Scalar response of the nucleon, Scalar response of the nucleon, Chiral symmetry Chiral symmetry

and nuclear matter propertiesand nuclear matter properties

G. Chanfray, IPN Lyon, IN2P3/CNRS, Université Lyon 1M. Ericson, IPN Lyon, IN2P3/CNRS, Université Lyon 1 and Theory division, CERN

Workshop in Honour of Tony Thomas's 60th Birthday Adelaide, February 2010

Relativistic models of nuclear binding (Walecka et al)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 condensatesConnection between nuclear background fields and QCD condensates

Many-body effects vs nucleon substructure response Many-body effects vs nucleon substructure response (lattice QCD)(lattice QCD)

Low energy QCDChiral sym/Confinement

The chiral invariant scalar background fieldThe chiral invariant scalar background fieldFields associated with the fluctuations of the chiral condensate.Go from cartesian (linear: σσ,,ππ) to polar (non linear: s, s, φφ) representation

Pion Pion φφ (Ξ orthoradial mode):phase fluctuationphase fluctuation

Chiral invariant scalar S field:Chiral invariant scalar S field: amplitude amplitude fluctuationfluctuation

The chiral invariant scalar background fieldThe 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 f0 (600): π π resonance (Un Chi.PT) Explicit model: NJL + confinement (Celenza-Shakin, Bentz-Thomas)

• We identify ss with the sigma meson of nuclear physics and relativistic 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 mediumNuclear medium Ξ « shifted vacuum »« shifted vacuum » with order parameter S=f+s.

BUT TWO MAJOR PROBLEMSBUT TWO MAJOR PROBLEMS

1-1-Nuclear matter stabilityNuclear matter stability: Unavoidable consequence of the chiral effective potential (mexican hat): attractive tadpoleattractive tadpole

Dropping of Sigma mass

Collapse ofCollapse ofnuclear matternuclear matter

s ss s

ss

2-2-Nucleon structureNucleon structure: the scalar susceptibility of the nucleonLattice data analysis (Leinweber, Thomas, Young, Guichon)

(Bentz, Thomas)

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

The two failures may have a common origin: the neglect of nucleon structure,i.e., confinementconfinement. Introduce the scalar responsescalar response of the nucleon, i.e., the nucleon gets polarized in the nuclear medium

The scalar susceptibility of the nucleon is modifiedThe scalar susceptibility of the nucleon is modified

Scalar nucleon response

s s

N

Cure: Nucleon structure effect and confinement mechanismCure: Nucleon structure effect and confinement mechanism

Nuclear matter can be stabilizedNuclear matter can be stabilized

ATTRACTIVE TADPOLEATTRACTIVE TADPOLE: destroys saturation + chiral mass dropping

SCALAR RESPONSE OF THE NUCLEONSCALAR RESPONSE OF THE NUCLEON: three body repulsive forcerestores matter stability and stabilizes the sigma and nucleon masses

s ss

First results: Two sets of parameters (before lattice analysis)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

Pion loops: correlation energy and chiral susceptibilitiesPion loops: correlation energy and chiral susceptibilities

On top of mean field:

VL=Pion + short range (g’) VT=Rho + short range (g’)

L,T: full (RPA) spin-isospin polarization propagators

Mean-field (Hartree)

TOTAL

Fock

Correlation energy

ms=850 MeV g=8 C=0.985+dep.

