Scalar response of the nucleon, Chiral symmetry and nuclear matter properties

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

2. Nuclear many-body problem Low energy QCD Chiral sym/Confinement Many-body effects vs nucleon substructure response (lattice QCD) 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 Connection between nuclear background fields and QCD condensates The chiral invariant scalar background field Fields associated with the fluctuations of the chiral condensate. Go from cartesian (linear: σ,π) to polar (non linear: s, φ) representation Pion φ(Ξ orthoradial mode):phase fluctuation Chiral invariant scalar S field:amplitude fluctuation

3. The chiral invariant scalar background field‘ (M. Ericson, P. Guichon, G.C) • We identify s with the sigma meson of nuclear physics and relativistic • (Walecka) theories, i.e., the background attractive scalar field at • the origin of the binding • Nuclear mediumΞ« shifted vacuum » with order parameter S=fp+s. • 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 f0 (600): π π resonance (Un Chi.PT) • Explicit model: NJL + confinement (Celenza-Shakin, Bentz-Thomas)

4. s s s s s s BUT TWO MAJOR PROBLEMS 1-Nuclear matter stability:Unavoidable consequence of the chiral effective potential (mexican hat): attractive tadpole Collapse of nuclear matter Dropping of Sigma mass (Bentz, Thomas) 2-Nucleon structure: the scalar susceptibility of the nucleon Lattice data analysis (Leinweber, Thomas, Young, Guichon) a2 : related to the non pionic piece of the sigma term with scalar field mass a4 : related to the scalar susceptibility of the nucleon: from lattice data essentially compatible with zero To be compared with

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

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

7. Mean-field (Hartree) TOTAL Fock Correlation energy Pion loops: correlation energy and chiral susceptibilities (M. Ericson,G.C) On top of mean field: PL,T: full (RPA) spin-isospin polarization propagators VL=Pion + short range (g’) VT=Rho + short range (g’) ms=850 MeV gw=8 C=0.985+r dep. Correl. energy L: -8 MeV T: -9 MeV

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

9. Relativistic Hartree-Fock (E. Massot, G.C) • One motivation: asymmetric nuclear matter; introducer,p, d • The (static) hamiltonian VDM: Strong rho: • Classical and fluctuating meson fields HARTREE EXCHANGE

10. « HARTREE » HAMILTONIAN Nuclear matter: assembly of nucleons (Y shaped color strings) moving in a self-consistent background fields (condensates) - Scalar (s, δ) pseudoscalar (p), 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 Generate together with Hartree terms, the Fock and rearrangement terms (Hugenholtz-Van Hove theorem)

12. Symmetric nuclear matter All parameters fixed (up to a fine tuning) by Hadron phenomenology + Lattice QCD • gS=MN/fp • ms=800 MeV (lattice) • C≈1.25 (lattice) • gρ=2.65, gω≈ 3 gρ (VDM) • Rho tensor: Kρ=3.7 (VDM) • Cut of contact pion and rho

13. Asymmetry energy Hartree (RMF) Fock Influence of rhe ρ tensor coupling: Kρ=5gives 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 • Take standard NJL • Semi bozonized, make the non linear transform • Make a low momentum expansion of the effective action (quark determinant) (Chan: PRL 87)) Scalar field, chiral effective potential pion Valid at low space-like momentum. Not for on-shell r,v,s (Shakin et al) Vectors (rho, omega)

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 • Seff effective scalar field normalized to Fp 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 r2 Non relativistic limit In vacuum: MN=1304 MeV gS=7.15 -365 MeV ( attributed to pion cloud) MD=400 MeV, K=(290 MeV)3 MN=HALF CONFINEMENT+ HALF Chi.SB Pion nucleon sigma term ** One of us (GC) would like to thank (25 years later) the Adelaide hospital for hospitality during the period of completion of this work ** Jameson, Thomas, GC

19. Quark, diquark, nucleon masses Nucleon Quark Diquark Nuclear matter saturation (M-M0)/M0=s/Fp Mean-field (Hartree) TOTAL Fock Correlation The saturation mechanism is there, but not sufficient binding, Add pion Fock+ correlation energy (M. Ericson, GC) r / r0

20. 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) 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 s(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 CONCLUSIONS

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

22. Neutron matter

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