Ab Initio Theory Outside the Box · Inside the Box Robert Roth – TU Darmstadt – 02/2014 this workshop has provided an impres- sive snapshot of the progress and perspectives in

Ab Initio Theory

Outside the Box

Robert Roth

1

Inside the Box

Robert Roth – TU Darmstadt – 02/2014

this workshop has provided an impres-sive snapshot of the progress andperspectives in ab initio nucleartheory and its links to experiment

definition: everything we have heardso-far is inside the ab initio box

2

Inside the Box


ab initio theory is entering new territory...

QCD frontiernuclear structure connected systematically to QCD via chiral EFT

accuracy frontiercontrol uncertainties, improve convergence, inform extrapolations

mass frontierab initio calculations up to heavy nuclei with quantified uncertainties

open-shell frontierextend to medium-mass open-shell nuclei and their excitation spectrum

continuum & clustering frontierinclude continuum & clustering effects for threshold states & nuclei

reaction frontierdescribe structure & reaction observables on the same footing

...providing a coherent theoretical framework for nuclearstructure & reactions and linking it to experiment

3

Outside the Box


two more things that are not yet insidethe ab initio box:

ab initio hypernuclear structurecan we describe the spectroscopy ofp-shell hypernuclei ab initio ?

perturbation theory — ab initio ?wouldn’t it be great if MBPT wouldqualify as ab initio approach ?

4

Ab Initio Hypernuclear Structure

with

Roland Wirth, Daniel Gazda, Petr Navrátil

5



precise data on ground states &spectroscopy of hypernuclei

ab initio few-body (A ≲ 4) andphenomenological shell modelor cluster calculations

chiral YN & YY interactions at(N)LO are available

constrain YN & YY interactionsby ab initio hypernuclear struc-ture calculations

6

YN Interaction — A Problem


Haidenbauer et al., NPA 915, 24 (2013), Polinder et al., NPA 779, 244 (2006), Haidenbauer et al., PRC 72, 044005 (2005)

100 200 300 400 500 600 700 800 900

plab (MeV/c)

0

100

200

300

σ (m

b)

Sechi-Zorn et al.Kadyk et al.Alexander et al.

Λp -> Λp

100 120 140 160 180

plab (MeV/c)

0

50

100

150

200

250

300

σ (m

b)

Engelmann et al.

Σ−p -> Λn

100 120 140 160 180

plab (MeV/c)

0

100

200

300

400

500

σ (m

b)

Engelmann et al.

Σ−p -> Σ0

n

100 120 140 160 180

plab (MeV/c)

0

50

100

150

200

250

300

σ (m

b)

Eisele et al.

Σ−p -> Σ−

p

Jülich’04 LO chiral NLO chiral

experimental YN scattering data is scarce and has large uncertainties

fit of interactions not well constrained (invoke symmetries)

scattering data cannot discriminate between different YN potentials

7

Ab Initio Toolbox


Hamiltonian from chiral EFT

NN: chiral N3LO by Entem & Machleidt, ΛNN = 500 MeV

3N: chiral N2LO by Navrátil, Λ3N = 500 MeV, A = 3 fit

YN: chiral LO by Polinder, Haidenbauer & Meißner, ΛYN = 600,700 MeVJülich’04 by Haidenbauer & Meißner

Similarity Renormalization Group

consistent SRG-evolution of NN, 3N, YN interactions

using particle basis and including Λ--coupling (larger matrices)

Λ- mass difference and p± Coulomb included consistently

Importance Truncated No-Core Shell Model

include explicit (p, n,Λ,+,0,−) with physical masses

larger model spaces easily tractable with importance truncation

all p-shell single-Λ hypernuclei are accessible

8

Application: Λ7Li


1+

3+

-40

-35

-30

-25

-20

.

E[M

eV]

1+

3+

4 6 8 10 12 Exp.Nmx

-1

0

1

2

3

.

E[M

eV]

6Li NN @ N3LOΛNN = 500 MeV

Entem&Machleidt

3N @ N2LOΛ3N = 500 MeV

NavratilA = 3 fit

Jülich’04Haidenbauer et al.scatt. & hypertriton

αN = 0.08 fm4

αY = 0.00 fm4

hΩ = 20 MeV

9

Application: Λ7Li


1+

3+

-40

-35

-30

-25

-20

.

E[M

eV]

1+

3+

4 6 8 10 12 Exp.Nmx

-1

0

1

2

3

.

E[M

eV]

12+32+

52+72+

12+

32+

52+

72+

4 6 8 10 12 Exp.Nmx

6Li 7ΛLi NN @ N3LO

ΛNN = 500 MeV

Entem&Machleidt


NavratilA = 3 fit


αN = 0.08 fm4

αY = 0.00 fm4

hΩ = 20 MeV

10

Application: Λ7Li


1+

3+

-40

-35

-30

-25

-20

.

E[M

eV]

1+

3+

4 6 8 10 12 Exp.Nmx

-1

0

1

2

3

.

E[M

eV]

12+32+

52+72+

12+

32+

52+

72+

4 6 8 10 12 Exp.Nmx

6Li 7ΛLi NN @ N3LO

ΛNN = 500 MeV

Entem&Machleidt


NavratilA = 3 fit


αN = 0.08 fm4

αY = 0.00 fm4

hΩ = 20 MeV

11

Application: Λ7Li


1+

3+

-40

-35

-30

-25

-20

.

E[M

eV]

1+

3+

4 6 8 10 12 Exp.Nmx

-1

0

1

2

3

.

E[M

eV]

12+32+

52+72+

12+

32+

52+

72+

4 6 8 10 12 Exp.Nmx

6Li 7ΛLi NN @ N3LO

ΛNN = 500 MeV

Entem&Machleidt


NavratilA = 3 fit

YN @ LOΛYN = 600 MeV

Polinder et al.scatt. & hypertriton

αN = 0.08 fm4

αY = 0.00 fm4

hΩ = 20 MeV

12

Application: Λ7Li


1+

3+

-40

-35

-30

-25

-20

.

E[M

eV]

1+

3+

4 6 8 10 12 Exp.Nmx

-1

0

1

2

3

.

E[M

eV]

12+32+

52+72+

12+

32+

52+

72+

4 6 8 10 12 Exp.Nmx

6Li 7ΛLi NN @ N3LO

ΛNN = 500 MeV

Entem&Machleidt


NavratilA = 3 fit



αN = 0.08 fm4

αY = 0.00 fm4

hΩ = 20 MeV

13

Application: Λ9Be


0+

2+

-65

-60

-55

-50

-45

-40

.

E[M

eV]

0+

2+

2 4 6 8 10 Exp.Nmx

0

1

2

3

4

.

E[M

eV]

12+

32+52+

12+

32+52+

2 4 6 8 10 Exp.Nmx

8Be 9ΛBe NN @ N3LO

ΛNN = 500 MeV

Entem&Machleidt


NavratilA = 3 fit


αN = 0.08 fm4

αY = 0.00 fm4

hΩ = 20 MeV

14

Application: Λ9Be


0+

2+

-65

-60

-55

-50

-45

-40

.

E[M

eV]

0+

2+

2 4 6 8 10 Exp.Nmx

0

1

2

3

4

.

E[M

eV]

12+

32+52+

12+

32+52+

2 4 6 8 10 Exp.Nmx

8Be 9ΛBe NN @ N3LO

ΛNN = 500 MeV

Entem&Machleidt


NavratilA = 3 fit



αN = 0.08 fm4

αY = 0.00 fm4

hΩ = 20 MeV

15

Application: Λ9Be


0+

2+

-65

-60

-55

-50

-45

-40

.

E[M

eV]

0+

2+

2 4 6 8 10 Exp.Nmx

0

1

2

3

4

.

E[M

eV]

12+

32+52+

12+

32+52+

2 4 6 8 10 Exp.Nmx

8Be 9ΛBe NN @ N3LO

ΛNN = 500 MeV

Entem&Machleidt


NavratilA = 3 fit



αN = 0.08 fm4

αY = 0.00 fm4

hΩ = 20 MeV

16

Application: Λ13C


0+

2+

-110

-105

-100

-95

-90

-85

-80

-75

.

E[M

eV]

0+

2+

2 4 6 8 10 Exp.Nmx

0

2

4

6

.

E[M

eV]

12+

32+

12+

32+

2 4 6 8 10 Exp.Nmx

12C 13ΛC NN @ N3LO

ΛNN = 500 MeV

Entem&Machleidt


NavratilA = 3 fit


αN = 0.08 fm4

αY = 0.00 fm4

hΩ = 20 MeV

17

Application: Λ13C


0+

2+

-110

-105

-100

-95

-90

-85

-80

-75

.

E[M

eV]

0+

2+

2 4 6 8 10 Exp.Nmx

0

2

4

6

.

E[M

eV]

12+

32+

12+

32+

2 4 6 8 10 Exp.Nmx

12C 13ΛC NN @ N3LO

ΛNN = 500 MeV

Entem&Machleidt


NavratilA = 3 fit



αN = 0.08 fm4

αY = 0.00 fm4

hΩ = 20 MeV

18

Application: Λ13C


0+

2+

-110

-105

-100

-95

-90

-85

-80

-75

.

E[M

eV]

0+

2+

2 4 6 8 10 Exp.Nmx

0

2

4

6

.

E[M

eV]

12+

32+

12+

32+

2 4 6 8 10 Exp.Nmx

12C 13ΛC NN @ N3LO

ΛNN = 500 MeV

Entem&Machleidt


NavratilA = 3 fit



αN = 0.08 fm4

αY = 0.00 fm4

hΩ = 20 MeV

19

SRG Evolution of YN Channels


6 8 10 12 14 16 18Nmx

-44

-42

-40

-38

-36

-34

.

E[M

eV]

αY [fm4]

0.000

0.005

0.010

0.0200.0400.080

7ΛLi

SRG evolution of YN chan-nels improves conver-gence as expected

significant αY dependenceindicates SRG-inducedYNN interactions


αN = 0.08 fm4

hΩ = 20 MeV

20

SRG Evolution of YN Channels


0 0.04 0.08 0.12 0.16

αY [fm4]

-44

-43

-42

-41

-40

-39

.

E[M

eV]

7ΛLi

SRG evolution of YN chan-nels improves conver-gence as expected

significant αY dependenceindicates SRG-inducedYNN interactions


αN = 0.08 fm4

hΩ = 20 MeV

3.4 MeVinduced YNN

8.0 MeVinitial YN

21



ab initio hypernuclear structure in the IT-NCSM now possible for allsingle-Λ p-shell hypernuclei

LO chiral YN interactions provide spectra that agree with experimentwithin cutoff uncertainties

hypernuclear structure sets tight constraints on YN interaction

significant SRG-induced YNN interactions, implications for mean-fieldtype models and the hyperon puzzle ?

NLO chiral YN interactions are expected to reduce cutoff dependence,but fit is difficult...(13 instead of 5 LECs in S/SD-waves assuming SU(3)ƒ and neglecting P-waves; fit to 36 data)

lots of applications are waiting...

22

Perturbation Theory — Ab Initio ?

with

Alexander Tichai, Christina Stumpf, Joachim Langhammer

23

Many-Body Perturbation Theory


low-order many-body perturbation theory is a cheap and sim-ple tool to access nuclear observables

wouldn’t it be great if low-order MBPT would qualify as ab initioapproach ?

problem: convergence behavior of perturbation series unclear

how to quantify uncertainties?

which factors influence the order-by-order convergence?

how to restore or accelerate the convergence?

strategy: study convergence behavior with explicit high-ordercalculations

24

Explicit High-Order MBPT


partitioning: definition of unperturbed basis ∣n⟩H(λ) = H0 + λW H0 ∣n⟩ = εn ∣n⟩

power-series ansatz for energy and eigenstates

En(λ) = ∞∑p=0

λp E(p)n ∣Ψn(λ)⟩ = ∞∑

p=0

λp ∣Ψ(p)n ⟩

recursive relations for energy E(p)n and states ∣Ψ(p)n ⟩ = ∑mC

(p)n,m ∣m⟩

E(p)n = ∑

m

⟨n∣W ∣m⟩ C(p−1)n,m

C(p)n,m = 1

εn − εm(∑m′⟨m∣W ∣m′⟩ C(p−1)n,m′ −p∑j=1

E(j)n C

(p−j)n,m )

easy to evaluate to ’arbitrary’ order with NCSM technology...

25

Summation and Resummation


partial sum: starting point for convergence study

Esum(p) = E(0) + λE(1) + λ2E(2) +⋯λpE(p)∣λ=1 Padé approximant: map power series of order p to a quotient ofpolynomials of orders M and N

EPadé(M/N) = A(0) + λA(1) + λ2A(2) +⋯λMA(M)B(0) + λB(1) + λ2B(2) +⋯λNB(N) ∣λ=1= Esum(M +N) +O(M +N + 1)

focus on Padé main sequence: EPadé(M/M) and EPadé(M/M − 1)

powerful convergence theory for special power series (e.g. Stieltjes)...

additional sequence transformations on top of Padé can further accel-erate convergence (Shanks, Levin-Weniger)...

26

16O: MBPT Convergence


HO – Partial Sum

0 5 10 15 20 25p

-200

-180

-160

-140

-120

-100

.

E(p)[M

eV]

partial sum exhibitsexponential oscillatory

divergence

emx = 2 emx = 4 emx = 6 ∎ emx = 8

NNonly , α = 0.08 fm4 , hΩ = 20 MeV , 2p2h truncated

27



HO – Partial Sum

0 5 10 15 20 25p

-200

-180

-160

-140

-120

-100

.

E(p)[M

eV]

HO – Padé Approx.

0 5 10 15 20 25 30p

Padé resummationleads to rapidly

converging series

converged resultsare identical to exact

eigenvalues

emx = 2 emx = 4 emx = 6 ∎ emx = 8


28



HO – Partial Sum

0 5 10 15 20 25p

-200

-180

-160

-140

-120

-100

.

E(p)[M

eV]

HF – Partial Sum

0 5 10 15 20 25 30p

HF as unperturbedbasis also yields rapid

convergence

wrong long-rangebehavior of HO drives

divergence ?

emx = 2 emx = 4 emx = 6 ∎ emx = 8


29




0 5 10 15 20 25p

-200

-180

-160

-140

-120

-100

.

E(p)[M

eV]

HF – Padé Approx.

0 5 10 15 20 25 30p

Padé always yieldsconvergence

emx = 2 emx = 4 emx = 6 ∎ emx = 8


30



HO – Partial Sum

0 5 10 15 20 25p

-200

-180

-160

-140

-120

-100

.

E(p)[M

eV]

HF – Partial Sum

0 5 10 15 20 25 30p

convergencein HF basis depends onsoftness of interaction

α = 0.02 fm4

α = 0.04 fm4

α = 0.08 fm4

NNonly , emx = 8 , hΩ = 20 MeV , 2p2h truncated

31



HO – pth Order

0 5 10 15 20 25p

10−3

10−2

0.1

1

10

102

103

104

105

.

∣E(p)∣[M

eV]

HF – pth Order

0 5 10 15 20 25 30p

convergence &divergence is always

exponential

α = 0.02 fm4

α = 0.04 fm4

α = 0.08 fm4


32




0 5 10 15 20 25p

-200

-180

-160

-140

-120

-100

.

E(p)[M

eV]

HF – Padé Approx.

0 5 10 15 20 25 30p

α = 0.02 fm4

α = 0.04 fm4

α = 0.08 fm4


33

MBPT Convergence Heuristics


in many cases partial sums do not converge, but there are sys-tematic exceptions

factors causing the non-convergence

unperturbed basis (partitioning) is the primary factor

softness of the interaction is a secondary factor

many-body truncation also has some influence

Padé resummation robustly provides convergence at intermedi-ate orders and agrees with exact diagonalization

can we rely on low-order MBPT ?

