The response functions for the extended second and third random phase
approximation are compared. A second order perturbation calculation shows that
the first-order amplitude for the direct $3p3h$ excitation from the ground
state cancels with those that are engendered by the $1p1h$-$3p3h$ coupling. As
a consequence nonvanishing $3p3h$ effects to the $1p1h$ response involve off
energy shell renormalization only. On shell $3p3h$ processes are absent.Comment: 12 pages text (LaTex) and 1 figure included, to be published in Phys.
  Rev. 

A. F. R. de Toledo Piza

A. Mariano

C. Gardee

C. Yannouleas

D. J. Rowe

F. Krmpotić

G. F. Bertsch

J. Hirsch

K. Takayanagi

M. H. Macfarlance

P. Barreau

S. Adachi

S. Drożdż

S. Nishizaki

W. M. Alberico

English

arXiv

Crossref

Physical Review C

Energy-shell contributions of the three-particle–three-hole excitations

The response functions for the extended second and third random phase approximation are compared. A second-order perturbation calculation shows that the first-order amplitude for the direct 3p3h (three-particle–three-hole) excitation from the ground state cancels with those that are engendered by the 1p1h-3p3h coupling. As a consequence nonvanishing 3p3h effects to the 1p1h response involve off energy shell renormalization only. On shell 3p3h processes are absent.Facultad de Ciencias Exacta

Mariano, Alejandro Edgardo

Krmpotić, Francisco

Toledo Piza, A. F. R. de

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

PHYSICAL REVIEW C VOLUME 49, NUMBER 4 APRIL 1994Energy-shell contributions of the three-particl" three-hole excitationsA. Mariano and F. KrmpoticDepartamento de Fssica, Facultad de Ciencias Ezactas, Universidad Nacional de Ia Plata, C.C. 6'7, 1900 Ia Plata, ArgentinaA. F. R. de Toledo PizaInstituto de Fasica, Universidade de Sao Paulo, Brazil(Received 27 April 1993)The response functions for the extended second and third random phase approximation arecompared. A second-order perturbation calculation shows that the Srst-order amplitude for thedirect 3p3h (three-particle-three-hole) excitation from the ground state cancels with those that areengendered by the 1plh-3p3h coupling. As a consequence nonvanishing 3p3h effects to the 1plhresponse involve off energy shell renormalization only. On shell 3p3h processes are absent.PACS number(s): 21.60.Jz, 21.60.CsMany efforts have been devoted during the last fewyears to developing generalized random phase approx-imations (RPA), which go beyond the standard one-particle —one-hole (1plh) approach [1]. This has beenaccomplished by including additional correlation effectsin both the ground state and the excited states [2—16].The reasons for that were mainly (i) the problem of themissing strength in the Gamow-Teller (GT) resonances,induced by (p, n) reactions [17,18], and (ii) the issue ofthe missing charge and missing dip strength in quasielas-tic electron scattering [19]. In particular, the extendedsecond RPA (ESRPA), which explicitly includes the 2p2hground state correlations (GSC), was extensively used todescribe the above-mentioned nuclear excitations [2,4—6,10—12,13,15]. Yet, it is self-evident that when the 2p2hadmixtures are present in the ground state, the exter-nal excitation field can lead, not only to the 1plh and2p2h states in the final nucleus, but also to the 3p3hstates. However, as the ESRPA does not involve the3p3h propagator these excitations cannot appear withinthe response function as real on the energy-shell pro-cesses. Recently the 3p3h degrees of freedom were explic-itly included within a Tamm-Dancoff approach (TDA),and their efFects on the non-energy-weighted GT sumrule were discussed [14]. Also an extended third RPA(ETRPA) [20], which possesses as the TDA limit the for-malism developed in Ref. [14],has been used to study theeKects of 3p3h excitations on the static strength functionfor quasielastic electron scattering [16].The purpose of this paper is to present some results foron the energy-shell 3p3h eff'ects in the response function.This is done in the context of the full ETRPA approachwhich is therefore reviewed below. The nature of the re-sulting response function is then con&onted to what oneobtains using the ESRPA by performing a perturbativeexpansion of the responses in each case. The possibil-ity of having a three nucleon ejection process is finallyanalyzed in this &amework.Let us start with the linear response to an external6eM I' de6ned aswhere F+(t) = e'+~Fe '+i, H = Ho + V, with Ho andV being, respectively, the Hartree-Fock (HF) mean fieldand the residual interaction. The spectral representationof the response function, in terms of a set (lv)) of eigen-states of the Hamiltonian II, readsE —E„+i' E+ E„—irIV(2)where rl is an infinitesimal positive number.Within the equation of motion method [1],the set (lv))is generated aslv) = ntlo);nt =) x,"ct —) y."c, ,n„l0) =0, for all v. (4)(ol[n- [»n,']llo) = E &o1[n- n', llo)b-, ~where E„stands for the excitation energy of the statelv), leads to the RPA-like eigenvalue problemwithXZ" = E„nlx",a ) „fx"~E~"&'(6)0The operators Ct and C; (with Ct = at&a, alai ag;) create and annihilate i particle-hole pairson the HF vacuuml—):—lOpOh), respectively.The equation of motion for O~R(E) = —i f (0(T[F t(t)P ( )](0)e' Ddt, (1) The submatrices A, B, and N given by0556-2813/94/49(4)/1949(6)/$06. 00 49 1949 1994 The American Physical Society1950 A. MARIANO, F. KRMPOTIC, AND A. F. R. de TOLEDO PIZA 49A, , = {ol[c;,[H, c,']]lo), a, ,, = {ol[c,, [H, c,]]lo(8)N;,, = (ol[c;,c,'. ]Io),and using Eqs. (2)—(6) it is possible to write the responsein representation independent form asR(E) = P'(EAf —A+irIZ') 'P,state 2p2h admixtures and a perturbatively suggestedtruncation of the dynamical matrices and excitation op-erator. It is obtained by the following.(i) Evaluating the matrix elements (8) and (10) for [ll]Io) = « Io) + ).c2. 12 )2pwherewhere T is de6ned asco —1 ) Ic2ol & c2o (16)',„F."= «I[C., F]lo),Fa I ' F*& = F~'(F- ~ Ft) (lo)&(E) = &~(E)gp(E)&p(E) + &Q(E)&qwheregp(E) = [Ehrp+irIIp —Ap—(A~Q &~QE)g—Q(E) (AQI &QI E)]—After splitting the Hilbert space of ipih states into a Pspace that includes only the lplh states and the Q spacethat spans on the rest of the states, the response functioncan be written as2o = (pip2hih2)o represents the 2p2h ground state ad-mixtures, E2, the corresponding unperturbed energy,and V2, o = {2olVI0).(ii) Keeping terms up to second order in V for the for-ward sector within the P space, terms linear in V for thebackward sector within the P space and for the couplingbetween the P and Q spaces, and only terms of zerothorder within the Q space. Under these conditions thenorm matrix elements read [5]N~ = b;i+ bN, ,with(12) where i = ipih and the nonzero AN;z arebNi, —) c2, c2, {2olDii I2o)) ANis —) c2, {1;2ol3))andgq(E) = [EJVQ +iraq —Aq] (13) 2p 2o 2p(18)PP(E) = PP —JVPQZQ + APqgq(E)FQ .In standard RPA the state Io) is approximated by theHF ground state and the Q space is absent, while the so-called extended RPA incorporates perturbative groundwhere Dii —[Ci, Ci, ) —bii and {1;2ol3) is the overlapbetween the lplh(2p2h)o and Sp3h final state configu-rations. (Note that within the quasiboson approximationDii~ = 0.) The explicit result for the matrix element(20IDii I2o) is{(plp2Iii~2)olDpg, y'g l(pip2hih2)o) = —[1+P(hi, h2)P(h'i) h2)]x[b„„b», h P (h, h2)P (pi, p2)bh, h, b~, h br, r b„,„]+ p++ h,where P (i, j) = [1 —P(i, j)], while the operator P(i, j)exchanges the arguments i and j.The forward going energy matrix elements are evalu-ated in the same way and one gets wherefi+ Qi, b, Nii fi for i = 1,P, c2, f;2, for i ) 1, (22)A,~ = b;~E~ + V~ + AA;~ ) (20) fi = (1IFlo) and f'2. =—('IFI2o)- (23)EAii —) (Ei —E2, )c2 c2 (2olD„ I2o),2p i2pAA$3 —ANg3E3 ~(21)The one-body matrix elements arewhere V~ = (ilVlj) and the nonzero matrix elementsAA;~ areBefore proceeding it is convenient to introduce the un-perturbed Green's function:t' ao(E) Oo a"(-E) Iwhere G (E) = [E+ A(H = Ho)] (with—E+ = E+iq)and rewrite the perturbed Green's function within thespace P in the formENERGY-SHELL CONTRIBUTIONS OF THE THREE-. . . 1951g~(E) = [(gp(E)) '-&~(E)] ' (25) andwhereKI (E)—:lC11 (E)«» +Z» (E)Bii,withBivV* +Z* ( E) I (26)BZiig (E) = EA11~ —AN»~E,AZ11, (E) = —[2V13 —AN13(G33(E)) ]b N31& . (28)In the above equations V~ I stands for the matrixrepresentation of the residual interaction within theipih(3 jpjh subspace.The response function now readsZiii (E) = AZ11, (E) + b, Z11, (E) + ) V1,G,;(E)V;i~,z 2)3&(E) = &1 (E)g» (E)&1 (E)+ ).& 4;(E)&' (29)i=2 3(27) whereFi (E) = f1 + BF1(E),Xi(E)—:I-i~ I, with AF1(E) = AF1 + b,F, + Q,. 23 Vi;Gp, (E)F;,AF, = AN»l fiI, EF1 = AN13F—3 .(30)Prom the expressions for AA~& and AN~~, given byEqs. (18) and (21), respectively, the matrix elementsb,Z11,(E) and b,F1 can be expressed as(2) (2)&Zii (E) = —) . c2, c2~ (2olDii I20)(E —Ei+ E2;),Xl(E) = fi + ) c2, c2 (2olD11' I2o) fi2Q, 20, 1) ~ V12f220c20E+ E2) 2Q(38)2p)2p(31)(ii) in the ETRPA (where the q space includes both the2p2h and 3p3h excitations)EF1 —) c2, c2,' (2o ID11' I 2o) fi' .2Q)2p(32) Zii (E) = ) c20c20 (2pldii& I2p) (E Ei + E2&&)2Q )2pMoreover, kom the relationshipsV13 = —) c2 E2 (1;2p]3),f32 = ) (3]1;2p)fi, (33)20one obtainsAZ„, = ) c,' c2, (1)2pll';20)(E —Ei+ E2,), (34)E+~ ~i=2,3+1(E) —fl ) c20 c20' (20 ldll' I20)fl'2Q )2pVlifi2oc2O+4—2)3 j2Q(40)2Q )20b,F, = —) c',,c2, (1;2pll'; 20) fi2p )20We can note here that(1;2pl1'i 2o) = (2ol(D11' + dii')12o)with dll' = bll' + Gi&G1 1 (36)and thus in summary we get the following.