Medium polarization e�e ts on pairing in stable andexoti nu lei and in neutron starsR.A.Broglia1;2;3, S.Baroni1;2, F.barran o4, P.F.Bortignon1;2 and E.Vigezzi21 Department of Physi s, University of Milan, Milan, Italy2 INFN, Sez. di Milano, Milan, Italy3 The Niels Bohr Institute, University of Copenaghen, Copenaghen, Denmark4 Es uela de Ingenieros Industriales, Universidad de Sevilla, Sevilla, SpainAbstra tMean �eld results are renormalized in an important fashion by the interweaving of olle tive vibrations and single-parti le motion. The strong isotopi dependen e of su h ouplings indi ate the need to al ulate expli itly the asso iated !-dependent ouplings.Nu lear Field Theory plus the Dyson equation provide a powerful tool to arry out su ha program.1. { Dedi ationThis paper is dedi ated to David Brink. David is a towering �gure in the �eld ofnu lear stru ture and rea tions. I (RAB) have been fortunate to enjoy, through theyears, the breadth of his physi al insight and the serenity of his friendship, elements whi hplayed a most important role during the many years whi h took us to write \Nu learSuper uidity".2. { Medium renormalization e�e tsMedium polarization e�e ts are a entral subje t in the study of systems with manydegrees of freedom (see e.g.. [1℄,[2℄,[3℄). As an example we show in Fig. 1 the di�erent So iet�a Italiana di Fisi a 1

2 R.A.Broglia1;2;3, S.Baroni1;2, F.barran o4, P.F.Bortignon1;2 and E.Vigezzi2

Fig. 1. { a) The bare Coulomb repulsion arising from virtual phonon ex hange, b) The s reeningof Coulomb intera tion resulting from the ex hange of plasmons (ep: ele tron plasmons), ) theindu ed pairing intera tion arising from the ex hange of phonons (ip: ion phonons). ontributions to the pairing intera tion a ting between ele trons moving in time-reversalstates lose to the Fermi energy: a) The bare Coulomb repulsion arising from virtualphonon ex hange, b) The s reening of Coulomb intera tion resulting from the ex hangeof plasmons (ep: ele tron plasmons), ) the indu ed pairing intera tion arising from theex hange of phonons (ip: ion phonons).

Medium polarization effe ts on pairing in stable and exoti nu lei and in neutron stars 3This theme was developed for the ase of �nite systems, within the framework ofatomi nu lei, leading to the Nu lear Field Theory (NFT) [4, 5, 6, 7, 8, 9℄. In Ref.[10, 11℄ (see also [12℄) it was systemati ally applied to the ase of single-parti le and olle tive nu lear motion (in parti ular to giant resonan es), and in Ref. [13℄ to the aseof plasmons in metalli lusters and fullerenes.From these results it was expe ted [12℄ that medium polarization e�e ts ould be animportant sour e of pairing orrelations in nu lei (see Fig. 2). In fa t, it was shown in(see [14℄ and refs) that the bare and the indu ed intera tions give essentially the same ontributions to the pairing gap in super uid nu lei (see Fig. 3).As shown in Ref. [1℄, a systemati dis ussion of the indu ed intera tion should alsotake self-energy and vertex renormalization e�e ts into a ount (see Figs. 4,5,6). Thiswas done [15℄ within the framework of NFT in the ase of 120Sn (see Fig. 7).The situation is even more extreme in the ase of halo, exoti nu lei, where theparti le-vibration oupling phenomenon leads to parity inversion as well as a ounts for80-90% of the full pairing orrelations (see Figs. 8,9).The role medium polarization e�e ts have in pairing orrelations in the inner rust ofneutron stars has been studied in a number of papers (see e.g. Fig. 10.19 of Ref. [14℄). No onsensus exists on erning the a tual value of the pairing gap in the di�erent Wigner ells (density) studied. In parti ular, re ent Monte Carlo al ulations [16℄ indi ate amarginal e�e t, in ontradi tion with all previous al ulations. Similar results havebeen obtained within the framework of uniform neutron matter [17℄. At the origin ofthis out ome is the almost omplete an ellation between spin (repulsive) and density(attra tive) modes, respe tively. The neutron star setup is thus expe ted to be a uniqueprobe to test our present understanding of the e�e t the interweaving of vibrations andsingle-parti le motion has on the phase shift asso iated with the 1S0 p-p hannel.3. { Con lusionAs Feynman for efully stated it (see Fig. 5), there is nothing free. Mean �eld pa-rameters are to be adjusted so that the dressed properties (mass, harge, intera tion)reprodu e the experimental data. This is one of the basi task lying at the forefront ofnu lear stru ture resear h.REFERENCES[1℄ R.P.Feynman, in The theory of Fundamental Pro esses, Benjamin, Reading, Mass. (1962).[2℄ J.R.S hrie�er, in Theory of Super ondu tivity, Benjamin, Reading, Mass. (1964).[3℄ G.D.Mahan, in Many-Parti le Physi s, Plenum Press, New York (1981).[4℄ D.R.Bes et.al, Nu l.Phys. A, 260 (1976) 1;[5℄ D.R.Bes et.al, Nu l.Phys. A, 260 (1976) 27.[6℄ D.R.Bes et.al, Nu l.Phys. A, 260 (1976) 77.[7℄ R.A.Broglia et.al, Phys.Lett. B, 64 (1976) 29.[8℄ D.R.Bes et.al, Phys.Lett. B, 52 (1974) 253.[9℄ P.F.Bortignon et.al, Phys.Rep. C, 30 (1977) 305.

