From universal nuclear forces to dense matter
Achim Schwenk
Indiana University
Mini-Symposium on Strong QCD, TU Darmstadt
Collaborators: Scott Bogner, Gerry Brown, Bengt Friman, Dick Furnstahl,
Tom Kuo, Andreas Nogga, Chris Pethick and Andres Zuker
Supported by DOE and NSF
Prologue - matter at the extremes
Chaco Canyon, NM
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T ~ 1 - 30 MeV
B ~ 1011 - 1015 g cm-3
p /B ~ 0.05 - 0.5
Supernova July 4, 1054
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Supernova explosions
fascinated mankind
over a millenium
Supernova SN1987a
Outline
1) Introduction and motivation
2) Unifying nuclear forces
3) From light nuclei to nuclear matter
4) Superfluidity in neutron stars and
neutron star cooling
5) Summary
1) Introduction and motivation
perturbative
T
(quarks, gluons)
s(T) << 1
QCD phase diagram
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Hadrons
Plasma
Nuclear
matter
Color
superconductivity
B
low T, B-mN
Effective theories of QCD
with nucleons and pions
perturbative
except for pairing
(quarks, gluons)
s(B) << 1
QCD @ T=0 + electromagnetic + weak interactions
Density Functional Theory A>100
Shell model, UCOM, coupled
cluster method A<100
Exact methods A≤12
GFMC, NCSM
“Soft” NN
interactions
Lattice
QCD
Nucleon-nucleon interaction
(Effective theory of QCD)
QCD Lagrangian
adapted from A. Richter @ INPC2004
Crab pulsar
Supernova July 4, 1054
QCD @ T=0 + electromagnetic + weak interactions
Density Functional Theory A>100
Crab pulsar
Supernova July 4, 1054
Shell model, UCOM, coupled
cluster method A<100
Exact methods A≤12
GFMC, NCSM
“Soft” NN
interactions
Lattice
QCD
Nucleon-nucleon interaction
(Effective theory of QCD)
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TIFF (LZW) decompressor
are needed to see this picture.
QCD Lagrangian
Feldmeier, Neff, Roth
adapted from A. Richter @ INPC2004
from 6He to
the drip lines
The neutron matter - cold Fermi gas connection
T
Normal phase
resonance
superfluidity
BCS
increasing
attraction
perturbative
except for pairing
O’Hara et al., Science 298 (2002) 2179.
The neutron matter - cold Fermi gas connection
T
Normal phase
resonance
superfluidity
BCS
increasing
attraction
perturbative
except for pairing
resonant and dilute
Fermi gases exhibit
universal properties
O’Hara et al., Science 298 (2002) 2179.
The neutron matter - cold Fermi gas connection
T
same equation of state
for low-density neutrons,
6Li or 40K
Normal phase
(Fermi momentum is the
only scale)
resonance
superfluidity
BCS
=0.51±0.04 Duke 2005
=0.27±0.12 Innsbruck 2004
=0.44±0.01 GFMC (Carlson
et al. 2003, Astrakharchik et al. 2004)
increasing
attraction
perturbative
except for pairing
The neutron matter - cold Fermi gas connection
resonant and dilute
Fermi gases exhibit
universal properties
First results for Tc
T
same equation of state
for low-density neutrons,
6Li or 40K
Tc ~ 0.27 TF Kinast et al., Science last week
Epair ~ 0.2 TF Bartenstein et al., cond-mat/0412712
Tc ~ 0.04 TF Lattice (Wingate, cond-mat/0502372)
(Fermi momentum is the
only scale)
Normal phase
resonance
superfluidity
BCS
=0.51±0.04 Duke 2005
=0.27±0.12 Innsbruck 2004
=0.44±0.01 GFMC (Carlson
et al. 2003, Astrakharchik et al. 2004)
increasing
attraction
perturbative
except for pairing
The neutron matter - cold Fermi gas connection
resonant and dilute
Fermi gases exhibit
universal properties
T
neutron matter
preliminary
same equation of state
for low-density neutrons,
6Li or 40K
AS, Pethick, in prep.
(low density
QCD @ Z=0)
(Fermi momentum is the
only scale)
resonance
superfluidity
BCS
=0.51±0.04 Duke 2005
=0.27±0.12 Innsbruck 2004
=0.44±0.01 GFMC (Carlson
et al. 2003, Astrakharchik et al. 2004)
increasing
attraction
perturbative
except for pairing
Comparing nuclear to atomic forces
Neutrons and protons are
composite objects with
interactions similar to
atom-atom interactions.
Differences:
nuclear forces have strong
spin dependences
(spin-orbit and tensor forces)
Comparing nuclear to atomic forces
liquid He
r12 3.9A
nuclear
matter
r12 1.8fm
Neutrons and protons are
composite objects with
interactions similar to
atom-atom interactions.
Differences:
nuclear forces have strong
spin dependences
(spin-orbit and tensor forces)
Comparing nuclear to atomic forces
liquid He
r12 3.9A
nuclear
matter
r12 1.8fm
Neutrons and protons are
composite objects with
interactions similar to
atom-atom interactions.
Differences:
nuclear forces have strong
spin dependences
(spin-orbit and tensor forces)
For scattering:
convenient to work in momentum space
k
k’
NN interactions well-constrained by NN scattering
Many different NN potentials fit
to data below
k
k
CD Bonn, Machleidt PR C63 (2001) 024001
NN interactions well-constrained by NN scattering
Many different NN potentials fit
to data below
k
k
but details not constrained for
momenta larger than
or distances
CD Bonn, Machleidt PR C63 (2001) 024001
Models of nuclear forces and Effective Field Theory
Argonne v18 =
CD Bonn =
Systematic EFT =
+ (1 exchange)2 + Woods-Saxon cores
+ one-boson exchange phenomenology (, , ,… )
Models of nuclear forces and Effective Field Theory
Argonne v18 =
CD Bonn =
Systematic EFT =
+ (1 exchange)2 + Woods-Saxon cores
+ one-boson exchange phenomenology (, , ,… )
Evidence for NNN=3N interactions
All exact calculations with only
NN forces miss A=3,4 nucleon
binding energies
Exact GFMC results with only
NN force miss A≤10 spectra
Nogga et al., PRL 85 (2000) 944.
Pieper, Wiringa, Ann. Rev. Nucl. Part. Sci. 51 (2001) 53.
Evidence for NNN=3N interactions
All exact calculations with only
NN forces miss A=3,4 nucleon
binding energies
Exact GFMC results with only
NN force miss A≤10 spectra
Nogga et al., PRL 85 (2000) 944.
Pieper, Wiringa, Ann. Rev. Nucl. Part. Sci. 51 (2001) 53.
Challenges
Is it possible to reduce the model dependence of nuclear forces?
Cores strongly restrict the use of nuclear forces - with potential
in harmonic oscillator basis:
<0|V|n>
Are nuclear interactions
without cores possible?
Connection to nuclei complicated
How are 3N forces tractable? Can one estimate the error due to
neglected many-body forces?
Reliable extrapolations to extremes: neutron/proton-rich nuclei,
dense matter, astrophysical temperatures,…
2) Unifying nuclear forces
Differences in NN interactions due to model assumptions at high
momenta (short distances)
n
scattering
amplitude
=
p
=
long-range
 exchange
+
+
Low-momentum strength of VNN depends
on how many intermediate states included
Low-energy universality:
Long-wavelength description should be
independent of high-momentum details
+…
+…
Integrating out high-momentum modes
In momentum space, scattering amplitude (T matrix) given by
k
in matrix form
VNN(k’,k)
Integrating out high-momentum modes leads
to effective interaction Vlow k (which reproduces
the low-momentum T matrix)
k’

Changes of effective interaction with cutoff 
are given by Renormalization Group (RG) eqn.
Vlow k

k’
k
Starting from any NN interaction:
Solutions to the RG eqn. evolve
to a “universal” interaction Vlow k
for cutoffs below
1S
0
Vlow k
Bogner, Kuo, AS, Phys. Rep. 386 (2003) 1.
Starting from any NN interaction:
Solutions to the RG eqn. evolve
to a “universal” interaction Vlow k
for cutoffs below
1S
0
Vlow k
Bogner, Kuo, AS, Phys. Rep. 386 (2003) 1.
Main effect: constant shift from VNN
(removes model-dependent cores)
Collapse in all partial waves
due to same long-distance physics and phase shift equivalence
small differences
related to spread
in phase shift fit
(Idaho misses 3F2)
Collapse of off-shell matrix elements as well
N2LO
N3LO
“Universal” result indicates that Vlow k effectively parameterizes
a high-order EFT interaction
Simplest EFT connection
Running of Vlow k(0,0) vs.
EFT contact interaction c0
fixed by scattering length as
and effective range re
1S
0
Shell-model matrix elements
1S
0
Low-momentum interactions are
tractable in a shell-model basis
<0|V|n>
3S
1
Vlow k similar to successful G matrix
pseudopotential (without drawbacks of G)
Plot of two-body Vlow k vs.
G matrix elements in MeV
AS, Zuker, nucl-th/0501038.
3) From light nuclei to nuclear matter
By construction: For Vlow k all NN observables are cutoffindependent and reproduced with same 2 ≈ 1
Nucleons are composite objects: All NN potentials have
corresponding 3N, 4N,… forces
If one omits the many-body forces, calculations of lowenergy 3N, 4N,… observables will be cutoff-dependent
Varying the cutoff gives an estimate for effects of omitted
3N, 4N,… forces Nogga, Bogner, AS, PR C70 (2004) 061002(R).
Important for extrapolations to dense and hot matter
Cutoff dependence due to omitted 3N forces
A=3,4 binding energies
Exact Faddeev
Vlow k only
Nogga, Bogner, AS, PR C70 (2004) 061002(R).
Cutoff dependence shows that three-body forces are inevitable
Cutoff dependence is almost linear (explains Tjon line), with 3N
contributions larger for core potentials
Low-momentum 3N forces with only pion exchanges
organized by chiral Effective Field Theory van Kolck, PR C49 (1994) 2932;
Epelbaum et al., PR C66 (2002) 064001.
operator structure of any 3N force collapses at low energies to
long (2π)
c-terms
intermediate (π) short-range
D-term
E-term
Motivation of adjusting chiral 3N force to Vlow k:
Low-energy universality indicates Vlow k effectively parameterizes a
high-order EFT interaction
Two couplings fit to 3H and 4He
Linear dependences in fits, consistent
with perturbative 3N contributions
3N forces become perturbative for
cutoffs
nonperturbative at larger cutoffs, with expectation values beyond
expected (Q/Λ)3 ~ (mπ/Λ)3 ≈ 0.1 of NN contribution
NN scattering in free-space vs. nuclear matter
Two sources of nonperturbative behavior:
1. Iterated interactions at low momenta are nonperturbative due to
near bound states
2. Cores scatter strongly to high-momentum states (overwhelms 1.)
For interactions with cores:
second-order contributions
larger than VNN
n
p
NN scattering in free-space vs. nuclear matter
Two sources of nonperturbative behavior:
1. Iterated interactions at low momenta are nonperturbative due to
near bound states
2. Cores scatter strongly to high-momentum states (overwhelms 1.)
For interactions with cores:
second-order contributions
larger than VNN
n
p
P1>kF
even with Pauli blocking
P2>kF
NN scattering in free-space vs. nuclear matter
Two sources of nonperturbative behavior:
1. Iterated interactions at low momenta are nonperturbative due to
near bound states
2. Cores scatter strongly to high-momentum states (overwhelms 1.)
For low-momentum interactions:
free-space nonperturbative due
to near bound states
n
p
NN scattering in free-space vs. nuclear matter
Two sources of nonperturbative behavior:
1. Iterated interactions at low momenta are nonperturbative due to
near bound states
2. Cores scatter strongly to high-momentum states (overwhelms 1.)
For low-momentum interactions:
free-space nonperturbative due
to near bound states
n
p
P1>kF
small in matter due to
occupied Fermi sea
P2>kF
NN scattering in free-space vs. nuclear matter
Two sources of nonperturbative behavior:
1. Iterated interactions at low momenta are nonperturbative due to
near bound states
2. Cores scatter strongly to high-momentum states (overwhelms 1.)
For low-momentum interactions:
leads to very good Hartree-Fock
neutron matter equation of state
P1>kF
P2>kF
n
p
Vision for nuclear forces
Lattice QCD determination
of low-energy parameters
Chiral EFT interactions with consistent
many-body forces and operators
RG evolution to lower cutoffs
for many-body applications
4) Superfluidity in neutron stars
BCS theory: any attractive
interaction renders Fermi
surface unstable to pairing
Pairing in nuclei:
excitation energies show
gap for N,Z even nuclei
Pairing in neutron stars:
some angular momentum
carried by superfluid vortices
trapped to neutron star crust
vortices unpin suddenly,
leading to glitches
Δ ~ 1 MeV, Tc ~ 1010 K
change of Crab pulsar period
Neutron star interior
From fastest pulsar (1.558 ms):
R < 75 km,  > 0 /2
Typical: R ~ 10 km, M ~ 1.4 Msun
Atmosphere (~10 cm)
Envelope (~10 m blanket)
from Horowitz, IU
Crust (~1 km n-rich pasta
superfluid neutrons, e )
Outer Core (superfluid n,
supercond. protons, e,  )
from Page, UNAM
“Inner” Core ( ? )
Benchmark BCS gaps in neutron matter
uses free-space NN interaction for pairing and free spectrum
low-density: 1S0 pairing
higher-density: 3P2 pairing
Baldo et al., PR C58 (1998) 1921.
phase-shift equivalent NN interactions give identical BCS gaps
Benchmark BCS gaps in neutron matter
uses free-space NN interaction for pairing and free spectrum
low-density: 1S0 pairing
higher-density: 3P2 pairing
np gaps
nn gaps
Baldo et al., PR C58 (1998) 1921.
phase-shift equivalent NN interactions give identical BCS gaps
resolves charge-dependence of NN force:
Beyond BCS theory: induced interactions
Simple case: dilute gases (perturbative
) Gorkov et al., JETP 13 (1961)
1018, Heiselberg et al., PRL 85 (2000) 2418.
screening/vertex corr.
benchmark + induced interactions
Induced interactions lead to a
significant reduction of the gap
[universal reduction for 2 spin states]
RG approach to include particle-hole
intermediate states away from Fermi surface
RG approach to Fermi systems
Shankar, RMP 66 (1994) 129.
Cutoff  around the Fermi
surface defines an effective
theory for low-momentum
particles and holes.
starting from
free-space Vlow k
integrate out
momentum shells
successively
effective interaction,
dressed quasiparticles
Condensed matter results from the RG approach
Zanchi, Schulz, PR B61 (2000) 13609, Halboth, Metzner, PRL 85 (2000) 5162, Salmhofer, Honerkamp,
PTP 105 (2001) 1, Binz et al., Ann. Phys. 12 (2003) 704.
2d Hubbard model:
U
t
Eliminate modes according to
phase diagram
Leading instabilities:
spin-density wave, d-wave SC
t‘
One-loop particle-hole RG
Change of effective 4-point vertex:
Intermediate states:
thin from momentum shells, thick from fast particles/holes
+ RG treats momentum dependences on equal footing
(in contrast to one-channel resummations, no momentum extrapolations needed)
+ RG maintains all symmetries
- Currently: treat pairing correlations explicitly after ph RG
Future: include BCS channel with RG equation for pairing gap
Efficacy of the RG method
Vlow k
Start from free-space
two-body interaction
After two decimations:
builds up many-body
correlations (~ parquet)
Nuclear physics results
BCS gaps: max. ∆ ≈ 3 MeV
With induced interactions:
max. S-wave gap ∆ ≈ 0.8 MeV
AS, Friman, Brown, NPA 713 (2003) 191.
spin fluctuations suppress
S-wave pairing
Extensions
Quasiparticle interations and n/p pairing in asymmetric matter AS,
Friman, Furnstahl, in prep.
Beyond one-loop RG: higher loops suppressed by 1/N ~ Λ/kF for
regular Fermi surfaces Shankar
Renormalization of
currents/operators
see Halboth, Metzner, PR B61 (2000) 7364.
Effective SM interactions
for heavy nuclei, see el.
excitations in atoms Murthy,
Kais, Chem. Phys. Lett. 290 (1998) 199.
Polarization effects on P-wave pairing
For P-waves: spin-orbit,
tensor forces crucial
Screening effects?
Spin fluctuations
attractive in S=1
Pethick, Ravenhall, Ann. NY Acad. Sci. 647
(1991) 503; Jackson et al., NPA 386 (1982) 125.
For P-wave pairing:
first perturbative assessment
AS, Friman, PRL 92 (2004) 082501.
(Note: 2nd order / Vlow k < 50%)
Induced spin-orbit forces
deplete P-wave pairing
P-wave gaps < 10 keV possible
Impact on neutron star cooling
951 years later
Cooling simulations provide a unique
probe of neutron star composition
from http://chandra.harvard.edu
Neutron star structure
Atmosphere (~10 cm) emission of photons, surface temperature from
spectral fits
Envelope (~10 m) controls heat leak
from interior, temperature gradient
Tinterior = 108 K ~ Tsurface = 106 K
from Horowitz, IU
Crust (~1 km n-rich pasta
superfluid neutrons, e )
relevant for early cooling
Outer Core (superfluid n,
from Page, UNAM
supercond. protons, e,  )
Cooling of neutron stars
for 105 yrs dominated by neutrino emission from interior nucleons
In superfluid matter:
neutrino emission suppressed due to gap, except for enhancement
just below superfluid Tc
surface temperatures
correspond to typical
Tint ~ 108 K ~ 10 keV
enhancement from
S-wave pairing early
Isolated neutron star data
from Yakovlev, Pethick, Ann. Rev. Astron. Astrophys. 42 (2004) 169.
