Few-Nucleon Systems in Chiral EFT TU München, Evgeny Epelbaum, Ruhr-Universität Bochum Outline Part 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 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAA Evgeny Epelbaum

Chiral Perturbation Theory Weinberg, Gasser, Leutwyler, Bernard, Kaiser, Meißner, … QCD and chiral symmetry SU(2) L x SU(2) R invariantbreaks chiral symmetry small is approximately chiral invariant vacuum invariant only under SU(2) V SU(2) L x SU(2) R spontaneous symmetry breaking Goldston Bosons (pions) Chiral perturbation theory Goldstone bosons + matter fields π (140) ρ (770) ω (782) mass gap 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 internucleon potential [MeV] 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 forces Use 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 forces The 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 forces Renormalization of the potentials 1π-exchange should factorize out 120 time-ordered graphs cannot renormalize the potential ! Solution (E.E.’06) unique (!) result for Similar to the large- N c 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 symmetries Slavnov ’71; Djukanovic et al. ’05,’07; also Donoghue, Holstein, Borasoy ’98,’99 Regularization of the LS equation

LO: NLO: N 2 LO: N 3 LO: Two-nucleon force Ordonez et al. ’94; Friar & Coon ’94; Kaiser et al. ’97; E.E. et al. ’98,‘03; Kaiser ’99-’01; Higa, Robilotta ’03; … V 2N = V 2N +V 2N + V 2N + V 2N + … Chiral expansion of the 2N force: (0) (2) (3) (4) renormalization of 1 π -exchange renormalization of contact terms 7 LECs leading 2 π -exchange 2 LECs subleading 2 π -exchangerenormalization of 1 π -exchange sub-subleading 2 π -exchange 3 π -exchange (small) 15 LECs renormalization of contact terms renormalization of 1 π -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 N 2 LO Ordonez, Ray, van Kolck ’96; Kaiser, Gerstendorfer, Weise ‘98; Krebs, E.E., Meißner ’07,‘08

AyAy dσ/dΩ [mb/sr] N 2 LO N 3 LO PWA Entem, Machleidt ’04; E.E., Glöckle, Meißner ‘05 EE, Glöckle, Meißner Entem, Machleidt Two nucleons up to N 3 LO Deuteron observables Neutron-proton phase shifts at N 3 LO Neutron-proton scattering at 50 MeV b EM + [Nijm78; 1π; 1π+2π] Energy-dependent boundary condition Rentmeester et al. ’99, ‘03 Evidence of the 2π -exchange from the partial wave analysis

Few-nucleon forces up to N 3 LO D E van Kolck ’94, E.E. et al.‘02 N 2 LO N 3 LO corrections to the 3NF Ishikawa, Robilotta ‘07 Bernard, 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 E α Rozpedzik et al.‘06; Nogga et al., in prep. first 4NF contributions first nonvanishing 3NF

Differential cross section in elastic Nd scattering NLO N 2 LO 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; … N 2 LO Three nucleons up to N 2 LO

No-Core-Shell-Model results for 10 B, 11 B, 12 C and 13 N 2 LO Navratil et al., PRL 99 (2007) He and 6 NLO and N 2 LO Nogga et al., NPA 737 (2004) 236 More nucleons

Hot topics (work in progress) Bridging different reactions with the D-term Hanhart et al.’00, Baru et al.‘09, Filin et al.‘09 Park et al.‘03; Nakamura et al.’07 Ando et al.‘02,‘03 Gardestig & Phillips ’06, Lensky et al.‘05,‘07 Gazit, Quaglioni, Navratil, ’09 Effects of the N 3 LO 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, (2007) EFT with explicit Δ(1232) DOF Improved convergence of the EFT expansion! Preliminary calculation of the ring diagrams yield rather strong potentials… Isoscalar central potential r 12 [fm] r 23 [fm] V [MeV] Krebs, E.E., to appear

