r. machleidt university of idaho
DESCRIPTION
Nuclear forces from chiral EFT: The unfinished business. R. Machleidt University of Idaho. Outline. Nuclear forces from chiral EFT: Overview & achievements Are we done? No! Sub-leading many-body forces Proper renormalization of chiral forces Outlook: Will we ever be done?. - PowerPoint PPT PresentationTRANSCRIPT
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?
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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
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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
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The Hierarchy of Nuclear
Forces
The Hierarchy of Nuclear
Forces
2N forces
Leading Order
Next-to Leading Order
Next-to-Next-to Leading Order
Next-to-Next-to-Next-to Leading Order
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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
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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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Applications of the chiral NN potential
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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
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The Hierarchy of Nuclear
Forces
The Hierarchy of Nuclear
Forces
2N forces
Leading Order
Next-to Leading Order
Next-to-Next-to Leading Order
Next-to-Next-to-Next-to Leading Order
4N forces
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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
The Hierarchy of Nuclear
Forces
The Hierarchy of Nuclear
Forces
2N forces 3N forces
Leading Order
Next-to Leading Order
Next-to-Next-to Leading Order
Next-to-Next-to-Next-to Leading Order
The focus:The 3NF
4N forces
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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
✗
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
N4LO
(Q5)
Power =2 + 2L + i all vertices∑
with i =di +ni2
−2
R. MachleidtR. MachleidtThe unfinished business The unfinished business
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
✗
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)
Coon & Friar
(1986)
Power =2 + 2L + i all vertices∑
with i =di +ni2
−2
R. MachleidtR. MachleidtThe unfinished business The unfinished business
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
✗
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)
The 3NFAt N2LO :
How well does it do?
Power =2 + 2L + i all vertices∑
with i =di +ni2
−2
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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
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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
R. MachleidtR. MachleidtThe unfinished business The unfinished business
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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
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The Ay puzzle is NOT solved
by the 3NF at NNLO.
AnalyzingPower, Ay
p-d
p-3He
2NF only
2NF+3NFCalculations bythe Pisa Group
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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 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?
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Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 2828R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 2828R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 2828
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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)
Power =2 + 2L + i all vertices∑
with i =di +ni2
−2
R. MachleidtR. MachleidtThe unfinished business The unfinished business
Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 2929R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 2929R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 2929R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 2929
✗
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)
Power =2 + 2L + i all vertices∑
with i =di +ni2
−2
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 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)
Power =2 + 2L + i all vertices∑
with i =di +ni2
−2
R. MachleidtR. MachleidtThe unfinished business The unfinished business
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
✗
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)
✗ ✗ ✗ ✗Time
derivative
Power =2 + 2L + i all vertices∑
with i =di +ni2
−2
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Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 3232R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 3232R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 3232
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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)
✗ ✗ ✗ ✗ 1-loop
lead. vert.
Power =2 + 2L + i all vertices∑
with i =di +ni2
−2
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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 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.
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What now?What now?
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Go to the next order!
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Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 3838R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 3838R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 3838R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 3838
✗
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)
✗ ✗ ✗ ✗
1-loopone ci vert.
1-looplead. vert.
Power =2 + 2L + i all vertices∑
with i =di +ni2
−2
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What to expect from those N4LO 1-loop diagrams?
Compare to “similar” 2NF diagrams.
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✗
1-looplead. vert.
1-loopOne ci vert.
Corresponding 2NF contributions
2π
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1-looplead. vert.
1-loopOne ci vert.
Corresponding 2NF contributions
2π 3π
Kaiser (2000)
Kaiser (2001)
Small
Large!!
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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)
✗ ✗ ✗ ✗
1-loopone ci vert.
1-looplead. vert.
Large!!
Power =2 + 2L + i all vertices∑
with i =di +ni2
−2
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.
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The 3NF in the “Δ-full” theory
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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 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 …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!
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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.
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The End
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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.
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The Hierarchy of Nuclear
Forces
The Hierarchy of Nuclear
Forces
Leading Order
Next-to Leading Order
Next-to-Next-to Leading Order
Next-to-Next-to-Next-to 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
N4LO
(Q5)
The Hierarchy of Nuclear
Forces
The Hierarchy of Nuclear
Forces
Ordonez ’94, Kaiser ‘98
Krebs ‘07
Δ less
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Either
In summary:
R. MachleidtR. MachleidtThe unfinished business The unfinished business
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
The Hierarchy of Nuclear
Forces
The Hierarchy of Nuclear
Forces
Leading Order
Next-to Leading Order
Next-to-Next-to Leading Order
Next-to-Next-to-Next-to 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
N4LO
(Q5)
The Hierarchy of Nuclear
Forces
The Hierarchy of Nuclear
Forces
Ordonez ’94, Kaiser ‘98
Krebs ‘07
Δ less
R. MachleidtR. MachleidtThe unfinished business The unfinished business
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
✗
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 3NF: Δ-less vs. Δ-fullLeading Order (Q0)
N4LO
(Q5)
Δ less Additional in Δ full
1-loops with Δ
e.g., Illinois
R. MachleidtR. MachleidtThe unfinished business The unfinished business
Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 5454
Or
R. MachleidtR. MachleidtThe unfinished business The unfinished business
Vietri, Italy, 22 May 2010 Vietri, Italy, 22 May 2010 5555R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5555R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 5555
The Hierarchy of Nuclear
Forces
The Hierarchy of Nuclear
Forces
Leading Order
Next-to Leading Order
Next-to-Next-to Leading Order
Next-to-Next-to-Next-to 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
N4LO
(Q5)
The Hierarchy of Nuclear
Forces
The Hierarchy of Nuclear
Forces
Ordonez ’94, Kaiser ‘98
Krebs ‘07
Δ less
R. MachleidtR. MachleidtThe unfinished business The unfinished business
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
✗
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 3NF: Δ-less vs. Δ-fullLeading Order (Q0)
N4LO
(Q5)
Δ less Additional in Δ full
1-loops with Δ
e.g., Illinois