Ab initio nuclear physics with chiral EFT · • Ab initio calculations usually carried out at several different oscillator energies hw to gauge the model dependence of the results

Andreas Ekström (UT/ORNL)

Progress in Ab Initio Techniques in Nuclear Physics February 23-26, 2016,TRIUMF, Vancouver, BC, Canada

Ab initio nuclear physics with chiral EFT

- Simultaneous optimization

- UQ applied to proton-proton fusion

- Chiral EFT tailored to the HO-basis

- Optimizing the 3NF at N3LO

Overview





Overview

~ 35 MeV

~ 2 MeV

B. D. Carlsson, A. Ekström, Christian Forssen et al. Phys. Rev. X (accepted)





Overview

1. Diversify and extend the statistical analysis and perform a sensitivity analysis of input data.

Robust parameter estimationThree-nucleon scattering data.The information content of heavy nuclei.……

~ 35 MeV

~ 2 MeV






Overview



2. Explore alternative strategies of informing the model about low-energy many-body observables.

~ 35 MeV

~ 2 MeV






Overview



2. Explore alternative strategies of informing the model about low-energy many-body observables.

3. Continue efforts towards higher orders of the chiral expansion, and possibly revisit the power counting.

Delta resonancesOther regulatorsLattice QCD dataModified PC…

~ 35 MeV

~ 2 MeV


Simultaneous optimization

NN NNN NNNN

LO (n=0)

NLO (n=2)

NNLO (n=3)

N3LO (n=4)

N4LO (n=5)

E. Epelbaum et al. Rev. Mod. Phys. 81, 1773 (2009)R. Machleidt et al. Phys. Rep. 503, 1 (2011)

chiral EFT is our tool to analyze the nuclear interaction

We optimize the LECs such that the chiral interactionsreproduces some calibration data, then we predict!


NN NNN NNNN

LO (n=0)

NLO (n=2)

NNLO (n=3)

N3LO (n=4)

N4LO (n=5)


two-nucleoninteraction

pion-nucleonscattering

three-nucleoninteraction

external probecurrent

the same LECs appear in the expressions for various

low-energy processese.g. the ci (green dot) and cD (blue square) three-nucleon

interaction




NN NNN NNNN

LO (n=0)

NLO (n=2)

NNLO (n=3)

N3LO (n=4)

N4LO (n=5)


two-nucleoninteraction

pion-nucleonscattering

three-nucleoninteraction

external probecurrent

the same LECs appear in the expressions for various

low-energy processese.g. the ci (green dot) and cD (blue square) three-nucleon

interaction

�2(p) ⌘X

i2MR2

i (p) =X

k2⇡N

R2k(p) +

X

j2NN

R2j (p) +

X

l2NNN

R2l (p)




UQ with N2LOsim

Simultaneous optimization is critical in order to

• find the optimal set of LECs.

• capture all relevant correlations between them.

• reduce the statistical uncertainty.

• attain order-by-order convergence.

Within such an approach we find that statistical errors are, in general, small, and that the total error budget is dominated by systematic errors.


UQ with N2LOsim







NNLOsep NNLOsim


UQ with N2LOsim







NNLOsep NNLOsim

compute the derivatives of your own observables wrt LECs, then explore:

•cutoff variations •order-by-order evolution •LEC UQ/correlations

All 42 different sim/sep potentials, as well as the respective covariance matrices are available as supplemental material.

LO-NLO-NNLO7 different cutoffs: 450,475,..,600 MeV6 different NN-scattering datasets

UQ applied to proton-proton fusion

In collaboration with B. Acharya, L. Platter, B. D. Carlsson, and C. Forssen

�(E) =

Zd3pe(2⇡)3

d3p⌫(2⇡)3

1

2Ee

1

2E⌫⇥

2⇡�

✓E + 2mp �md �

q2

2md� Ee � E⌫

◆

1

vrelF (Z,Ee)

1

4

X|hf |HW |ii|2

p+ p ! d+ e+ + ⌫eS(E) = �(E)Ee2⇡⌘

L. E. Marcucci et al PRL 110, 192503 (2013)R. Schiavilla et al PRC 58, 1263 (1998)J-W. Chen et al. PLB 720, 385 (2013)

In the core of the Sun, energy is released through sequences of nuclear reactions that convert hydrogen into helium. The primary reaction is thought to be the fusion of two protons with the emission of a low-energy neutrino and a positron.


