1 Nuclear Binding and QCD ( with G. Chanfray) Magda Ericson, IPNL, Lyon SCADRON70 Lisbon February 2008

2 Nuclear Binding and QCD The existence of a scalar meson coupled to nucleons has consequences for nuclear binding Basis of relativistic theories of nuclei (Walecka, Serot) :  (attraction) and  exchange (repulsion) What new developments and perspectives? -Quark-Meson-Coupling model (QMC) : introduction of nucleonic response to  (Guichon, Thomas et al.) -Link to QCD parameters (Chanfray, M.E.)

3 BUT : this identification is not allowed ! It violates chiral constraints! (Birse)  Additional exchanges needed to cancel violating terms. Feasible, but cumbersome Identification with  natural. Would make life simple ; mass would follow condensate evolution Nuclear scalar field in the   - model ( Here   is the nucleon sigma commutator and  s N the nucleon scalar density) Use of effective theories :  -model  and  chiral partners In vacuum = f . In medium = f  + 

4 SHORT CUT : introduce another scalar field (Chanfray, Guichon, M. E.) go from cartesian (linear representation) to polar coordinates (non-linear)  In nuclear medium : allow for a change in radius : Satisfies all chiral constraints Associate nuclear scalar field with the radial mode by the identification with Loss of previous simplicity : M N and evolve differently Pion cloud influence should be removed from to extract model dependence introduced

5 Nuclear Binding in  model The  model is not a viable theory of nuclear matter. The tadpole problem can be phenomenologically cured with introduction of the response,  N, of the nucleon to the scalar field The introduction of this nucleonic response is the basis of Quark-Meson-Coupling model, QMC (Guichon, Thomas,..). Large effect (about 30% decrease of m  at  0 ) Produces collapse instead of saturation (Kerman,Miller) s 3s coupling lowers  mass in medium : s s N Potential :

6 But phenomenology is not our aim! Where is the link to QCD?? It goes through the study of the For us, in a purely phenomenological description, saturation can be obtained, for  N > 0, with a cancellation of about 2/3 of the tadpole scattering amplitude QCD scalar susceptibility Definition : Scalar susceptibility = order parameter explicit symmetry breaking parameter Nuclear susceptibility defined as (vacuum value is subtracted)

7 is the propagator of the fluctuations of the order parameter,  In the  model, the simulation of by  (x) leads to D  (0) is the  propagator at q  =0 (vacuum value) For the nuclear medium In medium, modification of m  by tadpole diagram Expanding  s A in density :

8 The term linear in density of  s A represents the contribution of the individual nucleons,  s N  N s, to the nuclear response In the  model we thus find :  This contribution is proportional to the tadpole scattering amplitude Introducing Q s = scalar quark number of the nucleon Proportionality factor :

9 In summary : the  model predicts the existence of a non-pionic component in  s N linked to the scalar meson. Its sign is negative  Any indication in favor of its existence? Maybe! In lattice results on the evolution of M N with the quark mass (equivalently with m  2 ) : M N (m   ) Lattice data are available only for m   >0.1 (GeV) 2 Extrapolation needed to reach the physical M N M N (m    Nucleon mass GeV Lattice data for M N (m    versus m  

10 Lattice data of interest : successive derivatives of M N (m   ) provide Q s and  s N ! But these are total values which include the pionic contribution Fortunately, The pion loop contribution to M N (m  2 ) has been separated out (Thomas et al.) (It contains non-analytical terms in m q, which prevents a small m q expansion). The separation introduces model dependence

11  Method of Thomas, Leinweber et al. : a 2 = +1.5 GeV -1 ; a 4 = -0.5 GeV -3 The pionic loop contribution to M N depends on the  N form factor. Different forms are used (monopole, dipole, gaussian) with an adjustable parameter . The rest is expanded in m   m    pionic term + a 0 + a 2 m  2 + a 4 m  4 +… M N (m    pionic term + a 0 + a 2 m  2 + a 4 m  4 +… The parameters a are practically insensitive to the choice of the form factor (dominated by a 2 term) From this we deduce :

12 Sign of a 4 <0 is as predicted in the  model  This magnitude is more than 10 times too large!! The model is contradicted by the parameter The  model is contradicted by the parameter a 4 ! Are the magnitudes of the expansion parameters also compatible with the  model? In the model : In the  model :

13 Another failure of the  model ? No, the same one as before,  sN and T  N are related ! Our approach : M N is partly from condensate, partly from confinement. Keep assumption : The nuclear scalar field affects the quark condensate, as in the  model Need for compensating term ! Common cure found in confinement Introduces a positive nucleon response to the scalar field, as in QMC : Quark Meson Coupling Model : Bag model : M N totally from confinement No relation of nuclear field to chiral field With confinement : What is the link between  s N and T  N ?

14 Illustration in a hybrid model of the nucleon (introduced by Shen, Toki) -Scalar susceptibility : -Scalar nucleon charge : -Nucleon mass : M N = 3E(M) > 3M ; Three constituent quarks (mass M) kept together by a central harmonic potential V(r) Susceptibility of constituent quark <0Confinement term >0

15 Two terms of opposite sign contribute to  s N Compensation possible  Similar compensation in  N scattering amplitude T  N ? Two components contribute as well to T  i) Tadpole scattering amplitude on constituent quarks tadpole amplitude on a constituent quark = g  q =  -quark coupling constant = scalar number of constituent quarks Note : the coupling of the nuclear field to the constituent quarks is linked to the assumption that acts on the quark condensate, i. e., on the constituent quark mass :    q

16 ii) Amplitude  N from nucleon structure (confinement) Sign : >0 compensation possible ii) Chiral parts Compare term by term : i) Confinement part as in  model confinement chiral

17 In the NJL model which describes the constituent quark, we recover at the quark level the previous results of the  model : Note : numerically our simple model fails to produce enough compensation. But it is important to illustrate the role of confinement. Overall Same amount of compensation by confinement as in  s N and T  N The ratio r chiral becomes and

18 Our approach Use QCD lattice expansion to fix the scalar parameters of nuclear physics Expansion provides Q s and  N s From the expressions of Q s and  s N and the relation : instead of 3 for the tadpole alone we can write the in-medium  propagator : m  stabilized ! ii) i)

19 3-body forces repulsive 3-body forces m  stable. But the introduction of response,  N, is important Make field transformation : For C>1 overcompensation repulsive 3-body forces, important for saturation The chiral potential V(s) transforms : In our fit the energy per nucleon from the 3-body force is :

20 Our fit parameters  M*NM*N m*m* Mass MeV m *  with  N =0 Density dependence of M * N and m * s in our fit Pion contribution: not at Hartree level but through Fock term and correlation terms (with or without  ). Dependence on  N form factor. Short range interaction added through Landau-Migdal parameters g’. Vector potential : m  =783 MeV, g  : free (g  fit close to VDM value )  N form factor : dipole with  =0.98 GeV (  N pion =21 MeV or  N total =50 MeV ) g’ values : from spin-isospin physics g’ NN =0.7, g’ N  =0.3, g’  =0.5 g s /m  2 =a 2 /f  =15 GeV -2 (gives mean scalar field about 20 MeV at  0 ) with g s = M N /f  =10 (  model value), m  = 800 MeV C ( =  N f  /g s ) : allowed to vary near the lattice value C lattice =2.5 : C fit =2 Lead to successful description of nuclear binding !

21 Summary Full consistency between QCD lattice expansion and nuclear binding! linear  model fails for both proper description must include nucleon structure and confinement origin of nucleon mass mixed : - in part from condensate - in part from confinement nuclear scalar field identified with chiral invariant field, linked to quark condensate Description of nuclear binding successful with parameters close to those extracted from QCD Consistency favors existence of link between the scalar nuclear potential and the modification of the QCD vacuum! it is possible to link the parameters of QCD lattice expansion to the scalar parameters of the nuclear potential (Near model independence but separation of pion cloud effect necessary) With the assumptions: