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Contribution of boundness and motion of nucleons to the EMC effect

The kinematical corrections to the structure function of nucleon in nucleus due to the boundness and motion of nucleons arise from the excitation of the doorway states for one-nucleon transfer reactions in the deep inelastic scattering on nuclei.


Introduction
It is known more than 20 years that the cross section of deep inelastic scattering (DIS) on nuclear target is not equal to the sum of cross sections on free nucleons [1]. This means that the interaction inside the nucleus distorts the parton distribution in a nucleon. But at first it is necessary to single out the kinematical effects arising from the boundness and motion of nucleons in nuclei because otherwise it is hardly possible to conclude what actually happends with the nucleon structure functions in nuclear matter. This is the aim of our work rather than the explanation of the EMC effect. The above kinematical effect is due to the fact that the four-momentum of nucleon in nucleus is not equal to that of a free nucleon. Indeed, the heavy photon (γ * ) is absorbed by a single nucleon and the Deep Inelastic Scattering (DIS) proceeds via the following stage: where l and l ′ denote the incoming and outgoing leptons, X -the final hadronic state after the γ * -nucleon interaction and A is the target nucleus. Before absorbing the heavy photon (γ * ) the struck nucleon has a certain energy-momentum distribution in nucleus. Besides this the "residual" (A−1) nucleus is excited.
There were a few attempts to account for the Fermi motion, boundness and the change of the effective γ-nucleon flux factor inside the nucleus [2,3,4] (see [5] and references therein for more details). They all were based on a seemingly obvious assumption that the energy-momentum distribution of the struck nucleon is described by the ground-state one-nucleon spectral function.
where |A 0 is the ground state of target nucleus A, a(p) and a + (p) are operators of nucleon with the momentum p (the spin and isospin variables are omitted), H is the nuclear Hamiltonian in the second quantization and E 0 (A) is the ground-state binding energy of nucleus A. The calculations [3] were performed by using the following semiempirical model for the quantity (1): the nucleon energy distributions were described by the experimental data on the separation energies of protons from the (e, e ′ p) reactions [6] (the difference between the proton and neutron separation energies was neglected leading to about 10% error) and calculating the momentum distributions within the harmonic oscillator model with the parameterhω 0 = (45 A −1/3 − 25 A −2/3 ) MeV reproducing the observed rms radii of nuclei. However nobody realized in this connection that the DIS on nuclei is rapid process, and therefore the energy-momentum distribution of struck nucleon is described by the spectral function of nuclear mode which is excited via a sudden perturbation rather than that of the ground state. Our work is based on the fact that the relevant mode is provided by the doorway states for the one-nucleon transfer reactions. As demonstrated in [7]- [9] the above states are eigenstates of nucleon in the static nuclear field.
Recall that the microscopic nuclear models are based on certain approximations for the in-medium nucleon mass operator M. For instance the nuclear shell model potential is the approximation for the mass operator at nuclear Fermi-surface, the optical model potential is dealing with the mass operator at low and intermediate energies, etc. In all the available approaches the mass operator includes all the Feynman diagrams which are irreducible in the one-particle channel, and therefore it cannot be calculated. Instead it is described by a set of phenomenological parameters.
In contrast to the above models the nuclear static potential is the mass operator at the infinite value of the energy variable. Only the Hartree diagrams with the free space (i.e. vacuum) nuclear forces survive in this case thus permitting the model-independent calculation of the static field. So the doorway states (DS) under consideration appear to be the unique nuclear object, both model-independent and described by the exactly soluble problem.
The calculation [8] showed that the rms radii of the DS density distributions are appreciably less than those of the ground-state ones: for instance, the value of r 2 = A −1 ρ(r)r 2 d 3 r is 10.88 fm 2 for the ground state of 40 Ca being only 8.76 fm 2 for the DS. As a result the nucleon motion (i.e. the value of p 2 ) was underestimated in [3] by about 25% 1 .
In Sec. 2 we briefly describe the formalism of the DS. In Sec. 3 the DIS structure functions F 2 are calculated for different nuclei and deuteron; to single out the boundness and motion effects we disregarded the possible changes of the parton distribution inside the nucleon in nucleus. In the last Sec. 4 we compare the calculated ratios 2F 2A /AF 2D with the available experiments.
2 Doorway states for one-nucleon transfer reactions

