Unified description of lepton-nucleus scattering within the Spectral Function formalism

Transcription

1 Unified description of lepton-nucleus scattering within the Spectral Function formalism Scuola di dottorato Vito Volterra Dottorato di Ricerca in Fisica XXVIII Ciclo Candidate Noemi Rocco ID number Thesis Advisor Prof. Omar Benhar Co-Advisor Dr. Alessandro Lovato A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics December 2015

2 Thesis defended on 22 January 2016 in front of a Board of Examiners composed by: Prof. Luciano Maria Barone (chairman) Prof. Cinzia Da Vià Prof. Teresa Rodrigo Unified description of lepton-nucleus scattering within the Spectral Function formalism Ph.D. thesis. Sapienza University of Rome ISBN: Noemi Rocco. All rights reserved This thesis has been typeset by L A TEX and the Sapthesis class. Version: March 25, 2016 Author s noemi.rocco@roma1.infn.com

3 iii Abstract In this work we focused on the study of the electromagnetic response of nuclei in different kinematical regimes. The formalism we adopted is based on factorization and nuclear spectral functions. In particular we analyzed the mechanisms leading to the appearence of two particle - two hole final states, which include the interference between amplitudes involving one- and two-nucleon currents. The latter arise from processes in which the interaction with the beam particles involves a meson exchanged between the target nucleons. The inclusion of meson exchange currents is a necessary step to develop a quantitative description of the neutrino-nucleus cross section, required for the high precision determinations of the oscillation parameters. In scattering processes involving interacting many-body systems, two particle-two hole final states can be produced through the action of both one- and two-nucleon currents. However, in order for the matrix element of a one-body operator between the target ground state and a two particle-two hole final state not to be vanishing, the effects of dynamical nucleon-nucleon correlations must be included in the description of the nuclear wave functions. Correlations give rise to virtual scattering between nucleons in the target nucleus, leading to the excitation of the participating particles to continuum states. The initial state correlation contribution to the two particle-two hole amplitude arises from processes in which the beam particle couples to one of these high-momentum nucleons. The formalism based on factorization of the nuclear transition matrix elements allows to combine a fully relativistic description of the interaction vertex with an accurate treatment of the target, in which the effect of dynamical nucleon-nucleon correlations is properly taken into account. The nuclear cross section arising from interactions involving the one-nucleon currents will be given by the product of the cross section of the elementary scat- tering process and the nuclear spectral function derived within non relativistic nuclear many body theory describing the intrinsic property of the target nucleus. This scheme has been extended by means of the introduction of the two-nucleon spectral function, allowing for a consistent treatment of the amplitudes involving two-nucleon currents. The calculation of interference between the one- and twonucleon current operators has been carried out taking into account the product of the nuclear amplitudes entering the definition of both the one- and two-nucleon spectral functions. Our results show that the inclusion of the meson exchange currents provides a quantitative account of the experimental data in the dip region. The main achievement of the work discussed in this Thesis is the development of a novel approach, based on a generalisation of the factorisation ansatz, underlying the spectral function formalism. This scheme, allowing for a consistent treatment of one- and two-nucleon current contributions, appears to be quite promising for applications to neutrino scattering.

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5 v Contents Introduction 1 1 The lepton-nucleus cross section within the impulse approximation Nuclear response tensor Non relativistic regime Impulse approximation regime Nuclear spectral function Nuclear and nucleon structure functions Quasi elastic scattering Resonance production and deep inelastic scattering Comparison to experimental data The nucleon spectral function in nuclear matter Correlated basis function perturbation theory Lowest order approximation Significance of nucleon-nucleon correlations Reaction mechanisms beyond the impulse approximation y-scaling and scaling violations Role of meson-exchange currents in the nonrelativistic regime Meson-exchange currents in the extended factorization ansatz Factorization of the matrix element of the two-body current Contribution of initial state correlations Nuclear response tensor in the 2p2h sector Meson-Exchange Currents (MEC) MEC contribution to the transverse response: direct term MEC contribution to the transverse response: exchange term Contribution of interference terms to the transverse response Comparison to experimental data Non relativistic reduction of the response in the 2p2h sector Non relativistic limit of the two-nucleon current contributions Reduction of the interference term between one- and two-body currents 64 Conclusions 67

6 vi Contents A Nuclear overlaps relevant to the nucleon spectral function 69 A.1 Factorization of Slater determinants A.2 Correlation matrix elements B Calculation of the 2h nuclear overlap at two-body cluster level 75 C Relativistic 2p2h phase space 77 D Derivation of meson exchange currents in free space 79 E Matrix elements of the meson-exchange currents 85 E.1 π and contributions to the transverse response: direct term E.2 π and contributions to the transverse response: exchange term.. 88 E.3 π and contribution to the 2p2h interference term F Non relativistic reduction of current matrix elements 95 F.1 Non relativistic two-body transverse response F.2 Non relativistic 2p2h interference term Bibliography 99

7 1 Introduction Experimental searches of neutrino oscillations exploit neutrino-nucleus interactions to infer the properties of the beam particles, which are largely unknown. The use of nuclear targets as detectors, while allowing for a much needed increase of the event rate, entails non trivial problems, as the interpretation of the observed signal requires a quantitative understanding of neutrino-nucleus scattering. Given the present experimental accuracy, the treatment of nuclear effects is indeed acknowledged as one of the main sources of systematic uncertainty [1]. The issue of nuclear effects will be even more critical for future experiments, such as the Deep Underground Neutrino Experiment (DUNE) in the United States [2], designed to search for CP violation in the leptonic sector. Achieving this goal will in fact require measurements of both neutrino and antineutrino oscillations with percent accuracy. The authors of Ref. [3] argued that the uncertainty on neutrino energy reconstruction [1], arising from the limitations of the nuclear models employed in data analysis, may turn out to severely hamper the extraction of the CP violating phase from an oscillation result. Experimental studies of neutrino-nucleus interactions carried out over the past decade [4, 5, 6, 7] have provided ample evidence of the inadequacy of the relativistic Fermi gas model, routinely employed in event generators, to account for both the complexity of nuclear dynamics and the variety of reaction mechanisms other than single nucleon knock out contributing to the observed cross section. A striking manifestation of this problem is the large discrepancy between the predictions of Monte Carlo simulations and the double differential charged current quasi elastic cross section measured by the MiniBooNE Collaboration using a carbon target [7]. Over the past decade, a growing effort has been made to improve the existing models of neutrino-nucleus scattering using the knowledge of the nuclear response acquired from experimental and theoretical studies of electron scattering. Electron-nucleus scattering cross sections are usually analyzed at fixed beam energy E e, and electron scattering angle θ e as a function of the energy loss ω. As an example, Fig. 0.1 shows the typical behavior of the double differential cross sections of the inclusive process e + A e + X, (0.1) in which only the outgoing lepton is detected, at beam energy around 1 GeV. Here, A and X denote the target nucleus and the undetected hadronic final state, respectively. It is apparent that the different reaction mechanisms, yielding the dominant contributions to the cross section at different values of ω (corresponding to different values of the Bjorken scaling variable x = Q 2 /2Mω, where M is the nucleon mass

