Few-Nucleon Systems in Chiral EFT

1. EvgenyEpelbaum TU München, 31.05.2010 Few-Nucleon Systems in Chiral EFT EvgenyEpelbaum, 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

2. M [MeV] ω (782) 800 ρ (770) 600 mass gap 400 200 π (140) 0 Chiral Perturbation Theory Weinberg, Gasser, Leutwyler, Bernard, Kaiser, Meißner, … QCD and chiral symmetry SU(2)L xSU(2)Rinvariant breaks chiral symmetry small is approximately chiral invariant vacuum invariant only under SU(2)V SU(2)L x SU(2)R spontaneous symmetry breakingGoldston Bosons (pions) Chiral perturbation theory Goldstone bosons + matter fields most generalconsistent with the χ-symmetry of QCD powersofQ compute the amplitude via perturbative expansion in over (power counting): fix low-energy constants & make predictions...

3. Few nucleons: from ChPT to ChEFT Goldstone-bosonandsingle-nucleonsectors: weaklyinteractingsystemsChPT Two and more nucleons: strongly interacting systems Hierarchyofscales for non-relativistic ( ) nucleons: π-less EFT with local few-N interactions chiral EFT (cf. pNRQCD), instantaneous (nonlocal) potentials due to exchange of multiple Goldstone bosons rigorously derivable in ChPT Weinberg’s approach Weinberg ‘91,’92 zero-range operators irreducible contributions to be derived in ChPT internucleon potential [MeV] chiral expansion of multi-pion exchange enhanced reducible contributions must be summed up to infinite order separation between the nucleons [fm]

4. Derivation of nuclear forces Use canonical formalism to obtain the pion-nucleon Hamiltonian from the HB effective chiralLagrangian Decouple pions via a suitably chosen unitary transformation in Fock space: projectorontonucleonicstates energy-independent nuclear potentials projectorontostateswithmesons How to compute ? A convenient parametrizationin terms of (Okubo ’54): Requirethat The major problem is to solve the nonlinear decoupling equation.

5. with 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 Count powers of Q with Only non-renormalizableverices allowed (χ-symmetry) perturbative expansion Expansion in powers of Q/Λ: and Perturbative solution of the decoupling equation: The explicit form of the UT up to (Q/Λ)4isgiven in: E.E., EPJA 34(2007) 197 The same UT to be used to compute exchange currents, Kölling et al. ’09, to appear where

6. Derivation of nuclear forces Renormalization of the potentials 120 time-ordered graphs 1π-exchange should factorize out cannotrenormalizethe potential ! Solution(E.E.’06) unique (!) result for Similartothe large-Ncnuclear potential puzzle, Cohen et al. ‘02

7. Solving the Schrödinger equation Regularization of the LS equation DR difficult to implement numerically due to appearance of power-law divergencesPhillips 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 , grow with increasing momenta LS equation must be regularized & renormalized Renormalization à la Lepage Ordonez et al.’96; Park et al.’99; E.E. et al.’00,’04,’05; Entem, Machleidt ’02,’03 Choose & tune thestrengthsofto fit low-energy observables. generally, can only be done numerically; requires solving nonlinear equations for , self-consistencychecks via „Lepage plots“, residual dependence in observables survives

8. 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; … Chiralexpansionofthe 2N force: (0) (3) (2) (4) V2N = V2N+V2N+ V2N + V2N + … LO: 2 LECs NLO: renormalization of contact terms renormalization of 1π-exchange leading 2π-exchange 7 LECs N2LO: renormalization of 1π-exchange subleading 2π-exchange N3LO: renormalization of contact terms renormalization of 1π-exchange 15 LECs 3π-exchange (small) sub-subleading 2π-exchange + isospin-breakingcorrections… • van Kolck et al. ’93,’96; Friar et al. ’99,’03,’04; Niskanen ’02; Kaiser ’06; E.E. et al. ’04,’05,’07; … Resultsbased on EFT with explicit Δ(1232) degreesoffreedomavailableupto N2LO • Ordonez, Ray, van Kolck ’96; Kaiser, Gerstendorfer, Weise ‘98; Krebs, E.E., Meißner ’07,‘08

