r. machleidt, university of idaho
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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
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 33
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
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
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 55
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
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 66
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).
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 77R. 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).
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 88
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 99R. MachleidtR. Machleidt 99
Deuteron wave functionsof N3LO potentials
red = Idaho, green = Bochum/JuelichBlack lines = high-precision pots.
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 1010R. MachleidtR. Machleidt 1010
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
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 1111R. MachleidtR. Machleidt 1111
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
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 1212
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
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 1313R. MachleidtR. Machleidt 1313
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 …
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 1414
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 1515
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
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
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
✗
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. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
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
✗
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. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 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
(1/M)n corrections
N4LO
(Q5)
Power =2 + 2L + i all vertices∑
with i =di +ni2
−2
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 1919
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.
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
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
✗
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
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 2121R. MachleidtR. Machleidt 2121
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
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 2222
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 2323
Courtesy of M. Viviani;latest data from TUNL.
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 2424R. MachleidtR. Machleidt 2424
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.
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 2525
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?
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 2626
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 2727R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 2727R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 2727
✗
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. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 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 2828R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 2828
✗
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. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 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. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
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
✗
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
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 3131R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 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)
✗ ✗ ✗ ✗ 1-loop
lead. vert.
Power =2 + 2L + i all vertices∑
with i =di +ni2
−2
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
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
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 3333
The ring diagramsFrom: Bernard, Epelbaum, Krebs, Meissner, PRC 77, 064004 (2008)
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 3434
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.
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 3535
What now?What now?
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 3636
Go to the next order!
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 3737R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 3737R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 3737R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 3737
✗
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
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 3838
What to expect from those N4LO 1-loop diagrams?
Compare to “similar” 2NF diagrams.
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 3939
✗
1-looplead. vert.
1-loopOne ci vert.
Corresponding 2NF contributions
2π
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 4040
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 4141R. MachleidtR. MachleidtChiral 3NF beyond N2LO Chiral 3NF beyond N2LO PNuT Seminar, UA, 11-06-08 PNuT Seminar, UA, 11-06-08 4141
✗
1-looplead. vert.
1-loopOne ci vert.
Corresponding 2NF contributions
2π 3π
Kaiser (2000)
Kaiser (2001)
Small
Large!!
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
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
✗
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 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.
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 4343
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
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
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 4545R. MachleidtR. Machleidt 4545
The Hierarchy of Nuclear
Forces
The Hierarchy of Nuclear
Forces
Δ less
Leading Order
Next-to Leading Order
Next-to-Next-to Leading Order
Next-to-Next-to-Next-to Leading Order
Additional in Δ full
Ordonez ’94, Kaiser ‘98
Krebs ‘07
Chiral 2NF
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 4646R. MachleidtR. Machleidt 4646
Δ-full theoryRaises NLO,Better convergence
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 4747
The 3NF in the Δ-full theory
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
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
✗
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, 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?
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 4949
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 5050
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. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 5151
Either
In summary:
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 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. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
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
✗
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. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 5454
Or
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 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. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
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
✗
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
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.
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
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.
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
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.
R. MachleidtR. Machleidt
Missing 3NFs: Where are they? Missing 3NFs: Where are they? Mardi Gras, LSU, 02- Mardi Gras, LSU, 02-
18-0918-09 5959
The missing three-nucleon forces:Where are they?
They are
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