Theory
Evolution of the state arising from the one-nucleon transfer to the nuclear ground state |A 0 at the initial time moment t = 0 is described by the singleparticle propagator [11] S(x, By the same reason the kinematical effect was underestimated in Ref. [10] as well. = iθ(−τ ) At τ < 0 it describes the evolution of the hole state, when the nucleon is removed from the ground state A 0 , whereas at τ > 0 the evolution of the particle state is described, when nucleon is added to the ground state A 0 . The quantities E j (A − 1), E k (A + 1) and E 0 (A) are total binding energies of the states |(A − 1) j of the (A − 1) nucleus, the states |(A + 1) k of the (A + 1) nucleus and the ground state |A 0 of the A one. The Fourier transform of the propagator δ → +0 obeys the Dyson equation wherek x is the kinetic energy and the mass operator M(x, x ′ ; ε) includes all Feynman diagrams which are irreducible in the one-particle channel.
We are interested in the very beginning of the evolution, i.e. the τ → 0 limit. According to the time-energy Heisenberg relation this is equivalent to the limit ε → ∞. In this limit where (see the definition (1) of the propagator) the quantities I 0 , I 1 and I 2 thus describing the very beginning of the evolution Fig.3 of Ref. [9]) the higher-order diagrams describing the nuclear correlation effects (the lowest-order diagram of such kind was shown in Fig.4a of Ref. [9]) and the Fock ones ( Fig. 4b of Ref. [9]). The correlation diagrams include the propagators of intermediate states thus behaving as ε −1 in the ε → ∞ limit (see Ref. [12] for more stringent demonstration). The same is valid for the Fock diagrams. Indeed, the interaction between baryons proceeds via the exchange by some particles (they are quark-antiquark pairs and/or gluons in the QCD) and therefore both the momentum and the energy are transferred through the interaction. As a result the Fock diagrams also include the intermediate state propagators thus being of order of ε −1 in the ε → ∞ limit. (In Ref. [7] this is demonstrated for the meson-nucleon intermediate state). So the mass operator in this limit is

Introducing the static Hamiltonian
let us write down the high-energy limit Dyson equation in the form Putting into (13) the asymptotics (7) and equating coefficients at the same powers of ε −1 we get Equations (9), (12) and (15) may be written as As follows from the lhs of (17) the hamiltonian h st describes the very beginning of the one-nucleon transfer process the eigenstates of h st thus being the doorway states for one-nucleon transfer reactions. On the other hand the rhs of (17) shows that the hamiltonian h st describes the motion of nucleon in nuclear static field U st (x). Indeed, the latter is expressed through the freespace NN forces rather than the effective ones thus being the nucleon field rather than the quasiparticle one. So we proved that the doorway states for one-nucleon transfer fast reactions are the eigenstates of nucleon in nuclear static field.

Doorway eigenfunctions
Since the doorway states (DS) describe the motion of the nucleon in nuclear static field the corresponding eigenfunctions may be calculated in a model independent way. Indeed, the two-particle forces are determined from the experimental data on the elastic nucleon-nucleon scattering (i.e. from the phase shifts analysis) [13] and the deuteron properties. The necessary information about the multiparticle forces is obtained from the observed energy spectra of the doorway states [7]. So the only additional information needed for calculation of the static field in a given nucleus is that on the nucleon density distributions in this nucleus. In all the nuclei which are treated in the present paper these distributions are spherically symmetric thus leading to the static field with the same symmetry. Hence the quantum mechanical problem is the motion of a particle in a central field. This problem is solved with any desired accuracy and without any simplifications.
We have to emphasize that the doorway states are not the eigenfunctions of the total nuclear Hamiltonian thus being fragmented over the actual nuclear states owing to the correlation effects. The observed spreading width of the DS is about 20 MeV; that is the relaxation time ∼ 0.3 · 10 −22 sec. This is much larger than the time chracteristic for DIS which is of the order of 2q 0 /Q 2 ≃ 1/mx ∼ 3 · 10 −24 sec in the nucleus rest frame. So during the DIS process the DS do not have time to be distorted by the correlations thus permitting the exact account for the nucleon boundness and motion to the EMC effect.
The relevant energy-momentum distribution of nucleons for DIS is determined by the spectral function of the DS (rather than the ground state one): S DS (ε, p) = S p (ε, p) + S n (ε, p) , where the proton spectral function is The sum in the r.h.s. runs over the proton DS, λ stands for the angular momentum j and other quantum numbers of a particle state in central field, ν λ equal to 2j + 1 for the filled states and the actual number of nucleons on partly filled ones, ε λ are the DS energies and f λ (p) are found by solving the Dirac equation (see Ref.s [7,9] for details).
The function f λ (p) = u 2 λ (p) + w 2 λ (p), given by the sum of the upper and lower components square of the bi-spinor ψ λ (p) (in momentum space), is normalized by the condition f λ (p)p 2 dp = 1 .
The neutron spectral function obeys the same relation in which the proton DS are substituted by the neutron ones.
It is instructive to mention that the spectral functions S DS (ε, p) is evident Lorentz invariant obeying the following normalization: (here p 0 = m + ε, so dp 0 = dε).
The calculations were performed for 12 C, 14 N, 27 Al, 40 Ca, 56 F e and 63 Cu. The reason is as follows. As mentioned above the necessary information for the calculations is that about the proton and neutron density distributions. The former is available for throughout the whole periodic system [14], but it is not the case for the latter: the neutron densities are available only for doubly closed-shell nuclei 16 O, 40 Ca, 90 Zr and 208 P b [15]. That's why we confined ourselves by nuclei with a small neutron excess: the density distributions per nucleon are nearly the same for protons and neutrons in these nuclei [16].
To calculate the eigen functions the Bonn B [13] and OSBEP [17] NNpotentials were used 2 . In both cases the results are very close to each other. The difference never exceeds 0.5% for x < 0.6 and is less then the experimental error bars in the domain where the ratio (25) R th > 1.

Deep inelastic cross section on nuclear target
The DIS cross section is usually written in terms of the structure function F 2 (x, Q 2 ), that is the cross section of electron-nucleon interaction 2 For the deuteron the Bonn B wave function was used in both cases.
As a rule the data are taken at rather small y, where the coefficient y 2 /2 in front of the longitudinal part (F L ) is small. Next, the ratio R L = F L /F 2 ∼ 0.2 is not large. Moreover, unlike the F 2 , the function R L does not appear to depend on atomic number A [5].
Therefore the ratio of cross sections is given usually in terms of the ratio of structure functions F 2 .
In order to compare our results with the data, where experimentalists already accounted for the difference between proton and neutron, we write the structure function of nucleus as and select the isospin I=0 part of F 2 given by the first term of (24). The ratio which we will discuss reads The structure function of the proton in nucleus where ν pλ is the actual number of protons on the level λ (ν pλ = 2j + 1 for the completely ocupied shell). Note that in the experimental data the variable x was calculated assuming the proton momentum p N equal to the momentum of a free proton at rest, p N = (m N , 0, 0, 0). However to single out the precise "kinematics" one must account for the change of the nucleon structure function (parton distibutions) in medium caused by the change of the Bjorken variable x = Q 2 /2(p · q). In other words calculating the momentum fraction x ′ carried by the quark we need to use the precise four momentum of the nucleon in medium. That is where β = | q|/q 0 = (1 + 4m 2 x 2 Q 2 ) 1/2 and the variable t is the cosine of the angle between p and q.
Next we have to note that the structure function F 2 , which at the LO reads ( e f is the electric charge of the quark of flavour f ) contains two factors: the quark(antiquark) distribution q(x)(q(x)) and the kinematical factor x. The origin of this kinematical factor is as follows. The covariant quantity is not the cross section but discontinuity of the dimensionless interaction amplitute ImA ≃ sσ. Going from the amplitude A to the cross section σ ∝ 1/Q 2 we obtain the factor x A = Q 2 /2(pq) which corresponds to the true nucleus target and must be calculated as where m A is the mass of nucleus and ν = q 0 is the photon energy in the nucleus rest frame. Note that in the final expressions (29,30) we use the value of m A calculated within the doorway formalism, that is , which is about 3% less than the true mass of a nucleus in the ground state 3 . This is equivalent to the prescription given in [4], where the authors accounted for the relativistic flux factor and used the baryon charge conservation to normalize the spectral functions S p (ε, p) and S n (ε, p).

Thus in (26) we need to calculate the function
Here: p λ = ((1 − x)m + ε λ )/β, and f λ (p) was defined in sect.2.2 4 Exactly the same formulae is used for the neutron in nucleus. For the deuteron is just the sum of the squared monopole and quadrupole components of the deuteron wave function; m D is the deuteron mass. Note that denominator in the expression for x ′ D corresponds to the kinematics where the spectator nucleon is on mass-shell.
The F 2p (x, Q 2 ) and F 2n (x, Q 2 ) free nucleon structure functions were calculated using the MRST2002 NLO parametrization [18] obtained from the global parton analysis.