Constraints on P-wave gaps from neutron star cooling
With P-wave gaps ∆ > 30 keV (e.g., free-space BCS gaps):
Enhancement at intermediate stages and neutron stars cool too fast
∆ ~ 50 keV
∆ ~ 120 keV
Yakovlev, Pethick, ARAA 42 (2004) 169.
∆ ~ 100 keV
Blaschke et al., A&A 424 (2004) 979;
Grigorian et al., astro-ph/0501678.
Consistency with data requires P-wave gaps ∆ < 30 keV
Possibility: core neutrons only superfluid after 105 yrs
Implications of weak P-wave pairing
minimal neutron star cooling Page et al., astro-ph/0403657; Page, astro-ph/0405196.
most sensitive to P-wave gap, less severe dependence to size of gap
Magnetized H atmosphere fits:
1) RX J0822-4247 (in Puppis A)
2) 1E 1207.4-5209 (in PKS 1209-52)
3) PSR 0538+2817
4) RX J0002+6246 (in CTB 1)
5) PSR 1706-44
6) PSR 0933-45 (in Vela)
Blackbody atmosphere fits:
7) PSR 1055-52
P-wave gap
∆ < 10 keV
8) PSR 0656+14
9) PSR 0633+1748 "Geminga"
10) RX J1856.5-3754
11) RX J0720.4-3125
Upper limits:
a b c & d:
A) CXO J232327.8+584842 (in Cas A)
An X-Ray Search for Compact Central Sources in Supernova Remnants
Kaplan et al, ApJS 153, 269 (2004)
B) PSR J0205+6449 (in 3C58)
C) PSR J1124-5916 (in G292.0+1.8)
from Page @ MINS 2005.
D) RX J0007.0+7302 (in CTA 1)
Implications of weak P-wave pairing
minimal neutron star cooling Page et al., astro-ph/0403657; Page, astro-ph/0405196.
most sensitive to P-wave gap, less severe dependence to size of gap
Magnetized H atmosphere fits:
1) RX J0822-4247 (in Puppis A)
2) 1E 1207.4-5209 (in PKS 1209-52)
3) PSR 0538+2817
4) RX J0002+6246 (in CTB 1)
5) PSR 1706-44
6) PSR 0933-45 (in Vela)
Blackbody atmosphere fits:
7) PSR 1055-52
P-wave gap
∆ < 10 keV
8) PSR 0656+14
9) PSR 0633+1748 "Geminga"
10) RX J1856.5-3754
11) RX J0720.4-3125
Upper limits:
a b c & d:
A) CXO J232327.8+584842 (in Cas A)
An X-Ray Search for Compact Central Sources in Supernova Remnants
Kaplan et al, ApJS 153, 269 (2004)
B) PSR J0205+6449 (in 3C58)
C) PSR J1124-5916 (in G292.0+1.8)
Candidates for fast cooling, exotic cores
D) RX J0007.0+7302 (in CTA 1)
5) Summary
1. RG removes model dependences from nuclear forces,
results in “universal” low-momentum interaction
2. Low-momentum 3N forces are perturbative in this regime
3. Vlow k tractable with good results for nuclear structure and
nuclear matter AS, Zuker, nucl-th/0501038, Bogner, AS, Furnstahl, Nogga, in prep.
4. Cutoff variation estimates theoretical errors - important
for extrapolations to dense and hot matter
5. RG powerful for many-body applications: 2d Hubbard model,
electronic excitations in atoms, pairing in neutron stars,…
6. Weak P-wave pairing in neutron star cores likely,
P-wave gaps important to identify exotic neutron stars
Outlook
1. RG for Density Functional Theory with Polonyi
2. Neutrino transport in dense matter with Friman, Pethick
3. Effects of spin-violating forces in pA scattering with Stephenson
4. Error estimates for neutron star structure with Horowitz
5. Effects of 3N interactions in nuclear structure with Dean, Papenbrock
6. RG for effective operators
7. Interference of instabilities in high-density QCD
Perturbative nuclear matter?
Results indicate a perturbative behavior for the energy
Bogner, AS, Furnstahl, Nogga, in prep.
Hartree-Fock (Vlow k + V3N)
second-order
with largest 3N parts only and m*
3rd order < 1MeV
Simple phenomenology
obtained by filling lowest oscillator orbits AS, Zuker, nucl-th/0501038.
spin-orbit splittings
41Ca
17O
exp.: 0f5/2-0f7/2 ≈ 6.0 MeV
exp.: 0d3/2-0d5/2 ≈ 5.1 MeV
spectra in Ca region
low-lying levels depend
weakly on cutoff