The EMC effect for few-nucleon bound systems in Light-Front Hamiltonian Dynamics

The light-front formalism for a covariant description of the European Muon Collaboration (EMC) effect, already applied to $^3$He, is formally extended to any nucleus, and used for actually calculating the $^3$H and $^4$He cases. The realistic and accurate nuclear description of few-nucleon bound systems, obtained with both phenomenological and chiral potentials, has been properly combined with the Poincare' covariance and macroscopic locality, automatically satisfying both number of particles and momentum sum rule. While retaining the on-mass-shell nucleon structure functions, one is then able to predict a sizable EMC effect for $^4$He, as already observed for $^3$He. Moreover, the impact on the EMC effect of both i) the short-range correlations, such as those generated by modern nuclear interactions, and ii) the ratio between the neutron and proton structure functions has been studied. The short-range correlations generated by retaining only the standard nuclear degrees of freedom act on the depth of the minimum in the EMC ratio, while the uncertainties linked to the ratio of neutron to proton structure functions are found to be very small. These light-front results facilitates ascribing deviations from experimental data due to genuine QCD effects, not included in a standard nuclear description, and initiating unbiased investigations.


Introduction
After forty years of studies, the European Muon Collaboration (EMC) effect [1], i.e. the depletion of the ratio, R A EM C (x), between the electron-nucleus and the electrondeuterium deep inelastic scattering (DIS) cross sections, in the valence region of the Bjorken variable x, is still to be fully understood.Any attempt to explain the DIS electromagnetic (em) response of the nucleus as the incoherent response of the constituent nucleons, assumed to be the only active degrees of freedom (dofs), failed to succeed.A natural question has therefore arisen on how and to what extent the nucleus can be used as a QCD laboratory, with quark and gluon dofs playing a relevant role.A complete answer is still lacking, although progresses have been made, e.g. by evaluating the role of short-range correlations (SRCs) between the nucleons when the nucleus is probed by high-virtuality photons [2,3,4], and also by addressing medium modifications of the nucleons, given the large overlap between their wave functions (see, e.g.Ref. [3,5,6,7,8]).Hopefully, a wealth of complementary information should stem from the study of semi-inclusive and exclusive processes [9], planned for light nuclei in a new generation of experiments at existing [10] and future [11] facilities, like the Electron-Ion Collider (EIC), where measurements with (polarized) 3 He beams is under development.For now, recent 3 He and 3 H DIS data have been taken at JLab by the MARATHON Collaboration [12] allowing a fresh extraction of the neutron to proton structure functions ratio, (SFs) r(x) = F n 2 (x)/F p 2 (x), needed to obtain the flavor decomposition of parton distribution (see also Ref. [13]).This experimental scenario has motivated our efforts to quantitatively analyse the EMC effect.
In Refs.[14,15], we have recently proposed and applied to 3 He a new approach for evaluating the EMC ratio, R A EM C (x).The main features are i) a description of the interacting nuclear system that takes into account Poincaré covariance and macroscopic locality (see [16]) and ii) an accurate description of 3 He wave function, obtained from a realistic nuclear interaction.The chosen relativistic Hamiltonian dynamics is the light-front (LF) one, proposed by Dirac in his 1949 seminal paper [17], being the suitable framework for studying the hadron dynamics on the light-cone (such as the one experimentally investigated in the Bjorken limit).In this way, the correct support of the longitudinal-momentum distribution and both baryon number and momentum sum rules are automatically implemented (see Ref. [18] for a LF approach based on an Ansatz for the nuclear part), without introducing ad hoc normalization constants, and a ratio less than 1 is obtained in the typical interval 0.2 < x < 0.75.Our aim is to embed as many general principles as possible and adopt the most refined description of the nuclear dynamics, still keeping the on-mass-shell nucleon structure functions.As a result, one can fully describe the well-understood nuclear part and leave for further investigations the analysis of QCD effects to be unambiguously ascribed to partonic dofs.In summary, the rationale of our approach is to provide a baseline that considers only dofs adopted for decades to successfully describe the standard nuclear physics, so that one can address the study of genuine QCD effects, like nucleon swelling or exotic quark configurations, without biases generated by the lack of the Poincaré covariance and macroscopic locality.In other words, we re-propose the conventional description of the EMC effect, but with a degree of sophistication of the nuclear description appropriate to modern times.
The present work is mainly devoted to i) extending the LF formalism of our approach to any nucleus, and ii) providing a novel calculation of the 3 H and 4 He EMC effects.The α particle is a strongly bound system, and therefore represents a challenging test-bed.A second aim is the comparison of the EMC ratio for 3 He, 3 H and 4 He calculated by adopting different modern NN and NNN interactions.In particular, we have used i) the NN charge-dependent Argonne V18 [19] plus the NNN Urbana IX [20] (shortly AV18+3N) and ii) two versions of the Norfolk χEFT interactions, NVIa+3N and NVIb+3N, that generate three-nucleon forces within the same framework [21,22,23] (see also Refs.[24,25] for a general introduction to χEFT interactions in nuclear physics).Such a comparison sheds light on the dynamical effect of the SRCs, generated by retaining only the standard nuclear dofs, in the EMC region 0.2 < x < 0.75.In particular, as has long been known (see, e.g., the studies of the y-scaling in the widest range of A in Ref. [26]), the tail of the nucleon momentum distributions is strongly affected by the SRCs, and in principle they could manifest their relevance also at x < 1. Differently from other studies, both experimental and theoretical, which address SRCs in the region x > 1, (see, e.g., Ref. [27] and references quoted there in), here we are interested to see to which extent the EMC ratio is sensitive to the effect of the nuclear interaction, leaving room to an investigation of QCD effects, where partonic dofs not included in a standard nuclear description play the main role, to be presented elsewhere.Finally we have explicitly checked that the ratio of structure functions F n 2 (x)/F p 2 (x) we have extracted in Ref. [14] from the MARATHON Coll.data [12] and the one recently obtained in Ref. [28] by using statistical bootstrap, lead to basically equivalent results for the 4 He EMC effect.This outcome could suggest that the on-mass-shell neutron structure function is reliably related to the proton one, while the latter is shown to be affected by large uncertainties in the region x > 0.7 (see Sect. 18 in the recent Particle Data Group [29]).
In addition to the LF formalism, we exploit the Bakamjian-Thomas (BT) construction of the Poincaré generators [30] for implementing the Poincaré-covariant framework able to embed the successful phenomenology of nuclear physics.Actually, the BT construction allows one to make the nonrelativistic description of nuclei fully compliant with a Poincaré-covariant one, since in this way the Poincaré generators obey the standard commutations rules once the mass operator fulfills well-defined constraints.It turns out that the nonrelativistic mass operators satisfy these conditions, given their dependence on scalar products of the independent momenta at disposal (see Ref. [16]).
The main ingredient of our approach is the LF spectral function [31], P N (κ, ϵ), i.e. the probability distribution of finding a nucleon with LF momentum κ in the intrinsic reference frame of the cluster [1, (A − 1)] and the fully interacting (A − 1)-nucleon system with intrinsic energy ϵ.In particular, by using nonsymmetric intrinsic variables, the internal motion of the interacting (A−1)-nucleon system can be disentangled from that of the struck nucleon and hence one is able to implement the macroscopic locality.Notably, in the Bjorken limit the nuclear structure function can be obtained more directly from the nucleus wave function in momentum space, rather than through the cumbersome LF spectral-function [14].
In the next section of this Letter, we present the formalism adopted.Then, we present the numerical results and the conclusions in the following two sections.

The unpolarized structure function F A 2 (x) for a nucleus
As shown in Ref. [14], in our Poincaré covariant framework and in the Bjorken limit, the spin-independent structure function for the A-nucleon bound system is given by the following convolution where x = Q2 /2mν is the Bjorken variable, with Q 2 = −q 2 and q µ ≡ {ν, q} the virtual photon 4-momentum, ξ B min = xm/M A , F N 2 (mx/ξM A ) is the nucleon structure function, M A the mass of the nucleus and f N 1 (ξ) the light-cone momentum distribution, that reads with n N (ξ, k ⊥ ) the LF spin-independent nucleon momentum distribution, obtained from the LF spectral function, P N (κ, ϵ), after integrating over the energy ϵ and averaging on the spin direction.It can be written as follows Here above, is the squared nuclear wave function in momentum space, with the set of independent Cartesian 3-momenta describing the relative motion between the i-th nucleon and the A−i spectator system, i.e. k is the relative 3-momentum with respect to the A − 1 cluster, k 2 is relative to the A − 2 one, down to k A−1 that is the relative 3-momentum of two nucleons.Notice that in a Galilean-invariant framework, these momenta reduce to the familiar Jacobi momenta.Within the LF framework, recalling the properties of the LF boosts (see, e.g.Ref. [16]), one introduces the following intrinsic, nonsymmetric transverse momenta and LFlongitudinal ones (let us recall that the LF momenta 2 are conserved) for a system composed by on-shell, interacting A nucleons (see also Ref. [31]) where is the free-mass of the (A − i + 1)-nucleon system (see below for its definition).The nonsymmetric variables disentangle the motion of a particle from the motion of the rest of the system.These variables were introduced in Ref. [31] where a fully interacting spectator system was considered to obtain the nuclear spectral function and allow one to implement the macroscopic locality (see also Ref. [16]).