8 2 Introduction Meson exchange! currents Figure 0.1. Schematic representation of the inclusive electron-nucleus cross section as a function of energy loss. and Q 2 = 4E e (E e ω) sin 2 θ e /2) can be easily identified. The bump centered at ω Q 2 /2M, or x 1, the position and width of which are determined by the momentum and removal energy distribution of the struck particle, corresponds to single nucleon knockout, while the structure visible at larger ω reflects the onset of coupling to two-nucleon currents, arising from meson exchange processes, excitation of nucleon resonances and deep inelastic scattering. The available theoretical models of electron-nucleus scattering provide an overall satisfactory description of the data over a broad kinematical range. In particular, in the region in which quasi elastic scattering dominates, the data is generally reproduced with an accuracy of few percent (for a recent review on electron-nucleus scattering in the quasi elastic sector, see Ref. [8]). The main difficulty involved in the generalization of the approaches successfully employed to analyze electron scattering data to the case of neutrino interactions arises from the non monoenergetic nature of neutrino beams. Because neutrinos are always produced as secondary decay products, their energy, E ν, is not sharply defined, but broadly distributed according to a flux Φ(E ν ). For example, the observed charged current cross section for given kinetic energy and emission angle of the charged lepton picks up contributions from the cross sections corresponding to different beam energies and, consequently, to different values of the energy transfer ω and different reaction mechanisms with a weight dictated by the neutrino flux. Many authors have suggested that the excess charged current quasi elastic cross section observed by the MiniBooNE collaboration is to be ascribed to the occurrence of events with two particle-two hole final states, not taken into account by the relativistic Fermi gas model employed for data analysis [9, 10, 11]. As pointed out by the authors of Ref. [9], improving the treatment of nuclear effects in the analysis of neutrino experiments will require the development of a

9 Introduction 3 comprehensive and consistent description of neutrino-nucleus interactions, taking into account all relevant reaction mechanisms. As a first step in this direction, in this Thesis we discuss the three different reaction mechanisms leading to the appearance of two-particle two-hole final states in electron nucleus scattering. We will argue that final state interactions do not play a critical role, and focus on nucleon-nucleon correlations in the initial state and on processes involving two-nucleon currents. The main original contribution of our work is the generalization of the factorization ansatz underlying the spectral function formalism, which allows to treat the nuclear matrix elements involving one- and two-nucleon currents in a consistent manner. Within the proposed approach a fully relativistic treatment of the currents can be combined with the accurate description of the nuclear ground state provided by non relativistic nuclear many-body theory. The numerical results of our calculations of the electron-carbon cross sections are compared to the available data in a variety of kinematical conditions.

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11 5 Chapter 1 The lepton-nucleus cross section within the impulse approximation In this chapter, we discuss the lepton-nucleus cross section, focusing on the case of electron beams. The tensor describing the nuclear response is introduced in Section 1.1, whereas the dynamical model underlying nuclear many-body theory (NMBT), providing the framework for its calculation in the non relativistic regime, is outlined in Section 1.2. In the kinematical region in which the non relativistic approximation is no longer applicable, the approach based on the impulse approximation and the spectral function formalism, discussed in Section 1.3, allows for a consistent treatment of the lepton-nucleus interactions including relativistic effects. The structure of the nuclear spectral function is analyzed in in Section 1.4. while in Section 1.6 the calculated electron-carbon double-differential cross section is compared to experimental data. 1.1 Nuclear response tensor Consider electron-nucleus scattering. The double differential differential cross section of the inclusive (e, e ) process e + A e + X, (1.1) in which an electron of initial four-momentum k e = (E e, k e ) scatters off a nuclear target to a state of four-momentum k e = (E e, k e ), the target final state being undetected, can be written in the one-photon-exchange approximation as ( d 2 σ de e dω e ) A = α2 q 4 E e E e L µν W µν A, (1.2) where α 1/137 is the fine structure constant, dω e is the differential solid angle in the direction specified by k e, and q = k e k e = (ω, q) is the four momentum transfer.

12 6 1. The lepton-nucleus cross section within the impulse approximation The lepton tensor L µν is fully specified by the measured electron kinematical variables. Neglecting the lepton mass it can be cast in the form L µν = 2[k µ e k ν e kν e k µ e g µν (k e k e )], (1.3) with g µν diag(1, 1, 1, 1) and (k e k e ) = E e E e k e k e. All information on target structure and dynamics is comprised in the nuclear response tensor W µν A, defined as W µν A (q, ω) = 0 J A µ X X JA 0 δ ν (4) (P 0 + q P X ), (1.4) X where 0 and X denote the initial and final hadronic states, the four-momenta of which are P 0 (E 0, p 0 ) and P X (E X, p X ), and J µ A is the nuclear electromagnetic current operator. Note that the target ground state 0 is obviously independent of momentum transfer, while the state X includes at least one particle carrying a momentum of order q, and the current operator explicitly depends on q. As a consequence, in the absence of a comprehensive relativistic description of nuclear structure and dynamics, a fully consistent theoretical calculation of the response tensor is only possible in the kinematical regime corresponding to q /m 1, with m being the nucleon mass, where the non relativistic approximation is applicable. 1.2 Non relativistic regime The non relativistic description of nuclear dynamics is based on the tenet that nucleons can be treated as point-like particles, the dynamics of which are described by the Hamiltonian H = i p 2 i 2m + v ij + V ijk. (1.5) j>i k>j>i In the above equation, p i is the momentum of the i-th nucleon, while the potentials v ij and V ijk describe two- and three-nucleon interactions, respectively. The first theoretical description of the nucleon-nucleon (NN) interaction, based on the one-pion-exchange mechanism, was proposed by Yukawa in the 1930s [12]. More realistic potential models have been developed within extensions of of Yukawa s theory, including exchange of the vector bosons ρ and ω whose masses are m ρ = 770 MeV and m ω = 782 MeV as well as of a scalar boson with mass 500 MeV, that can be interpreted as a narrow two-pion resonance [13, 14]. Alternative models of nuclear forces have been also derived from purely phenomenological analyses [15] and, more recently, from approaches inspired to chiral perturbation theory [16]. Phenomenological potentials are obtained from an accurate fit to the available data on the two-nucleon system, in both bound and scattering states, and reduce to the Yukawa one-pion-exchange potential at large distances. The state-of-the-art parametrization of v ij referred to as Argonne v 18 potential [15], is written in the form v ij = 18 n=1 v n (r ij )O n ij, (1.6)

