Few-Nucleon Systems in Chiral EFT TU München, 31.05.2010 Evgeny Epelbaum, Ruhr-Universität Bochum...
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Few-Nucleon Systems in Chiral EFT TU München, 31.05.2010 Evgeny Epelbaum, Ruhr-Universität Bochum Outline Part I: Foundations Introduction Chiral expansion
Few-Nucleon Systems in Chiral EFT TU Mnchen, 31.05.2010 Evgeny
Epelbaum, Ruhr-Universitt 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
Slide 2
Chiral Perturbation Theory Weinberg, Gasser, Leutwyler,
Bernard, Kaiser, Meiner, 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 200 400 600
800 0 (140) (770) (782) mass gap M [MeV] powers of Q most general
consistent with the -symmetry of QCD compute the amplitude via
perturbative expansion in over (power counting): fix low-energy
constants & make predictions...
Slide 3
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 -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 Weinbergs approach
Slide 4
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.
Slide 5
Derivation of nuclear forces The decoupling equation can be
solved recursively utilizing chiral power counting (NDA): Powers of
can only be generated through LECs with Only non-renormalizable
verices allowed ( -symmetry) perturbative expansion Count powers of
Q with Expansion in powers of Q/: and Perturbative solution of the
decoupling equation: The explicit form of the UT up to (Q/) 4 is
given in: E.E., EPJA 34(2007) 197 The same UT to be used to compute
exchange currents, Klling et al. 09, to appear where
Slide 6
Derivation of nuclear forces Renormalization of the potentials
1-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
Slide 7
, grow with increasing momenta LS equation must be regularized
& renormalized Solving the Schrdinger equation Renormalization
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
Slide 8
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
-exchange renormalization of contact terms 7 LECs leading 2
-exchange 2 LECs subleading 2 -exchangerenormalization of 1
-exchange sub-subleading 2 -exchange 3 -exchange (small) 15 LECs
renormalization of contact terms renormalization of 1 -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 (1232) degrees of freedom available up
to N 2 LO Ordonez, Ray, van Kolck 96; Kaiser, Gerstendorfer, Weise
98; Krebs, E.E., Meiner 07,08
Slide 9
AyAy d/d [mb/sr] N 2 LO N 3 LO PWA Entem, Machleidt 04; E.E.,
Glckle, Meiner 05 EE, Glckle, Meiner 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; 1; 1+2]
Energy-dependent boundary condition Rentmeester et al. 99, 03
Evidence of the 2 -exchange from the partial wave analysis
Slide 10
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, Meiner 07; E.E. 06,08 parameter-free
-symmetry essential nontrivial constraints trough renormalizability
effects in 3N scattering observables in progress... parameter-free
contributes a few 100 keV to E Rozpedzik et al.06; Nogga et al., in
prep. first 4NF contributions first nonvanishing 3NF
Slide 11
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
Slide 12
No-Core-Shell-Model results for 10 B, 11 B, 12 C and 13 C @ N 2
LO Navratil et al., PRL 99 (2007) 042501 4 He and 6 Li @ NLO and N
2 LO Nogga et al., NPA 737 (2004) 236 More nucleons
Slide 13
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, 014006 (2007) EFT with explicit (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
Slide 14
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)
(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 +
Slide 15
p-wave -production and the D-term Loops start to contribute at
N 3 LO Simultaneous description of pn pp -, pp pn + and pp d +
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 +, pp d +
; 1 S 0 for pn pp - Hanhart, van Kolck, Miller 00; Baru, EE,
Haidenbauer, Hanhart, Kudryavtsev, Lensky, Meiner 09 1 S 0 for pp
pn +, pp d + ; 3 S 1 for pn pp - Reaction pp d + Near threshold:
Natural units for D : dimensionless coefficient ~ 1
Slide 16
p-wave -production and the D-term Reaction pn pp - 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 + 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, Meiner
09
Slide 17
Isospin breaking & few-N systems isospin-breaking hard /
soft 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) 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
Slide 18
np d 0 & the np mass difference Niskanen 99; van Kolck et
al. 00; Bolton, Miller 09; Filin, Baru, E.E., Haidenbauer, Hanhart,
Kudryavtsev, Meiner 09 gives rise to A fb, nonzero only for pn d 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 measurement @ 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
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
Slide 21
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
Slide 22
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
)
Slide 23
Slater-det. of single-nucleon MEs (path integral calculated by
Monte Carlo) Transfer matrix with auxilliary fields
Slide 24
Two-particle scattering: spherical wall method interacting
system free system amplitude Borasoy, E.E., Krebs, Lee, Meiner,
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
Slide 25
294912 processors, overall peak performance 1 petaflops Blue
Gene/P supercomputer @ Jlich Supercomputing Centre (JSC), FZ Jlich
Computational equipment: JUGENE
Slide 26
Nucleon-nucleon phase shifts E.E., Krebs, Lee, Meiner 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
Slide 27
NNLO: Inclusion of the three-nucleon force E.E., Krebs, Lee,
Meiner 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
Slide 28
3 H- 3 He binding energy difference E.E., Krebs, Lee, Meiner 10
Infinite-volume extrapolations via: Lscher 86
Slide 29
More nucleons E.E., Krebs, Lee, Meiner 10 Simulations for 6 Li,
L=9.9 fm Simulations for 12 C, L=13.8 fm
Slide 30
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 0 consistent with the value
obtained using the Cottingham sum rule