9. Two nucleons up to N3LO Entem, Machleidt ’04; E.E., Glöckle, Meißner ‘05 Neutron-proton scattering at 50 MeV Neutron-proton phase shifts at N3LO Ay dσ/dΩ [mb/sr] EE, Glöckle, Meißner Entem, Machleidt Deuteron observables N2LO N3LO PWA Evidence of the 2π-exchange from the partial wave analysis Rentmeester et al. ’99, ‘03 Energy-dependent boundary condition b EM + [Nijm78; 1π; 1π+2π]

10. Few-nucleon forces up to N3LO N2LO • first nonvanishing 3NF E D • van Kolck ’94, E.E. et al.‘02 N3LO corrections to the 3NF Ishikawa, Robilotta ‘07 Bernard, E.E., Krebs, Meißner’07; • parameter-free • χ-symmetry essential • nontrivialconstraintstrough • renormalizability • effects in 3N scattering • observables in progress... first 4NF contributions • E.E. ’06,’08 • parameter-free • contributes a few 100 keVto Eα Rozpedzik et al.‘06; Nogga et al., in prep.

11. Three nucleons up to N2LO E.E. et al.’02; Kistryn et al.’05; Witalaet al.’06; Ley et al.‘06; Stephan et al.’07; … Differential cross section in elastic Nd scattering N2LO NLO Polarization observables in elastic Ndscatering N2LO

12. More nucleons 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

13. Hot topics (work in progress) Bridging different reactionswiththe D-term Hanhart et al.’00, Baru et al.‘09, Filin et al.‘09 Ando et al.‘02,‘03 Gardestig & Phillips ’06, Lensky et al.‘05,‘07 Park et al.‘03; Nakamura et al.’07 Gazit, 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) Isoscalar central potential EFT with explicit Δ(1232) DOF Improved convergence of the EFT expansion! V [MeV] Preliminary calculation of the ring diagrams yield rather strong potentials… r23 [fm] Krebs, E.E., toappear r12 [fm]

14. Pion production in NN collisions Considerablymorechallenging due totheappearanceof a new „soft“ scale slowerconvergenceofthechiralexpansion (expansionparametervs ) State-of-the-art Hybrid approach (EFT descriptionofthe 2N systemfor not yetavailable) Δ(1232) isobar plays an importantrole must beincludedas an explicit DOF s-wavepionproductionworked out upto NLO Cohen et al.’96; Dmitrasinovic et al.’99; da Rocha et al.’00; Hanhart et al.’01,’02 results for pp → dπ+ NLO LO Proper separationofirred. contributionscrucial! Lensky et al. ’01 Nearthreshold: with

15. p-wave π-production and the D-term Hanhart, van Kolck, Miller ’00; Baru, EE, Haidenbauer, Hanhart, Kudryavtsev, Lensky, Meißner ‘09 Loops starttocontributeat N3LO D Upto N2LO, Distheonlyunknown LEC N2LO Simultaneousdescriptionof pn → ppπ-, pp → pnπ+andpp → dπ+nontrivialconsistency check ofchiral EFT In thefuture: implicationsforthe 3NF andforweakreactionswithlightnuclei 1S0forpp → pnπ+, pp → dπ+ ; 3S1forpn → ppπ- 3S1forpp → pnπ+, pp → dπ+ ; 1S0forpn → ppπ- Reactionpp → dπ+ Nearthreshold: Natural unitsforD: ← dimensionless coefficient ~ 1

16. p-wave π-production and the D-term Baru, EE, Haidenbauer, Hanhart, Kudryavtsev, Lensky, Meißner ‘09 Reactionpn → ppπ- The final pp relative mo-mentum is restricted to be: pp p-waves suppressed Data only available at expect only qualitative description... Data from TRIUMF and PSI New data at lower energies will be taken at COSY. Reactionpp → pnπ+ The relevant amplitude (1S0 → 3S1p) is suppressed compared to the dominant 1D2 → 3S1p amplitude minor sensitivity to the D-term… Overall bestresultsfor d ~ 3 Flammang et al.’98

17. Isospin breaking & few-N systems isospin-breaking hard / soft γ‘s + terms IB 2NF, 3NF worked out uptohighorders, long-rangecontributionslargelydrivenby , 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-breakingnuclearforcesandBE differences in 3He – 3H Friar et al. PRC 71 (2005) 024003 measuredatIUCF: @ 228.5 / 231.8 MeV Stephenson et al. ’03 Theoreticalanalysischallenging; firstestimationsyieldtheright order ofmagnitude. Gardestig et al. ’04; Nogga et al.’06 CSB forward-backwardasymetry in @ 279.5 MeVatTRIUMF (Opper et al. ’03)

18. np→ dπ0 & the np mass difference Niskanen ‘99; van Kolck et al. ’00; Bolton, Miller ‘09; Filin, Baru, E.E., Haidenbauer, Hanhart, Kudryavtsev, Meißner ‘09 The goal: use Afb measured at TRIUMF to extract the strong/em contributions to the neutron-to-proton mass shift. Gasser, Leutwyler ’82 (based on the Cottingham sum rule) gives rise to Afb, nonzero only for pn → dπ0 due to interference of IB and IC amplitudes A0 can be determined from the pionic deuterium lifetime measurement @ PSI: A1 at LO in chiral EFT: IC amplitudes calculated at NLO Baru et al.’09 Our result: Lattice: Beane et al.’07

19. NuclearLatticeSimulations 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) 142501, arXiv:1003.5697 [nucl-th]

20. Lattice QCD vs lattice chiral EFT Chiral EFT on thelattice Lattice QCD fundamental, theonlyparametersare canaccessfew- andmany-nucleonsystems can probe biggervolumes hardtogobeyondthe 2N system, e.g. for : LECs ( ) tobedeterminedfromthedataor LQCD signal/noise

21. Transfer matrix method Correlation-functionfor A nucleons: Slater determinantsforAfreenucleons Groundstateenergy: Expectationvaluesof a normal orderedoperator : where:

22. Leading-order action Transfer matrixatleading order: wherethe Hamilton densityreads: freenucleons freepions (instantaneouspropagation) pion-nucleoncoupling nucleon-nucleoncontactinteractions Contactinteractionscanbereplacedbyauxilliaryfieldsinteractingwith a singlenucleonusingtheidentities: (for ) (for )

23. Transfer matrix with auxilliary fields Slater-det. ofsingle-nucleon MEs (path integral calculatedby Monte Carlo)

24. Two-particle scattering: spherical wall method Borasoy, E.E., Krebs, Lee, Meißner, EPJA 34 (2007) 185 Place a wall atsufficiently large R. Phase shifts & mixinganglescanbeextractedbymeasuringenergyshiftsfromfree-particlevalues. interactingsystem freesystem interactingsystem freesystem Phase shiftsfor a toy model potential amplitude

25. Computational equipment: JUGENE Blue Gene/P supercomputer @ Jülich SupercomputingCentre (JSC), FZ Jülich 294912 processors, overallpeakperformance 1 petaflops

26. Nucleon-nucleon phase shifts E.E., Krebs, Lee, Meißner ‘10 9 LECs fittedto S- and P-wavesandthedeuteronquadrupolemoment Coulomb repulsionandisospin-breakingeffectstakenintoaccount Accurateresults, deviationsconsistentwiththeexpectedsizeofhigher-order terms , pp , np

27. NNLO: Inclusion of the three-nucleon force E.E., Krebs, Lee, Meißner ‘09 The new LECs D and E fixedfromthe3H bindingenergy & nddoublet S-wave. D E 3H binding energy Neutron-deuteron spin-1/2 channel

28. 3H-3He binding energy difference E.E., Krebs, Lee, Meißner ‘10 Infinite-volume extrapolations via: Lüscher ’86

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

30. Summary & outlook Part I: Modern theory of nuclear forces qualitative & quantitative understandingofnuclearforcesandfew-N dynamics accuratedescriptionof 2N data, effectsofthe N3LO 3NF tobeexplored Part II: Selected applications pion s/p-waveproduction in NN collisionsanalyzedat NLO/N2LO; variousreactionaredescribedsimultaneouslybyadjusting a singlecounterterm extractedfromAfb in np → dπ0 consistentwiththevalueobtainedusingtheCottinghamsumrule Part III: Nuclear lattice simulations formulatedcontinuum EFT on space-time lattice promising resultsfor NN scattering, lightnucleiandthediluteneutron matter upto N2LO Future: hypernuclei, electroweakreactions, heaviersystems, higherprecision, …