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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
R. MachleidtR. Machleidt 33Sub-Leading 3NFs Sub-Leading 3NFs
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• 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 44Sub-Leading 3NFs Sub-Leading 3NFs
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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
R. MachleidtR. Machleidt 66
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
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. Machleidt 77
N3LO Potential by Entem & Machleidt, PRC 68, 041001 (2003).NNLO and NLO Potentials by Epelbaum et al., Eur. Phys. J. A19, 401 (2004).
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. Machleidt 88
Applications of the chiral NN potential
at N3LO
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. Machleidt 99
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
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R. MachleidtR. Machleidt 1010Sub-Leading 3NFs Sub-Leading 3NFs
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R. MachleidtR. Machleidt 1111
… including the chiral 3NF
at N2LO
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. Machleidt 1212Sub-Leading 3NFs Sub-Leading 3NFs
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R. MachleidtR. Machleidt 1313
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
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. Machleidt 1414
2N (N3LO) force only
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
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. Machleidt 1515R. MachleidtR. Machleidt 1515
2N (N3LO) force only
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
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. Machleidt 1616
The Ay puzzle is NOT solved
by the 3NF at NNLO.
AnalyzingPower
Ay
p-d
p-3He
2NF only
2NF+3NFCalculations bythe Pisa Group
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. Machleidt 1717
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.
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
So, let’s take a close look at the developmentof chiral 3NFs.
R. MachleidtR. Machleidt 1818Sub-Leading 3NFs Sub-Leading 3NFs
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R. MachleidtR. Machleidt 1919
The 3NFat NNLO;
used so far.
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. Machleidt 2020
Ishikawa & Robilotta,PRC 76, 014006 (2007)
Bernard,Epelbaum,Krebs,Meissner,PRC 77, 064004 (2008)
In progress
The 3NF at N3LO explicitlyOne-loop, leading vertices
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. Machleidt 2121
The ring diagramsFrom: Bernard, Epelbaum, Krebs, Meissner, PRC 77, 064004 (2008)
R. MachleidtR. Machleidt 2121Sub-Leading 3NFs Sub-Leading 3NFs
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R. MachleidtR. Machleidt 2222
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 2222Sub-Leading 3NFs Sub-Leading 3NFs
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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.
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. Machleidt 2424R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 2424R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 2424
Small?
The 3NFat NNLO;
used so far.
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. Machleidt 2525
What to expect from those N4LO 1-loop diagrams?
Compare to “similar” 2NF diagrams.
R. MachleidtR. Machleidt 2525Sub-Leading 3NFs Sub-Leading 3NFs
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R. MachleidtR. Machleidt 2626
✗
1-looplead. vert.
1-loopOne ci vert.
Corresponding 2NF contributions
2π
R. MachleidtR. Machleidt 2626Sub-Leading 3NFs Sub-Leading 3NFs
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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. Machleidt 2727Sub-Leading 3NFs Sub-Leading 3NFs
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R. MachleidtR. MachleidtThe Nuclear Force Problem The Nuclear Force Problem Mumbai, 22 November 2010 Mumbai, 22 November 2010 2828R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 2828R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 2828
Small?
The 3NFat NNLO;
used so far.
Large!!R. MachleidtR. Machleidt 2828
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
By the way, the full story is more complicated,See next slide.
R. MachleidtR. Machleidt 2929Sub-Leading 3NFs Sub-Leading 3NFs
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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
✗
The Hierarchy of Nuclear
Forces
The Hierarchy of Nuclear
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)
✗ ✗ ✗ ✗
R. MachleidtR. Machleidt 3030Sub-Leading 3NFs Sub-Leading 3NFs
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But to keep a long story short,Let’s not bother with those 1/M corrections.
R. MachleidtR. Machleidt 3131Sub-Leading 3NFs Sub-Leading 3NFs
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R. MachleidtR. MachleidtThe Nuclear Force Problem The Nuclear Force Problem Mumbai, 22 November 2010 Mumbai, 22 November 2010 3232R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 3232R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 3232
Small?
The 3NFat NNLO;
used so far.
Large!!R. MachleidtR. Machleidt 3232
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
A variation of the theory is possible:Introduce Δ degrees of freedom
R. MachleidtR. Machleidt 3333Sub-Leading 3NFs Sub-Leading 3NFs
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R. MachleidtR. MachleidtThe Nuclear Force Problem The Nuclear Force Problem Mumbai, 22 November 2010 Mumbai, 22 November 2010 3434R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 3434R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 3434
Small?
The 3NFat NNLO;
used so far.
R. MachleidtR. Machleidt 3434Sub-Leading 3NFs Sub-Leading 3NFs
ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. Machleidt 3535
pi-N Lagrangian with two derivativespi-N Lagrangian with two derivatives(“next-to-leading” order)(“next-to-leading” order)
Bernard et al. ‘97
R. MachleidtR. Machleidt 3535Sub-Leading 3NFs Sub-Leading 3NFs
ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. MachleidtThe Nuclear Force Problem The Nuclear Force Problem Mumbai, 22 November 2010 Mumbai, 22 November 2010 3636R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 3636R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 3636
Small?
The 3NFat NNLO;
used so far.
R. MachleidtR. Machleidt 3636Sub-Leading 3NFs Sub-Leading 3NFs
ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. MachleidtThe Nuclear Force Problem The Nuclear Force Problem Mumbai, 22 November 2010 Mumbai, 22 November 2010 3737R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 3737R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 3737
Small?
Large!!
The 3NFat NNLO;
used so far.
R. MachleidtR. Machleidt 3737Sub-Leading 3NFs Sub-Leading 3NFs
ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
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.
R. MachleidtR. Machleidt 3838Sub-Leading 3NFs Sub-Leading 3NFs
ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. Machleidt 3939Sub-Leading 3NFs Sub-Leading 3NFs
ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
Summarize: Δ-full vs. Δ-less (2NF and 3NF).
R. MachleidtR. Machleidt 4040Sub-Leading 3NFs Sub-Leading 3NFs
ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. MachleidtThe Nuclear Force Problem The Nuclear Force Problem Mumbai, 22 November 2010 Mumbai, 22 November 2010 4141R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 4141R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 4141R. MachleidtR. Machleidt 4141Sub-Leading 3NFs Sub-Leading 3NFs
ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
R. MachleidtR. MachleidtThe Nuclear Force Problem The Nuclear Force Problem Mumbai, 22 November 2010 Mumbai, 22 November 2010 4242R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 4242R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 4242R. MachleidtR. Machleidt 4242Sub-Leading 3NFs Sub-Leading 3NFs
ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
What’s the better approach is a matter of taste
•Bochum pursues Δ-full
• Idaho Δ-less
R. MachleidtR. Machleidt 4343Sub-Leading 3NFs Sub-Leading 3NFs
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R. MachleidtR. MachleidtThe Nuclear Force Problem The Nuclear Force Problem Mumbai, 22 November 2010 Mumbai, 22 November 2010 4444R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 4444R. MachleidtR. Machleidt
Nuclear forces from chiral EFT Nuclear forces from chiral EFT EFB21, Salamanca, 08-31- EFB21, Salamanca, 08-31-
20102010 4444
Small?
The 3NFat NNLO;
used so far.
Large!!R. MachleidtR. Machleidt 4444
Sub-Leading 3NFs Sub-Leading 3NFs ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
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
R. MachleidtR. Machleidt 4545Sub-Leading 3NFs Sub-Leading 3NFs
ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
Pisa
Power 2 2L i all vertices
with i di ni2
2
e_i
The N4LO 3NF 2PE diagrams
R. MachleidtR. Machleidt 4646Sub-Leading 3NFs Sub-Leading 3NFs
ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
The N4LO 3NF 2PE-1PE diagrams
R. MachleidtR. Machleidt 4747Sub-Leading 3NFs Sub-Leading 3NFs
ECT* Trento 07/11/2011 ECT* Trento 07/11/2011
Many new isospin/spin/momentum structures,some similar to N3LO 2PE-1PE 3NF.
ConclusionsConclusions
R. MachleidtR. Machleidt 4848
• 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.
R. MachleidtR. Machleidt 4848Sub-Leading 3NFs Sub-Leading 3NFs
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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
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