On the role of the ∆(1232) on the transverse nuclear response in the (e, e′) reaction. E. Bauer∗ 8 9 Departamento de F´ısica, Facultad de Ciencias Exactas, 9 1 Universidad Nacional de La Plata, La Plata, 1900, Argentina. n a February 9, 2008 J 6 1 v 6 Abstract 0 0 1 The transverse nuclear response to an electromagnetic probe which is lim- 0 8 ited to create (or destroyed) a particle-hole (ph) or delta-hole (∆h) pair is 9 / analyzed. Correlations of the random phase approximation (RPA) type and h t self energy insertions are considered. For RPA correlations we have developed - l c a scheme which includes explicitly the ∆ and the exchange terms. Self energy u n insertions over ph and ∆h bubbles are studied. Several residual interactions : v based on a contact plus a (π+ρ)-meson exchange potential are used. All cal- i X culations are performed in non-relativistic nuclear matter. The main effect of r a the ∆ is to reduce the intensity over the nuclear quasi-elastic peak. Exchange RPA terms are very important, while the self energy depends strongly on the residual interaction employed. We compare our final result with data for 40Ca at momentum transfer q = 410 and q = 550 MeV/c. PACS number: 21.65, 25.30.Fj, 21.60.Jz. Keywords: Nuclear Electron Scattering. Delta resonance. ∗ Fellow of the Consejo Nacional de Investigaciones Cient´ıficas y T´ecticas, CONICET. 1 1 Introduction. Quasi elastic electron scattering is a powerful tool to study the atomic nucleus. Since theexperimentalseparationoftheinclusivelongitudinalandtransverseresponsefunc- tion [1]-[4], a great deal of theoretical effort was developed to understand these re- sponses. More recently [5], the extraction of the experimental points was re-analyzed. Even though, till now there is no theoretical frame which is able to account for both longitudinal and transverse response functions at any momentum transfer. Let us resume some of the theoretical efforts. Some works assume that the nucleus is described by a Fermi gas with a modified charge radius for individual nucleons [6]. But much of the works are based on a many body theory [7]-[29]. The present work belongs to this second group. Within this group some works deal with relativistic effects [7], the correlated basis function [8], meson exchange currents (MEC) [9], [11]- [13], RPA correlations [14]-[18], Second RPA [19, 20], Extended RPA [21]-[23], the Green function approach [24], and the ∆ degree of freedom [25]-[29]. In fact this list is not complete, we just wanted to mention the most relevant approaches related with the present work. From all these references, we learn that each effect which is considered in them, is important. In addition, there is a strong dependence on the residual interaction employed. The residual interaction is usually picked from the literature, which in general corresponds to a parameterization fixed for low energy calculations. This procedure is questionable because an effective interaction depends on the theory where it was adjusted [18, 27, 29]. That means that the search for one simple mechanism to explain data seems to be hopeless. Many correlations like RPA, MEC, the ∆ degree of freedom and so on, are all equally important. Also, the nuclear residual interaction is unknown. Fortunately, some simplification occurs as non relativistic nuclear matter describe reasonable well the properties of medium mass nuclei in the energy momentum region of interest, once a proper Fermi momentum is used [13]. The delta play an important role in the transverse nuclear response. In this work we have developed a method to account for RPA correlations with the explicit inclusion of the ∆ degree of freedom and we have also analyzed self energy insertions. This is done for several residual interactions. As mentioned, these contributions should be seen as part of a set of calculations which aim should be to reproduce both 2 the longitudinal and the transverse responses. The paper is organized as follows. In Section 2 we present the formalism for RPA and self energy insertions which includes the ∆. In Section 3, we make a numerical analysis of the different contributions. Finally, Section 4 contains the conclusions. 2 Formalism In this section we will show first the nuclear response to an external electromagnetic probe in a general way. Then in two sub-sections RPA correlations and final state interactions (FSI) of the self-energy type will be analyzed in detail. Let us start by introducing the nuclear response function as, 1 R(q,h¯ω) = − Im < |O†G(h¯ω)O| > , (1) π where q represents the magnitud of the three momentum transfer by the electromag- netic probe, h¯ω the excitation energy and | > is the Hartree-Fock nuclear ground state. Ground state correlations beyond RPA are not analyzed in this work. The polarization propagator is given by, 1 1 G(h¯ω) = − , (2) h¯ω −H +iη h¯ω +H −iη where H is the nuclear Hamiltonian. As usual, H is separated into a one-body part, H , and a residual interaction V. In Eq. (1) O represents the external probe, given 0 by a one body excitation operator which will be defined soon. We present now two projection operators P and Q. The action of P is to project into the ground state, the one particle-one hole (ph) and one delta-one hole (∆h) configurations. While Q projects into the residual n particle-n hole-n delta con- p h ∆ figurations. More explicitly, P = | >< |+P +P , (3) N ∆ with P = |ph >< ph|, (4) N X ph P = |∆h >< ∆h| (5) ∆ X ∆h 3 and Q = |n p (n −n )∆ n h >< n p (n −n )∆ n h|, (6) p h p h p h p h X n ≥ 2 h 0 ≤ n ≤ n p h where we have introduced P and P for convenience. It is easy to verify that N ∆ P + Q = 1, P2 = P, Q2 = Q, and PQ = QP = 0 and also, P P = δ P and i j ij i P Q = QP = 0 (i = N,∆). i i By inserting the identity into eq. (1) and noting that the external one body oper- ator can connect the Hartree-Fock ground state only to the P space, we have, 1 R (q,h¯ω) = − Im < |O† P G (h¯ω) P O| > , (7) PP PP π where G ≡ PGP. It is easy to see that, PP 1 1 G (h¯ω) = − , (8) PP h¯ω −H −ΣPQP +iη h¯ω +H +ΣPQP −iη PP PP where, 1 1 ΣPQP = V V −V V , (9) PQ QP PQ QP h¯ω −H +iη h¯ω +H −iη QQ QQ withobviousdefinitionsforH ,etc. Asourmainconcernistheeffectofthe∆(1232), PP we analyze the nuclear transverse response. The matrix elements for the external operator are then [30], i µ +µ τ < ph|O| >= G (q,h¯ω) s v 3 q ×(σ ×q) (10) E 2mq 2 and i < ∆h|O| >= G (q,h¯ω) µ T q ×(S ×q) (11) ∆ N∆ 3 2mq where m is the nucleonic mass, we have used µ = 0.88, µ = 4.70 and µ = 3.756. s v N∆ In eq. (10) we have neglected the convection contribution. In eq. (11) the Pauli matrices σ and τ were replaced by the corresponding transitions matrices S and T 3 3 [31]. The electromagnetic form factors are, (h¯cq)2 −(h¯ω)2 G (q,h¯ω) = (1+ )−2. (12) E (839MeV)2 (h¯cq)2 −(h¯ω)2 (h¯cq)2 −(h¯ω)2 G (q,h¯ω) = (1+ )−2 (1+ )−1/2. (13) ∆ (1196MeV)2 (843MeV)2 4 The residual interaction in the ph sector is given by, f2 V(l) = πNNΓ2 (l)(g σ · σ′+g˜′ (l)τ · τ′σ · σ′ +h˜′ (l)τ · τ′σ · lσ′ · l) µ2 πNN NN NN NN π b b (14) with Γ2 (l) l2 g˜′ (l) = g′ − ρNN C , (15) NN NN Γ2 (l) ρNNl2 +µ2 πNN ρ l2 Γ2 (l) l2 h˜′ (l) = − + ρNN C , (16) NN l2 +µ2 Γ2 (l) ρNNl2 +µ2 π πNN ρ where µ h¯c (µ h¯c ) is the pion (rho) rest mass and the Landau Migdal parameters π ρ g and g′ account for short range correlations. We have used the static limit for NN NN the interaction, where l represents the momentum transfers. For the form factor of the πNN (ρNN ) vertex we have taken Λ2 −(µ h¯c)2 πNN,ρNN π,ρ Γ (l) = , (17) πNN,ρNN Λ2 +(h¯cl)2 π,ρ Numerical values for the coupling constants, masses and form factors will be given in the next section. Analogous expressions are obtained when deltas are involved. In this case no Landau Migdal g parameter is considered. All the other NN constants and parameters should be replace by their corresponding N∆ and ∆∆ values. Also Pauli matrices must be replaced by the corresponding transitions matrices S and T or S and T , the 3/2-3/2 spin matrices (see ref. [27]); depending on the character of the mesonic vertex. Just as an example, we consider the interaction where in one mesonic vertex there is an incoming and outgoing ∆ and in the other vertex there is a hole. In that case the interaction reads, f f V′(l) = πNN π∆∆ Γ (l)Γ (l)(g˜′ (l)τ · T σ · S +h˜′ (l)τ · T σ · lS · l) µ2 πNN π∆∆ ∆∆ ∆∆ π b b (18) Equations (7)-(9) generates the standard RPA and self energy contributions. We analyze separately now these two kind of correlations. 2.1 RPA correlations: The aim of this subsection is to present a RPA formalism in nuclear matter with the ∆(1232) and which explicitly includes exchange terms. As a first step we neglect self 5 energy insertions (or equivalently we turn off the Q-space). Eq. (8) becomes, 1 1 G (h¯ω) = − , (19) PP h¯ω −H −V +iη h¯ω +H +V −iη 0 0 where we have split the nuclear Hamiltonian. The presence of V, the residual inter- action, makes G to be nondiagonal in P-space. To treat this, the standard Dyson PP equation is employed, G = G0 + G0 V G PP PP PP PP = G0 + G0 V G0 + G0 V G0 V G0 + ..., (20) PP PP PP PP PP PP whereG0 resultsfromreplacingthetotalHamiltonianbyitsonebodypart. Eq.(20) PP contains both direct and exchange terms. If one keeps only direct terms or if one uses a contact interaction, then Eq. (20) can be easily sum up to infinite order, leading to the ring series. This sum can not be done when exchange terms for a finite range interaction are included. Even this is a well know fact, let us show it in a rather elementary way, as it will simplified the further discussion. We consider the firsts two terms in the second line of Eq. (20) and we replace P by it definition of Eq. (3). In addition, let us analyzed ph configurations only. Taking the matrix elements for the firsts perturbative terms and inserting them into Eq. (7), the response function becomes, 1 R = − Im{ < |O†|ph >< ph|G0|ph >< ph|O| > + PP π X ph < |O†|ph >< ph|G0|ph >< ph|V|p′h′ > × D+E phX,p′h′ < p′h′|G0|p′h′ >< p′h′|O| > + ... }. (21) using momentum conservation as shown in Fig. 1, direct and exchange matrix ele- ments of the residual interaction can be draw as, < (h + q),h|V|(h′ +q′),h′ > ≡ V (q) (22) D D and < (h + q),h|V|(h′ +q′),h′ > ≡ V (|h− h′|). (23) E E 6 Finally, the response function becomes, 1 R = − Im{ < |O†|ph >< ph|G0|ph >< ph|O| > + PP π X ph ( < |O†|ph >< ph|G0|ph >) V (q) D X ph ( < p′h′|G0|p′h′ >) < p′h′|O| >) + pX′h′ < |O†|ph >< ph|G0|ph > V (|h − h′|) E phX,p′h′ < p′h′|G0|p′h′ >< p′h′|O| > + ... }. (24) it is trivial to extend the procedure to higher orders or ∆h configurations. As seen in the second term of this equation, direct terms split into common factors. This is not the case of the third term due to the presence of V (|h −h′|), except if one uses E a contact interaction 1. Exchange terms of the RPA type happens to be important (see refs. [16] and [17]) and as shown, they can not be sum up to infinite order. One has to evaluate each exchange term explicitly and in practice this can be done up to second order. In the next section, we will show that keeping exchange terms up to second order is not in general a good approximation. Evidently if one choose an arbitrarily small residual interaction a fast convergence to the RPA series will be obtained from its firsts perturbative terms. Let us go back to our scheme which accounts for RPA correlations in nuclear matter with the explicit inclusion of exchange terms. The scheme is an extension of the one developed in ref. [17] to include ∆h excitations and it is based on three elements. First, it is possible to sum up to infinite order exchange terms for a contact interaction. Second, it is possible to sum up to infinite order direct terms for any interaction and for some particular interactions the first two perturbative terms ac- counts for the full sum. Finally, even the exchange terms of a finite range interaction 1Also the same holds when VE(|h−h′|) is a separable interaction which is not the case of a (π+ρ)-meson exchange potential. 7 can be evaluated up to second order, it is plausible to expect that higher order terms will be negligible small if it is the case for theirs corresponding direct ones and they keep smaller than them. Thus, we divide the residual interaction as follows, V = V +V , (25) 1 2 where V is a contact interaction and V contains a contact plus the exchange of the 1 2 (π+ρ)-mesons (or any finite range interaction). The contact term in V is chosen to 2 fulfill the second and third conditions mentioned above. An additional constrain is that the remaining contact term (V ) allows a perturbative treatment. 1 The polarization propagator of Eq. (19) can now be written as G = G + G + G , (26) PP 1PP 2PP 12PP where, G = G0 + G0 V G , (27) 1PP PP PP 1 1PP G = G0 V G0 + G0 V G0 V G0 , (28) 2PP PP 2 PP PP 2 PP 2 PP G = G0 V G0 V G0 + G0 V G0 V G0 V G0 + ... (29) 12PP PP 2 PP 1 PP PP 2 PP 1 PP 1 PP Inserting now Eq. (26) into Eq. (7) one can define three different contributions to the response function, R , R and R , associated to G , G and G , respectively. Let 1 2 12 1 2 12 us analyzed each contribution separately. The R contribution is simply the ring approximation (RA), with the inclusion of 1 the ∆h space. The solution of Eq. (27) is given by (see Ref. [25] and [32]), G = (I − G0 V )−1 G0 , (30) 1PP PP 1PP PP where G0 0 V V G0 =  NN , V =  1NN 1∆N  (31) PP 0 G0 1PP V V  ∆∆   1N∆ 1∆∆  8 and P 0 N I =  . (32) 0 P ∆   We have split up the projection operator into its components P and P . Finally, N ∆ the contact contribution to the response becomes, 1 G G O R (q,h¯ω) = − Im < | (O† ,O† ) 1NN 1∆N  N  | > (33) 1 N ∆ π G G O 1N∆ 1∆∆ ∆    where we have defined O =< ph|O| > and O =< ∆h|O| >. N ∆ As mentioned V is a contact interaction and contains both direct and exchange 1 contributions. How to build this direct plus exchange interaction is described in Appendix A. Obviously Eq. (33) is also valid for direct terms of any interaction. A graphical representation of the firsts perturbative terms stemming from this equation is given in Fig. 2. Also in Fig. 2 we show the R contribution. In Appendix B, we list analytical 2 expressions for the main terms contributing to R , given by the standard rules for 2 Golstone diagrams. Finally, also some of the lower order contributions to R are presented in Fig. 2. 12 In our scheme V is included up to second order and V up to infinite order. From 2 1 the three contributions, R has the most complex structure. Formally, the analysis 12 is simplified due to the fact that a direct plus exchange contact interaction can not connect the P and P spaces (see Appendix A). N ∆ In Fig. 3 we show in detail the contributions to R limiting V up to first order. 12 2 It was further split up into three contributions, (R ) , (R ) and (R ) ; de- 12 NN 12 N∆ 12 ∆∆ pending on the configuration where the external operator is attached. Each line in Fig. 3 represents a sum up to infinite order in V . Let us call by x the ph bubble 1 and by z the corresponding ∆h one. Both are defined in Appendix B. Also we de- note by B , B and B (B B , B and B ) the direct plus 1NN 1N∆ 1∆∆ 2NNN 2NN∆ 2N∆∆ 2∆∆∆ exchange response functions which are first order in V (second order in V ). In each 2 2 case, subindex N or ∆ refers to the particular P space which builds the contribution. Forinstance, B isthesumofgraphs(B ) plus(B ) ofFig.1. Expressions 1NN 1NN D 1NN E for each of these contributions are given in Appendix B. 9 We show now R . As mentioned, 12 R = (R ) + (R ) + (R ) , (34) 12 12 NN 12 N∆ 12 ∆∆ where, 1 1 x (2−V′ x) (R ) = − Im { V′ (B +3 B ) 1N }, (35) 12 NN π 2 1N 1NN 2NNN (1−V′ x)2 1N 1 1 1 (R ) = − Im {V′ V′ [B ( −1) + 12 N∆ π 1N 1∆ 1N∆ 1−V′ x 1−V′ z 1N 1∆ x (2−V′ x) 1 1N + B + 2NN∆ (1−V′ x)2 1−V′ z 1N 1∆ z (2−V′ z) 1 1∆ + B }, (36) 2N∆∆ (1−V′ z)2 1−V′ x 1∆ 1N 1 1 z (2−V′ z) (R ) = − Im { V′ (B +3 B ) 1∆ } (37) 12 ∆∆ π 2 1∆ 1∆∆ 2∆∆∆ (1−V′ z)2 1∆ and mc2 f2 1 V′ = 8 πNN g′ Γ2 (Q) (38) 1N (2π)2 4π µ2k 1NN πNN π F 32 mc2 f2 1 V′ = π∆N g′ Γ2 (Q) (39) 1∆ 9 (2π)2 4π µ2k 1∆∆ π∆N π F where g′ and g′ are the Landau Migdal direct plus exchange terms for the con- 1NN 1∆∆ tact interaction V . Some third order contributions (B in Eq. (35) and B ) 1 2N∆N 2∆N∆ in Eq. (37)), where neglected as they are negligible small. We write now the RPA contribution to the response function as, RRPA = R˜ + R + R , (40) PP 1 12 2 where we have redefined R˜ ≡ R − R0 , R0 being the free response. This was 1 1 PP PP done for convenience because the free response will be included within the self energy contribution. 10