1. Scalar response of the nucleon, Chiral symmetry and nuclear matter properties G. Chanfray, IPN Lyon, IN2P3/CNRS, Université Lyon 1 M. Ericson, IPN Lyon, IN2P3/CNRS, Université Lyon 1 and Theory division, CERN Workshop in Honour of Tony Thomas's 60th Birthday Adelaide, February 2010

2. Relativistic models of nuclear binding (Walecka et al) Nucleon in attractive scalar (σ) and repulsive vector (ω) background fields Economical saturation mechanism + magnitude of spin-orbit splitting Nuclear many-body problem Connection between nuclear background fields and QCD condensates Many-body effects vs nucleon substructure response (lattice QCD) Low energy QCD Chiral sym/Confinement The chiral invariant scalar background field Fields associated with the fluctuations of the chiral condensate. σ,πs, φ Go from cartesian (linear: σ,π) to polar (non linear: s, φ) representation Pion φ phase fluctuation Pion φ (Ξ orthoradial mode):phase fluctuation Chiral invariant scalar S field:amplitude fluctuation Chiral invariant scalar S field: amplitude fluctuation

3. The chiral invariant scalar background field The chiral invariant scalar background field ‘ (M. Ericson, P. Guichon, G.C) It decouples from the low energy pion dynamics: (S frozen in chiral perturbation theory). This s field relevant in nuclear physics at low space-like momentum possibly not related to the f 0 (600): π π resonance (Un Chi.PT) Explicit model: NJL + confinement (Celenza-Shakin, Bentz-Thomas) sthe sigma meson of nuclear physics and relativistic We identify s with the sigma meson of nuclear physics and relativistic (Walecka) theories (Walecka) theories, i.e., the background attractive scalar field at the origin of the binding Nuclear medium« shifted vacuum » Nuclear medium Ξ « shifted vacuum » with order parameter S=f  +s.

4. BUT TWO MAJOR PROBLEMS 1- Nuclear matter stability 1- Nuclear matter stability: Unavoidable consequence of the chiral attractive tadpole effective potential (mexican hat): attractive tadpole Dropping of Sigma mass Collapse of nuclear matter s s s s ss 2- Nucleon structure 2- Nucleon structure : the scalar susceptibility of the nucleon Lattice data analysis (Leinweber, Thomas, Young, Guichon) (Bentz, Thomas) a 2 : related to the non pionic piece of the sigma term with scalar field mass a 4 : related to the scalar susceptibility of the nucleon: from lattice data essentially compatible with zero To be compared with

5. The two failures may have a common origin: the neglect of nucleon structure, confinementscalar response i.e., confinement. Introduce the scalar response of the nucleon, i.e., the nucleon gets polarized in the nuclear medium The scalar susceptibility of the nucleon is modified Scalar nucleon response s s N Cure: Nucleon structure effect and confinement mechanism Nuclear matter can be stabilized ATTRACTIVE TADPOLE ATTRACTIVE TADPOLE: destroys saturation + chiral mass dropping SCALAR RESPONSE OF THE NUCLEON SCALAR RESPONSE OF THE NUCLEON: three body repulsive force restores matter stability and stabilizes the sigma and nucleon masses s s s

6. First results: Two sets of parameters (before lattice analysis) (green line) (red line + density dep.) Nucleon structure effects compensates the chiral dropping EOS SIGMA MASS Chiral dropping

7. Pion loops: correlation energy and chiral susceptibilities On top of mean field: V L =Pion + short range (g’) V T =Rho + short range (g’)  L,T : full (RPA) spin-isospin polarization propagators Mean-field (Hartree) TOTAL Fock Correlation energy m s =850 MeV g  =8 C=0.985+  dep. Correl. energy L: -8 MeV T: -9 MeV (M. Ericson,G.C)

8. PSEUDOSCALAR SCALAR SUSCEPTIBILITIES Pion loop enhancement Downwards shift ot the strength TAPS data Valencia group calculation

9. Relativistic Hartree-Fock One motivation:One motivation: asymmetric nuclear matter; introduce   The (static) hamiltonianThe (static) hamiltonian VDM: Strong rho: Classical and fluctuating meson fieldsClassical and fluctuating meson fields HARTREE EXCHANGE (E. Massot, G.C)

