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Nuclear Binding and QCD

( with G. Chanfray)

Magda Ericson, IPNL, Lyon

SCADRON70 Lisbon February 2008

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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.)

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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 sN the nucleon scalar density)

Use of effective theories : -model andchiral

partners

In vacuum <> = f . In medium <> = f +

4SHORT 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 : MN and evolve differently

Pion cloud influence should be removed from to extract

model dependence introduced

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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 :

ss

N

Potential :

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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 mby tadpole diagram

Expanding sA in density :

8The term linear in density of sA represents the contribution of the

individual nucleons, sN N

s , to the nuclear response

In the model we thus find :

This contribution is proportional to the tadpole scattering amplitude

Introducing Qs = scalar quark number of the nucleonProportionality factor :

9In 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 MN with the quark mass (equivalently with m

2) : MN (m)

Lattice data are available only for m >0.1 (GeV)2

Extrapolation needed to reach the physical MN

MN(m

Nucleon mass GeV

Lattice data for MN(m versus m

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Lattice data of interest : successive derivatives of MN (m

) provide Qs and sN !

But these are total values which include the pionic contribution

Fortunately,

The pion loop contribution to MN(m2) has been separated out (Thomas et al.)

(It contains non-analytical terms in mq , which prevents a small mq expansion) .The separation introduces model dependence

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Method of Thomas, Leinweber et al. :

a2 = +1.5 GeV-1 ; a4 = -0.5 GeV-3

The pionic loop contribution to MN depends on the N form factor.Different forms are used (monopole, dipole, gaussian)

with an adjustable parameter .The rest is expanded in m

MN(mmpionic term + apionic term + a0 0 + a+ a22mm

2 2 + a+ a44mm4 4 +… +…

The parameters a are practically insensitive to the choice of the form factor

(dominated by a(dominated by a22 term) term)

From this we deduce :

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Sign of a4<0 is as predicted in the model

This magnitude is more than 10 times too large!!This magnitude is more than 10 times too large!!The The model is contradicted by the parameter model is contradicted by the parameter a4 !

Are the magnitudes of the expansion parameters also compatible with the model?

In the In the model : model :

13Another failure of the model ?No, the same one as before,

sN and TN are related !

Our approach : MN 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 !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 : MN totally from confinementNo relation of nuclear field to chiral field

With confinement :What is the link between s

N and TN ?

14Illustration in a hybrid model of the nucleon (introduced by Shen, Toki)

-Scalar susceptibility :

-Scalar nucleon charge :

-Nucleon mass : MN = 3E(M) > 3M ;

Three constituent quarks (mass M) kept together by a central harmonic potential V(r)

Susceptibility of constituent quark <0Confinement term >0

15Two terms of opposite sign contribute to sN

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 quarkg

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

16ii) 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

17In 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 sN and TN

The ratio rchiral becomes

and

18Our approach

Use QCD lattice expansion to fix the scalar parameters of nuclear physicsExpansion provides Qs and N

s

From the expressions of Qs and sN and the relation :

instead of 3 for the tadpole alone

we can write the in-medium propagator :

m stabilized !

ii)

i)

193-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 :

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Our fit parameters

M*N

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 (gfit close to VDM value)

• N form factor : dipole with =0.98 GeV (Npion=21 MeV or N

total =50 MeV )

• g’ values : from spin-isospin physics g’NN=0.7, g’N=0.3, g’=0.5

• gs/m2 =a2 /f =15 GeV-2

(gives mean scalar field about 20 MeV at 0)with gs= MN/f =10 ( model value), m= 800 MeV

• C ( = N f/gs ) : allowed to vary near the lattice value Clattice =2.5 :

Cfit =2

Lead to successful description of nuclear binding !

21Summary

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: