Effective%20Theory%20of%20Shallow%20Nuclei

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Effective TheoryofShallow Nuclei
U. van Kolck
University of Arizona
Supported in part by US DOE
Background by S. Hossenfelder
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Outline

• Introduction
• Effective (Field) Theories
• Few-nucleon systems
• Halo nuclei
• Outlook

c
N.B. On wiki page

• suggested reading
• homework!

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MPI-Heidelberg
Nuclear Chart
see Bertulanis lecture
What are the nucleosynthesis reaction rates? What
are the limits of nuclear existence?
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Nuclear scales
perturbative QCD
1 GeV
see Schwenks lecture
Chiral EFT
Typical nuclei
150 MeV
10 MeV
Contact and Halo/cluster EFTs
Shallow nuclei
30 MeV
1 MeV
TODAY
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In classical mechanics
bound-state size
range of force
e.g.
square well
reduced mass
But not always true in quantum mechanics!
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(center-of-mass frame)
In quantum mechanics
elastic scattering (for simplicity)
(N.B. )
(conservation of energy)
given by certain probability amplitude the
scattering amplitude
Legendre polynomial
angular momentum
partial-wave amplitude
Two important properties

1. Poles
2. Low-energy expansion

bound states
effective range
shape parameter
resonances
(before any singularity)
Effective-Range Expansion
scattering length
b.s.
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square well, S-wave
e.g.
when
generic
fine-tuning
etc.
new scale emerges
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In the quantum world, one can have a b.s.
with size much larger than the range of the
force provided there is fine-tuning
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Feshbach resonance
Chiral EFT (incomplete) NLO
Beane, Bedaque, Savage v.K. 02
cf. Beane Savage. 03
Fukugita et al. 95
Lattice QCD quenched
cf. Beane, et al. 06
Regal Jin 03
unitarity limit
NN triplet scattering length
Large deuteron size because
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c
EFT the basic idea
more generally same argument for any short-range
potential
same
same wf tail
similar observables
systematic improvement
in the limit,
just like multipole expansion
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Effective Hamiltonian
fitted to
2-body data
3-body data
etc.
3-body interaction
no calculation in physics (except for TOE, if it
exists) is ever exact
Whatever can happen will happen
Wikipedia
quantum field theory
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Chen, Rupak Savage 99
fitted
LO EFT
NNLO EFT
predicted
NLO EFT
Nijmegen PSA
LO EFT
fitted
NLO EFT
Nijmegen PSA
predicted
NNLO EFT
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Rupak 01
fitted
NNNNLO EFT
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Bedaque, Hammer v.K. 99 00 Hammer Mehen
01 Bedaque et al. 03
Bedaque v.K. 97 Bedaque, Hammer v.K. 98
no 3-body force up to NNNNLO
3-body force already at LO
fitted
predicted
v.Oers Seagrave 67
NNLO EFT
Dilg et al. 71
v.Oers Seagrave 67
NLO EFT
Kievsky et al. 96
LO EFT
LO EFT
fitted nothing
predicted
Dilg et al. 71
QED-like precision!
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Light nuclei
Stetcu, Barrett v.K., 06
fitted to d, t, a ground-state binding energies
LO EFT
Harmonic-oscillator basis
fits
works within 10 !
see Rotureaus lecture
works within 30
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new scale leads to proliferation of shallow
states (near driplines)
loosely bound nucleons around tightly bound cores
Halo/Cluster states
separation energy
core excitation energy
core
p
n
n
p
p
p
n
n
p
n

e.g.
resonance at
bound state at
resonance at
resonance at
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Bertulani, Hammer v.K. 02
fitted
fitted
LO
NLO
NLO EFT
LO EFT
Arndt et al. 73
NLO EFT
fitted
scatt length only
NNNLO EFT
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Higa, Hammer v.K. 08
see Higas lecture
Extra fitting parameters
Bohr radius
none
fitted with
and
More fine-tuning!!!
fine-tuning of 1 in 10
fine-tuning of 1 in 1000!
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What next

• Coulomb interaction in higher waves
• e.g.
• three-body states
• e.g. 1)
• 2)
• reactions
• e.g.

Bertulani, Higa v.K., in progress
c.f.
Kong Ravndal 99
Rotureau, in progress
c.f.
Bedaque, Hammer v.K. 99
Higa Rupak, in progress
c.f.
Rupak 01
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SM
Forecast
QCD
lattice
Extrapolates to realistically small
Chiral EFT
Faddeev eqs,
Extrapolate to larger and larger
Contact EFT
NCSM,
Halo/cluster EFT
Low-energy reactions