RG Methods for Nuclei and Neutron Stars Achim Schwenk University of - - PowerPoint PPT Presentation

RG Methods for Nuclei and Neutron Stars

Achim Schwenk University of Washington / TRIUMF (2006-)

supported by US DOE and GSI

1) Motivation and introduction 2) Renormalization group approach to nucleonic matter 3) Applications to electronic systems 4) Some low-momentum results 5) Summary

Outline

adapted from A. Richter @ INPC2004 Lattice QCD QCD Lagrangian Exact methods A≤12 GFMC, NCSM Chiral EFT interactions (low-energy theory of QCD) Coupled Cluster, Shell Model A<100 Low-mom. interactions Density Functional Theory A>100

1) Motivation and introduction

Supernova SN1987a

Crab pulsar Supernova July 4, 1054

Many-body physics of nucleonic matter Nuclei with extreme compositions of n,p “Femtoscience”

structural dep. on N/Z

Connections to QCD

quark mass dep. of shell structure

Nuclear superfluidity

Cold atoms in the vicinity of Feshbach resonances

Fermi gases with resonant interactions exhibit universal properties

for large scattering lengths, no scales associated with interaction at low energy

can constrain properties of extreme low-density neutron matter in lab experiments with resonantly-tuned atomic 6Li or 40K universal energy

ξ=0.51±0.04 Duke 2005, (2002) ξ=0.27±0.12 Innsbruck 2004

Thermodynamics continuous across resonance! Bourdel et al., PRL 91 (2003) 020402.

Feshbach resonances in 6Li

Nuclear matter under extreme conditions in supernovae

sun in neutrinos with SuperK

desire systematic approach that includes bound states: d,α,… and resonances: nn,pp(1S0), nα(P3/2), αα(0+,2+),…

• n equal footing

Virial Equation of State of low-density nuclear matter composed of n,p,α

with second virial coefficients directly from NN, Nα, αα phase shifts

Horowitz, AS, nucl-th/0507033, nucl-th/0507064.

gives model-independent results for neutrinosphere in supernovae,

n ~ 1011-1012 g/cm3, T ~ 4 MeV

resulting α particle mass fraction disagrees with all EoS models used in supernova simulations

Supernova SN1987a

Extreme conditions in neutron stars

Yakovlev, Pethick, ARAA 42 (2004) 169. Blaschke et al., A&A 424 (2004) 979.

∆ ~ 50 keV ∆ ~ 120 keV ∆ ~ 100 keV Neutron star cooling is dominated by ν emission from interior nucleons

beta decay in 1N or 2N reactions, ν bremsstrahlung,…

ν emission suppressed due to superfluidity (costs pairing energy to break nn,

pp pair) except for enhancement of ν emission below superfluid Tc

found too rapid cooling for P-wave superfluid gaps ∆P > 30 keV Need reliable, microscopic predictions for nucleonic pairing gaps!

Pairing in halo nuclei and ν emission

neutron stars: ν emission suppressed due to superfluidity, costs pairing

energy to break nn, pp pair

beta decay of nn halo suppressed due to pairing, as for ν emission in neutron star cooling

from P. Garrett

3/2- 8.81

Sarazin et al., PR C70 (2004) 031302(R).

different nucleon-nucleon (NN) interactions fit to NN scattering below (for low momenta k < 2.1 fm-1) Details not constrained for momenta k > 2.1 fm-1 or distances r < 0.5 fm This is where NN models are strong and thus difficult to handle.

Many-body calculations are traditionally based on

k k

Argonne v18

Scattering amplitude (T matrix) in mom. space Integrating out high-momentum modes leads to effective interaction Vlow k (which reproduces

the low-momentum T matrix)

Changes of effective interaction with cutoff Λ given by RG equation VNN(k’,k)

Unifying nuclear forces with the Renormalization Group

in matrix form

k k’ k’ k

Vlow k

Λ Λ

for low-energy phenomena, desire low-momentum interactions, no need

for model-dependent high-momentum modes integrate out high-mom. modes with k >Λ using the Renormalization Group (RG)

Bogner, Kuo, AS, Phys. Rep. 386 (2003) 1.

leads to universal low-momentum interaction Vlow k for all VNN(k’, k)

in matrix form

k’ k

Low-momentum interactions

k k’

Vlow k

Λ Λ Vlow k

uses free-space NN interaction for pairing and free spectrum low-density (crust): 1S0 pairing higher-density (core): 3P2 pairing phase-shift equivalent NN interactions give identical BCS gaps resolves charge-dependence of NN force:

Benchmark BCS gaps in neutron matter

nn gaps np gaps

Baldo et al., PR C58 (1998) 1921.

Simple case: dilute gases (perturbative ) Gorkov et al., JETP 13 (1961)

1018, Heiselberg et al., PRL 85 (2000) 2418.

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 benchmark + induced interactions screening/vertex corr.

Beyond BCS theory: induced interactions

Shankar, RMP 66 (1994) 129.

Cutoff Λ around the Fermi surface defines an effective theory for low-mom. particles and holes Simpler problem in truncated space, RG focuses on low-lying modes

2) RG approach to interacting Fermi systems

start from full space with vacuum Vlow k effective interaction, dressed quasiparticles integrate out

• mom. shells

successively

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

One-loop particle-hole RG

[ ]

ZS ZS’

At one-loop, renormalization due to change in ph phase space On the Fermi surface keep only quasi-local a(q2,q’2) in RGE to extrapolate for

• ff the

Fermi surface, on Vlow k level dep. small [similar to N-patch scheme for 2d Hubbard:

• const. 4-point vertex for momenta in patch]

Patch k wave vect. k Fermi surface

ZS ZS’

Start from free-space two-body interaction After two decimations:

builds up many-body correlations (~ parquet)

Vlow k

Efficacy of the RG method

Phase space argument in RG equation: In long-wavelength limit can calculate RGE analytically q=0 eliminates ZS flow, define qp interaction + forward amplitude in RGE

cutoffs fast ph

ZS ZS’

Spinless fermions with simple interaction

in units of bare dos, coupling can be related to P-wave scattering volume AS, Friman et al., nucl-th/0207004.

angles ~ momentum transfers Flow of A(q,q’ fixed)

• interplay of ZS and ZS’channels

at one-loop

• large q amplitude runs first
• ZS and ZS’ channels counteract

flow large q small q

Illustration for a schematic model

Flow according to

large q’ small q’ l>1 Landau parameters generated

in RG, no truncations or model for mom. dependences needed

[m*/m = 6/5]

RG flow of qp interaction and Fermi liquid parameters

Should be solving RGE for 2-pt function Use approximate method for quadratic dispersion

• Fermi liquid relation for running of eff. mass
• static/adaptive method for qp strength

resulting factor agrees well with self-consistent in-medium ladders for

• 1. static density-independent mean value

in RGE

• 2. start with

adapt self-consistently

approximates k-mass by Vlow k contribution

Renormalization of effective mass and qp strength

static z factor

Find increase of effective mass at low densities see Wambach et al., NP A555 (1993) 128. Fermi liquid parameters

Results for neutron matter

Vlow k ph RG decrease of F0

small effects on G0

triangles: adaptive z factor squares: static z factor

With induced interactions:

• max. S-wave gap ∆ ≈ 0.8 MeV

AS, Friman, Brown, NPA 713 (2003) 191.

spin fluctuations suppress S-wave pairing in neutron matter BCS gaps: max. ∆ ≈ 3 MeV

S-wave pairing in neutron matter

RG also yields non-forward scattering amplitude for transport, pairing,…

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 Confirmed in above neutron star cooling simulations

P-wave gaps < 10 keV possible

Quasiparticle interations, n/p pairing, phase diagram of 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 interactions among valence nucleons in finite nuclei

see el. excitations in atoms Murthy, Kais, Chem. Phys. Lett. 290 (1998) 199.

Extensions

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: N-patch scheme: 4-point V(ki) constant for ki in same patch RG approach studies interference of infrared instabilities via flow towards the Fermi surface

Patch k wave vect. k Fermi surface

k1 k2 k4 k3

U t t‘

3) Applications to electronic systems

Eliminate momentum shells according to typical flow of couplings due to build-up and interference of instabilities identify leading instabilities: spin-density wave, d-wave SC unbiased RG phase diagram

flow V=U

Electronic excitations in atoms

Murthy, Kais, Chem. Phys. Lett. 290 (1998) 199.

