Effective Field Theory (EFT)and

Density Functionals

Dick FurnstahlDepartment of PhysicsOhio State University

Extremes of the Nuclear LandscapeSeptember 1, 2014

State-of-the-art Skyrme EDFs

Is there a limit to improvement ofSkyrme rms energy residual?

Recently many advances byUNEDF/NUCLEI, FIDIPRO, andothers to improve/test EDFs

Extra observables and ab initiocalculations in neutron drops forconstraints (e.g., on isovector)

Sophisticated fit and correlationanalysis implies the EDF is notlimited by the parameter fitting

But still don’t beat the energybarrier (and not nearly as goodenergy rms as mass models)

=⇒ limit of Skyrme EDF strategy?

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

0 1 2 3 4 5 6 7 8

rm

s re

sid

ual

(M

eV)

dimension of parameter space

Bertsch,)Sabbey,)))))and)Uusnakki,)Phys.)Rev.)C)71,)))054311)(2005))

State-of-the-art Skyrme EDFs

Is there a limit to improvement ofSkyrme rms energy residual?

Recently many advances byUNEDF/NUCLEI, FIDIPRO, andothers to improve/test EDFs

Extra observables and ab initiocalculations in neutron drops forconstraints (e.g., on isovector)

Sophisticated fit and correlationanalysis implies the EDF is notlimited by the parameter fitting

But still don’t beat the energybarrier (and not nearly as goodenergy rms as mass models)

=⇒ limit of Skyrme EDF strategy?

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

0 1 2 3 4 5 6 7 8

rm

s re

sid

ual

(M

eV)

dimension of parameter space

Bertsch,)Sabbey,)))))and)Uusnakki,)Phys.)Rev.)C)71,)))054311)(2005))

4

6

8

10

E tot/N4/3(MeV)

0 10 20 30 40 50 60N

=10MeV

=5MeV

AFDMCUNEDF0UNEDF1UNEDF2

SLy4SLy4 adj.

hω

hω

M.#Kortelainen#et#al.,#PRC#89,#054314#(2014)#

State-of-the-art Skyrme EDFs

Is there a limit to improvement ofSkyrme rms energy residual?

Recently many advances byUNEDF/NUCLEI, FIDIPRO, andothers to improve/test EDFs

Extra observables and ab initiocalculations in neutron drops forconstraints (e.g., on isovector)

Sophisticated fit and correlationanalysis implies the EDF is notlimited by the parameter fitting

But still don’t beat the energybarrier (and not nearly as goodenergy rms as mass models)

=⇒ limit of Skyrme EDF strategy?

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

0 1 2 3 4 5 6 7 8

rm

s re

sid

ual

(M

eV)

dimension of parameter space

Bertsch,)Sabbey,)))))and)Uusnakki,)Phys.)Rev.)C)71,)))054311)(2005))

radiimasses OES FI s.p.e.

State-of-the-art Skyrme EDFs

Is there a limit to improvement ofSkyrme rms energy residual?

Recently many advances byUNEDF/NUCLEI, FIDIPRO, andothers to improve/test EDFs

Extra observables and ab initiocalculations in neutron drops forconstraints (e.g., on isovector)

Sophisticated fit and correlationanalysis implies the EDF is notlimited by the parameter fitting

But still don’t beat the energybarrier (and not nearly as goodenergy rms as mass models)

=⇒ limit of Skyrme EDF strategy?

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

0 1 2 3 4 5 6 7 8

rm

s re

sid

ual

(M

eV)

dimension of parameter space

Bertsch,)Sabbey,)))))and)Uusnakki,)Phys.)Rev.)C)71,)))054311)(2005))

Masses (def)

Masses (sph) Radii OES FI spe

State-of-the-art Skyrme EDFs

Is there a limit to improvement ofSkyrme rms energy residual?

Recently many advances byUNEDF/NUCLEI, FIDIPRO, andothers to improve/test EDFs

Extra observables and ab initiocalculations in neutron drops forconstraints (e.g., on isovector)

Sophisticated fit and correlationanalysis implies the EDF is notlimited by the parameter fitting

But still don’t beat the energybarrier (and not nearly as goodenergy rms as mass models)

=⇒ limit of Skyrme EDF strategy?

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

0 1 2 3 4 5 6 7 8

rm

s re

sid

ual

(M

eV)

dimension of parameter space

Bertsch,)Sabbey,)))))and)Uusnakki,)Phys.)Rev.)C)71,)))054311)(2005))

0 20 40 60 80 100 120 140 160Neutron Number N

-6

-4

-2

0

2

4

6

E th

- Eexp

(MeV

)

UNEDF2

(a)

M.#Kortelainen#et#al.,#PRC#89,#054314#(2014)#

Questions for empirical energy density functionals (EDFs)

Are density dependencies too simplistic? How do you know?

How should we organize possible terms in the EDF?

Where are the pions? Where is chiral symmetry?

What is the connection to many-body forces?

How do we estimate a priori theoretical uncertainties?

What is the theoretical limit of accuracy?

and so on . . .

=⇒ Extend or modify EDF forms in controlled way

=⇒ Use microscopic many-body theory for guidance=⇒ E(F)T

There are multiple paths to a nuclear EDF =⇒ consider EFT for all

Effective theories [H. Georgi, Ann. Rev. Nucl. Part. Sci. 43, 209 (1993)]

One of the most astonishing things about the world in which we live is thatthere seems to be interesting physics at all scales.

To do physics amid this remarkable richness, it is convenient to be able toisolate a set of phenomena from all the rest, so that we can describe it withouthaving to understand everything. . . . We can divide up the parameter space ofthe world into different regions, in each of which there is a differentappropriate description of the important physics. Such an appropriatedescription of the important physics is an “effective theory.”The common idea is that if there are parameters that are very large or verysmall compared to the physical quantities (with the same dimension) that weare interested in, we may get a simpler approximate description of the physicsby setting the small parameters to zero and the large parameters to infinity.Then the finite effects of the parameters can be included as smallperturbations about this simple approximate starting point.

E.g., non-relativistic QM: c →∞E.g., pionless effective field theory (EFT): mπ,MN →∞E.g., chiral effective field theory (EFT): mπ → 0, MN →∞Goals: model independent, improvable, and with error estimates

Features: limited domain, breakdown is predicted internally

Effective theories [H. Georgi, Ann. Rev. Nucl. Part. Sci. 43, 209 (1993)]

One of the most astonishing things about the world in which we live is thatthere seems to be interesting physics at all scales.To do physics amid this remarkable richness, it is convenient to be able toisolate a set of phenomena from all the rest, so that we can describe it withouthaving to understand everything. . . . We can divide up the parameter space ofthe world into different regions, in each of which there is a differentappropriate description of the important physics. Such an appropriatedescription of the important physics is an “effective theory.”

The common idea is that if there are parameters that are very large or verysmall compared to the physical quantities (with the same dimension) that weare interested in, we may get a simpler approximate description of the physicsby setting the small parameters to zero and the large parameters to infinity.Then the finite effects of the parameters can be included as smallperturbations about this simple approximate starting point.

E.g., non-relativistic QM: c →∞E.g., pionless effective field theory (EFT): mπ,MN →∞E.g., chiral effective field theory (EFT): mπ → 0, MN →∞Goals: model independent, improvable, and with error estimates

Features: limited domain, breakdown is predicted internally

Effective theories [H. Georgi, Ann. Rev. Nucl. Part. Sci. 43, 209 (1993)]

One of the most astonishing things about the world in which we live is thatthere seems to be interesting physics at all scales.To do physics amid this remarkable richness, it is convenient to be able toisolate a set of phenomena from all the rest, so that we can describe it withouthaving to understand everything. . . . We can divide up the parameter space ofthe world into different regions, in each of which there is a differentappropriate description of the important physics. Such an appropriatedescription of the important physics is an “effective theory.”The common idea is that if there are parameters that are very large or verysmall compared to the physical quantities (with the same dimension) that weare interested in, we may get a simpler approximate description of the physicsby setting the small parameters to zero and the large parameters to infinity.Then the finite effects of the parameters can be included as smallperturbations about this simple approximate starting point.

E.g., non-relativistic QM: c →∞E.g., pionless effective field theory (EFT): mπ,MN →∞E.g., chiral effective field theory (EFT): mπ → 0, MN →∞

Goals: model independent, improvable, and with error estimates

Features: limited domain, breakdown is predicted internally

Effective theories [H. Georgi, Ann. Rev. Nucl. Part. Sci. 43, 209 (1993)]

One of the most astonishing things about the world in which we live is thatthere seems to be interesting physics at all scales.To do physics amid this remarkable richness, it is convenient to be able toisolate a set of phenomena from all the rest, so that we can describe it withouthaving to understand everything. . . . We can divide up the parameter space ofthe world into different regions, in each of which there is a differentappropriate description of the important physics. Such an appropriatedescription of the important physics is an “effective theory.”The common idea is that if there are parameters that are very large or verysmall compared to the physical quantities (with the same dimension) that weare interested in, we may get a simpler approximate description of the physicsby setting the small parameters to zero and the large parameters to infinity.Then the finite effects of the parameters can be included as smallperturbations about this simple approximate starting point.

E.g., non-relativistic QM: c →∞E.g., pionless effective field theory (EFT): mπ,MN →∞E.g., chiral effective field theory (EFT): mπ → 0, MN →∞Goals: model independent, improvable, and with error estimates

Features: limited domain, breakdown is predicted internally

Principles common to all E(F)Ts (but can be in different forms)

In coordinate space, define ato separate short and longdistance

In momentum space, use Λ toseparate high and low momenta

Much freedom how this is done(e.g., different regulator forms)=⇒ different scales / schemes

Principles common to all E(F)Ts (but can be in different forms)

In coordinate space, define ato separate short and longdistance

In momentum space, use Λ toseparate high and low momenta

Much freedom how this is done(e.g., different regulator forms)=⇒ different scales / schemes

Long distance solved explicitly (symmetries);short-distance captured in some LECs.Naturalness =⇒ error estimates

Model independence comes fromcompleteness. All terms allowed bysymmetries should be present, up toredundancies. One way: QFT!

Power counting from small parameters(e.g., ratio of scales: Q/Λ) =⇒ systematic

Classical analogy to EFT: Multipole expansionIf we have a localized charge distributionρ(r) within a volume characterized bydistance a, the electrostatic potential is

φ(R) ∝∫

d3rρ(r)

|R− r|

If we expand 1/|R− r| for r � R, we get the multipole expansion∫

d3rρ(r)

|R− r| =qR

+1

R3

∑

i

RiPi +1

6R5

∑

ij

(3RiRj − δijR2)Qij + · · ·

=⇒ pointlike total charge q, dipole moment Pi , quadrupole Qij :

q =

∫d3r ρ(r) Pi =

∫d3r ρ(r) ri Qij =

∫d3r ρ(r)(3ri rj − δij r2)

Hierarchy of terms from separation of scales =⇒ a/R expansion

Or: include known long-distance structure or reference distribution

Can determine coefficients (LECs) by matching to actual distribution(if known; cf. LQCD) or comparing to experimental measurements