The free mass of A nucleons is given by and it can be decomposed in a contribution from a nucleon and a (A − 1) cluster, which depends upon (A − 2) relative 3-momenta, viz.
once the LF-momenta in Eq. ( 4) are adopted.Plainly, this decomposition holds for the free-mass of any cluster, that can be further decomposed in the contribution from a nucleon and a sub-cluster, etc.In general the cluster free-mass can be written for (A − 1) ≥ i ≥ 2 as follows To proceed, we adopt the frame where P ⊥ = 0, and P + = M A .Instead of the plus components, k + and k + i , given in Eq. ( 4), we introduce a new set of variables to match the Cartesian expression for each 3-momentum.As is well-known [16], for on-mass-shell particles one can write Then, from Eq. ( 8), we get with the Cartesian 3-momentum k ≡ {k z , k ⊥ }, and analogously for the i-th LF-momentum.
With the above variables one gets another expression of the free mass M 0 [1, 2, 3, . . ., A] decomposed in a contribution from a nucleon and a (A − 1) cluster, viz.
where the (A − 1) cluster has to be considered as an on-mass-shell system, with total mass M 0 [2, 3, . . ., A] and recoiling with 3-momentum (−k).Let us recall that the ± components of the cluster momentum are In general the cluster free-mass can be written for (A − 1) ≥ i ≥ 2 as follows For A = 2, 3 one has, respectively, with and For A = 4 one has with and Eventually, in Eq. ( 3), the Jacobian ∂k z /∂ξ can be determined from the definitions in Eqs. ( 8), ( 4) and (6), i.e.
with E 2,3...A (k, k 2 , . . ., k A−1 ) = M 2 0 [2, 3, . . ., A] + |k| 2 In summary (cf.Refs.[31] and [32] for the three-body case) one writes the explicit expression of the LF spin-independent nucleon momentum distribution for an A-nucleon bound system as follows where {σ ℓ } ≡ σ 2 , σ 3 , . . ., σ A are canonical spins and the same notation is adopted for both isospins and Cartesian 3-momenta, with k z and k iz given in Eq. ( 8).The amplitude ⟨σ, {σ ℓ }; τ, {τ ℓ }; k, {k i }|A; J A , π A , M, T A z ⟩ is the wave function of an A-nucleon bound system, in momentum space, that is evaluated in the instant form (IF) of the Dirac relativistic Hamiltonian dynamics (see Ref. [31] for A=3).One can demonstrate that the amplitude we need within the LF framework is related to the IF one by the proper number of Melosh rotations [33] that transform canonical spins into LF ones.Due to the sum over the nucleon spins in Eq. ( 19), the Melosh rotations reduce to unity, and we remain with canonical spins.This is a very important result, since the canonical spins allow the use of the standard Clebsch-Gordan machinery.
The key point in our approach is: which is the wave function of an A-nucleon bound system one has to use in Eq. ( 19) ?It is fundamental for our scope that it is obtained in a framework that fulfills the constraints imposed on the interacting mass operator by the BT construction of interacting generators of the Poincaré group.The necessary requirement is the dependence of the interactions on only scalar products among intrinsic 3-momenta at disposal.One should notice that this is the case for Hamiltonians routinely adopted in a non relativistic framework, once we reinterpret the Cartesian 3-momenta there present, as the ones we have formally introduced through Eqs. ( 4) and (8).Moreover, as it is well-known for the two-nucleon case, which is relevant for extracting information on the two-nucleon interaction from the phase-shifts (see, e.g., Ref. [34]), one can formally embed the non relativistic Schrödinger equation and the Lippmann-Schwinger one in a Poincaré covariant framework, once the BT construction is exploited.In conclusion, within the BT construction, one has for the interacting mass operator or equivalently For the two-body case one adopts the second expression.What we can do in practice is to use the non relativistic wave function, expressed in terms of Jacobi momenta of the A-nucleon bound system (for the three-body case see [14,15]), and obtain a Poincaré covariant description.Lets us emphasize that this is the merit of the BT construction and of the fit to the phase-shifts, which clearly incorporate the whole (relativistic) nature of the NN scattering.Since we are considering DIS processes, the issue of macroscopic locality has to be taken into account, i.e., if the sub-clusters which compose a system are brought to infinity, the Poincaré generators of the system must reduce to the sum of the sub-cluster Poincaré generators.It is worth noting that the packing operators which cooperate for implementing the macroscopic locality [16,35], are not considered in the present approximation for the description of the bound state.However, when we build up the LF spectral function we implement macroscopic locality through the tensor product of a plane wave for the knocked-out constituent times the fully interacting intrinsic state for the (A − 1) spectator system, as allowed by the choice of LF nonsymmetric intrinsic momenta (see Ref. [31]).

