Few-Nucleon Systems in Chiral EFTTU München, 31.05.2010

Evgeny Epelbaum, Ruhr-Universität Bochum

OutlinePart I: Foundations Introduction Chiral expansion of nuclear forces Few-nucleon dynamics

Part II: Selected applications Pion production in NN collisions Isospin breaking & few-N systems

Part III: Nuclear lattice simulations

Introduction Anatomy of calculation Results for light nuclei

Summary and outlook

Evgeny Epelbaum

Chiral Perturbation TheoryWeinberg, Gasser, Leutwyler, Bernard, Kaiser, Meißner, …

QCD and chiral symmetry

SU(2)L x SU(2)R invariantbreaks chiral symmetry

small is approximately chiral invariantvacuum invariant only under SU(2)V SU(2)L x SU(2)R

spontaneous symmetry breaking Goldston Bosons (pions)

Chiral perturbation theoryGoldstone bosons + matter

fields

200

400

600

800

0

π (140)

ρ (770)ω (782)

mas

s ga

p

M [MeV]

powers of Q

most general consistent with the χ-symmetry of QCD

compute the amplitude via perturbative expansion in over (power counting):fix low-energy constants & make predictions...

Two and more nucleons: strongly interacting systems Hierarchy of scales for non-relativistic ( )

nucleons:

Goldstone-boson and single-nucleon sectors: weakly interacting systems ChPT

Weinberg ‘91,’92

chiral EFT (cf. pNRQCD), instantaneous (nonlocal) potentials due to exchange of multiple Goldstone bosons rigorously

derivable in ChPT

π-less EFT with local few-N interactions

inte

rnuc

leon

pot

entia

l [M

eV]

separation between the nucleons [fm]

chiral expansion of multi-pion exchange

zero-range operators

Few nucleons: from ChPT to ChEFT

irreducible contributions to be derived in ChPT

enhanced reducible contributions must be summed up to infinite order

Weinberg’s approach

projector onto states with mesons

Derivation of nuclear forcesUse canonical formalism to obtain the pion-nucleon Hamiltonian from the HB effective chiral Lagrangian

Decouple pions via a suitably chosen unitary transformation in Fock space:

energy-independent nuclear potentials

projector onto nucleonic states

How to compute ?

A convenient parametrization in terms of (Okubo ’54):

Require that

The major problem is to solve the nonlinear decoupling equation.

Derivation of nuclear forcesThe decoupling equation can be solved recursively utilizing chiral power counting (NDA):

Powers of Λ can only be generated through LECs

with

Only non-renormalizable verices allowed (χ-symmetry) perturbative expansion

Count powers of Q

with

Expansion in powers of Q/Λ: and

Perturbative solution of the decoupling equation:

The explicit form of the UT up to (Q/Λ)4 is given in: E.E., EPJA 34(2007) 197

The same UT to be used to compute exchange currents, Kölling et al. ’09, to appear where

Derivation of nuclear forcesRenormalization of the potentials

1π-exchange shouldfactorize out

120 time-ordered graphs

cannot renormalize the potential !

Solution (E.E.’06)

unique (!) result for

Similar to the large-Nc nuclear potential puzzle, Cohen et al. ‘02

, grow with increasing momenta LS equation must be regularized & renormalized

Solving the Schrödinger equation

Renormalization à la Lepage

Choose & tune the strengths of to fit low-energy observables. generally, can only be done numerically; requires solving nonlinear equations for , self-consistency checks via „Lepage plots“,residual dependence in observables survives

Ordonez et al.’96; Park et al.’99; E.E. et al.’00,’04,’05; Entem, Machleidt ’02,’03

DR difficult to implement numerically due to appearance of power-law divergences Phillips et al.’00 Cutoff (employed in most applications)

— needs to be chosen to avoid large artifacts (i.e. large -terms) — can be employed at the level of in order to preserve all relevant symmetriesSlavnov ’71; Djukanovic et al. ’05,’07; also Donoghue, Holstein,

Borasoy ’98,’99

Regularization of the LS equation

LO:

NLO:

N2LO:

N3LO:

Two-nucleon forceOrdonez et al. ’94; Friar & Coon ’94; Kaiser et al. ’97; E.E. et al. ’98,‘03; Kaiser ’99-’01; Higa, Robilotta ’03; …

V2N = V2N +V2N + V2N + V2N + … Chiral expansion of the 2N force: (0) (2) (3) (4)

renormalization of 1π-exchange renormalization of contact terms7 LECs leading 2π-exchange

2 LECs

subleading 2π-exchangerenormalization of 1π-exchange

sub-subleading 2π-exchange 3π-exchange (small)

15 LECsrenormalization of contact termsrenormalization of 1π-exchange

+ isospin-breaking corrections…van Kolck et al. ’93,’96; Friar et al. ’99,’03,’04; Niskanen ’02; Kaiser ’06; E.E. et al. ’04,’05,’07; …

Results based on EFT with explicit Δ(1232) degrees of freedom available up to N2LOOrdonez, Ray, van Kolck ’96; Kaiser, Gerstendorfer, Weise ‘98; Krebs, E.E., Meißner ’07,‘08

Ay

dσ/d

Ω [

mb/

sr]

N2LO N3LO PWA

Entem, Machleidt ’04; E.E., Glöckle, Meißner ‘05

EE, Glöckle, MeißnerEntem, Machleidt

Two nucleons up to N3LO

Deuteron observables

Neutron-proton phase shifts at N3LO Neutron-proton scattering at 50 MeV

bEM + [Nijm78; 1π; 1π+2π]

Energy-dependent boundary condition

Rentmeester et al. ’99, ‘03Evidence of the 2π-exchange from the partial wave analysis

Few-nucleon forces up to N3LOD E

van Kolck ’94, E.E. et al.‘02

N2LO

N3LOcorrections to the 3NF

Ishikawa, Robilotta ‘07Bernard, E.E., Krebs, Meißner ’07;

E.E. ’06,’08

parameter-free χ-symmetry essential nontrivial constraints

trough renormalizability effects in 3N scattering observables in progress...

parameter-free contributes a few 100 keV to Eα

Rozpedzik et al.‘06; Nogga et al., in prep.

first 4NF contributions

first nonvanishing 3NF

Differential cross section in elastic Nd scattering

NLON2LO

Polarization observables in elastic Nd scatering

E.E. et al.’02; Kistryn et al.’05; Witala et al.’06; Ley et al.‘06; Stephan et al.’07; …

N2LO

Three nucleons up to N2LO

EN = 22:7 MeV

EN = 90 MeV

No-Core-Shell-Model results for 10B,11B, 12C and 13C @ N2LO

Navratil et al., PRL 99 (2007) 042501

4He and 6Li @ NLO and N2LO

Nogga et al., NPA 737 (2004) 236

More nucleons

Hot topics (work in progress)Bridging different reactions with the D-term

pp! de+ºe

Hanhart et al.’00, Baru et al.‘09, Filin et al.‘09

Park et al.‘03; Nakamura et al.’07

ºed ! e¡ pp; ºd ! ºpn; ¹ ¡ d ! º¹ nn¼¡ d ! °nn

Ando et al.‘02,‘03Gardestig & Phillips ’06, Lensky et al.‘05,‘07 DGazit, Quaglioni, Navratil, ’09

Effects of the N3LO 3NF in Nd scattering Preliminary calculations

(incomplete) indicate that effects of the N3LO cor-rections to the 3NF in Nd scattering at low energy are small…

Ishikawa & Robilotta, PRC 76, 014006 (2007)

EFT with explicit Δ(1232) DOFImproved convergence of the EFT expansion!

