SLIDE 1 Jason D son D. H . Holt
Medium-Mass Nuclei from First Principles
2v 1q1v 2q 3p1h 4p2h 2v 1q1v 2q 3p1h 4p2h
SLIDE 2 protons
82! 50! 28! 28! 50! 82! 20! 8! 2! 2! 8! 126!
neutrons
Drip Lines and Magic Numbers: The Evolving Nuclear Landscape
20!
stable nuclei known nuclei Undiscovered Physics of exotic nuclei – era of coming decades What are the limits of nuclear existence? How do magic numbers form and evolve?
SLIDE 3 82! 50!
protons
50! 28! 20! 8! 2! 2!
8!
20! 126!
neutrons
Drip Lines and Magic Numbers: 3N Forces in Medium-Mass Nuclei
Heavie viest o st oxy xygen isotope n isotope
Otsuka, Suzuki, JDH, Schwenk, Akaishi, PRL (2010)
28!
π π π
Exploring the frontiers of nuclear science:
Worldwide joint experimental/theoretical effort What are the properties of proton/neutron-rich matter? What are the limits of nuclear existence? How do magic numbers form and evolve?
82!
−60 −40 −20 8 16 20 14
Neutron Number (N) Neutron Number (N) Neutron Number (N) Energy (MeV)
(a) Energies calculated from phenomenological forces SDPF-M USD-B Exp. (b) Energies calculated from G-matrix NN + 3N (∆) forces
NN NN + 3N (∆) 8 16 20 14 8 16 20 14
Exp. (c) Energies calculated from V NN + 3N (∆,N LO) forces
low k
2
NN NN + 3N (N LO) NN + 3N (∆)
Exp.
2
SLIDE 4 protons
82! 50! 28! 28! 50! 82! 20! 8! 2! 2! 8!
20!
126!
neutrons
Limits of Empirical Approach
N=3 =34 m magic gic n num umbe ber in c r in calc lcium ium?
42 44 46 48 50 52 54 56 58 1 2 3 4 5 6
2
+ Energy (MeV)
Experiment GXPF1 KB3G
(a) Phenomenological Forces
1
Mode Model de l depe pende ndent e nt extr xtrapola polations tions Dif iffic icult to e ult to extr xtract phy t physic sics: c s: contin
uum, de , deform
tion Aim im: pr : predic dictiv tive ab ab initio the initio theory f
r from
stability bility
What are the properties of proton/neutron-rich matter? What are the limits of existence of matter? How do magic numbers form and evolve? Worldwide joint experimental/theoretical effort!
SLIDE 5 protons
82! 50! 28! 28! 50! 82! 20! 8! 2! 2! 8!
20!
126!
neutrons
Drip Lines and Magic Numbers: 3N Forces in Medium-Mass Nuclei
New m w magic gic n num umbe bers in c s in calc lcium ium
JDH, Otsuka, Schwenk, Suzuki, JPG (2012) JDH, Menendez, Schwenk, JPG (2013)
44 48 52 56 60 64 68
Mass Number A
1 2 3 4 5 6
2
+ Energy (MeV)
NN NN+3N NN+3N (MBPT)
1
π π π
Exploring the frontiers of nuclear science:
Worldwide joint experimental/theoretical effort What are the properties of proton/neutron-rich matter? What are the limits of nuclear existence? How do magic numbers form and evolve?
SLIDE 6 protons
82! 50! 28! 28! 50! 82! 20! 8! 2! 2! 8!
20!
126!
neutrons
Drip Lines and Magic Numbers: 3N Forces in Medium-Mass Nuclei
New m w magic gic n num umbe bers in c s in calc lcium ium
JDH, Otsuka, Schwenk, Suzuki, JPG (2012) JDH, Menendez, Schwenk, JPG (2013)
44 48 52 56 60 64 68
Mass Number A
1 2 3 4 5 6
2
+ Energy (MeV)
NN NN+3N NN+3N (MBPT)
1
π π π
Exploring the frontiers of nuclear science:
Worldwide joint experimental/theoretical effort What are the properties of proton/neutron-rich matter? What are the limits of nuclear existence? How do magic numbers form and evolve?
SLIDE 7 protons
82! 50! 28! 28! 50! 82! 20! 8! 2! 2! 8!
20!
126!
neutrons
Drip Lines and Magic Numbers: 3N Forces in Medium-Mass Nuclei
New m w magic gic n num umbe bers in c s in calc lcium ium
JDH, Otsuka, Schwenk, Suzuki, JPG (2012) JDH, Menendez, Schwenk, JPG (2013)
44 48 52 56 60 64 68
Mass Number A
1 2 3 4 5 6
2
+ Energy (MeV)
NN NN+3N NN+3N (MBPT)
1
π π π
Exploring the frontiers of nuclear science:
Worldwide joint experimental/theoretical effort What are the properties of proton/neutron-rich matter? What are the limits of nuclear existence? How do magic numbers form and evolve?
LETTER
doi:10.1038/nature12522
Evidence for a new nuclear ‘magic number’ from the level structure of 54Ca
- D. Steppenbeck1, S. Takeuchi2, N. Aoi3, P. Doornenbal2, M. Matsushita1, H. Wang2, H. Baba2, N. Fukuda2, S. Go1, M. Honma4,
- J. Lee2, K. Matsui5, S. Michimasa1, T. Motobayashi2, D. Nishimura6, T. Otsuka1,5, H. Sakurai2,5, Y. Shiga7, P.-A. So
¨derstro ¨m2,
- T. Sumikama8, H. Suzuki2, R. Taniuchi5, Y. Utsuno9, J. J. Valiente-Dobo
´n10 & K. Yoneda2
LETTER
doi:10.1038/nature12226
Masses of exotic calcium isotopes pin down nuclear forces
- F. Wienholtz1, D. Beck2, K. Blaum3, Ch. Borgmann3, M. Breitenfeldt4, R. B. Cakirli3,5, S. George1, F. Herfurth2, J. D. Holt6,7,
- M. Kowalska8, S. Kreim3,8, D. Lunney9, V. Manea9, J. Mene
´ndez6,7, D. Neidherr2, M. Rosenbusch1, L. Schweikhard1,
- A. Schwenk7,6, J. Simonis6,7, J. Stanja10, R. N. Wolf1 & K. Zuber10
SLIDE 8
Approaches to Nuclear Structure
“The first, the basic approach, is to study the elementary particles, their properties and mutual interaction. Thus one hopes to obtain knowledge of the nuclear forces. If the forces are known, one should, in principle, be able to calculate deductively the properties of individual nuclei. Only after this has been accomplished can one say that one completely understands nuclear structure… The other approach is that of the experimentalist and consists in obtaining by direct experimentation as many data as possible for individual nuclei. One hopes in this way to find regularities and correlations which give a clue to the structure of the nucleus... The shell model, although proposed by theoreticians, really corresponds to the experimentalist’s approach.” –M. Goeppert-Mayer, Nobel Lecture Ab initio approach vs. phenomenological Theories of medium-mass nuclei largely empirical Purpose of lectures is to show how shell model can be based on the first approach!
SLIDE 9
To understand the properties of complex nuclei from first principles
The Challenge of Ab Initio Nuclear Theory
Two significant issues: Interaction Not well understood Not obtainable from QCD Too “hard” to be useful Multiple energy scales Many-body Problem Not ‘exactly’ solvable above Here we focus on shell model
A ∼ 20
SLIDE 10
To understand the properties of complex nuclei from first principles Two significant issues: Interaction Not well understood Not obtainable from QCD Too “hard” to be useful Multiple energy scales Many-body Problem Not ‘exactly’ solvable above Here we focus on shell model
A ∼ 20
The Challenge of Ab Initio Nuclear Theory
SLIDE 11 Two significant issues: Interaction Not well understood Not obtainable from QCD Too “hard” to be useful Multiple energy scales Many-body Problem Not ‘exactly’ solvable above Here we focus on shell model
How will we approach this problem: QCD ! NN (3N) forces ! Renormalize ! “Solve” many-body problem ! Predictions
To understand the properties of complex nuclei from first principles
A ∼ 20
The Challenge of Ab Initio Nuclear Theory
SLIDE 12 How will we approach this problem: QCD ! NN (3N) forces ! Renormalize ! “Solve” many-body problem ! Predictions
Nucleon-nucleon interaction Some history Anatomy of an NN interaction Construction from QCD? Ideas of Effective Field Theory Chiral EFT for nuclear forces Constraint by data To understand the properties of complex nuclei from first principles
The Challenge of Ab Initio Nuclear Theory
SLIDE 13 How will we approach this problem: QCD ! NN (3N) forces ! Renormalize ! “Solve” many-body problem ! Predictions