13 1.2 Non relativistic regime 7 with r ij = r i r j and O n 6 ij = [1, (σ i σ j ), S ij ] [1, (τ i τ j )], (1.7) where σ i and τ i are Pauli matrices acting in spin and isospin space, respectively, and S ij is the tensor operator given by S ij = 3 rij 2 (σ i r ij )(σ j r ij ) (σ i σ j ). (1.8) The operators corresponding to n = 7,..., 14 are associated with non static components of the NN interaction, while those corresponding to n = 15,..., 18 account for small violations of charge symmetry. Being fit to the full Nijmegen phase-shift database, as well as to low energy scattering parameters and deuteron properties, the Argonne v 18 potential provides an accurate description of the two-nucleon system by construction. The inclusion of the additional three-body term, V ijk, is needed to explain the binding energies of the three-nucleon systems and nuclear matter saturation properties [17]. The results of Quantum Monte Carlo (QMC) calculations of the energies of both the ground state and the low-lying excited states of nuclei with mass number A 12, shown in Fig. 1.1, turn out to be in excellent agreement with the experimental values, thus showing the remarkable predictive power of NMBT [18]. 17 Energy (MeV) He 6 He Li 7 Li 7/2 5/2 5/2 7/2 1/2 3/ He Argonne v 18 with Illinois-7 GFMC Calculations 8 Li AV18 8 Be AV18 +IL7 9 Li 5/2 1/2 3/2 Expt. 9 Be 7/2 + 5/2 + 7/2 7/2 3/2 3/2 + 5/2 + 1/2 5/2 1/2 + 3/2 10 Be , B C FIG. 3 GFMC energies of light nuclear ground and excited states for the AV18 and AV18+IL7 Hamiltonians compared to experiment. See Table I for references. Figure 1.1. Comparison between the energies of the ground and low-lying excited states of nuclei with A 12 computed within the GFMC scheme using a realistic nuclear TABLE I AV18+IL7 GFMC results for A apple 12 nuclear ground states (Brida et al., 2011;Lovatoet al., 2013;McCutchanet al., 2012; Hamiltonian Pastore et al., 2013, and2014; experimental Pieper and Carlson, data 2015; (taken Wiringa from et al., 2013), Ref. compared [18]). to experimental values (Amroun et al., 1994; NNDC, 2014; Nörtershäuser and et al., 2009; Nörtershäuser et al., 2011; Purcell et al., 2010; Shineret al., 1994; Tilley et al., 2002, 2004). Numbers in parentheses are statistical errors for the GFMC calculations or experimental errors; errors of less than one in the last decimal place are not shown. A fundamental feature of the description of electron- and, more generally, leptonnucleus interactions within NMBT is the possibility of employing a set of charge and current operators consistent with the Hamiltonian of Eq.(1.5). The nuclear electromagnetic current, J µ (J 0, J) is constrained by H through the continuity equation (see, e.g., Ref [19]) A Z(J ; T ) E (MeV) r p [r n] (fm) µ (µ N ) Q (fm 2 ) GFMC Expt. GFMC Expt. GFMC Expt. GFMC Expt. 2 H(1 + ;0) H( 1 ; 1 ) (1) [1.73] (1) He( 1 ; 1 ) 7.72(1) [1.60] (1) He(0 + ;0) 28.42(3) (6) 6 He(0 + ;1) 29.23(2) (3) [2.88] 1.93(1) 6 Li(1 + ;0) 31.93(3) (4) 0.835(1) (2) 0.082(2) 7 He( 3 2 ; 3 ) (3) [3.32(1)] 7 Li( 3 2 ; 1 ) (3) [2.44] 2.31(5) 3.24(1) (2) 4.06(8) 7 Be( 3 ; 1 ) 37.54(3) [2.32] 2.51(2) 1.42(1) 1.398(15) 6.6(2) 2 2 He(0 + ;2) 31.42(3) (2) [2.73] 1.88(2) 8 Li(2 + ;1) 41.14(6) [2.46] 2.20(5) 1.48(2) (2) 3.27(6) 8 Be(0 + ;0) 56.5(1) (1) 8 B(2 +, 1) 37.51(6) [2.10] 1.11(2) (4) 6.83(21) 8 C(0 + ;2) 24.53(3) [1.85] 9 Li( 3 2, 3 ) (4) [2.33] 2.11(5) 3.39(4) (1) 2.74(10) 9 Be( 3 2, 1 ) (2) [2.46] 2.38(1) 1.29(1) (1) 5.29(4) 9 C( 3, 3 ) 38.88(4) [1.99] 1.35(4) (4) 2 2 Be(0 + ;1) 64.4(2) [2.44] 2.22(2) 10 B(3 + ;0) 64.7(3) (1) 1.76(1) (3) 8.47(6) 10 C(0 + ;1) 60.2(2) [2.25] 12 C(0 + ;0) 93.3(4) J + i[h, J 0 ] = 0. (1.9)

14 8 1. The lepton-nucleus cross section within the impulse approximation Since the NN potential v ij does not commute with the charge operator J 0, the above relation implies that J µ A involves both one- and two-nucleon contributions. As a consequence, it can be written in the form J µ A = i j µ i + j µ ij. (1.10) j>i In the above equation, j µ i describes interactions involving a single nucleon, while j µ ij accounts for processes in which the beam particle couples to the meson-exchange currents (MEC), i.e. currents arising from meson exchange between two interacting nucleons. Within the conventional meson-exchange formalism [20, 21], the dominant static component of the two-nucleon potential arises from exchange of pseusoscalar (πlike) and vector (ρ-like) mesons, and the corresponding charge and current operators are projected out of the static components of the potential. As a consequence, the resulting current is conserved by construction. Nonrelativistic meson-exchange currents (MEC) have been used in analyses of a variety of electroweak processes in light nuclei at low and intermediate values of energy and momentum transfer. Using Monte Carlo computational techniques and taking MEC into account, a good agreement has been achieved between theoretical predictions and experimental data for the M 1 and E2 radiative transition rates between low-lying states [22, 23], β-decays and electron- and muon-capture rates [24, 25], elastic and inelastic form factors extracted from (e, e ) cross sections [26, 27, 28, 29] and low-energy radiative and weak capture reactions [20]. 1.3 Impulse approximation regime At momentum transfer exceeding 400 MeV, the non relativistic approximation is expected to break down. However, in the kinematical region corresponding to λ π/ q d, d being the average NN distance in the target nucleus, nuclear scattering can be approximated with the incoherent sum of scattering processes involving individual nucleons, as illustrated in Fig This is the conceptual basis of the impulse approximation (IA), which obviously amounts to neglecting the contribution of the two-nucleon current, i.e. to setting J µ A i j µ i. (1.11) 2 2 i i Figure 1.2. Schematic representation of the IA scheme, in which the nuclear cross section is replaced by the incoherent sum of cross sections describing scattering off individual bound nucleons, the recoiling (A-1)-nucleon system acting as a spectator. Under the further assumption that the struck nucleon is decoupled from the spectator particles, the final state X can be written in a factorized form according

15 1.3 Impulse approximation regime 9 to X x, p x R, p R, (1.12) where x denotes the hadronic state produced at the electromagnetic vertex, carrying momentum p x, and R the state describing the residual system, carrying momentum p R. Using Eq. (1.12), we can replace X X d 3 p x x, p x p x, x d 3 p R R, p R p R, R. (1.13) x R X Substitution of Eq.(1.12) and insertion of a complete set of free nucleon states, satisfying d 3 k N, k k, N = 11, (1.14) N=p,n where the index N = p, n labels either a proton or a neutron, leads to factorization of the nuclear transition matrix element of Eq.(1.4). The resulting expression is ( 0 J µ A X = m ) 1/2 p pr 2 + m N=p,n 0 R, 2 R ; N, p R p R, N j µ i x, p x, i (1.15) where the factor (m/ p R 2 + m 2 ) 1/2, takes into account the implicit covariant normalization of the state p R, N in the matrix element of j µ i. Using the above relations, the hadronic tensor can be written in the form W µν A = ( ) d 3 p R d 3 p x 0 R, p R ; N, p R 2 m (1.16) pr R,x N=p,n 2 + m 2 i p R, N j µ i x, p x p x, x j ν i N, p R δ (3) (q p R p x )δ(ω + E 0 E R E x ), (1.17) where E 0 is the target ground state energy, E R = p R 2 + MR 2, M R being the mass of the recoiling system, and E x is the energy of the hadronic state x produced at the interaction vertex. Equation (1.17) can be cast in the more concise form W µν A where = N=p,n d 3 k de P N (k, E) m k, N j µ i E x, k + q x, k + q jν i k, N k i x δ( ω + k 2 + m 2 E x ), (1.18) ω = E x k 2 + m 2 = ω +m E k 2 + m 2 = ω +E 0 E R k 2 + m 2. (1.19) The spectral function P N (k, E) appearing in Eq. (1.18), defined as P N (k, E) = R 0 R, k; N, k 2 δ(e m + E 0 E R ), (1.20)

16 10 1. The lepton-nucleus cross section within the impulse approximation yields the probability of removing a nucleon with momentum k from the target ground state leaving the residual system with excitation energy E. Finally, the right hand side of Eq.(1.18) can be further rewritten as w µν N = x k, N j µ N x, k + q x, k + q j ν N k, N δ( ω + k 2 + m 2 E x ). (1.21) Assuming that the current operators are not modified by the nuclear environment, the quantity defined by Eq. (1.21) can be identified with the tensor describing electron scattering off a free nucleon carrying momentum k, at four momentum transfer q ( ω, q). The effect of nuclear binding is taken into account through the replacement q (ω, q) q ( ω, q), (1.22) reflecting the fact that a fraction δω of the energy transfer goes into excitation energy of the spectator system. Therefore, the elementary scattering process is described as if it took place in free space, but with energy transfer ω = ω δω. Note that, exploiting the requirements of Lorentz covariance, conservation of parity and gauge invariance, the right hand side of Eq.(1.21) can also be written in terms of two structure functions, w N 1 ( q2, (k q)) and w N 2 ( q2, (k q)), according to ( w µν N = wn 1 g µν + qµ q ν ) q 2 + wn ( 2 m 2 k µ k q q 2 qµ)( k ν (k q) q 2 qν). (1.23) Collecting together the above results, the nuclear response tensor can be finally cast in the form = W µν A d 3 k de m E k [ZP p (k, E) w µν p + (A Z)P n (k, E) w µν n ]. (1.24) In the case of isoscalar nuclei with Z = A/2, we can make the further assumption that the proton and neutron spectral functions be the same, i.e. that P p (k, E) = P n (k, E) = P (k, E), and Eq.(1.24) simplifies to = A d 3 k de m P (k, E) w µν. (1.25) E k W µν A where w µν = (w p µν + w n µν )/2. It should be noted that the nuclear tensor obtained inserting w µν N of Eq. (1.23) into Eq. (1.24) does not fulfill the requirement of current conservation implying in turn q µ W A µν = 0, (1.26) q µ w µν = 0. (1.27) This problem is in fact inherent in the IA scheme, which does not allow energy and current to be simultaneously conserved. A simple and effective prescription to restore current conservation, extensively employed in the analysis of (e, e p) experiments, is based on the use of an off-shell extrapolations of the tensor w µν, that will be denoted w µν, developed by de Forest in the early 80s [30]. It essentially amounts to assuming that the time components of w µν are the same as those of w µν, while the

17 1.4 Nuclear spectral function 11 longitudinal components can be obtained from the continuity equation, requiring that w 3ν = ω q w0ν. (1.28) The above prescription, referred to as CC1, is obviously not unique. However, in quasi elastic kinematics the struck nucleon is almost on the mass shell and the effect of the violation of current conservation turns out to be small. The nuclear cross section obtained within the IA takes the transparent form where ( d 2 σ de e dω e )A = A ( d 2 σ de e dω e ( d 3 d 2 σ ) k de P (k, E), (1.29) de e dω e ) = α2 q 4 E e E e L µν W µν A. (1.30) Equations (1.29) and (1.30) show that within the IA scheme it is possible to trace back the nuclear cross section to the cross section describing the elementary electronnucleon interaction which, at least in principle, can be measured using proton and deuteron targets provided the four momentum transfer q is replaced with q and an integration on the nucleon momentum and removal energy is performed, with a weight given by the nuclear spectral function. 1.4 Nuclear spectral function The spectral function defined in the previous section is trivially related to the two-point Green s function G(k, E), a fundamental quantity of many-body theory describing nucleon propagation in the nuclear ground state [31]. Let us consider, for the sake of simplicity, the case uniform nuclear matter. The definition of G(k, E) can be cast in the form [32] G(k, E) = 0 a 1 k H E 0 E iη a 1 k 0 0 a k H E 0 E iη a k 0n =G h (k, E) + G p (k, E), (1.31) where H is the nuclear Hamiltonian and 0 denotes its ground state, satisfying the Schrödinger equation H 0 = E 0 0. (1.32) The contribution denoted G p describes the propagation of a particle state, and is therefore defined for ε = E > e F, e F being the Fermi energy. On the other hand, the contribution corresponding to hole states, G h, is defined for ε = E < e F 1. 1 Note that the second argument of G h (k, E) of Eq.(1.31) is the nucleon removal energy E, not its binding energy ε = E. This choice, is consistent with the interpretation of the spectral function discussed in Section 1.3.

18 12 1. The lepton-nucleus cross section within the impulse approximation The spectral function defined in the previous section is proportional to the imaginary part of the Green s function describing nucleons occupying hole states within the Fermi sea. It can be written in the form [33] P (k, E) = n n(a 1) a k 0 2 δ(e E n + E 0 ) (1.33) where n(a 1) is a (A 1)-particle state with energy E n. Accurate calculations of the spectral function have been carried out for threenucleon systems [34, 35, 36], oxygen [37] and symmetric nuclear matter [33, 38]. The nuclear matter calculation of Ref. [33] will be described in Chapter 2. Exploiting the Källén-Lehman representation of the two-point Green s function, the spectral function of Eq. (1.33), can be conveniently split into two components, displaying distinctly different energy dependences [39]. The single-particle, or meanfield, component, P 1h (k, E), which includes the contributions of bound one hole (1h) states only, exhibits a pole at E = e k, e k being the energy of a nucleon in the hole state of momentum k. The width of the peak provides a measure of the lifetime of the hole state and becomes vanishingly small as e k e F. Integration of P 1h (k, E) over energy yields the normalization of the hole state, referred to as spectroscopic factor, Z(k), which is reduced with respect to unity due to NN correlations. A detailed analysis of the properties of single-particle states in nuclei has been performed measuring (e,e p) cross sections with a variety of nuclear targets [40]. The continuum, or correlation, component, on the other hand, is smooth, and extends to large values of energy and momentum. It accounts for the contributions of n-hole (n-1)-particle states, its leading term corresponding to two-hole oneparticle (2h1p) states. Note that this term would be vanishing in the absence of NN correlations. The spectral function of finite nuclei have been modeled within the local density approximation (LDA) [41], which is expected to be applicable to any functions F (r 1, r 2 ) when the dependence on the center-of-mass coordinate R = (r 1 + r 2 )/2 is weak. In nuclei, this condition is satisfied if the range in the relative distance r = r 1 r 2 is small compared to the surface thickness. In the absence of correlations the nuclear spectral function reduces to the mean field component, which can be written in the form P MF (k, E) = α φ α (k) 2 δ(e e α ). (1.34) where the sum includes all occupied single particle states, the momentum-space wave functions and energies of which are denoted φ α (k) and e α, respectively. In the presence of NN correlations the normalization the shell model states is reduced to Z α < 1. In addition, interactions lead to a broadening of the δ-function, reflecting the finite lifetime of the single-particle states. As a consequence P MF (k, E) of Eq. (1.34) is replaced by P MF (k, E) = α Z α φ α (k) 2 F α (E e α ), (1.35) with the function F α being sharply peaked at E = e α. The missing normalization strength is accounted for by the appearance of a non vanishing correlation component.

19 1.5 Nuclear and nucleon structure functions 13 Within the approach developed in Ref [41], the nuclear matter spectral function computed for a set of different densities is split into single-particle and correlation components. As mentioned above, the latter arises from short-range dynamics, giving rise to the occurence of strongly correlated nucleon pairs, and is expected to be nearly unaffected by finite size and shell effects. As a consequence, it can be used to obtain the corresponding quantity in finite nuclei within LDA. The resulting expression is P corr (k, E) = d 3 R ρ A (R)P NM corr (k, E; ρ = ρ A (R)), (1.36) where ρ A is the nuclear density distribution and P NM corr (k, E; ρ) is the correlation component of the spectral function of infinite nuclear matter at density ρ. The carbon spectral function employed in this thesis has been obtained combining Eqs.(1.35) and (1.36), to obtain [41] P (k, E) = P MF (k, E) + P corr (k, E). (1.37) The spectroscopic factors and the widths of the s and p states have been taken from the analysis of (e, e p) data carried out in Ref. [42]. As pointed out above, the LDA is based on the premise that short-range nuclear dynamics is largely unaffected by surface and shell effects. The validity of this assumption is confirmed by theoretical calculations of the nucleon momentum distribution, defined as n(k) = de P (k, E), (1.38) showing that for A 4 the quantity n(k)/a becomes nearly independent of A in the region of large momentum ( k ( 350 MeV), which is most sensitive to short-range correlations. This feature, illustrated in Fig.1.3, suggests that the correlation part of the spectral function exhibit the same scaling behavior. 1.5 Nuclear and nucleon structure functions The most general expression of the target tensor of Eq.(1.4) fulfilling the requirements of Lorentz covariance, conservation of parity and gauge invariance, can be written in terms of two-structure functions W 1 and W 2 depending on the scalars q 2 and (P 0 q) as ( W µν A = W 1 A g µν + qµ q ν ) q 2 + W 2 A ( m 2 P µ 0 (P 0q) q 2 q µ)( P0 ν (P 0 q) q 2 q ν). (1.39) In the target rest frame (P 0 q) = M A ω and W1 A and W 2 A become functions of the measured momentum and energy transfer q and ω. Substituting Eq. (1.39) in Eq. (1.2) and carrying out the contraction with the lepton tensor L µν one finds ( d 2 σ de e dω e )A = ( dσ dω e ) M [ W2 A ( q, ω) + 2W1 A ( q, ω)tan 2 θ ], (1.40) 2

