r. machleidt, university of idaho university of idaho the missing three-nucleon forces: where are...

R. Machleidt,R. Machleidt,

University of IdahoUniversity of Idaho

The missing three-nucleon forces:Where are they?

2009 Mardi Gras “Special Symmetries and Ab Initio Methods for Light Nuclei”

WorkshopLouisiana State University, 18-20 February, 2009

R. MachleidtR. Machleidt

Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-

18-0918-09 22

OutlineOutline

• Nuclear forces from Chiral Perturbation Nuclear forces from Chiral Perturbation Theory (ChPT): Current statusTheory (ChPT): Current status

• The 3NF at N2LOThe 3NF at N2LO

• The 3NF at N3LO: Weak expectationsThe 3NF at N3LO: Weak expectations

• N4LO or Δ-isobars?N4LO or Δ-isobars?

• SummarySummary

The ChPT approach to nuclear The ChPT 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



18-0918-09 33



18-0918-09 44R. 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



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Forces


Forces

2N forces

Leading Order






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Phase shifts up to 300 MeVRed Line: N3LO Potential by Entem & Machleidt, PRC 68, 041001 (2003).Green dashed line: NNLO Potential, and blue dotted 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).



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Deuteron wave functionsof N3LO potentials

red = Idaho, green = Bochum/JuelichBlack lines = high-precision pots.




Results from applications of chiral two-Results from applications of chiral two-nucleon forcesnucleon forces

• Latest Coupled Cluster results for A≤48 Latest Coupled Cluster results for A≤48 using N3LOusing N3LO




Nucleus E / A [MeV]

4He 1.08 (0.73FY)

16O 1.25

40Ca 0.84

48Ca 1.27

48Ni 1.21

Chiral NN potential at N3LO underbinds by ~1MeV/nucleon. (Size extensivity at its best.)

?V_low-k



18-0918-09 1212


Forces


Forces

2N forces

Leading Order




4N forces




4NF at N3LO (leading order)

(2) (1) (0)

Note that only vertices from

, and are involved,

no new parameters,

weak.

First rough estimate:

0.1 MeV to binding.

N NN

L L L

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 …



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3NF


Forces


Forces

2N forces 3N forces

Leading Order




The focus:The 3NF

4N forces



18-0918-09 1616R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 1616R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 1616

✗


Forces


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

N4LO

(Q5)

Power =2 + 2L + i all vertices∑

with i =di +ni2

−2



18-0918-09 1717R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 1717R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 1717R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 1717R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 1717

✗


Forces


Forces





(1/M)n corrections

N4LO

(Q5)

Coon & Friar

(1986)


with i =di +ni2

−2




✗


Forces


Forces





(1/M)n corrections

N4LO

(Q5)


with i =di +ni2

−2



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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.



18-0918-09 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


Forces





(1/M)n corrections

N4LO

(Q5)

The 3NFAt N2LO :

How well does it do?


with i =di +ni2

−2




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

2N (N3LO) +3N (N2LO)

forces

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

LLNL

2N (N3LO) force only



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18-0918-09 2323

Courtesy of M. Viviani;latest data from TUNL.




Recent Adv. in Th. of Nucl. Recent Adv. in Th. of Nucl. Forces EXOCT07, Forces EXOCT07,

Catania, 14-June-07Catania, 14-June-07

n-d at 3 MeV

Ay puzzle at NLO and NNLO (2NF+3NF)

NLO

NNLO

The

Ay puzz

le is

NOT

solv

ed

by th

e 3NF

at N

NLO.



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

What 3NFs are generated What 3NFs are generated by ChPT beyond NNLO?by ChPT beyond NNLO?



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✗


Forces


Forces





(1/M)n corrections

N4LO

(Q5)


with i =di +ni2

−2



18-0918-09 3030R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 3030

✗


Forces


Forces





(1/M)n corrections

N4LO

(Q5)

✗ ✗ ✗ ✗Time

derivative


with i =di +ni2

−2




✗


Forces


Forces





(1/M)n corrections

N4LO

(Q5)

✗ ✗ ✗ ✗ 1-loop

lead. vert.


with i =di +ni2

−2



18-0918-09 3232

Ishikawa & Robilotta,PRC 76, 014006 (2007)

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

In progress

The 3NF at N3LO explicitlyOne-loop, leading vertices



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



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Proton-deuteron elastic scatteringBlack line: 2NF only

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

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

The

2πN3LO 3

NF

is n

eglig

ible

.

