4 induced by the interaction with the probe. The target ground and final states, | 0 〉 and | n 〉 , are eigenstates of the nuclear Hamiltonian H belonging to the eigenvalues E$_{0}$ and E$_{n}$ , respectively. Note that scattering cross sec- tion of a scalar probe (e.g. thermal neutrons scattering off liquid helium) can be written in the simple form where ( dσ/d Ω)$_{0}$ is the elementary cross section describ- ing scattering off individual target constituents, while the response defined in Eq. (10) is an intrinsic property, fully determined by internal target dynamics. In the case of electron- or neutrino-nucleus scattering, Eq. (10) can be readily generalized, replacing the density fluctuation operator with the operators describing the electromagnetic and weak nuclear currents. The resulting response tensor reads dσ/dQ 2 shrinks, we observe an increase of the allowed range of Q 2 $_{rec}$; the behavior in the two variables is thus completely different. where the hole and particle spectral functions, P$_{h}$ and P$_{p}$ , describe the energy and momentum distributions of the struck particle in the initial and final states, respectively. The simplest implementation of the IA (usually referred to as plane wave impulse approximation, or PWIA) is based on the further assumption that final state inter- actions (FSI) between the knocked out nucleon and the spectator particles can be neglected. As a consequence, the nuclear matter particle spectral function reduces to In Fig. 4 we compare the lines of constant Q 2 = 0 . 607 GeV 2 and Q 2 rec = 0 . 306 GeV 2 corresponding to E$_{ν}$ = 0.8 GeV. The results clearly show that the two variables, in spite of leading to very similar distributions of events, are quite different. It clearly appears that the Q 2 and Q 2 rec distributions are determined by the nuclear response in different regions of the ( ω, | q | ) plane, the latter being sensitive to significantly lower values of the momentum transfer. This feature implies that the Q 2 rec distribu- tion receives contributions from the kinematical region in which the validity of the IA may become questionable. where n ( p + q ) is the occupation probability of the single particle state of momentum p + q and E$_{p}$ denotes the en- ergy of a free nucleon carrying momentum p . Note that, in the above definition, Pauli blocking of the phase space available to the struck nucleon is treated in a consistent fashion, using the momentum distribution obtained from the hole spectral function through In order to pin down the boundary of the kinemati- cal region in which the IA is applicable, we have studied the response of nuclear matter (a translationally invari- ant system consisting of equal number of protons and neutrons subject to strong interactions only) to a scalar probe delivering momentum q and energy ω : In most calculations of electron- and neutrino-nucleus scattering cross sections, 4 πp 3 $_{F}$n ( p ) / 3 is replaced with the RFG result θ ( p$_{F}$ − | p | ), yielding In the above equation, the operator ρ$_{q}$ = ∑ $_{p}$a † p + $_{q}$a$_{p}$ , a † p and a$_{p}$ being nucleon creation and annihilation op- erators, describes the fluctuations of the target density TABLE I. Position of the knees or bumps of the distributions shown in Fig. 2. E$_{ν}$ (GeV) ω (GeV) | q | (GeV) Q 2 (GeV $^{2}$) Q 2 rec (GeV $^{2}$) 0.4 0.2536 0.5013 0.187 0.107 0.8 0.5923 0.9788 0.607 0.306 1.2 0.9576 1.4181 1.094 0.494 FIG. 4. (color online). Comparison of curves of fixed Q and Q 2 rec corresponding to the knee (bump) of the differential cross section dσ/dQ 2 ( dσ/Q 2 rec ). Q 2 rec = 0 . 306 GeV 2 Q 2 = 0 . 607 GeV 2 hadronic leptonic Conditions: 12 $_{6}$C( ν$_{µ}$, µ $^{−}$), RFG, E$_{ν}$ = 0 . 8 GeV ω (GeV) | q | (GeV) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 dσ d Ω = ( dσ d Ω )$_{0}$ S ( q , ω ) , (11) W$_{νµ}$ = ∑ n 〈 0 | J † $_{ν}$| n 〉〈 n | J$_{µ}$ | 0 〉 δ $^{4}$( q + p$_{0}$ − p$_{n}$ ) . (12) S ( q , ω ) = ∫ d $^{3}$p dE P$_{h}$ ( p , E ) P$_{p}$ ( p + q , ω − E ) , (13) In the IA regime, Eq. (10) can be rewritten in the form P$_{p}$ ( p + q , ω − E ) = [1 − 4 3 πp 3 F n ( p + q )] × δ ( ω − E + M − E$_{p}$$_{+}$$_{q}$ ) , (14) III. BREAKDOWN OF THE IMPULSE APPROXIMATION n ( p ) = ∫ dE P$_{h}$ ( p , E ) . (15) S ( q , ω ) = ∑ n 〈 0 | ρ † $_{q}$| n 〉〈 n | ρ$_{q}$ | 0 〉 δ ( ω + E$_{0}$ − E$_{n}$ ) . (10) P$_{p}$ ( p $^{′}$, E ) = [ 1 − θ ( p$_{F}$ − | p $^{′}$| ) ] δ ( E + M − E$_{p}$ ′ ) = θ ( | p $^{′}$|  − p$_{F}$ ) δ ( E + M − E$_{p}$ ′ ) . (16)