Partial Chiral Symmetry restoration and. QCD Scalar Susceptibilities in Nuclear Matter

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1 Partial Chiral Symmetry restoration and QCD Scalar Susceptibilities in Nuclear Matter M. Ericson, P. Guichon, M. Martini, D. Davesne, G.C Chiral restoration in hadronic matter Scalar and pseudoscalar susceptibilities Susceptibility in a chiral effective theory Some consequences

2 A- CHIRAL RESTORATION IN HADRONIC MATTER A1 -IN-MEDIUM SPECTRAL FUNCTIONS Restructuring of the QCD vacuum Dropping of the condensates Observable consequences in the excitation spectrum In-medium hadronic spectral functions ρ h ω, q = 1 ImD hω, q Evolution of the centroids masses : no universal links between masses and condensates i.e. masses vs f Shape of the hadronic spectral functions and related quantities χ Softening, sharpening σ or broadening K, ρ, dissolving of the resonances

3 A2- CHIRAL RESTORATION AND HADRON STRUCTURE Dropping of the quark condensate at finite ρ B,T qq µ B,T =1 qq vac h ρ sµ B,T σ h f 2 m2 σ h = m q M h m q = m 2 M h m 2 ρ s µ B,T= Ωµ B,T M h Dominated by lightest hadrons, heavy hadrons decouple from the condensate Nuclear matter, leading order qq ρ qq vac 1.35 ρ ρ T 2 8 f 2 Extrapolation of lattice data Leinweber,Thomas M N, = α + βm 2 + P ion loopλ The specific contribution of the pion cloud is very important and depends on a scale : the nucleon size hidden in χp T σ N 5MeV, half of it from the pion cloud

4 A3- IN-MEDIUM MASS SPLITTING Pattern of symmetry breaking generated by chiral dynamics Mass MeV 18 5 D +, D + Κ, Κ + Κ Κ D + D 5 MeV 1 MeV Deeply bound pionic states : In medium increase of the isovector scattering length b 1 M exp =23 27MeV in the center of Pb +, MeV ρ/ρ Chiral effective theory Vertex correction : pion cloud effect b 1 b 1 =1+ σ N ρ f 2 m Σ N ρ f 2 m ρ ρ Energy dependance of rescattering in presence of Pauli correlations Everything together b 1 b 1 =1.19 at ρ =.5ρ resc δπ = ω M q p - q < k p<k F M ω

5 A4-2 PROCESSES AND CHIRAL SYMMETRY A, CHAOS, CB, Aγ,TAPS Downwards shift of the invariant mass distribution in the scalar-isoscalar channel I = J = What is the role of dσ/dm /A nb/mev γ, H C 28 Pb dσ/dm /A nb/mev γ, H 12 C 28 Pb T * A -ChiralDynamics - Chiral restoration? M MeV M M MeV M Roca et al, γ In-medium I = J = interaction In-medium pions Schuck-Chanfray Dropping of m σ,f Hatsuda-Kunihiro σ σ σ σ N

6 Dropping of the sigma mass Associated with partial chiral restoration Linear sigma model Hatsuda : 3% dropping at ρ = ρ m 2 σ m 2 σ =1+3 s f qq qq vac =1+ s f φ2 2 f 2 But constraints from nuclear matter stability Closely related to the p h contribution to the nuclear scalar susceptibility In-medium modified two-pion interaction Closely related to to the pion cloud contribution to the nuclear scalar susceptibility Also linked to chiral restoration In-medium Scalar and Pseudoscalar susceptibilities Partial Chiral symmetry restoration?

7 B- SCALAR AND PSEUDOSCALAR SUSCEPTIBILITIES B1- FLUCTUATIONS OF THE CONDENSATE Scalar susceptibility : quark density fluctuations from the scalar correlator i.e. the correlator of the scalar χ s = qq m =2 dt d r δ qq,, δ qq r, t Thermal susceptibilty on Lattice Karsch.6 χ /2 SCALAR PSEUDOSCALAR a [δ] f [σ] m =.2 L = Finite density : effective chiral theory M. Ericson, G. C SUSCEPTIBILITY PSEUDOSCALAR SCALAR DENSITY / NORMAL DENSITY Chiral Restoration : χ S χ PS

8 B2- SCALAR SUSCEPTIBILITY χ S = qq =2 dt dr Θt t i [ qq, qqr t ] m q χ S = 2 ω m 2 q µ = ReG S ω =, q = dω 2 ω ImG S ω, q = A strong contribution of low energy nuclear excitations is expected Linear sigma model qq qq vac f σ : The nuclear susceptibility is relatedtothein-medium σ propagator B3- PSEUDOSCALAR SUSCEPTIBILITY [ χ PS =2 dt dr Θt t τ α i qiγ 5 2 q, qiγ 5 using µ A α µ x =2m q qiγ 5 τ α2 qx and soft pion theorem ] τ α 2 qr t = qq ρ m q χ PS behaves like the chiral condensate

9 B4- LEADING ORDER ESTIMATE Grand Potential : ω = 4 d 3 p 2 3 E p µ θ µ E p Chiral condensate : qq ρ qq vac = 1 2 ω m q µ = 1 2 m M ω q M µ σ N 2 m q ρ S σ N : nucleon sigma term, ρ S : scalar density Nuclear susceptibility : χ S ρ =χ S vac + ρ S σn m q 2 m q + σ N ρs 2 m q m q µ ρ S χ N S + χnuclear S χ N S : nucleonic scalar susceptibility

10 Nucleonic contribution : dominated by the pion cloud through the Leading non analytical contribution χ N S = 2 qq 2 vac f 4 m 9 64 ga f 2 4 Λ Λ+m LNAC : ρ S χ N S =.8 χ PS vac With +FF+Pauli : ρ S χ N S.4.5 χ PS vac N N, Nuclear p-h excitation contribution : Compressibility K χ nuclear S = σ2 N 2 m 2 q 2 p F M 2 N 2 ρs M µ σ2 N 2 m 2 q 2 p F M 2 N 2 =Π ph ω =, q = 9 ρ K ρ=ρ σ N p h σ N Sigma model : χ nuclear S σ N = M N m 2 m 2 σ =.35 χ PS vac + σ N σ INTERFERENCE σn σ N

11 C- SUSCEPTIBILITY IN A CHIRAL EFFECTIVE THEORY C1- BASICS Order parameter : M = qq 2 + i τ i qγ5 τ 2 q = σ + i τ From Cartesian to Polar coordinates [ σ + i τ = f + s exp i τ φ ] f G φ 2 f 2 φ S Pion φ : phase fluctuation Scalar field s = S f : amplitude fluctuation f σ qq medium qq vacuum =1 φ2 ρ 2 f 2 sρ f Effective Lagrangian L = i Nγ µ µ N MN s NN µ s µ s m2 σ m2 s 2 +2f 8f 2 s + 2f 2m2 m 2 σ m2 + L ω + L pwave NN + L + L WT MN s = M N 1+ sf BUTNOMATTERSTABILITY 2

12 C2- MATTER STABILITY IN CHIRAL THEORIES Shifted vacuum with chiral order parameter S <f Energy density : ɛρ, S = p<p F p 2 + M N S +V S +C V ρ 2 NJL Bentz, Thomas, Birse : Nucleon qq q state. M q = g q S Chiral QHD : S = f + s chiral invariant scalar field : M N = g snn S g snn S = M N S drops Needed to stabilize nuclear matter NJL : Infrared cutoff : simulate confinement QMC : Polarization of the quark WF : nucleon structure Nuclear saturation vs Nucleon structure and QCD lattice

13 C3- MEAN FIELD : IN-MEDIUM SIGMA MASS AND NUCLEAR SCALAR SUSCEPTIBILITY Include the scalar response κ NS of the nucleon to a scalar field M N = M N 1+ s f κ NS s 2 Minimization : ε s = g S ρ S + V s =, g S s = M N s drops with ρ In-medium sigma mass : m 2 σ = 2 ε s 2 = V s +κ NS ρ S χ S related to in-medium sigma propagator dressed by ph excitations χ S = 2 qq 2 vac f 2 1 m 2 σ + 1 m 2 σ Π SS 1 m 2 σ Π SS is the full scalar polarization propagator related to K Π SS = g 2 S M N E F Π [ 1 g 2 ω m 2 ω E F M N g 2 S m 2 σ MN E F Π ] 1 Parameters : m σ phase shifts, g ω and κ NS

14 2 sets of parameters In-medium sigma mass E/A MeV 5 1 SIGMA MASS MeV ρ/ρ Nuclear susceptibility 1 2 ρ/ρ NUCLEAR SUSC VACUUM SCALAR SUSC PSEUDOSCALAR SCALAR Nucleon structure effect compensates the chiral dropping! κ NS Probably too large Introduce pion loops CHIPT or pionic correlation energy ρ/ρ

15 With pionic contribution to the nucleon sigma term and pion Fock term χ S = 2 qq 2 vac f 2 1 m 2 σ + 1 m 2 σ σ N + σσ N σ σ N 2 eff Π SS 1 m 2 σ BINDING ENERGY CHI/CHIPS vac SIGMA MASS=85 MeV GV=4.9 C= PSEUDOSCALAR SCALAR SCALAR DENSITY DENSITY

16 C4- PION CLOUD CONTRIBUTION TO THE IN-MEDIUM SUSCEPTIBILITY : i.e., pion loop or in-medium modified pion cloud contribution to the nucleonic susceptibility Correlation energy to be incorporated numerically in the EOS 1 dλ id 4 q E corr+fock = λ 2 4 λ2 D qπ L q; λ +λ 2 m2 12 S 2 φ2 2 Π L q; λ =Π q/ 1 λ 2 g + V exch Π q is the full longitudinal spin-isospin polarization propagator and Π q is the Lindhardt function including a slightly in-medium modified g NN φ 2 =3 id 4 q 2 4 q2 D 2 qπ Lq is the in-medium modified pion scalar density. + g E corr = Π L + Π L T + g Π L

17 In-medium quark condensate Feynman-Hellman theorem qq cloud Medium = m2 E corr+fock S φ 2 = qq 2 m q m 2 Vac 1+ φ2 1 H f 2 S 2 12 S 2 χsb cloud Medium 2m q Compatible with χp T at finite T Gasser-Leutwyler, GC-Ericson-Wambach Susceptibility from the pion cloud χ cloud S = qq cloud Medium m 2 GE = dq 2 3 pion dressed by p h and h = 2 qq 2 Vac f 2 S f 3 2 S 2 G G vac 1 V 2 G idq 2 D q,q D q, E q is the in-medium two-pion propagator RESULTS - Vacuum : χ S vac =.4 χ PS vac - First order one p-h insertion : δχ nuclear S = ρ S χ N S =.64 χ PS vac - Full calculation : full pion dressing + rescatt. δχ nuclear S =.1 χ PS vac

18 D- SOME CONSEQUENCES D1- THE SCALAR SUSCEPTIBILITY AND PRODUCTION The Tmatrix T E =V E +V E GE T E GE = dq 2 3 idq 2 D q,q D q, E q T E calculable in unitarized χp T Valencia or in sigma model Lyon-Darmstadt but the medium effects near the 2 threshold are actually independant of the sigma mass if the sigma mass remains stable. The medium effects seen in production experiment are embedded in GE 2 m, the in-medium two-pion propagator pion dressed by p h and h The in-medium modification of the pionic susceptibility is embedded in GE = Sigma model phase shifts T E = 6 λe2 m 2 1 3λGE D σe D σ E = E 2 m 2 σ 6λ2 f 2 GE 1 3λGE / Im D σ ρ=ρ E MeV Dropping condensate, sigma mass In medium pions Vacuum

19 The full nuclear scalar susceptibility The σ Chiral partner of the vs the s Chiral invariant Strong σ coupling weak derivative s coupling Express the pion-pion T matrix in terms of either the σ propagator or the s propagator χ S = 2 qq 2 vac f 2 D σ D σ E =D s E S E2 m 2 E 2 m 2 σ G G is the full two-pion propagator : G = G GV G The chiral invariant s mode is the exchanged meson in the scalar NN potential and is free of pionic many-body effect : weak coupling to in-medium modified 2 states. It couples to p-h low energy nuclear excitations and gives the nuclear piece of the susceptibility. This is the sigma meson of nuclear physics. The medium effects in the σ propagator and in two-pion processes come from the second 2 modes. It gives the pion cloud piece of the scalar susceptibility.

20 D2- NUCLEAR PHYSICS IMPLICATIONS : THE NN SCALAR POTEN- TIAL Nucleons exchange not only σ but also 2 states with delicate compensations M. Birse a b σ σ σ σ c = s + T 2 + Om THE NONLINEAR σ FORMULATION IS MUCH MORE ECONOMICAL THE MEDIUM EFFECT IN THE 2 PROCESSES ARE MOSTLY ABSENT IN THE NN SCALAR EXCHANGE

21 D3- CORRELATOR MIXING Axial-vector mixing At finite temperature Π µν V q; T =1 ɛπµν V q; T =+ɛπµν A q; T = A A V V Π µν A q; T =1 ɛπµν A q; T =+ɛπµν V q; T = V A ɛ = T 2 = 2 dk nω k = 2 << Φ 2 >> 6 f 2 f ω k 3 f 2 Also at finite density G.C, M. Ericson, Rapp-Wambach, Wolf : Dilepton production Scalar-pseudoscalar mixing D σ E =D s E E2 m 2 G 2 S 2 E 2 m 2 σ For soft thermal pions, q<<e : 3 GE,T =3 dq 2 3 D σ E,T = idq 2 D q,q D q, E q <<Φ2 >> E 2 m 2 1 <<Φ2 >> S 2 D σ E,T = D s E 1/E 2 m 2 σ D σ E,T =+ <<Φ2 >> 2 S 2 D E,T =

22 CONCLUSION Sizeable effects in the nuclear scalar susceptibility Scalar and pseudoscalar susceptibilities becomes closer partial chiral restoration Nucleon structure effect compensates the chiral dropping of the sigma mass ; related to the p-h piece of the scalar susceptibility σ 2 medium effects in two-pion processes related to the pionic nuclear scalar susceptibility. Linked to a scalar-pseudoscalar mixing effect associated with partial chiral restoration. The chiral invariant s mode couples to p-h low energy nuclear excitations and gives the nuclear piece of the susceptibility. Extension at higher density and/or temperature. How to constrain the effective theories? Renormalization group?