The New World Out There:
Beyond the Two-Body Interaction
in Many-Body Systems
• Nuclear Models
• Choice of Interaction
- The 2-body interaction problem
- QCD derived nuclear interaction to the rescue
• Are A-body interactions important?
- Extended pairing interaction
- Application to heavy nuclei
• Q&A about CD-CE curve, Λ, and the
interaction terms…
Vesselin Gueorguiev
NNN
UC Merced
Simple Models, when applicable, Help
Understand Complex Systems
Liquid Drop Model (key: nuclear surface R(θ,ϕ)=R0(1+αλµYλµ(θ,ϕ))
Fermi Gas Model (key: particle statistics)
Energy-Density Functional (key: framework reformulation)
Independent Particles in a Mean Field
• Harmonic Oscillator, Square Well (key: exactly solvable)
• Nilsson Model (key: nuclear deformation & spin-orbit interaction)
• Wood-Saxon (key: finite range, diffuse surface, spin-orbit term)
 Microscopic Shell Models - “Deriving the Mean Field’’




• Self-consistent Mean Field (Hartree-Fock)
• Algebraic Models (key: exactly solvable -- Lie algebra based)
• Spherical Shell-Model with Realistic Interactions (key: matrix
elements are adjusted to reproduce the available nuclear structure data)
 Ab-initio No-Core Shell Model (key: highly accurate interactions fitted
to nucleon-nucleon data, properties of A-body systems are derived from the
few-body interactions)
• Other methods for few- and many-body systems (MonteCarlo methods, Coupled-clusters, cluster models …)
Nuclear Shell-Model Hamiltonian
H = #" iai+ai +
i
#V
+ +
a
ijkl i a j ak al
i, j,k ,l
+
i
where a and
"i and Vijkl
ai are fermion creation and annihilation operators,
are real and Vijkl = Vklij = $V jikl = $Vijlk
 Spherical shell-model basis states are eigenstates of the onebody part of the Hamiltonian - single-particle states.
 The two-body part of the Hamiltonian H is dominated by the
quadrupole-quadrupole interaction Q·Q ~ C2 of SU(3).
 SU(3) basis states - collective states - are eigenstates of H for
degenerate single particle energies ε and a pure Q·Q interaction.
!
pre-XXI century NN-Potentials
Usual NN-potentials are combination of:
Central scalar potential + spin-orbit (LS) + tensor force Sij
Argonne V18 potential, Phys. Rev. C 51, 38 (1995), v ij =
"v
p
(r
)
O
p ij
ij ,
p=1,18
Oijp=1,8 = 1, # i $ # j ,% i $ % j ,(# i $ # j )(% i $ % j ),Sij ,Sij (# i $ # j ),L $ S,L $ S(# i $ # j )
Oijp= 9,14 = L2 ,L2 (# i $ # j ),L2 (% i $ % j ),L2 (# i $ # j )(% i $ % j )),(L $ S) 2 ,(L $ S) 2 (# i $ # j )
Oijp=15,18 = Tij ,(% i $ % j )Tij ,Sij Tij ,(# iz + # jz ), with isospin breaking Tij = 3# iz# jz - (# i $ # j )
 Common phenomenological choices for ν(rij) are of Yukawa or Gaussian form.
 Some Modern interaction are giving up locality, e. g. CD-Bonn is nonlocal. In general, however, the form of ν(rij) (ν(pij) ) is derived from chiral
perturbation theory or meson exchange theory:
LN" i N = #g" i $ N (i% 5& i' " i )$ N
π ρ ω σ1 σ2
(
3
3 + exp(#m" r)
v 5 (r) = Tm " (r) = *1+
+
2m
r
(m
r)
m" r
)
,
"
"
The 2-body to A-body
problem!
2<A<5 Nuclear Systems
The highly accurate modern
NN-interactions fail for 2<A<5
Experiment
Entem& Machleidt, Phys. Rev. C 68, R041001(2003)
-8.482
-7.718
-28.295
Wiringa et. al, Phys. Rev. C 68,054006 (2003)
Nucleon Interaction from QCD
(Chiral Perturbation Theory)
Chiral perturbation theory (χPT) allows for controlled power series expansion
$ Q '*
Expansion parameter : &&
)) , Q + momentum transfer,
"
% #(
" # , 1 GeV, # - symmetry breaking scale
NNN - potentials such as Tucson-Melbourne & Urbana
!
Terms suggested within the
Chiral Perturbation Theory
R. Machleidt, D. R. Entem, nucl-th/0503025
Regularization is essential, which is quite
obvious within the Harmonic Oscillator
wave function basis.
Would the NNN interaction be
sufficient to describe nuclear
structure?
NN versus NN+NNN in 10B & 11B
Petr Navratil, V. G. Gueorguiev, J. P. Vary, W. E. Ormand, and A. Nogga,
Phys. Rev. Lett. 99, 042501 (2007), (nucl-th-0701038).
NN versus NN+NNN in 12C & 13C
Petr Navratil, V. G. Gueorguiev, J. P. Vary, W. E. Ormand, and A. Nogga,
Phys. Rev. Lett. 99, 042501 (2007), (nucl-th-0701038).
What about the
A-body interaction in
heavy nuclei?
Pb
Yb
Sn
V. G. Gueorguiev, Feng Pan and J. P. Draayer, (nucl-th-0403055)
“Application of the extended pairing model to heavy isotopes”,
The European Physical Journal A, Vol. 25 No. Supplement 1 (September 2005) p.515.
Effective Interactions in a
Finite Model Space
U=eiS
H
Heff=UHU-1
A
pi 2 A
H ="
+ "Vij (q)
i=1 2m
i< j
A
H eff
U=eiS
1 and 2-body
interaction terms
!
A
A
pi 2
="
+ "Vi1 i2 (q) + ...+ "Vi1 ...iA (q)
i=1 2m
i1 <i2
i1 <...<iA
1, 2, 3 … up to A-body
interaction terms
Effective Hamiltonian
in Second Quantized Form
A
H eff
1
+
+
= $" a a + L + $
V
a
Ka
$
#1 K# 2k #1
# k a# k+1 Ka# 2 k
2
(k!) #1 %...%# k
#
k=1
+
# # #
# k+1 %...%# 2k
 Calculating 3-body effective interactions is difficult, however
doable, and essential in understanding the structure of light
nuclei! P. Navratil and W. E. Ormand, Phys. Rev. C 68, 034305(2003).
 In principle, A-body effective Hamiltonians can be calculated
from realistic interactions, but their applications are yet ahead!
 Need for exactly solvable models to understand possible
implications of the A-body effective interactions!
Extended Pairing Problem
The Standard Pairing Problem
"
H = %# j n j $ % c jj' A +j A$j' ,
j
jj'
A +j = % ($) j$m a +jm a +j$m .
Exactly solvable cases:
m>0
 Constant pairing cjj’=G with non-degenerate single particle energies
εj ≠εj’ R. W. Richardson, Phys. Lett. 5, 82 (1963).
!
 Separable pairing cjj’=cjcj’ with degenerate single particle energies εj
=εj’ F. Pan, J. P. Draayer, W. E. Ormand, Phys. Lett. B 422, 1 (1998).
 Arbitrary Simple Lie algebra based Richardson-Gaudin models with
non-degenerate single particle energies, including proton-neutron pairing, J.
Dukelsky, V.G. Gueorguiev, P. Van Isacker (PRL96,072503(2006)).
Algebraic Models




Standard pairing is actually an SU(2) RG-model.
Proton-neutron T=1 pairing is SO(5) RG-model.
One can “easily” solve such RG-models exactly.
Possible applications:
 Nuclear physics,
 Condensed matter (superconductivity),
 High energy physics (pairing in quarks).
 Generalized Richardson-Gaudin models are new interesting set of
exactly solvable algebraic models.
rank
An su(n+1)
Bn so(2n+1)
Cn sp(2n)
Dn so(2n)
1
su(2) pairing
so(3)~su(2)
sp(2) ~su(2)
so(2) ~u(1)
2
su(3) Elliott
so(5) pnpairing
sp(4) ~so(5)
so(4)
~su(2)⊕su(2)
3
su(4) Wigner
so(7)⊂so(8)
FDSM
sp(6) Rowe &
Rosensteel
so(6)~su(4)
4
su(5)
so(9)
sp(8)
so(8) Evans,
FDSM
T=1 Proton-Neutron Pairing
SO(5) RG-model
 Nucleon pairs:
+
b"1+ = n + n + , b+1
= p + p + , b0+ = (n + p + + p + n + ) / 2
 u(1)xsuT(2) algebra:
!
T+1 = ( p + n + p + n ) / 2, T"1 = (n + p + n + p) / 2
 Pairing Hamiltonian:
H=
T0 = ( p + p + p + p) /2 " (n + n + n + n ) /2
N = p+ p + p + p + n + n + n + n
$ $" ( #
!
# = p,n jm
j
+
jm
# jm + # +jm # jm ) % g
 Generalized !
Richarsdon equations:
!
N
N #T
1
"i
1
1
='
+2 '
+'
,
g
z
#
e
e
#
e
w
#
e
i
$
%
(
$
i
% (&$ ) $
( =1
N
N #T
N
1
1
0 ='
+'
, E = ' e$
$ =1 e$ # w(
) (&( ) w( # w)
$ =1
$ $b
+
µ , jm µ , jm
µ =%1,0,1 j,m
b
Extended Pairing Model
A
H eff
1
+
+
= $" a a + L + $
V
a
Ka
$
#1 K# 2k #1
# k a# k+1 Ka# 2 k
2
(k!) #1 %...%# k
#
k=1
+
# # #
# k+1 %...%# 2k
 Nilsson single particle energies εjm play the role of 1-body
effective Hamiltonian that takes into account the nuclear
deformation due to the Q·Q interaction.
Simplifying assumption: equal coupling between different
configurations:
V"1" 2 = V"1" 2" 3" 4 = V"1 K" 2A = G
 The RESULT is Exactly solvable Hamiltonian:
F. Pan, V. G. Gueorguiev, J. P. Draayer, Phys. Rev. Lett. 92, 112503 (2004).
A
Hˆ = #"i n i $ G# bi+b j $ G# (k!)1 2
i
!
ij
k= 2
#b
+
i1
i1 %...%i2k
Kbi+k bik+1 Kbi2k , bi+ = ai&+ ai'+
Bethe Ansatz
Wavefunction:
k, " ; j1 K j m #
(" )
+
+
C
b
Kb
$ i1i2 ...ik i1 ik j1, j2 K jm
i1 <L<ik
C
!
(" )
i1 Kik
=
1
k
bi j1 K j m = 0 if i " j s
n=1
bi+ j1 K j m = 0 if i = j s
z(" ) # 2 % $in
Bethe Ansatz Equation:
!
#
! i1 <L<ik
G
k
#" in $z (% )
2
n=1
Eigenenergy: E k(" ) = z(" ) # G(k #1)
!
=1
Solving the Equations
E k(" ) = z(" ) # G(k #1)
1=
$
i1 <L<ik
k "1
= $
(# )
(# )
z " Ek
i1 <L<ik
G
E i1Lik #z ( " )
(# )
(# )
z
"
E
k
1
(# ) , G =
E i1Lik "z
k "1
k
E i1 Lik = 2 $ %in
n=1
1
G
!
min
Ei
1Lik
(1)
Ei
1Lik
( 2)
Ei
1Lik
( 3)
E i!
Li
1
k
5 pairs in
10 levels
e1=1.179
e2=2.650
e3=3.162
e4=4.588
e5=5.006
e6=6.969
e7=7.262
e8=8.687
e9=9.899
e10=10.20
Extended Pairing for Nuclei
Zero BE Reference
Pb
Yb
Sn
Zero BE Reference
• Nilsson levels using nuclear deformation (Audi & Wapstra (1995)).
• Pauli blocking for odd A nuclei.
• Set the scale of the single particle energies from near closed
shell system… (Nilsson BE is 3/4 E filling, Ring & Schuck)
Binding Energy of the Yb Isotopes
300
Pairing Strength
Relative BE (MeV)
250
Log(G)
0
200
Odd A Fit
-1
Exact value
-2
Even A Fit
-3
-4
152
162
172
182
A
150
BE Nilsson
BE theory
BE experiment
100
50
even A : G(A) = exp(662.2247 " 7.7912A + 0.0226A 2 )
odd A : G(A) = exp( 716.3279 " 8.4049A + 0.0244 A 2 )
0
152
157
!
162
167
A
172
177
182
Binding Energy of the Sn Isotopes
350
Relative BE (MeV)
300
250
Log(G)
1
200
Pairing Strength
Log[G] Exact
Log[G] fit -odd A
Log[G] fit -even A
0
-1
-2
-3
100
110
120
130
A
BE Nilsson
BE theory
150
BE experiment
100
even A : G(A) = exp( 365.0584 " 6.4836A + 0.0284 A 2 )
50
odd A : G(A) = exp( 398.2277 " 7.0349A + 0.0307A 2 )
0
100
105
!
110
115
A
120
125
130
Binding Energy of the Pb Isotopes
0
Relative BE (MeV)
-50
Log(G)
Pairing Strength
-100
1
0
-1
-2
-3
-4
180
Log[G] Exact
Log[G] fit -odd A
Log[G] fit -even A
A 200
BE Nilsson
BE theory
BE experiment
-150
-200
even A : G(A) = exp( 382.3502 " 4.1375A + 0.0111A 2 )
odd A : G(A) = exp( 391.6113 " 4.2374 A + 0.0114 A 2 )
-250
180
185
!