(i) In the ESRPA (where the q space includes only the2p2h excitations)The results (37) and (38) are in essence those obtainedpreviously by Arima and collaborators [5,11] and by theJiilich group [6,12]. On the other hand, when terms con-taining the xnatrix elements (20ldii I20) are neglected inEqs. (39) and (40), one finds the results derived in ourprevious works [16].1In order to elucidate some of the content of these equa-tions we turn next to a perturbative expansion of the re-sponse function and examine the leading corrections tothe unperturbed 1plh response B (E) = pi I fi I /(E+-Ei). To achieve maximum simplicity we first omit theresidual interaction within the 1plh sector and backwardcontributions, so that to second order the Bethe-Salpeterz„(E)= —) c*,.c2. (2olDii 12o)(E —Ei+20)20) V12V21I- E+ (37)These terms give rise to disconnected graphs, which arenonphysical, as weH as to double connected graphs repre-sented in Figs. 1(d) and 1(e), respectively. As seen fromrelations (43) and (47) below, they do not contribute to theresponse function.1952 A. MARIANO, F. KRMPOTIC, AND A. F. R. de TOLEDO PIZA 49equation Eq. (25) readsG» (E) —= G»(E) + &»(E)~» (E)&11 (E) ( )which substituted into Eq. (11) leads to the desired ap-proximation for the response function. Within the ES-RPA one gets22p 2p' 1,2, 2p+ ) E+ ) c2 c2' (20ID11'l20)(E E1 E2o) + )/ 1 / 2 2(42)and in the ETRPAR(E) =- R'(E) +i=2)3)2p)2pfi2o fi2o ~ +(f1 fi2o c2o ) Vlii=2,3 1 2p—) ' ) c, c,.(20ld„ I2', )(E —E, —E2. ) —)'=2, 3(43)Now the two expressions (42) and (43) can be shown to be equivalent. This results in fact from explicitly performingthe sums over 3p3h states in Eq. (43). To do that one first rewrites these sums making use of relations (33) and (36)asf32, f32,' ~ - „„(20I (D11' + d11')120)32p 2p' 3 3)2p )2p 2o(44)2- &(f1f32oc2o) V13E+ E E+ g1,3,2pE2, (20 I (D11' + d11') I 20)2o f'* (E+'E,) (E+ E, E(45)andf1 V13V31' f1'E+ E E+ E E+1,1',3f1 c2o E2o (20 I (D11' + d11' ) I 20) E2o~ f1' c2o(E+ E1)(E+ E1 E2 )(E+ E,)1,1')2p)2p(46)The result of performing the sum is). (, ' ",.".(2.l(d ~ +D-)I2'.)(E-E -E..), , '1 IE+ E /)1,1,2p, 2p (47)which substituted into Eq. (43) gives the expression (42)also for the ETRPA response.The cancellation among the 3p3h on the energy-shellcontributions can be exhibited also making use of theRayleigh-Schrodinger perturbation expansion, i.e. ,I') = I ) + I')" + «' = E'+ E,' '+z = ipih, (48)where the perturbed wave functions and energies are in-dicated by the symbol and the superscript points theorder of the correction introduced by the residual interac-tion V on the unperturbed quantities Ii) and E; The am-.plitude for the I" excitation from the correlated groundstate to the perturbed 3p3h states reads3 p o (3IIR +IIo) (31[R +ll»E.—E0 E3 —Eowith&3l[H F]lo) = ) (3lt I1)&1IFlo) —):(31+12)(20lvlo)2p(50)where the last equivalence is a direct consequence of theENERGY-SHELL CONTRIBUTIONS OF THE THREE-. . .relations (33), i.e.,2) &31f1 = —) czp&zp(1 20]3)fl = ).f3,2pvzpo .1,20 20Thus we see once more that, up to the second order inV, the 3p3h final states do not contribute to the responsefunction and that ](3~F~0)~= O(V ). The Goldstone di-agrams for the fourth order 3p3h on the mass-shell con-tributions to the response function are shown in Figs.1(e) and l(f).At first glance it might look as if the connected Gold-stone diagrams associated with the terms (44), (45), and(46) of the ETRPA response [illustrated in Figs. 1(a),1(b), and 1(c), respectively] should give rise to on themass-shell 3p3h contributions, through the imaginarypart of the propagator (E+ —Es) i. However, Eq. (47)shows that these contributions in fact cancel out so thatthe 3p3h sector only affects the 1plh excitations by cou-pling them with the virtual intermediate states [1;2o).Thus in spite of including the 3p3h propagator in theGreen's function, three nucleon ejection does not occurin the leading order processes. The above-mentioned di-agrams also explain the physical meaning of the fourthterm in the expression (42). The cancellation of on shell3p3h contributions results &om the destructive interfer-ence between amplitudes involving creation of the 3p3hstate from a ground state correlation and &om V31 cou-pling, respectively. A similar calculation in which thebackward part of Eq. (25) and/or the residual interac-tion within 1plh space are kept up to the relevant orderleads again to the same result. It is worth stressing that(~) (b) (c) (~) (e)FIG. 1. Graphical representation of the second- andfourth-order contribution to the response function. The dot-ted circles (0) denote the one-body vertices and the filledones (~ ) indicate the two-body matrix elements. The dia-grams (a), (b), snd (c) correspond, respectively, to the termsgiven by Eqs. (44), (45), and (46). Second-order unlinkedand double-linked graphs analogous to the diagram (c) areshown in (d) and (e), respectively. The last ones, althoughcontained in Eqs. (39) and (40), do not contribute to theresponse function. Finally, (f) illustrates the fourth order onthe energy-shell 3p3h processes.this does not depend on the form of the two-body forceused as residual interaction or on the size of single parti-cle space.Note that Hp does not contribute since (3~[Hp, P] ~0) = 0.F.K. acknowledges the financial support of theCONICET &om Argentina.[1] D. J. Rowe, Nuclear Collective Motion (Methuen, Lon-don, 1970).[2] G. F. Bertsch and I. Hamamoto, Phys. Rev. C 26, 1323(1982).[3] C. Yannouleas, M. D. Worzecka, snd J. J. GrHFin, Nucl.Phys. A397, 239 (1983).[4] W. M. Alberico, M. Ericson, and A. Molinari, Ann. Phys.(N.Y.) 154, 3M (1984).[5] K. Tsksyanagi, K. Shimizu, and A. Arima, Nucl. Phys.A444, 436 (1985).[6] S. Drozdz, V. Klemt, J. Speth, and J. Wambach, Phys.Lett. 166B, 253 (1986).[7] M. H. Macfarlance, Phys. Lett. B 182, 265 (1986).[8] J. Hirsch, A. Mariano, M. Faig, and F. Krmpotic, Phys.Lett. B 210, 55 (1988).[9] S. Adachi and E. Lipparini, Nucl. Phys. A489, 445(1988).[10] S. Nishizaki, S. Drozdz, J.Wambach, and J. Speth, Phys.Lett. B 215, 231 (1988).[11] K. Tskayansgi, K. Shimizu, and A. Arima, Nucl. Phys.A477, 205 (1988); A481, 313 (1988).[12] S. Drozdz, S. Nishizski, J. Speth, and J. Wambach, Phys.Rep. 197, 1 (1990).[13] W. M. Alberico, A. De Pace, A. Drsgo, and A. Molinari,Universita di Torino Report No. DFTT 7/90, 1990.[14] A. Msriano, J. Hirsch, and F. Krmpotic, Nucl. Phys.A518, 523 (1990).[15] K. Taksyansgi, Phys. Lett. B 230, ll (1989); Nucl. Phys.A510, 162 (1990);A516, 276 (1990);A522, 494 (1991).[16] A. Mariano, E. Bauer, F. Krmpotic, snd A. F. R. deToledo Pizs, Phys. Lett. B 268, 332 (1991).[17] C. Gardee, J. Rapaport, T. N. Taddeucci, C. D. Good-man, C. Foster, D. E. Bainum, C. A. Goulding, M. G.Green6eld, D. J. Horen, and E. Sugarbaker, Nucl. Phys.A369, 258 (1981).[18] J. Rapaport, in The Interaction Between Medium EnergyNucleons in Nuclei, Proceedings, edited by H. O. Meyer,American Institute of Physics, AIP Conf. Proc. No. 97(AIP, New York, 1982), p. 365.[19] P. Barreau, M. Bernheim, J. Duclos, J. M. Finn, Z.1954 A. MARIANO, F. KRMPOTIC, AND A. F. R. de TOLEDO PIZA 49Meziani, J.Morgenstern, J. Mougey, D. Royer, B.Saghai,D. Tarnowski, S. Turck-Chieze, M. Brussel, G. P. Capi-tani, E. De Sanctis, S. Frullani, F. Garibaldi, D. B. Is-abelle, E. Jaus, I. Sick, and P. D. Zimmerman, Nucl.Phys. A402, 515 (1983).[20] A. Mariano, Proceedings of the XV Workshop on NuclearPhysics, Buenos Aires, Argentina (World Scientific, Sin-gapore, 1993), p. 216.

http://sedici.unlp.edu.ar/bitstream/handle/10915/123272/Documento_completo.pdf?sequence=1

On the energy-shell contributions of the three-particle~-~ three-hole
  excitations

On the energy-shell contributions of the three-particle~-~ three-hole excitations

Abstract

Similar works

Full text

Available Versions

Crossref

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas