Effective Field Theory Applied to Nuclei

Evgeny Epelbaum, Jefferson Lab, USA

PN12, 4 Nov 2004

OutlineIntroductionFew nucleons at very low energyGoing to higher energies: chiral EFT

2 nucleons

3,4 and 6 nucleons Selected further topicsOutlook

Nuclear A-body problem: QCD pn

π

atom

icnu

clei

How can effective (field) theory contribute?

Provides dynamical input (systematic, consistent, QCD-based). Simplifies calculations in some cases (effective degrees of freedom).

Major difficulties:

Quantum mechanical many-body problem. - microscopic ab initio calculations:

solved for any and ;bound state problem solved for any and . First results for the continuum available;spectra of nuclei using Green’s Function Monte Carlo method (restricted to local ) and the No-Core Shell Model including .

- Shell Model; - Density Functional Theory.

Underlying dynamics (i. e.: ).

Effective (Field) Theoryand the nuclear

many-body problem

We cannot (yet) solve QCD at low E use chiral EFT to derive and to be applied in microscopic many-body calculations

“Hybrid” approach: - from chiral EFT, - phenomenologically

At very low E: pion-lessEFT (i.e. nucleons inter-acting via )

In-medium chiral EFT

Shell Model (SM) as an effective theory

Use effective theory to get rid of the high-momentum components of no need for -matrix in SM calculations

models (Urbana-IX, Tuscon-Melbourne, …).

Dynamical input:

Works good in many cases but problems remain.

AV 18, CD-Bonn, … (all with: χ2

datum~1)

meson exchange currents via Siegert theorem or Riska prescription.

Also conceptual problems:

Relation to QCD? , inconsistent with each other!Structure of .Theoretical uncertainty?How to improve?

Chiral EFT can help to solve these problems! Chiral EFT can help to solve these problems!

Linked to QCD.Consistent and systematic framework.Theoretical uncertainly can be estimated.Straightforward to improve.

Conventional approach to few-body systems

Tensor analyzing powers for dd -> pt at Ed=6.1 MeV

Ay versus θCM for p 3He reaction

ECM=1.2 MeV ECM=1.69 MeV

(from: www.unitn.it/convegni/download/FFLEEP.pdf)

Effective field theory

identify the relevant degrees of freedom and symmetries,construct the most general Lagrangian consistent with ,do standard quantum field theory with this Lagrangian.

“if one writes down the most general possible Lagrangian, including all terms consistent with the assumed symmetry principles, and then calculates S-matrix elements with this Lagrangian to any order in perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition and the assumed symmetry princi-ples”

S.Weinberg, Physica A96 (79) 327

Few nucleons at very low energy (expansion in Q/Mπ)

Shallow (virtual) bounds states in S-waves:1S0 channel: 3S1 channel:

Nonperturbative problem, resummation is needed!Nonperturbative problem, resummation is needed!

Power counting (Kaplan, Savage, Wise ‘97):

; where:

S-matrix in the 1S0 channel:

equivalent to effective range expansion in the pure 2N case

(using DR & Power Divergence Subtraction)

3S1 phase shift (Chen, Rupak, Savage 99)

LO

Nijmegen PSA

NLO

NNLO

Applications and extensions

Chen, Rupak & Savage ’99; Chen & Savage ’99; Rupak ’00(at N4LO accurate to 1% for )

M1E1

M1+E1

(from: Chen & Savage ’99)

Kong & Ravndal ’99, ’01; Butler & Chen ‘01

Chen, Rupak & Savage ‘99

Butler & Chen ’00; Butler, Chen & Kong ’01; Chen ‘01

Bedaque, Hammer, van Kolck ‘98; Gabbiani, Bedaque, Grieβhammer ‘00; Blankleider, Gegelia ‘01, …

Platter, Hammer & Meißner ‘04

halo-nuclei:Bertulani, Hammer & van Kolck ’02; Bedaque, Hammer & van Kolck ‘03

Going to higher energies: chiral EFT

If typical nucleon momenta , pions should be included as explicit degrees of freedom.

chiral EFTchiral EFT (expect to work for ) ma

ss

ga

p

Chiral symmetry of QCD

Define:

Chiral group SU(Nf)L X SU(Nf)R = group of independent rotations of in the flavor space.

chiral invariant not chiral invariant

strong interactions are approximately chiral invariant

strong interactions are approximately chiral invariant

QCD vacuum is only invariant under spontaneous symmetry

breaking Goldstone bosons (pions, due to ).

(Leutwyler ’96)

Notice: chiral symmetry has to be realized nonlinearly.

(worked out by: Weinberg ’68; Coleman, Callan, Wess & Zumino ’69)

Degrees of freedom: Goldstone bosons (pions) and matter fields (N, Δ, …).Symmetries: Lorentz invariance, spontaneously broken chiral symmetry, …

coefficients fixed by chiral symmetry

ChPT = simultaneous expansion in energy and around the chiral limit (mq=Mπ=0)

ChPT = simultaneous expansion in energy and around the chiral limit (mq=Mπ=0)

ππ,πN: perturbation theory (Goldstone bosons do not interact at E~0)

LO, ~(Q/Λ)2 NLO, ~(Q/Λ)4

Soft scale:

Q~p~Mπ; Hard

scale: Λ~Λχ~Mρ.

NN: perturbation theory does not work (deuteron, large aNN, …)

Weinberg’s idea:

Use chiral EFT to calculate . (Irreducible diagram =

diagram that is not generated through iterations in the dynamical equation.)

is not unique and can be derived in various ways, see e.g. Ordonez, Ray & van Kolck ‘94; Friar & Coon ‘94; Kaiser, Brockmann & Weise ‘97; Epelbaum, Glöckle & Meißner ‘98, ‘00; Higa & Robilotta 03, … .

Generate observables by solving the dynamical equation:

Notice: as a consequence of chiral symmetry; is bounded from below and for any there is a finite number of graphs to be calculated.

Two nucleons

LO (Q0):

NLO (Q2):

N2LO (Q3):

N3LO (Q4):

3π exchange (small), Kaiser ‘99, ‘00 2π exchange, Kaiser ‘01

Timeline

1990: Formulation by Weinberg.

1994: N2LO, energy-dependent, by Ordonez et al.

1998: N2LO, energy-independent, by Epelbaum et al.2003-2004: N3LO

by:- Entem, Machleidt;- Epelbaum et al.

Important work by:Kaiser, vanKolck, Friar,Robilotta, …

Low-energy constants:

known from the πN system fixed from NN data

Valid at low momenta. Wrong behavior (grows) at large momenta needs to be regularized.

We use the finite momentum cutoff Λ.

(see P.Lepage, nucl-th/9706029 for more details)

We use the novel regularization scheme for loop integrals introduced in E.Epelbaum et al., EPJA 19 (04) 125 (quicker convergence compared to DR).

Selected NN phase shifts at NLO, N2LO and N3LO

1S03S1

3P0

1D23P1

3D1

1F31G4ε2

N2LO

NLO

N3LO

(from E.Epelbaum, W.Glöckle, Ulf-G.Meißner, nucl-th/0407037, to appear in Nucl. Phys. A)

Λ=450…600 MeV

Elab=25 MeV Elab=50 MeV

Differential cross section for np scattering

NLO N2LO N3LO Exp

Ed [MeV] -2.171…-2.186 -2.189…-2.202 -2.216…-2.223 -2.225

AS [fm-1/2] 0.868…0.873 0.874…0.879 0.882…0.883 0.8846(9)

η 0.0256…0.0257

0.0255…0.0256

0.0254…0.0255

0.0256(4)

Deuteron observables

At large r :

3,4,… nucleons

Hierarchy of nuclear forces

No 3NF parameter-free

(Epelbaum et al. ‘01)

First 3NF:

D

E

LECs D, E fixed from 3H BE and aNd.

(Epelbaum et al. ‘02)

in progress…

In collaboration with:A.Nogga, W.Glöckle, H.Kamada, Ulf-G.Meißner and H.Witala

Elastic Nd scattering at EN = 65 MeV

Deuteron break up at EN = 65 MeV

NLO

NNLO

3N and 4N binding energies

Predictions for 6Li ground and excited states

(Calculation performed by A. Nogga, University of Washington, USA)

Selected further topics: chiral extrapolation in the NN system

EFTdatalattic

e gauge theor

yToday’s lattice calculations adopt large mq (or Mπ, since

),Chiral EFT might be used to extrapolate to physical values of Mπ.Beane & Savage ’03; Epelbaum, Meißner & Glöckle ‘03.see:

Chiral extrapolation of the NN observables at NLO

physical point

uncertainty due to d16

uncertainty due to D

1/a1S0 [fm-1] 1/a3S1 [fm-1]

M.Fukugita et al., PRD 52 (95)

(from E.Epelbaum, U.-G.Meißner, W.Glöckle NPA 714 (03) 535)

πN scattering length from πd scattering(in collaboration

with:S.R.Beane, V.Bernard, Ulf-G.Meißner and D.R.Phillips)

In the limit of exact isospin symmetry at threshold:

No πN data at very low energy.Extractions of a+ and a- from the level shifts and lifetime of pionic hydrogen have large error bars.πd scattering length aπd measured with high accuracy.

use chiral EFT to extract a+ and a- from aπd

(from Ulf-G.Meiβner et al., nucl-th/0301079)

J.Gasser et al., EPJC 26 (02) 13

LO ChPT

our calculationNovel power counting:

where .

Isospin violation in nuclear reactions

chiral invariant break chiral (and isospin) symm.

includes in addition to isospin conserving terms:strong isospin breaking terms ,electromagnetic isospin breaking terms (due to hard photons) ,coupling to (soft) photons .2NF

3NFN2LØ

van Kolck et al. ‘96 van Kolck et al. ‘98Friar et al. ‘99,‘03,‘04; Niskanen ‘02

N2LØNLØLØ

em str em

The 3NF depends on (δm)str, (δm)em, δMπ and f1. (Epelbaum et al. ‘04; J.L.Friar et al. ‘94)

N3LØ

f1

Summary

Few-nucleon systems can be studied in chiral EFT approach in a systematic and model independent way.The 2N system has been analyzed at N3LO. Accurate results for deuteron and scattering observables at low energy.3N, 4N and 6N systems have been studied at N2LO including the chiral 3NF. The results look promising.Many other applications have been performed.

Outlook

Few-nucleon systems at N3LO need V3N, V4N at N3LO.

Electroweak probes in nuclear environment need currents!

Reactions with pions.

Going to higher energies: inclusion of the Δ-resonance.

Perspectives:

Few-nucleon scattering

Properties of

light nuclei

Electroweak reactionswith nuclei

Chiral VNN provides abasis for applications

to other systems

Reactions withpionic probes

3He as neutron target

Nuclear parity violation

Astrophysicalapplications

Effective (field) theory and the nuclear many-body problem

We cannot (yet) solve QCD at low energy

Use chiral EFT to derive and to be applied in microscopic many-body calculations (see: S.Weinberg 90, 91; C.Ordóñez, L.Ray, U.van Kolck 96; U.van Kolck 94;

E.E., W.Glöckle, U.-G.Meißner 98, 00,04; D.R.Entem, R.Machleidt 03; S.R.Beane et al 03; …).

“Hybrid” approach: from chiral EFT, - phenomenologically.(see: S.Weinberg 92; T.-S.Park et al. 93,96,98,00,01,03; C.H.Hyun, T.-S.Park, D.-P.Min 01; S.R.Beane 98,99,04; V.Bernard, H.Krebs, U.-G.Meißner 00; L.E.Marcucci et al. 01; S.Ando et al.

02,03; …)

At very low even π’s can be treated as heavy particles

Use pion-less EFT [nucleons interacting via ] to describe few-nucleon systems, also in the presence of external sources (see: U. van Kolck 99; J.W.Chen, G.Rupak, M.J.Savage 99; X.Kong, F.Ravndal 99,00; G.Rupak 00; M.Butler et al. 00, 01; J.W.Chen 01; P.F.Bedaque, H.-W.Hammer, U. van Kolck 00; Gabbiani, Bedaque, Grieβhammer 00; Blankleider, Gegelia 01, … ).

Use in-medium chiral EFT to describe nuclear structure properties (see: M.Lutz 00, M.Lutz, B.Friman, Ch.Appel 00; N.Kaiser, S.Fritsch, W.Weise 02, 03, 04).

Shell Model (SM) as an effective theory (see: W.C.Haxton, C.-L.Song 00).

Use effective theory to get rid of the high-momentum components of . The resulting has no hard core and can be used as input in SM calculations (no need for -matrix).(see: E.E. et al. 98,99; S.K.Bogner et al. 01,02,03; S.Fujii et al. 04; A.Nogga, S.K.Bogner, A.Schwenk

04)

Status of the few-body problemStatus of the few-body problem

Both bound state and scattering problems can be accurately solved for any and . Coulomb problem in the continuum can be handled for 2 charged par-ticles (in configuration space only for local ).

Properties of the ground and low-lying excited states are studied using the Green’s Function Monte Carlo method (restricted to local ) and the No-Core Shell Model including .

Bound state problem can be accurately solved for any and . First re-sults for the continuum spectrum become available. Most advanced calculations are performed in configuration space only local . not yet included.

3N:

4N:

5…13N:

models (Urbana-IX, Tuscon-Melbourne, …).

Dynamical input in most of the calculations:

high-precision potentials (i.e.: χ2datum~1)

like AV 18, CD-Bonn, Nijm I,II, …

Proton Ay for elastic pd scattering

Proton Ay for pd -> γ 3He at Ep=150 MeV

(from: J.Golak et al., PRC 62 (00) 054005)

single nucleon

Siegert theorem

Riska prescription

meson exchange currents via Siegert theorem or Riska prescription.

Works good in many cases but problems remain.

Also conceptual problems:

Relation to QCD? , inconsistent with each other!Structure of .Theoretical uncertainty?How to improve?

Chiral EFT can help to solve these problems! Chiral EFT can help to solve these problems!

Linked to QCD.Consistent and systematic framework.Theoretical uncertainly can be estimated.Straightforward to improve.

A natural consequence of the chiral power counting:

Hierarchy of nuclear forces