φ (0) = 0.01, or equivalently The φ − f trajectory is shown in Fig. 5. One can also compute the intercept in LFHQCD with effective quark masses, see Appendix B, the value is α φ (0) = 0.00 ± 0.04. The value of ∆M 2 φ is significantly larger than ∆M 2 ρ due to the presence of the more massive strange quarks in the φ meson.
Since the light-front holographic framework is inherently relativistic, the LFWF for a state with twist-τ automatically incorporates Fock state components with two different orbital angular momenta L z and L z + 1, in analogy to the upper and lower components of a Dirac 4-component spinor. For example, the valence quark distributions of a nucleon correspond to a leading twist-3 effective LFWF with orbital angular momentum L z = 0, plus a twist-4 term corresponding to a three-quark effective LFWF with L z = 1. Note that Fock states with both L z and L z + 1 are needed in order that a baryon can have a nonzero Pauli form factor and a nonzero anomalous magnetic moment [39].
The five-quark state |uudss is the lowest Fock state which contains strangeness. Therefore, the leading contributions to the strange form factor are terms with twist-5 and twist-6. Using the constraint F s 1 (0) = 0 from the sum rule (1), the analytic structure of F s 1 (Q 2 ) is uniquely determined by the holographic structure up to twist-6: where we have allowed for a small φ − ω mixing η in the strange form factor [40]. N s is a normalization factor and F ω,φ τ (Q 2 ) is the twist-τ form factor (40) with Regge trajectory α ω,φ (t) given by (49) and (51) respectively. The form factor can also be expressed as a product of τ − 1 poles located at t = −Q 2 = 4λ n + 1 2 + ∆M 2 ω and t = −Q 2 = 4λ n + 1 2 + ∆M 2 φ , n = 0, 1, 2 · · · τ − 2.
One thus obtains in this case the form factor poles at the mass of the ω and φ vector meson and its radial excitations. To illustrate the effect of the φ − ω mixing we show in Fig. 6 the effect of a 10 % mixing in . The effect of the small mixing turns out to be negligible for F s 1 (Q 2 ). We also show in Fig. 6 the chiral limit for massless quarks. Since the quark mass effect is very small in the ω trajectory, this chiral limit corresponds to a pure ω trajectory.
Note that the normalization factor N s in (52) is not the intrinsic strange/antistrange quark number I s , since the strange and antistrange distributions can both have twist-5 and twist-6 contributions. However, the shape of F s 1 (Q 2 ) is completely determined from the structure of LFHQCD. The result is shown in Fig. 2, together with predictions from the fluctuation model and lattice QCD.
The value of √ λ = 0.534 GeV and the mass shift ∆M 2 φ = 1.96 λ are obtained from the φ trajectory depicted in Fig. 5. The value of N s = 0.047 in Fig. 2 is determined by a best fit to lattice QCD predictions. As in the case of the fluctuation model, we also fit the lattice QCD data, taking √ λ and N s as free parameters. The result is shown in Fig. 3 with parameter values √ λ = 0.52 (17) GeV and N s = 0.046 (17). This value of √ λ agrees with that determined from the Regge trajectory. The conformal limit results, ∆M 2 = 0, are also shown in the figures for comparison. The strange form factor (52) has the large-Q 2 behavior Q 8 F s 1 (Q 2 ) → Const, with Const = 1680 N s λ 4 0.5 GeV 8 , consistent with the scaling predicted from the hard-scattering counting rules [64,65].

B. Strange quark distribution functions
To describe the quark distribution functions in the holographic formalism it is convenient to express the Beta function (41) in a reparametrization invariant form provided that w(x) satisfies the constraints [14] w(0) = 0, Therefore, using (53) and the Regge trajectory, (48), (49) or (51), the EM form factor (40) for twist-τ can be written in the invariant form The EM form factor can also be expressed by the exclusive-inclusive connection as the integrated expression of the t-evolved PDF, namely, the generalized parton distribution (GPD) at zero where f (x) is the profile function and q τ (x) is the collinear PDF of twist-τ . Comparing (56) with the holographic expression (55) we find that both functions, f (x) and q τ (x), are determined in terms of the reparametrization function of the Beta function, w(x), by where q τ (x) is normalized by 1 0 dx q τ (x) = 1. In the conformal limit where the quark masses vanish, ∆M 2 → 0, we recover the results given in Ref. [14].
The specific function w(x), taken from Ref. [14], is effectively determined by Regge behavior at small-x and the local power-law counting rule at x → 1. At x → 0, w(x) scales as w(x) ∼ x to recover Regge behavior [67]. At x → 1 the additional constraints yield the Drell-Yan counting rule q τ (x) ∼ (1 − x) 2τ −3 at large x [41]. Since w(1) = 1, it follows that log 1 w(x) → 0 in the limit x → 1, which implies that the local counting rules at large-x are unmodified by the introduction of quark masses in the holographic structural framework. However, the squared mass shift induced by finite quark masses does modify the small-x behavior by a factor x ∆M 2 /4λ , therefore softening the Regge behavior of the PDFs at small-x since w(x) in (58) scales as w(x) ∼ x at small-x. Since ∆M 2 is considerably larger for strange quarks than for the up and down quarks, the predicted behavior of the strange sea distributions is less singular at x → 0 than the nonstrange light quarks.
It has been noted in the pre-QCD era that the behavior of parton distributions near x → 0 is governed by the Regge intercept [60]. This is again in agreement with LFHQCD even including the finite quark mass correction. The t-dependence of GPDs, instead, is not influenced by the introduction of quark masses, since the Regge slope is universal for light hadrons [33].
The expression for the strange-antistrange PDF asymmetry s(x) −s(x) corresponding to (52) is with q ω,φ τ (x) given by (58) for ∆M 2 ω and ∆M 2 φ respectively. For the universal reparametrization function w(x) we use the form in Ref. [14], with a = 0.531 determined from the first moment of proton valence quark distributions. The effect of the φ − ω mixing for the s(x) −s(x) asymmetry also turns out to be negligible for a mixing of the order of 10% and will be neglected.
The PDF predictions for the asymmetry s(x) −s(x) are shown in Fig. 4 and compared with the fluctuation model and global fits for N s = 0.046 (17) and √ λ = 0.52 (17) GeV obtained from the lattice form factor results. The actual computations are carried out with the universal function w(x) given by (62). In contrast to the baryon-meson fluctuation model, which has the small-x behavior s(x) −s(x) → 0, the holographic model has the Regge behavior s(x) −s(x) −0.044x −0.01 in the limit x → 0. This can be compared with the global data fit results, shown in Fig. 4 at the initial scale µ = 1 GeV.
From our analysis, the lattice QCD result favors a positive S − . However, the fits with the fluctuation model and LFHQCD yield S − = 0.0011(4), which is not sufficient to solely explain the NuTeV anomaly; other sources are needed. Although the value for S − is model dependent, we emphasize that more precise determinations of F s 1 (Q 2 ) from first-principle lattice QCD calculations and/or future experiments will provide important constraints on the strange-antistrange asymmetry.

C. Separation of strange and antistrange asymmetric quark distributions
Light-front holographic QCD predicts the structural behavior of the strange asymmetry (61) up to twist-6, but it does not directly predict the individual distributions s(x) ands(x) which together determine the intrinsic strange contribution to the quark sea in the nucleon dx s(x) = dxs(x) = I s .
We will show, however, how one can uniquely determine the minimum strange probability I s in the proton and then give constraints on the separate s(x) ands(x) distributions.
We expand the longitudinal quark distributions s(x) ands(x) into their twist-5 and twist-6 corresponding to L z = 0 and L z = 1, respectively. Comparing with (61) and using the sum rule (1), we find with the general solution We can thus write with α an arbitrary parameter constrained by the conditions s(x) ≥ 0 ands(x) ≥ 0. Since the twist-5 term dominates at large-x we require α ≥ 0 and γ ≥ 0. For positive N s , the positivity constraints lead to α ≥ N s . At small-x we have the behavior with N τ defined in (40). In the conformal limit, ∆M 2 = 0, we have R = 8 9 . Incorporating quark masses, ∆M 2 φ = 1.96 λ, we have R = 0.80. This small-x behavior leads to the condition I s ≥ (1 − R)α from Eq. (70). Together with α ≥ N s we have the condition Because the ratio q τ =5 (x)/q τ =6 (x) is monotonically increasing, the condition (76) ensures s(x) ≥ 0 ands(x) ≥ 0 over the full range of x.