To understand the properties of complex nuclei from first principles Renormalizing NN Interactions Basic ideas of RG Low-momentum interactions Similarity RG interactions Benefits of low cutoffs G-matrix renormalization
The Challenge of Ab Initio Nuclear Theory
SLIDE 14 How will we approach this problem: QCD ! NN (3N) forces ! Renormalize ! “Solve” many-body problem ! Predictions
To understand the properties of complex nuclei from first principles Microscopic Valence- Space Interactions Model spaces Many-body perturbation theory (MBPT) Calculating effective interaction In-medium Similarity RG Monopole part of interaction Deficiencies of this approach
The Challenge of Ab Initio Nuclear Theory
SLIDE 15 How will we approach this problem: QCD ! NN (3N) forces ! Renormalize ! “Solve” many-body problem ! Predictions
To understand the properties of complex nuclei from first principles Three-Nucleon Forces Basic ideas – why needed? 3N from chiral EFT Implementing in shell model Relation to monopoles Predictions/new discoveries Connections beyond structure
a b c
The Challenge of Ab Initio Nuclear Theory
SLIDE 16 How will we approach this problem: QCD ! NN (3N) forces ! Renormalize ! Solve many-body problem ! Predictions
Nucleon-nucleon interaction Some history Anatomy of an NN interaction Construction from QCD? Ideas of Effective Field Theory Chiral EFT for nuclear forces Constraint by data To understand the properties of complex nuclei from first principles
Part I: The Nucleon-Nucleon Interaction
SLIDE 17 Interaction Between Two Nucleons
“In the past quarter century physicists have devoted a huge amount of experimentation and mental labor to this problem – probably more man- hours than have been given to any other scientific question in the history
–H. Bethe So let’s burn a few more man-hours of mental labor on this! To start, think to yourself what this should look like, and write it down… Ok, the nuclear potential as a function of the distance between nucleons… Got it!
SLIDE 18 Meson-Exchange Potentials: Yukawa
- First field-theoretical model of nucleon interaction proposed by Yukawa 1935
- Postulated nuclear force mediated by (NEW!) particle exchange
- Short range (~1fm) of nuclear force
= ⇒
SLIDE 19 Meson-Exchange Potentials: Yukawa
- First field-theoretical model of nucleon interaction proposed by Yukawa 1935
- Postulated nuclear force mediated by (NEW!) particle exchange
- Short range (~1fm) of nuclear force
New particle must be massive: Hint: r ∼ 1/m; m =?
= ⇒ ~c ≈ 197 MeV · fm
SLIDE 20 Meson-Exchange Potentials: Yukawa
- First field-theoretical model of nucleon interaction proposed by Yukawa 1935
- Postulated nuclear force mediated by (NEW!) particle exchange
- Short range (~1fm) of nuclear force
New particle must be massive:
r ∼ 1/m; m =?
= ⇒
V (~ r) = − f 2
π
m2
π
⇢ ~ 1 · ~ 2 + CT ✓ 1 + 3 mαr + 3 (mαr)2 ◆ S12(r) e−mπr mπr
SLIDE 21 Meson-Exchange Potentials: Yukawa
- First field-theoretical model of nucleon interaction proposed by Yukawa 1935
- Postulated nuclear force mediated by (NEW!) particle exchange
- Short range (~1fm) of nuclear force
New particle must be massive:
- Pion discovered 1947!
- Attractive, “long” range
r ∼ 1/m; m =?
= ⇒
V (~ r) = − f 2
π
m2
π
⇢ ~ 1 · ~ 2 + CT ✓ 1 + 3 mαr + 3 (mαr)2 ◆ S12(r) e−mπr mπr
SLIDE 22 V (~ r) = − f 2
π
m2
π
⇢ ~ 1 · ~ 2 + CT ✓ 1 + 3 mαr + 3 (mαr)2 ◆ S12(r) e−mπr mπr
Meson-Exchange Potentials: Yukawa
- First field-theoretical model of nucleon interaction proposed by Yukawa 1935
- Postulated nuclear force mediated by (NEW!) particle exchange
- Short range (~1fm) of nuclear force
New particle must be massive:
- Pion discovered 1947!
- Attractive, “long” range, spin dependent
r ∼ 1/m; m =?
= ⇒
SLIDE 23 Meson-Exchange Potentials: Yukawa
- First field-theoretical model of nucleon interaction proposed by Yukawa 1935
- Postulated nuclear force mediated by (NEW!) particle exchange
- Short range (~1fm) of nuclear force
New particle must be massive:
- Pion discovered 1947!
- Attractive, “long” range, spin dependent, non-central (tensor) part
Depends on spin, isospin, orientation of nucleons Does not conserve L2, S2, but does conserve parity Mixes different L states (but only differing by 2 units) r ∼ 1/m; m =?
= ⇒ = ⇒
V (~ r) = − f 2
π
m2
π
⇢ ~ 1 · ~ 2 + CT ✓ 1 + 3 mαr + 3 (mαr)2 ◆ S12(r) e−mπr mπr
SLIDE 24 Meson-Exchange Potentials: Yukawa
- First field-theoretical model of nucleon interaction proposed by Yukawa 1935
- Postulated nuclear force mediated by (NEW!) particle exchange
- Short range (~1fm) of nuclear force
New particle must be massive:
- Pion discovered 1947!
- Attractive, “long” range, spin dependent, non-central (tensor) part
- Successful in explaining scattering data, deuteron
- One pion is good, therefore more pions are better…
- Advanced to multi-pion theories in 1950’s – FAILED! Now what??
r ∼ 1/m; m =?
= ⇒
V (~ r) = − f 2
π
m2
π
⇢ ~ 1 · ~ 2 + CT ✓ 1 + 3 mαr + 3 (mαr)2 ◆ S12(r) e−mπr mπr
SLIDE 25 One-Boson Exchange Potentials
- Heavy mesons discovered in late 1950s – formed basis for new theories
- Intermediate range – attractive central, spin-orbit
~ ki ≡ 1 2 (~ p 0
i + ~
pi) ~ qi ≡ ~ p 0
i − ~
pi
V σ = g2
σNN
1 k2 + m2
σ
−1 + q2 2M 2
N
− k2 8M 2
N
− ~ L · ~ S 2M 2
N
!
SLIDE 26 One-Boson Exchange Potentials
- Heavy mesons discovered in late 1950s – formed basis for new theories
- Short range; repulsive central force, attractive spin-orbit
V ω = g2
ωNN
1 k2 + m2
ω
1 − 3 ~ L · ~ S 2M 2
N
!
SLIDE 27 One-Boson Exchange Potentials
- Heavy mesons discovered in late 1950s – formed basis for new theories
- Short range; tensor force opposite sign of one-pion exchange
V ρ = g2
ρNN
k2 k2 + m2
ρ
⇣ −2 ~ 1 · ~ 2 + S12(ˆ k) ⌘ ~ ⌧1 · ~ ⌧2
SLIDE 28 Parameterizing the NN Interaction
Starting from any NN-interaction, first solve: Lipmann-Schwinger scattering T-matrix equation: where Parameterized in partial waves α – in relative/center of mass frame (K,L) Fully-on-shell T-matrix directly related to experimental data tan δ(k) = −kT(k, k)
T α
ll0 (k, k0; K) = V α ll0(k, k0) + 2
π X
l00
Z 1 q2dq V α
ll00(k, q)
q k2 − q2 + iεT α
l00l0(q, k0; K)
T α
ll0 (k, k0; K) = hkK, lL; JST | T | k0K, l0L; JSTi
SLIDE 29
Phase shift is a function of relative momentum k; Figure shows s-wave Scattering from an attractive well potential Scattering from repulsive core: phase shift opposite sign
Constraining NN Scattering Phase Shifts
SLIDE 30 Parameterizing the NN Interaction
Starting from any NN-interaction, first solve: Lipmann-Schwinger scattering T-matrix equation: where Parameterized in partial waves α – in relative/center of mass frame (K,L) Fully-on-shell T-matrix directly related to experimental data tan δ(k) = −kT(k, k)
T α
ll0 (k, k0; K) = V α ll0(k, k0) + 2
π X
l00
Z 1 q2dq V α
ll00(k, q)
q k2 − q2 + iεT α
l00l0(q, k0; K)
T α
ll0 (k, k0; K) = hkK, lL; JST | T | k0K, l0L; JSTi
Note intermediate momentum allowed to infinity (but cutoff by regulators) Coupling of low-to-high momentum in V
x
SLIDE 31 Form of NN Interactions
Textbook nuclear potentials in coordinate (r) space (distance between nucleons)
2S+1LJ
SLIDE 32
Form of NN Interactions
Textbook nuclear potentials in coordinate (r) space (distance between nucleons) Hard core, intermediate-range 2π, long-range 1π exchange (OPE)
SLIDE 33
Form of NN Interactions
Textbook nuclear potentials in coordinate (r) space (distance between nucleons) Hard core, intermediate-range 2π, long-range 1π exchange Transform to momentum space via Fourier Transformation Strong high-momentum repulsion, low-momentum attraction Vl(k, k0) = 2 π Z 1 r2dr jl(kr)V (r)jl(k0r)
Fourier Trans
SLIDE 34 Form of NN Interactions
1 0.5 – 0.5 –1 (fm) 1 2 3 4
k (fm–1)
5 1 2 3 4 5
a b
Textbook nuclear potentials in coordinate (r) space (distance between nucleons) Hard core, intermediate-range 2π, long-range 1π exchange Transform to momentum space via Fourier Transformation Strong high-momentum repulsion, low-momentum attraction Vl(k, k0) = 2 π Z 1 r2dr jl(kr)V (r)jl(k0r)
Fourier Trans
SLIDE 35 Form of NN Interactions
Wait a minute… these potentials can’t really go to zero range/infinitely high energies; that would be QCD?