Preliminary calculation of the ring diagrams yield rather strong potentials…

Isoscalar central potential

r12 [fm] r23 [fm]

V [M

eV]

Krebs, E.E., to appear

Pion production in NN collisionsConsiderably more challenging due to the appearance of a new „soft“ scale

slower convergence of the chiral expansion(expansion parameter vs )

State-of-the-artHybrid approach (EFT description of the 2N system for not yet available)Δ(1232) isobar plays an important role must be included as an explicit DOFs-wave pion production worked out up to NLOCohen et al.’96; Dmitrasinovic et al.’99; da Rocha et al.’00; Hanhart et al.’01,’02

Proper separation of irred. contributions crucial!Lensky et al. ’01

Near threshold: with

LO

NLOresults for pp → dπ+

p-wave π-production and the D-termLoops start to contribute at N3LO

Simultaneous description of pn → ppπ-, pp → pnπ+ and pp → dπ+ nontrivial consistency check of chiral EFT

Up to N2LO, D is the only unknown LECD

N2LO

In the future: implications for the 3NF and for weak reactions with light nuclei

3S1 for pp → pnπ+, pp → dπ+ ; 1S0 for pn → ppπ-

Hanhart, van Kolck, Miller ’00; Baru, EE, Haidenbauer, Hanhart, Kudryavtsev, Lensky, Meißner ‘09

1S0 for pp → pnπ+, pp → dπ+ ; 3S1 for pn → ppπ-

Reaction pp → dπ+

Near threshold:

Natural units for D:← dimensionless coefficient ~ 1

p-wave π-production and the D-termReaction pn → ppπ-

The final pp relative mo-mentum is restricted to be: pp p-waves suppressed Data only available at expect only qualitative description...

Reaction pp → pnπ+

The relevant amplitude (1S0 → 3S1p) is suppressed compared to the dominant 1D2 → 3S1p amplitude minor sensitivity to the D-term…

New data at lower energies will be taken at COSY.

Overall best results for d ~ 3

Data from TRIUMF and PSI

Flammang et al.’98

Baru, EE, Haidenbauer, Hanhart, Kudryavtsev, Lensky, Meißner ‘09

Isospin breaking & few-N systemsisospin-breaking hard / soft γ‘s + terms

IB 2NF, 3NF worked out up to high orders, long-range contributions largely driven by , and van Kolck et al. ’93,’96; Friar et al. ’99,’03,’04; Niskanen ’02; Kaiser ’06; E.E. et al. ’04,’05,’07; …

Charge-symmetry-breaking nuclear forces and BE differences in 3He – 3H

Friar et al. PRC 71 (2005) 024003

CSB forward-backward asymetry in @ 279.5 MeV at TRIUMF

(Opper et al. ’03)

measured at IUCF: @ 228.5 / 231.8 MeV Stephenson et al. ’03

Theoretical analysis challenging; first estimations yield the right order of magnitude.Gardestig et al. ’04; Nogga et al.’06

np → dπ0 & the np mass differenceNiskanen ‘99; van Kolck et al. ’00; Bolton, Miller ‘09; Filin, Baru, E.E., Haidenbauer, Hanhart, Kudryavtsev, Meißner ‘09

gives rise to Afb, nonzero only for pn → dπ0

due to interference of IB and IC amplitudes

The goal: use Afb measured at TRIUMF to extract the strong/em contributions to the neutron-to-proton mass shift.

A0 can be determined from the pionic deuterium lifetime measurement @ PSI:

Gasser, Leutwyler ’82 (based on the Cottingham sum rule)

A1 at LO in chiral EFT: IC amplitudes calculated at NLOBaru et al.’09

Our result:

Lattice:Beane et al.’07

Nuclear Lattice SimulationsBorasoy, E.E., Krebs, Lee, Meißner, Eur. Phys. J. A31 (07) 105, Eur. Phys. J. A34 (07) 185, Eur. Phys. J. A35 (08) 343, Eur. Phys. J. A35 (08) 357, E.E., Krebs, Lee, Meißner, Eur. Phys. J A40 (09) 199, Eur. Phys. J A41 (09) 125, Phys. Rev. Lett 104 (10) 142501, arXiv:1003.5697 [nucl-th]

» 0:1fm

Lattice QCD Chiral EFT on the lattice

fundamental, the only parameters are

hard to go beyond the 2N system,e.g. for :

can access few- and many-nucleon systems

LECs ( ) to be determined from the data or LQCD

®S ; mq

gA ; F¼; ci ; di ; Ci ; : : :

can probe bigger volumes

Lattice QCD vs lattice chiral EFT

» e¡ (2mN ¡ 3M ¼)tsignal/noisehN (t)N (t)N y(0)N y(0)i

Correlation-function for A nucleons:

