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Nuclear Forces / DFT for Nuclei I

Dick Furnstahl

Department of PhysicsOhio State University

August, 2008

I. Overview of EFT/RG.

II. Chiral effective field theory.

III. RG for nuclear forces. EFT for many-body systems.

IV. EFT/DFT for dilute Fermi systems.

V. Ab initio nuclear DFT.

Overview Potential Resolution EFT 3-Body

Outline

Overview: The Big Picture

Nuclear Potential in Momentum Space

Quantum Resolution

A Simple EFT for Short-Range Repulsion

Three-Body (and Higher-Body) Forces

Dick Furnstahl Nuclear Forces/DFT


Getting the Most From these Lectures

Ask questions, ask questions, ask questions!Many opportunities before/during/after lectures, afternoonsessions, lunch, . . .

Vocabulary and notationIf you are unsure of what a word or phrase means(“correlations”, “effective interaction”, “renormalize”, . . . ),or what a symbol stands for, please ask!

Please try as many exercises as you canWarning: They are aimed at different levels!

References with details (and more references)For EFT (including chiral EFT), see list by H. Griesshammer

For EFT/DFT, see “EFT for DFT” by rjf, nucl-th/0702040

Did I mention about asking questions?



My Plan for these Lectures

Give you an idea of the modern methods and current statusin the areas of nuclear forces and density functional theory.

Not enough time for a detailed course, but we can coverenough to get you started (and you can ask questions aboutother topics).

Illustrate general principles with concrete (but simplified)examples.

Discuss various subtleties and give you my prejudices.



Outline



Quantum Resolution





The Big Picture: From QCD to Nuclei

Lattice

QCD

QCD

Lagrangian

Exact methods A!12

GFMC, NCSM

Chiral EFT interactions

(low-energy theory of QCD)

Coupled Cluster, Shell Model

A<100

Low-mom.

interactions

Density Functional Theory A>100

Anytime there is a hierarchy of energy scales, think EFT!


Overview Potential Resolution EFT 3-Body Core k-Space

Outline



Quantum Resolution





What Does the Nuclear Potential Look Like?Textbook answer (for 1S0 [what’s that?]) — cf. force between atoms:

Momentum units (~ = c = 1): typical relative momentum inlarge nucleus ≈ 1 fm−1 ≈ 200 MeV [Elab ≈ 83 MeV–fm2k2

rel]Dick Furnstahl Nuclear Forces/DFT


Consequences of a Repulsive Core

0 2 4 6r [fm]

0

0.05

0.1

0.15

0.2

|ψ(r)

|2 [fm

−3]

Argonne v18

3S1 deuteron probability density

0 2 4 6r [fm]

-0.1

0

0.1

0.2

0.3

0.4

|ψ(r)

|2 [fm

−3]

uncorrelatedcorrelated

0 2 4 6r [fm]

−100

0

100

200

300

400

V(r)

[MeV

]

Probability at short separations suppressed =⇒ “correlations”

Greatly complicates expansion of many-body wave functions

Short-distance structure ⇔ high-momentum components



Local vs. Nonlocal Potentials

Consider the operator Hamiltonian:

H =P2

2M+ V

Matrix elements of wave function |ψ〉 in coordinate space

〈ψ|H|ψ〉 =

∫d3r

∫d3r ′ 〈ψ|r′〉〈r′|H|r〉〈r|ψ〉 with 〈r|ψ〉 ≡ ψ(r)

〈r′| P2

2M|r〉 = δ3(r′−r)

−~2∇2

2M, 〈r′|V |r〉 =

V (r)δ3(r′ − r) if localV (r′, r) if nonlocal

r and r′ are relative distances (e.g., r = r1 − r2)

Local S-eqn: −~2∇2

2M ψ(r) + V (r)ψ(r) = Eψ(r)

Non-local S-eqn: −~2∇2

2M ψ(r) +∫

d3r ′ V (r, r′)ψ(r′) = Eψ(r)



Potentials in Momentum SpaceConsider the same abstract Hamiltonian:

H =P2

2M+ V

Matrix elements of wave function |ψ〉 in momentum space

〈ψ|H|ψ〉 =

∫d3k

∫d3k ′ 〈ψ|k′〉〈k′|H|k〉〈k|ψ〉 with 〈k|ψ〉 ≡ ψ(k)

〈k′| P2

2M|k〉 = δ3(k′−k)

~2k2

2M, 〈k′|V |k〉 =

V (k′ − k) if localV (k′, k) if nonlocal

k and k′ are relative momenta

Yukawa pot’l: e−µ|r|

4π|r| ↔1

(k′−k)2+µ2 depends on k′ − k only

Partial wave expansion:

〈k′|V |k〉 =1

2π2

∑L

(2L + 1) VL(k ′, k) PL(k · k′)



S-Wave (L = 0) NN Potential in Momentum Space

Fourier transform in partial waves (Bessel transform)

VL=0(k , k ′) =

∫d3r j0(kr) V (r) j0(k ′r) = 〈k |VL=0|k ′〉

Repulsive core =⇒ big high-k (> 2 fm−1) components


Overview Potential Resolution EFT 3-Body QM Low-Energy Filter Decoupling

Outline



Quantum Resolution





Diffraction and Resolution

Resolution of Small Apertures

Two point sources far from the aperture each produce a diffrac-tion pattern.

If the angle subtended by the sources at the aperture is largeenough, the diffraction patterns are distinguishable as shownin Fig. (a).

If the angle is small, the diffraction patterns can overlap so thatthe sources are not well resolved as shown in Fig. (b).



Diffraction and Resolution



Intensity as a Function of Angle

Single-slit diffraction =⇒ first minimum when sin θmin = λ/a=⇒ Pattern widens as λ increases



Wavelength and Resolution



Quantum Mechanics and Resolving Power

de Broglie relation: λ = h/p

λ ≈ 10−10 m =⇒ probe atoms

λ ≈ 10−14 m =⇒ probe nucleus

λ ≈ 10−18 m =⇒ probe quarks



Nuclear Diffraction with Pions

Warm-up exercise: From this figure and some reasonableassumptions, estimate the radius of a lead nucleus.



What if your theory has too much resolution?



What if your theory has too much resolution?

Claim: Nuclear physics with usual V (r) is like using beer coasters



Low Resolution Makes Physics EasierWeinberg’s Third Law of Progress in Theoretical Physics:

“You may use any degrees of freedom you like to describea physical system, but if you use the wrong ones, you’ll besorry!”

There’s an old vaudeville joke about a doctor and patient . . .

Patient: Doctor, doctor, it hurts when I do this!Doctor: Then don’t do that.




“You may use any degrees of freedom you like to describea physical system, but if you use the wrong ones, you’ll besorry!”There’s an old vaudeville joke about a doctor and patient . . .

Patient: Doctor, doctor, it hurts when I do this!

Doctor: Then don’t do that.




“You may use any degrees of freedom you like to describea physical system, but if you use the wrong ones, you’ll besorry!”There’s an old vaudeville joke about a doctor and patient . . .

Patient: Doctor, doctor, it hurts when I do this!Doctor: Then don’t do that.



Less Painful to use a Low-Resolution Version!

High resolution Low resolution

Lower resolution by “block spinning” or low-pass filter

Choose a resolution appropriate to the problem at hand!



Try a Low-Pass Filter on V (r)

=⇒ Set to zero high momentum (k > 2 fm−1) matrix elementsand see the effect on low-energy observables



Use Phase Shifts to Test

0 R

sin(kr+δ)

r

0 100 200 300Elab (MeV)

−20

0

20

40

60

phas

e sh

ift (d

egre

es)

AV18

1S0

Here: 1S0 (spin-singlet, L = 0, J = 0) neutron-proton scattering

Different phase shifts in each partial wave channel



A Low-Pass Filter on V (r) Fails At Low Energy

0 R

sin(kr+δ)

r

0 100 200 300Elab (MeV)

−20

0

20

40

60

phas

e sh

ift (d

egre

es)

AV18AV18 [kmax = 2.2 fm-1]

1S0



What Should We Conclude About the Potential?Does it mean that the physics of repulsive cores andcorrelations is necessary even for low-energy observableslike phase shifts or bound-state energies?

Are we stuck with keeping high momentum contributions forlow-energy physics?Many say yes!

Is the (short-range) potential an observable (i.e.,measurable)?

Intuition from Coulomb,classically and quantummechanically and QED

E.g., measure the energy at afixed separation r

Isn’t V (r) = − e2

r always?Answer to all questions: No!



Coulomb Potential from QEDUsual: match QED and potential calculations of scatteringQED Lagrangian including gauge-fixing

LQED =12

Aµ[gµν∂λ∂λ − (ξ−1 − 1)∂µ∂ν ]Aν − jµe Aµ + ψ(i/∂ −m)ψ

=12

Aµ[DµνF ]−1Aν − jµe Aµ + ψ(i/∂ −m)ψ

with electromagnetic current (charge e) jµe = eψγµψ

Physics of e±’s and photons from functional (path) integral:

Z =

∫DψDψDA exp[iS(ψ,ψ,A)]

But suppose we “integrate out” the photon field Aµ:

exp[iSeff(ψ,ψ)] =

∫DA exp[iS(ψ,ψ,A)]



Coulomb Potential from QED (cont.)

Completing the square, we get [jµe = eψγµψ]

Seff =

∫d4x ψ(x)(i/∂−m)ψ(x) +

12

∫d4x d4y jµe (x)DFµν(x − y)jνe (y)

Can we identify the last term as (particle density ρ = ψ†ψ)

−12

∫dt

∫d3x d3y ρ(x, t)V (x − y)ρ(y, t) ?

If we consider a classical static distribution jµe → e(ρ,0),

V (x−y) = −e2∫

dt ′∫

d4k(2π)4

e−ikµ(x−y)µ

k20 − k2 + iε

= e2∫

d3k(2π)3

eik·(x−y)

k2 =e2

4π|x − y|

Really the current density is a quantum-mechanical operator=⇒ V (x−y) is not uniquely defined quantum mechanically!



Why Did Our Low-Pass Filter Fail?

Basic problem: low k and high kare coupled

E.g., perturbation theoryfor (tangent of) phase shift:

〈k |V |k〉+∑k ′

〈k |V |k ′〉〈k ′|V |k〉(k2 − k ′2)/m

+ · · ·

We can’t blindly changeshort-distance structure withoutchanging the low-energy resultsbecause of coupling

0 100 200 300Elab (MeV)

−20

0

20

40

60

phas

e sh

ift (d

egre

es)

AV18AV18 [kmax = 2.2 fm-1]

1S0



Why Did Our Low-Pass Filter Fail?

Sensitivity tohigh-energy/short-distance physicsfrom “virtual” intermediate states

〈k |V |k〉+∑k ′

〈k |V |k ′〉〈k ′|V |k〉(k2 − k ′2)/m

+ · · ·

But the effect of high-energyphysics on low-energy physics canbe absorbed by adjustments in thebasic forces

=⇒ “Renormalization Group”

α(0) ≈ 1137 ; α(MW ) ≈ 1

128

This is the basis of both our RGand EFT discussions



How Can a Low-Pass Filter Succeed?

Basic problem: low k and high kare coupled with usual V

But V can be transformed

Solution: Unitary transformationof the H matrix (U†U = 1)=⇒ choose U to decouple!

En = 〈Ψn|H|Ψn〉= (〈Ψn|U†)UHU†(U|Ψn〉)= 〈Ψn|H|Ψn〉

Various methods: Vlow k , SRG,UCOM, . . .