1 0.5 – 0.5 –1 (fm) 1 2 3 4
k (fm–1)
5 1 2 3 4 5
a b
Textbook nuclear potentials in coordinate (r) space (distance between nucleons) Hard core, intermediate-range 2π, long-range 1π exchange Transform to momentum space via Fourier Transformation Strong high-momentum repulsion, low-momentum attraction Vl(k, k0) = 2 π Z 1 r2dr jl(kr)V (r)jl(k0r)
SLIDE 36
NN Interaction from QCD?
Meson exchange in principle described in Quantum Chromodynamics (QCD) Low-energy region non-perturbative – treat in the context of Lattice QCD Directly from QCD Lagrangian, solve numerically on discretized space-time
SLIDE 37
NN Interaction from QCD?
Meson exchange described in QCD Low-energy region non-perturbative – treat in the context of Lattice QCD Directly from QCD Lagrangian, solve numerically on discretized space-time Lattice results give long-range OPE tail, hard core
SLIDE 38
NN Interaction from QCD?
Meson exchange described in QCD Low-energy region non-perturbative – treat in the context of Lattice QCD Directly from QCD Lagrangian, solve numerically on discretized space-time Lattice results give long-range OPE tail, hard core Not yet to physical pion mass – work in progress – so we’re done, right?
SLIDE 39
Unique NN Potential?
What does this tell us in our quest for an NN-potential? Expected form seems to be confirmed by QCD
SLIDE 40
OBE Potentials: Summary/Problems
First generation (1960-1990): Paris, Reid, Bonn-A,B,C High-precision potentials (1990s): Focus on precision ~40 parameters fit NN data ArgonneV18, Reid93, Nijmegen, CD-Bonn NN problem “solved” !! χ2/dof ≈ 1 χ2/dof ≈ 2
SLIDE 41
First generation (1960-1990): Paris, Reid, Bonn-A,B,C High-precision potentials (1990s): Focus on precision ~40 parameters fit NN data ArgonneV18, Reid93, Nijmegen, CD-Bonn NN problem “solved” !!
OBE Potentials: Summary/Problems
Hmmm… what if we were to go beyond two nucleons? χ2/dof ≈ 1 χ2/dof ≈ 2
SLIDE 42 OBE Potentials: Summary/Problems
First generation (1960-1990): Paris, Reid, Bonn-A,B,C High-precision potentials (1990s): Focus on precision ~40 parameters fit NN data ArgonneV18, Reid93, Nijmegen, CD-Bonn NN problem “solved” !! Many successes, but…
1) Difficult (impossible) to assign theoretical error 2) 3N forces (what are those??) not consistent with NN forces 3) No clear connection to QCD 4) Clear model dependence…
Hmmm… what if we were to go beyond two nucleons? χ2/dof ≈ 1 χ2/dof ≈ 2
SLIDE 43
Meson-Exchange Potentials and Phase Shifts
Further model dependence: scattering phase shifts of NN potentials
SLIDE 44
Meson-Exchange Potentials and Phase Shifts
Further model dependence: scattering phase shifts of NN potentials Remember, all have
That’s strange… why do they only agree to 350MeV? χ2/dof ≈ 1
SLIDE 45
Meson-Exchange Potentials and Phase Shifts
Further model dependence: scattering phase shifts of NN potentials Agree well up to pion-production threshold ~350MeV Most models don’t fit phase shifts above this energy – unconstrained
That’s strange… why do they only agree to 350MeV?
SLIDE 46
From QCD to Nuclear Interactions
How do we determine interactions between nucleons? Old view: Multiple scales complicate life No meaningful way to connect them Modern view: Ratio of scales – small parameters Effective field theory at each scale connected by RG Choose convenient resolution scale
Increased Resolution
H (Λ) = T + VNN (Λ) + V3N (Λ) + V4N (Λ) + · · ·