Slater determinants for A free nucleons

Ground state energy:

E 0A = lim

t! 1

h¡ d

dt lnZA (t)i

Expectation values of a normal ordered operator :O

hª A jOjª A i = limt! 1

ZOA (t)

ZA (t)where:

ZA (t) = hª 0A j exp(¡ tH )jª 0

A i

ZOA (t) = hª 0

A j exp(¡ tH=2)O exp(¡ tH=2)jª 0A i

Transfer matrix method

Leading-order actionTransfer matrix at leading order:e¡ H L O ¢ t = e¡

Rd3r H L O ¢ t

where the Hamilton density reads:

HLO = 12mN y~r 2N + 1

2(~r ¼)2 ¡ 12M 2

¼¼2 + gA2F¼

N y¿~¾N ¢¢~r ¼

+ 12C(N yN)(N yN ) + 1

2CI (N y¿N ) ¢(N y¿N )free nucleons free pions (instantaneous propagation) pion-nucleon coupling

nucleon-nucleon contact interactions

Contact interactions can be replaced by auxilliary fields interacting with a single nucleon using the identities:

e¡ 12 C N y N N y N = 1p 2¼

Zds e¡ 1

2 s2+sN yN p ¡ C ¢ t C < 0

e¡ 12 C I N y¿N ¢N y¿ N = 1p 2¼

ZdsI e¡ 1

2 sI ¢sI +isI ¢N y¿ N p C I ¢ t CI > 0

(for )

(for )

ZA (t) /Z

DsY

IDsI D¼I hª 0

A jT exp(¡ tH (s;sI ;¼I ))jª 0A i

Slater-det. of single-nucleon MEs

(path integral calculated by Monte Carlo)

Transfer matrix with auxilliary fields

Two-particle scattering: spherical wall method

interacting systemfree system

ampl

itud

e

Borasoy, E.E., Krebs, Lee, Meißner, EPJA 34 (2007) 185

Place a wall at sufficiently large R. Phase shifts & mixing angles can be extracted by measuring energy shifts from free-particle values.

free system interacting system

Phase shifts for a toy model potential

294912 processors, overall peak performance 1 petaflops

Blue Gene/P supercomputer @ Jülich Supercomputing Centre (JSC), FZ Jülich Computational equipment: JUGENE

Nucleon-nucleon phase shiftsE.E., Krebs, Lee, Meißner ‘10

9 LECs fitted to S- and P-waves and the deuteron quadrupole moment

Accurate results, deviations consistent with the expected size of higher-order terms

, np , pp

Coulomb repulsion and isospin-breaking effects taken into account

NNLO: Inclusion of the three-nucleon forceE.E., Krebs, Lee, Meißner ‘09

D E

The new LECs D and E fixed from the 3H binding energy & nd doublet S-wave.

Neutron-deuteron spin-1/2 channel 3H binding energy

3H-3He binding energy differenceE.E., Krebs, Lee, Meißner ‘10

Infinite-volume extrapolations via:Lüscher ’86

More nucleonsE.E., Krebs, Lee, Meißner ‘10

Simulations for 6Li, L=9.9 fm

Simulations for 12C, L=13.8 fm

Summary & outlookPart I: Modern theory of nuclear forces

qualitative & quantitative understanding of nuclear forces and few-N dynamicsaccurate description of 2N data, effects of the N3LO 3NF to be explored

Part III: Nuclear lattice simulationsformulated continuum EFT on space-time latticepromising results for NN scattering, light nuclei and the dilute neutron matter up to N2LO

Future:hypernuclei, electroweak reactions, heavier systems, higher precision, …

Part II: Selected applicationspion s/p-wave production in NN collisions analyzed at NLO/N2LO; various reaction are described simultaneously by adjusting a single counterterm extracted from Afb in np → dπ0 consistent with the value obtained using the Cottingham sum rule