20 14 1. The lepton-nucleus cross section within the impulse approximation Figure 1.3. Momentum distribution per nucleon in 2 H, 4 He, 16 O, and uniform nuclear matter (NM) (taken from Ref. [8]). where (dσ/dω e ) M = α 2 cos 2 (θ/2)/4e e sin 4 (θ/2) is the Mott cross section. Using de Forest s prescription of Eq. (1.28) [30], the explicit expressions of the nuclear structure functions W1 A and W 2 A can be written in terms of the nucleon structure functions of Eq.(1.24) according to W1 A ( q, ω) = [ { ( ) m d 3 kde ZP p (k, E) Ek w p 2 ( q, ω) k q 2 ] m 2 q 2 w p 1 ( q, ω) }, (1.41) and { ( ) [ m W2 A ( q, ω) = d 3 kde ZP p (k, E) w p q2 1 ( q, ω) Ek q 2 ( + wp 2 ( q, ω) q 4 ( m 2 q 4 E k ω E k ω k q q 2 ) ( q 2 q 2 1 ) q 2 k q 2 q 2 q 2 )] +... (1.42) }, where the ellipses denote the neutron contributions. It has to be emphasized that the formalism based on the IA provides a unified framework, suitable to describe electron-nucleus scattering in different kinematical regimes. The form of the nucleon structure functions w N 1 and w N 2 in the quasi elastic (QE), resonance production (RES) and deep-inelastic (DIS) channels will be discussed in the next section.

21 1.5 Nuclear and nucleon structure functions Quasi elastic scattering In the QE channel, the structure functions involve the energy conserving δ-function enforcing the condition that the scattering process be elastic. They are conveniently cast in the form wi N = w i N δ ( ω + q2 ), (1.43) 2m where the functions w i N can be written in terms of the measured electric and magnetic nucleon form factors, G E and G M, as w 1 N = τ G 2 M N, w 2 N 1 = (1 + τ) ( G 2 EN + τg2 M N ), (1.44) with τ = q 2 /4m 2. The Sachs form factors G E and G M are related to the vector form factors appearing in definition of the vector current, F 1 (q 2 ) and F 2 (q 2 ), through the relations F 1 (q 2 ) = 1 (1 τ) [G E(q 2 ) τg M (q 2 )], F 2 (q 2 ) = 1 (1 τ) [ G E(q 2 ) + G M (q 2 )]. (1.45) While more refined parametrizations of the large body of data are available (for a review, see, e.g., Ref. [43]), the form factors G E and G M are often written in the simple dipole form G E (q 2 ) = ( 1 q2 MV 2 ( G M (q 2 ) =(µ p µ n ) ) 2, 1 q2 M 2 V ) 2, (1.46) where µ p and µ n are the proton and neutron magnetic moments, respectively, and M 2 V = 0.71 GeV Resonance production and deep inelastic scattering To take into account the possible production of hadrons other than protons and neutrons one has to replace w N 1 and w N 2 given by Eq.(1.44) with the inelastic nucleon structure functions extracted from the analysis of electron-proton and electron-deuteron scattering data [44, 45]. The resulting IA cross section can still be written as in Eq.(1.40), with the two nuclear structure functions W A 1 and W A 2 given bt Eqs. (1.41) and (1.42), respectively. The main advantage of using the somewhat outdated parametrization of the proton and neutron inelastic structure functions of Bodek and Ritchie[44, 45], describing both the resonance production and and deep inelastic scattering, is that it

22 16 1. The lepton-nucleus cross section within the impulse approximation allows to avoid double counting in the kinematical region in which both mechanisms provide appreciable contributions to the cross section. However, calculations carried out using the structure functions of Refs. [44, 45] sizably underestimate the data in the production region at low Q 2. The authors of Ref.[46] argued that this problem is to be largely ascribed to the fact that Bodek and Ritchie performed a global fit of the SLAC data set spanning the kinematical domain 1 < Q 2 < 20 GeV 2, and their use at lower values Q 2 involves an extrapolation which unavoidably increases the theoretical uncertainty. 1.6 Comparison to experimental data The approach based on the IA has been systematically applied to the study of the inclusive nuclear cross section for a variety of targets. As an example, in Fig. 1.4 we report the cross section of the process e + 12 C e + X (1.47) at beam energy E e = 961 MeV and electron scattering angle θ e = 37.5 deg. The calculation has been carried out using the formalism described in Sec.1.3, with the spectral function of Ref. [41], the parametrization of the nucleon form factors of Ref. [47] and the inelastic structure functions of Refs.[44, 45]. The solid line corresponds to the full result, including the QE, RES and DIS channels. The experimental data are taken from Ref.[48]. The integrations involved in Eq.(1.29) have been carried out using the Monte Carlo approach, according to which ( d 2 σ dωdω e )A = dk de d cos θ k G(k e, k e ; k, E, cos θ k )F (k, E) 1 N G(k e, k e ; {k, E, cos θ k } n ), (1.48) N n=1 where θ k is the polar angle specifying the direction of the nucleon momentum, k, and we introduced the quantities F (k, E) = 4π k 2 P (k, E), (1.49) and ( d G(k e, k e ; 2 σ ) k, E, cos θ k ) = A. (1.50) dωdω e The above expressions have been evaluated using Monte Carlo configurations { k, E, cos θ k } n with n = 1,..., N and N = 20, 000. The values of k and E have been sampled from the distribution of Eq. (1.49), while cos θ k has been sampled from a uniform distribution. It clearly appears that theory and data are in remarkably good agreement in the region of the quasi elastic peak, while significant discrepancies are visible at larger values of the energy transfer ω. As pointed out above, the disagreement thoroughly

23 1.6 Comparison to experimental data 17 analyzed in Ref.[46] is likely to arise from the inadequacy of the Bodek-Ritchie parametrization to describe the region of momentum transfer Q GeV 2, corresponding to the -production peak in the kinematical setup of Fig Processes involving two-nucleon currents, to be discussed in the following chapters, are also known to provide a sizable contribution in the dip region, between the quasi elastic and -production peaks. Figure 1.4. Electron-carbon cross section at E e = 961 MeV and θ e = 37.5 deg, computed using Eq. (1.29) with the spectral function of Ref. [41]. The experimental data are taken from Ref. [48] It is important to keep in mind that the nuclear tensor given by Eq. (1.4) can be written as a sum of contributions corresponding to different hadronic final states X. Consider, for example, the case of QE scattering, in which the final state particles are nucleons only. For a carbon target we find X = 11 B, p, 11 C, n, 10 B, pn, 10 Be, pp..., (1.51) where the residual nucleus can be in any bound state. The states X are usually classified according to the number of nucleons excited to the continuum, and referred to as 1p1h, 2p2h, etc. In Eq. (1.51), 11 B, p and 11 C, n are 1p1h states, while 10 B, pn and 10 Be, pp are 2p2h states. Neglecting the contributions of final states involving more than two nucleons in the continuum, the cross section can be written as dσ A = dσ 1p1h + dσ 2p2h L µν (W µν 1p1h + W µν 2p2h ). (1.52) The calculation of the nuclear cross section requires the evaluation of the transition amplitudes X J µ A 0, involving both one- and two-nucleon current operators, as well

24 18 1. The lepton-nucleus cross section within the impulse approximation as all possible final states. In principle, in correlated many-body systems both the one- and two-nucleon currents can excite 1p1h and 2p2h final states, giving rise to a complex pattern of inteference effects that need to be carefully studied. Figure 1.5. Cross section of the process e + 12 C e + X at beam energy E e = 961 MeV and electron scattering angle θ e = 37.5 deg, computed using Eq. (1.29) with the spectral function of Ref. [41]. The solid line shows the results of the full calculation, while the breakdown into 1p1h and 2p2h contributions is illustrated by the dot-dash and dashed lines, respectively. At the level of IA, in which the two-nucleon current is neglected, the 2p2h contribution arises from the correlation component of the spectral function. Figure 4.1 shows the energy dependence of dσ A, dσ 1p1h and dσ 2p2h in the kinematical setup of Fig It clearly appears that the cross section in the 2p2h sector, accounting for 10% of the total strength, provides a smooth background extending to large values of ω, well beyond the region in which single-nucleon knock out is the dominant reaction mechanism.

25 19 Chapter 2 The nucleon spectral function in nuclear matter In this chapter, we outline the calculation of the spectral function in uniform and isospin-symmetric nuclear matter within the framework of Correlated Basis Function (CBF) perturbation theory. In this case, the pole component of P (k, E) is associated with bound one-hole (1h) states of the (A 1)-nucleon system, while the leading contribution to the continuum component arises from two-hole one-particle (2h1p) states, in which one of the nucleons is excited to the continuum. The calculation of the continuum component of P (k, E), whose results will be employed in Chapter 4, is described in detail. 2.1 Correlated basis function perturbation theory In Ref. [33], nuclear overlaps involving 1h states, 0 N = k, as well as 2h1p states, 0 N = hh p, are evaluated using the formalism of CBF perturbation theory, based upon a set of correlated states defined as n = G n) (n G, (2.1) G n) 1/2 where n) is a generic eigenstate of the Fermi gas (i.e. kinetic energy) Hamiltonian and the correlation operator G is defined as A G = S. (2.2) F ij j>i=1 The two-body correlation operator F ij, whose structure reflects the complexity of the NN potential, is written in the form F ij = 6 n=1 F n ij = 6 f n (r ij )Oij n, (2.3) n=1 with the O n 6 ij given in Eq. (1.7). Because the spin-isospin structure of the two-body correlation operator implies that [F ij, F ik ] 0 the symmetrisation S is needed to

26 20 2. The nucleon spectral function in nuclear matter fulfill the requirement that the wave function be antisymmetric. Note that the correlated states form a complete set, but are not orthogonal to each other. However, they can be orthogonalised following the procedure derived in Ref. [49]. The scalar, f c = f 1, and operator, f n>1, pair correlation functions describe the effects of the repulsive core of the NN potential and satisfy the boundary conditions implied by the requirement of cluster separability lim f n (r) = const 1, lim f 1 (r) = 1, lim f n>1 (r) = 0. (2.4) r 0 r r The correlation functions are generated solving a set of Euler-Lagrange equations, obtained from the functional minimization of the expectation value of the Hamiltonian in the correlated ground state, E v = 0 H 0 /A. This procedure results in the derivation of eight differential equations, involving a set of variational parameters. Since the radial functions f n are generally smaller than one for n 1, and close to one for n = 1, it is convenient to introduce the functions g n, defined as { g 1 (r) = f 1 (r) 1 g n>1 (r) = f n>1, (r) which can be seen as smallness parameters of the cluster expansion of E v E v = T F + n ( E) n. (2.5) In the above equation, T F is the energy per particle of the Fermi gas, while the terms denoted ( E) n account for contributions to the energy arising from subsystems (clusters) consisting of n nucleon. The starting point for the development of perturbation theory in the basis of correlated states (2.1) is the separation of the nuclear Hamiltonian H into unperturbed and interaction terms, according to H = H 0 + H I, (2.6) with H 0 and H I defined in terms of their matrix elements between correlated states i H 0 j = δ ij i H j = δ ij H ii, (2.7) i H I j = (1 δ ij ) i H j = H ij. (2.8) If the correlation operator, determined from the minimisation of E v, is such that the correlated states have large overlaps with the true eigenstates of the hamiltonian, the off-diagonal matrix elements of H are small. In this case, the perturbative expansions in powers of H ij will be rapidly convergent. To see how this approach works for the spectral function, let us rewrite its definition, Eq. (1.20), in the form [33] P (k, E) = 1 [ 0 a π Im k (H E 0 E iη) 1 a k 0 ], (2.9) 0 0 where n denotes an eigenstate of the Hamiltonian. The quantity H E 0 in Eq.(2.9) can be written as (H 0 H 00 )+(H I E 0 ), where E 0 = E 0 H 00 is the perturbative

27 2.2 Lowest order approximation 21 correction to variational, i.e. 0-th order, estimate of the ground-state expectation value of the Hamiltonian. Expanding in powers of (H I E 0 ) yields (H E 0 E iη) 1 1 = H 0 E0 v E iη (2.10) ( 1) n{ 1 } n (H I E 0 ). n H 0 E0 v E iη Similarly, the ground state 0 is expanded in terms of orthogonalised correlated states according to 0 = n ( 1) n{ 1 n 0 H 0 E0 v (H I E 0 )}. (2.11) Combining the above results and exploiting completeness of the set of correlated states, we can write the first order approximation to the spectral function in the form P (k, E) 1 [ π Im 0 a k n 1 n En v E0 v E iη n a k 0 0 a k n 1 n,m En v E0 v E iη H 1 nm Em v E0 v E iη m a k 0 1 H 0N E v N,m N N a Ev k m 1 0 Em v E0 v E iη m a k 0 0 a k n 1 E v n,m n E0 v E iη n a 1 ] k M EM v H M0, (2.12) Ev 0 where the indices N, M, and n, m run over the A- and (A 1)-particle states, respectively. 2.2 Lowest order approximation At lowest order of the perturbative expansion, off-diagonal matrix elements of the hamiltonian are not involved, and the spectral function is given by P (0) (k, E) = 1 π Im n = n χ k (n) 2 1 E v n E v 0 E iη χ k (n) 2 δ(e v n E v 0 E), (2.13) with the overlap matrix element χ k (n) defined as χ k (n) = 0 a k n, (2.14) n being a generic state of the (A 1)-particle system. Both 1h and 2h1p states contribute to the sum. We will discuss these two terms in detail. At zero-th order of CBF the 1h and 2h1p terms can be readily identified as the mean field and correlation component of the spectral function, respectively. The

