Dense Matter for ``Dummies'' (Non-Experts) - Physicsntg/talks/2006/furnstahl... · Dense Matter for “Dummies” (Non-Experts) ... Chiral EFT offers systematic low-energy alternative

Dense Matter for “Dummies” (Non-Experts)

Dick Furnstahl

Department of PhysicsOhio State University

July, 2006

Pictures have been freely borrowed from online sources;I apologize in advance for any omitted citations. Also, inclusion ofparticular calculations does not imply that they are the “best”.No animals were harmed in creating this talk.

Overview Internucleon Many-Body EOS Summary

Aspects of Dense Matter in Subsequent Talks

Internucleon interactions =⇒ inputhadronic vs. quark/gluon (QCD) DOF’s

boson-exchange (OBE) vs. chiral effective field theory (EFT)

relativistic field theory vs. nonrelativistic potentials

low-momentum potentials: renormalization group −→ “Vlow k ”

three-body forces

Many-body methods for dense mattervirial expansion

microscopic “ab initio” (e.g., variational)

lattice EFT calculations

density functional theory (DFT) [cf. mean-field models]

Equation of state (EOS) constraints and featuresdensity dependence of symmetry energy

pairing (of nucleons and quarks)

Dick Furnstahl Dense Matter


Neutron StarAnatomy (from D. Page)

Dilute neutrons at surface=⇒ neutron superfluid(cf. cold atoms?)

Outer core: bulk EOScalibrated by nuclei?

High density inner core:color superconductor fromquark Cooper pairs?



Schematic of possibilities (from Fridolin/Weber)



Mass-Radius Diagram (from Lattimer/Prakash)

Where do all these EOS’s come from? Theoretical error bars?Dick Furnstahl Dense Matter


EOS Scorecard [EOS is P(n) vs. E(n)]Integrate TOV equations from r = 0 to r = R

dPdr

= −GE(r)M(r)

r2

[1 +

P(r)E(r)

] [1 +

4πr3P(r)M(r)

] [1− 2GM(r)

r

]−1

dMdr

= 4πr2E(r) with P(0) = P0 → P(R) = 0, M(0) = 0 → M(R) = Mns

EOS dof’s approach commentAPn n,p variational AV18 NN + NNN + rel.PALn n,p phenomenological nonrel. potentialMSn n,p,σ,ω,ρ rel. MFT (DFT) isovector parametersGMn n,p,Λ rel. MFT (DFT) hyperonsGSn n,p,K rel. MFT (DFT) kaon condensation

SQMn Q(u,d,s) pQCD + bag strange quark matter

TOV equations don’t know EOS origin

Different physics (e.g., phases) vs. approximations



The QCD Phase Diagram [from Mark Alford]



Nuclear and Neutron Matter Energy vs. Density

0 0.05 0.1 0.15 0.2 0.25-20

0

20

40

APRNRAPRRAPR

)-3n (fm

E/A

(MeV

)

Nuclear matter

Neutron matter

E(n, x) = E(n, x = 1/2) + S2(n)[1− 2x ]2

Nuclear matter =⇒ x = 1/2; Neutron matter =⇒ x = 0Symmetry energy S2(n) not pinned down by past fits to nuclei



Landscape of Finite Nuclei: Earth to Heaven?

Large extrapolations to neutron stars in density and proton fraction!Dick Furnstahl Dense Matter


Nuclei by the Numbers (adapted from Richter by Schwenk)


Overview Internucleon Many-Body EOS Summary Basics EFT 3-Body

Principle of Any Effective Low-Energy Description

If system is probed at low energies, fine details not resolved

use low-energy variables for low-energy processesshort-distance structure can be replaced by something simplerwithout distorting low-energy observables

Could be a model or systematic (e.g., effective field theory)physics can change with resolution!

Low density ⇔ low interaction energy ⇔ low resolution




If system is probed at low energies, fine details not resolved

use low-energy variables for low-energy processesshort-distance structure can be replaced by something simplerwithout distorting low-energy observables






If system is probed at low energies, fine details not resolveduse low-energy variables for low-energy processesshort-distance structure can be replaced by something simplerwithout distorting low-energy observables






If system is probed at low energies, fine details not resolveduse low-energy variables for low-energy processesshort-distance structure can be replaced by something simplerwithout distorting low-energy observables


Low density ⇔ low interaction energy ⇔ low resolutionDick Furnstahl Dense Matter


Plan: Use DOF’s to Simplify and Make EfficientWeinberg’s Third Law of Progress in Theoretical Physics:“You may use any degrees of freedom you like to describe aphysical system, but if you use the wrong ones, you’ll besorry!”

There’s an old vaudeville joke about a doctor and patient . . .

Patient: Doctor, doctor, it hurts when I do this!Doctor: Then don’t do that.



Plan: Use DOF’s to Simplify and Make EfficientWeinberg’s Third Law of Progress in Theoretical Physics:“You may use any degrees of freedom you like to describe aphysical system, but if you use the wrong ones, you’ll besorry!”There’s an old vaudeville joke about a doctor and patient . . .





Patient: Doctor, doctor, it hurts when I do this!

Doctor: Then don’t do that.







Quark ( QCD) vs. Hadronic NN Interaction

Old goal: replace hadronic descriptions at ordinary nucleardensities with quark description (since QCD is the theory)New goal: use effective hadronic dof’s systematically

Seek model independence and theory error estimatesFuture: Use lattice QCD to match via “low-energy constants”

Need quark dof’s at higher densities (resolutions) wherephase transitions happen



Quark ( QCD) vs. Hadronic NN Interaction

Old goal: replace hadronic descriptions at ordinary nucleardensities with quark description (since QCD is the theory)New goal: use effective hadronic dof’s systematically

Seek model independence and theory error estimatesFuture: Use lattice QCD to match via “low-energy constants”

Need quark dof’s at higher densities (resolutions) wherephase transitions happen



Brief Aside: Can We Use Scattering Data Directly?

If we want model independence, what could be better thanusing only data?

EOS for a dilute gas based on virial expansion to 2nd orderdoes this!

controlled expansion in small parameter (fugacity eµ/T ) withwell-defined range of validity

for neutron matter, applies for n ≤ 4 · 1011(T/MeV)3/2 g/cm3

virial EOS provides benchmark for all nuclear EOS’s at lowdensity and temperature

See A. Schwenk’s talk for details (and assumptions)



Nuclear and Cold Atom Many-Body ProblemsInter-atom and nucleon-nucleon potentials [Dobaczewski]

-10

0

10

20

30

40

50

-100

0

100

200

300

400

500

0 1 2 3

0 0.4 0.8 1.2

n-n distance (fm)

n-n

pote

ntia

l (M

eV)

O -O

pot

entia

l (m

eV)

22

O -O distance (nm)22

O -O system22

n-n system

Are there universal features of such many-body systems?How can we deal with “hard cores” in many-body systems?

How does it change if we lower the resolution? (“Vlow k ”)



Nuclear and Cold Atom Many-Body ProblemsInter-atom and nucleon-nucleon potentials [Schwenk]





Nucleon-Nucleon Interaction

Potential for nonrelativistic many-body Schrodinger equation

Depends on spins and isospins of nucleons; also non-centrallongest-range part is one-pion-exchange potential

Vπ(r) ∝ (τ1·τ2)

[(3σ1 · r σ2 · r − σ1 · σ2)(1 +

3mπr

+3

(mπr)2 ) + σ1 · σ2

]e−mπr

r

Characterize operator structure of shorter-range potentialcentral, spin-spin, non-central tensor and spin-orbit

1,σ1 · σ2,S12,L · S,L2,L2σ1 · σ2, (L · S)2 ⊗ 1, τ1 · τ2

Argonne v18 is VEM + Vπ + Vshort range(all cut off at small r )

Fit to NN scattering data up to 350 MeV (or krel ≤ 2.05 fm−1)

Alternative characterization is one-boson-exchange



One-Boson-Exchange Model Scorecard (Machleidt)T = 0 T = 1 Central Spin-Spin Tensor Spin-Orbit

coupling [1] [τ1 · τ2] [1] [σ1 · σ2] [S12] [L · S]pseudoscalar η π — weak strong —

scalar σ δ strong — — adds toattractive vector

vector ω ρ strong weak opposes ps strongrepulsive adds to s

tensor ω ρ — weak opposes ps —



Boson Exchange vs. Chiral Effective Field Theory

Cut-off OBE potentials =⇒ model of short-distance physicsused as nonrelativistic potential or in DBHF (or RBHF)

model dependent =⇒ not a complete description

Chiral EFT offers systematic low-energy alternative

meson exchanges unresolved in chiral EFT (except for pion)

instead encoded in coefficients of contact termsρ, ω, σ, . . . −→ + ∇2 + · · ·

g2

q2 + m2

g2

m2− g2

m2

q2

m2

treat multiple pion exchanges systematically

general and complete characterization (infinite # of terms!)

breakdown when q ≈ m (how high density?)



