Effective Field Theory (EFT) and Density Functionals

Transcription

1 Effective Field Theory (EFT) and Density Functionals Dick Furnstahl Department of Physics Ohio State University Extremes of the Nuclear Landscape September 1, 2014

2 State-of-the-art Skyrme EDFs Is there a limit to improvement of Skyrme rms energy residual? Recently many advances by UNEDF/NUCLEI, FIDIPRO, and others to improve/test EDFs Extra observables and ab initio calculations in neutron drops for constraints (e.g., on isovector) Sophisticated fit and correlation analysis implies the EDF is not limited by the parameter fitting rms residual (MeV) Bertsch,)Sabbey,))) ))and)uusnakki,) Phys.)Rev.)C)71,) ))054311)(2005)) dimension of parameter space But still don t beat the energy barrier (and not nearly as good energy rms as mass models) = limit of Skyrme EDF strategy?

3 State-of-the-art Skyrme EDFs Is there a limit to improvement of Skyrme rms energy residual? Recently many advances by UNEDF/NUCLEI, FIDIPRO, and others to improve/test EDFs Extra observables and ab initio calculations in neutron drops for constraints (e.g., on isovector) Sophisticated fit and correlation analysis implies the EDF is not limited by the parameter fitting But still don t beat the energy barrier (and not nearly as good energy rms as mass models) = limit of Skyrme EDF strategy? rms residual (MeV) E tot /N 4/3 (MeV) Bertsch,)Sabbey,))) ))and)uusnakki,) Phys.)Rev.)C)71,) ))054311)(2005)) dimension of parameter space hω =10MeV hω =5MeV AFDMC M.#Kortelainen#et#al.,#PRC#89,#054314#(2014)# UNEDF0 UNEDF1 UNEDF2 SLy4 SLy4 adj N

4 State-of-the-art Skyrme EDFs Is there a limit to improvement of Skyrme rms energy residual? Recently many advances by UNEDF/NUCLEI, FIDIPRO, and others to improve/test EDFs Extra observables and ab initio calculations in neutron drops for constraints (e.g., on isovector) Sophisticated fit and correlation analysis implies the EDF is not limited by the parameter fitting But still don t beat the energy barrier (and not nearly as good energy rms as mass models) = limit of Skyrme EDF strategy? rms residual (MeV) Bertsch,)Sabbey,))) ))and)uusnakki,) Phys.)Rev.)C)71,) ))054311)(2005)) dimension of parameter space masses radii OES FI s.p.e.

5 State-of-the-art Skyrme EDFs Is there a limit to improvement of Skyrme rms energy residual? Recently many advances by UNEDF/NUCLEI, FIDIPRO, and others to improve/test EDFs Extra observables and ab initio calculations in neutron drops for constraints (e.g., on isovector) Sophisticated fit and correlation analysis implies the EDF is not limited by the parameter fitting But still don t beat the energy barrier (and not nearly as good energy rms as mass models) = limit of Skyrme EDF strategy? rms residual (MeV) Masses (def) Bertsch,)Sabbey,))) ))and)uusnakki,) Phys.)Rev.)C)71,) ))054311)(2005)) dimension of parameter space Masses (sph) Radii OES FI spe

6 State-of-the-art Skyrme EDFs Is there a limit to improvement of Skyrme rms energy residual? Recently many advances by UNEDF/NUCLEI, FIDIPRO, and others to improve/test EDFs Extra observables and ab initio calculations in neutron drops for constraints (e.g., on isovector) Sophisticated fit and correlation analysis implies the EDF is not limited by the parameter fitting But still don t beat the energy barrier (and not nearly as good energy rms as mass models) = limit of Skyrme EDF strategy? rms residual (MeV) E th - E exp (MeV) (a) Bertsch,)Sabbey,))) ))and)uusnakki,) Phys.)Rev.)C)71,) ))054311)(2005)) dimension of parameter space UNEDF2 M.#Kortelainen#et#al.,#PRC#89,#054314#(2014)# Neutron Number N

7 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

8 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 that there seems to be interesting physics at all scales.

