Scaling function and nucleon momentum distribution

PHYSICAL REVIEW C 81, 055502 (2010) Scaling function and nucleon momentum distribution J. A. Caballero,1,*M. B. Barbaro,2A. N. Antonov,3M. V. Ivanov,3and T. W. Donnelly4 1Departamento de F´ ısica At´ omica, Molecular y Nuclear, Universidad de Sevilla, ES-41080 Sevilla, Spain 2Dipartimento di Fisica Teorica, Universit` a di Torino and INFN, Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy 3Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, BG-1784 Sofia, Bulgaria 4Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 10 March 2010; published 18 May 2010) Scaling studies of inclusive quasielastic electron scattering reactions have been used in the past as a basic tool to obtain information on the nucleon momentum distribution in nuclei. However, the connection between the scaling function, extracted from the analysis of cross-section data, and the spectral function only exists assuming very restricted approximations. We revisit the basic expressions involved in scaling studies and how they can be linked to the nucleon momentum distribution. In particular, the analysis applied in the past to the so-called scaling region, that is, negative values of the scaling variable y, is extended here to positive y, as a “universal” superscaling function has been extracted from the analysis of the separated longitudinal data. This leads to results that clearly differ from those based solely on the negative-yscaling region, providing new information on how the energy and momentum are distributed in the spectral function. DOI: 10.1103/PhysRevC.81.055502 PACS number(s): 25.30.Fj, 21.60.Cs, 24.10.Jv, 21.10.Ft I. INTRODUCTION: BASIC ASPECTS OF SCALING Scaling studies of inclusive quasielastic (QE) electronnucleus scattering have largely been considered to provide a powerful tool for extracting the momentum distribution of nucleons inside nuclei [1–7]. Such analyses have been applied to few-body systems, complex nuclei, and nuclear matter, with an important effort devoted to estimating binding corrections and, in particular, the high-momentum components of the nucleon momentum distribution that are governed by shortrange correlations [8,9]. However, caution should be borne in mind for the conclusions reached, as a close relationship between the momentum distribution and the scaling function only emerges after some approximations are made. These are linked not only to the general description of the electron scattering reaction mechanism, but also to the integration limits involved and the behavior of the spectral function [1]. The phenomenon of yscaling emerges from the analysis of QE (e,e) reactions. The scaling function, defined as the QE (e,e) differential cross section divided by an appropriate factor involving the single-nucleon cross section [1,10–12], is shown to depend only on a single variable, y, given as a particular combination of the two independent variables in the process, namely, the energy and momentum transfers, ωand q.Inthe QE domain and for values of ωand qlarge enough, the basic mechanism in (e,e) reactions on nuclei corresponds to elastic scattering from individual nucleons in the nuclear medium with “quasifree” ejection of a nucleon from the nuclear system. This implies that the inclusive (e,e) cross section is mainly constructed from the exclusive (e,eN) process, including the contribution of all nucleons in the target and integrating over all (unobserved) ejected nucleon variables. In other words, QE scattering off a nucleus is simply described as an incoherent *[email protected] sum of single-nucleon scattering processes. This approach, which constitutes the basis of the impulse approximation (IA), although being an oversimplified description of (e,e) reactions, has demonstrated its validity under appropriate kinematic conditions. Mechanisms beyond the IA (correlations, meson exchange currents, rescattering processes, etc.) may play a significant role in electron scattering and, hence, may lead to non-negligible scaling violations. The IA provides an intuitive explanation of how the scaling behavior emerges from the analysis of data. In this case the QE (e,e) cross section is given by dσ dd(e,e)= A  i=1 (ω,q) pdpdEdφNiENi qp2 Ni ×dσ dddpNidNi(e,eNi) ,(1) where the sum extends to all nucleons in the target and {, } refer to the scattered electron variables. The integration over the ejected (unobserved) nucleon variables {pNi,E Ni, Ni} has been expressed in terms of the excitation energy Eof the residual nucleus and the missing momentum p.The significance of these variables as well as the kinematically allowed integration region denoted (ω,q)isdiscussedin detail in next section. Within the IA, evaluation of (e,eNi) cross sections for both proton and neutron knockout determines the inclusive QE cross section. The study of exclusive (e,eN) reactions has been presented in previous work [13–18], focusing on different aspects of the problem: final-state interactions (FSIs), relativity, correlations, etc. Although such ingredients have been proven to be essential to fit experimental (e,eN) cross sections, in what follows we restrict our attention to the plane-wave IA (PWIA), where the knocked-out nucleon has no interaction with the residual nucleus. Being the simplest 0556-2813/2010/81(5)/055502(12) 055502-1 ©2010 The American Physical Society CABALLERO, BARBARO, ANTONOV, IVANOV, AND DONNELLY PHYSICAL REVIEW C 81, 055502 (2010) approach to (e,eN) processes, PWIA retains important relativistic effects that are essential in describing reactions at high qand ω. Moreover, the (e,eN) differential cross section in PWIA factorizes in two basic terms: the electron-nucleon cross section for a moving, off-shell nucleon and the spectral function that gives the combined probability of finding a nucleon of certain momentum and energy in the nucleus [16–18]. In general we can write dσ dddpNdNPWIA (e,eN)=KσeN (q,ω;p, E,φ N)S(p, E), (2) with Ka kinematical factor [19] and where pis the missing momentum and Ethe excitation energy, essentially the missing energy minus the separation energy. It is important to point out that the factorization property shown in Eq. (2) no longer persists if dynamical relativistic effects in the bound nucleons are incorporated, that is, effects from the lower components in the relativistic wave functions, even in the plane-wave limit [20,21]. Note that both the eN cross section and the spectral function depend on the two integration variables in Eq. (1), pand E. To show how the scaling function emerges from PWIA, further assumptions are needed. First, the spectral function is assumed to be isospin independent, and second, σeN is assumed to have a very mild dependence on the missing momentum and excitation energy, which is supported by the most commonly used off-shell cross sections [1]. Hence the eN cross section can be evaluated at fixed values of pand E: typically the differential cross section for inclusive QE (e,e) processes is written in the form dσ dd(e,e)∼ =σe(q,ω;p=|y|,E=0) ·F(q,ω),(3) where the single-nucleon cross section is evaluated at the special kinematics p=|y|(with ythe scaling variable; see the next section) and E=0 (the residual nucleus in its ground state). This corresponds to the lowest value of the missing momentum occurring when E=0. The term σerefers to the azimuthal-angle-averaged single-nucleon cross section and it also incorporates the kinematical factor Kin Eq. (2) and the contribution of all nucleons in the target, that is, σe≡KA i=1dφNiσeNi/2π. The function F(q,ω)inEq.(3) is known as the scaling function and it is given in PWIA in terms of the spectral function: F(q,ω)=2π (q,ω) pdpdES(p, E).(4) A detailed study of the scaling function and its connection with the momentum distribution is presented in the next section. However, let us start by pointing out some general interesting features of this basic result. First, only in the case in which it was possible to extend the kinematically allowed region (q,ω) to infinity in the excitation energy plane, that is, Emax →∞, would the scaling function be directly linked to the true momentum distribution of the A-nuclear system: n(p)≡∞ 0 dES(p, E).(5) Second, guided by the PWIA result in Eq. (3), an experimental scaling function can also be defined by dividing the experimental QE (e,e) cross section by the single-nucleon function, σe. At high enough values of the momentum transfer q, the function Fexp(q,ω) has been shown to satisfy scaling in the region below the QE peak; that is, Fexp becomes only a function of the scaling variable y(see Refs. [1,11,12], and [22] for details). Note that Eq. (4) does not apply to Fexp(q,ω), which incorporates ingredients not included in the simple PWIA approach: FSIs, meson exchange currents, rescattering processes, etc. The contribution of these effects and their impact on the scaling phenomenon depend on the kinematical region explored, leading, in particular, to a significant scaling breaking in the region above the QE peak. Furthermore, based on the analysis performed with the relativistic Fermi gas (RFG) model, and making use of the separate longitudinal (L) and transverse (T)(e,e) data, experimental superscaling functions have been introduced: fexp(q,ω)≡kFFexp(q,ω),(6) fL(T) exp (q,ω)≡kFFL(T) exp (q,ω),(7) where kFis the Fermi momentum. In particular, the L response is thought to have very little contribution from meson production and from meson-exchange currents and thus should be the place where the underlying nuclear dynamics can cleanly be resolved. It has been shown to superscale; that is, the function fL exp shows only a very mild dependence on the momentum transfer q(first-kind scaling) and the nuclear system considered (second-kind scaling). This has led to the introduction of a universal experimental superscaling function that constitutes a strong constraint for any theoretical model describing QE electron scattering. Not only should the superscaling behavior be fulfilled, but also the specific shape of fL exp must be reproduced. This subject has been studied in detail in previous work showing the importance of FSI and relativity [23–27], and those studies clearly show that any conclusion about the momentum distribution based on Eq. (4) should be made with caution. Being aware of this, it is illustrative, however, to analyze in detail the basic approaches on which the “link” between the momentum distribution and the scaling (superscaling) function is based. Moreover, the usual analysis, restricted in the past to the region below the QE peak, is now extended to the region above the peak, as the superscaling function fL exp is defined for both negative and positive values of the scaling variable (see discussion in the next section). II. THE SCALING FUNCTION As already shown, in PWIA the scaling function can be expressed as an integral of the spectral function Sin the (p, E) plane [Eq. (4)], with pthe struck nucleon’s momentum, E(p)≡M∗2 B+p2−M02 B+p2⩾0(8) the excitation energy of the recoiling system B,M0 Bthe ground-state mass of the residual nucleus, and M∗ Bthe general invariant mass of the daughter final state. The integration in Eq. (4) is extended to the kinematically allowed region in 055502-2 SCALING FUNCTION AND NUCLEON MOMENTUM ... PHYSICAL REVIEW C 81, 055502 (2010) FIG. 1. (Color online) Excitation energy corresponding to negative (left) and positive (right) values of y. the (p, E) plane at fixed values of the momentum and energy transfer, (q,ω). This is represented by (q,ω). The general kinematics corresponding to QE (e,e) processes leads to the E-integration range [1,10] max{0,E+}⩽E⩽E−,(9) where E±(p;q,ω)=M0 A+ω −(q±p)2+m2 N+M02 B+p2(10) and where M0 Ais the target nuclear mass and mNthe nucleon mass. The intercepts between the curve E−and the paxis are denoted −yand Y; that is, E−(−y;q,ω)=E−(Y;q,ω)=0. The integration region (q,ω) is shown in Fig. 1for fixed values of the transferred energy and momentum for ω<ω QE (left) and ω>ω QE (right), with ωQE the energy at which the quasielastic peak (QEP) occurs. In the region below the QEP, yis negative and p=−yrepresents the minimum value for the struck nucleon’s momentum. Above the QEP yis positive and the curve E+cuts the integration region when p<y. In terms of the independent variables qand ω, the intercepts ±yand Yare given by y(q,ω)=M0 A+ω2−M02 BW2−qW2,(11) Y(q,ω)=M0 A+ω2−M02 BW2+qW2,(12) with W≡√(M0 A+ω)2−q2the center-of-mass energy and ≡(M02 B−m2 N+W2)/2. Then the scaling function in Eq. (4) can be recast as follows: 1 2πF(q,y)=Y(q,y) −y pdp E−(p;q,y) 0 dES(p, E)ify<0, (13) 1 2πF(q,y)=y 0 pdp E−(p;q,y) E+(p;q,y) dES(p, E)+Y(q,y) y pdp ×E−(p;q,y) 0 dES(p, E)ify>0,(14) for negative and positive values of y, respectively. The analysis presented in the previous work has been restricted to the negative-yregion, that is, below the QEP, as this is the region where cross-section data fulfill y-scaling properties. The function Fexp does not scale for positive values of ybecause of the significant scaling violations introduced by effects beyond the IA, namely, inelastic processes and contributions from meson-exchange currents. However, these contributions mostly reside in the purely transverse response and are negligible in the Lchannel. The “universal” superscaling function extracted from the analysis of the separated Ldata, and defined for both negative and positive values of the scaling variable, explains our interest in extending the study to the region above the QEP. This strategy, which forces us to employ the superscaling function fL exp to determine FL exp =fL exp/kFinstead of the usual y-scaling function Fexp, can lead to significant effects concerning the momentum and energy distribution in the spectral function, as discussed here. In the preceding expressions we have chosen (p, E;q,y)as independent variables. In terms of these we can also express the energy transfer, ω(q,y)=(q+y)2+m2 N+M02 B+y2−M0 A,(15) the limits of the excitation energy, E±(p;q,y)=(q+y)2+m2 N−(q±p)2+m2 N +M02 B+y2−M02 B+p2,(16) and the upper limit of p, Y(q,y)= M02 B(2q+y)+2(q+y)M02 B+y2(q+y)2+m2 N+y2(q+y)2+m2 N M02 B+2M02 B+y2(q+y)2+m2 N+2y(q+y)+m2 N .