Nucleon parton distributions from hadronic quantum fluctuations

A physical model is presented for the non-perturbative parton distributions in the nucleon. This is based on quantum fluctuations of the nucleon into baryon-meson pairs convoluted with Gaussian momentum distributions of partons in hadrons. The hadronic fluctuations, here developed in terms of hadronic chiral perturbation theory, occur with high probability and generate sea quarks as well as dynamical effects also for valence quarks and gluons. The resulting parton momentum distributions $f(x,Q_0^2)$ at low momentum transfers are evolved with conventional DGLAP equations from perturbative QCD to larger scales. This provides parton density functions $f(x,Q^2)$ for the gluon and all quark flavors with only five physics-motivated parameters. By tuning these parameters, experimental data on deep inelastic structure functions can be reproduced and interpreted. The contribution to sea quarks from hadronic fluctuations explains the observed asymmetry between $\bar{u}$ and $\bar{d}$ in the proton. The strange-quark sea is strongly suppressed at low $Q^2$, as observed.

A physical model is presented for the non-perturbative parton distributions in the nucleon. This is based on quantum fluctuations of the nucleon into baryon-meson pairs convoluted with Gaussian momentum distributions of partons in hadrons. The hadronic fluctuations, here developed in terms of hadronic chiral perturbation theory, occur with high probability and generate sea quarks as well as dynamical effects also for valence quarks and gluons. The resulting parton momentum distributions f (x, Q 2 0 ) at low momentum transfers are evolved with conventional DGLAP equations from perturbative QCD to larger scales. This provides parton density functions f (x, Q 2 ) for the gluon and all quark flavors with only five physics-motivated parameters. By tuning these parameters, experimental data on deep inelastic structure functions can be reproduced and interpreted. The contribution to sea quarks from hadronic fluctuations explains the observed asymmetry betweenū andd in the proton. The strange-quark sea is strongly suppressed at low Q 2 , as observed.

I. INTRODUCTION
The parton distribution functions (PDFs) of the nucleon are of great importance. One reason is that they provide insights in the structure of the proton and neutron as bound states of quarks and gluons, which is still a largely unsolved problem due to our limited understanding of strongly coupled QCD. Another reason is their use for calculations of cross-sections for high-energy collision processes, factorized in a hard parton level scattering process, calculated in perturbation theory, and the flux of incoming partons given by the PDFs.
This involves the factorization of processes that occur at momentum-transfer scales of significantly different magnitudes. Of particular importance here is that the PDFs f (x, Q 2 ) have the property that for Q 2 > Q 2 0 ∼ 1 GeV 2 the dependence on Q 2 can be calculated by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equations (DGLAP) [1][2][3] derived from perturbative QCD (pQCD), which is well-established theoretically and experimentally confirmed. However, the x-dependence needed at the starting scale Q 2 0 is not known from fundamental principles and instead parametrized to reproduce proton structure function data. This typically requires x-shapes given in terms of five parameters for each parton flavor, resulting in ∼ 30 free parameters to account for valence quarks, gluons and sea quarks (u, d, g,ū,d, s,s). There are different collaborations [4][5][6][7] performing such PDF parametrizations with DGLAP-based Q 2 -evolution that give good fits of proton structure data and are excellent tools for cross-section calculations. However, the basic xdependence at Q 2 0 originating from the bound-state proton is here only parametrized, but not understood. * andreas.ekstedt@physics.uu.se † hazhar.ghaderi@physics.uu.se ‡ gunnar.ingelman@physics.uu.se § stefan.leupold@physics.uu.se To understand the basic shape of the parton momentum distributions in physical terms, we here follow up on earlier studies [8][9][10] giving phenomenologically successful results. The first basic idea is to use the uncertainty relation in position and momentum, ∆x∆p ∼ /2, to give the basic momentum scale of partons confined in the length scale ∆x given by the hadron diameter D. In the hadron rest frame it is natural to assume a spherically symmetric Gaussian momentum distribution with a typical width σ ∼ /(2D). The Gaussian form is not only a convenient mathematical form which cuts off large momenta that correspond to rare fluctuations. It can also be motivated as a natural distribution resulting from many soft interactions within the hadron that, via the law of large numbers, add up to a Gaussian. The strength of this approach lies in its simplicity and its small number of parameters.
The second basic idea is that whereas the valence quark and gluon distributions are essentially given by the bare nucleon, the sea quark distributions are given by the hadronic fluctuations of the nucleon. For example, the proton quantum state |P = α bare |P bare + α P π 0 |P π 0 + α nπ + |nπ + + · · · contains not only the bare proton but also nucleon-pion fluctuations with probability amplitudes α N π . The point is that one should consider the dominant quantum fluctuations in terms of least energy fluctuation and thereby most long-lived [11,12]. It is expected that pionic fluctuations dominate due to the small mass of the pion. In turn, its smallness compared to a typical hadronic scale ∼ 1 GeV is a consequence of spontaneous chiral symmetry breaking, which leads to the identification of pions as Goldstone bosons [13,14]. From these dominant fluctuations in the proton, with the presence of π + but lack of π − , one expects an asymmetry in the proton sea such thatd >ū [8,10,[15][16][17][18], as is also observed in data [19].