9 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 that there seems to be interesting physics at all scales. To do physics amid this remarkable richness, it is convenient to be able to isolate a set of phenomena from all the rest, so that we can describe it without having to understand everything.... We can divide up the parameter space of the world into different regions, in each of which there is a different appropriate description of the important physics. Such an appropriate description of the important physics is an effective theory.

10 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 that there seems to be interesting physics at all scales. To do physics amid this remarkable richness, it is convenient to be able to isolate a set of phenomena from all the rest, so that we can describe it without having to understand everything.... We can divide up the parameter space of the world into different regions, in each of which there is a different appropriate description of the important physics. Such an appropriate description of the important physics is an effective theory. The common idea is that if there are parameters that are very large or very small compared to the physical quantities (with the same dimension) that we are interested in, we may get a simpler approximate description of the physics by setting the small parameters to zero and the large parameters to infinity. Then the finite effects of the parameters can be included as small perturbations about this simple approximate starting point. E.g., non-relativistic QM: c E.g., pionless effective field theory (EFT): m π, M N E.g., chiral effective field theory (EFT): m π 0, M N

11 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 that there seems to be interesting physics at all scales. To do physics amid this remarkable richness, it is convenient to be able to isolate a set of phenomena from all the rest, so that we can describe it without having to understand everything.... We can divide up the parameter space of the world into different regions, in each of which there is a different appropriate description of the important physics. Such an appropriate description of the important physics is an effective theory. The common idea is that if there are parameters that are very large or very small compared to the physical quantities (with the same dimension) that we are interested in, we may get a simpler approximate description of the physics by setting the small parameters to zero and the large parameters to infinity. Then the finite effects of the parameters can be included as small perturbations about this simple approximate starting point. E.g., non-relativistic QM: c E.g., pionless effective field theory (EFT): m π, M N E.g., chiral effective field theory (EFT): m π 0, M N Goals: model independent, improvable, and with error estimates Features: limited domain, breakdown is predicted internally

12 Principles common to all E(F)Ts (but can be in different forms) In coordinate space, define a to separate short and long distance In momentum space, use Λ to separate high and low momenta Much freedom how this is done (e.g., different regulator forms) = different scales / schemes

13 Principles common to all E(F)Ts (but can be in different forms) In coordinate space, define a to separate short and long distance In momentum space, use Λ to separate 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 from completeness. All terms allowed by symmetries should be present, up to redundancies. One way: QFT! Power counting from small parameters (e.g., ratio of scales: Q/Λ) = systematic

14 Classical analogy to EFT: Multipole expansion If we have a localized charge distribution ρ(r) within a volume characterized by distance a, the electrostatic potential is φ(r) d 3 r ρ(r) R r If we expand 1/ R r for r R, we get the multipole expansion d 3 ρ(r) r R r = q R + 1 R 3 R i P i + 1 6R 5 (3R i R j δ ij R 2 )Q ij + i = pointlike total charge q, dipole moment P i, quadrupole Q ij : q = d 3 r ρ(r) P i = d 3 r ρ(r) r i Q ij = d 3 r ρ(r)(3r i r j δ ij r 2 ) ij

15 Classical analogy to EFT: Multipole expansion If we have a localized charge distribution ρ(r) within a volume characterized by distance a, the electrostatic potential is φ(r) d 3 r ρ(r) R r If we expand 1/ R r for r R, we get the multipole expansion d 3 ρ(r) r R r = q R + 1 R 3 R i P i + 1 6R 5 (3R i R j δ ij R 2 )Q ij + i = pointlike total charge q, dipole moment P i, quadrupole Q ij : q = d 3 r ρ(r) P i = d 3 r ρ(r) r i Q ij = d 3 r ρ(r)(3r i r j δ ij r 2 ) Hierarchy of terms from separation of scales = a/r expansion ij

16 Classical analogy to EFT: Multipole expansion If we have a localized charge distribution ρ(r) within a volume characterized by distance a, the electrostatic potential is φ(r) d 3 r ρ(r) R r If we expand 1/ R r for r R, we get the multipole expansion d 3 ρ(r) r R r = q R + 1 R 3 R i P i + 1 6R 5 (3R i R j δ ij R 2 )Q ij + i = pointlike total charge q, dipole moment P i, quadrupole Q ij : q = d 3 r ρ(r) P i = d 3 r ρ(r) r i Q ij = d 3 r ρ(r)(3r i r j δ ij r 2 ) Hierarchy of terms from separation of scales = a/r expansion Or: include known long-distance structure or reference distribution ij

