1. Chiral Nuclear Forces with Delta Degrees of Freedom
Hermann Krebs
FZ Jülich & Universität Bonn
November 10, 2008, PANIC 08, Eilat, Israel
In collaboration with E. Epelbaum and U. Meißner
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2. Outline ChPT and low energy QCD
Δ-less few nucleon forces within chiral EFT
Expectation from inclusion of Δ explicitely
Isospin-conserving nuclear forces with explicit Δ
Isospin-violating nuclear forces with explicit Δ
Summary & Outlook

3. ChPT and low energy QCD
Spontaneous + explicit (by small quark masses) breaking of chiral symmetry in QCD
Existence of light weakly interacting Goldstone bosons
Systematic description of QCD by ChPT in low energy sector
Free parameters of QCD
Free parameters of ChPT
Computable by Lattice QCD / fit to data

4. Nucleon-nucleon force up to N3LO
Ordonez et al. ’94; Friar & Coon ’94; Kaiser et al. ’97; Epelbaum et al. ’98,‘03; Kaiser ’99-’01; Higa et al. ’03; …
Chiral expansion for the 2N force:
V2N = V2N +V2N + V2N + V2N + …
(0)
(2)
(3)
(4)
LO:
2 LECs
NLO:
renormalization of 1π-exchange
7 LECs
renormalization of contact terms
leading 2π-exchange
N2LO:
subleading 2π-exchange
renormalization of 1π-exchange
N3LO:
renormalization of 1π-exchange
15 LECs
renormalization of contact terms
sub-subleading 2π-exchange
3π-exchange (small)
+ 1/m and isospin-breaking corrections…

5. Neutron-proton phase shifts up to N3LO np scattering at 50 MeV
dσ/dΩ [mb/sr]
N3LO
N2LO
NLO Ay
N2LO
N3LO
PWA
Deuteron binding energy & asymptotic normalizations As and ηd
Entem & Machleidt ‘03; Epelbaum, Glöckle & Meißner ‘05

6. Delta-less effective potential
Standard chiral expansion:
Delta contributions encoded in LECs
(Bernard, Kaiser & Meißner ’97)
Enlargement due to
Delta contribution
Delta-resonance saturation
Convergence of EFT potential
Expectation from inclusion of Δ explicitely
more natural size of LECs better convergence applicability at higher energies

7. Ordonez et al.’96, Kaiser et al. ‘98
Few-nucleon forces with the Delta
Isospin-symmetric contributions
Ordonez et al.’96, Kaiser et al. ‘98
H.K., Epelbaum & Meißner ‘07

8. Delta excitations and the three-nucleon force
Epelbaum, H.K., Meißner, Nucl. Phys. A806 (2008) 65
standard chiral EFT
Including Δ as an explicit DOF LO
van Kolck’96
Fujita & Miyazawa ‘57
NLO
N2LO
The LO NNNΔ contact interaction vanishes due to the Pauli principle
the LECs and are not saturated by the delta.
No contributions from subleading 2π –exchange due to at the vertex.
The entire effect of the Δ is given by a partial shift of the N2LO TPE 3NF to NLO…

9. Fit of LECs to πN scattering data
Epelbaum, H.K., Meißner, Eur. Phys. J. A32 (2007) 127
+ crossed graphs
Two possible values for
fit 1:
(SU(4), large Nc)
fit 2:
(Fettes & Meißner ‘01)
Results of the fit
S- and P-wave threshold parameters
Improved description of P-wave parameters when is included
Strongly reduced values for
Resulting depend strongly on
while the thresh. param. do not

10. NN potential with explicit Δ
Chiral 2π- exchange potential up to NNLO
with Δ
without Δ
with Δ
without Δ
NLO
NNLO
NLO
NNLO
Advantages when Δ is included explicitly
Dominant contributions already at NLO
Much better convergence in all potentials

11. 3F3 partial waves up to NNLO with and without Δ
Virginia Tech
Nijmegen
NNLO-Δ
NNLO-Δ
NLO-Δ
NLO-Δ
LO (OPE)
(calculated in the first Born approximation)

12. Δ-mass splitting in chiral EFT
Epelbaum, H.K., Meißner, Nucl. Phys. A806 (2008) 65
Equidistant splitting: strong & em
Non-equidistant splitting: em
Most recent data from the PDG:
In addition, PDG‘s recommended value for the average mass:
On the other hand:
(Arndt et al. ’06)
use:
Alternatively, use & the QM relation:
(Rubinstein et al. ’67)

13. Isospin-breaking NN potential
Epelbaum, H.K., Meißner, Phys. Rev. C77 (2008)
2π – exchange contributions with expicit Δ
Charge-symmetry-breaking 2π–exchange potential
Δ-full EFT, LO
Δ-less EFT, NLO
full
Similar contr. to
in the Δ-less and Δ-full EFT
Sizeable deviation in
contr. for
Strong cancelations between
and terms
Big contributions beyond the subleading corrections in the Δ-less EFT

14. Summary
Nuclear forces with explicit Δ analyzed up to NNLO in chiral EFT
The unknown LECs fixed from πN S- & P-waves; ci strongly reduced
A more natural convergence pattern for nuclear forces.
The most important effects of the Δ are well approximated by the saturation of LECs ci in the Δ-less theory
Leading isospin-violating Δ-contr. to nuclear forces are worked out
Delta mass splitting analyzed at LO
Found effects beyond resonance saturation of ci
Outlook
Few nucleon observables within the Δ-full chiral EFT at NNLO
Extension to N3LO: important effects
expected for 3NF and 4NF