Correl. energyCorrel. energy

L: -8 MeV T: -9 MeV

(M. Ericson,G.C)

PSEUDOSCALAR

SCALAR

SUSCEPTIBILITIES

Pion loop enhancementPion loop enhancement

Downwards shift ot the strength

TAPS dataValencia group calculation

Relativistic Hartree-FockRelativistic Hartree-Fock

•One motivationOne motivation:: asymmetric nuclear matter; introduce

•The (static) hamiltonianThe (static) hamiltonian

VDM:

Strong rho:

•Classical and fluctuating meson Classical and fluctuating meson fieldsfields

HARTREE EXCHANGE

(E. Massot, G.C)

« HARTREE » HAMILTONIAN« 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«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

Hartree-Fock equationsHartree-Fock equations

Generate together with Hartree terms, the Fock Fock and rearrangementrearrangement terms

(Hugenholtz-Van Hove theorem)(Hugenholtz-Van Hove theorem)

Symmetric nuclear matterSymmetric nuclear matter

All parameters fixedAll parameters fixed (up to a fine tuning) by Hadron phenomenology + Lattice QCD

•gS=MN/f m=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

Asymmetry energyAsymmetry energy

Hartree (RMF)

Fock

Influence of rhe Influence of rhe ρρ tensor coupling tensor coupling:

Kρ=5 gives interesting result

Isovector splitting of nucleon effective massesIsovector splitting of nucleon effective masses

Neutron rich matterNeutron rich matter

Dirac mass

Effective mass in nuclear physics (Landau mass)

neutron

protonIn agreement with Dirac-BHFIn agreement with Dirac-BHFand DDRHFand DDRHF

Two questionsTwo questions : : 1- Status of the background scalar field1- Status of the background scalar field 2- Nucleon structure and scalar response of the 2- Nucleon structure and scalar response of the nucleonnucleon•Take standard NJL

•Semi bozonized, make the non linear transform

•Make a low momentum expansionlow 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))

Use delocalized NJLUse delocalized NJL

Momentum dependant quark mass (lattice)

Pion decay constant(q=0)

Zero momentum masses

Equivalent linear sigma modelEquivalent linear sigma model

•Chiral effective potential

•Expansion around the vacuum expectation value of S

•Seff effective scalar field normalized to Fin vacuum

A toy model for the A toy model for the nucleonnucleon

Introduce scalar diquark

Decrease with S, i.e., withnuclear 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 cutoffPresent 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)

Pion nucleon sigma term

MN=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

MD=400 MeV, K=(290 MeV)3

Nuclear matter saturationNuclear matter saturation

The saturation mechanism is there, but not sufficient binding,Add pion Fock+ correlation energy (M. Ericson, GC)

Quark, diquark, nucleon massesQuark, diquark, nucleon masses

Mean-field (Hartree)

TOTAL

Fock

Correlation

Nucleon

Quark

Diquark

(M-M0)/M0=s/F

CONCLUSIONS

The scalar attractive background field at the origin of nuclear The scalar attractive background field at the origin of nuclear binding is identified with the radial fluctuation of the chiral binding is identified with the radial fluctuation of the chiral condensatecondensate

The stability of nuclear matter is linked to the response The stability of nuclear matter is linked to the response (susceptibility)(susceptibility)of the nucleon to this scalar field and depends on the confinement 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,Relativistic Hartree-Fock (+pion+rho) good, almost parameter free,description of symmetric and asymmetric matter. Pions loop description of symmetric and asymmetric matter. Pions loop correlation energy helps to saturate ( building of a functionnal for correlation energy helps to saturate ( building of a functionnal for finite nuclei)finite nuclei)

The scalar field (sigma meson of low momentum nuclear physics) The scalar field (sigma meson of low momentum nuclear physics) not necessarily related to the not necessarily related to the (600)(600)

The scalar response of the nucleon particularly sensitive to the The scalar response of the nucleon particularly sensitive to the balance between chiral symmetry breaking and confinement in the balance between chiral symmetry breaking and confinement in the origin of the nucleon massorigin of the nucleon mass

What about the LNA and

NLNA contributions to the sigma

term?

HAPPY BIRTHDAY TONYHAPPY BIRTHDAY TONY

ADELAIDE 1985

Neutron matterNeutron matter

Hugenholtz-Van Hove theoremHugenholtz-Van Hove theorem

μ without rearrangement

with rearrangement

Binding energy

Very important for finite nuclei (position of the fermi energy displaced by 5 MeV)