Results
The main ingredient for calculating the EMC ratio R A EM C (x) is the light-cone momentum distributions for proton and neutron.In the actual application of our general formalism to 4 He, we used the α-particle wave functions, obtained from i) the NN charge-dependent AV18+3N [19,20] and ii) two versions of the Norfolk χEFT interactions, NVIa+3N and NVIb+3N, that generate three-nucleon forces within the same framework [21,22,23].It should be recalled that both NVIa and b interactions fit the NN phase shifts up to 125 MeV, with χ 2 ∼ 1 (see Refs. [21,22,23]), and the calculated 4 He binding energy is −28.298 and −28.247MeV for NVIa+3N and b, respectively.The short-range component of the 3N interactions has two adjusted parameters, C E and c D .The first parameter, c E , is fixed by reproducing the 3 He and 3 H binding energies, then the 4 He energy follows; the second one is fixed by reproducing the matrix element of the Gamow-Teller transition of 3 H.Let us recall that the 4 He wave functions corresponding to NVIa+3N and b interactions have been presented in Refs.[36] and [21], respectively.
In addition to the light-cone distribution of 4 He, we have evaluated the ones for: i) the deuteron and ii) 3 He and 3 H (the result of 3 He corresponding to AV18+3N is given in Ref. [14]).The calculations, obtained by using Eqs.( 2) and ( 19), are shown in Fig. 1: in the left panel one finds 4 He and 2 H results (Z = N ), while in the right panel the 3 He case (Z ̸ = N ).Notice that the support, 0 ≤ ξ ≤ 1, is naturally obtained within our approach, and both normalization and momentum sum-rule, that amounts to ⟨ξ⟩ = 1/A, have been numerically checked.The height of the peaks are almost the same, and their position is at ξ = 1/A.The tail of 4 He, 3 He and deuteron light-cone distributions corresponding to the AV18+3N interaction is bigger than the ones obtained from the χEFT potentials.Indeed, this feature is generated by the different content of SRCs, which affect the tail of the nucleon momentum distribution for the respective nuclei (see, e.g., https://www.phy.anl.gov/theory/research/momenta/ for detailed calculations of both single and two-nucleon momentum distributions for light nuclei), and in turn the tail of the light-cone distributions, since low or high values of ξ imply high values of |k z | [37].Moreover, the 4 He tail for NVIb+3N is greater than the one for NVIa+3N, and the same ordering can be found also for the A=3 nuclei (in the right panel of Fig. 1, only the results for AV18+3N and NVIb+3N are shown, for the sake of readability), as well as for the deuteron.Finally, for increasing A the f 1 (ξ) widths decrease, while the asymmetry with respect to the position of the peak increases.This can be understood by recalling that k + in Eq. ( 8) increases with A, for fixed values of ξ (cf.Eq. ( 4)).Hence, for ξ > 1/A one has an increasing k z for a growing A, and more and more large values of |k| come into play making faster the f 1 (ξ) decreasing.Differently, for ξ < 1/A the variable k z becomes negative and decreases in modulo for growing A, resulting in a f 1 (ξ) fall-off different from the one seen for ξ > 1/A.Interestingly, the asymmetry of f 1 (ξ) has an impact on the description of the EMC effect for the studied light nuclei, and one could surmise that the same happens for heavier nuclei (cf. the extrema of integration in Eq. ( 1)).
In the following study of the EMC effect, it will be interesting to look for a possible signature of the differences induced by the SRCs, as shown in the tails of the light-cone distributions.