34

Low-Order MBPT


-25

-20

-15

-10

-5

.

E/A[M

eV]

4He16O

24O34Si

40Ca48Ca

48Ni56Ni

78Ni88Sr

90Zr100Sn

114Sn132Sn

146Gd208Pb

-6

-5

-4

-3

-2

-1

.

Ecorr/A[M

eV]

NNonly

α = 0.08 fm4

HF basisemx = 10hΩ = 20 MeV

Exp.

EHF

∎

EHF + E(2)

EHF + E(2)+ E(3)

35

Low-Order MBPT vs. Coupled-Cluster


-25

-20

-15

-10

-5

.

E/A[M

eV]

4He16O

24O34Si

40Ca48Ca

48Ni56Ni

78Ni88Sr

90Zr100Sn

114Sn132Sn

146Gd208Pb

-6

-5

-4

-3

-2

-1

.

Ecorr/A[M

eV]

NNonly

α = 0.08 fm4


Exp.

EHF

∎

EHF + E(2)

EHF + E(2)+ E(3)

CR-CC(2,3)

36

Low-Order MBPT vs. Coupled-Cluster


-16

-14

-12

-10

-8

-6

-4

-2

.

E/A[M

eV]

4He16O

24O34Si

40Ca48Ca

48Ni56Ni

78Ni88Sr

90Zr100Sn

114Sn132Sn

146Gd208Pb

-6

-5

-4

-3

-2

-1

.

Ecorr/A[M

eV]

NNonly

α = 0.02 fm4


Exp.

EHF

∎

EHF + E(2)

EHF + E(2)+ E(3)

CR-CC(2,3)

low-order MBPTseems to work nicely,

but...

37

Perspectives: Degenerate MBPT


PRC 86, 054315 (2012), PLB 683, 272 (2010)

-40

-30

-20

-10

0

.

E[M

eV]

d = 0 d = 1 d = 2 d = 3 d = 4

0 4 8 12 16 20 24 28pth order

-40

-30

-20

-10

E[M

eV]

d = 5

0 4 8 12 16 20 24 28pth order

6LiNmax = 8~Ω = 20 MeV

d = 6

0 4 8 12 16 20 24 28pth order

d = 7

0 4 8 12 16 20 24 28pth order

d = 8

0 4 8 12 16 20 24 28pth order

d = 9

-35

-30

-25

-20

-15

.

E∗

[MeV

]

d = 0 d = 1 d = 2 d = 3 d = 4

0 4 8 12 16 20 24 28L + M

-35

-30

-25

-20

.

E∗

[MeV

]

d = 5

0 4 8 12 16 20 24 28

6LiNmax = 8~Ω = 20 MeV

L + M

d = 6

0 4 8 12 16 20 24 28L + M

d = 7

0 4 8 12 16 20 24 28L + M

d = 8

0 4 8 12 16 20 24 28L + M

d = 9

38

Epilogue


thanks to my group & my collaborators

S. Binder, J. Braun, A. Calci, S. Fischer,E. Gebrerufael, H. Spiess, J. Langhammer, S. Schulz,C. Stumpf, A. Tichai, R. Trippel, R. Wirth, K. VobigInstitut für Kernphysik, TU Darmstadt

P. NavrátilTRIUMF Vancouver, Canada

J. Vary, P. MarisIowa State University, USA

S. Quaglioni, G. HupinLLNL Livermore, USA

P. PiecuchMichigan State University, USA

H. HergertOhio State University, USA

P. PapakonstantinouIBS/RISP, Korea

C. ForssénChalmers University, Sweden

H. Feldmeier, T. NeffGSI Helmholtzzentrum

JUROPA LOEWE-CSC HOPPER

COMPUTING TIME

39

Documents

Ab Initio Theory Outside the Box · Inside the Box Robert Roth – TU Darmstadt – 02/2014 this workshop has provided an impres- sive snapshot of the progress and perspectives in