The solution which minimizes the strange sea probability corresponds to α = N s and I s = We show in Fig. 7 the holographic results for the individual quark distributions s(x) ands(x).
The results correspond to the lower bound I s = 0.92%. As we discussed in Sec. II, the strange distribution s(x) should have its support for larger values of the longitudinal momentum x, as compared withs(x), to lead to negative s(x) −s(x) asymmetry at small-x and to a positive asymmetry at large-x. This important property is verified for the holographic quark distributions shown in Fig. 7. One can observe in Fig. 7 (left) that the high-twist suppression at large-x from local counting rules is significant for the s(x) leading-twist-5 distribution above x ∼ 0.7 and for thes(x) twist-6 distribution above x ∼ 0.6.
The positive form factor F s 1 (Q 2 ) obtained from the lattice calculations [5,6], shown in Fig. 2, requires that the strange quarks are more concentrated at small transverse separation compared with the antistrange quarks (See Sec. II). As shown in Fig. 8 this is indeed the case for the LFHQCD results computed from the coordinate space transverse distribution given by Eq. (18).

DISCUSSIONS AND CONCLUSIONS
In this article, we have demonstrated that a nonzero strangeness contribution to the spacelike electromagnetic form factor of the nucleon F s 1 (Q 2 ) = 0 implies a strange-antistrange asymmetry in the nucleon's light-front wave function and thus in the nucleon PDF.
A lattice QCD calculation predicts a positive strange quark form factor, which indicates that the strange quark distribution is more centralized in coordinate space than the antistrange quark distribution. Consequently, the strange quark distribution is more spread out in momentum space.
The lattice result thus indicates a negative s(x) −s(x) longitudinal momentum distribution at small-x and a positive distribution at large-x.
We have shown how the baryon-meson fluctuation model leads to a nonzero strange quark form factor of the nucleon, and a s(x) −s(x) asymmetry. Imposing the lattice QCD data, we have analyzed the constraints on the model, leading to 1.06(51)% intrinsic strange sea quark probability in the nucleon.
We have also discussed a new model for the intrinsic sea-quark distributions based on light-front holographic QCD. The strange quark form factor and the s(x) −s(x) asymmetry are determined in this framework up to a normalization factor, which can be constrained by the lattice prediction.
Effects from the finite quark masses of the vector mesons which couples to the quark current in the nucleon have also been discussed. Remarkably, the holographic structure of form factors and PDFs allows the introduction of quark masses without modifying the hard scattering counting rules, the local counting rules, or the t-dependence of GPDs. The small-x behavior modified by quark masses is still governed by the Regge intercept. Since the strange quark mass is much greater than up and down quark masses, the strange quark distributions at small-x in LFHQCD is less singular than up and down quark distributions. By incorporating the positivity bound on quark distribution functions, we have derived a lower bound for the intrinsic strange sea probability using the holographic approach. The lower bound is 0.92%, compatible with the value found in the fluctuation model; however, the intrinsic strangeness probability contributing to s(x) +s(x) can be significantly larger. We have also evaluated the individual s(x) ands(x) distributions and the coordinate-space transverse distributions for the strange and antistrange quarks in the nucleon for the intrinsic strange quark probability determined by the lower bound. The result supports the qualitative analysis that the strange quark is more concentrated at small transverse separation than the antistrange quark. This novel nonperturbative approach to sea quark distributions presented here, based on the light-front holographic framework, complements the physical picture inherent to the meson-baryon fluctuation model, and gives new insights into both the structure of the strangeantistrange asymmetry and the strange form factor of the nucleons. This approach can also be extended to the study of intrinsic charm and bottom.
the largest momentum transfer available on the 48I ensemble is Q 2 ∼ 0.31 GeV 2 . We note that the uncertainties in the extrapolation of the nucleon strange electromagnetic form factor become very large and the form factors are consistent with zero above Q 2 ∼ 0.7 GeV 2 for the 48I ensemble and therefore the extrapolations of the 48I ensemble electromagnetic form factor data were constrained up to Q 2 = 0.5 GeV 2 in the global fit (a simultaneous fit in lattice spacing, volume and pion mass) in Ref. [6]. It is important to note that the lattice QCD estimate of G s E,M (Q 2 ) in Ref. [6] is the most precise and accurate first-principles calculation of s-quark EMFFs to date. This is the only calculation at the physical pion mass where the quark mass dependence, as well as finite lattice spacing (a), volume corrections, and partial quenching effect (when the valence and sea quark masses are not the same in lattice QCD simulation) were considered.
After obtaining Q 2 -dependence from the z-expansion fit to the lattice data, for a given Q 2 -value, we obtain 24 data points corresponding to different valence quark masses from 3 different lattice spacings and volumes and 4 sea quark masses including one at the physical point. We use the chiral extrapolation formula from Ref. [78] and volume correction from Ref. [79], yielding a global fit in different quark masses, lattice spacings, volumes of the strange quark Sachs electric form factor at a given Q 2 . It is given by G s E (m π , m K , m π,vs , a, L) = A 0 + A 1 m 2 K + A 2 m 2 π + A 3 m 2 π,vs + A 4 a 2 + A 5 √ Le −mπL , where m π /m K is the valence pion/kaon mass and m π,vs is the partially quenched pion mass m 2 π,vs = 1/2(m 2 π + m 2 π,ss ) with m π,ss the pion mass corresponding to the sea quark mass. The χ 2 /d.o.f. for different Q 2 global fits ranges between 0.7-1.13. For example, in the continuum limit, the global fit for Q 2 = 0.25 GeV 2 provides the physical value of G s E | phys = 0.0024(8), A 1 = 0.58 (30), A 2 = −0.29 (15), A 3 = −0.003(9), A 4 = 0.001 (2), and A 5 = −0.001(3) with χ 2 /d.o.f. = 1.1. One could also consider a log(m K )-term in the chiral extrapolation of G s E as shown in [78], however our analysis shows that this term does not have any effect on the global fit for our lattice data. A similar vanishing difference has been observed if one considers e −mπL instead of a √ Le −mπL term in the volume correction, where L is the finite box size of a lattice. For example, including the factor log(m K ) and e −mπL instead of √ Le −mπL , one obtains G s E | phys = 0.0026 in comparison with G s E | phys = 0.0024. We include these small effects in the systematics of the global fit results. We also assign a 20% systematic uncertainty from the model-independent z-expansion interpolation coming from adding a higher order term a 3 while fitting the G s E (Q 2 ) data. These uncertainties are added in quadrature to the systematics discussed in [5].
Similarly, we calculate the strange Sachs magnetic form factor G s M at a particular Q 2 using the global fit formula G s M ( m π , m K , m π,vs , a, L) = A 0 + A 1 m π + A 2 m K + A 3 m 2 π,vs + A 4 a 2 + A 5 m π (1 − 2 m π L )e −mπL , where we have used a chiral extrapolation linear in m π and m loop = m K [78,[80][81][82]. For the volume correction we refer to Ref. [83]. From the global fit formula (A3), for example, in the continuum limit at Q 2 = 0.25 GeV 2 , we obtain G s M | phys = −0.018(4), A 1 = 0.04(3), A 2 = −0.18 (12), A 3 = −1.27 (84), A 4 = 0.008 (6), and A 5 = 0.04(5) with χ 2 /d.o.f. = 1.13. From the values of the parameters in the global fit formula (A3), it is seen that the quark mass dependencies play an important role in calculating G s M (Q 2 ) at the physical point. A 9% systematic uncertainty from the model-independent z−expansion and an uncertainty from the empirical fit formula have been included as discussed in [5]. We obtain systematics from the global fit formula by replacing the volume correction by e −mπL only and also by adding a m π,vs term in the fit and include the difference in the systematics of the global fit results.
More details about the lattice analysis can be found in Refs. [5,7].

Appendix B: The vector meson trajectories in LFHQCD
The meson spectrum in LFHQCD is given by [13,33] where the squared mass shift ∆M 2 [m 1 , m 2 ] incorporates the effect from finite light quark masses.
Following the procedure discussed in Refs. [13,33], one can add a correction term of the invariant mass i m 2 i x i to the LF kinetic energy in the LF Hamiltonian, and leave, as a first approximation, the LF transverse potential unchanged. The resulting LF eigenfunction is then modified by the factor e where the m i are effective quark masses.
The longitudinal confinement dynamics in presence of quark masses has also been discussed in Refs. [85,86]. In [86] a specific longitudinal confinement potential is introduced by extending the transverse holographic potential while maintaining rotational invariance in the heavy quark limit.