(17) In the thermodynamic limit M0 B→∞, we get E±(p;q,y)→(q+y)2+m2 N−(q±p)2+m2 N ≡Eq+y−Eq±p,(18) Y(q,y)→2q+y, (19) where we have introduced the nucleon energies Ek≡ √k2+m2 N. Moreover, note that in the limit of a very high momentum transfer, that is, q|y|and qmN,the preceding limiting values reduce to Y→2qand E±→y∓p. Following previous arguments presented in Refs. [1] and [4], it is instructive to split the spectral function into two 055502-3 CABALLERO, BARBARO, ANTONOV, IVANOV, AND DONNELLY PHYSICAL REVIEW C 81, 055502 (2010) terms, corresponding to zero and finite excitation energy, respectively: S(p, E)=n0(p)δ(E)+S1(p, E),(20) with S1(p, E=0) =0, which, inserted into Eqs. (13) and (14), yields 1 2πF(q,y < 0) =Y(q,y) −y pdp n0(p)+Y(q,y) −y pdp ×E−(p;q,y) 0 dES1(p, E),(21) 1 2πF(q,y > 0) =Y(q,y) y pdpn0(p) +y 0 pdp E−(p;q,y) E+(p;q,y) dE +Y(q,y) y pdp E−(p;q,y) 0 dES1(p, E). (22) To analyze how the scaling function and the nucleon momentum distribution are connected, we proceed by evaluating the derivatives of the scaling function Fwith respect to yand q. Making use of Leibniz’s formula and choosing (p;q,y)as the three remaining independent variables, after some algebra we finally get the following results. A. Negativeyregion 1 2π ∂F ∂y =Yn 0(Y)∂Y ∂y −yn 0(−y) +Y −y pdp ∂E− ∂y S1(p, E−),(23) 1 2π ∂F ∂q =Yn 0(Y)∂Y ∂q +Y −y pdp ∂E− ∂q S1(p, E−). (24) Making use of the limits in Eq. (16) and assuming the residual mass M0 Bto be much larger than the momenta, |y|,p,q,wesimplyhave ∂E− ∂y ≃q+y Eq+y ,∂E− ∂q ≃q+y Eq+y−q−p Eq−p .(25) Likewise, the derivatives of Yreduce to ∂Y/∂y ≃1 and ∂Y/∂q ≃2. Introducing these results in the general expressions in Eqs. (23) and (24), we get 1 2π ∂F ∂y =Yn 0(Y)−yn 0(−y) +q+y Eq+yY −y pdpS 1(p, E−),(26) 1 2π ∂F ∂q =2Yn 0(Y)+Y −y pdp ×q+y Eq+y−q−p Eq−pS1(p, E−),(27) with E−and Ygiven in the thermodynamic limit by Eqs. (18) and (19). Note that the excited-state contribution in the spectral function, that is, S1, is evaluated at energies along the curve E−. For qsufficiently large, q−y, the upper limit Ycan be safely taken to ∞, and as limY→∞ Yn 0(Y)=0, the expressions for the derivatives simplify to 1 2π ∂F ∂y =−yn 0(−y)+q+y Eq+y∞ −y pdpS 1(p, E−),(28) 1 2π ∂F ∂q =∞ −y pdpq+y Eq+y−q−p Eq−pS1(p, E−).(29) If we further assume that S1is small for large values of p, so that the main contribution to the integral Eq. (29) comes from p≃−y, then we get lim q→∞ ∂F ∂q =0,(30) namely, scaling of the first kind (the scaling function Floses its dependence on q). We also observe that, because at a fixed value of y the integration region in Eq. (27) increases with qand the integrand is a positive function, the asymptotic value F(y) is reached from below (i.e., monotonically increasing as a function of q) in any PWIA approach, in contrast with what experimental data seem to indicate [11,12,22]. This is clearly illustrated in Fig. 2, where the integration region is shown for different values of the momentum transfer at fixed y, and it is also consistent with results shown in Figs. 3(a) and 4(a). In Fig. 3we present the superscaling function f(ψ) evaluated within the framework of the relativistic PWIA (RPWIA) (see Refs. [24] and [25] for details) for different qvalues and plotted 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 p [GeV/c] 0.1 0.2 0.3 0.4 ε [GeV] q=0.5 GeV/c q→∞ q=0.7 GeV/c q=1.0 GeV/c -y FIG. 2. (Color online) Integration region in the (E,p)plane for y=−0.1 GeV/c and 12C as the target selected. Each curve corresponds to E−for a different momentum transfer. 055502-4 SCALING FUNCTION AND NUCLEON MOMENTUM ... PHYSICAL REVIEW C 81, 055502 (2010) -0.7-0.6 -0.5-0.4 -0.3-0.2 -0.1 0 ψ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 f(ψ) q=0.5 GeV/c q=0.6 q=0.7 q=0.8 q=0.9 q=1.0 0 0.2 0.4 0.6 0.8 11.2 ψ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (a) (b) FIG. 3. (Color online) Superscaling function f(ψ) for negative (a) and positive (b) values of the scaling variable ψ. Results correspond to 12C(e,e) evaluated in RPWIA for different momentum transfers. against the superscaling variable ψin the negative-ψregion (below the QEP). This variable is given by [10,12] ψ=1 √ξF λ−τ (1 +λ)τ+κ√τ(1 +τ) ,(31) where λ≡ω/2mN,κ≡q/2mN, and τ≡|Q2|/4m2 N=κ2− λ2. The scaling variables yand ψare closely connected [12]: ψ=y kF⎡ ⎣1+1+m2 N q2 1 2ηFy kF+Oη2 F⎤ ⎦≃y kF , (32) where ηF=kF/mN, and, as noted above, the superscaling function fis connected to Fvia f≡kF×F, with kFthe -0.7-0.6 -0.5-0.4 -0.3-0.2 -0.1 0 y/kF 0.2 0.3 0.4 0.5 0.6 0.7 0.8 f(y/kF) q=0.5 GeV/c q=0.6 q=0.7 q=0.8 q=0.9 q=1.0 0 0.2 0.4 0.6 0.8 1 y/kF 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (a) (b) FIG. 4. (Color online) Superscaling function for negative (a) and positive (b) values of the dimensionless scaling variable y/kF. Results correspond to 12C(e,e) evaluated in RPWIA for different momentum transfers. Fermi momentum. The curves in Fig. 3may be compared with the RPWIA results for the superscaling function, now for negative and positive values of the dimensionless