chiral eft for nuclear forces - forschungszentrum...
TRANSCRIPT
Modern Theory of Nuclear ForcesModern Theory of Nuclear Forces
PAX Meeting, Stockholm, 15.06.2010
Evgeny Epelbaum, Ruhr-Universität Bochum
OutlineOutlineOutlineOutline
Evgeny Epelbaum
Chiral EFT for nuclear forcesChiral EFT for nuclear forcesChiral EFT for nuclear forcesChiral EFT for nuclear forces
Some hot topics (work in progress)Some hot topics (work in progress)Some hot topics (work in progress)Some hot topics (work in progress)
Deuteron breakup at COSYDeuteron breakup at COSYDeuteron breakup at COSYDeuteron breakup at COSY
SummarySummarySummarySummary
Chiral EFT Chiral EFT forfor NuclearNuclear ForcesForcesE.E., Hammer, Meißner, Rev. Mod. Phys., in press
Kalantar-Nayestanaki, E.E., Nucl. Phys. News. 17 (07) 22
E.E., Prog. Part. Nucl. Phys. 57 (06) 654
Intro & motivation
Two nucleons at N3LO
Three nucleons at N2LO
Light nuclei
One-pion exchange as the longest-range contribution(Yukawa ’35)
Phenomenological parametrization of the shorter-range terms.Available vectors and isovectors . Hermiticity + isospin conservation + invariance under rotations, spacereflections and time reversal only 12 structures left for :
The corresponding scalar functions of are to be fit to data.
np scattering
at 96 MeV
nearly perfect description of pp and np
data below ~ 350 MeV with χ2/datum ~ 1
typically about 40 - 50 parameters
Modern phenomenological potentials: AV18, CD Bonn, Nijm I/II, Reid93, …
Conventional approach: NN forceConventional approach: NN force
spin-momentum isospin
Conventional approach: 3N forceConventional approach: 3N force
„…replacement of field interactions by two-body action-at-a-distance potentials isa poor approximation in nuclear physics.“
Indeed, 3N calculations based on NN potentials show evidence for missing 3N forces(e.g. the underbinding of 3H by about 1 MeV).
Phenomenological parametrization of the 3NF seems not feasible: too many possible structures (> 100)too scarce data base availabletoo involved calculations
need guidance from a theory
Fujita-Miyazawa, Brazil,
Tucson-Melbourne,
Urbana IX,
Illinois, …
Three-nucleon force models
SuccessesSuccesses and and failuresfailures
V2N
V2N+V3N
65 MeV
V2N
V2N+V3N
V2N, V2N+V3N
Space STAR, 13 MeV
Inclusion of the 3NF sometimes leads to improvements, sometimes — not. Situati-on, in part, chaotic.
Elastic scattering observables Deuteron breakup
Spectra of light nuclei
135 MeV
Need a theoretical approach which would:be based on QCD, yield consistent many-body forces,be systematically improvable,allow for error estimation
chiral chiral chiral chiral effectiveeffectiveeffectiveeffective fieldfieldfieldfield theorytheorytheorytheory
Chiral Perturbation TheoryChiral Perturbation TheoryWeinberg, Gasser, Weinberg, Gasser, LeutwylerLeutwyler, , BijnensBijnens, Bernard, Bernard, Kaiser, , Kaiser, MeiMeißßnerner, , EckardEckard, , PichPich, , ……
asymptotically observed sta-tes as effective DOF → EFT
spontaneously broken appro-ximate χ-symmetry of QCD plays a crucial role
At very low energy, the only relevant DOF are Goldstone bosons.
low-energy constants
Gasser, Leutwyler ‘84
200
400
600
800
0
ππππ (140)
ρρρρ (770)ωωωω (782)
mas
s g
ap
M [MeV]
Derivative expansion:
Scattering amplitude can be calculated via a perturbative expansion in and
soft (Mππππ) and hard (Mρρρρ) scaleswell separated
HierarchyHierarchyHierarchyHierarchy of of of of scales for nonscales for nonscales for nonscales for non----relativistic ( ) nucleonsrelativistic ( ) nucleonsrelativistic ( ) nucleonsrelativistic ( ) nucleons::::
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
pote
ntia
l [M
eV]
separation between the nucleons [fm]
chiral expansion of multi-pion exchange
zero-range operators
Few nucleons: from ChPT to Few nucleons: from ChPT to ChEFTChEFT
irreducible contributions to be derived in ChPT
enhanced reducible contributions must be summed up to infinite order
Weinberg’s approach
Meson & Meson & Meson & Meson & singlesinglesinglesingle----baryonbaryonbaryonbaryon sectorssectorssectorssectors: chiral : chiral : chiral : chiral perturbationperturbationperturbationperturbation theorytheorytheorytheoryWeinberg, Gasser, Leutwyler, Bijnens, Bernard, Kaiser, Meißner, Ecker, Pich, …
FewFewFewFew////manymanymanymany baryonbaryonbaryonbaryon systemssystemssystemssystems:::: non-perturbative physics (2H, 3H, 3He, …).
Explains the observed hierarchy of nuclear forces:
(values from: Pudliner et al., Phys. Rev. Lett. 74 (95) 4396)
Two-nucleon force Three-nucleon force Four-nucleon force
Q0
Q2
Q3
Q4
Hierarchy of nuclear forcesHierarchy of nuclear forces
LO:LO:LO:LO:
NLO:NLO:NLO:NLO:
NNNN2222LO:LO:LO:LO:
NNNN3333LO:LO:LO:LO:
TwoTwo--nucleon forcenucleon 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 Chiral Chiral Chiral expansionexpansionexpansionexpansion of of of of thethethethe 2N force:2N force:2N force: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 LECs renormalization 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
24 LECs fit to np data
Ay
dσ/d
Ω[m
b/sr
]
N2LO N3LOPWA
Entem, Machleidt ’04; E.E., Glöckle, Meißner ‘05
EE, Glöckle, MeißnerEntem, Machleidt
Two nucleons up to NTwo nucleons up to N33LOLO
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, ‘03
Evidence of the Evidence of the Evidence of the Evidence of the 2ππππ----exchange from the partial wave analysisexchange from the partial wave analysisexchange from the partial wave analysisexchange from the partial wave analysis
ThreeThree--nucleon force at Nnucleon force at N22LOLOvan Kolck ’94; E.E. et al. ’02
to be determined from >2N data such as e.g. 3H BE and Nd scattering length
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 NThree nucleons up to N22LOLO
EN = 22:7 MeV
EN = 90 MeV
Deuteron breakup at NDeuteron breakup at N22LOLOd + p → p + p + n @ 19 MeV
p dpp
n
α
The so-called Symmetric-Constant-Energy geometry
Ley et al.’06; data taken at the Cologne FN Tandem accelerator
d + p → p + p + n @ 130 MeV
Kistryn et al.’05, KVI data
N2LO
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 nucleonsMore nucleons
Hot Hot topicstopics
Three-nucleon force at N3LOThree-nucleon force and the D-termChiral EFT with explicit Δ(1232) isobarHeavier systems: nuclear lattice simulations
ThreeThree--nucleon force at Nnucleon force at N33LOLOIshikawa, Robilotta ’07; Bernard, E.E., Krebs, Meißner ’07; to appear
no unknown LECs contribute N3LO corrections to the 3NF are parameter-free!analytic expressions available both in coordinate and momentum space
ThreeThree--nucleon force at Nnucleon force at N33LOLOIshikawa, Robilotta ’07; Bernard, E.E., Krebs, Meißner ’07; to appear
For example, for ring diagrams one obtains:
Complicated expressions in momentumspace (3-point function), much morecompact expressions in r-space
1
2
3
Bernard, E.E., Krebs, Meißner, PRC 77 (08) 064004
spin-space operators Yukawa-like functionsknown constants isospin operators
Next step: partial-wave decomposition
Analytical calculations tedious (more than 100 operators!) not feasible…
Work in progress: numerical partial wave decomposition
can be reduced to 5 dim. integral
matrix, ~ 105x 105 depends on spin & isospin
a feasible task for modern supercomputers, work in progress…
Golak et al.’09
FFFFirst irst irst irst nonvanishingnonvanishingnonvanishingnonvanishing 3N3N3N3N----force contributionforce contributionforce contributionforce contributionappears at nextappears at nextappears at nextappears at next----totototo----nextnextnextnext----totototo----leading orderleading orderleading orderleading order
ThreeThree--nucleon force and the Dnucleon force and the D--termterm
D Evan Kolck ’94, E.E. et al.‘02
Nd →→→→ NdNN →→→→ NNππππ
dππππ →→→→ γγγγNN NN →→→→ d l ννννl
DDDD----term figures prominently in various reactionsterm figures prominently in various reactionsterm figures prominently in various reactionsterm figures prominently in various reactions
E.E. et al.’02,
Nogga et al.’05,
Navratil et al.‘07
Hanhart et al.’00,
Baru et al.’09,
Filin et al.‘09,
…
Park et al. ’03,
Ando et al.‘02,‘03,
Nakamura et al.‘07
Lensky et el.‘05,‘07
Gardestig et al.‘06
pp--wave wave ππ--production and the Dproduction and the D--termterm
Loops start to contribute at N3LO
Simultaneous description of pn → ppππππ----, pp → pnππππ++++ and pp → dππππ++++ nontrivial consistency check of chiral EFT
Up to N2LO, D is the only unknown LECD
N2LO
3S1 for pp → pnππππ++++, pp → dππππ+ + + + ; 1S0 for pn → ppππππ----
Hanhart, van Kolck, Miller ’00; Baru, EE, Haidenbauer, Hanhart, Kudryavtsev, Lensky, Meißner ‘09
1S0 for pp → pnππππ++++, pp → dππππ+ + + + ; 3S1 for pn → ppππππ----
pp → dππππ++++: angular asymmetry pn → ppππππ----: diff. cross section pp → pnππππ++++: angular asymmetry
overall best results for d ~ 3
Δ-less theory
Two-nucleon force in EFT with and without ΔΔ-full theory: additional graphs
NLO
N2LO
LO
Chiral EFT with explicit Chiral EFT with explicit ΔΔ(1232)(1232)Δ couples strongly to the πN-system and has low excitation energy which sets thescale of certain LECs in
expect much better convergence in the formulation based on Preis to pay: calculations much (!) more involved, more LECs…
Ordonez, Ray & van Kolck.’96,Kaiser, Gerstendorfer & Weise ‘98
Krebs, E.E., Meißner EPJA 32 (2007) 127
ΔΔ--isobarisobar & & thethe twotwo--nucleonnucleon forceforceKrebs, E.E., Meißner EPJA 32 (2007) 127
2222ππππ----exchangeexchangeexchangeexchange up to up to up to up to NNNN2222LOLOLOLOwith Δ without Δ
LO (OPE)
NLO
NNLONLO-Δ
NNLO-Δ
3333FFFF3333 partial partial partial partial wavewavewavewave up to up to up to up to NNNN2222LOLOLOLO
a much better convergence for the potential when Δ is included explicitly
clearly visible in NN peripheral waves
NLO
NNLO
ΔΔ--isobarisobar & & thethe threethree--nucleonnucleon forceforceE.E., Krebs, Meißner NPA 806 (2008) 65
NLO
N2LO
Δ-less theory Δ-full theory: additional graphs
N3LO
Δ contributions at N3LO are large!
Long-range part is parameter free
Much richer spin/isospin structurecompared to the Illinois model
Complete analysis still to be done
isoscalar central potential
Illinois model
Ring (Δ), chiral EFT
NuclearNuclear LatticeLattice SimulationsSimulations
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]
Pions and nucleons as point-like particles on the lattice (typical lattice size ~ 20 fm)
systematic ab initio approach to few- & many-nucleon systems
Use Monte Carlo to evaluate path integral E 0
A = limt ! 1
h¡
ddt
ln ZA (t)i
ZA (t) = hª 0A j exp(¡ tH )jª 0
A i
Lattice simulations of light nucleiLattice simulations of light nucleiE.E., Krebs, Lee, Meißner ‘10
Simulations for 6Li, L=9.9 fm
Simulations for 12C, L=13.8 fm
Deuteron Deuteron breakupbreakupat COSYat COSY
Deuteron breakup at COSYDeuteron breakup at COSY
Nd breakup as a laboratory for 3N forces
For the first time, we now have the theory & machinery to derive 3NF in a systematic way, calculate corrections and test their effects in 3N continuum.
Energy range 30…50 MeV is ideal to test chiral 3NF, no systematic breakup measure-ments exist!
New, independent way to fix the LECs D and E
interesting tests of the theory (e.g. MuSun experiment@ PSI)
implications for spectra of (light) nuclei, pion production and electroweak reactions;
PWD, Faddeev equations
Deuteron breakup at COSYDeuteron breakup at COSY
Find phase space regions sensitive to particular isospin-spin-space operators
guidence for the measurementtests for the theory
from: Kuros-Zolnierczuk et al., Few-Body Syst. 34 (2004) 259
3NF effects in pd breakup at EN=135 MeV
More details in the talk by Pia
Thörngren Engblom
Combined with measurements at higher energies, will allow to test theoretical pre-dictions for energy dependence of polari-zation observables
SummarySummary
Chiral EFT allows for quantitative understanding of the 3NF and predictsrich isospin-spin-space structure
The proposed Nd breakup measurement @ COSY will allow to test novelchiral 3NF and provide a new way to determine the LECs D and E
Implications for spectra of light nuclei, pion production, electroweakreactions, etc.
Important step towards precision nuclear physics