17 Classical analogy to EFT: Multipole expansion If we have a localized charge distribution ρ(r) within a volume characterized by distance a, the electrostatic potential is φ(r) d 3 r ρ(r) R r If we expand 1/ R r for r R, we get the multipole expansion d 3 ρ(r) r R r = q R + 1 R 3 R i P i + 1 6R 5 (3R i R j δ ij R 2 )Q ij + i = pointlike total charge q, dipole moment P i, quadrupole Q ij : q = d 3 r ρ(r) P i = d 3 r ρ(r) r i Q ij = d 3 r ρ(r)(3r i r j δ ij r 2 ) 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) ij

18 Different E(F)Ts depending on scale/observables of interest LQCD constituent quarks scale& separa)on& ab initio CI Resolution DFT collective models

19 Different E(F)Ts depending on scale/observables of interest LQCD constituent quarks ab initio CI DFT scale& separa)on& Resolution There is not just one EFT! Chiral EFT: nucleons, [ s,] pions: {q, m π }/Λ χ m ρ Pionless EFT: nucleons only (low-energy few-body) or nucleons and clusters (halo) ET for deformed nuclei: systematic collective dofs (Papenbrock, Weidenmueller) EFT at Fermi surface (Landau-Migdal theory): quasi-nucleons collective models Scale and scheme dependence also means non-uniqueness (e.g., of EDFs)

20 Comparing dilute EFT local density and Skyrme functionals Skyrme EDF (for N = Z and without pairing) { τ E[ρ, τ, J] = dr 2M t 0ρ (3t 1 + 5t 2 )ρτ (9t 1 5t 2 )( ρ) W 0ρ J + 1 } 16 t 3ρ 2+α + where ρ(r) = i ψ i(r) 2, τ(r) = i ψ i(r) 2,...

21 Comparing dilute EFT local density and Skyrme functionals Skyrme EDF (for N = Z and without pairing) { τ E[ρ, τ, J] = dr 2M t 0ρ (3t 1 + 5t 2 )ρτ (9t 1 5t 2 )( ρ) W 0ρ J + 1 } 16 t 3ρ 2+α + where ρ(r) = i ψ i(r) 2, τ(r) = i ψ i(r) 2,... Pionless zero-range EFT = dilute LDA ρτj EDF (with V external = 0) { τ E[ρ, τ, J] = dr 2M C 0ρ (3C 2 + 5C 2)ρτ (9C 2 5C 2)( ρ) C 2 ρ J + c 1 2M C2 0ρ 7/3 + c 2 2M C3 0 ρ8/3 + 1 } 16 D 0ρ 3 +

22 Comparing dilute EFT local density and Skyrme functionals Skyrme EDF (for N = Z and without pairing) { τ E[ρ, τ, J] = dr 2M t 0ρ (3t 1 + 5t 2 )ρτ (9t 1 5t 2 )( ρ) W 0ρ J + 1 } 16 t 3ρ 2+α + where ρ(r) = i ψ i(r) 2, τ(r) = i ψ i(r) 2,... Pionless zero-range EFT = dilute LDA ρτj EDF (with V external = 0) { τ E[ρ, τ, J] = dr 2M C 0ρ (3C 2 + 5C 2)ρτ (9C 2 5C 2)( ρ) C 2 ρ J + c 1 2M C2 0ρ 7/3 + c 2 2M C3 0 ρ8/3 + 1 } 16 D 0ρ 3 + Looks like same functional as LDA dilute Fermi gas with t i C i! 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?

