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
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