Pion production in NN collisions Considerably more challenging due to the appearance of a new „soft“ scale slower convergence of the chiral expansion (expansion parameter vs ) State-of-the-art Hybrid approach (EFT description of the 2N system for not yet available) Δ(1232) isobar plays an important role must be included as an explicit DOF s-wave pion production worked out up to NLO Cohen 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 NLO results for pp → d π +

p-wave π -production and the D-term Loops start to contribute at N 3 LO Simultaneous description of pn → pp π -, pp → pn π + and pp → d π + nontrivial consistency check of chiral EFT Up to N 2 LO, D is the only unknown LEC D N 2 LO In the future: implications for the 3NF and for weak reactions with light nuclei 3 S 1 for pp → pn π +, pp → d π + ; 1 S 0 for pn → pp π - Hanhart, van Kolck, Miller ’00; Baru, EE, Haidenbauer, Hanhart, Kudryavtsev, Lensky, Meißner ‘09 1 S 0 for pp → pn π +, pp → d π + ; 3 S 1 for pn → pp π - Reaction pp → d π + Near threshold: Natural units for D : ← dimensionless coefficient ~ 1

p-wave π -production and the D-term Reaction pn → pp π - The final pp relative mo- mentum is restricted to be: pp p-waves suppressed Data only available at expect only qualitative description... Reaction pp → pn π + The relevant amplitude ( 1 S 0 → 3 S 1 p) is suppressed compared to the dominant 1 D 2 → 3 S 1 p 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 systems isospin-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 3 He – 3 H Friar et al. PRC 71 (2005) CSB forward-backward asymetry MeV at TRIUMF (Opper et al. ’03) measured at / 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 → d π 0 & the np mass difference Niskanen ‘99; van Kolck et al. ’00; Bolton, Miller ‘09; Filin, Baru, E.E., Haidenbauer, Hanhart, Kudryavtsev, Meißner ‘09 gives rise to A fb, nonzero only for pn → d π 0 due to interference of IB and IC amplitudes The goal: use A fb measured at TRIUMF to extract the strong/em contributions to the neutron-to-proton mass shift. A 0 can be determined from the pionic deuterium lifetime PSI: Gasser, Leutwyler ’82 (based on the Cottingham sum rule) A 1 at LO in chiral EFT: IC amplitudes calculated at NLO Baru et al.’09 Our result: Lattice: Beane et al.’07

Nuclear Lattice Simulations Borasoy, 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) , arXiv: [nucl-th]

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 can probe bigger volumes Lattice QCD vs lattice chiral EFT signal/noise

Correlation-function for A nucleons: Slater determinants for A free nucleons Ground state energy: Expectation values of a normal ordered operator : where: Transfer matrix method

Leading-order action Transfer matrix at leading order: where the Hamilton density reads: 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: (for )

Slater-det. of single-nucleon MEs (path integral calculated by Monte Carlo) Transfer matrix with auxilliary fields

Two-particle scattering: spherical wall method interacting system free system amplitude 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

processors, overall peak performance 1 petaflops Blue Gene/P Jülich Supercomputing Centre (JSC), FZ Jülich Computational equipment: JUGENE

Nucleon-nucleon phase shifts E.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 force E.E., Krebs, Lee, Meißner ‘09 D E The new LECs D and E fixed from the 3 H binding energy & nd doublet S-wave. Neutron-deuteron spin-1/2 channel 3 H binding energy

3 H- 3 He binding energy difference E.E., Krebs, Lee, Meißner ‘10 Infinite-volume extrapolations via: Lüscher ’86

More nucleons E.E., Krebs, Lee, Meißner ‘10 Simulations for 6 Li, L=9.9 fm Simulations for 12 C, L=13.8 fm

Summary & outlook Part I: Modern theory of nuclear forces qualitative & quantitative understanding of nuclear forces and few-N dynamics accurate description of 2N data, effects of the N 3 LO 3NF to be explored Part III: Nuclear lattice simulations formulated continuum EFT on space-time lattice promising results for NN scattering, light nuclei and the dilute neutron matter up to N 2 LO Future: hypernuclei, electroweak reactions, heavier systems, higher precision, … Part II: Selected applications pion s/p-wave production in NN collisions analyzed at NLO/N 2 LO; various reaction are described simultaneously by adjusting a single counterterm extracted from A fb in np → d π 0 consistent with the value obtained using the Cottingham sum rule