scaling variable y/kFobtained using the quadratic form of Eq. (32); see Fig. 4.Asshown,atfixedψ(or y/kF) the function f(ψ) increases with qin accordance with the previous discussion. The basic results shown in Figs. 3and 4demonstrate that ψ and y/kFcan be used interchangeably as long as one does not focus on the few percent differences seen in the figures, namely, for large magnitudes of the scaling variables. In showing the results we choose 12C as an illustrative example. Indeed this nucleus is relevant for many neutrino oscillation experiments, where superscaling ideas can be used to make reliable predictions of neutrino-nucleus cross sections [28]. Moreover, the analysis of the world data performed in Ref. [11] points to an excellent superscaling in the so-called scaling region (ψ<0) for nuclei with A⩾12. Note, however, that even the 4He data display a very good superscaling behavior for large negative values of the scaling variable (ψ<−0.2), while at the QEP there is a 10% violation owing to the very different spectral function of the lightest nuclei. B. Positive-yregion In this case, as shown in Fig. 1(right), the integration region in the (p, E) plane is limited by the two curves, E+ and E−, in the missing momentum region [0,y]. This makes the derivative analysis somewhat more complicated. Moreover, the experimental data show that scaling arguments of the first kind do not apply to the function F(q,ω) in this region; that is, Fdoes not become a function dependent only on the scaling variable y. On the contrary, it shows a strong dependence on the momentum transfer q. As already mentioned, this is because of the important contributions beyond the IA contained in the transverse channel. Therefore, although the analysis that follows is applied to F(q,y), it should be clearly stated that only the use of the “universal” (namely, longitudinal) superscaling function fL, in particular, the study of its derivative with respect to the scaling variable in the positive-yregion, can reveal important effects not accounted for by the results obtained in the negative-yscaling region. After some algebra, the derivatives of the scaling function F(q,y)aregivenby 1 2π ∂F ∂y =Yn 0(Y)∂Y ∂y −yn 0(y) +Y(q,y) 0 pdpS 1(p, E−)∂E− ∂y  −y 0 pdpS 1(p, E+)∂E+ ∂y ,(33) 1 2π ∂F ∂q =Yn 0(Y)∂Y ∂q  +Y(q,y) 0 pdpS 1(p, E−)∂E− ∂q  −y 0 pdpS 1(p, E+)∂E+ ∂q .(34) 055502-5 CABALLERO, BARBARO, ANTONOV, IVANOV, AND DONNELLY PHYSICAL REVIEW C 81, 055502 (2010) As in the previous case, from the general expressions for E±given in Eq. (16) and assuming the thermodynamic limit, we get ∂E± ∂y ≃q+y Eq+y ,∂E± ∂q ≃q+y Eq+y−q±p Eq±p ,(35) and the derivatives reduce to 1 2π ∂F ∂y =Yn 0(Y)−yn 0(y)+q+y Eq+y ×Y 0 pdpS 1(p, E−)−y 0 pdpS 1(p, E+), (36) 1 2π ∂F ∂q =2Yn 0(Y)+q+y Eq+yY 0 pdpS 1(p, E−) −y 0 pdpS 1(p, E+) +y 0 pdpq+p Eq+p S1(p, E+) −Y 0 pdpq−p Eq+p S1(p, E−).(37) Moreover, in the limit of the momentum transfer large enough, qy, so that the condition limY→∞ Yn 0(Y)=0 holds, the expressions of the derivatives result: 1 2π ∂F ∂y =−yn 0(y)+q+y Eq+y∞ 0 pdpS 1(p, E−) −y 0 pdpS 1(p, E+),(38) 1 2π ∂F ∂q =∞ 0 pdp q+y Eq+y−q−p Eq−pS1(p, E−) −y 0 pdp q+y Eq+y−q+p Eq+pS1(p, E+). (39) Note that in the limit in which ycan be neglected compared with q, that is, (q+y)/Eq+y→q/Eq, the same comment applies to the ratio (q+p)/Eq+pinvolved in the second integral in Eq. (39), as pis limited within the range [0,y]. Thus, in such a limiting case, y 0 pdp q+y Eq+y−q+p Eq+pS1(p, E+)≃0forqy, (40) and only the first integral in Eq. (39) survives. Furthermore, if the spectral function is such that we can neglect pcompared with qinside the integral, we again get scaling of the first kind: limq→∞(∂F/∂q)=0. This is strictly valid only for very large values of qand it is entirely based on the approximations leading to the expression in Eq. (4) that connects the scaling function to the spectral function. As shown in Fig. 4(b) (positive-yregion), the RPWIA scaling function shows a negligible dependence on the momentum transfer for 0.3y/kF0.8(qy), whereas for larger y/kF, scaling of the first kind begins to be slightly violated. The experimental scaling function extracted from the analysis of data at intermediate qvalues (less than or of the order of the nucleon mass) shows very important scaling violations in the region above the QEP (positive values of y). With regard to the dependence of the scaling function F with qat fixed y, we get different behaviors for small and large values of y. Indeed from Eq. (39) we observe that in the case of ybeing very small (in the vicinity of 0), the second integral in Eq. (39) can be neglected. As the integrand in the remaining integral is positive, we get ∂F/∂q > 0; that is, the scaling function grows with q. This behavior is in accordance with that already shown