Probing short-range NN-correlations in the reaction 12C + p → p + pN + 10A

Abstract. The exclusive reaction C(p, ppN)A of quasi-elastic knockout of the nucleon from the short-range correlated nucleon pair <NN> in the C nucleus by proton with an energy of few GeV is considered in the plane-wave approximation.The two-nucleon spectroscopic factors and momentum distributions of the center of mass motion of NN-pairs are considered by analogy with theory of quasi-elastic knockout of fast deuteron clusters from nuclei (p,Nd) using the translationally-invariant shell model.


Introduction
Quasi-elastic knock-out of fast deuterons from 12 C nucleus by protons at 670 MeV discovered in Dubna in 1957 [1] had demonstrated surprisingly high cross section. The transfer of high kinetic energy to such a weakly bound system as a deuteron lead D. Blokhintsev to a key idea about fluctuations of nuclear matter density in nuclei [2] later on called as fluctons [3]. Now fluctons are considered either like multiquark configurations or as few-nucleon correlations. Two nucleons being at short relative distances r NN ∼ 0.5 fm due to the uncertainties principle have a high relative momentum q rel ∼ 0.4 GeV/c and the repulsive core in the NN-interactions which nature is still not quite clear, enhances this effect. Two-nucleon short-range correlations (SRC) in nuclei are actively studied during few last decades [4]. Under the SRC pair is assumed a pair of nucleons with small momentum of their center mass motion k c.m. and large as compared the Fermi momentum in heavy nuclei p F = 250 − 300 MeV/c and oppositely directed individual momenta of the nucleons p 1 ≈ −p 2 . Experiments with electron and proton beams show that such correlations do exist in nuclei and the probability to find in nucleus the SRC pn pair is about 20 times higher than the pp-and nn-pair [5]. The dominance of the pn SRC pair is connected with the tensor forces acting in spin-triplet NN pair and vanishing in the spin-singlet 1 S 0 states of the pp-and nn pairs. Furthermore, factorization over the k c.m. and q rel momenta takes the place at enough large q rel and low k c.m. in momentum distribution of the SRC: n(p 1 , p 2 ) ≈ C A n c.m. (k c.m. )n rel (q rel ), where C A is a smooth function of number of nucleons A. For broad class of nuclei from 4 He to 208 Pb the distribution over internal momentum q rel at high |q rel | is an universal function close to the deuteron wave function squared. The measured distribution n c.m. (k c.m. ) is compatible with the tree-dimensional Gaussian with the parameter σ = 140 − 160 MeV/c (see Ref. [6] and references therein).
A new experiment for study of the SRC in the 12 C nucleus was performed at the BM@N in JINR [7] and analysis of the collected data is in progress now. In this experiment an inverse kinematics is used, where the 12 C beam with momentum 4 GeV/c per nucleon interacts with the liquid hydrogen target providing the transition 12 C + p → pp + N + 10 A. All free final nucleons and residual nucleus 10 A are registered at conditions when the missing momentum is large, p miss > 250 MeV/c, where p miss is the momentum of the nucleon which is knocked out by the target proton from the SRC pair. The scattering angle in the center mass frame of the subprocess pN → pN is chosen to be close to 90 • , θ cm = 90 • ± 30 • , in order to minimize the off-mass-shell effects in the pp-scattering. From theory side, this reaction was analyzed in Ref. [8] within the plane-wave approximation corresponding to the pole diagrams depicted in Fig.1 and the corresponding formalism was presented. In the developed formalism the two-nucleon spectroscopic factors in the 12 C nucleus are calculated using the translationallyinvariant shell model with mixing configurations. The short-range correlations are introduced by replacement of the internal wave function of the two-nucleon cluster by the deuteron wave function ψ d (q rel ) for the spin-triplet states. For the spin-singlet NN-pairs the t-matrix of NN scattering in the 1 S 0 can be used for realistic NN-interaction potentials. Since the internal momentum q rel in the SRC pair is large, the relativistic effects are necessary to take into account. These effects are considered in [8] within the light-front dynamics.
Here we are focused mainly on the distribution on the momentum of the center mass motion of the SCR pair in the initial nucleus. A necessary part of the mathematical formalism [8] is given in the next section together with the obtained numerical results. Concluding remarks are given in the last section.

The model
Considering the contribution of the SRC pairs to the reaction in question we use as a basis the results of analysis of the reactions of quasi-elastic knock-out of fast nucleon clusters from nuclei by protons. For our purposes the most important are data on the reactions of qusielastic knock-out of the deuterons A(p, pd)B from light nuclei A = 6 Li, [9], A = 7 Li, 12 C [10] and on the reaction 6,7 Li(p, nd)B obtained in JINR with the proton beam of 670 MeV. The reaction (p, nd) presents a special interest since it corresponds to the kinematics of the backward quasielastic scattering on the di-neutron <nn>, p+<nn>→ n + d, where the spin-isospin state of the <nn> system differs from that for the deuteron. Theory of these reactions was developed in the framework of the shell-model model for nuclear structure using realistic mechanisms for the p<NN>-backward elastic or quasi-elastic scattering [9][10][11][12] and the inverse reaction pd →<pp>+n [13,14]. The spectroscopic factors obtained on this way were found to be compatible with shell-model predictions. For this reason in our study of the SRC we use the translationally-invariant shell model for calculation of the two-nucleon spectroscopic factors. In order to account for the short-range NN-correlations we replace the shell model wave function of the NN-pair at large relative momenta between nucleons by the deuteron (or dineutron/diproton) wave function.

Elements of formalism
The transition matrix element corresponding to the Feynman diagram in Fig. 1a has the following form where M(A → B+<NN>) is the amplitude of the virtual decay of the nucleus A into <NN> pair and the residual nucleus B in certain internal states and definite state of their relative motion, p 2 <NN> − m 2 <NN> + iε −1 is the propagator of the <NN>-pair with four-momentum p <NN> and the mass m <NN> , M(p<NN>→ pNN) is the matrix element of the subprocess of quasielastic knock-out of the nucleon N from the <NN> pair. All tree factors in Eq. (1) are relativistic invariant and therefore can be calculated in any suitable reference frame. The amplitude M(A → B+<NN>) can be calculated in the rest frame of the nucleus A and in this case it can be expressed as a product of the spectroscopic factor S x A of the <NN> cluster, which we denote below as x, and the wave function of the relative motion of the center mass of the cluster and the center mass of the residual nucleus Ψ νΛM Λ (p B ) in the following form where p B is the 3-momentum of the residual nucleus B in the rest frame of the nucleus A, ν is the number of oscillator quanta, Λ is the orbital angular momentum and M Λ is its zprojection, m i is the mass of the the nucleus i (i = A, B). The spectroscopic factor S x A is determined by the overlap integral between the wave function of the nucleus A, ψ A , and the product of the wave functions ψ B , ψ x and ψ νΛ as (fo detail see, for example, [15]) where the combinatoric factor A x 1/2 takes into account identity of the nucleons. The wave function of the nucleus A in the translationally-invariant shell model (TISM) [16] is where N is the total number of oscillator quanta, [ f ] is the Young scheme determining the permutational symmetry of the orbital part of the wave function, (λµ) is the Elliott symbol defining the S U(3) symmetry, L, S , J and T are the orbital, spin, isospin and total angular and isospin momenta, respectively; M J and M T are the z-projections of the total angular momentum J and of the isospin T ; the α is required for a complete labeling of the nuclear states. The TISM wave function with mixing configurations with different [ f ], L, and S but the same J and T is the following superposition of the wave functions given in Eq. (4): In the TISM the parentage expansion of the antisymmetric wave function |AN A α > over the product of the antisymmetric wave functions of the A − x(≡ B) nucleus, |BN B β >, and the antisymmetric wave function of the cluster x, |xN x γ >, and the wave function of their relative motion ψ νΛM Λ has the following form whereÂ is the properly normalized antisymmetrization operator in respect of permutation of the nucleons between the systems A − x and x, < AN A α|A − xN B β, νΛM Λ , xN x γ > is the parentage coefficient. The following relation takes the place in Eq.(6) for numbers of the oscillator quanta: Using the TISM model for the nuclear wave functions of nuclei A, B(≡ A − x) and the cluster x the spectroscopic factor (3) can be calculated analytically and after that the transition matrix element (1) can be written as Here we used the standard notations for the Clebsch-Gordan and Racah coefficients and 9jsymbols of the rotation group. The wave function of the relative motion of the c.m. of the cluster x and the center mass of the residual nucleus B = A − x contains the space part R νΛ (k) that is the harmonic oscillator wave function, and angular wave function Y ΛM Λ (θ k , φ k ) that is the spherical harmonic:  (8). Using these simplifications for the case of the ground state of the 12 C nucleus with J i = 0, T i = 0, the spin averaged matrix element (1) can be written as When deriving Eq. (9) we neglect the spin dependence of the pN elastic scattering amplitude in the upper vertex of the diagram in Fig. 1 b.