This kind of hadronic fluctuations has earlier [8,10] been handled by having the different baryon-meson (BM ) fluctuation probabilities |α BM | 2 as free parameters fitted to data. Here, we use the leading-order La-grangian of three-flavor chiral perturbation theory [20][21][22][23] to develop a theoretical model for the proton state (1) The different terms are here theoretically well defined and related to each other with only three coupling constants that are known from hadronic processes and weak decays of baryons. In addition to the probability for the different hadronic fluctuations, the theoretical formalism gives the hadron momentum distribution of the fluctuations. Incorporating the hadronic momentum distributions with the above parton momentum distributions in a hadron provides an improved model for the parton momenta of the proton quantum state.
The PDFs are closely related to the proton structure functions that are measured in deep inelastic scattering (DIS) of leptons on protons. The most precise data are from electron and muon scattering, where the exchanged virtual photon has high resolution power and couples to quarks in the proton. The photon may therefore couple to a quark in the bare proton or in either the baryon or the meson in a baryon-meson fluctuation.
In this paper we present the complete model we have constructed based on these basic ideas. Section II presents the formalism for DIS on the proton with its hadronic fluctuations, where some more technical details are provided in appendices at the end of the paper. In Section III we present our model for the parton distributions in a probed hadron, i.e. the x-shape at the starting scale Q 2 0 for pQCD evolution. Results are then presented in Section IV in terms of obtained parton momentum distributions and their ability to reproduce data on proton structure functions and quark sea asymmetries. We give our conclusions in Section V.

II. DIS ON A NUCLEON WITH HADRON FLUCTUATIONS
The cross-section for deep inelastic lepton-nucleon scattering is theoretically well known as a product of the leptonic and hadronic tensors, dσ ∝ l µν W µν . The leptonic tensor is straightforward to calculate and well known for photon exchange, l µν = tr[ / p l γ µ / p l γ ν ]/2, as well as for W or Z exchange. We consider both electromagnetic and weak interactions.
The hadronic tensor W µν is a much more complex object and is of prime interest here. In order to take into account the proton target with its hadronic fluctuations, as illustrated in Fig. 1, we decompose the hadronic tensor to include the possibilities to probe either the bare proton or the meson or baryon in a fluctuation as follows (2) where the notation M B and BM denotes probing the meson and baryon, respectively. The general form of the hadronic tensor is [24] in terms of the hadronic current J µ (ξ) as a function of the spacetime coordinate ξ. Using light-cone time-ordered perturbation theory [25] we calculate the here introduced part corresponding to the hadronic fluctuations giving where the first term is for DIS probing the meson (M ) and the second term for probing the baryon (B). Ex-pressions equivalent to (4) can be found in the literature [18].
The integration variable is the fraction y of the proton's energy-momentum carried by the meson or baryon. Following common practice in DIS theory we use lightcone momenta p + = p 0 + p 3 and p − = p 0 − p 3 , and thereby y i = p + i p + (i = B, M ). This has the advantage of being independent of longitudinal boosts, e.g. from the proton rest frame to the commonly used infinitemomentum frame. The light-cone momenta p − i are given by the on-shell condition which in the p ⊥ = 0 frame be- In (4) the sum runs over all baryon-meson pairs, with helicity λ of the baryon. We have included all baryons in both the octet and decuplet of flavor SU (3), and all the Goldstone bosons represented by the mesons in the spinzero octet. Naturally, the fluctuations with a pion will dominate due to its exceptionally low mass. Kaons are needed to get the leading contribution for the strangequark sea. Table I shows the relative strengths of different fluctuations due to the couplings to be discussed further below.
The dynamical behavior depends on the hadronic dis-tribution functions which are probability distributions for the physical proton to fluctuate to a baryon-meson pair. The baryon carries a light-cone fraction y and the meson the remaining momentum fraction, i.e. satisfying the relation giving flavor and momentum conservation for each particular hadronic contribution. This ensures that all parton momentum sum rules come out correctly [26]. The hadronic distribution functions are explicitly given in Eqs. (12,13). Their explicit form depends on the Lagrangian used for the hadronic fluctuations, to which we now turn. The relevant part of the leading-order chiral Lagrangian describing the interaction of spin 1/2 and spin 3/2 baryons with spin 0 mesons (as Goldstone bosons) is given by [20][21][22][23] where 'tr' refers to flavor trace. Here, the B ab are the matrix elements of the matrix B representing the octet baryons. The decuplet baryons are represented by the totally symmetric flavor tensor T µ abc . Similarly, the spin 0 octet mesons are represented by a matrix Φ appearing in the Lagrangian through u µ given by u µ ≡ iu † (∇ µ U )u † = u † µ where u 2 ≡ U = exp(iΦ/F π ). For further details see Appendix A.