23 Naturalness in Skyrme coefficients as EFT signatures? Georgi (1993): f π for strongly interacting fields; rest is Λ χ m ρ ; c lmn O(1) ( N ) l ( ) m ( N π µ ) n, m π L χ eft = c lmn fπλ 2 f χ f π Λ πλ 2 2 χ f π 100 MeV χ Chiral NDA analysis for EDFs: [Friar et al., rjf et al.] [ N ] l [ ] n N c fπλ 2 f χ Λ πλ 2 2 χ χ = ρ N N τ N N J N N Density expansion? 1000 Λ χ 500 = 1 7 ρ 0 f 2 πλ χ 1 4 Also gradient expansion Applied to RMF, Skyrme EDFs energy/particle (MeV) What does this tell us about accuracy limits? ε 0 natural (Λ=600 MeV) Skyrme ρ n RMFT-II ρ n net RMFT-I ρ n net k F = 1.35 fm power of density

24 Naturalness in Skyrme coefficients as QCD signatures? Georgi (1993): f π for strongly interacting fields; rest is Λ χ m ρ ; c lmn O(1) ( N ) l ( ) m ( N π µ ) n, m π L χ eft = c lmn fπλ 2 f χ f π Λ πλ 2 2 χ f π 100 MeV χ Check chiral naturalness for large set of Skyrme EDFs: = 687 MeV, iv. scaled Coupling constant C ρ C ρ D C τ C ρ C J C J Kortelainen*et*al.*(2010)* ~50*scaled*sets*of*Skyrme*coefficients* Looks like natural distribution = Does this mean pionful EFT is needed?

25 Conceptual basis of (chiral) effective field theory Separate the short-distance (UV) from long-distance (IR) physics = scale Λ χ m ρ Exploit chiral symmetry = hierarchical treatment of long-distance physics Use complete basis for short-distance physics = hierarchy à la multipoles

26 Conceptual basis of (chiral) effective field theory Separate the short-distance (UV) from long-distance (IR) physics = scale Λ χ m ρ Exploit chiral symmetry = hierarchical treatment of long-distance physics Use complete basis for short-distance physics = hierarchy à la multipoles From QCD to nuclear physics QCD T T T = ChPT From QCD to nuclear physics QCD ChPT T = NN'interac/on'is'strong:(resumma-ons/nonperturba-ve(methods(needed 1/m N J(expansion:(nonrela-vis-c(problem(((((((((((((((((((((((((((((()( pi M mn the(qm(ajbody(p NN'interac/on'is'strong:(resumma-ons/nonperturba-ve(methods(needed Generate a nonrelativistic apple X A 1/mpotential N J(expansion:(nonrela-vis-c(problem(((((((((((((((((((((((((((((()( for many-body 2 m i + O(mN 3 ) + V 2N + V 3N + V p i M m N the(qm(ajbody(problem 4N +... = E i=1 2m N derived&within&chpt methods (controversies!) apple X A 2 (((((Construct(effec-ve(poten-al(perturba-vely Possible m ieft 2m + O(m for nuclear N 3 ) + V 2N EDF: + V 3N + Use V 4N chiral +... Hamiltonian = E in MBPT.... [H. Krebs]

27 Adaptation of chiral EFT MBPT to Skyrme HFB form E Skyrme = τ 2M t 0ρ t 3ρ 2+α (3t 1 + 5t 2 )ρτ (9t 1 5t 2 ) ρ 2 + = E DME = τ 2M + A[ρ] + B[ρ]τ + C[ρ] ρ 2 + Kohn Sham Potentials Skyrme energy functional t 0, t1, t 2,... HFB solver Orbitals and Occupation # s V KS (r) = δe int[ρ] δρ(r) [ 2 2m +V KS(x)]ψ α = ε α ψ α = ρ(x) = α n α ψ α (x) 2

28 Adaptation of chiral EFT MBPT to Skyrme HFB form E Skyrme = τ 2M t 0ρ t 3ρ 2+α (3t 1 + 5t 2 )ρτ (9t 1 5t 2 ) ρ 2 + = E DME = τ 2M + A[ρ] + B[ρ]τ + C[ρ] ρ 2 + Kohn Sham Potentials DME energy functional A[ ρ], B[ ρ],... HFB solver Orbitals and Occupation # s V KS (r) = δe int[ρ] δρ(r) [ 2 2m +V KS(x)]ψ α = ε α ψ α = ρ(x) = α n α ψ α (x) 2