Numerical results and discussion
A sensitivity of the cross section of this reaction to the high-momentum part of the ψ <NN> (q rel ) was studied in Ref. [8] using the Light-Front dynamics for the matrix element of the process p+<NN>→ p + N + N. The ratio of squared deuteron wave function R = |ψ d (q LFD )| 2 /|ψ d (q nr )| 2 at the relativistic q LFD and non-relativistic q nr relative momentum is shown in Fig. 2 versus the momentum of the spectator nucleon p r in the rest frame of the 12 C at zero momentum of the residual nucleus 10 B for different type of the NN-interaction potential [17][18][19]. The scattering angles of the nucleon p r are θ r = 6 • and φ r = 0. Below we concentrated on the center mass momentum distribution of the SCR pair. In the considered model this distribution is determined by the (squared) wave function ψ νΛM Λ (k c.m. ), where k cm = p B . The radial parts of this wave functions R νΛ (p/p 0 ) corresponding to the s 4 p 6 , s 3 p 7 and s 2 p 8 configurations of the residual nucleus 10 B are, respectively, the following where C = 4/ √ πp 3 0 , x = p/p 0 and p 0 is the oscillator parameter. For calculation of the parentage coefficients we use the formalism of Ref. [15]. The coefficients of the mixing configurations α A,JT [ f ]LS in Eq. (5) for the ground state of the 12 C nucleus and for the lowest levels of the 10 B nucleus corresponding to s 4 p 6 configuration are taken from Ref. [20]. For the states with the destroyed s-shell corresponding to the s 2 p 8 and s 3 p 7 configurations, the TISM model with mixing configurations, for our knowledge, was not performed. Therefore, in this case for estimate we use the ordinary TISM model. For the oscillator parameter p 0 in Eqs. (10) we use the values found in Ref. [10] from the analysis of the data on the reaction 12 C(p, pd) 10 B, i.e. p 0 = 155.3 MeV/c (144.7 MeV/c) for transition to the s 4 p 6 (s 2 p 8 ) configuration of the 10 B. In that experiment the resolution over the excitation energy E * B of the residual nucleus was high enough to separate the transitions to the s 4 p 6 and s 2 p 8 states. It was found in [10] that the p B momentum distribution for transitions to the ground state and lowest excited states of the 10 B with E * B ≤ 5 MeV is well described by the 2S -type distribution (corresponding to νΛ = 20), whereas for the transitions to the excited states with the s 2 p 8 configuration the 0S -type distribution (νΛ = 00) is well suitable, as was expected within this model. On the other hand, it was concluded in [10] that the 1D (νΛ = 22) transitions to the s 4 p 6 states are strongly suppressed as well as the knockout of the sp pairs is suppressed as compared to the p 2 and s 2 pairs. The latter result can be understand qualitatively, since the size of the averaged radius of the trajectory for the s nucleon differs from that for the p nucleon and therefore the probability to find these nucleons at short distances in the quasi-deuteron cluster is lower than for the p 2 and s 2 configurations. A reason for suppression of the 1D transitions in the (p, pd) reaction is not quite clear.