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 involved, which are known to be weak (e.g., from NN).weak (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.



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What now?What now?



18-0918-09 3636

Go to the next order!




✗


Forces


Forces





(1/M)n corrections

N4LO

(Q5)

✗ ✗ ✗ ✗

1-loopone ci vert.

1-looplead. vert.


with i =di +ni2

−2



18-0918-09 3838

What to expect from those N4LO 1-loop diagrams?

Compare to “similar” 2NF diagrams.



18-0918-09 3939

✗

1-looplead. vert.

1-loopOne ci vert.

Corresponding 2NF contributions

2π



18-0918-09 4040




✗

1-looplead. vert.

1-loopOne ci vert.

Corresponding 2NF contributions

2π 3π

Kaiser (2000)

Kaiser (2001)

Small

Large!!



18-0918-09 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


Forces





(1/M)n corrections

N4LO

(Q5)

✗ ✗ ✗ ✗

1-loopone ci vert.

1-looplead. vert.

Large!!


with i =di +ni2

−2

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



18-0918-09 4343



18-0918-09 4444

pi-N Lagrangian with two derivativespi-N Lagrangian with two derivatives(“next-to-leading” order)(“next-to-leading” order)

Bernard et al. ‘97





Forces


Forces

Δ less

Leading Order




Additional in Δ full

Ordonez ’94, Kaiser ‘98

Krebs ‘07

Chiral 2NF




Δ-full theoryRaises NLO,Better convergence



18-0918-09 4747

The 3NF in the Δ-full theory



18-0918-09 4848R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4848R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4848R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4848R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4848R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4848

✗


Forces


Forces




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

N4LO

(Q5)

Δ less Additional in Δ full

1-loops with Δ

e.g., Illinois

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

• For reasons of consistency, the 2NF For reasons of consistency, the 2NF must then also include Δ’s---up to must then also include Δ’s---up to N3LO!N3LO!

• So, what does that look like?So, what does that look like?



18-0918-09 4949



18-0918-09 5050


Forces


Forces

Leading Order





+ 1/M corrections, Kaiser ’98: large?

2pi & 3pi 2-loops with Δ

The chiral 2NF: Δ-less vs. Δ-full

N4LO

(Q5)


Forces


Forces


Krebs ‘07

Δ less



18-0918-09 5151

Either

In summary:





Forces


Forces

Leading Order








N4LO

(Q5)


Forces


Forces


Krebs ‘07

Δ less



18-0918-09 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


Forces





N4LO

(Q5)


1-loops with Δ

e.g., Illinois



18-0918-09 5454

Or





Forces


Forces

Leading Order








N4LO

(Q5)


Forces


Forces


Krebs ‘07

Δ less



18-0918-09 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


Forces





N4LO

(Q5)


1-loops with Δ

e.g., Illinois

Whatever we decide (Δ-less or Δ-full), Whatever we decide (Δ-less or Δ-full), it will be impossible to do a complete it will be impossible to do a complete calculation anytime soon; so what’s a calculation anytime soon; so what’s a good strategy?good strategy?

• Assume “resonance saturation” for Assume “resonance saturation” for the 2NF, i.e., stay with the present Δ-the 2NF, i.e., stay with the present Δ-less N3LO 2NF (this has flaws!)less N3LO 2NF (this has flaws!)

• Add the Δ-full 3NF at N3LO.Add the Δ-full 3NF at N3LO.



18-0918-09 5757

ConclusionsConclusions

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

• The 3NF at N3LO of the Δ-less theory may The 3NF at N3LO of the Δ-less theory may be weak and useless.be weak and 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 ChPT of nuclear forces has been fairly easy and straightforward, so far; but what’s left and straightforward, so far; but what’s left won’t be easy, no matter how you do it.won’t be easy, no matter how you do it.



18-0918-09 5858

• at N4LO in Δ-less orat N4LO in Δ-less or

• at N3LO in Δ-full.at N3LO in Δ-full.



18-0918-09 5959

The missing three-nucleon forces:Where are they?

They are

r. machleidt, university of idaho university of idaho the missing three-nucleon forces: where are...

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