Preview: RG “flow equations” 0 100 200 300Elab (MeV)

−20

0

20

40

60

phas

e sh

ift (d

egre

es)

AV18AV18 [kmax = 2.2 fm-1]

1S0



Flow Equations (“SRG”) in Action: NN OnlyIn each partial wave with εk = ~2k2/M and λ2 = 1/

√s

dVλ

dλ(k , k ′) ∝ −(εk − εk ′)2Vλ(k , k ′) +

∑q

(εk + εk ′ − 2εq)Vλ(k ,q)Vλ(q, k ′)



Unitary Transformation: Bare vs. SRG phase shifts

-25

0

25

50

δ [d

eg]

bare psvsrg ps

-60

-40

-20

0

-30

-15

0

15

0 200 400 600E

lab [MeV]

-45

-30

-15

0

δ [d

eg]

0 200 400 600E

lab [MeV]

-10

-5

0

0 200 400 600E

lab [MeV]

-5

0

5

10

1S

01P

13P

0

3P

1

3F

33G

4



Low-Pass Filters Work! [nucl-th/0701013]

Phase shifts with Vs(k , k ′) = 0 for k , k ′ > kmax

0 500 1000-100

-50

0

50

AV18AV18 [kmax = 2.2 fm-1]

Vs [kmax = 2.2 fm-1]

0 500 1000

-50

0

50

100

0 500 1000

-30

-20

-10

0

0 500 1000

-40

-20

0

20

0 500 1000

Elab [MeV]

-10

-5

0

5

phas

e sh

ift [d

eg]

1S03S1

3P03F3

3D1



Consequences of a Repulsive Core Revisited

0 2 4 6r [fm]

0

0.05

0.1

0.15

0.2

0.25

|ψ(r)

|2 [fm

−3]

Argonne v18


0 2 4 6r [fm]

-0.1

0

0.1

0.2

0.3

0.4

|ψ(r)

|2 [fm

−3]


0 2 4 6r [fm]

−100

0

100

200

300

400

V(r)

[MeV

]

Probability at short separations suppressed =⇒ “correlations”

Greatly complicates expansion of many-body wave functions

Short-distance structure ⇔ high-momentum components



Consequences of a Repulsive Core Revisited

0 2 4 6r [fm]

0

0.05

0.1

0.15

0.2

0.25

|ψ(r)

|2 [fm

-3]

Argonne v18

λ = 4.0 fm-1

λ = 3.0 fm-1

λ = 2.0 fm-1


0 2 4 6r [fm]

-0.1

0

0.1

0.2

0.3

0.4

|ψ(r)

|2 [fm

−3]


0 2 4 6r [fm]

−100

0

100

200

300

400

V(r)

[MeV

]

Transformed potential =⇒ no short-range correlations in wf

Potential is now non-local: V (r)ψ(r) −→∫

d3r′ V (r, r′)ψ(r′)Also few-body forces. Problems for many-body methods?

But what about the initial phenomenological potential?



Problems with Phenomenological PotentialsThe best potential models can describe with χ2/dof ≈ 1 all ofthe NN data (about 6000 points) below the pion productionthreshold. So what more do we need?Some drawbacks:

Usually have very strong repulsive short-range part =⇒requires special (non-systematic) treatment in many-bodycalculations (e.g. nuclear structure).Difficult to estimate theoretical error and range of applicability.Three-nucleon forces (3NF) are largely under-constrained andnon-systematic models. How to define consistent 3NF’s andoperators (e.g., meson exchange currents)?Models are largely unconnected to QCD (e.g., chiral symmetry).They don’t connect NN and other strongly interacting processes(e.g., ππ and πN). Lattice QCD will be able to predict NN, 3Nobservables for high pion masses. How to extrapolate tophysical pion masses?

Answer to problems =⇒ Use EFT and RGDick Furnstahl Nuclear Forces/DFT

Overview Potential Resolution EFT 3-Body Low Energy Perturbative EFT

Outline



Quantum Resolution





“Simple” Few-Body Problem: Hard Spheres

Infinite potential at radius R

0 R

sin(kr+δ)

r

Do two-body now, many-bodylater (Ref.: nucl-th/0004043)

Two-body S-eqn easy!

Phase shift: δ0(k) = −kR

Scattering length: a0 = R

k F

R

1/~



Quick Review of Scattering

P/2− k

P/2 + k

P/2− k′

P/2 + k′

0 R

sin(kr+δ)

r

Relative motion with total P = 0: ψ(r) r→∞−→ eik·r + f (k , θ)eikr

r

where k2 = k ′2 = MEk and cos θ = k · k ′

Differential cross section is dσ/dΩ = |f (k , θ)|2

Central V =⇒ partial waves:f (k , θ) =

∑l(2l + 1)fl(k)Pl(cos θ)

where fl(k) =eiδl (k) sin δl(k)

k=

1k cot δl(k)− ik

and the S-wave phase shift is defined by

u0(r)r→∞−→ sin[kr + δ0(k)] =⇒ δ0(k) = −kR for hard sphere



Quick Review of Scattering

P/2− k

P/2 + k

P/2− k′

P/2 + k′

0 R

sin(kr+δ)

r

Relative motion with total P = 0: ψ(r) r→∞−→ eik·r + f (k , θ)eikr

r

where k2 = k ′2 = MEk and cos θ = k · k ′

Differential cross section is dσ/dΩ = |f (k , θ)|2Central V =⇒ partial waves:f (k , θ) =

∑l(2l + 1)fl(k)Pl(cos θ)

where fl(k) =eiδl (k) sin δl(k)

k=

1k cot δl(k)− ik

and the S-wave phase shift is defined by

u0(r)r→∞−→ sin[kr + δ0(k)] =⇒ δ0(k) = −kR for hard sphere



At Low Energies: Effective Range Expansion

As first shown by Schwinger, k l+1 cot δl(k) has a power seriesexpansion. For l = 0:

k cot δ0(k) = − 1a0

+12

r0k2 − Pr30 k4 + · · ·

defines the scattering length a0 and the effective range r0

While r0 ∼ R, the range of the potential, a0 can be anythingif a0 ∼ R, it is called “natural”

|a0| R (unnatural) is particularly interesting =⇒ cold atoms

The effective range expansion for hard sphere scattering is:

k cot(−kR) = − 1R

+13

Rk2 + · · · =⇒ a0 = R r0 = 2R/3

so the low-energy effective theory is natural



In Search of a Perturbative Expansion

If a0 is natural, then low-energy scattering simplifies further

For scattering at momentum k 1/R, we should recover aperturbative expansion in kR for scattering amplitude:

f0(k) =1

k cot δ(k)− ik=

1

− 1a0

+ 12 r0k2 + · · · − ik

−→ −a0[1− ia0k − (a2

0 − a0r0/2)k2 +O(k3a30)

]

Can we reproduce this simple expansion for the hard-sphere?

Perturbation theory in the hard-sphere potential won’t work:

0 R

=⇒ 〈k|V |k′〉 ∝∫

dx eik·x V (x) e−ik′·x −→∞



In Search of a Perturbative Expansion

If a0 is natural, then low-energy scattering simplifies further

For scattering at momentum k 1/R, we should recover aperturbative expansion in kR for scattering amplitude:

f0(k) =1

k cot δ(k)− ik=

1

− 1a0

+ 12 r0k2 + · · · − ik

−→ −a0[1− ia0k − (a2

0 − a0r0/2)k2 +O(k3a30)

]Can we reproduce this simple expansion for the hard-sphere?

Perturbation theory in the hard-sphere potential won’t work:

0 R

=⇒ 〈k|V |k′〉 ∝∫

dx eik·x V (x) e−ik′·x −→∞



In Search of a Perturbative Expansion (cont.)

Standard solution: Solve the scattering problemnonperturbatively, then expand in kR

For our example, this is easy =⇒ use δ0(k) = −kR:

f0(k) =1

k cot δ0(k)− ik−→ −a0

[1− ia0k − (a2

0 − a0r0/2)k2 +O(k3a30)

]−→ −R[1− ikR − 2k2R2/3 +O(k3R3)]

Easy for 2–2 scattering, but not for the many-body problem!

EFT approach: k 1/R means we probe at low resolution=⇒ replace potential with a simpler but general interaction



EFT for “Natural” Short-Range Interaction

A simple, general interaction is a sum of delta functions andderivatives of delta functions. In momentum space,

〈k|Veft|k′〉 = C0 +12

C2(k2 + k′2) + C′2k · k′ + · · ·

Or, Left has the most general local (contact) interactions:

Left = ψ†[i∂

∂t+

−→∇ 2

2M

]ψ − C0

2(ψ†ψ)2 +

C2

16

[(ψψ)†(ψ

↔∇2ψ) + h.c.

]+

C′2

8(ψ

↔∇ψ)† · (ψ

↔∇ψ)− D0

6(ψ†ψ)3 + . . .

Dimensional analysis =⇒ C2i ∼ 4πM R2i+1 , D2i ∼ 4π

M R2i+4



Feynman Rules for EFT Vertices

Left = ψ†[i∂

∂t+

−→∇ 2

2M

]ψ − C0

2(ψ†ψ)2 +

C2

16

[(ψψ)†(ψ

↔∇2ψ) + h.c.

]+

C′2

8(ψ

↔∇ψ)† · (ψ

↔∇ψ)− D0

6(ψ†ψ)3 + . . .

P/2− k

P/2 + k

P/2− k′

P/2 + k′

= + + + · · ·

−i〈k′|VEFT|k〉 − iC0 −iC2k2 + k′2

2−iC ′2 k · k′

= + · · ·

−iD0



Renormalization Fixes High-Energy Contribution

Reproduce f0(k) in perturbation theory (Born series):

f0(k) ∝ a0 − ia20k − (a3

0 − a20r0/2)k2 +O(k3a4

0)

Consider the leading potential V (0)EFT(x) = C0δ(x) or

〈k|V (0)eft |k

′〉 =⇒ =⇒ C0

Choosing C0 ∝ a0 gets the first term. Now 〈k|VG0V |k′〉:





f0(k) ∝ a0 − ia20k − (a3

0 − a20r0/2)k2 +O(k3a4

0)


〈k|V (0)eft |k

′〉 =⇒ =⇒ C0


=⇒ C0M∫

d3q(2π)3

1k2 − q2 + iε

C0 −→∞!

=⇒ Linear divergence!





f0(k) ∝ a0 − ia20k − (a3

0 − a20r0/2)k2 +O(k3a4

0)


〈k|V (0)eft |k

′〉 =⇒ =⇒ C0


=⇒∫ Λc d3q

(2π)3

1k2 − q2 + iε

−→ − ik4π

+Λc

2π2 +O(k2

Λc)

=⇒ If cutoff at Λc , then absorb linear Λc dependence in C0





f0(k) ∝ a0 − ia20k − (a3

0 − a20r0/2)k2 +O(k3a4

0)


〈k|V (0)eft |k

′〉 =⇒ =⇒ C0


=⇒∫

dDq(2π)3

1k2 − q2 + iε

D→3−→ − ik4π

Dimensional regularization with minimal subtraction=⇒ cleaner since only one power of k per diagram!



Dim. reg. + minimal subtraction =⇒ simple power counting:Each propagator: M/k2, each loop: k5/M, every n-body vertexwith 2i derivatives: k2iR2i+3n−5/M. A diagram with E externallines and V n

2i vertices scales as kν with

ν = 5− 32

E +∞∑

n=2

∞∑i=0

(2i + 3n − 5)V n2i

P/2− k

P/2 + k

P/2− k′

P/2 + k′

= +

iT (k, cos θ) − iC0 − M

4π(C0)

2k

+ + + + O(k3)

+i

(M

4π

)2

(C0)3k2 − iC2k

2 − iC ′2k2 cos θ

Matching (to underlying theory or data):

C0 = 4πM a0 = 4π

M R , C2 = 4πM

a20r02 = 4π

MR3

3 , · · ·



Effective Field Theory Ingredients[See H. Griesshammer references for more details/examples]

1 Use the most general L with low-energy dof’s consistent withglobal and local symmetries of underlying theoryLeft = ψ†

[i ∂∂t + ∇ 2

2M

]ψ − C0

2 (ψ†ψ)2 − D06 (ψ†ψ)3 + . . .

