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