29 Density matrix expansion (DME) revisited [Negele/Vautherin] Dominant chiral EFT MBPT contributions can be put into form V dr dr 12 dr 34 ρ(r 1, r 3 )K (r 12, r 34 )ρ(r 2, r 4 ) ρ(r 1,r 3 ) r 1 r 2 K(r 1 -r 2, r 3 -r 4 ) ρ(r 2,r 4 ) r 3 r 4 finite range and non-local resummed vertices K (+ NNN)

30 Density matrix expansion (DME) revisited [Negele/Vautherin] Dominant chiral EFT MBPT contributions can be put into form V dr dr 12 dr 34 ρ(r 1, r 3 )K (r 12, r 34 )ρ(r 2, r 4 ) ρ(r 1,r 3 ) r 1 r 2 K(r 1 -r 2, r 3 -r 4 ) ρ(r 2,r 4 ) r 3 r 4 finite range and non-local resummed vertices K (+ NNN) DME: Expand KS ρ in local operators w/factorized non-locality ρ(r 1, r 2 ) = ψ r α(r 1 )ψ α (r 2 ) = Π n (r) O n (R) 1 r 2 ɛ α ɛ F n -r/2 R +r/2 with O n (R) = {ρ(r), 2 ρ(r), τ(r), } maps V to Skyrme-like EDF! Adds density dependences, isovector,... missing in Skyrme

31 Density matrix expansion (DME) revisited [Negele/Vautherin] Dominant chiral EFT MBPT contributions can be put into form V dr dr 12 dr 34 ρ(r 1, r 3 )K (r 12, r 34 )ρ(r 2, r 4 ) ρ(r 1,r 3 ) r 1 r 2 K(r 1 -r 2, r 3 -r 4 ) ρ(r 2,r 4 ) r 3 r 4 finite range and non-local resummed vertices K (+ NNN) DME: Expand KS ρ in local operators w/factorized non-locality ρ(r 1, r 2 ) = ψ α(r 1 )ψ α (r 2 ) = Π n (r) O n (R) ɛ α ɛ F n r 1 r 2 with O n (R) = {ρ(r), 2 ρ(r), τ(r), } maps V to Skyrme-like EDF! -r/2 R +r/2 Adds density dependences, isovector,... missing in Skyrme Original DME expands about nuclear matter (k-space + NNN) ρ(r+r/2, R r/2) 3j 1(sk F ) ρ(r)+ 35j 3(sk F ) ( 1 sk F 2skF ρ(r) τ(r)+ 3 ) 5 k F 2 ρ(r)+

32 Full ab-initio: Is Negele-Vautherin DME good enough? Try best nuclear matter with RG-softened χ-eft NN/NNN A V srg λ = 2.0 fm 1 (N 3 LO) B C DME Sly4 total <V> <V> 40 Ca HFBRAD DME total E Sly4 E NN + NNN scaled to "fit" NM density [fm 3 ] HFBRAD 16 O 40 Ca Skyrme SLy4 V srg DME λ = 2 fm 1 ("fit") r [fm] Do densities look like nuclei from Skyrme EDF s? Yes! Are the error bars competitive? No! 1 MeV/A off in 40 Ca

33 Improved DME for pion exchange [Gebremariam et al.] Phase-space averaging for finite nuclei (symmetries, sum rules) E (MeV) Exact NVDME PI-DME I PI-DME II E (MeV) Exact NVDME PI-DME I PI-DME II Cr neutron number Pb neutron number

34 Long-range chiral EFT = extended Skyrme Add long-range (π-exchange) contributions in the density matrix expansion (DME) NN/NNN through N 2 LO [Gebremariam et al.] Refit Skyrme parameters for short-range parts Test for sensitities and improved observables (e.g., isotope chains) [NUCLEI] Spin-orbit couplings from 2π 3NF particularly interesting Can we see the pion in medium to heavy nuclei?

35 DFT development with hybrid Skyrme/DME [M. Kortelainen et al.] utrons in a Trap atter? U ext Can bind neutrons by applying an external trap lated with Coupled-Cluster Warm-up calculations: theory Neutron drops with Minnesota potential l, varying!" ext

36 DFT development with hybrid Skyrme/DME [M. Kortelainen et al.] utrons in a Trap atter? U ext Can bind neutrons by applying an external trap lated with Coupled-Cluster Warm-up calculations: theory Neutron drops with Minnesota potential l, varying!" ext

37 DFT development with hybrid Skyrme/DME [M. Kortelainen et al.] eutrons in a Trap matter? U ext Can bind neutrons by applying an external trap ulated with Coupled-Cluster Warm-up calculations: theory Neutron drops with Minnesota potential al, varying!" ext

38 DFT development with hybrid Skyrme/DME [M. Kortelainen et al.] eutrons in a Trap matter? U ext Can bind neutrons by applying an external trap ulated with Coupled-Cluster Warm-up calculations: theory Neutron drops with Minnesota potential al, varying!" Results ext are promising! On-going work (Bogner et al.): Extend fit comparisons to new local chiral EFT interactions Iterate with EDF optimization technology Validate DME and extend to higher-order contributions

39 Chiral Dynamics of Nuclear Matter Munich Group (Kaiser, Fritsch, Holt, Weise,... ) Basic idea: ChPT loop expansion becomes EOS expansion: E(k F ) = kf n f n (k F /m π, /m π ) n=2 [ = M M N 300 MeV] 1st pass: N s and π s = count k F s by medium insertions Saturation from Pauli-blocking of iterated 1π-exchange Problems with single-particle and isospin properties and... 2nd pass: include πn dynamics:

40 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

41 Effective theory for Nuclear EDFs J. Dobaczewski, K. Bennaceur, F. Raimondi, J. Phys. G 39, (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 2m π or k F, consider δk to dissociate a nucleon: δe kin = 2 k F δk/m 0.25 c δk 8 MeV = δk 32 MeV/ c And describe nuclear excitations and shell-effects at the 1 MeV energy, which implies δk 4 MeV/ c and below So from this perspective the pion is a high-energy dof

42 Effective theory for Nuclear EDFs J. Dobaczewski, K. Bennaceur, F. Raimondi, J. Phys. G 39, (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 2m π or k F, consider δk to dissociate a nucleon: δe kin = 2 k F δk/m 0.25 c δk 8 MeV = δk 32 MeV/ c And describe nuclear excitations and shell-effects at the 1 MeV energy, 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 folding with 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

43 Effective theory for Nuclear EDFs J. Dobaczewski, K. Bennaceur, F. Raimondi, J. Phys. G 39, (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 2m π or k F, consider δk to dissociate a nucleon: δe kin = 2 k F δk/m 0.25 c δk 8 MeV = δk 32 MeV/ c And describe nuclear excitations and shell-effects at the 1 MeV energy, 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

44 Regularized pseudopotential: pionless-eft-like expansion Central two-body regularized pseudopotential (also s.o. and tensor) V (r 1, r 2; r 1, r 2 ) = 4 P i Ô i (k, k )δ(r 1 r 1 )δ(r 2 r 2 )g a (r 1 r 2 ) i=1 with operators P i (spin,isospin exchange), Ôi (derivative), k, k (relative momentum), while a sets the resolution scale: g a (r) = 1 (a 2 π) 3 e r /a 2 δ(r) a 0 Simplified special case: If Ôi = Ôi(k + k ), then V (r) = 4 P i Ô i (k)g a (r) = i=1 4 n max P i V (i) 2n 2n g a (r) where V (i) 2n are the coupling constants to be fit EDF as functional of the one-body density matrix (cf. Gogny) E eff [ρ(r, r )] = dr dr V (r r )[ρ(r)ρ(r ) ρ(r, r )ρ(r, r)] i=1 n=0

45 Does it work like an effective theory? Proof of principle Order-by-order convergence test against pseudo-data (from a Gogny functional) factor of 4 at each order can fine-tune couplings N 2 LO regulator independent; N 3 LO converged energy/radius B (%) (a) NLO (c) N 2 LO (e) N 3 LO (b) NLO (d) N 2 LO (f) N 3 LO R (%) Independence of the regulator scale a (i.e., flatness ) and independent of reference nucleus Regularization scale a (fm) Error plots vs. A shows convergence patterns Fixed a = 0.85 fm; exponential decrease of constants with n with Λ 700 MeV