10. « HARTREE » HAMILTONIAN Nuclear matter: assembly of nucleons (Y shaped color strings) moving in a self-consistent background fields (condensates) - Scalar (s, δ) pseudoscalar (  ), vectors (ρ, ω) - The nucleon gets polarized in the nuclear scalar field «EXCHANGE» HAMILTONIAN Nucleons interact through the propagation of the fluctuations of these meson fields - Scalar fluctuation propagates with the in-medium modified scalar (« sigma ») mass

11. Hartree-Fock equations Fock rearrangement Generate together with Hartree terms, the Fock and rearrangement terms (Hugenholtz-Van Hove theorem)

12. Symmetric nuclear matter All parameters fixed All parameters fixed (up to a fine tuning) by Hadron phenomenology + Lattice QCD g S =M N /f  m  =800 MeV (lattice) C≈1.25 (lattice) g ρ =2.65, g ω ≈ 3 g ρ (VDM) Rho tensor: K ρ =3.7 (VDM) Cut of contact pion and rho

13. Asymmetry energy Hartree (RMF) Fock Influence of rhe ρ tensor coupling Influence of rhe ρ tensor coupling: Kρ=5 gives interesting result

14. Isovector splitting of nucleon effective masses Neutron rich matter Dirac mass Effective mass in nuclear physics (Landau mass) neutron proton In agreement with Dirac-BHF and DDRHF

15. Two questions : 1- Status of the background scalar field 2- Nucleon structure and scalar response of the nucleon 2- Nucleon structure and scalar response of the nucleon Take standard NJL Semi bozonized, make the non linear transform low momentum expansionMake a low momentum expansion of the effective action (quark determinant) Vectors (rho, omega) Scalar field, chiral effective potential pion Valid at low space- like momentum. Not for on-shell   Shakin et al) (Chan: PRL 87))

16. Use delocalized NJL Momentum dependant quark mass (lattice) Pion decay constant(q=0) Zero momentum masses

17. Equivalent linear sigma model Chiral effective potential Expansion around the vacuum expectation value of S S eff effective scalar field normalized to F  in vacuum

18. A toy model for the nucleon Introduce scalar diquark Decrease with S, i.e., with nuclear density Nucleon as a quark-diquark system. But confinement has to be included in some way to generate a sizeable scalar response of the nucleon and to prevent nuclear matter collapse Bentz, Thomas: infrared cutoff Present work: confining potential between quark (triplet)-diquark (anti-triplet) V=K r 2 Non relativistic limit In vacuum: M N =1304 MeV g S =7.15 -365 MeV ( attributed to pion cloud) Pion nucleon sigma term M N =HALF CONFINEMENT+ HALF Chi.SB ** Jameson, Thomas, GC ** One of us (GC) would like to thank (25 years later) the Adelaide hospital for hospitality during the period of completion of this work M D =400 MeV, K=(290 MeV) 3

19. Nuclear matter saturation The saturation mechanism is there, but not sufficient binding, Add pion Fock+ correlation energy (M. Ericson, GC) Quark, diquark, nucleon masses Mean-field (Hartree) TOTAL Fock Correlation Nucleon Quark Diquark (M-M 0 )/M 0 =s/F   

20. CONCLUSIONS The scalar attractive background field at the origin of nuclear binding is identified with the radial fluctuation of the chiral condensate The stability of nuclear matter is linked to the response (susceptibility) of the nucleon to this scalar field and depends on the confinement (quark structure) (quark structure) mechanism Relativistic Hartree-Fock (+pion+rho) good, almost parameter free, description of symmetric and asymmetric matter. Pions loop correlation energy helps to saturate ( building of a functionnal for finite nuclei) The scalar field (sigma meson of low momentum nuclear physics) not necessarily related to the  (600) The scalar response of the nucleon particularly sensitive to the balance between chiral symmetry breaking and confinement in the origin of the nucleon mass

21. What about the LNA and NLNA contributions to the sigma term? HAPPY BIRTHDAY TONY ADELAIDE 1985

22. Neutron matter

24. Hugenholtz-Van Hove theorem μ without rearrangement  with rearrangement Binding energy Very important for finite nuclei (position of the fermi energy displaced by 5 MeV)