In collaboration with B. Acharya, L. Platter, B. D. Carlsson, and C. Forssen

�(E) =

Zd3pe(2⇡)3

d3p⌫(2⇡)3

1

2Ee

1

2E⌫⇥

2⇡�

✓E + 2mp �md �

q2

2md� Ee � E⌫

◆

1

vrelF (Z,Ee)

1

4

X|hf |HW |ii|2

3.928 3.930 3.932 3.934 3.936 3.938c4 (GeV�1)

3.8

3.9

4.0

4.1

4.2

4.3

4.4

S(0

)(1

0�23

MeV

fm2)

NNLOSpline

p+ p ! d+ e+ + ⌫eS(E) = �(E)Ee2⇡⌘

L. E. Marcucci et al PRL 110, 192503 (2013)R. Schiavilla et al PRC 58, 1263 (1998)J-W. Chen et al. PLB 720, 385 (2013)

In the core of the Sun, energy is released through sequences of nuclear reactions that convert hydrogen into helium. The primary reaction is thought to be the fusion of two protons with the emission of a low-energy neutrino and a positron.

460 480 500 520 540 560 580 600⇤EFT (MeV)

4.06

4.08

4.10

4.12

4.14

S(0

)(1

0�23

MeV

fm2 )

cutoff variation

TLab

statistical uncertainty


Preliminary

460 480 500 520 540 560 580 600⇤EFT (MeV)

4.06

4.08

4.10

4.12

4.14

S(0

)(1

0�23

MeV

fm2 )

cutoff variation

TLab



PreliminaryS(0)

⇤2

�2B

E(2H)

rpt�p(2H)

Q(2H)

D(2H)

E(3H)

rpt�p(3H)

E(3He)

rpt�p(3He)

E1A(3He)

E(4He)

rpt�p(4He)

S(0

)

⇤2

�2

B

�1.0

�0.8

�0.6

�0.4

�0.2

0.00.2

0.40.6

0.81.0

Correlation

correlation analysis:

S(0) - E(2H): phase spaceL2-r(2H): radial overlap L2-rQ(2H): radial overlap d2B-E1A: 2B-current operator

460 480 500 520 540 560 580 600⇤EFT (MeV)

4.06

4.08

4.10

4.12

4.14

S(0

)(1

0�23

MeV

fm2 )

cutoff variation

TLab



PreliminaryS(0)

⇤2

�2B

E(2H)

rpt�p(2H)

Q(2H)

D(2H)

E(3H)

rpt�p(3H)

E(3He)

rpt�p(3He)

E1A(3He)

E(4He)

rpt�p(4He)

S(0

)

⇤2

�2

B

�1.0

�0.8

�0.6

�0.4

�0.2

0.00.2

0.40.6

0.81.0

Correlation

correlation analysis:

S(0) - E(2H): phase spaceL2-r(2H): radial overlap L2-rQ(2H): radial overlap d2B-E1A: 2B-current operator

- Insofar most consistent xEFT-study of this reaction- Correlation study indicates sound statistical analysis- Cutoff variation not large source of error- Statistical error in S(0) is 3 times larger than what was

previously thought- Central value is most likely also larger due to

previously neglected systematic uncertainties.

converging heavy nuclei with ab initio methods

dimensionality of problem increases

Nuclei in the Harmonic Oscillator basis

⇤UV ⇡r2N

max

m!

~

L ⇡r2N

max

~m!

r

p (Nmax

,!) ! (⇤UV

, L)basis parameters

⇤UV > ⇤�

L > Rnucleus

Nucleus needs to fit into the modelspace

Interaction must be captured


• Converging calculations beyond ~calcium becomes truly expensive, or even impossible.

• Ab initio calculations usually carried out at several different oscillator energies hw to gauge the model dependence of the results

• To facilitate calculations in heavy nuclei we propose to tailor the EFT interaction to a finite HO basis

⇤UV ⇡r2N

max

m!

~

L ⇡r2N

max

~m!

r

p (Nmax

,!) ! (⇤UV


⇤UV > ⇤�

L > Rnucleus




• Converging calculations beyond ~calcium becomes truly expensive, or even impossible.

• Ab initio calculations usually carried out at several different oscillator energies hw to gauge the model dependence of the results

• To facilitate calculations in heavy nuclei we propose to tailor the EFT interaction to a finite HO basis

⇤UV ⇡r2N

max

m!

~

L ⇡r2N

max

~m!

r

p (Nmax

,!) ! (⇤UV


⇤UV > ⇤�

L > Rnucleus



Similar ideas in nuclear physics already exist:

Arizona group developed pionless-EFT in HO basis and studied UV/IR cutoff dependencies. Coupling constants depend on the size of the basis.

Haxton et al. proposed HOBET (HO-based effective theory). “Shell-Model” (Bloch-Horowitz) plus resummed kinetic energy and physics beyond a cutoff absorbed by contact-gradient expansion (like EFT contact potential)

We propose to choose (and fix) an oscillator space and evaluate the existing chiral EFT interaction operators in this space. This projection requires us to refit the LECs of chiral EFT

EFT interaction in a truncated HO basis

n

n’

S. Binder, A.Ekstrom, G. Hagen, T Papenbrock, and K .A. Wendt arXiv:1512.03802 [nucl-th]


n

n’


N =

N

max

� `

2

�


n

n’


(Nmax

,!) ! (⇤UV

, L)

N =

N

max

� `

2

�


n

n’


(Nmax

,!) ! (⇤UV

, L)

N =

N

max

� `

2

�


n

n’

KINETIC ENERGY

ONLY


(Nmax

,!) ! (⇤UV

, L)

N =

N

max

� `

2

�


n

n’

KINETIC ENERGY

ONLY

simple picture: choose e.g. Nmax

= 10, ~! = 40MeV

set hn|V |n0i to zero outside Nmax

“thinking inside the box”


(Nmax

,!) ! (⇤UV

, L)

N =

N

max

� `

2

�


n

n’

KINETIC ENERGY

ONLY


= 10, ~! = 40MeV



Optimize the LECs to reproduce selected fit-datawe can compute phase shifts using the J-matrix formalism, and finite nuclei using e.g. NCSM, CC, IM-SRG, …..


(Nmax

,!) ! (⇤UV

, L)

N =

N

max

� `

2

�


n

n’

KINETIC ENERGY

ONLY


= 10, ~! = 40MeV



However, in the longer perspective, we seek to develop an EFT. So it is important to understand the squared momentum operator

hn`|p2|n0`0i



(Nmax

,!) ! (⇤UV

, L)

N =

N

max

� `

2

�


n

n’

KINETIC ENERGY

ONLY


= 10, ~! = 40MeV



We start by solving the eigenvalue problem of the squared momentumoperator in a finite oscillator space truncated at an energy (Nmax +3/2)hw.It turns out that momenta k=kµ such that yN+1(k)=0 solve this eigenvalue problem.


hn`|p2|n0`0i



(Nmax

,!) ! (⇤UV

, L)

N =

N

max

� `

2

�


n

n’

KINETIC ENERGY

ONLY


= 10, ~! = 40MeV





hn`|p2|n0`0i



(Nmax

,!) ! (⇤UV

, L)

n`(k) =

s2n!b3

�(n+ `+ 32 )

(kb)` e�12k

2b2L`+ 1

2n

�k2b2

�

This has pleasant consequences for most analytical and numerical evaluations!

N =

N

max

� `

2

�


n

n’

KINETIC ENERGY

ONLY


= 10, ~! = 40MeV





hn`|p2|n0`0i

In harmonic oscillator EFT we compute matrix elements of the chiral interaction V as:

hn`|V |n0`0i =NX

µ⌫=0

c2µ` n,`(kµ,`) hkµ,`, `|V |k⌫,`0 , `0i| {z }�EFT

c2⌫,`0 n0`0(k⌫,`0) +O(k2N+2)



(Nmax

,!) ! (⇤UV

, L)

n`(k) =

s2n!b3

�(n+ `+ 32 )

(kb)` e�12k

2b2L`+ 1

2n

�k2b2

�


N =

N

max

� `

2

�


n

n’

KINETIC ENERGY

ONLY


= 10, ~! = 40MeV





hn`|p2|n0`0i

In harmonic oscillator EFT we compute matrix elements of the chiral interaction V as:

hn`|V |n0`0i =NX

µ⌫=0

c2µ` n,`(kµ,`) hkµ,`, `|V |k⌫,`0 , `0i| {z }�EFT

c2⌫,`0 n0`0(k⌫,`0) +O(k2N+2)



(Nmax

,!) ! (⇤UV

, L)

n`(k) =

s2n!b3

�(n+ `+ 32 )

(kb)` e�12k

2b2L`+ 1

2n

�k2b2

�


N =

N

max

� `

2

�

Idaho-N3LO in the Harmonic Oscillator basis



selected proton-neutron phase shifts of Idaho-N3LO(500) projected onto an Nmax=10, hw=40 MeV oscillator space

LUV = 700 MeV > Lc = 500 MeV




LUV = 700 MeV > Lc = 500 MeV

OBSERVATIONS:

• There are oscillations in the phase shifts

• The period of this oscillation is approximately proportional to the IR cutoff

• Gauss-Laguerre (“HO-EFT”) phases exhibits sligthly smaller oscillations than the “exact” phases

• At the energies corresponding to the eigenenergies of the truncated Hamiltonian (solid dots), computed phases are closest to the true N3LO value

(N, ~!) = (10, 40MeV)

(N, ~!) = (10, 40MeV)

(N, ~!) = (10, 40MeV)




LUV = 700 MeV > Lc = 500 MeV

OBSERVATIONS:

• There are oscillations in the phase shifts

• The period of this oscillation is approximately proportional to the IR cutoff

• Gauss-Laguerre (“HO-EFT”) phases exhibits sligthly smaller oscillations than the “exact” phases

• At the energies corresponding to the eigenenergies of the truncated Hamiltonian (solid dots), computed phases are closest to the true N3LO value

(N, ~!) = (10, 40MeV)

(N, ~!) = (10, 40MeV)

(N, ~!) = (10, 40MeV)

(N, ~!) = (20, 23MeV)

(N, ~!) = (20, 46MeV)(N, ~!) = (10, 80MeV)

(N, ~!) = (10, 40MeV)

Increasing N and/or hw gives “smaller” oscillations


Refitting the NLO interaction to CD-Bonn phasesWith goal of computing heavy nuclei, we design a HO-NLO interaction with:

small number of oscillator shells Nmax=10

Lower frequencies, rapid IR convergence

Lower frequencies, lower UV cutoffs

We choose hw=24 MeV, which gives:LUV=550 MeV, L=9.6 fm. Considering the tail of the chiral regulatorfunction, we set Lc=450MeV.














See. G. Hagen’s talk for HO-EFT in Coupled Cluster

Open questions, and next steps

Some open questions that we are currently addressing



Why do we need to fiddle with ⇤UV > ⇤�

Insofar, we observed that the oscillations in the phase shifts are significantly reduced if the chiral cutoff falls below the UV cutoff of the harmonic oscillator.





Why do we need the chiral cutoff to begin with? ⇤�

The projection to a smaller HO-space should act as a regulator. But taking the chiral cutoff to infinity induced rather large oscillations



The HO-EFT approach introduces two new scales: ⇤UV,�IR

The infrared scale is rather undesirable in a low-energy EFT. Can we get rid of it? It seem like it. At least we can sacrifice the highest momentum point kN, to better approximate the IR-physics of xEFT.













What correction does HOEFT introduce..

Or, put differently, how much uncertainty do we bring in by resolving the chiral EFT in HOEFT? Insofar, we have not observed any catastrophes when studying light or heavy nuclei with HOEFT, nor have any LECs become unnatural.

⇠"1 +

1

N

✓k

⇤UV

◆2#

?









Next step: HO-NNLOsat

What correction does HOEFT introduce..

Or, put differently, how much uncertainty do we bring in by resolving the chiral EFT in HOEFT? Insofar, we have not observed any catastrophes when studying light or heavy nuclei with HOEFT, nor have any LECs become unnatural.

⇠"1 +

1

N

✓k

⇤UV

◆2#

?

NNN-N3LO: simple cD/cE scan with Idaho-N3LO

+ rel. corr.

2π-1π2π rings 2π-contact

efficient NNN-PWD enabled by K. Hebeler, H. Krebs, et al PRC 91, 044001 (2015)


+ rel. corr.




+ rel. corr.


3H 4HeEgs -8.48 -28.32rpt-p 1.61 1.49

<c1> -0.14 -0.69<c3> -1.29 -6.82<c4> 0.35 2.16<cD> -0.39 -2.16<cE> 0.02 0.08<2pi> -0.40 -2.54

<2pi1pi> 1.22 6.48<rings> -0.57 -3.38

<2pi-cont> 0.20 1.25<rel. corr> 0.24 1.28

N2LO -1.45 -7.44N3LO 0.68 3.09

Expectation values (in MeV and fm)


N3LO optimizations are challenging

Phase shifts from Granada analysis: Navarro Pérez et al PRC 88, 064002 (2013)

Initialize by computing phase shifts for 105 random contact LEC values for each partial wave and select the ~1000 best values and optimize. This leads to 192 different optima (for cutoff 450 MeV) with respect to phase shifts. (pi-N LECs from sep-optimization)

The A=3 observables weed out several ofthe S-wave minima, but many P-waveminima remain. Things improve whenA=4 is included. But still, several local minima remain.

This is where we stand right now.

Work led by B. D. Carlsson (Chalmers)












AD DERIVATIVES

BETTER INFORMED OPTIMIZATION


Summary and conclusions

Covariance matrices for optimized LO-NLO-NNLO potentials available for download

Small variations in the nuclear interaction renders large fluctuations in predictions for heavier nuclei

Harmonic Oscillator EFT could be a promising approach for ab initio studies of heavy atomic nuclei

Non-local 3NF at N3LO is not constrained by A=2,3 data

N3LO optimizations benefit from gradients

⇤UV ⇡r2N

max

m!

~

L ⇡r2N

max

~m!

r

Thank you for your attention

and thanks to all collaborators!

Bijaya Acharya Sven Binder

Boris D. CarlssonChristian Forssen

Gaute HagenGustav Jansen

Oskar LiljaMattias LindbyBjörn Mattsson

Thomas PapenbrockLucas Platter

Dag Fahlin StrömbergKyle Wendt

Documents

Ab initio nuclear physics with chiral EFT · • Ab initio calculations usually carried out at several different oscillator energies hw to gauge the model dependence of the results