Effective Coulomb interactions at Fermi surface of single C60

large reduction due to intraball screening

Berdenis, Murthy, PR B52 (1995) 3083.

RG approach to quantum dots Murthy, Shankar, PRL 90 (2003) 066801.

4) Some low-momentum results

• Connection to chiral EFT interactions
• 3N interactions, cutoff variation as a tool
• Coupled cluster results

Chiral Effective Field Theory (EFT) interactions

NN 3N 4N

systematic expansion in low-momenta over breakdown scale

Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Meissner, Nogga, Machleidt,… Epelbaum, Gloeckle, Meissner, NP A474 (2005) 362.

Collapse of off-shell matrix elements

N2LO N3LO RG evolution of chiral EFT interactions (Λ0~3 fm-1) and NN models (Λ0~5-20 fm-1) generates higher-order short-range operators (Λ to Λ0) necessary to preserve NN observables (NN phase shifts, Edeuteron) Therefore, universal result indicates that Vlow k parameterizes a higher-order EFT interaction

supported by simple EFT connection

LO: c0 fixed by scattering length as + RG invariance NLO: + effective range re

1S0

Running of Vlow k(0,0) vs. EFT contact interaction c0

Low-momentum interactions in few-nucleon systems

Exact Faddeev Vlow k only

A=3,4 binding energies

Nogga, Bogner, AS, PR C70 (2004) 061002(R).

Vlow k(Λ) defines NN interactions with cutoff-indep. NN observables Therefore, cutoff variation of many-body observables estimates errors due to neglected many-body interactions Cutoff dependence explains “Tjon line” due to neglected 3N interactions

Nogga et al., PRL 85 (2000) 944.

leading-order 3N interaction has only 2 new couplings

van Kolck, PR C49 (1994) 2932; Epelbaum et al., PR C66 (2002) 064001.

2 couplings fit to 3H and 4He for Vlow k(Λ) found that 3N interactions become perturbative for cutoffs and scales as (Q/Λ)3 ~ (mπ/Λ)3 relative to NN interaction, as expected from EFT power counting Vlow k + low-momentum 3N interaction make model-indep. predictions with theoretical error estimates

long (2π) intermed. (π) short-range c-terms D-term E-term

Low-momentum 3N interactions from chiral EFT

Radii with 3N interaction are

• approx. cutoff-independent,

agree reasonably with expt

Radii and p-shell nuclei

+3N

Λ=500…600 MeV

6Li

Theoretical errors in nuclear structure! Nuclei important to constrain 3N interaction

NCSM results with chiral EFT interactions from A. Nogga

Coupled cluster results for Vlow k

with D. Dean et al.

nearly converged

N=8, hω=18 MeV: -145.85 MeV

• calc. with 3N interaction in

progress, is repulsive also have 40Ca results CC method can provide exact answer for low-momentum interactions without resummations or similarity trafo. Can test other many-body methods for same interaction Build DF from low-momentum interactions and compare to CC results

Λ=1.9 fm-1

16O expt

As for binding energies, interplay of NN and 3N contributions to LS splitting, individual parts depend on cutoff CC results for different NN interactions, N=8, Gour et al., nucl-th/0507049 3N contrib. to LS for N3LO or Vlow k smaller, agrees with Q/Λ expectation

Spin-orbit strength of low-momentum interactions

similar observation in 0hω with Vlow k

AS, Zuker, nucl-th/0501038.

17O exp.: 0d3/2-0d5/2 ≈ 5.1 MeV 41Ca exp.: 0f5/2-0f7/2 ≈ 6.0 MeV

• Exciting era in nuclear theory: ISAC/FAIR/RIA and astrophys.

challenges, many-body connections to cold atoms, cm,…

• RG is a powerful tool for complex systems:

* leads to universal low-momentum NN interaction * effective interactions in the vicinity of the Fermi surface * superfluidity in neutron stars * extension to asymmetric matter, effective operators, valence-nucleon interactions for nuclei and to DFT * cutoff variation estimates theoretical uncertainties

• Low-momentum 3N interactions are weaker and thus tractable

will perform first calcs. with microscopic 3N interactions for A>12 nuclei

• Coupled cluster method provides converged results for medium-

mass nuclei, can be used as many-body test cases

5) Summary