R. Machleidt University of Idaho

R. MachleidtR. Machleidt

University of IdahoUniversity of Idaho

Nuclear forces from chiral EFT:The unfinished business

OutlineOutline

•Nuclear forces from chiral EFT: Nuclear forces from chiral EFT:

Overview & achievementsOverview & achievements

• Are we done? No!Are we done? No!

• Sub-leading many-body forcesSub-leading many-body forces

• Proper renormalization of chiral Proper renormalization of chiral forcesforces

• Outlook: Will we ever be done?Outlook: Will we ever be done?

R. MachleidtR. Machleidt 22The unfinished business The unfinished business

Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010

R. MachleidtR. MachleidtThe unfinished business The unfinished business

Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 33

The chiral EFT approach to nuclear The chiral EFT approach to nuclear forcesforces

• Clear connection to QCD: via Clear connection to QCD: via symmetriessymmetries

• Degrees of freedom relevant to (low-Degrees of freedom relevant to (low-energy) nuclear physics: pions and energy) nuclear physics: pions and nucleonsnucleons

• Systematic expansion: Q/ΛSystematic expansion: Q/Λ

• Controlled error, predictive power.Controlled error, predictive power.

• Explains the empirically known Explains the empirically known hierarchy of nuclear forceshierarchy of nuclear forces

R. MachleidtR. Machleidt 44

2N forces

3N forces 4N forces

Leading Order

Next-to-Next-to Leading Order

Next-to-Next-to-Next-to Leading Order

Next-to Leading Order

The Hierarchy of Nuclear

Forces

The unfinished business The unfinished business Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010

Forces

2N forces

Leading Order

Phase shifts up to 300 MeVRed Line: N3LO Potential by Entem & Machleidt, PRC 68, 041001

(2003).Green dash-dotted line: NNLO Potential, and

blue dashed line: NLO Potential by Epelbaum et al., Eur. Phys. J. A19, 401 (2004).

N3LO Potential by Entem & Machleidt, PRC 68, 041001 (2003).

NNLO and NLO Potentials by Epelbaum et al., Eur. Phys. J. A19, 401 (2004).

Applications of the chiral NN potential

Chiral NN potential at N3LO underbinds by ~1MeV/nucleon.

(Size extensivity at its best.)

Nucleus E / A [MeV]

4He 1.08 (0.73FY)

16O 1.25

40Ca 0.84

48Ca 1.27

48Ni 1.21

Forces

2N forces

Leading Order

4N forces

Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 1515R. MachleidtR. Machleidt 1515

4NF at N3LO (leading order)

(2) (1) (0)

Note that only vertices from

, and are involved,

no new parameters,

First rough estimate:

0.1 MeV to binding.

Essentially negligible,as to be expected.

Interim summaryInterim summary

• 2NF o.k.2NF o.k.

(except for the issue of non-(except for the issue of non-perturbative renormalization)perturbative renormalization)

• 4NF (negligibly) small …4NF (negligibly) small …

Forces

2N forces 3N forces

Leading Order

The focus:The 3NF

4N forces

Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 1818R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 1818R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 1818

Forces

Next-to Leading Order (Q2)

Next-to-Next-to Leading Order (Q3)

Next-to-Next-to-Next-to Leading Order (Q4)

The chiral 3NFLeading Order (Q0) Basic Diagrams

Power =2 + 2L + i all vertices∑

with i =di +ni2

Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 1919R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 1919R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 1919R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 1919R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 1919

Forces

(1/M)n corrections

Coon & Friar

(1986)

with i =di +ni2

Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 2020R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 2020R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 2020R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 2020

Forces

(1/M)n corrections

The 3NFAt N2LO :

How well does it do?

with i =di +ni2

Calculating the properties of light nuclei usingCalculating the properties of light nuclei usingchiral 2N and 3N forces chiral 2N and 3N forces

“No-Core Shell Model “ Calculations by P. Navratil et

al., LLNL

2N (N3LO) force only

al., LLNL

2N (N3LO) force only

2N (N3LO) +3N (N2LO)

forces

al., LLNL

The Ay puzzle is NOT solved

by the 3NF at NNLO.

AnalyzingPower, Ay

2NF only

2NF+3NFCalculations bythe Pisa Group

Why do we need 3NFs beyond Why do we need 3NFs beyond NNLO?NNLO?

• The 2NF is N3LO;The 2NF is N3LO;

consistency requires that all consistency requires that all contributions are at the same order.contributions are at the same order.

•There are unresolved problems in 3N, There are unresolved problems in 3N, 4N scattering and nuclear structure.4N scattering and nuclear structure.

What’s there beyond What’s there beyond NNLO?NNLO?

Forces

(1/M)n corrections

with i =di +ni2

Forces

(1/M)n corrections

with i =di +ni2

Forces

(1/M)n corrections

with i =di +ni2

Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 3131R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 3131

Forces

(1/M)n corrections

✗ ✗ ✗ ✗Time

derivative

with i =di +ni2

Forces

(1/M)n corrections

✗ ✗ ✗ ✗ 1-loop

lead. vert.

with i =di +ni2

Ishikawa & Robilotta,PRC 76, 014006 (2007)

Bernard,Epelbaum,Krebs,Meissner,PRC 77, 064004 (2008)

In progress

The 3NF at N3LO explicitlyOne-loop, leading vertices

The ring diagramsFrom: Bernard, Epelbaum, Krebs, Meissner, PRC 77, 064004 (2008)

Proton-deuteron elastic scatteringBlack line: 2NF only

Red dashed line: 2NF + 3NF(N2LO+2πN3LO) ≈ 2NF + 3NF(N2LO)

From:Ishikawa &Robilotta,PRC 76, 014006(2007)

2πN3LO 3

Interim balanceInterim balance

• The 3NF at N3LO may be generally The 3NF at N3LO may be generally weak.weak.

• Reason: only “leading” vertices are Reason: only “leading” vertices are involved, which are known to be weak involved, which are known to be weak (e.g., from NN).(e.g., from NN).

• Thus, the 3NF at N3LO may not solve Thus, the 3NF at N3LO may not solve any of the outstanding problems.any of the outstanding problems.

What now?What now?

Go to the next order!

Forces

(1/M)n corrections

✗ ✗ ✗ ✗

1-loopone ci vert.

1-looplead. vert.

with i =di +ni2

What to expect from those N4LO 1-loop diagrams?

Compare to “similar” 2NF diagrams.

1-looplead. vert.

1-loopOne ci vert.

Corresponding 2NF contributions

1-looplead. vert.

1-loopOne ci vert.

Corresponding 2NF contributions

2π 3π

Kaiser (2000)

Kaiser (2001)

Large!!

Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 4242R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4242R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4242R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4242R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4242

Forces

(1/M)n corrections

✗ ✗ ✗ ✗

1-loopone ci vert.

1-looplead. vert.

Large!!

with i =di +ni2

So, we will have to go to N4LO!?So, we will have to go to N4LO!?

• No, not necessarily.No, not necessarily.

• There is an alternative: There is an alternative:

go to N3LO, but include Δ(1232) go to N3LO, but include Δ(1232) isobars explicitly.isobars explicitly.

The 3NF in the “Δ-full” theory

Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 4545R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4545R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4545R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4545R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4545R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4545

Forces

The chiral 3NF: Δ-less vs. Δ-fullLeading Order (Q0)

Δ less Additional in Δ full

1-loops with Δ

e.g., Illinois

However, there is a however …However, there is a however …

• For reasons of consistency, the For reasons of consistency, the 2NF must then also include Δ’s, 2NF must then also include Δ’s,

up to N3LO!up to N3LO!

• It will be a big mess!It will be a big mess!

ConclusionsConclusions• Substantial advances in chiral nuclear forces during Substantial advances in chiral nuclear forces during

the past decade. The major milestone of the decade: the past decade. The major milestone of the decade: “high precision” NN pots. at N3LO, successful in “high precision” NN pots. at N3LO, successful in nuclear structure.nuclear structure.

But there are still issues:But there are still issues:

• 3NFs beyond N2LO are needed!3NFs beyond N2LO are needed!

• The 3NF at N3LO of the Δ-less theory may be weak and The 3NF at N3LO of the Δ-less theory may be weak and useless.useless.

• This calls for further 3NF contributions: This calls for further 3NF contributions:

- Δ-less N4LO or - Δ-less N4LO or

- Δ-full N3LO, - Δ-full N3LO,

they will be large, but none will be easy.they will be large, but none will be easy.

• ChPT of nuclear forces has been fairly easy and ChPT of nuclear forces has been fairly easy and straightforward, so far; but what’s left won’t be easy, straightforward, so far; but what’s left won’t be easy, no matter how you do it.no matter how you do it.

The End

Three-nucleon forces at N2LOThree-nucleon forces at N2LO

TPE-3NF

OPE-3NF

Contact-3NF

No new parameters!

One new parameter, D.

One new parameter, E.

Strategy: Adjust D and E to two few-nucleon observables, e.g., the triton and alpha-particle binding energies. Then

predict properties of other light nuclei.

Forces

Leading Order

Additional in Δ full

+ 1/M corrections, Kaiser ’98: large?

2pi & 3pi 2-loops with Δ

The chiral 2NF: Δ-less vs. Δ-full

Forces

Ordonez ’94, Kaiser ‘98

Krebs ‘07

Δ less

Either

In summary:

Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 5252R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5252

Forces

Leading Order

Forces

Krebs ‘07

Δ less

Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 5353R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5353R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5353R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5353R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5353R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5353R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5353

Forces

1-loops with Δ

e.g., Illinois

Forces

Leading Order

Forces

Krebs ‘07

Δ less

Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 5656R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5656R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5656R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5656R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5656R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5656R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5656R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5656

Forces

1-loops with Δ

e.g., Illinois

R. Machleidt University of Idaho

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