Completeness =⇒ model independent (cf. model of distribution)

Classical analogy to EFT: Multipole expansionIf we have a localized charge distributionρ(r) within a volume characterized bydistance a, the electrostatic potential is

φ(R) ∝∫

d3rρ(r)

|R− r|

If we expand 1/|R− r| for r � R, we get the multipole expansion∫

d3rρ(r)

|R− r| =qR

+1

R3

∑

i

RiPi +1

6R5

∑

ij

(3RiRj − δijR2)Qij + · · ·

=⇒ pointlike total charge q, dipole moment Pi , quadrupole Qij :

q =

∫d3r ρ(r) Pi =

∫d3r ρ(r) ri Qij =

∫d3r ρ(r)(3ri rj − δij r2)

Hierarchy of terms from separation of scales =⇒ a/R expansion

Or: include known long-distance structure or reference distribution

Can determine coefficients (LECs) by matching to actual distribution(if known; cf. LQCD) or comparing to experimental measurements

Completeness =⇒ model independent (cf. model of distribution)

Classical analogy to EFT: Multipole expansionIf we have a localized charge distributionρ(r) within a volume characterized bydistance a, the electrostatic potential is

φ(R) ∝∫

d3rρ(r)

|R− r|

If we expand 1/|R− r| for r � R, we get the multipole expansion∫

d3rρ(r)

|R− r| =qR

+1

R3

∑

i

RiPi +1

6R5

∑

ij

(3RiRj − δijR2)Qij + · · ·

=⇒ pointlike total charge q, dipole moment Pi , quadrupole Qij :

q =

∫d3r ρ(r) Pi =

∫d3r ρ(r) ri Qij =

∫d3r ρ(r)(3ri rj − δij r2)

Hierarchy of terms from separation of scales =⇒ a/R expansion

Or: include known long-distance structure or reference distribution

Can determine coefficients (LECs) by matching to actual distribution(if known; cf. LQCD) or comparing to experimental measurements

Completeness =⇒ model independent (cf. model of distribution)

Classical analogy to EFT: Multipole expansionIf we have a localized charge distributionρ(r) within a volume characterized bydistance a, the electrostatic potential is

φ(R) ∝∫

d3rρ(r)

|R− r|

If we expand 1/|R− r| for r � R, we get the multipole expansion∫

d3rρ(r)

|R− r| =qR

+1

R3

∑

i

RiPi +1

6R5

∑

ij

(3RiRj − δijR2)Qij + · · ·

=⇒ pointlike total charge q, dipole moment Pi , quadrupole Qij :

q =

∫d3r ρ(r) Pi =

∫d3r ρ(r) ri Qij =

∫d3r ρ(r)(3ri rj − δij r2)

Hierarchy of terms from separation of scales =⇒ a/R expansion

Or: include known long-distance structure or reference distribution

Can determine coefficients (LECs) by matching to actual distribution(if known; cf. LQCD) or comparing to experimental measurements

Completeness =⇒ model independent (cf. model of distribution)

Different E(F)Ts depending on scale/observables of interest

Res

olut

ion

scale&separa)on&

DFT

collective models

CI

ab initio

LQCD

constituent quarks

There is not just one EFT!

Chiral EFT: nucleons, [∆’s,]pions: {q,mπ}/Λχ ≈ mρ

Pionless EFT: nucleons only(low-energy few-body) ornucleons and clusters (halo)

ET for deformed nuclei:systematic collective dofs(Papenbrock, Weidenmueller)

EFT at Fermi surface(Landau-Migdal theory):quasi-nucleons

Scale and scheme dependencealso means non-uniqueness(e.g., of EDFs)

Different E(F)Ts depending on scale/observables of interest

Res

olut

ion

scale&separa)on&

DFT

collective models

CI

ab initio

LQCD

constituent quarks

There is not just one EFT!

Chiral EFT: nucleons, [∆’s,]pions: {q,mπ}/Λχ ≈ mρ

Pionless EFT: nucleons only(low-energy few-body) ornucleons and clusters (halo)

ET for deformed nuclei:systematic collective dofs(Papenbrock, Weidenmueller)

EFT at Fermi surface(Landau-Migdal theory):quasi-nucleons

Scale and scheme dependencealso means non-uniqueness(e.g., of EDFs)

Comparing dilute EFT local density and Skyrme functionalsSkyrme EDF (for N = Z and without pairing)

E [ρ, τ, J] =

∫dr{

τ

2M+

38

t0ρ2 +1

16(3t1 + 5t2)ρτ+

164

(9t1 − 5t2)(∇ρ)2

− 34

W0ρ∇ · J +116

t3ρ2+α + · · ·}

where ρ(r) =∑

i |ψi (r)|2 , τ(r) =∑

i |∇ψi (r)|2 , . . .

Pionless zero-range EFT =⇒ dilute LDA ρτJ EDF (with Vexternal = 0)

E [ρ, τ, J] =

∫dr{

τ

2M+

38

C0ρ2 +

116

(3C2 + 5C′2)ρτ+1

64(9C2 − 5C′2)(∇ρ)2

− 34

C′′2 ρ∇ · J +c1

2MC2

0ρ7/3 +

c2

2MC3

0ρ8/3 +

116

D0ρ3 + · · ·

}

Looks like same functional as LDA dilute Fermi gas with ti ↔ Ci !

Also corresponding pairing terms (with renormalization)Is Skyrme missing important gradient, non-analytic, NN· · ·N,long-range (pion) terms?What about the natural scaling of coefficients?

Comparing dilute EFT local density and Skyrme functionalsSkyrme EDF (for N = Z and without pairing)

E [ρ, τ, J] =

∫dr{

τ

2M+

38

t0ρ2 +1

16(3t1 + 5t2)ρτ+

164

(9t1 − 5t2)(∇ρ)2

− 34

W0ρ∇ · J +116

t3ρ2+α + · · ·}

where ρ(r) =∑

i |ψi (r)|2 , τ(r) =∑

i |∇ψi (r)|2 , . . .

Pionless zero-range EFT =⇒ dilute LDA ρτJ EDF (with Vexternal = 0)

E [ρ, τ, J] =

∫dr{

τ

2M+

38

C0ρ2 +

116

(3C2 + 5C′2)ρτ+1

64(9C2 − 5C′2)(∇ρ)2

− 34

C′′2 ρ∇ · J +c1

2MC2

0ρ7/3 +

c2

2MC3

0ρ8/3 +

116

D0ρ3 + · · ·

}

Looks like same functional as LDA dilute Fermi gas with ti ↔ Ci !

Also corresponding pairing terms (with renormalization)Is Skyrme missing important gradient, non-analytic, NN· · ·N,long-range (pion) terms?What about the natural scaling of coefficients?

Comparing dilute EFT local density and Skyrme functionalsSkyrme EDF (for N = Z and without pairing)

E [ρ, τ, J] =

∫dr{

τ

2M+

38

t0ρ2 +1

16(3t1 + 5t2)ρτ+

164

(9t1 − 5t2)(∇ρ)2

− 34

W0ρ∇ · J +116

t3ρ2+α + · · ·}

where ρ(r) =∑

i |ψi (r)|2 , τ(r) =∑

i |∇ψi (r)|2 , . . .

Pionless zero-range EFT =⇒ dilute LDA ρτJ EDF (with Vexternal = 0)

E [ρ, τ, J] =

∫dr{

τ

2M+

38

C0ρ2 +

116

(3C2 + 5C′2)ρτ+1

64(9C2 − 5C′2)(∇ρ)2

− 34

C′′2 ρ∇ · J +c1

2MC2

0ρ7/3 +

c2

2MC3

0ρ8/3 +

116

D0ρ3 + · · ·

}

Looks like same functional as LDA dilute Fermi gas with ti ↔ Ci !

Also corresponding pairing terms (with renormalization)Is Skyrme missing important gradient, non-analytic, NN· · ·N,long-range (pion) terms?What about the natural scaling of coefficients?

Naturalness in Skyrme coefficients as EFT signatures?Georgi (1993): fπ for strongly interacting fields; rest is Λχ ≈ mρ; clmn ∼ O(1)

Lχ eft = clmn

(N†Nf 2πΛχ

)l (π

fπ

)m (∂µ,mπ

Λχ

)n

f 2πΛ2

χ fπ ∼ 100 MeV

Chiral NDA analysis for EDFs:[Friar et al., rjf et al.]

c[

N†Nf 2πΛχ

]l [ ∇Λχ

]n

f 2πΛ2

χ

=⇒ρ←→ N†Nτ ←→ ∇N† · ∇NJ←→ N†∇N

Density expansion?1000 ≥ Λχ ≥ 500 =⇒ 1

7≤ ρ0

f 2πΛχ

≤ 14

Also gradient expansion

Applied to RMF, Skyrme EDFs2 3 4 5

power of density1

5

10

50

100

500

1000

ener

gy/p

artic

le (M

eV)

ε0

natural (Λ=600 MeV)Skyrme ρn

RMFT-II ρn netRMFT-I ρn net

kF = 1.35 fm−1

What does this tell us about accuracy limits?

Naturalness in Skyrme coefficients as QCD signatures?

Georgi (1993): fπ for strongly interacting fields; rest is Λχ ≈ mρ; clmn ∼ O(1)

Lχ eft = clmn

(N†Nf 2πΛχ

)l (π

fπ

)m (∂µ,mπ

Λχ

)n

f 2πΛ2

χ fπ ∼ 100 MeV

Check chiral naturalness for large set of Skyrme EDFs:

10-3

2

5

10-2

2

5

10-1

2

5

1

2

5

10

2

Co

up

lin

gco

nst

ant

= 687 MeV, iv. scaled

0.0

0.4

0.8

1.2

rms,

Lo

g1

0

01

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

C!

C!D

C"

C!!

C!J

CJ

Kortelainen*et*al.*(2010)*

~50*scaled*sets*of*Skyrme*coefficients*

Looks like natural distribution =⇒ Does this mean pionful EFT is needed?

Conceptual basis of (chiral) effective field theory

Separate the short-distance(UV) from long-distance (IR)physics =⇒ scale Λχ ≈ mρ

Exploit chiral symmetry =⇒hierarchical treatment oflong-distance physics

Use complete basis forshort-distance physics=⇒ hierarchy a la multipoles

Conceptual basis of (chiral) effective field theory

Separate the short-distance(UV) from long-distance (IR)physics =⇒ scale Λχ ≈ mρ

Exploit chiral symmetry =⇒hierarchical treatment oflong-distance physics

Use complete basis forshort-distance physics=⇒ hierarchy a la multipoles

QCD

TChPT

T =...

⇡

1

NN'interac/on'is'strong:(resumma-ons/nonperturba-ve(methods(needed

1/mN

1

J(expansion:(nonrela-vis-c(problem(((((((((((((((((((((((((((((()(|�pi | �M� ⇥ mN

1

the(QM(AJbody(problem

(-15*-9/�F�

✓ AX

i=1

�⇥2i

2mN

+ O(m�3N )

◆+ V2N + V3N + V4N + . . .

�|�� = E|��

1

derived&within&ChPT

✓ AX

i=1

�⇥2i

2mN

+ O(m�3N )

◆+ V2N + V3N + V4N + . . .

�|�� = E|��

1

From QCD to nuclear physics

(((((Construct(effec-ve(poten-al(perturba-vely

V …

((((Solve(LippmannJSchwinger(equa-on(nonperturba-vely

V TT V

[H. Krebs]

Generate a nonrelativisticpotential for many-bodymethods (controversies!)

QCD

TChPT

T =...

⇡

1

NN'interac/on'is'strong:(resumma-ons/nonperturba-ve(methods(needed

1/mN

1

J(expansion:(nonrela-vis-c(problem(((((((((((((((((((((((((((((()(|�pi | �M� ⇥ mN

1

the(QM(AJbody(problem

(-15*-9/�F�

✓ AX

i=1

�⇥2i

2mN

+ O(m�3N )

◆+ V2N + V3N + V4N + . . .

�|�� = E|��

1

derived&within&ChPT

✓ AX

i=1

�⇥2i

2mN

+ O(m�3N )

◆+ V2N + V3N + V4N + . . .

�|�� = E|��

1

From QCD to nuclear physics

(((((Construct(effec-ve(poten-al(perturba-vely

V …

((((Solve(LippmannJSchwinger(equa-on(nonperturba-vely

V TT V

Possible EFT for nuclear EDF: Use chiral Hamiltonian in MBPT

Adaptation of chiral EFT MBPT to Skyrme HFB form

ESkyrme =τ

2M+

38

t0ρ2 +116

t3ρ2+α +116

(3t1 + 5t2)ρτ+1

64(9t1 − 5t2)|∇ρ|2 + · · ·

=⇒ EDME =τ

2M+ A[ρ] + B[ρ]τ + C[ρ]|∇ρ|2 + · · ·

Orbitals and Occupation #’s

Kohn−Sham Potentials

t , t0 1 , ..., t2

Skyrmeenergy

functionalHFB

solver

VKS(r) =δEint[ρ]

δρ(r)⇐⇒ [−∇2

2m+VKS(x)]ψα = εαψα =⇒ ρ(x) =

∑

α

nα|ψα(x)|2

Adaptation of chiral EFT MBPT to Skyrme HFB form

ESkyrme =τ

2M+

38

t0ρ2 +116

t3ρ2+α +116

(3t1 + 5t2)ρτ+1

64(9t1 − 5t2)|∇ρ|2 + · · ·

=⇒ EDME =τ

2M+ A[ρ] + B[ρ]τ + C[ρ]|∇ρ|2 + · · ·

Orbitals and Occupation #’s

Kohn−Sham Potentials

energyfunctional

HFBsolver

DME

ρρA[ ], B[ ], ...

VKS(r) =δEint[ρ]

δρ(r)⇐⇒ [−∇2

2m+VKS(x)]ψα = εαψα =⇒ ρ(x) =

∑

α

nα|ψα(x)|2

Density matrix expansion (DME) revisited [Negele/Vautherin]

Dominant chiral EFT MBPT contributions can be put into form

〈V 〉 ∼∫

dR dr12 dr34 ρ(r1, r3)K (r12, r34)ρ(r2, r4)

r1r2

ρ(r1,r3)ρ(r2,r4)

r3 r4

K(r1-r2, r3-r4)

finite range and non-local resummed vertices K (+ NNN)

DME: Expand KS ρ in local operators w/factorized non-locality

ρ(r1, r2) =∑

εα≤εF

ψ†α(r1)ψα(r2) =∑

n

Πn(r)〈On(R)〉 r1r2

R-r/2 +r/2

with 〈On(R)〉 = {ρ(R),∇2ρ(R), τ(R), · · · } maps 〈V 〉 to Skyrme-like EDF!

Adds density dependences, isovector, . . . missing in SkyrmeOriginal DME expands about nuclear matter (k -space + NNN)

ρ(R+r/2,R−r/2) ≈ 3j1(skF)

skFρ(R)+

35j3(skF)

2sk3F

(14∇2ρ(R)−τ(R)+

35

k2Fρ(R)+· · ·

)

Density matrix expansion (DME) revisited [Negele/Vautherin]

Dominant chiral EFT MBPT contributions can be put into form

〈V 〉 ∼∫

dR dr12 dr34 ρ(r1, r3)K (r12, r34)ρ(r2, r4)

r1r2

ρ(r1,r3)ρ(r2,r4)

r3 r4

K(r1-r2, r3-r4)

finite range and non-local resummed vertices K (+ NNN)

DME: Expand KS ρ in local operators w/factorized non-locality

ρ(r1, r2) =∑

εα≤εF

ψ†α(r1)ψα(r2) =∑

n

Πn(r)〈On(R)〉 r1r2

R-r/2 +r/2

with 〈On(R)〉 = {ρ(R),∇2ρ(R), τ(R), · · · } maps 〈V 〉 to Skyrme-like EDF!

Adds density dependences, isovector, . . . missing in Skyrme

Original DME expands about nuclear matter (k -space + NNN)

ρ(R+r/2,R−r/2) ≈ 3j1(skF)

skFρ(R)+

35j3(skF)

2sk3F

(14∇2ρ(R)−τ(R)+

35

k2Fρ(R)+· · ·

)

Density matrix expansion (DME) revisited [Negele/Vautherin]

Dominant chiral EFT MBPT contributions can be put into form

〈V 〉 ∼∫

dR dr12 dr34 ρ(r1, r3)K (r12, r34)ρ(r2, r4)

r1r2

ρ(r1,r3)ρ(r2,r4)

r3 r4

K(r1-r2, r3-r4)

finite range and non-local resummed vertices K (+ NNN)

DME: Expand KS ρ in local operators w/factorized non-locality

ρ(r1, r2) =∑

εα≤εF

ψ†α(r1)ψα(r2) =∑

n

Πn(r)〈On(R)〉 r1r2

R-r/2 +r/2

with 〈On(R)〉 = {ρ(R),∇2ρ(R), τ(R), · · · } maps 〈V 〉 to Skyrme-like EDF!

Adds density dependences, isovector, . . . missing in SkyrmeOriginal DME expands about nuclear matter (k -space + NNN)

ρ(R+r/2,R−r/2) ≈ 3j1(skF)

skFρ(R)+

35j3(skF)

2sk3F

(14∇2ρ(R)−τ(R)+

35

k2Fρ(R)+· · ·

)

Full ab-initio: Is Negele-Vautherin DME good enough?

Try best nuclear matter with RG-softened χ-EFT NN/NNN

−1600

−1400

−1200

−1000

−800

−600

−400

−200

0

200

400

600Vsrg λ = 2.0 fm−1 (N3LO)

A

B

C

DME Sly4total

40Ca

<V> <V>

DMEtotal

E

Sly4E

NN + NNN scaled to "fit" NM

HFBRAD

0 1 2 3 4 5 6r [fm]

0

0.02

0.04

0.06

0.08

0.1

dens

ity [f

m−3

]

Skyrme SLy4Vsrg DME

16O40Ca

HFBRAD

λ = 2 fm−1 ("fit")

Do densities look like nuclei from Skyrme EDF’s? Yes!

Are the error bars competitive? No! 1 MeV/A off in 40Ca

Improved DME for pion exchange [Gebremariam et al.]

Phase-space averaging for finite nuclei (symmetries, sum rules)

20 30 40 50 60Cr neutron number

0

1

2

3

4

5

6

7

E (

MeV

)

ExactNVDMEPI-DME IPI-DME II

96 104 112 120 128Pb neutron number

0

2

4

6

8

10

E (

MeV

)ExactNVDMEPI-DME IPI-DME II

Long-range chiral EFT=⇒ extended Skyrme

Add long-range (π-exchange)contributions in the densitymatrix expansion (DME)

NN/NNN through N2LO[Gebremariam et al.]

Refit Skyrme parameters forshort-range parts

Test for sensitities andimproved observables (e.g.,isotope chains) [NUCLEI]

Spin-orbit couplings from 2π3NF particularly interesting

Can we “see” the pion inmedium to heavy nuclei?

DFT development with hybrid Skyrme/DME [M. Kortelainen et al.]

Neutron Matter: Neutrons in a Trap

What are the properties of neutron-rich matter?

Protons and neutrons form self-bound system

Can bind neutrons by applying an external trap

!

Uext

Neutron Drops (mini neutron stars) calculated with Coupled-Cluster theory Use external harmonic oscillator potential, varying

!

!" ext

Warm-up calculations: Neutron drops with Minnesota potential

Results are promising! On-going work (Bogner et al.):

Extend “fit” comparisons to new local chiral EFT interactionsIterate with EDF optimization technologyValidate DME and extend to higher-order contributions

DFT development with hybrid Skyrme/DME [M. Kortelainen et al.]

Neutron Matter: Neutrons in a Trap

What are the properties of neutron-rich matter?

Protons and neutrons form self-bound system

Can bind neutrons by applying an external trap

!

Uext

Neutron Drops (mini neutron stars) calculated with Coupled-Cluster theory Use external harmonic oscillator potential, varying

!

!" ext

Warm-up calculations: Neutron drops with Minnesota potential

Results are promising! On-going work (Bogner et al.):

Extend “fit” comparisons to new local chiral EFT interactionsIterate with EDF optimization technologyValidate DME and extend to higher-order contributions

DFT development with hybrid Skyrme/DME [M. Kortelainen et al.]

Neutron Matter: Neutrons in a Trap

What are the properties of neutron-rich matter?

Protons and neutrons form self-bound system

Can bind neutrons by applying an external trap

!

Uext

Neutron Drops (mini neutron stars) calculated with Coupled-Cluster theory Use external harmonic oscillator potential, varying

!

!" ext

Warm-up calculations: Neutron drops with Minnesota potential

Results are promising! On-going work (Bogner et al.):

Extend “fit” comparisons to new local chiral EFT interactionsIterate with EDF optimization technologyValidate DME and extend to higher-order contributions

DFT development with hybrid Skyrme/DME [M. Kortelainen et al.]

Neutron Matter: Neutrons in a Trap

What are the properties of neutron-rich matter?

Protons and neutrons form self-bound system

Can bind neutrons by applying an external trap

!

Uext

Neutron Drops (mini neutron stars) calculated with Coupled-Cluster theory Use external harmonic oscillator potential, varying

!

!" ext

Warm-up calculations: Neutron drops with Minnesota potential

Results are promising! On-going work (Bogner et al.):

Extend “fit” comparisons to new local chiral EFT interactionsIterate with EDF optimization technologyValidate DME and extend to higher-order contributions

Chiral Dynamics of Nuclear MatterMunich Group (Kaiser, Fritsch, Holt, Weise, . . . )

Basic idea: ChPT loop expansion becomes EOS expansion:

E(kF) =∞∑

n=2

knF fn(kF/mπ,∆/mπ) [∆ = M∆ −MN ≈ 300 MeV]

1st pass: N’s and π’s =⇒ count kF’s by medium insertions

Saturation from Pauli-blocking of iterated 1π-exchangeProblems with single-particle and isospin properties and . . .

2nd pass: include πN∆ dynamics:

Chiral Dynamics of Nuclear Matter (cont.)Munich Group (Kaiser, Fritsch, Holt, Weise, . . . )

3-Loop: Fit nuclear matter saturation, predict neutron matter

Substantial improvement in s.p. properties, spin-stability, . . .Issues for perturbative chiral expansion of nuclear matter:

higher orders, convergence? power counting?relation of LEC’s to free space EFT

Apply DME to get DFT functional

Effective theory for Nuclear EDFsJ. Dobaczewski, K. Bennaceur, F. Raimondi, J. Phys. G 39, 125103 (2012)

Seek spectroscopic quality functional (including single-particle levels)

Consider non-ab-initio formulation but with firm theoretical basis

Claim: resolution scale of chiral EFT is higher than needed

Rather than k . 2mπ or kF, consider δk to dissociate a nucleon:δEkin = ~2kF δk/M ≈ 0.25~c δk ≈ 8 MeV =⇒ δk ≈ 32 MeV/~c

And describe nuclear excitations and shell-effects at the 1 MeVenergy, which implies δk ≈ 4 MeV/~c and below

So from this perspective the pion is a high-energy dof

Effective theory for Nuclear EDFsJ. Dobaczewski, K. Bennaceur, F. Raimondi, J. Phys. G 39, 125103 (2012)

Seek spectroscopic quality functional (including single-particle levels)

Consider non-ab-initio formulation but with firm theoretical basis

Claim: resolution scale of chiral EFT is higher than needed

Rather than k . 2mπ or kF, consider δk to dissociate a nucleon:δEkin = ~2kF δk/M ≈ 0.25~c δk ≈ 8 MeV =⇒ δk ≈ 32 MeV/~c

And describe nuclear excitations and shell-effects at the 1 MeVenergy, which implies δk ≈ 4 MeV/~c and below

So from this perspective the pion is a high-energy dof

Strategy: expand “pseudopotential”, which specifies the EDF by foldingwith an uncorrelated Slater determinant, found self-consistently

Spirit of mean-field approaches (and technology)

Gives full functional within HF approximation (completeness)

Self-interaction problem solved by deriving EDF in HF form

Effective theory for Nuclear EDFsJ. Dobaczewski, K. Bennaceur, F. Raimondi, J. Phys. G 39, 125103 (2012)

Seek spectroscopic quality functional (including single-particle levels)

Consider non-ab-initio formulation but with firm theoretical basis

Claim: resolution scale of chiral EFT is higher than needed

Rather than k . 2mπ or kF, consider δk to dissociate a nucleon:δEkin = ~2kF δk/M ≈ 0.25~c δk ≈ 8 MeV =⇒ δk ≈ 32 MeV/~c

And describe nuclear excitations and shell-effects at the 1 MeVenergy, which implies δk ≈ 4 MeV/~c and below

So from this perspective the pion is a high-energy dof

Regulated zero-range interaction =⇒ introduces resolution scale

Gaussians smear away details of nuclear densities

Describe residual smooth variations within a controlled expansion

Fit coupling constants to data with constraints (continuity equation)

Check for independence of expansion scale (renormalized?),convergence, and naturalness

Regularized pseudopotential: pionless-EFT-like expansionCentral two-body regularized pseudopotential (also s.o. and tensor)

V (r′1, r′2; r1, r2) =

4∑

i=1

PiOi (k,k′)δ(r′1 − r1)δ(r′2 − r2)ga(r1 − r2)

with operators Pi (spin,isospin exchange), Oi (derivative), k,k′(relative momentum), while a sets the resolution scale:

ga(r) =1

(a√π)3 e−r2/a2 −→

a→0δ(r)

Simplified special case: If Oi = Oi (k + k′), then

V (r) =4∑

i=1

PiOi (k)ga(r) =4∑

i=1

Pi

nmax∑

n=0

V (i)2n∇2nga(r)

where V (i)2n are the coupling constants to be fit

EDF as functional of the one-body density matrix (cf. Gogny)

Eeff[ρ(r, r′)] =

∫dr∫

dr′ V (r− r′)[ρ(r)ρ(r′)− ρ(r, r′)ρ(r′, r)]

Does it work like an effective theory? Proof of principle

Order-by-order convergencetest against pseudo-data(from a Gogny functional)

factor of 4 at each ordercan fine-tune couplings

N2LO regulator independent;N3LO converged energy/radius

Independence of the regulatorscale a (i.e., flatness ) andindependent of referencenucleus

Error plots vs. A showsconvergence patterns

Fixed a = 0.85 fm; exponentialdecrease of constants with nwith Λ ≈ 700 MeV

-3

-2

-1

0

0

2

4

6

-4

0

4

-0.3

0.0

0.3

0.9 1.0 1.1

-10

0

10

-0.5

0.0

0.5

0.9 1.0 1.1

Regularization scale a (fm)

∆B

(%) ∆

R(%

)

NLO NLO

(a) (b)

(c) (d)

(e) (f)

N3LO N3LO

N2LO N2LO

16404856

78100132208

Does it work like an effective theory? Proof of principle

Order-by-order convergencetest against pseudo-data(from a Gogny functional)

factor of 4 at each ordercan fine-tune couplings

N2LO regulator independent;N3LO converged energy/radius

Independence of the regulatorscale a (i.e., flatness ) andindependent of referencenucleus

Error plots vs. A showsconvergence patterns

Fixed a = 0.85 fm; exponentialdecrease of constants with nwith Λ ≈ 700 MeV

0.9 1.0 1.1

0.0

0.5

1.0

0.9 1.0 1.1

0.0

0.5

1.0

0.0

0.5

1.0

Regularization scale a (fm)

∆B

/ ∆

B (

208P

b)

(a)

(c)

(e)

(b)

(d)

(f)

∆R

/ ∆R

(208P

b)

40 80 120 160 200

NLO

N2LO

N3LO

0.4

0.6

0.8

1.0

40 80 120 160 200

Mass number A

∆b

/ ∆

b(2

08P

b)

(a) (b)

∆r

/ ∆r (

208P

b)

Does it work like an effective theory? Proof of principle

Order-by-order convergencetest against pseudo-data(from a Gogny functional)

factor of 4 at each ordercan fine-tune couplings

N2LO regulator independent;N3LO converged energy/radius

Independence of the regulatorscale a (i.e., flatness ) andindependent of referencenucleus

Error plots vs. A showsconvergence patterns

Fixed a = 0.85 fm; exponentialdecrease of constants with nwith Λ ≈ 700 MeV

1

10

100

1000

0 2 4 6

|V2n| (M

eV

fm3+

2n)

Order of expansion 2n

Λ -2n

Wigner

Bartlett

Heisenberg

Majorana

0.9 1.0 1.1

-2

-1

0

1

0.9 1.0 1.1

-2

-1

0

1

Regularization scale a (fm)

v2

n(n

atu

ral

un

its)

(a) n=0

(c) n=2

(b) n=1

(d) n=3

Wigner

Bartlett

Heisenberg

Majorana

Outlook: Two strategies for nuclear EDFs from EFT

Both extend or modify conventional EDF forms in controlled ways

1 Long-distance chiral physics from an EFT expansionDensity matrix expansion (DME) applied to NN and NNN diagramsRe-fit residual Skyrme parameters and test descriptionPerturbative expansion justified by phase-space-based powercounting =⇒ rethink chiral EFT formulation

2 Extend existing functionals following EFT principlesNon-local regularized pseudo-potential [Raimondi et al., 1402.1556]

Optimize pseudo-potential to experimental dataTest with correlation analysis technology

Stay tuned for many developments in the near future!

Paths to a nuclear EDF =⇒Where can EFT contribute?1 Improve empirical energy functional (Skyrme, Gogny or RMF)2 Emulate Coulomb DFT: LDA based on precision calculation of

uniform system E [ρ] =∫

dr E(ρ(r)) plus constrained gradientcorrections (∇ρ factors) =⇒ “non-empirical EDF”Fayans and collaborators(e.g., nucl-th/0009034)

Ev = 23εFρ0

[av

+1−hv

1+x1/3+

1−hv2+x1/3

+

x2+

+ av−

1−hv1−x1/3

+

1−hv2−x1/3

+

x2−

]

where x± = (ρn ± ρp)/2ρ0(also Baldo et al., 1005.1810)

SLDA+ (Bulgac et al.)

Neutron drops in traps0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

(fm-3)

-200

204060

80100120140160

E/A

(MeV

)

neutron matter

nuclear matter

FP81WFF88FaNDF0

3 EDF from perturbative chiral interactions + DME (Kaiser et al.)

4 Kohn-Sham DFT from EFT-based, RG-softened V ’s (Bogner et al.)

5 RG approach (S. Kemler, J. Braun [1304.1161], from Polonyi and Schwenk)

Paths to a nuclear EDF =⇒Where can EFT contribute?1 Improve empirical energy functional (Skyrme, Gogny or RMF)2 Emulate Coulomb DFT: LDA based on precision calculation of

uniform system E [ρ] =∫

dr E(ρ(r)) plus constrained gradientcorrections (∇ρ factors) =⇒ “non-empirical EDF”Fayans and collaborators(e.g., nucl-th/0009034)

Ev = 23εFρ0

[av

+1−hv

1+x1/3+

1−hv2+x1/3

+

x2+

+ av−

1−hv1−x1/3

+

1−hv2−x1/3

+

x2−

]

where x± = (ρn ± ρp)/2ρ0(also Baldo et al., 1005.1810)

SLDA+ (Bulgac et al.)

Neutron drops in traps0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

(fm-3)

-200

204060

80100120140160

E/A

(MeV

)

neutron matter

nuclear matter

FP81WFF88FaNDF0

3 EDF from perturbative chiral interactions + DME (Kaiser et al.)

4 Kohn-Sham DFT from EFT-based, RG-softened V ’s (Bogner et al.)

5 RG approach (S. Kemler, J. Braun [1304.1161], from Polonyi and Schwenk)

What is needed for ab initio Kohn-Sham DFT =⇒ EDF?1 Need MBPT to work with tuned U [H = (T + U) + (V − U)]

0.8 1.0 1.2 1.4 1.6

kF [fm

−1]

−20

−15

−10

−5

0

5

En

erg

y/n

ucl

eon

[M

eV] Λ = 1.8 fm

−1

Λ = 2.0 fm−1

Λ = 2.2 fm−1

Λ = 2.8 fm−1

0.8 1.0 1.2 1.4 1.6

kF [fm

−1]

0.8 1.0 1.2 1.4 1.6

kF [fm

−1]

Hartree-Fock

Empiricalsaturationpoint 2nd order

Vlow k

NN from N3LO (500 MeV)

3NF fit to E3H and r4He

3rd order pp+hh

2.0 < Λ3NF

< 2.5 fm−1

Lower resolution Λ from Λχ with RG to make more perturbativeIf convergence insensitive to U =⇒ choose so KS density exact

2 Need to calculate VKS(r) from δE [ρ, τ, . . .]/δρ(r), etc.,but diagrams depend non-locally on KS orbitals

Density matrix expansion (DME) =⇒ explicit densitiesOr use chain rule =⇒ “optimized effective potential” (OEP)