From this Lagrangian we derive the non-zero terms when applied to our cases of a proton fluctuating into a meson together with an octet or decuplet baryon and respectively. The effective nature of the hadronic theory -manifested by the appearance of the derivative couplings of the form ∼ γ 5 γ µ ∂ µ M (z) in the Lagrangians-introduces a slight ambiguity for the meson momentum p M appearing in the numerators in the application of the light-cone time-ordered framework. In the literature, there are two common choices for the meson momentum appearing in the numerators [15,18], We find that these two choices give nearly identical results concerning the extracted values of our model's parameters and hence both choices yield similar conclusions. But even though choice (10) gives a slightly better shape for the flavor asymmetry, to be discussed in Section IV C, we will use choice (11) since this choice is in line with the Goldstone theorem [27] whereas choice (10) is not, as explicitly shown in Appendix B.
As discussed in Appendix A, the parameter values are as follows [28]. The pion decay constant F π = 92.4 MeV and the couplings D = 0.80, F = 0.46 [29] and h A = 2.7 ± 0.3 with an uncertainty range to include partial decay width data on ∆ → N π and Σ * → Λπ as well as the large-N C limit [30,31] where g A = F + D = 1.26 is well constrained by the beta decay of the neutron [32].
Using light-cone time-ordered perturbation theory, the Lagrangians (8,9) lead to the hadronic distribution functions for the baryon and meson, respectively, probed in the fluctuation. As required, they satisfy f BM (y) = f M B (1− y). The various hadronic couplings g BM are provided in Table I and the vertex functions S λ (y, k ⊥ ) are given in Appendix B. The suppression of the energy fluctuation is seen as the propagator with the difference of the squared masses of the proton and the baryon-meson system given by The function G(y, k 2 ⊥ , Λ 2 H ) is a cut-off form factor, which is used to avoid the integral getting an unphysical divergence. The physics issue to account for is the fact that the description in terms of hadronic degrees of freedom is only valid at hadronic scales, whereas for higher momentum-transfer scales parton degrees of freedom should be used. To phase out the hadron formalism it is convenient to introduce a suitably constructed form factor.
In practice, it is conceivable to cut on the virtuality of the fluctuation [15] or on the modulus of the threemomentum (in a proper reference frame). While the first option sounds plausible from a point of view of Heisen-berg's uncertainty relation (or Fermi's Golden Rule), this quantum-mechanical aspect is already accounted for by the just mentioned propagator in Eqs. (12,13). An additional such cut is therefore artificial. Instead we choose to cut off the three-momentum of the hadrons in the fluctuation as seen in the rest frame of the proton. If relevant at all, high-momenta fluctuations should be of partonic not hadronic nature.
To conserve the condition f BM (y) = f M B (1 − y) it is necessary to use a symmetric combination of the meson/baryon three-momentum and a natural choice is to use the average of the squares of the three-momenta of the meson and baryon. To make this manifestly frameindependent we write its value in the proton rest frame expressed in a Lorentz-invariant form and take the form factor to be where Λ H is the parameter that regulates the suppression of larger scales. Since this is related to the switch to partonic degrees of freedom, one would expect it to be of the same order as the starting scale Q 0 of the pQCD formalism. The function A 2 in the form factor is given by  where light-cone momenta p + B = yp + and p + M = (1−y)p + have been used to obtain the last expression. This form factor regularizes any potential end-point (y = 0, 1) singularities. Furthermore, high values of k ⊥ are largely suppressed which renders the integrals in Eqs. (12,13) finite and restricts the hadronic fluctuations to the lowmomentum scales where the hadronic language is applicable.
Using this theoretical formalism we illustrate the total fluctuation probability for a proton to a BM pair by calculating for both momentum choices, Eqs. (10,11), giving the result shown in Fig. 2. One observes that the probability for a proton to fluctuate into a baryon-meson state is quite sizable. Notably, for a cut-off Λ H around 1 GeV, the contribution from the baryon-decuplet members (mainly from the ∆'s) is comparable in size to the nucleon-pion fluctuations.
Due to the hadronic fluctuations, the PDFs for the proton are given by a convolution of the hadronic distributions, Eqs. (12,13), and the PDFs for the hadron being probed. Thus, the PDF for a parton i in the proton can be written in the form [15,18,33] taking into account the contributions from the bare proton and the BM fluctuations. In our approach, the PDF for the 'bare' part in any of these contributions (bare proton, baryon or meson in a fluctuation) is obtained from a Gaussian as mentioned in the Introduction and to be discussed in Section III. These bare distributions contain constituent quarks and gluons, but no sea quarks.