Discussion
The results of calculations are presented in Table 1-5 and Fig.1 Recall that here we assume the parton distributions inside the nucleon in nucleus to be the same as that for the free nucleon and evaluate the pure kinematical effect of the boundness and the motion of nucleon in nuclear matter. Using the doorway states, which are the correct eigen functions to describe the fast interaction with one nucleon, we account for the full 4-momemtum of the (target) nucleon and for the exitation of the "residual" nucleus (A − 1).
Thus the difference between the calculated value of R kin and the data indicates the distortion of the parton wave function of a nucleon placed in nuclear medium.
As expected the account of the boundness and Fermi motion of nucleons in nuclei diminishes the cross section in the x = 0.2 − 0.63 interval. Indeed, due to the boundness (and the fact that about 24 -27 MeV is spent for the exitation of the residual (A − 1) nucleus) the mean value of shifted argument x ′ (27) is larger than the value of x on a free nucleon. On the other hand in this domain the free nucleon structure function F 2 falls down with x. Therefore we get R kin < 1.
At a large x, close to 1, the details of angular integration (over t in (29)) become important. For a negative t, due to a Fermi motion, there is a region where x ′ < x (see (27)  the value of R kin becomes larger than 1 for x > 0.65 − 0.7.
Clearly, besides the Fermi motion there should be some dynamical effects. At a large x the growth of the ratio R(x, Q 2 ) with x is usually atributed to a short range nucleon-nucleon correlations [25] or to a multiquark bags [26] (see for a details the reviews [5,25] and reference therein). However, contrary to the conventional expectations, the theoretical value of R kin resulting after account of the Fermi motion in the doorway states is even larger than the value R exp measured experimentally 5 .
This means that in nuclear medium the (one nucleon) parton distribution becomes softer, that is the probability to find a parton with x > 0.45 inside the in-medium-nucleon is less than that in a free nucleon. In other words in medium the quark distribution is shifted towards a lower x, leading to the decrease of quark density at x > 0.45 and a larger quark density at a lower Next at small x < 0.2 the partons from different (neighbouring) nucleons start to overlap and to interact with each other. Indeed, according to un-  certainty principle the characteristic size of localization is ∆r ∼ 1/mx and for x < 0.2 the value of ∆r > 1fm becomes comparable with the nucleonnucleon separation. At a very low x the partons screen each other and this shadowing correction results in decreasing of R(x, Q 2 ). Another way to describe this effect is to say that two low-x partons from two different nucleons recombine into one parton. However the whole energy must be conserved. This leads to the antishadowing (growth of the parton density) [27] (see the reviews [5,25] for more details) just in the region (x ∼ 0.1 − 0.2) of the begining of recombination. On the other hand this antishadowing effect is expected to reveal itself more in the gluon distributions than in the quark structure function.
Thus it is not surprising that in the interval 0.2 < x < 0.45 the ratio given by the pure kinematical effects R kin (25)is close (within the error bars) to that observed experimentally R exp .
Note that, at large x, the Fermi motion is not negligible, even for the deuteron. The ratio R D,kin = F 2D /(F 2p + F 2n ) is close to one for x < 0.65, but it noticeably differs from one for x > 0.75, reaching values of R D,kin =1.07 (1.42) at x =0.75 (0.85); see Table 6.
An analysis performed by the MRST group shows that if this effect is included then one obtains practically the same partons, but the description  of the high x deuteron data is much improved; with χ 2 reduced by 20 for the 12 deuteron data points that are fitted at x = 0.75 6 .
After the present work was completed we have read the recent paper of A.Molochkov [28] where a little bit another (but not quite different from that used here) prescripton was proposed to account for the boundness and momentum distribution of nucleons. A.Molochkov had considered the ratio of the 4 He to deuteron structure functions. The shortness of his prescription is the assumption that both the nucleon structure function F 2 and the momentum distribution of the nucleons in nucleus f N (P A , p) are regular (i.e. have no singularities) with respect to p 0 . Besides this some terms, coming from the differentiation of the nucleus (A − 1) propagator and the factor 1/(p 0 + √ m 2 + p 2 ) 2 (corresponding to the antinucleon pole) in the nucleon propagator, which are proportional to the binding (or nuclear exitation) energy, were omitted in [28]. We hope that our approach, based on the 'doorway' formalism is more precise. Moreover, in terms of the Molochkov's integral our result may be obtained by closing the integration contour over p 0 in the upper half-plane (on the pole corresponding to the residue (A − 1) Figure 1: Fig.1 The ratio of structure function F A 2 measured on nucleus A to that on deuteron. Q 2 = 5 GeV 2 . The data are taken from [24]. The empty bar is the ratio R kin calculated using the Bonn-B potential.  nucleus) instead of the lower one as it was done in [28].
However we are planning to compare both approaches in the forthcoming paper, using the doorway eigen functions to describe the distributions of nucleons in a heavier nuclei.