190
A
195
200
Binding Energy of the Pb Isotopes
0
Pairing Strength
Relative BE (MeV)
-50
Log(G)
1
366.7702
G(A) =
Dim(A) 0.9972
Log[G] Exact
0
Log[G] fit
-1
-2
-3
-4
-100
5
10
Log(Dim)
!
15
-150
-200
-250
180
BE Nilsson
BE theory
BE experiment
185
190
A
195
200
Binding Energy of the Sn Isotopes
25
Relative BE (MeV)
-75
Log(G)
2
-25
259.436
G(A) =
Dim(A) 0.9985
Pairing Strength
Log[G] Exact
1
Log[G] fit
0
-1
-2
1
3
!
5
Log(Dim)
-125
-175
BE Nilsson
BE theory
BE experiment
-225
-275
100
105
110
115
A
120
125
130
Extended Pairing
log(G(A))
Pb
Yb
!
Sn
G(A) ~
 Has nice and interesting systematic behavior!
The Extended Pairing model gives reasonable results…
!
Many-body interactions beyond two- & threebody!
"
Dim(A)
A-body interaction in nuclei
 Start getting used to the idea that we may
need A-body interactions to understand nuclear
properties across the nuclear chart…
Write your codes with A-body interactions in
mind!
Q&A about CD-CE curve, Λ, and the
NNN interaction terms…
Nucleon Interaction from QCD
(Chiral Perturbation Theory)
Chiral perturbation theory (χPT) allows for controlled power series expansion
$ Q '*
Expansion parameter : &&
)) , Q + momentum transfer,
"
% #(
" # , 1 GeV, # - symmetry breaking scale
NNN - potentials such as Tucson-Melbourne & Urbana
!
Terms suggested within the
Chiral Perturbation Theory
R. Machleidt, D. R. Entem, nucl-th/0503025
Regularization is essential, which is quite
obvious within the Harmonic Oscillator
wave function basis.
Lambda=550 MeV
Lambda=500 MeV
3H, 3He,
and 4He Binding Energy
First values for {CD,CE}=({-1.11,-0.66}, {8.14,-2.02}) by A. Nogga et al, Nucl. Phys. A737, 236 (2004)
in momentum space with Exp[-((p2+q2)/Λ2)2] cut-off function. We keep the same regulator for the
contact terms but use Exp[-((Q2)/Λ2)2] cut-off function for the V2π and work in coordinate space.
A
A
B
B
Along the appropriate curve, all CD-CE points reproduce BE of 3H & 3He
within 0.1 keV. Our {CD,CE} points for N3LO were deduced in Nmax=16 model
space for 4He: {0.476,-0.167}A and {7.686,-0.949 }B.
Along the CD-CE Curve (average)
BE deviation is within the accuracy
of the kinetic energy for equal mass
nucleons mn=mp=2µpn.
"T = T
"m
# 26keV (# 37 $ 0.7 $10%3 )
m
3N-contact 〈VE〉 >0 (less binding)
2N contact 1π−exchange, 〈VD〉 {-,+,-}
2π−exchange 〈V
2π〉 < 0 (more binding)
T=3/2 channel has been neglected in the calculations.
!
3H
Rch
2
and 3He Charge Radii
"N% 2
= rp + R p + $ ' Rn
#Z&
2
2
Rp={0.854, 0.875, 0.895} [fm]
R2n={-0.12,-0.11} [fm2]
!
Relating the theoretical point
proton radius rp to the
experimental charge radius Rch
is very sensitive to the chosen
proton charge radius Rp.
4He
A
Charge Radius
B
 Point A gives definitely better charge radius for 4He.
Along the CD-CE Curve (average)
BE deviation is within the accuracy
of the kinetic energy for equal mass
nucleons mn=mp.
"T = T
"m
# 26keV (# 37 $ 0.7 $10%3 )
m
m=(Zmp+(A-Z)mn)/A
PRC60, 044304(1999) G. P. Kamuntavicius, P.
Navratil, B. R. Barrett,G. Sapragonaite, and R.
K. Kalinauskas
T=3/2 channel has been neglected in the calculations.
!
ab-initio no core with
QCD derived nuclear
interaction
• Pushing the limits using the full N3LO,
• We may need codes with explicit mp & mn.
• We will need efficient A-body codes.