# Presentation: 932501

## Medium-modified                           NN interactions

- Jeremy W. Holt*
- Nuclear Theory Group

** ****State University of New York**

- * with G.E. Brown, J.D. Holt, and T.T.S. Kuo

- J.W. Holt

- LBL 5/21/2007

**Notes:**

Okay, so my talk is motivated by the large number of theoretical and experimental efforts to determine the properties of hadrons in a hot or dense medium.  Now even at normal nuclear matter density and zero temperature, a number of studies suggest that hadronic masses could be modified on the level of 10-20%, and I would like to discuss what consequences this would have for the NN interaction in medium and studies in low energy nuclear physics.
[Well, a large part of this conference is devoted to the study of medium modifications to hadronic properties due to the partial restoration of chiral symmetry at finite temperature or density.]

## Can such interactions provide a good description of nuclear systems?

- J.W. Holt

- LBL 5/21/2007

**Notes:**
Alright, well the main idea, I think, is pretty straightforward.  All realistic models of the NN interaction are based on meson exchange. Now there’s no question that some aspects of these potentials are purely phenomenological, but nevertheless the essential idea is that the masses, coupling constants, and form factor cutoffs that appear in these models do have physical meaning.  Therefore, we should expect that as the density of some nuclear system increases, the basic NN interaction should change according to the modifications in the meson masses, coupling constants, and cutoffs.  Now during the course of this talk I would like to focus on several questions: First, are the modifications proposed by theory and experiment consistent with what is known from nuclear structure calculations in dense systems.  Second, are there specific observables that are particularly sensitive to this density dependence.  And finally, are mass modifications sufficient or must we also modify the coupling constants or cutoffs in the potentials?

[For a time it was thought that meson exchange was the fundamental theory of the nuclear force, but of course nowadays we understand that it provides only a low energy effective theory of QCD.]
We will see that to some extent the NN interaction remains reasonable provided that both the vector and scalar meson masses drop.  In this talk I would like to discuss the consequences for the NN interaction in-medium.  In particular we will study the properties of nuclear matter.  , such studies should in principle be able to provide some recent work that I’ve been doing with both from a theoretical and experimental point of view.  In this talk I would like to discuss the consequences of medium-modified hadronic masses for the NN interaction in-medium.  Now there will be lots of other talks this week that study the theoretical and experimental state of affairs at the moment.  

One might expect that there should be evidence for this be revealed in the properties of actual nuclei.  Of course the problems with low energy nuclear structure are well-known and are twofold: (1) the NN interaction is not KNOWN, (2) many-body theory is not PERFECT (especially since the methods are generally based on perturbation theory—and the NN interaction is not SMALL).
In QCD hadronic masses are generated dynamically.  This is understood and non-negotiable: starting from bare quarks with masses on the order of a few MeV or less, lattice QCD is able to predict the masses of 

## Double Decimation

- _Double Decimation_

- Match *T* and  dependence with QCD Sum Rule correlation functions

- Traditional OBE potentials (e.g. Bonn, Nijmegen, etc.)

- EFT with intrinsic           ***T*** and **** dependence

- QCD

- Chiral decimation

- Fermi liquid decimation

- Describe in a unified way

- G. Brown and M. Rho, Phys. Rept. 396 (2004)

- (1) Finite nuclei     (2) Nuclear matter (3) Hot/dense matter

- Low-momentum interaction

- Introduce medium modifications “by hand”

- J.W. Holt

- LBL 5/21/2007

**Notes:**
This approach has been advocated by Gerry Brown and Mannque Rho in a slightly different formulation known as “Double Decimation”.  The idea is to start from QCD and perform the standard chiral decimation down to an effective theory incorporating the spontaneous breaking of chiral symmetry (in their case a HLS theory that includes dynamical vector mesons as gauge bosons).  By matching the current correlators in the theory to those from QCD sum rules, one obtains parameters in the theory that include the temperature and density dependence.  With this new effective theory, one then performs the second decimation down to the region of the Fermi surface to obtain a low momentum interaction.
In this way the idea is to describe in a unified way finite nuclei, nuclear matter, and hot/dense matter.  Our approach will be a bit more practical: we start from one of the well-known realistic NN interactions and include the density dependence “by hand”, relying on a broad set of results from experiments and theoretical analysis.

It was shown by Shankar that one can obtain Fermi liquid theory by integrating out the momentum modes around the fermi surface.  In practice it is realized by integrating out the high momentum modes that are unconstrained by two nucleon scattering data.  Let me now say more about our decimation procedure.
How should one actually carry out this procedure?  HLS proposes to be able to access the density and temperature dependence through Wilsonian matching.