In this work we include all the admissible octetbaryon-meson and decuplet-baryon-meson pairs in the fluctuations, i.e. the |N π , |∆π , |ΛK , |ΣK , and |Σ * K fluctuations. The |P η contribution can be neglected due to mass suppression and its very small coupling to the proton state, see Table I. The nucleon-pion and the Delta-pion fluctuations give the largest contributions, while the |ΛK , |ΣK and |Σ * K fluctuations act as small corrections. However, since neither |N π nor |∆π contribute to the strange sea, other fluctuations like |ΛK while being small are the leading hadronic contributions to the strange sea. It is found that the |ΛK fluctuation is most important, while the |ΣK and |Σ * K fluctuations are suppressed due to a small coupling and the larger masses involved respectively, see Table I.
Once the starting distributions have been obtained from the convolution model at a particular starting scale Q 2 0 , the PDFs are obtained for higher Q 2 by DGLAP evolution. The DGLAP evolution is performed at next-toleading order (NLO) using the QCDNUM package [34].

III. GENERIC MODEL FOR PARTON DISTRIBUTIONS IN A HADRON
For any bare hadron we are considering the parton momentum distributions for its valence quarks/antiquarks and a gluon component. This applies for both the above considered bare proton as well as for the baryons and mesons in a hadronic fluctuation. In the rest frame of such a generic bare hadron there is no preferred direction. Therefore the spherical symmetry motivates the assumption that the parton's momentum distributions in k x , k y and k z are the same. Assuming a Gaussian momentum distribution for these components provides a convenient mathematical form which suppresses large momenta that should correspond to rare momentum fluctuations. It can furthermore be motivated as the natural distribution arising from many soft interactions within the bound-state hadron that produce an additive effect such that, via the law of large numbers, a Gaussian distribution results. Based on the above discussion, the fourmomentum distribution for a parton of type i and mass m i is assumed to be given by [8,10] where N is a normalization factor. The width σ of this Gaussian is expected to be physically given by the uncertainty relation ∆x∆p ∼ /2 that enforces increasing momentum fluctuations for a particle confined in a smaller spatial range. Thus, for a hadron of size D (diameter) one expects σ ∼ /(2D) and therefore being typically of order 0.1 GeV.
Using light-cone momenta, x = k + /p + H will be the convenient energy-momentum fraction (independent of longitudinal boosts) carried by a parton in a hadron. The PDF for a parton i = q,q, g of mass m i in the hadron H is given by The physical conditions of having a kinematically allowed final state impose the constraint m 2 i ≤ j 2 = (k+q) 2 < (p H +q) 2 for the scattered parton to be on-shell or have a timelike virtuality (causing final-state QCD radiation) limited by the mass of the hadronic system. Likewise, the hadron remnant must have a four-vector Fig. 1a. These constraints ensure that 0 < x < 1 and f (x) → 0 for x → 1.
The normalizations N q/H (σ q , m q ) are fixed by the flavor sum rules, i.e. the integrals giving the correct numbers of different valence quark flavors. N g/H (σ g , 0) is fixed by the momentum sum rule, i.e. to get the sum of x-weighted integrals to be unity.
Thus, the only free parameters are the Gaussian widths σ g , σ 1 , σ 2 , where the indices refer to the widths of the distributions for the gluon and for the quark flavors represented by one quark ( With this model we have chosen a minimalistic approach with the same Gaussian distributions for all partons, having a width that only depends on the number of same-flavor quarks, but not on the particular hadron considered. Of course, one could introduce more complexity requiring more parameters, but we find it more interesting to see what insights this minimal physics-motivated model can give. The above parametrization automatically conserves isospin (e.g. f bare u/P (x) = f bare d/n (x) and similarly for the other hadrons). With the above widths for all possible hadrons, the distributions only depend on mass effects via the mentioned kinematical constraints.
It should be noted that these PDFs can be analytically evaluated in terms of error functions [10], but in practice it is more convenient to evaluate them numerically. As discussed, these bare distributions will only contain valence quarks and gluons, whereas the sea distributions will be entirely generated by hadronic fluctuations. All the resulting PDFs are at the low hadronic scale to be used as starting distributions at Q 2 0 for DGLAP evolution to large scales Q 2 .

IV. MODEL RESULTS BASED ON DATA COMPARISON
A. The few adjustable parameters The model introduced above has few parameters which are expected to lie in a limited range in order for the model to make sense.
The description of hadronic fluctuations is controlled by three coupling strengths with values already fixed by data from various hadronic processes. As discussed in connection with Table I above, the coupling g A = F + D = 1.26 is constrained to the 1% level from the beta decay of the neutron [32] whereas D = 0.80 and F = 0.46 may vary independently by ∼ ±5% as long as their sum is fixed [29]. Since it is their sum that appears in the most probable fluctuations, a variation in D and F has a negligible effect on the results. For the decuplet coupling we take h A = 2.7 ± 0.3. Since h A /m R , with m R the resonance mass (basically m ∆ ), appears as the effective coupling in the decuplet Lagrangian (9), we vary the ratio to see the resulting sensitivity on this uncertainty (see Appendix A for details). The only newly introduced parameter in the hadron fluctuation model is the regulator for the highmomentum suppression, Λ H . This parameter is constrained to have a value large enough to allow hadronic fluctuations of some baryon-meson configurations, i.e. energy fluctuations of at least a few hundred MeV. On the other hand, it must be small enough to ensure a separation between the hadronic and partonic degrees of freedom. Thus, a reasonable expectation is a value in the range 0.
The proton structure function F2 as a function of Q 2 for different x-bins. Our model curve compared to data on fixed target µP scattering from the New Muon Collaboration (NMC) [35] and BCDMS [36].
value on the order of 1 GeV. The former is given, as discussed above, by the inverse size of hadrons and the latter by the factorization scale of non-perturbative bound hadron state dynamics from the pQCD description using DGLAP equations for the Q 2 -evolution of parton density functions. In addition, it is reasonable to expect that Q 0 ∼ Λ H . But since Q 0 and Λ H are defined in two different formalisms, the partonic and hadronic respectively, and there is no theoretically well-defined link between these two descriptions, one cannot a priori take them as being the same parameter. Still, as will be seen below they do come out to have the same value within their uncertainties.

B. Comparison with proton structure function data
The values of the just discussed parameters are obtained from inclusive deep inelastic lepton-proton scattering giving the proton structure functions F 2 and xF 3 . Figures 3-5 show µP data from NMC [35] and BCDMS [36], neutrino data from CDHSW, NuTeV and CHORUS [37][38][39] and eP data from H1 [40] in comparison to our model results.
Not all parameters affect the fit to all data sets. The parameters σ 1 , σ 2 , Λ H , and Q 0 can be nailed down using F 2 and xF 3 data. Q 0 and σ g are given by the small-x F 2 data: With Q 0 given, it's always possible to fit data by varying σ g . We find that the following parameter values give the best overall result σ 1 = 0.11 GeV, σ 2 = 0.22 GeV, σ g = 0.028 GeV, Λ H = 0.87 GeV, Q 0 = 0.88 GeV. (22) Notice that the fit results in Λ H and Q 0 being practically the same, confirming our expectation that this scale constitutes the transition from hadron to parton degrees FIG. 4. The proton structure functions (a) xF3 and (b) F ν 2 as function of Q 2 for different x-bins with data from neutrinoscattering experiments CDHS [37], NuTeV [38] and CHORUS [39] compared to our model curves.
of freedom in the model. Moreover, the Gaussian widths are found to be of the expected magnitude ∼ 0.1 GeV. The gluon distribution is particularly soft, which may seem surprising. However, the above argument based on the uncertainty relation gives σ ∼ /(2D) = 56 MeV for the proton charge radius 0.875 fm [32]. In view of the symmetry properties of two-particle wave functions of indistinguishable states it should not be surprising that the momentum distribution for quark flavors that appear singly in the hadron differ from the one for quark flavors that appear pairwise.
Considering the fact that the model has effectively only four parameters, which are also constrained by the physics assumptions of the model, it is remarkable that such a large amount of structure function data can be x H1 x H1 x H1 x H1 1 FIG. 5. The proton structure function F2 as a function of x for various Q 2 -bins. Our model curve compared to data from the H1 eP collider experiment [40].
reasonably well described. Admittedly, there are some kinematical regions of some experimental data sets where deviations do occur, but the general behavior is reproduced and substantial (x, Q 2 ) ranges are well fitted. It is therefore of interest to look into some details on the x-shapes of individual parton densities as they emerge from the model including both the hadronic fluctuations and the probed hadron's generic parton density description, but without any pQCD evolution. This is shown in the top panel of Fig. 6, where the overall shape of the valence quark distributions is quite similar to conventional PDF parametrizations. The gluon is quite large for smaller x. The sea quarks are suppressed, but not at all negligible. So there is a non-trivial contribution of non-perturbatively generated sea quarks in the boundstate proton. Examining the sea quark distributions one notes the different distributions forū andd, on the one hand, and for s ands, on the other. This is the basis for asymmetries in the light sea and strange sea, as will be further discussed below.
The effect on the PDFs from pQCD evolution using the DGLAP equations is shown in the middle and lower panels of Fig. 6. Due to the log Q 2 -dependent evolution there is a quick increase from Q 2 0 so that already at Q 2 = 1.3 GeV 2 the perturbatively generated sea quarks and gluons dominate at small x over the originally nonperturbative sea.
The PDFs obtained at the starting scale Q 2 0 are evaluated numerically. However, for illustrative purposes the starting distributions for a parton i can be parametrized in the convenient form xf i (x) = a x b (1 − x) c . The fitted coefficients for the various distributions are given in Table II.

C. Thed-ū asymmetry
From a pQCD point of view, the momentum distribution of thed andū sea in the proton should be similar since m u , m d Λ QCD , Q 0 . This is, however, not the case as seen in data from e.g. [19] where a clear asymmetry is seen (cf. Fig. 7). Such an asymmetry arises naturally from hadronic fluctuations of the proton where the nonperturbative sea distributions are dominantly generated by the pions [8,10,[15][16][17][18]. The energy-wise lowest fluctuations are P π 0 and nπ + , where the former does not contribute to thed-ū asymmetry since the π 0 is symmetric indd andūu. Taking only these nucleonic fluctuations into account gives already decent agreement with data on the difference xd − xū as shown by the dotted curve in Fig. 7 (upper panel). However, these nucleonic fluctuations are not sufficient to explain the ratiod(x)/ū(x) as shown by the dotted curve in the lower panel of Fig. 7.
The results become better when also including fluctuations with other baryons. In particular the |∆ ++ π − state, having the largest decuplet coupling (see Table I) and havingū in the π − , contributes significantly to bring the curves down to the data points. The full octet and decuplet contribution is shown in Fig. 7 where the band represents a variation in the decuplet coupling h A /m ∆ , with the largest (smallest) value in Eq. (21) corresponding to the solid (dashed) curve. As seen in the figure an n% variation in the coupling results in an n% variation in the difference xd − xū for small x 0.15. The variation has a slightly smaller impact on the ratiod/ū, but the variation is essentially of the same order of magnitude as that of xd − xū. Due to the possibility of the proton to fluctuate into |ΛK + , |ΣK and |Σ * K states a non-perturbative strange sea will arise, as shown in Fig. 6. It is suppressed relative to the light-quark sea partly due to the kinematical suppression of these fluctuations with higher-mass hadrons, but also due to the smaller hadronic couplings shown in Table I. Moreover, the x-distributions of s ands are not the same, but s has a harder momentum distribution thans [9]. This is a kinematical effect arising from the fact that the s quark is in the baryon which, due to its higher mass, gets a harder y-spectrum in the hadronic fluctuation than the lighter meson containing thes. The dominance of kaons in the low-x region and similarly the dominance of strange baryons in the higher-x region is clearly seen in the ratio (s −s)/(s +s) in Fig. 8. Here one can also see how the additional symmetric ss from g → ss in pQCD reduces this ratio with increasing Q 2 . Since pQCD fills up the low-x region to a higher degree, the kaon effect is more depleted than the 'baryon peak', which is, however, shifted to lower x. The symmetric ss sea from the log Q 2 DGLAP evolution builds up quickly and dominates at small-x already for Q 2 = 1.3 GeV 2 , as shown in the middle panel of Fig. 8. Thus, the asymmetry is only expected to be visible at quite low Q 2 and therefore hard to observe experimentally.
The extraction of the strange sea from data is not at all trivial since it requires some additional observable to signal that an s ors has been probed. In Fig. 9 our model is compared to data on the total strange sea (xs(x) + xs(x)) /2. The CCFR data [41] are obtained from neutrino-nucleon scattering producing a charm quark decaying semileptonically giving an opposite sign dimuon signature, i.e. ν µ + N → µ − + c + X where c → s + µ + + ν µ orν µ + N → µ + +c + X wherec →s + µ − +ν µ . The charged-current subprocess W + s → c or W −s →c is here the essential point. Other sources of charm production, such as W + g → cs or W − g →cs, or other sources of dimuon production from other decays must be taken into account to extract a proper measure of the strange sea, as discussed in [41]. The result shows that although the shape difference between the xs(x) and xs(x) distributions is consistent with zero, it has large uncertainties. CCFR assumed xs(x) = xs(x) for extracting the data points shown in  [41,42]. The CCFR analysis assumes xs(x) = xs(x). Fig. 9.
The more recent result of HERMES [42] is obtained from data on the multiplicities of charged kaons in semiinclusive deep-inelastic electron-proton scattering. This requires a detailed and non-trivial analysis of the fragmentation function into kaons to extract the contribution from initial-state strange quarks in the basic DIS process γs → s or γs →s. As seen in Fig. 9 the CCFR and HERMES results differ substantially and do not provide a clear result on the strange sea. Our model result agrees reasonably well with the HERMES result, but compared to CCFR it has a too small strange sea at low Q 2 . Since the strange-quark sea is not yet well determined, we contribute with some further investigations.
The strange-quark content of the proton can be characterized by the momentum fraction carried by the strange sea relative to the light-quark sea or the non-strange quark content [41] where κ = 1 would mean a flavor SU(3) symmetric sea. These ratios are shown in Fig. 10 versus Q 2 , where the qualitative behavior is understandable within our model. At Q 2 0 there is, as discussed, only a small nonperturbative strange-quark sea from hadron fluctuations. With increasing Q 2 the perturbative log Q 2 evolution first builds up the ss sea quickly and then flattens off at larger scales (note the logarithmic Q 2 scale in the figure).
The proton sea is, however, not flavor SU(3) symmetric as indicated by the value of κ and quantified by the strange-sea suppression factor Our results for this quantity are shown in Fig. 11 together with ATLAS data [43,44]. As seen for Q 2 slightly larger than the starting value for the QCD evolution Q 2 0 , the suppression factor is constant and near unity for x 0.01. For low x this is in agreement with the eP W Z-fit of [43]. For larger x our model gives r s 0.023, 1.9 GeV 2 ≈ 0.62, which is consistent within uncertainties of experimental observations: r s (0.023, 1.9 GeV 2 ) = 0.56 ± 0.04 [45], r s (0.023, 1.9 GeV 2 ) = 1.00 +0. 25 −0.28 [43] and r s (0.023, 1.9 GeV 2 ) = 0.96 +0. 26 −0.30 [44]. As seen in Fig. 11, r s → 1 as x → 0, this supports the hypothesis that the quark sea at low x is flavor symmetric. For completeness we show in Fig. 12 (top panel) the dependence of the strange-sea ratios κ and η on the hadron fluctuation regulator Λ H . Whereas κ strongly depends on Λ H , η is almost independent of Λ H . This can be understood from the plot in the lower panel of the same figure which compares the non-strange fluctuation probability P ns (e.g. for |N π and |∆π ) and the probability P s that the proton fluctuates into a hadron pair that does contain strangeness. Not only is P s P ns , but also its slope is much smaller implying that for increasing Λ H the rate of population growth is much larger for those fluctuations that containū andd quarks, than those containing s ands quarks (∆P s /∆Λ H ≈ 5% GeV −1 and ∆P ns /∆Λ H ≈ 90% GeV −1 between 0.5 GeV ≤ Λ H ≤ 1.0 GeV). Hence κ depends much more strongly on Λ H than does η due to appearance ofū andd distributions in its definition. As shown in the lower panel of Fig. 12, at the regulator value of Λ H = 0.87 GeV, roughly 1% of the fluctuations contain strangeness. This can be compared to the result obtained in Ref. [10], where the strangeness fluctuations had to constitute 5% in order to reproduce the then available CCFR data.
If it turns out to be a need for a larger non-perturbative strange-quark sea than in our present model, this might be remedied by a minor modification of the model. One option could be a flavor-dependent momentum cutoff Λ H , but to keep our model as simple as possible we refrained from introducing more parameters. An alternative explanation might come from the importance of additional degrees of freedom not considered so far. In the strangeness S = −1 sector there are four baryonic states below the antikaon-nucleon threshold: Λ, Σ, Σ * (1385) and Λ * (1405). The first three have been taken into account in our approach as the strangeness counterparts of the nucleon and the ∆(1232) considered in the pion-baryon fluctuations. But we have not included the Λ * (1405) in our framework. On the one hand, we found that the comparatively heavy KΣ * (1385) fluctuation is much less important than the lighter KΛ. This suggests that also KΛ * (1405) is negligible. On the other hand, the negative-parity Λ * (1405) couples with an swave to nucleon-antikaon while all our interactions are of p-wave nature. This can enhance the importance of the Λ * (1405). The ultimate reason why we have not explored its influence in the present work is the absence of unambiguous experimental information about the coupling strength between a nucleon and KΛ * (1405). This is related to the long-standing question about the nature of the Λ * (1405). Being lighter than all non-strange baryons with negative parity, it has been speculated since a long time [46] that the Λ * (1405) is merely an antikaon-nucleon bound state instead of a three-quark state; see, for instance [47] for further discussion and references. This would point to a relatively large coupling strength. Yet in view of these theoretical uncertainties we have not pursued a detailed analysis of the KΛ * (1405) fluctuation as long as there is no clear need for an enhancement of the strange sea.

V. CONCLUSIONS
This study has demonstrated that the momentum distribution of partons in the proton, and thereby the observed proton structure functions, can be understood in terms of basic physical processes. We thereby obtain new knowledge regarding the poorly understood nonperturbative dynamics of the bound-state proton. Using the well-established pQCD DGLAP equations for the Q 2 -dependence above the scale Q 2 0 , our model developed here addresses the basic shape in the distribution of the energy-momentum fraction x carried by different parton species at Q 2 0 . Thus, the model treats the physics at the transition from bound-state hadron degrees of freedom to the internal parton degrees of freedom. It does so by convoluting hadronic quantum fluctuations with partonic fluctuations. To describe the former we use the leading-order Lagrangian of chiral perturbation theory, the low-energy effective theory that respects the symmetries of QCD as the underlying theory. The partonic fluctuations arise quantum mechanically due to confinement within the small size of a hadron, as given by the uncertainty relation in position and momentum.
Interestingly the fit that gives best agreement with the structure functions F 2 and xF 3 data yields a value where the hadronic language ends and QCD evolution begins to be Λ H = Q 0 = 0.87 GeV.
Thus, having a model with effectively only four dimensionful parameters with physically meaningful values, it is highly non-trivial that we obtain a very satisfying reproduction of a large amount of data. In particular we find that the nπ + and the ∆π fluctuations generate the flavor asymmetry xd−xū > 0 in the proton sea, to a large extent consistent with experimental data. This shows that the model captures the essential physics observed.
Regarding the strange-quark sea of the proton which arises from proton fluctuations into strange hadrons, we find that it is substantially suppressed due to the larger masses of strange hadrons. An asymmetry in terms of different x-distribution for s ands, with s(x) being harder, is found. However, this effect is reduced at larger Q 2 due to the development of the symmetric ss sea from g → ss in pQCD. The remaining asymmetry at observed Q 2 is too small to be seen in present data. Further details of the non-perturbative strange sea of our model are given to promote future studies, including the potentially interesting inclusion of the Λ * (1405) in the hadronic fluctuations.
We have here considered PDFs of the proton where most experimental information is available for testing our model. The model is, however, quite general and can give the parton momentum distributions in any hadron. Based on the phenomenological success of the model and its theoretical basis, spin degrees of freedom and the proton spin puzzle are studied in another paper [48].

ACKNOWLEDGMENTS
We acknowledge helpful discussions with C. G. Granados and U. Aydemir at an early stage of this project. This work was supported by the Swedish Research Council under contract 621-2011-5107.

Appendix A: The relevant Lagrangians
For the metric we use g = diag(+1, −1, −1, −1) and 0123 = +1. The relevant part of the leading-order chiral Lagrangian describing the interaction of Goldstone bosons with nucleons and spin 3/2 baryons is given by [20][21][22][23] (A1) Relativistic Rarita-Schwinger fields exhibit some problematic features related to how to handle its spin-1/2 components. Apart from exchanging spin-3/2 resonances, the Lagrangian (A1) induces an additional unphysical contact interaction. This can be cured by subscribing to the Pascalutsa prescription, which in our case means making the substitution [21,22] where m R refers to the resonance mass (m R = m ∆ , m Σ * ). Note that this substitution induces an explicit flavor breaking but these effects are beyond leading order. In (A1) B ab is the entry in the ath row, bth column of the matrix representing the octet baryons The Goldstone bosons are contained in and u µ is essentially given by (A5) Finally, the decuplet is represented by a totally symmetric flavor tensor (A6) The couplings we use [28] are F π = 92.4 MeV, D = 0.80, F = 0.46 [29] and h A can be determined from the partial decay width Σ * → Λπ or from ∆ → N π to be h Σ * →Λπ A = 2.4 and h ∆→N π A = 2.88. (A7) In the large-N C limit [30,31], one also gets (N C = number of colors) where g A = F + D = 1.26. We will explore the range h ± A = 2.7 ± 0.3.
Notice that after the substitution (A2) it is the ratio h A /m R that appears with each decuplet-baryon-meson term [cf. Eq. (7)]. That is, on the probability level one has schematically T Σ * (Λ H ) + · · · (A10) so that one could vary h A by ∼ 10% for each of the separate decuplet terms keeping the masses as shown but numerically it makes not much of a difference to instead use m R = m ∆ for both terms and vary the ratio between its smallest and largest values 1. (A11) The effect of this variation is shown in Fig. 7 and its effects on the probabilities are studied in Ref. [49].

Appendix B: The vertex functions
The functions S λ (y, k ⊥ ) are the amplitudes for a particular hadronic fluctuation of a proton with positive helicity and we calculate them using (on-shell) light-front spinors and the Lagrangian of Eq. (7). These amplitudes were calculated for a wide variety of hadronic fluctuations in [15]. Our results are basically the same with minor differences due to a different choice of Lagrangian. Apart from different normalizations, our results for the functions S λ (y, k ⊥ ) agree with those found in [33]. We now present the vertex functions for both choices of the meson's 'derivative momentum'.