2 Declaration of regularization and renormalization schemenatural a0 =⇒ dimensional regularization and min. subtractionare most efficient but can use cutoff as well

3 Well-defined power counting =⇒ small expansion parameteruse separation of scales =⇒ k

Λ with Λ ∼ 1/R =⇒ ka0 1, etc.

Recovers expansion order-by-order with diagrams

f0(k) ∝ 1k cot δ0(k)− ik

−→ a0[1− ia0k − (a2

0 − a0r0/2)k2 +O(k3a30)

]−→ R[1− ikR − 2k2R2/3 +O(k3R3)] [hard sphere]

with DR/MS, one power of k per diagram, natural coefficientsestimate truncation error from dimensional analysisvalid for any natural short-range interaction!



Outline



Quantum Resolution





What About 3-Body Forces?

Feynman rules and power counting predict a three-body force

At low resolution, don’t resolve series of two-body scatterings=⇒ three-body (even if underlying two-body only)!

New logarithmic divergences in 3–3 scattering

+ ∝ (C0)4 ln(k/Λc)

Changes in Λc must be absorbed by 3-body coupling D0(Λc)=⇒ D0(Λc) ∝ (C0)

4 ln(a0Λc) + const. [Braaten & Nieto]

ddΛc

[

]= 0 =⇒ fixes coefficient!



Atomic 3-Body Forces: Axilrod-Teller Term (1943)

Three-body potential for atoms/molecules from triple-dipolemutual polarization (3rd-order perturbation correction)

V (i , j , k) =ν(1 + 3 cos θi cos θj cos θk )

(rij rik rjk )3

Usually negligible in metals and semiconductors

Can be important for ground-state energy of solids bound byvan der Waals potentials

Bell and Zuker (1976): 10% of energy in solid xenon



Preview of Nuclear Three-Body Forces

Three-body forces arise fromeliminating dof’s

excited states of nucleon

relativistic effects

high-momentumintermediate states

Omitting 3-body forces leadsto model dependence

observables depend on λ

e.g., Tjon line

3-body contributionsincrease with density

saturates nuclear matter

how large is 4-body? 7.6 7.8 8 8.2 8.4 8.6 8.8Eb(

3H) [MeV]

24

25

26

27

28

29

30

31

E b(4 He)

[MeV

]NN potentialsSRG N3LO (500 MeV)

N3LOλ=1.0

λ=3.0λ=1.25 λ=2.5

λ=2.25λ=1.5 λ=2.0

λ=1.75

Expt.

A=3,4 binding energiesSRG NN only, λ in fm−1



Principle of Effective Low-Energy Theories

If system is probed at low energies, fine details not resolved

use low-energy variables for low-energy processes

short-distance structure can be replaced by something simplerwithout distorting low-energy observables

systematically achieved by effective field theory



Principle of Effective Low-Energy Theories

If system is probed at low energies, fine details not resolveduse low-energy variables for low-energy processes

short-distance structure can be replaced by something simplerwithout distorting low-energy observables

systematically achieved by effective field theory



Take-Away Points

Low-energy means low resolution.

It’s often painful to use a theory with more resolution thannecessary; RG transformations help by lowering resolution!

An EFT is model independent because it has a complete setof operators at each order in a well-defined expansion.

You can always match to data if the underlying theory isunsolvable.

An EFT may be easier to calculate with even if the underlyingtheory is known.

A low-energy theory with “natural” scattering length isperturbative; conversely shallow bound states (largescattering length) requires a non-perturbative EFT!

In a low-energy theory, three- (and higher) body forces areinevitable!


nuclear forces / dft for nuclei i - physicsntg/talks/furnstahl_tsi_1.pdf · nuclear forces / dft...

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