4 R.A.Broglia1;2;3, S.Baroni1;2, F.barran o4, P.F.Bortignon1;2 and E.Vigezzi2

Fig. 2. { Quotation from Ref. [12℄.[10℄ C.Mahaux et.al, Phys.Rep., 120 (1985) 1-274.[11℄ P.F.Bortignon, A.Bra o and R.A.Broglia, Giant Resonan es: Nu lear Stru ture at FiniteTemperature, Harwood A ademi Publishers, Amsterdam (1998).[12℄ A.Bohr and B.R.Mottelson, Nu lear Stru ture, Vol. II, Benjamin, Reading, Mass. (1975).[13℄ R.A.Broglia, G.Col�o, G.Onida and H.E.Roman, Solid State Physi s of Finite Systems:Metal Clusters, Fullerens, Atomi Wires, Berlin, Heidelberg (2004).

Medium polarization effe ts on pairing in stable and exoti nu lei and in neutron stars 5

Fig. 3. { (a) The state-dependent pairing gap of 120Sn for the levels lose to the Fermi energyobtained using BCS theory with the v14 Argonne potential (see Ref. [15℄) (b) Average valueof the state dependent pairing gap asso iated with levels lying lose to the Fermi energy ofSn isotopes, al ulated making use of the indu ed pairing intera tion, in omparison with theempiri al pairing gap.[14℄ R.A.Broglia and D.Brink, Nu lear Super uidity: Pairing in Finite Systems, CambridgeUniversity Press, Cambridge (2005).[15℄ F.Barran o et.al, Eur.Phys.J. A, 21 (2004) 57.[16℄ A.Fabro ini et.al, Phys.Rev.Lett., 95 (2005) 192501.

6 R.A.Broglia1;2;3, S.Baroni1;2, F.barran o4, P.F.Bortignon1;2 and E.Vigezzi2

Fig. 4. { Sele ted pages of Ref. [1℄.[17℄ S.Baroni and A.Gualandris, private ommuni ation.

Medium polarization effe ts on pairing in stable and exoti nu lei and in neutron stars 7

Fig. 5. { Sele ted pages of Ref. [1℄.

Fig. 6. { Sele ted page of Ref. [1℄.

8 R.A.Broglia1;2;3, S.Baroni1;2, F.barran o4, P.F.Bortignon1;2 and E.Vigezzi2

Fig. 7. { The state-dependent pairing gap for the levels lose to the Fermi energy obtained usingBCS theory with the v14 Argonne potential ( ir les) is ompared with the result obtained byalso in luding renormalization e�e ts (squares) (see Ref. [15℄).

Medium polarization effe ts on pairing in stable and exoti nu lei and in neutron stars 9

Fig. 8. { NFT low energy spe trum of 10Li (I) and 11Li (II).

10 R.A.Broglia1;2;3, S.Baroni1;2, F.barran o4, P.F.Bortignon1;2 and E.Vigezzi2

Fig. 9. { NFT solution of the low energy spe trum of 11Be (I) and 12Be (II).