46 Does it work like an effective theory? Proof of principle Order-by-order convergence test against pseudo-data (from a Gogny functional) factor of 4 at each order can fine-tune couplings N 2 LO regulator independent; N 3 LO converged energy/radius Independence of the regulator scale a (i.e., flatness ) and independent of reference nucleus Error plots vs. A shows convergence patterns Fixed a = 0.85 fm; exponential decrease of constants with n with Λ 700 MeV B / B ( 208 Pb) b / b ( 208 Pb) (a) (b) (c) (d) (e) (f) Regularization scale a (fm) NLO N 2 LO (a) N 3 LO Mass number A (b) r / r ( R / R ( 208 Pb) 208 Pb)

47 Does it work like an effective theory? Proof of principle Order-by-order convergence test against pseudo-data (from a Gogny functional) factor of 4 at each order can fine-tune couplings N 2 LO regulator independent; N 3 LO converged energy/radius Independence of the regulator scale a (i.e., flatness ) and independent of reference nucleus Error plots vs. A shows convergence patterns Fixed a = 0.85 fm; exponential decrease of constants with n with Λ 700 MeV V 2n (MeV fm 3+2n ) v 2n (natural units) Wigner Bartlett (a) n=0 (c) n=2 Wigner Bartlett Heisenberg Majorana Λ -2n Order of expansion 2n Heisenberg Majorana (b) n=1 (d) n= Regularization scale a (fm)

48 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 expansion Density matrix expansion (DME) applied to NN and NNN diagrams Re-fit residual Skyrme parameters and test description Perturbative expansion justified by phase-space-based power counting = rethink chiral EFT formulation 2 Extend existing functionals following EFT principles Non-local regularized pseudo-potential [Raimondi et al., ] Optimize pseudo-potential to experimental data Test with correlation analysis technology Stay tuned for many developments in the near future!

49 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 gradient corrections ( ρ factors) = non-empirical EDF Fayans and collaborators (e.g., nucl-th/ ) E v = 2 3 ɛ Fρ 0 [a v + + a v 1 h v 1+ x 1/3 + x 1 h2+ v x 1/ h v 1 x 1/3 + x 1 h2 v x 1/3 2 + where x ± = (ρ n ± ρ p )/2ρ 0 (also Baldo et al., ) SLDA+ (Bulgac et al.) Neutron drops in traps ] E/A (MeV) FP81 WFF88 FaNDF 0 neutron matter nuclear matter (fm -3 )

50 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 gradient corrections ( ρ factors) = non-empirical EDF Fayans and collaborators (e.g., nucl-th/ ) E v = 2 3 ɛ Fρ 0 [a v + + a v 1 h v 1+ x 1/3 + x 1 h2+ v x 1/ h v 1 x 1/3 + x 1 h2 v x 1/3 2 + where x ± = (ρ n ± ρ p )/2ρ 0 (also Baldo et al., ) SLDA+ (Bulgac et al.) Neutron drops in traps ] E/A (MeV) FP81 WFF88 FaNDF 0 neutron matter nuclear matter (fm -3 ) 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 [ ], from Polonyi and Schwenk)

51 What is needed for ab initio Kohn-Sham DFT = EDF? Energy/nucleon [MeV] Need MBPT to work with tuned U [H = (T + U) + (V U)] Hartree-Fock Empirical saturation point k F [fm 1 ] V low k NN from N 3 LO (500 MeV) 3NF fit to E 3H and r 4He 2nd order 3rd order pp+hh Λ = 1.8 fm 1 Λ = 2.0 fm 1 Λ = 2.2 fm 1 Λ = 2.8 fm < Λ 3NF < 2.5 fm k F [fm 1 ] k F [fm 1 ] Lower resolution Λ from Λ χ with RG to make more perturbative If convergence insensitive to U = choose so KS density exact 2 Need to calculate V KS (r) from δe[ρ, τ,...]/δρ(r), etc., but diagrams depend non-locally on KS orbitals Density matrix expansion (DME) = explicit densities Or use chain rule = optimized effective potential (OEP)