28 22 2. The nucleon spectral function in nuclear matter former can be written in the form (for the rest of this chapter we will use the notation k = k) χ k (h) = ( 1) h+1 ] A 0 [ k 1 h A 1, (2.15) the corresponding coordinate-space expression being χ k (h) = ( 1) h+1 A dx 1... dx A (2.16) ] 0 x 1... x A x 1... x A [ k 1 h A 1. Introducing the definition reported in Eq. (2.1) we obtain the expression of the 1h overlap, in which we include the appropriate normalization factor ] χ k (h) = ( 1) h+1 (0 G G [ k) 1 h) A 1 A [ ] 1/2, (2.17) (0 G G 0) 1/2 1(k A 1 (h G G k) 1 h) A 1 ] where it is understood that when G operates on the factorized state [ k) 1 h) A 1 there are no correlation between k) 1 and h) A 1. To begin with, let us consider the numerator of Eq. (2.17), which will be denoted N k (h). At zero-th order in G ij = n g n (r ij )O n ij, (2.18) we recover the Fermi gas result N (0) k (h) = ( 1)h+1 A dx 1... dx A Φ 0(x 1,..., x A )φ k (x 1 )Φ h (x 2,..., x A ). (2.19) Using the property Φ 0 (x 1,..., x A ) = 1 ( 1) n+1 φ n (x 1 )Φ n (x 2,..., x A ), (2.20) A n where Φ n (x 2,..., x A ) denotes the Fermi gas wave function describing A 1 nucleons occupying the states m n, Eq. (2.19) yields N (0) A k (h) = ( 1)h+1 ( 1) n+1 dx 1 φ n(x 1 )φ k (x 1 ) (2.21) A n dx 2... dx A Φ n(x 2,..., x A )Φ h (x 2,..., x A ). Orthogonality of the Fermi gas states requires dx 2... dx A Φ n(x 2,..., x A )Φ h (x 2,..., x A ) = δ nh, (2.22) implying in turn ( 1) 2h+2 = 1. It follows that N (0) k (h) = dx 1 φ h(x 1 )φ k (x 1 ). (2.23)

29 2.2 Lowest order approximation 23 Using the explicit expression of the the single particle wave functions in terms of plane waves and Pauli spinors { φh (x i ) = η αh (i)e ih r i / V, φ k (x i ) = η αk (i)e ik r i / V, (2.24) we finally recover the Fermi gas overlap N (0) k (h) = 1 V dr 1 e i(h k) r 1 η α h (1)η αk (1) = δ h k η α h (1)η αk (1). (2.25) Let us derive, as a pedagogical example, the expression of the Fermi gas spectral function. In this case, the only contribution comes from 1h states, and the properly normalized the spectral function reads P 1h (k, E) = 1 A h η αh A 0 e ik r 1 h 2 δ(e + e h ), (2.26) where, according to Eq. (2.25) A 0 e ik r 1 h = V δ h k η α h (1)η αk (1). (2.27) Substitution of Eq. (2.27) in Eq. (2.26) yields the result P F G (k, E) = 4 V δ h k δ(e + e h ) = 4 A ρ θ(k F k)δ(e + e h ), (2.28) h normalized in such a way as to have d 3 k (2π) 3 de P F G(k, E) = 4 ρ d 3 k (2π) 3 θ(k F k) = 4 kf 3 ρ 6π 2 = 1, (2.29) where ρ = 2k 3 F /(3π2 )), is the matter density. Going beyond the Fermi gas result, let us consider the contribution of first-order in G ij to the overlap N (1) k (h) = ( 1)h+1 A A 1 (2.30) dx 1... dx A Φ 0(x 1,..., x A ) G 12 φ k (x 1 )Φ h (x 2,..., x A ). We start extracting the wave functions of particles 1 and 2 from Φ 0 (x 1,..., x A ) using the relation 1 { Φ 0 (x 1,..., x A ) = ( 1) n 1+n 2 +1 A[φ n1 (x 1 )φ n2 (x 2 )] A(A 1) n 1 <n 2 } Φ 0 n 1,n 2 (x 3,..., x A ), (2.31) where A[φ n1 (x 1 )φ n2 (x 2 )] denotes the antisymmetrized product and Φ 0 n 1,n 2 is the Fermi gas wave function describing A 2 nucleons occupying the single-particle

30 24 2. The nucleon spectral function in nuclear matter states specified by the index m n 1, n 2. Because G 12 operates on particles 1 and 2, we also rewrite the state Φ h (x 2,..., x A ) in the form Φ h (x 2,..., x A ) = 1 A 1 m 2 ( 1) m 2+1 φ m2 (x 2 )Φ h,m2 (x 3,..., x A ). (2.32) Exploiting again orthogonality of the Fermi gas basis, implying dx 3... dx A Φ n 1,n 2 (x 3,..., x A )Φ h,m2 (x 3,..., x A ) = δ n1 hδ n2 m 2, (2.33) we find N (1) A(A 1) k (h) = A(A 1) dx 1 dx 2 m 2 A[φ h(x 1 )φ m 2 (x 2 )] G 12 φ k (x 1 )φ m2 (x 2 ). (2.34) Finally, using the explicit expression of the single particle wave functions, Eq. (2.24), we obtain N (1) k (h) = 1 V 2 dr 1 dr 2 m 2 [e i(h k) r 1 e i(m 2 m 2 ) r 2 η α h (1)η α m2 (2)G 12 η αk (1)η αm2 (2) ] e i(m 2 k) r 1 e i(h m 2) r 2 ηα m2 (1)ηα h (2)G 12 η αk (1)η αm2 (2). (2.35) We can now express N (1) k (h) in terms of the center-of-mass and relative coordinates { R12 = (r 1 + r 2 )/2,, r 12 = r 1 r 2 with the result N (1) k (h) = 1 V 2 dr 12 dr 12 m 2 [e i(h k) R 12 e i(h k) r 12/2 η α h (1)η α m2 (2)G 12 η αk (1)η αm2 (2) e i(m 2+h k m 2 ) R 12 e i(m 2 k h+m 2 ) r 12 /2 ] ηα m2 (1)ηα h (2)G 12 η αk (1)η αm2 (2) = 1 V δ [ h k dr 12 ηα h (1)ηα m2 (2)G 12 η αk (1)η αm2 (2) m 2 ] e i(m 2 k) r 12 ηα m2 (1)ηα h (2)G 12 η αk (1)η αm2 (2). (2.36) The above equation can be conveniently written introducing the Fermi gas density matrix l(x), defined as l(k F r 12 ) = 4 e im 2 r 12. (2.37) A m 2

31 2.2 Lowest order approximation 25 The final result reads N (1) k (h) = A V δ h k dr 12 α m2 [ η α h (1)η α m2 (2)G 12 η αk (1)η αm2 (2) 1 ] eik r 12 ηα 4 m2 (1)ηα h (2)G 12 η αk (1)η αm2 (2), (2.38) where the sum is performed on discrete degrees of freedom only. The calculation of the overlap matrix element φ k (k) which we have outlined focusing on the evaluation of the numerator of Eq. (2.17) at two body cluster level involves severe difficulties. It has been performed by the authors of Refs. [33] using the FHNC/SOC summation scheme to take into account the relevant terms of the cluster expansion to all orders. The contribution from 1h intermediate states to the 0-th order (i.e. variational) spectral function of Ref. [33] can be cast in the form P (0) (k, E) = φ k (k) 2 θ(k F k)δ(e v (k) + E), (2.39) where φ k (k) is the overlap defined in Eq. (2.17) and e v (k) = E v 0 Ev k = H 00 H kk is the variational single-particle energy associated with the hole state k. The corresponding variational estimate of the momentum distribution is obtained from the spectral function of Eq. (2.39) by integrating over energy n(k) = e v(k F ) de P (0) (k, E) = φ k (k) 2. (2.40) The above n(k) exhibits a discontinuity at the Fermi surface, whose magnitude Z = φ kf (k F ) 2 = lim ε 0 [n v (k F + ε) n v (k F ε)], (2.41) can be identified with the residue of the two-point Green s function at the pole E = e v (k). In Ref. [39], it has been shown that the 1h overlap is the only one contributing to the discontinuous component of the spectral function Let us now focus on the 2h1p overlap χ k (hh p) = ( 1) h+1 ] A 0 [ k 1 hh p A 1, (2.42) whose coordinate-space expression reads χ k (hh p) = ( 1) h+1 A ] dx 1... dx A 0 x 1,..., x A x 1,..., x A [ k hh p A 1. (2.43) Neglecting the orthogonalization corrections, the above equation can be written in terms of correlated states in the form ] φ k (hh p) = ( 1) h+1 (0 G G [ k) 1 hh p) A 1 A [ ] 1/2, (0 G G 0) 1/2 1(k A 1 (hh p G G k) 1 h) A 1 (2.44)

32 26 2. The nucleon spectral function in nuclear matter where, again, we are not taking into account correlations between k) 1 and hh p) A 1. At 0-th order in G ij the numerator of Eq. (2.44) is vanishing, while at first order it reads N (1) k (hh p) =( 1) h+1 ( A dx 1... dx A Φ 0(x 1,..., x A ) G ij + ) G kl j>i l>k 1 φ k (x 1 )Φ hh p(x 2,..., x A ). (2.45) We can isolate particle 1 and 2 from Φ 0 (x 1,..., x A ) recovering the result of Eq. (2.31). In order to isolate particle 2 from Φ hh p(x 2,..., x A ), we need to distinguish the case h < h, in which 1 Φ hh p(x 2,..., x A ) = ( 1) m2+1 φ m2 (x 2 )Φ hh p m A 1 2 (x 3,..., x A ), (2.46) m 2 from the case h > h, in which 1 Φ hh p(x 2,..., x A ) = ( 1) m 2 φ m2 (x 2 )Φ hh p m A 1 2.(x 3,..., x A ). (2.47) m 2 Exploiting orthogonality of the Fermi gas states, implying dx 3... dx A Φ 0 n 1,n 2 (x 3,..., x A )Φ hh p m 2 (x 3,..., x A ) = δ n1 hδ n2 h δ n 1 hδ m2 p (2.48) we find N (1) k (hh p) = dx 1 dx 2 A[φ h(x 1 )φ h (x 2)]G 12 φ k (x 1 )φ p (x 2 ). (2.49) Finally, substituting the explicit form of the single particle states, we obtain for the direct term the expression N (1,d) k (hh p) = 1 V 2 dr 1 dr 2 e i(h k) r 1 e i(h p) r 2 ηα h (1)ηα (2)G h 12η k (1)η p (2), (2.50) which can be rewritten in terms of relative and center-of-mass coordinates as N (1,d) k (hh p) = 1 V 2 dr 12 dr 12 e i(h k+h p) R 12 e i(h k h +p) r 12 /2 ηα h (1)ηα (2)G h 12η k (1)η p (2) = 1 V δ h k+h p η α h (1)η α h (2)G 12(h k)η k (1)η p (2), (2.51) with G 12 (q) = dr 12 e iq r 12 G 12 (r 12 ). (2.52) The exchange term is obtained from Eq. (2.51) by replacing h h and α h α h, with the result N (1,e) k (hh p) = 1 V δ h k+h p η α h (1)η α h (2)G 12 (h k)η k (1)η p (2), (2.53)

33 2.2 Lowest order approximation 27 The continuous, or background, component of P (0) (k, E) is almost completely exhausted by the terms in which the 2h1p states hh p appear as intermediate states. Therefore, we can write P (0) B (k, E) = h,h,p 0 a k hh p 2 δ(e + e v (h) + e v (h ) e v (p)). (2.54) Moreover, it can be shown that P (0) B (k, E) is dominated by two-body cluster terms. Let us now analyze the 2h1p contribution to the spectral function, neglecting the corrections arising from the orthogonalization of correlated states. At two-body cluster level its definition P 2h1p (k, E) 1 A 0 e ik r 1 hh p 2 δ(e + e(h) + e(h ) e(p)) (2.55) A hh p can be rewritten in the form P 2h1p (k, E) = 1 N hh p 2 k δ(e + e(h) + e(h ) e(p)) ρ hh p = 1 { N (1,d) k (hh p) 2 + N (1,e) k (hh p) 2 2Re[N (1,d) k (hh p)n (1,e) } k (hh p)] ρ hh p δ(e + e(h) + e(h ) e(p)). (2.56) After carrying out the integrations over all momenta, and using the results reported in Eqs. (2.51),(2.53), we can perform the sums over spin and isospin degrees of freedom for each term of the above equation. For the direct term the result is α h α h α p N (1,d) k (hh p) 2 = (2.57) = 1 α p V 2 η α k (1)ηα p (2)G 12 (h k) G 12 (h k)η αk (1)η αp (2) = 1 ( α p V 2 η α k (1)ηα p (2) g n (h k)o n) ( g m (h k)o m) η αk (1)η αp (2). The exchange term gives a similar result n m (2.58) α h α h α p N (1,e) k (hh p) 2 = (2.59) = 1 α p V 2 η α k (1)ηα p (2)G 12 (h k) G 12 (h k)η αk (1)η αp (2) = 1 ( α p V 2 η α k (1)ηα p (2) g n (h k)o n) ( g m (h k)o m) η αk (1)η αp (2), n m

34 28 2. The nucleon spectral function in nuclear matter while the mixed interference term reads { N (1,d) k (hh p)n (1,e) } k (hh p) α h α h α p Re (2.60) = 1 { α h α h α p V 2 ηα k (1)ηα p (2)G 12(h k)η αh (1)η αh (2) } ηα h (1)ηα (2)P h 12G 12 (h k)η αk (1)η αp (2) = 1 ( α p V 2 η α k (1)ηα p (2) g n (h k)o n) ( P12 g m (h k)o m) η αk (1)η αp (2), n where P 12 denotes the spin and isospin exchange operator, i.e. P 12 = (1 + σ 1 σ 2 ) 2 m (1 + τ 1 τ 2 ) 2 (2.61) (2.62) Note that in all the above equations the Kronecker delta arising from conservation of momentum, δ h k+h p is understood. In the continuum limit δ h k+h p (2π) 3 δ(h k + h p)/v. 2.3 Significance of nucleon-nucleon correlations The typical energy scale associated with NN correlations can be estimated considering a pair of correlated nucleons carrying momenta k 1 and k 2 whose magnitude is much larger than the Fermi momentum ( 250 MeV/c). In the nucleus rest frame, assuming that the remaining A 2 particles carry low momenta, k 1 k 2 = k. Hence, knock-out of a nucleon of large momentum k leaves the residual system with a particle in the continuum, and requires an energy E E thr + k2 2m, (2.63) much larger than the typical energies of shell model states ( 30 MeV). The above equation, where E thr denotes the threshold for two-nucleon removal, shows that large removal energy and large momentum are strongly correlated. Figure 2.1 shows the nucleon spectral function of isospin-symmetric nuclear matter at equilibrium density, computed within the CBF approach by the authors of Ref. [33]. It clearly appears that, in addition to the peaks corresponding to single particle states, i.e. to bound one-hole states of the (A 1)-nucleon system, the resulting P (k, E) exhibits a broad background, extending up to E 200 MeV and k 800 MeV/c, associated with n-hole (n 1)-particle (A 1)-nucleon states, in which at least one nucleon is excited to the continuum. This component provides 20% of the spectral function normalisation. Note that the correlation ridge at E k 2 /2m (see Eq. (2.63)) is clearly visible. In the absence of interactions, the surface shown in Fig. 2.1 collapses to a collection of δ-function peaks distributed along the line E = k 2 /2m, with k < k F.

35 2.3 Significance of nucleon-nucleon correlations 29 Figure 2.1. Nucleon spectral function of isospin-symmetric nuclear matter at equilibrium density, computed in Ref. [33] using CBF perturbation theory.