r. machleidt, university of idaho d. r. entem, university of salamanca sub-leading three-nucleon...

R. Machleidt, University of IdahoR. Machleidt, University of Idaho

D. R. Entem, University of D. R. Entem, University of SalamancaSalamanca

Sub-Leading Three-Nucleon Forcesin Chiral Perturbation Theory

OutlineOutline

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

Overview & achievementsOverview & achievements

• Are we done? No!Are we done? No!

• Sub-leading three-nucleon forcesSub-leading three-nucleon forces

• OutlookOutlook

R. MachleidtR. Machleidt 22Sub-Leading 3NFs Sub-Leading 3NFs

ECT* Trento 07/11/2011 ECT* Trento 07/11/2011

From QCD to nuclear physics via chiral From QCD to nuclear physics via chiral EFT (in a nutshell)EFT (in a nutshell)

• QCD at low energy is strong.QCD at low energy is strong.

• Quarks and gluons are confined into Quarks and gluons are confined into colorless hadrons.colorless hadrons.

• Nuclear forces are residual forces Nuclear forces are residual forces (similar to van der Waals forces)(similar to van der Waals forces)

• Separation of scalesSeparation of scales



• Calls for an EFT Calls for an EFT

soft scale: Q ≈ msoft scale: Q ≈ mπ π ,, hard scale: Λhard scale: Λχ χ ≈ m ≈ mρ ρ ; ; pions and nucleon relevant d.o.f. pions and nucleon relevant d.o.f.

•Low-energy expansion: (Q/ΛLow-energy expansion: (Q/Λχχ))ν ν

with ν bounded from below.with ν bounded from below.

•Most general Lagrangian consistent with Most general Lagrangian consistent with all symmetries of low-energy QCD.all symmetries of low-energy QCD.

•π-π and π-N perturbativelyπ-π and π-N perturbatively

•NN has bound states:NN has bound states:

(i) NN potential perturbatively(i) NN potential perturbatively

(ii) apply nonpert. in LS equation. (ii) apply nonpert. in LS equation.

(Weinberg) (Weinberg)



R. MachleidtR. Machleidt 55

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

Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011


NN 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).

LO

NLO

NNLO

N3LO



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

at N3LO



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 1.21Sub-Leading 3NFs Sub-Leading 3NFs



… including the chiral 3NF

at N2LO



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



LLNL


R. MachleidtR. Machleidt 1515R. MachleidtR. Machleidt 1515

2N (N3LO) force only


2N (N3LO) +3N (N2LO)

forces


LLNL



The Ay puzzle is NOT solved

by the 3NF at NNLO.

AnalyzingPower

Ay

p-d

p-3He

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.


So, let’s take a close look at the developmentof chiral 3NFs.




The 3NFat NNLO;

used so far.



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)

The

2πN3LO 3

NF

is n

eglig

ible

.



R. MachleidtR. Machleidt 2323R. MachleidtR. Machleidt

Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-

20102010 2323

Small?

The 3NFat NNLO;

used so far.


R. MachleidtR. Machleidt 2424R. MachleidtR. Machleidt


20102010 2424R. MachleidtR. Machleidt


20102010 2424

Small?

The 3NFat NNLO;

used so far.



What to expect from those N4LO 1-loop diagrams?

Compare to “similar” 2NF diagrams.




✗

1-looplead. vert.

1-loopOne ci vert.

Corresponding 2NF contributions

2π



R. MachleidtR. Machleidt 2727R. MachleidtR. Machleidt 2727

✗

1-looplead. vert.

1-loopOne ci vert.

Corresponding 2NF contributions

2π 3π

Kaiser (2000)

Kaiser (2001)

Small

Large!!



R. MachleidtR. MachleidtThe Nuclear Force Problem The Nuclear Force Problem Mumbai, 22 November 2010 Mumbai, 22 November 2010 2828R. MachleidtR. Machleidt




20102010 2828

Small?

The 3NFat NNLO;

used so far.

Large!!R. MachleidtR. Machleidt 2828


By the way, the full story is more complicated,See next slide.



R. MachleidtR. MachleidtThe unfinished business The unfinished business

Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 3030R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 3030R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 3030R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 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

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

(1/M)n corrections

N4LO

(Q5)

✗ ✗ ✗ ✗



But to keep a long story short,Let’s not bother with those 1/M corrections.







20102010 3232

Small?

The 3NFat NNLO;

used so far.



A variation of the theory is possible:Introduce Δ degrees of freedom







20102010 3434

Small?

The 3NFat NNLO;

used so far.




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

Bernard et al. ‘97







20102010 3636

Small?

The 3NFat NNLO;

used so far.







20102010 3737

Small?

Large!!

The 3NFat NNLO;

used so far.



So, with Δ’s you have to go “only” to N3LO(to catch the large 3NFs),But notice that there are many more diagramsAt N3LO with Δ’s as compared to Δ-less.So, it’s not necessarily simpler.

Notice further that in the Δ-full theory the 2NFGets also more complicated.



Summarize: Δ-full vs. Δ-less (2NF and 3NF).







20102010 4141R. MachleidtR. Machleidt 4141Sub-Leading 3NFs Sub-Leading 3NFs






20102010 4242R. MachleidtR. Machleidt 4242Sub-Leading 3NFs Sub-Leading 3NFs


What’s the better approach is a matter of taste

•Bochum pursues Δ-full

• Idaho Δ-less







20102010 4444

Small?

The 3NFat NNLO;

used so far.



The Δ-less N4LO (ν=5) 3NF in more detail:

Tree diagrams and 1-loop graphs

1-loop graphs: 5 topologies

2PE 2PE-1PE Ring Contact-1PE Contact-2PE



Pisa

Power 2 2L i all vertices

with i di ni2

2

e_i

The N4LO 3NF 2PE diagrams



The N4LO 3NF 2PE-1PE diagrams



Many new isospin/spin/momentum structures,some similar to N3LO 2PE-1PE 3NF.

ConclusionsConclusions


• Substantial advances in chiral nuclear forces Substantial advances in chiral nuclear forces during the past decade. The major milestone of during the past decade. The major milestone of the decade: “high precision” NN pots. at N3LO; the decade: “high precision” NN pots. at N3LO; good for nuclear structure.good for nuclear structure.

• But there are still substantial open issues But there are still substantial open issues concerning three-nucleon forces (3NF), concerning three-nucleon forces (3NF), which is which is why we are here.why we are here.

• Large 3NFs with many new structures to be Large 3NFs with many new structures to be expected at N3LO of expected at N3LO of ΔΔ-full or N4LO at -full or N4LO at ΔΔ-less. -less. Construction is under way.Construction is under way.

• There will be many new 3NFs to check out in the There will be many new 3NFs to check out in the near future. near future. Stay tuned.Stay tuned.



And a specific request to 3NF appliersAnd a specific request to 3NF appliers

• In the 2PE 3NF at NNLO, consider the c_i In the 2PE 3NF at NNLO, consider the c_i LECs as free parameters: Is there a LECs as free parameters: Is there a parameter set that substantially improves parameter set that substantially improves the Nd Ay?the Nd Ay?

• Is there any combination of isospin, spin, Is there any combination of isospin, spin, and momentum operators of 3 nucleons and momentum operators of 3 nucleons that particularly improves Ay?that particularly improves Ay?

R. MachleidtR. MachleidtSub-Leading 3NFs Sub-Leading 3NFs

ECT* Trento 07/11/2011 ECT* Trento 07/11/2011 4949

r. machleidt, university of idaho d. r. entem, university of salamanca sub-leading three-nucleon...

Documents