When calculating the distribution over the momentum of the center of mass motion of the NN-pair k cm = p B we use Eq.(9) and put the factors |M pN (s, t)| 2 |Ψ x (q rel )| 2 to be constant. For the s 4 p 6 configuration of the 10 B nucleus we put ν = 2 and perform summation over the lowest states of the 10 B nucleus from the region of E * B = 0 − 5 MeV (see [8] for details) taking into account the mixing configurations [20]. As follows from Eqs. (10), for this case The transitions to the s 2 p 8 states of the 10 B nucleus correspond to νΛ = 00 and therefore the p B distribution in this case is of the 0S type (dot-dashed line in Fig.3). If the resolution on E * B is rather poor corresponding, for example, to the range E * B = 0 − 30 MeV, one should make sum over transitions to the states with all configurations s 4 p 6 , s 3 p 7 , s 2 p 8 . Keeping in mind the above mentioned results for suppression of the sp pairs knock-out in the (p, pd) reaction [10], we neglect the transitions to the s 3 p 7 configuration and make the sum of the s 4 p 6 and s 2 p 8 configurations. The obtained result is shown in Fig. 3 by the full thin line which is neither 0S nor 1D type. For comparison, the pure 1S type distribution is also shown in Fig.3 by the dotted line that corresponds to the transition to the 10 B state with quantum numbers J f = 0, T f = 1 , E exp = 1.74 MeV (with E = 1.51 MeV from the theoretical model [20]) corresponding to νΛ = 20. One can find from Fig. 3 that both 0S and 1S distributions, which well describe the p B -distributions in the 12 C(p, pd) 10 B reaction [10], are considerably narrower than the Gaussians n(p B ) ∼ exp [−p 2 B /2σ 2 ] with σ = 140 − 160 MeV/c found from the (e, epp) reaction in Ref. [6].
If the experimental data on the p B distribution in the reaction 12 C+p → ppN + 10 B will be obtained and found to be similar to that from the (e, epp) reaction with electron beam [6], then one will have to conclude that the formalism which allows one to explain data on the reactions of quasi-elastic knock-out of fast deuterons from the light nuclei [10,21,22] and used here for describing the nuclear structure of the 12 C, has to be modified in the region of SRC. Presumably, antisymmetrization effects in realistic nuclear wave functions with cluster configurations are of importnace here. This issue will be considered in a separate publication. In any case, it is important to get together with the results of the SRC experiment [7] some data at the same beam energy on the reaction of quasi-elastic knockout of the fast deuterons 12 C(p, pd) 10 B at high transferred momenta corresponding to the p<NN>→ pd scattering at ≈ 90 • in the center mass of the pd system.

Concluding remarks
The exclusive reaction 12 C(p, ppN) 10 A of quasi-elastic knockout of the nucleon from the short-range correlated nucleon pair <NN> in the 12 C nucleus by proton with an energy of few GeV is considered in the plane-wave impulse approximation by analogy with theory of reactions of quasi-elastic knockout of fast deuterons (p, pd). The nuclear structure is considered within the translationally-invariant shell model with mixing configurations. Short-range correlation effects in the wave function of the 12 C nucleus ψ A are taken into account by modification of the internal wave function of the two-nucleon cluster at high relative momenta in the parentage expansion of the ψ A . Within this assumption the "spectator" A − 2 system and its center of mass motion are not modified by the SRC. Relativistic effects in the subprocess p+<NN>→ p + N + N are taken into account within the light-front dynamics and found to be very important at the spectator momenta (in the rest frame of 12 C nucleus) p r ≥ 0.5 − 0.6 GeV/c. In this region of p r a sensitivity to details of the NN-interaction potential at short NNdistances is very high. The distributions over the center of mass momentum of the <NN>-pair are calculated for transitions to different states of the residual nucleus 10 B and compared with similar results for the reaction (p, pd) [10,21,22] and with available data on distribution of the center mass of SRC pairs from experiments with electron beam [6]. The next step of our study will be an account for initial and final state interactions in the considered reaction.