in the negative-yregion. On the contrary, for increasing values of ythe first integral in Eq. (39) is expected to diminish significantly, as the excitation energy curve E− along which S1is evaluated lies much higher than E+(see Fig. 5), and it is reasonable to expect that S1(p, E) gets its main contribution for values of the momentum and energy that are not too large. For ylarge enough, only the second integral in Eq. (39) survives, and because its integrand is also positive, the minus sign in front of it leads to ∂F/∂q < 0; that is, the scaling function Fdecreases with q, changing its behavior with respect to the previous cases. It is interesting to point out that this result is consistent with the integration regions shown in Fig. 5where, for increasing momentum transfer, the curve E+moves to higher excitation energies in the (E,p) plane. This means that as qgoes up, regions at low (E,p) values, where the spectral function mostly resides, are not kinematically accessible anymore. A similar argument can be applied to the case of very small values of y[see Fig. 5(a)]. However, here the integration region lost as E+goes up with increasing qis less important than the effects introduced by the growing integration region attached to E−. This general behavior is also in accordance with the RPWIA results for the superscaling function f shown in Fig. 4(b) (positive values of y), or, alternatively, Fig. 3(b). One sees that fincreases with qup to ψ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 ε [GeV] 00.20.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 p [GeV/c] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ε [GeV] q=0.5 GeV/c q→∞ q=0.7 GeV/c q=1.0 GeV/c q=0.5 GeV/c q→∞ q=0.7 GeV/c q=1.0 GeV/c (a) (b) FIG. 5. (Color online) As Fig. 2, but now for positive values of y:(a)y=0.1 GeV/c;(b)y=0.5 GeV/c. 055502-6 SCALING FUNCTION AND NUCLEON MOMENTUM ... PHYSICAL REVIEW C 81, 055502 (2010) 0.4; that is, y/kF∼0.364 (q=0.5GeV/c), y/kF∼0.375 (q=1.0GeV/c), and y/kF∼0.382 (q=∞GeV/c), with kF=1.2fm −1the Fermi momentum. This corresponds to y∼ 0.1GeV/c, which is the situation represented in Fig. 5(a).Also note that the qdependence of fin the region where y/kF> 0.3 shown in Fig. 4is very weak. Examination of Figs. 5(a) and 5(b) shows that for large yvalues, the energy curves E± lie very high, and hence, as qincreases, the integrals involved incorporate only additional contributions that are very small, leading to a very weak variation with momentum transfer. III. NUCLEON MOMENTUM DISTRIBUTION AND THE SCALING FUNCTION In the previous section we have derived general integrodifferential equations connecting the derivatives of the scaling function, ∂F/∂y and ∂F/∂q, with the spectral function. Based on these results applied to both negative and positive values of y, in what follows we revisit the “usual” procedure to obtain the nucleon momentum distribution function from the analysis of QE (e,e) data. Because the kinematics of electron scattering lead to finite integration limits, we may not a priori draw any strong conclusions about the “true” momentum distribution given as n(p)≡∞ 0dES(p, E), namely, the integral of the spectral function up to infinite excitation energy. However, assuming the spectral function to reside mostly in the (p, E) plane at values of pand Ethat are not too large, the previous analyses applied to negative- and positive-yregions lead to different results, thus providing important and complementary information on how the energy and momentum are distributed within the spectral function. The usual procedure considered in previous work [3,4]to generate the nuclear momentum distribution from the scaling function has been based on the expression n(k)=−1 2πy ∂F ∂y |y|=k ,(41) which has been widely applied in the negative-yregion. In what follows we extend this study to the positive-yregion based on the universal superscaling function introduced from the analysis of the separated longitudinal data. Making use of the general expressions given by Eqs. (26) and (36) and assuming the limiting case limY→∞ Yn 0(Y)=0, which is valid if the momentum transfer qis sufficiently large, the momentum distribution functions can be written as follows: ny<0(q,k)=n0(−y)−q+y yEq+y∞ −y pdpS 1(p, E−)−y=k =n0(k)+q−k kEq−k∞ k pdpS 1(p, E−),(42) ny>0(q,k)=n0(y)−q+y yEq+y∞ 0 pdpS 1(p, E−) −y 0 pdpS 1(p, E+)y=k=n0(k)−q+k kEq+k ×∞ 0 pdpS 1(p, E−)−k 0 pdpS 1(p, E+). (43) As observed, both expressions receive contributions from the A−1 system ground state, n0(k), as well as from the excited states described through S1(p, E). Although using the same notation for the excitation energy E−, note that the Ecurves that enter in the spectral function S1in Eqs. (42) and (43) are very different (see Figs. 2and 5). Conclusions about the particular behavior of the previous expressions can only be drawn based on a specific model for the spectral function; however, it is illustrative to discuss some general, “model-independent” properties. For negative ythe function in Eq. (42) exceeds the purely ground-state contribution, that is, ny<0(q,k)>n 0(k) for all q,k values. This means that the contribution from the excited states adds to the ground-state momentum distribution. Concerning the specific role played by each one of the two terms in Eq. (42), it is difficult to drawstringent conclusions without having control over S1. As the momentum kgrows, the contribution of the integral in Eq. (42) is expected to diminish significantly (S1 mostly residing at momenta and excitation energies that are not too large). A similar comment also applies to the ground-state contribution, which decreases as kgets larger. The analysis of Eq. (43) in the positive-yregion differs because of the relative contributions provided by the two integrals linked to the excited states. In this case the global response ny>0(q,k) can be smaller and/or larger than the purely ground-state contribution, n0(k), depending on the specific missing momentum value. In what follows we discuss some particular situations in detail, thereby drawing some preliminary conclusions on the general behavior shown by ny≶0(q,k). Let us start by considering the value of the nucleon momentum kto be in the vicinity of 0. Thus, neglecting kcompared with the momentum transfer q(kq) and assuming ∞ 0pdpS 1(p, E−) k 0pdpS 1(p, E+)→0, we can write ny<0(q,k)≃n0(k)+q kEq∞ k pdpS 1(p, E−)>n 0(k), (44) ny>0(q,k)≃n0(k)−q kEq∞ 0 pdpS 1(p, E−)<n 0(k). (45) From these results the following relation (valid for ksmall enough) occurs: ny>0(q,k)⩽n0(k)⩽ny<0(q,k).(46) Moreover, from Eqs. (44) and (45) the ground-state contribution is roughly given as n0(k)≃[ny<0+ny>0]/2. As the nucleon momentum kgrows, the two functions ny<0(q,k) and ny>0(q,k)inEqs.(42) and (43) get closer, crossing each other at some specific k, such that ny>0(q,k)> ny<0(q,k) for larger k. From the integration region in the (E-p) plane shown in Fig. 5, and assuming most of the strength in the spectral function to be located at not too high pand E, we can conclude that for intermediate to high missing momentum values the main contribution in ny>0(q,k) comes from the second integral in Eq. (43); that is, ny>0(q,k)≃[(q+k)/(kEq+k)] k 0pdpS 1(p, E+). 055502-7 CABALLERO, BARBARO, ANTONOV, IVANOV, AND DONNELLY PHYSICAL REVIEW C 81, 055502 (2010) FIG. 6. (Color online) Average fexp L(ψ) compared with the Gumbel distribution in Eq. (47) (solid curve) and a fit of the experimental data (dashed curve). To prove these general properties, in what follows we present results based on the derivative analysis making use of the superscaling function f(ψ). To simplify the calculations we represent f(ψ) by means of the Gumbel probability density function (i.e., the derivative of the Gumbel distribution): fG(ψ)=1 σexp −(ψ−µ) σexp −exp −(ψ−µ) σ. (47) In our case the values of the parameters are µ=0 and σ=0.67 [fmax G=fG(0) =0.55]. In Fig. 6we compare the Gumbel distribution [Eq. (47)] with fL exp(ψ) and a fit of the experimental data [22]. As shown, the Gumbel distribution nicely fits the data. Moreover, it fulfills the unitarity condition +∞ −∞ f(ψ)dψ =1. The nucleon momentum distribution is evaluated through the derivative of the scaling function by using Eq. (41) and recalling that f=kFF, thus getting n(k)=−1 2πy 1 kF df (ψ(y)) dy |y|=k ,(48) which, using the approximate relation ψ≃y/kF, can be presented in the form n(k)=− 1 2πk 1 kFdf (ψ) d(kF|ψ|)kF|ψ|=k .(49) Note that if the superscaling function is not symmetric with respect to ψ, as is the case for the experimental data, Eq. (49) yields different momentum distributions for negative and positive values of ψ, which are denoted n<and n>, respectively. On the contrary, symmetric scaling functions, like the RFG one, lead to n<=n>. In the case of the Gumbel distribution, we get (setting µ=0) dfG(ψ) dψ =1 σ(e−ψ/σ −1)fG(ψ),(50) FIG. 7. (Color online) Nucleon momentum distribution extracted through the derivative of the superscaling function given by the Gumbel probability density in Eq. (47). Results corresponding to negative (solid line) and positive (dashed line) values of the scaling variable are compared. which leads to n< G(k)=1 2πσk2 Fk[ek/(σkF)−1]fG(−k/kF),(51) n> G(k)=1 2πσk2 Fk[1 −e−k/(σkF)]fG(k/kF).(52) In Fig. 7we present the results for nψ<0(k)=n< G(k)/2 (solid line) and nψ>0(k)=n> G(k)/2 (dashed line), with n< G(k) and n> G(k)giveninEqs.(51) and (52)(atkF=1.2fm −1). As expected, n< G(k) and n> G(k) (and nψ<0and nψ>0, respectively) coincide in the limiting case k=0: n> G(0) =n< G(0) =1 2πσ3k3 Fe.(53) For missing momenta up to k∼1fm −1the main contribution resides in n<, which is in accordance with Eq. (46) and the general discussion presented above. At k≃1.3–1.4 fm−1, that is, kclose to the Fermi momentum, n<and n>cross each other, with n>being much higher for larger kvalues. In fact, whereas n<shows a steep slope when kincreases, which is in accordance with results based on independent-particle model descriptions, n>presents a high momentum tail very far from n<and, hence, from shell-model results (see next section). As already explained, this tail at intermediate to high kis linked to the much larger contribution given by the spectral function S1when evaluated along the curve E+instead of E−.This general behavior is illustrated in Fig. 8, where the contour curves E±corresponding to positive and negative yvalues are presented. The presence of the tail at high momentum values in the nucleon momentum distribution is a clear signature of the importance of nucleon-nucleon correlations. Because the spectral function maps very different regions in the (E−k) plane for negative and positive y(Fig. 8), the joint analysis of the two kinematical regions can provide important clues in the knowledge of NN correlations. It should be pointed out that the functions nψ<0(k) and nψ>0(k), evaluated through Eq. (49) 055502-8 SCALING FUNCTION AND NUCLEON MOMENTUM ... PHYSICAL REVIEW C 81, 055502 (2010) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 p [GeV/c] 0 0.1 0.2 0.3 0.4 0.5 ε [GeV] y=-0.2 GeV/c y=+0.2 GeV/c y=-0.5 GeV/c y=+0.5 GeV/c y=0 ε+ ε- FIG. 8. (Color online) Integration region in the (E,p)plane for q=1 GeV/c and different, negative and positive, values of the scaling variable y. The contour curves E±in both regions are represented. and presented in Fig. 7, are normalized to different values connected with the different areas subtended by the Gumbel distribution function fG(ψ) at negative and positive ψ, that is, 0.37 (for ψ<0) and 0.63 (ψ>0). In particular, it has been shown in Refs. [24] and [26]in the framework of relativistic nuclear models that the large positive-ψtail of the scaling function is closely related to FSIs, while the negative-ψregion is more affected by initial-state correlations, as also shown in the next section using the CDFM model. The possibility of connecting different aspects of the momentum distribution to initial- and final-state physics will be explored further in future work. IV. NUCLEON MOMENTUM DISTRIBUTION WITHIN THE COHERENT DENSITY FLUCTUATION MODEL In this section we give, as an example, the results for the nucleon momentum distribution extracted from the scaling function, obtained within the framework of a particular nuclear model, namely, the coherent density fluctuation model (CDFM) [29,30]. The latter is a natural extension to finite nuclei of the RFG model within which the scaling variable ψwas introduced.1The CDFM is based on the generator coordinate method [31] and includes long-range NN correlations (LRC) of collective type. In [32,33] the scaling function was defined within the CDFM using the RFG scaling function [10,34–36] and applied it to various processes [32,33,37–40]. In the CDFM model [29,30], the one-body density matrix ρ(r,r) is an infinite superposition of one-body density matrices ρx(r,r) corresponding to single Slater determinant wave functions of systems of Afree nucleons homogeneously distributed in a sphere with radius x, density ρ0(x)≡3A/(4πx3), and Fermi momentum kF(x)≡[3π2 2ρ0(x)]1/3≡α x[with α≡ 1The scaling variable ψdiffers from ψby a phenomenological energy shift Es≃20 MeV (for 12C) introduced to reproduce the experimental position of the QEP: ψ(q,ω)=ψ(q,ω −Es). (9π 8A)1/3∼ =1.52A1/3]: ρ(r,r)=∞ 0|F(x)|2ρx(r,r)dx. (54) The weight function |F(x)|2can be expressed in an equivalent way either by means of the density distribution [29,30,33], |F(x)|2=− 1 ρ0(x) dρ(r) dr r=x at dρ(r) dr ⩽0,(55) or by means of the nucleon momentum distribution [33], |F(x)|2=−3π2 2 α x5 dn(k) dk k=α/x at dn(k) dk ⩽0.(56) In Eqs. (55) and (56) ρ(r)dr=A, n(k)dk=A, ∞ 0|F(x)|2dx =1. (57) In the version of the CDFM approach suggested in Refs. [32] and [33], the scaling function has the form f(ψ)=α/(kF|ψ|) 0|F(x)|2fRFG(x,ψ)dx, (58) where the RFG scaling function is fRFG(x,ψ)=3 41−kFx|ψ| α2 ×⎧ ⎨ ⎩ 1+!xmN α"2kFx|ψ| α2 ×⎡ ⎣2+α xmN2 −21+α xmN2⎤ ⎦⎫ ⎬ ⎭ . (59) In the CDFM the Fermi momentum kFis calculated for each nucleus by kF=∞ 0 kF(x)|F(x)|2dx =∞ 0 α x|F(x)|2dx (60) and is not a fitting parameter, as it is in the RFG model. By using Eqs. (55) and (56)inEqs.(58) and (60), the CDFM scaling function f(ψ) and kFcan be expressed equivalently by the density and momentum distributions [33]: f(ψ)=4π Aα/(kF|ψ|) 0 ρ(x)x2fRFG(ψ,x) +x3 3 dfRFG(ψ,x) dx dx, (61) where fRFG(ψ,x) is given by Eq. (59), and f(ψ)=4π A∞ kF|ψ| n(k)k2fRFG(ψ,k) +k3 3 dfRFG(ψ,k) dk ,(62) 055502-9