Effective Field Theory Ingredients

From “Crossing the Border” [nucl-th/0008064]

1 Use the most general L with low-energy dof’s consistent withthe global and local symmetries of the underlying theory

Left = Lππ + LπN + LNN

chiral symmetry =⇒ systematic long-distance pion physics

2 Declaration of regularization and renormalization scheme

momentum cutoff and “Weinberg counting”

use cutoff sensitivity as measure of uncertainties!

3 Well-defined power counting =⇒ small expansion parameters

use the separation of scales =⇒ p,mπΛχ

, pMN

with Λχ ∼ mρ

Weinberg: power-count VNN and solve S-equation

chiral symmetry =⇒ VNN =∑∞

ν=νmincνQν with ν ≥ 0



Effective Field Theory Ingredients: Chiral NN


1 Use the most general L with low-energy dof’s consistent withthe global and local symmetries of the underlying theoryLeft = Lππ + LπN + LNN


2 Declaration of regularization and renormalization scheme

momentum cutoff and “Weinberg counting”




, pMN

with Λχ ∼ mρ










2 Declaration of regularization and renormalization schememomentum cutoff and “Weinberg counting”




, pMN

with Λχ ∼ mρ










2 Declaration of regularization and renormalization schememomentum cutoff and “Weinberg counting”




, pMN

with Λχ ∼ mρ






Chiral EFT for Two Nucleons as Prototype

Epelbaum, Meißner, et al.

Also Entem, Machleidt

LπN + match at low energy

1π 2π contact

LO —

NLO

N2LO

N3LO −→∇

4 (15)

-40

0

40

80

0 0.1 0.2 0.3

1S0

0

50

100

150

0 0.1 0.2 0.3

3S1

0

40

80

0 0.1 0.2 0.3

3P0

0

4

8

12

0 0.1 0.2 0.3

1D2

-12

-6

0

6

0 0.1 0.2 0.3

3D3

-1.2

-0.8

-0.4

0

0 0.1 0.2 0.3

3G5







1π 2π contact

LO —

NLO

N2LO

N3LO −→∇

4 (15)

-40

0

40

80

0 0.1 0.2 0.3

1S0

0

50

100

150

0 0.1 0.2 0.3

3S1

0

40

80

0 0.1 0.2 0.3

3P0

0

4

8

12

0 0.1 0.2 0.3

1D2

-12

-6

0

6

0 0.1 0.2 0.3

3D3

-1.2

-0.8

-0.4

0

0 0.1 0.2 0.3

3G5







1π 2π contact

LO —

NLO

N2LO

N3LO −→∇

4 (15)

-40

0

40

80

0 0.1 0.2 0.3

1S0

0

50

100

150

0 0.1 0.2 0.3

3S1

0

40

80

0 0.1 0.2 0.3

3P0

0

4

8

12

0 0.1 0.2 0.3

1D2

-12

-6

0

6

0 0.1 0.2 0.3

3D3

-1.2

-0.8

-0.4

0

0 0.1 0.2 0.3

3G5







1π 2π contact

LO —

NLO

N2LO

N3LO −→∇

4 (15)

-40

0

40

80

0 0.1 0.2 0.3

1S0

0

50

100

150

0 0.1 0.2 0.3

3S1

0

40

80

0 0.1 0.2 0.3

3P0

0

4

8

12

0 0.1 0.2 0.3

1D2

-12

-6

0

6

0 0.1 0.2 0.3

3D3

-1.2

-0.8

-0.4

0

0 0.1 0.2 0.3

3G5



State of the Art: N 3LO (Epelbaum, nucl-th/0509032)

-20

0

20

40

60

Phas

e Sh

ift [

deg] 1S0

0

50

100

1503S1

-30

-20

-10

01P1

-20

-10

0

10

Phas

e Sh

ift [

deg] 3P0

-30

-20

-10

03P1

0

20

403P2

-2

0

2

4

Phas

e Sh

ift [

deg] ε1

0

5

10

151D2

-30

-20

-10

03D1

0 50 100 150 200 250Lab. Energy [MeV]

0

10

20

30

40

Phas

e Sh

ift [

deg] 3D2

0 50 100 150 200 250Lab. Energy [MeV]

-5

0

53D3

0 50 100 150 200 250Lab. Energy [MeV]

-4

-3

-2

-1

0

ε2



State of the Art: N 3LO (Epelbaum, nucl-th/0509032)

100

200

300400500

dσ/dΩ0

0.02

0.04

0.06Ay

10

100

1000

dσ/dΩ0

0.05

0.1

0.15

0.2 Ay

0 60 120 180θ [deg]

1

10

100

dσ/dΩ

0 60 120 180θ [deg]

-0.8

-0.4

0

0.4

Ay

100 110

90

100

110

115 120 12516

18

20



Many-Body Forces are Inevitable in EFT!

What if we have threenucleons interacting?

Successive two-bodyscatterings with short-livedhigh-energy intermediatestates unresolved =⇒ mustbe absorbed intothree-body force

A feature, not a bug!

How do we organize(3,4, · · · )–body forces?EFT! [nucl-th/0312063]

2N forces 3N forces 4N forces



Observations on Three-Body Forces

Three-body forces arise fromeliminating dof’s

excited states of nucleon

relativistic effects

high-momentumintermediate states

Omitting 3-body forces leadsto model dependence

but different for eachHamiltonian

3-body contributionsincrease with density

uncertain extrapolation if notconstrained



Observations on Three-Body Forces

Three-body forces arise fromeliminating dof’s

excited states of nucleon

relativistic effects

high-momentumintermediate states

Omitting 3-body forces leadsto model dependence

but different for eachHamiltonian

3-body contributionsincrease with density

uncertain extrapolation if notconstrained


Overview Internucleon Many-Body EOS Summary DFT Relativistic

Given an Interaction, Why Not Just Solve?



The Many-Body Schr odinger Wave Function

[adapted from Joe Carlson]

How to represent the wave function for an A-body nucleus?

Consider 8Be (Z = 4 protons, N = 4 neutrons)

|Ψ〉 =∑σ,τ

χσχτφ(R) where R are the 3A spatial coordinates

χσ =↓1↑2 · · · ↓A (2A terms) = 256 for A = 8

χτ = n1n2 · · ·pA ( A!N!Z ! terms) = 70 for 8Be

So for 8Be there are 17,920 complex functionsin 3A− 3 = 21 dimensions!

Suppose you represent this for a nucleus of size 10 fm witha mesh spacing of 0.5 fm. You would need 1027 grid points!

Obviously this is not the way it is done!







|Ψ〉 =∑σ,τ


χσ =↓1↑2 · · · ↓A (2A terms) = 256 for A = 8











|Ψ〉 =∑σ,τ


χσ =↓1↑2 · · · ↓A (2A terms) = 256 for A = 8







Argonne v18 NN plus Illinois-2 NNN

Green’s Function Monte Carlo for nuclei to 12C (note 3-body!)Variational for neutron/nuclear matter (see later talks)



Density Functional Theory (DFT)

Hohenberg-Kohn: There existsan energy functional Evext[ρ] . . .

Evext[ρ] = FHK [ρ] +

∫d3x vext(x)ρ(x)

FHK is universal (same for anyexternal vext) =⇒ H2 to DNA!

Introduce orbitals and minimizeenergy functional =⇒ Egs, ρgs

Useful if you can approximatethe energy functional

Construct microscopicallyor fit a “general” form =⇒ EOS



Skyrme Hartree-Fock Energy Functionals

Skyrme energy density functional (for N = Z ):

E [ρ, τ , J] =1

2Mτ +

38

t0ρ2 +1

16t3ρ2+α +

116

(3t1 + 5t2)ρτ

+1

64(9t1 − 5t2)(∇ρ)2 − 3

4W0ρ∇ · J +

132

(t1 − t2)J2

where ρ(x) =∑

i |φi(x)|2 and τ(x) =∑

i |∇φi(x)|2 (and J)

Minimize E =∫

dx E [ρ, τ, J] by varying the (normalized) φi ’s

Fix t0–t3, W0, and isovector parameters by fits to nucleimany parameter sets in literature from past fits



Table of the Nuclides



Problems with ExtrapolationsMass formulas and energy functionals do well where there

is data, but elsewhere . . .

16

25

20

15

10

10

5

0

50 60 70 80 80 90 100 110 120 130

Neutron Number

S2n(M

eV)

Experiment

HFB-SLy4

HFB-SkP

HFB-D1S

SkX

RHB-NL3

LEDF

Sn

two-neutron separation energies

data exist data do not exist

0

10

20

60 80 100 120

Neutron Number

exp

FRDMCKZ

T+MJJM

Mass Formulae

Neutron Number

Figure 6: Predicted two-neutron separation energies for the even-even Sn isotopes using several microscopic models based on effective nucleon-nucleon interactions and obtained with phenomenological mass formulas (shown in the inset). While calculations agree well in the region where experimental data are available, they diverge for neutron-rich isotopes with N>82. It is seen that the position of the neutron drip line is uncertain. Unknown nuclear deformations or as yet uncharacterized phenomena, such as the presence of neutron halos or neutron skins, make theoretical predictions highly uncertain. Experiments for the Sn isotopes with N=80–100 will greatly narrow the choice of viable models.

In progress: “ab initio” DFT from microscopic interactions



“Old” View of Relativistic Mean-Field ModelsQHD Lagrangian with one-boson-exchange meson fields

Covariant Walecka model + φ3 and φ4 terms to get K0

+ gs + gv + gρ + gsκ

+ gsλ

Mean meson fields 〈φ〉 = φ0, 〈Vµ〉 = δµ0V0

Parameters: gs,gv ≈ 10, κ ≈ 5000 MeV, λ ≈ −200

Unexplained features:

justification of mean-field, “no-sea” approximation

how to deal with large couplings and loop corrections

truncation at φ4; why not V 40 ?

minimal isovector physics (chiral symmetry?)

Reinterpret as covariant density functional



Power Counting: Naive Dimensional AnalysisMass scales in low-energy QCD: [Georgi & Manohar, 1984]

fπ ≈ 93 MeV , Λ ≈ 500 to 800 MeV

NDA rules for a generic term in energy functional:

c [f 2πΛ2]

[(NNf 2πΛ

)` 1m!

(gφΛ

)m 1n!

(gV0

Λ

)n(∇Λ

)p ]

“Naturalness” =⇒ dimensionless c is of order unity

ratio Λ/fπ → g ≈ 5–10 is origin of strong couplings=⇒ gs,gv ∼ g κ ∼ gΛ λ ∼ g2

Provides expansion parameters at finite density:

gsφ

Λ≈ gv V0

Λ≈ 1/2 ,

ρs

f 2πΛ

≈ ρB

f 2πΛ

≈ 1/5 at ρ0B

Allows truncation and calibration with quantitativelyaccurate fits to bulk nuclear observables



Estimates in Finite Nuclei

ρs, ρB →〈ρB〉f 2π Λ

∇ρB →〈∇ρB〉f 2π Λ2 s → 〈s〉

f 2π Λ

ρ3 → Z − N2A

〈ρB〉f 2π Λ

≈ 1 isovector parameter constrained by energy fit


Overview Internucleon Many-Body EOS Summary Symmetry Pairing Vlowk

Adding New Terms to the Energy Functional

As density increases, more terms become significant

Mueller/Serot =⇒ isoscalar (V0) and isovector (b0) vector

add ζ4!

g4v V 4

0 +ξ4!

g4ρb4

0

explore natural size ζ and ξ impact on EOS =⇒ MSn EOS’s

Horowitz/Piekarewicz

add ζ4!

gv V 40 + g2

ρb20[Λ4g2

sφ2 + Λv g2

v V 20 ]

knobs for symmetry energy and high-density EOS

QuestionsIs the energy functional general enough? (E.g., are nonanalyticor nonlocal terms needed?)

Can we derive/constrain the functional more microscopically?

How can we better constrain parameters?



Neutron Skin in 208Pb vs. Symmetry EnergyE(n, α) = E(n,0) + S2(n)α2 + S4(n)α4 + · · · α ≡ N − Z

A= 2x − 1

E(n,0) = −av +K0

18n20

(n − n0)2 + · · ·

S2(n) = a4 +p0

n20

(n − n0) +∆K0

18n20

(n − n0)2 + · · ·

0 0.05 0.1 0.15 0.2 0.25-20

0

20

40

APRNRAPRRAPR

)-3n (fm

E/A

(MeV

)

Nuclear matter

Neutron matter




A= 2x − 1

E(n,0) = −av +K0

18n20

(n − n0)2 + · · ·

S2(n) = a4 +p0

n20

(n − n0) +∆K0

18n20

(n − n0)2 + · · ·

24 26 28 30 32 34 36 38 40 42 44symmetry energy a4 (MeV)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

r n−

r p (fm

)

vmd eftpc eftmeson eft


0.05

0.1

0.15

0.2

0.25

0.3

0.35

r n−

r p (fm

)

skyrme standardrmf standardpc standard


0.05

0.1

0.15

0.2

0.25

0.3

0.35

r n−

r p (fm

) skyrme 7 fitsskyrme 9 fitsvmd eft fitsddrh

0.08 0.1 0.12 0.14 0.16

n (fm−3)

1

2

3

4

5

6

∆S2 (n

) (M

eV)

rjf [NPA 707, 85 (2002)]

Smallest spreadin symmetry energy




A= 2x − 1

E(n,0) = −av +K0

18n20

(n − n0)2 + · · ·

S2(n) = a4 +p0

n20

(n − n0) +∆K0

18n20

(n − n0)2 + · · ·

0 2 4 6 8 10

p0 (MeV/fm3)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

r n−

r p (fm

)

rmf standardvmd eftpc eftmeson eftskyrme standardskyrme 7 fitsskyrme 9 fitsddrh

0.08 0.1 0.12 0.14 0.16

n (fm−3)

1

2

3

4

5

6

∆S2 (n

) (M

eV)

rjf [NPA 707, 85 (2002)]

Smallest spreadin symmetry energy



Symmetry Energy and EOS (from Steiner et al.)

Isospin Dependence of Strong Interactions

Nuclear MassesNeutron Skin Thickness

Isovector Giant Dipole ResonancesFission

Heavy Ion FlowsMulti-Fragmentation

Nuclei Far from StabilityRare Isotope Beams

Many-Body TheorySymmetry Energy

(Magnitude and Density Dependence)

SupernovaeWeak Interactions

eνEarly Rise of LBounce Dynamics

Binding Energy

Proto-Neutron Stars Opacitiesν

EmissivitiesνSN r-ProcessMetastability

Neutron StarsObservational

Properties

Binary MergersDecompression/Ejectionof Neutron-Star Matter

r-Process

QPO’sMass

Radius

NS CoolingTemperature

, z∞RDirect Urca

Superfluid Gaps

X-ray Bursters, z∞R

Gravity WavesMass/Radius

dR/dM

PulsarsMasses

Spin RatesMoments of Inertia

Magnetic FieldsGlitches - CrustMaximum Mass, Radius

Composition:Hyperons, Deconfined Quarks

Kaon/Pion Condensates



Nuclear and Cold Atom Many-Body ProblemsInter-atom and nucleon-nucleon potentials [Dobaczewski]

-10

0

10

20

30

40

50

-100

0

100

200

300

400

500

0 1 2 3

0 0.4 0.8 1.2

n-n distance (fm)

n-n

pote

ntia

l (M

eV)

O -O

pot

entia

l (m

eV)

22

O -O distance (nm)22

O -O system22

n-n system





Quick Review of Scattering

P/2− k

P/2 + k

P/2− k′

P/2 + k′

0 R

sin(kr+δ)

r

Relative motion with total P = 0: ψ(r) r→∞−→ eik·r + f (k , θ)eikr

r

where k2 = k ′2 = MEk and cos θ = k · k ′

Differential cross section is dσ/dΩ = |f (k , θ)|2

Central V =⇒ partial waves:f (k , θ) =

∑l(2l + 1)fl(k)Pl(cos θ)

where fl(k) =eiδl (k) sin δl(k)

k=

1k cot δl(k)− ik

and the S-wave phase shift is defined by

u0(r)r→∞−→ sin[kr + δ0(k)] =⇒ δ0(k) = −kR for hard sphere



Quick Review of Scattering

P/2− k

P/2 + k

P/2− k′

P/2 + k′

0 R

sin(kr+δ)

r

Relative motion with total P = 0: ψ(r) r→∞−→ eik·r + f (k , θ)eikr

r

where k2 = k ′2 = MEk and cos θ = k · k ′

Differential cross section is dσ/dΩ = |f (k , θ)|2Central V =⇒ partial waves:f (k , θ) =

∑l(2l + 1)fl(k)Pl(cos θ)

where fl(k) =eiδl (k) sin δl(k)

k=

1k cot δl(k)− ik

and the S-wave phase shift is defined by

u0(r)r→∞−→ sin[kr + δ0(k)] =⇒ δ0(k) = −kR for hard sphere



Effective Range Expansion

Total cross section: σtotal =4πk2

∞∑l=0

(2l + 1) sin2 δl(k)

What happens at low energy (λ = 2π/k 1/R)?

k cot δ0(k)k→0−→ − 1

a0+

12

r0k2 + . . .

a0 is the “scattering length” and r0 is the “effective range”



Near-Zero-Energy Bound States

Bound-state or near-bound state at zero energy=⇒ large scattering lengths (a0 → ±∞)

For kR → 0, the total cross section is

σtotal = σl=0 =4πa2

0

1 + (ka0)2 =

4πa2

0 for ka0 14πk2 for ka0 1 (unitarity limit)



Near-Zero-Energy Bound States

Bound-state or near-bound state at zero energy=⇒ large scattering lengths (a0 → ±∞)

For kR → 0, the total cross section is

σtotal = σl=0 =4πa2

0

1 + (ka0)2 =

4πa2

0 for ka0 14πk2 for ka0 1 (unitarity limit)



Trapped Fermion Atoms vs. QCD

Atoms: Change magnetic field =⇒ resonant scattering

QCD: Adjust quark mass theoretically =⇒ mπ changes

Universal properties! Scale invariant!



Trapped Fermion Atoms vs. QCD

Atoms: Change magnetic field =⇒ resonant scattering

QCD: Adjust quark mass theoretically =⇒ mπ changes

Universal properties! Scale invariant!



Large Scattering Length Problem

Attractive with a0 →∞If R 1/kF a0, then expectscale invariance

Energy and gap are pure

numbers times EFG = 35

k2F

2M

Non-perturbative many-body

Discussion here:virial expansion [A. Schwenk]

lattice [D. Lee]

k F

a

1/~

R

0



GFMC Results [J. Carlson et al.]Extrapolate to large numbers of fermions

10 20 30 40N

0

10

20

E/E

FG

E = 0.44 N EFG

Pairing gap (∆) = 0.9 EFG

odd N

even N

Energy per particle: E/N = 0.44(1)EFG



Observation of Vortices by W. Ketterle Group (MIT)



When Does Cooper Pairing Occur?If there is an attractive interaction at the Fermi surface,back-to-back fermions condense as Cooper pairs

The excitation spectrum (energy vs. momentum relation)develops a gap ∆, which affects transport, cooling

For dilute fermions, ∆ ∝ e−π/2kF|a0|



Conjectured QCD Phase Diagrams (from M. Alford)

CFLliq

QGP

T

µnuclear

gas

compact star

RHIC

CFLliq

QGP

T

µnuclear

gas

superconducting= color

RHIC

non−CFL

compact star

QGP =⇒ “Quark-Gluon Plasma”

CFL =⇒ “Color-Flavor-Locked” phase(see S. Reddy)



Are Quark Stars Ruled Out? [See F. Ozel]

From Alford, Blaschke, Drago, Klhn, Pagliara, Schaffner-Bielich.

7 8 9 10 11 12 13 14 15Radius R [km]

0.4

0.8

1.2

1.6

2.0

2.4M

ass

M [M

]Neutron star(DBHF)

Quark star(QCD O(αs

2)) Hybrid(2SC+HHJ)

Hybrid(2SC+DBHF)

EXO 0748-676

1σ

2σ

Hybrid, mixed(CFL+APR)

Where are the theoretical error bars?Dick Furnstahl Dense Matter


Sources of Nonperturbative Physics for NN

1 Strong short-range repulsion(“hard core” or singular V2π)

2 Iterated tensor (S12) interaction

3 Near zero-energy bound states

Consequences:In Coulomb DFT, Hartree-Fock gives dominate contribution

=⇒ correlations are small corrections =⇒ DFT works!cf. NN interactions =⇒ correlations HF =⇒ DFT fails??

However . . .the first two depend on the resolution =⇒ changed by RGall three are affected by Pauli blocking



Use Variable Cutoff Potential from RG

Bogner, Kuo, Schwenk

T

−k +k

−k′ +k′

= VΛ

−k +k

−k′ +k′

+ q ≤ Λ

VΛ

T

Require dTdΛ

= 0=⇒ RG equation for VΛ

Integrated =⇒ Lee-Suzuki

Run cutoff to Λ ≈ 2 fm−1

Same long distance physics=⇒ coalesce to “Vlow k ”

New: smooth cutoffs!

0 1 2 3 4

k [fm-1

]

-2

-1

0

1

2

3

VN

N (

k,k)

[fm

]

ParisBonn AArgonne v18Idaho ACD BonnNijmegen94 IINijmegen94 I



The Deuteron at Different Resolutions

0 2 4 6 8 10r [fm]

0

0.1

0.2

0.3

0.4

0.5

ψ(r)

[fm

-3/2

] Argonne v18

3S1 deuteron w.f.

0 1 2 3 4 5k [fm-1]

0.01

0.1

1

10

|ψ0(k

)| [fm

3/2 ] Argonne v18

3S1 deuteron w.f.

Repulsive core =⇒ short-distance suppression=⇒ high-momentum components

Low-momentum potential =⇒ much simpler wave function!



The Deuteron at Different Resolutions

0 2 4 6 8 10r [fm]

0

0.1

0.2

0.3

0.4

0.5

ψ(r)

[fm

-3/2

]

Λ = 2.0 fm-1

Λ = 3.0 fm-1

Λ = 4.0 fm-1

Argonne v18

3S1 deuteron w.f.

0 1 2 3 4 5k [fm-1]

0.01

0.1

1

10

|ψ0(k

)| [fm

3/2 ] Λ = 2 fm-1

Λ = 3 fm-1

Λ = 4 fm-1

Argonne v18

3S1 deuteron w.f.

Repulsive core =⇒ short-distance suppression=⇒ high-momentum components

Low-momentum potential =⇒ much simpler wave function!



Quantities that vary with convention=⇒ not observables

deuteron D-state probability[Friar, PRC 20 (1979)]

off-shell effects[Fearing/Scherer]

occupation numbers[Hammer/Furnstahl]

wound integrals

short-range part of wavefunctions

short-range potentials; e.g.,contribution of short-range3-body forces

0.5 1 2 3 4 5 10Λ (fm-1)

0

0.01

0.02

0.03

0.04

0.05

0.06

PD

D-state probability −2.23

−2.225

−2.22

EDBinding energy (MeV)

0.02

0.025

0.03

ηsd

Asymptotic D-S ratio

Vlow k from av18 using e−(p2/Λ2)8



(Approximate) Nuclear Matter with NN and NNNHartree-Fock

0.8 1 1.2 1.4 1.6kF [fm-1]

-15

-10

-5

0

5

E/A

[MeV

]

Λ = 1.6 fm-1

Λ = 1.9 fm-1

Λ = 2.1 fm-1

Λ = 2.3 fm-1

Λ = 2.1 fm-1 [no V3N]

“≈ 2nd Order”

0.8 1 1.2 1.4 1.6kF [fm-1]

-15

-10

-5

0

5

E/A

[MeV

]

Λ = 1.6 fm-1

Λ = 1.9 fm-1

Λ = 2.1 fm-1

Λ = 2.3 fm-1

Λ = 2.1 fm-1 [no V3N]

Looks perturbative! =⇒ note decrease in cutoff dependence3-body forces drive saturation (!) but not unnatural



Some Final Thoughts

Dense matter in astophysical systems spans many poorlyknown regions of the QCD phase diagram

Needed for robust EOS’sreliable calibration =⇒ experimental input on earth (e.g.,symmetry energy) and in heavens (neutron star observables)

reliable extrapolations =⇒ reduced model dependence plustheoretical error estimates

better constraints on many-body forces

Some ways to get thereexploit model independent results (e.g., virial expansion)

use full arsenal of many-body methods to crosscheckpredictions of observables

systematic effective field theory (EFT)

exploit cut-off dependence



Some Final Thoughts (cont.)

Some new and upcoming developments that could impactextensions of EFT methods

from low (pionless neutron matter, e.g., on lattice [D. Lee]), tointermediate densities (chiral EFT), to very high density(EFT’s of quark systems)

application of renormalization group (RG) to soften hard coressee L. Tolos’ talk

microscopic (“ab initio”) energy functionals

On to Dense Matter!



(Nuclear) Many-Body Physics: “Old” Approach

One Hamiltonian for allproblems and energy/lengthscales (not QCD!)

For nuclear structure, protonsand neutrons with a localpotential fit to NN data

Find the “best” potential NN potential with χ2/dof ≈ 1up to ∼ 300 MeV energy

Two-body data may besufficient; many-body forcesas last resort

Let phenomenology dictatewhether three-body forces areneeded (answer: yes!)

Avoid (hide) divergences

Add “form factor” to suppresshigh-energy intermediatestates; don’t consider cutoffdependence

Choose approximationsby “art”

Use physical arguments (oftenhandwaving) to justify thesubset of diagrams used



(Nuclear) Many-Body Physics: “Old” vs. “New”

One Hamiltonian for allproblems and energy/lengthscales

Infinite # of low-energypotentials; differentresolutions =⇒ different dof’sand Hamiltonians

Find the “best” potential There is no best potential=⇒ use a convenient one!

Two-body data may besufficient; many-body forcesas last resort

Many-body data needed andmany-body forces inevitable

Avoid divergences Exploit divergences (cutoffdependence as tool)

Choose diagrams by “art” Power counting determinesdiagrams and truncation error


Extrapolate to large numbers of fermions

10 20 30 40N

0

10

20

E/E

FG

E = 0.44 N EFG

Pairing gap (∆) = 0.9 EFG

odd N

even N

Energy per particle: E/N = 0.44(1)EFG

Documents

Dense Matter for ``Dummies'' (Non-Experts) - Physicsntg/talks/2006/furnstahl... · Dense Matter for “Dummies” (Non-Experts) ... Chiral EFT offers systematic low-energy alternative