## In-medium modifications (T = 0,   0)

| CD-Bonn potential | Nijmegen potentials |
| --- | --- |
| 0(135), ±(140) | 0(135), ±(140), (549), '(958) |
| 1(500), 2(900) | (760), a00(983), a0±(983), f0(976) |
| 0(770), ±(770), (782) | 0(767), ±(768), (782), (1020) |

- _In-medium modifications_ (*T *= 0,   0)

- _Change in mass at ____ = ___0         _Theory_    _Experiment_    _Us_
- ,  ~ 10 – 15% decrease           Yes             Yes          15%
-       ~     small increase              Yes             Yes           0%
-       ~     10% decrease               Yes             No             7%

- Assume linear scaling:

- J.W. Holt

- LBL 5/21/2007

**Notes:**
The two potentials that are most suitable for this purpose are the CD-Bonn and Nijmegen potentials.  In the Bonn potential only the most important mesons are included explicitly.  The scalar mesons are listed in blue because the masses and coupling constants are treated essentially as free parameters in different partial waves.  The Nijmegen potential includes many more mesons, but the idea of using the properties of the sigma meson as free is the same.
Now the following list is by no means intended to be complete, but rather I wanted to give an approximate summary on both the theory and experimental side.  The most widely studied are the vector mesons, which are expected to decrease in mass by 10 to 15 percent.  In our calculations we typically scale the vector meson masses and their cutoffs by 15% at nuclear matter density.  The pion mass is expected to increase slightly in NJL model calculations and in experiments with deeply bound pionic atoms, but we will assume the increase is negligible.  The sigma meson has been predicted to decrease in NJL calculations and BR scaling, but there is no experimental evidence at the moment.  These are the expected modifications at nuclear matter density, and at other densities we employ a linear scaling of the masses.
[NJL models, QCD sum rules, BR scaling, KEK, CBELSA, deeply bound pionic atoms]

## RG and EFT

_**RG and EFT**_

- Traditional problem: Strong short distance repulsion

- (a) ***G*****-matrix**

- (b) **V****low-k**

- J.W. Holt

- LBL 5/21/2007

**Notes:**
Now one of the traditional difficulties associated with using these realistic interactions in many-body perturbation theory calculations is that they have a strong short-distance repulsion in S-waves.  For a long time this was treated by including particle-particle scattering to all orders, giving rise to a well-behaved G-matrix.  Nowadays, one takes a renormalization group/effective field theory perspective and explicitly integrates out the intermediate states above some momentum cutoff Lambda in such a way that the low energy physics is preserved.  This can be accomplished by defining a low-momentum T-matrix, which includes only intermediate states below Lambda.  The low momentum interaction called Vlowk is then defined by requiring the low-momentum T-matrix to equal the original T-matrix for momenta below the cutoff.  In this way the low energy physics is preserved, since the T-matrix is directly related to the free particle scattering phase shifts.

## Universal interaction for  = 2.1 fm-1

**Universal interaction for ******** = 2.1 fm****-1**

*k* = 2.1 fm-1  *E*lab = 350 MeV

- Integrate out the _experimentally_ _unconstrained_ part of the NN interaction

- J.W. Holt

- LBL 5/21/2007

**Notes:**
Under this construction all high precision NN interactions flow to a universal interaction as the momentum cutoff Lambda is lowered to 2.1 inverse fermi.  Here we show this convergence in the 1S0 partial wave, but a similar collapse happens in all partial wave channels.  The interesting point is that 2.1 inverse fermi is actually the center of mass momentum beyond which no experimental phase shift data is incorporated into the NN interactions.  Therefore, one can also view this procedure as integrating out the unconstrained parts of the NN interaction.

## Nuclear matter

-  Easier than finite nuclei (only one density)
-  Saturation (*E/A*, *k*F, K )
-  Hartree-Fock (preliminary)
-  Ring diagram expansion (preliminary)
-  Fermi liquid theory [JWH et al., NPA 785 (2007) 322.]

- _Outline of Results_

- Finite nuclei
-  Diminishing tensor force
-  Beta decay of 14C

- J.W. Holt

- LBL 5/21/2007

**Notes:**
So I have just described for you the basic theoretical setup, it’s a novel framework for calculations in nuclear structure physics.  Now let me briefly discuss our strategy for the applications.  Symmetric nuclear matter provides a convenient starting ground since there is uniform density and the single-particle states are simply plane waves.  We discuss two features of nuclear matter: the saturation properties (in both Hartree-Fock theory and in a higher-order ring diagram calculation) and also the bulk equilibrium properties around the saturation point with the formalism of Fermi liquid theory.  Finally, if there is time I will discuss some ideas that we have for finite nuclei, including the anomalously long lifetime of carbon-14.

## (no saturation)

- Fixed cutoff in Hartree-Fock approximation

**(no saturation)**

- S. Bogner et al.,  NPA **763** (2005) 59.

- Add leading order chiral 3N force

- Empirical saturation energy and density

- J.W. Holt

- LBL 5/21/2007

**Notes:**
Recall that our novel program is to use medium-modified low-momentum interactions.  It turns out that in a Hartree-Fock calculation of the binding energy per nucleon, V(low-k) does not saturate, that is, the binding energy keeps increasing as the density increases.  It is well-known, however, that the most important 3N force is becomes strongly repulsive at high densities, and a calculation of Scott Bogner and collaborators using the leading order chiral 3N force confirms this.  We will look at this problem from the perspective of dropping masses.

## Tensor force with dropping masses

- J.W. Holt

- LBL 5/21/2007

**Notes:**
The second order tensor force is known to give a strong contribution to the binding energy.  Now the pion and rho meson give contributions that act opposite to each other in the tensor force.  Since the rho meson mass drops but the pion mass remains relatively unchanged, the tensor force will be cut down as the rho mass decreases.  Thus, as the density increases the binding energy should decrease, since the strongly attractive tensor force is diminished.  Here we show the both the effect on both the tensor force and the second-order tensor force.  We therefore expect that saturation might be achieved by including these medium effects.

## Hartree-Fock

- Novel saturation mechanism

**Hartree-Fock**

**+**

- Preliminary!

- J.W. Holt

- LBL 5/21/2007

**Notes:**
In the first-order Hatree-Fock calculation at each density we use a scaled NN interaction assuming a linear decrease in the masses up to nuclear matter density.  We see that saturation is indeed achieved for a fixed cutoff of 2.1 inverse fermi in Vlowk.

## +

- Introduce model space m ~ 3.0 fm –1

- Choose vlowk = m

- J.W. Holt

- LBL 5/21/2007

**Notes:**
We can extend this calculation by including not only the first order term, but also the higher order ring diagrams shown here to second and third order.  In the standard linked diagram expansion for the binding energy, each term can be written as an integral over the frequency variable omega.  Here F refers to the free two-particle or two-hole propagator, which depends on omega, and the antisymmetrized interaction V.  This can be simplified by introducing a strength parameter lambda and rewriting the infinite summation in terms of the exact particle-particle Green function.  The final ingredient is to use an effective interaction formalism by introduction of a model space which restricts the intermediate particles and holes to lie below a cutoff lambda_m, which makes the calculation practicable.  In previous calculations a model space cutoff of 3.0 inverse fermi was found to be suitable, so in our calculations we match the low momentum cutoff to the model space cutoff.

## Preliminary!

**[fm****–1****]**

**[MeV]**

- J.W. Holt

- LBL 5/21/2007

**Notes:**
Our preliminary calculations suggest that medium modifications help to saturate nuclear matter at close to the correct saturation density and binding energy.  The ring diagram calculation using Vlowk saturates at too high a density and at too large a binding energy.  However, the dropping masses help with both of these problems.  We also calculated the compression modulus of nuclear matter and find the result of 220 MeV to be in good agreement with the empirical value of 200 to 300 MeV.

## Hartree-Fock

_**Hartree-Fock**_

- J.W. Holt

- LBL 5/21/2007

**Notes:**
Now that we are confident that our interactions lead to nuclear saturation, we would like to study the bulk equilibrium properties of nuclear matter.  The formalism we use is due originally to Landau, but which was brought to nuclear physics by Migdal.  Fermi liquid theory is typically used to describe strongly interacting normal Fermi systems at zero temperature in terms of weakly interacting quasiparticles.  It is important to keep in mind that the theory is valid only for small excitations of the system because the weakly interacting quasiparticles are defined only near the fermi surface, where they have very long lifetimes.
Expanding the energy to second order in the quasiparticle distribution function, we obtain this expression.  Here epsilon-k-zero is the quasiparticle energy in the absence of other quasiparticles, and f describes the interaction between quasiparticles, which plays a key role in Landau’s theory, as we will see.  The simplest example is Hartree-Fock theory in which the ground state energy is given by this expression.  Then the quasiparticle interaction is just the antisymmetrized interaction matrix element.

## Slide 14

- J.W. Holt

- LBL 5/21/2007

**Notes:**
The quasiparticle interaction is typically parameterized as a pure exchange interaction.  For two momentum states on the fermi surface, the quasiparticle interaction can depend only on the relative angle between the momenta, and so we can perform a Legendre polynomial expansion for f, g, f’, and g’.  It is convenient to factor out the density of states at the fermi surface, which yields dimensionless Fermi liquid parameters.  One of the advantages to using Fermi liquid theory is that there is a direct correspondence between these dimensionless Fermi liquid parameters and observables.  For instance, the effective mass, compression modulus, symmetry energy, and anomalous orbital gyromagnetic ratio are each related to one of the FLP’s.  So in the end, we want to compare the known experimental values to those obtained from our microscopic calculation.

## Slide 15

- J.W. Holt

- LBL 5/21/2007

**Notes:**
It can be shown that the scattering amplitude is related to the quasiparticle interactionto, and by enforcing the scattering amplitude to satisfy the Pauli principle, one can show that the Fermi liquid parameters must satisfy two sum rules, one of them being shown here.
In practice, we cannot calculate all diagrams contributing to the f-function.  But this presents a problem.  If we include only a finite set of diagrams for the f-function, we will not satisfy the Pauli principle sum rules, because many exchange terms will be missing.  For example, if we include only the two lowest order diagrams for f, we will generate this diagram in the scattering amplitude but the corresponding exchange term will be missing, which should be included in f.

## Largest effect is to cut down the strong attraction in F channel

|  | Full |  |  |  | Driving |  |  |  | Induced |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| l | F | G | F’ | G’ | F | G | F’ | G’ | F | G | F’ | G’ |
| 0 | -0.48 | 0.03 | 0.22 | 0.78 | -1.28 | 0.14 | 0.37 | 0.64 | 0.80 | -0.12 | -0.15 | 0.14 |
| 1 | -0.34 | 0.26 | 0.27 | 0.17 | -0.53 | 0.26 | 0.28 | 0.13 | 0.20 | 0.01 | 0.00 | 0.05 |

- Largest effect is to cut down the strong attraction in *F* channel

- J.W. Holt

- LBL 5/21/2007

**Notes:**
F0 cut down by the rescattering in the denominator

## How to improve?

|  | Nijmegen I | Nijmegen II | CD-Bonn | Expt. |
| --- | --- | --- | --- | --- |
| m*/m | 0.887 | 0.930 | 0.888 |  |
| K [MeV] | 136 | 102 | 136 | 200-300 |
|  [MeV] | 18.1 | 20.5 | 17.6 | 25-35 |
| gl[N] | 0.682 | 0.452 | 0.685 | 0.20-0.26 |
| Sl | 0.20 | 0.16 | 0.27 |  |
| S2 | -0.04 | -0.02 | -0.04 |  |

- 1.  Explicit three-body forces
- 2.  In-medium modifications to NN interaction

- Medium modifications equivalent to a type of 3N force

- Extend Walecka mean field theory to constituent quarks

- J.W. Holt

- LBL 5/21/2007

**Notes:**
Nuclear matter is too compressible.

## Decreasing g2NN by 20% cuts K by 50% but changes other observables by ~ 5%

|  | VNI | VNII | VN93 | VCDB | Exp. |
| --- | --- | --- | --- | --- | --- |
| m*/m | 0.721 | 0.763 | 0.696 | 0.682 |  |
| K [MeV] | 218 | 142 | 190 | 495 | 200-300 |
|  [MeV] | 20.4 | 25.5 | 23.7 | 19.2 | 25-35 |
| gl[N] | 0.246 | 0.181 | 0.283 | 0.267 | 0.20-0.26 |

- quark model

- Decreasing *g*2***NN* by 20% cuts K by 50% but changes other observables by ~ 5%

- J.W. Holt

- LBL 5/21/2007

## Finite nuclei

- Decreasing tensor force necessary!

- Decay of 14C
- 0– T=1 and 0– T=0 splitting in 16O
- E2 and M1 moments of 6Li

**Traditional many-body effects or novel scaling mechanism?**

- J.W. Holt

- LBL 5/21/2007

**Notes:**
In general finite nuclei are more difficult.  The density is not constant, and 

## Slide 20

- J.W. Holt

- LBL 5/21/2007

## 14C

- 14C

- 14N

- 0+

- 1+

- 2 holes in 0*p*-shell

- –

- Nijmegen I

- Preliminary

- J.W. Holt

- LBL 5/21/2007

## Summary

-  Saturation of nuclear matter and Fermi liquid parameters improved with dropping masses

- _Summary_

-  Full analysis of ring diagrams
-  Look for nuclear observables where tensor force plays dominant role
-  Understand the connection between medium modifications and three-body forces

- _Outlook_

- J.W. Holt

- LBL 5/21/2007