The EMC ratio has been calculated in the Bjorken limit and compared to the experimental data.Although the latter ones have finite values of the momentum transfer, one  2) and ( 19).Left panel.Solid lines: 4 He, peaked at ξ = 1/4 and deuteron peaked at ξ = 1/2.Calculations have been obtained by using wave functions corresponding to the AV18+3N potential [19,20].Dashed lines: the same as the solid line, but corresponding to the χEFT interaction NVIb+3N [21,22,23].Dot-dashed lines: the same as the dashed lines, but corresponding to the χEFT interaction NVIa+3N.Right panel.The same as the left panel but for proton (solid lines) and neutron (dashed lines) in 3 He, peaked at ξ = 1/3.Thick lines: results obtained from AV18+3N in Ref. [14].Thin lines: results obtained from the χEFT interaction NVIb+3N [21,22,23].
can safely adopt the Bjorken limit given the tiny dependence upon Q 2 [6,7,38].Once F A 2 (x), Eq. ( 1), is evaluated for 4 He, one obtains the following EMC ratio where The results of 4 He and, for the sake of comparison, of 3 He are shown in the left and the right panels of Fig. 2, respectively.For both nuclei, it has been used the proton structure function F p 2 (x) obtained by the SMC Coll.[39], while F n 2 (x) = r(x) F p 2 (x), with r(x) extracted from the MARATHON Coll.data [12] in Ref. [14].It has to be pointed out that almost overlapping lines are obtained by using the ratio r(x) provided by Ref. [28], as well as F n 2 (x) from the SMC Coll.[39].This result strongly suggests that the dependence on the ratio F n 2 (x)/F p 2 (x) is largely under control.The lines shown in Fig. 2 have been calculated by exploiting three sets of 4 He and 3 He wave functions, corresponding to i) the AV18+3N [19,20] and ii) the χEFT interactions NVIa+3N and NVIb+3N [21,22,23].It is clear that a formally correct use of the LF Hamiltonian dynamics plus BT construction and the reliable description of nuclei lead to predict a ratio less than 1 in the range 0.2 < x < 0.75, still taking into account only the dofs of the standard nuclear physics.This result appears quite encouraging for achieving a quantitative separation between hadronic dofs and partonic ones, and helps to progress in the path toward understanding the transition from one realm to another.It is interesting to notice that the AV18+3N interaction, which gives rise to a higher momentum content and then more SRCs, produces a deeper minimum in the EMC ratio both for 4 He and 3 He with respect to the two χEFT interactions which give almost the same results.However, from Fig. 2, one can see that the differences between the calculations with these potentials are definitely less than the difference between data and theoretical predictions around the dip, i.e. x ∼ 0.7, leaving space for other effects.
One can notice that the dip displacement of our calculations is not greatly affected by either the nuclear interaction or the ratio F n 2 (x)/F p 2 (x), but it should be driven by the treatment of the off-shellness of the interacting nucleon stricken by the virtual photon.
, where r(x) has been determined in Ref. [14] (see Eq. ( 12)) by using the experimental data of the MARATHON Collaboration [12] and iii) the light-cone distribution corresponding to the wave function calculated in Ref. [36] with AV18+3N [19,20].Dashed line: the same as the solid one, but using the NVIa+3N interaction [21,22,23].Dash-dotted line: the same as the dashed line but for NVIb+3N.N.B.The results obtained by using the two χEFT interactions are hardly distinguishable.Black squares: Jlab data of Ref. [40].Right panel.The EMC ratio for 3 He, using the wave functions from AV18+3N and NVIa+3N potentials (the results from NVIa+3N and NVIb+3N fully overlap).Black squares: Jlab data of Ref. [40], as reanalyzed in Ref. [41].
In view of future experimental data and for the sake of completeness, the EMC ratio for 3 H is shown in Fig. 3 for all the nuclear potentials we have discussed.It is noteworthy that even for 3 H the dependence of our uncertainties on the ratio r(x) is very weak, as confirmed by the result obtained by using r(x) given in Ref. [28], that overlaps the one from r(x) of Ref. [14], shown in Fig. 3.
In Fig. 4, the impact of the proton structure function on the EMC ratio is analyzed by using two choices: i) F p 2 (x) of Ref. [39] and ii) F p 2 (x) of Ref. [42], the one which includes target mass corrections, keeping the same ratio r(x) of Ref. [14].This study puts in evidence the lack of knowledge on the proton structure function for x > 0.7 (see Sect. 18 in the recent Particle Data Group [29]), that however does not affect the typical range of x where the EMC effect manifests.

Conclusions
We have extended our Poincaré covariant approach, based on the LF Hamiltonian dynamics supplemented by the Bakamjian-Thomas construction of the Poincaré genera- The EMC ratio for 3 H.Left panel.Solid line: our calculation with i) F p 2 (x) of Ref. [39], ii) F n 2 (x) = r(x) F p 2 (x), where r(x) has been determined in Ref. [14] (see Eq. ( 12)) by using the experimental data of the MARATHON Collaboration [12] and iii) the light-cone distribution corresponding to the wave function calculated in Ref. [36] with AV18+3N [19,20].Dashed line: the same as the solid one, but with the outcome from NVIa+3N.Right panel.Comparison with two different determination of F n 2 (x), using the 3 H wave function from AV18+3N.Solid line: F n 2 (x) from the SMC Coll.[39].Dashed line: F n 2 (x) obtained through r(x) of Ref. [14].
tors (needed for determining a suitable mass operator for A interacting nucleons), to the calculation of the EMC effect for any nuclei.Actual calculations have been performed for the first time for 4 He and 3 H, by using wave functions obtained from modern nuclear interactions: i) Argonne V18 plus the 3N Urbana IX and ii) two χEFT interactions: NVIa+3N and NVIb+3N.Moreover, also the 3 He has been re-analyzed by using wave functions corresponding to the χEFT interactions.This is a substantial step forward for providing a reliable baseline, which retains only the dofs of the standard nuclear physics, but within a Poincaré covariant framework.For the very light nuclei, the impact on R A EM C (x) of the different short-range correlation content, generated by the realistic nuclear potentials we have considered, has been quantitatively studied, and it results quite smaller than the EMC effect by itself.Moreover, the uncertainty on the ratio between the neutron and proton structure functions appears to be negligible, while the lack of knowledge on F p 2 (x) for x > 0.7 clearly calls for new experimental campaigns.
In principle, our analysis could help in ascribing the deviation of the EMC ratio from the proposed baseline, to exotic phenomena involving necessarily partonic dofs, not included in a standard nuclear description, and possibly initiates a systematic and quantitative analysis of the transition from hadronic to QCD dofs in light nuclei.  He evaluated with different proton structure functions and F n 2 (x) = r(x) F p 2 (x), where r(x) has been determined in Ref. [14].Solid line: F p 2 (x) from the SMC Coll.[39].Dashed line: F p 2 (x) from Ref. [42], the one which includes target mass corrections.

Figure 2 :
Figure 2: Left panel.The EMC ratio for4 He.Solid line: our calculation with i) F p 2 (x) of Ref.[39], ii)F n 2 (x) = r(x) F p 2 (x), where r(x) has been determined in Ref.[14] (see Eq. (12)) by using the experimental data of the MARATHON Collaboration[12] and iii) the light-cone distribution corresponding to the wave function calculated in Ref.[36] with AV18+3N[19,20].Dashed line: the same as the solid one, but using the NVIa+3N interaction[21,22,23].Dash-dotted line: the same as the dashed line but for NVIb+3N.N.B.The results obtained by using the two χEFT interactions are hardly distinguishable.Black squares: Jlab data of Ref.[40].Right panel.The EMC ratio for 3 He, using the wave functions from AV18+3N and NVIa+3N potentials (the results from NVIa+3N and NVIb+3N fully overlap).Black squares: Jlab data of Ref.[40], as reanalyzed in Ref.[41].
Figure3: The EMC ratio for 3 H.Left panel.Solid line: our calculation with i) F p 2 (x) of Ref.[39], ii)F n 2 (x) = r(x) F p 2 (x), where r(x) has been determined in Ref.[14] (see Eq. (12)) by using the experimental data of the MARATHON Collaboration[12] and iii) the light-cone distribution corresponding to the wave function calculated in Ref.[36] with AV18+3N[19,20].Dashed line: the same as the solid one, but with the outcome from NVIa+3N.Right panel.Comparison with two different determination of F n 2 (x), using the 3 H wave function from AV18+3N.Solid line: F n 2 (x) from the SMC Coll.[39].Dashed line: F n 2 (x) obtained through r(x) of Ref.[14].

Figure 4 :
Figure4: EMC ratio in4 He evaluated with different proton structure functions and F n 2 (x) = r(x) F p 2 (x), where r(x) has been determined in Ref.[14].Solid line: F p 2 (x) from the SMC Coll.[39].Dashed line: F p 2 (x) from Ref.[42], the one which includes target mass corrections.