r. machleidt collaborators: e. marji, ch. zeoli university of idaho

R. Machleidt R. Machleidt

Collaborators: E. Marji, Ch. ZeoliCollaborators: E. Marji, Ch. Zeoli

University of IdahoUniversity of Idaho

The nuclear force problem:Have we finally reachedthe end of the tunnel?

474-th International Heraeus Seminar474-th International Heraeus Seminar

Bad Honnef, Germany, February 12 – 16, 2011Bad Honnef, Germany, February 12 – 16, 2011

OutlineOutline

•Historical perspectiveHistorical perspective

• 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 forcesProper renormalization of chiral forces

• The end of the tunnel?The end of the tunnel?

R. MachleidtR. Machleidt 22

The Nuclear Force Problem The Nuclear Force Problem Bad Honnef, 14 February Bad Honnef, 14 February

20112011

R. MachleidtR. Machleidt


20112011 33

The circle

of history is

closing!

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



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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)



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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


20112011



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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

NNLON3LO



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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

at N3LO



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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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… including the chiral 3NF

at N2LO



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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



LLNL



20112011 1414R. MachleidtR. Machleidt 1414

2N (N3LO) force only


2N (N3LO) +3N (N2LO)

forces


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.



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The 3NFat NNLO;

used so far.



20112011 1818R. MachleidtR. Machleidt

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

20102010 1818R. MachleidtR. Machleidt

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

20102010 1818

The 3NFat NNLO;

used so far.

Small?

Large!!

See contribution to This SeminarBy H. Krebs.

So, we are obviously not done! So, we are obviously not done!

• Subleading few-nucleon forces: Subleading few-nucleon forces: N4LO in Δ-less or N3LO in Δ-full.N4LO in Δ-less or N3LO in Δ-full.

• Renormalization of chiral nuclear forcesRenormalization of chiral nuclear forces



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I will focus now on this one.

Some of the more crucial open issues:



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“I about got this one renormalized”



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The issue has produced lots and lots of papers; this is just a small sub-selection.



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So, what’s the problem with this renormalization?



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The EFT approach is not just another

phenomenology. It’s field theory.

The problem in all field theories are

divergent loop integrals.

The method to deal with them in field theories:

1. Regularize the integral (e.g. apply a “cutoff”) to make it finite.2. Remove the cutoff dependence

by Renormalization (“counter

terms”).



20112011 2424

For calculating pi-pi and pi-NFor calculating pi-pi and pi-Nreactions no problem.reactions no problem.

However, the NN case is tougher,However, the NN case is tougher,because it involves because it involves two kinds two kinds of (divergent) loop integrals.of (divergent) loop integrals.



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The first kind:The first kind:

• ““NN Potential”: NN Potential”:

irreducible diagrams calculated perturbatively. irreducible diagrams calculated perturbatively.

Example:Example:

Counterterms

perturbative renormalization (order by order)




The first kind:The first kind:

• ““NN Potential”: NN Potential”:

irreducible diagrams calculated perturbatively. irreducible diagrams calculated perturbatively.

Example:Example:

Counterterms

perturbative renormalization (order by order)

This is fine.

No

problems.



20112011 2727

The second The second kind:kind:• Application of the NN Pot. in the Schrodinger or Application of the NN Pot. in the Schrodinger or

Lippmann-Schwinger (LS) equation: non-Lippmann-Schwinger (LS) equation: non-perturbative summation of ladder diagrams perturbative summation of ladder diagrams (infinite sum):(infinite sum):

27272727

In diagrams: T = + + + …



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• Divergent integral.Divergent integral.

• Regularize it:Regularize it:

• Cutoff dependent results.Cutoff dependent results.

• Renormalize to get rid of the cutoff dependence:Renormalize to get rid of the cutoff dependence:

2828

Non-perturbative renormalization






• Divergent integral.Divergent integral.

• Regularize it:Regularize it:

• Cutoff dependent results.Cutoff dependent results.

• Renormalize to get rid of the cutoff dependence:Renormalize to get rid of the cutoff dependence:

2929

Non-perturbative renormalization 2929

With what to renormalize this time?

Weinberg’s silent assumption:

The same counter terms as before.

(“Weinberg counting”)

Weinberg counting fails already in Leading Weinberg counting fails already in Leading OrderOrder

(for (for Λ Λ ∞ renormalization) ∞ renormalization)

•

• 3S1 and 1S0 (with a caveat) renormalizable with 3S1 and 1S0 (with a caveat) renormalizable with LO counter terms.LO counter terms.

• However, where OPE tensor force attractive:However, where OPE tensor force attractive:

3P0, 3P2, 3D2, …3P0, 3P2, 3D2, …

a counter term a counter term

must be added.must be added.



20112011 3030

“Modified Weinberg counting” for LO

Nogga, Timmermans, v. Nogga, Timmermans, v. Kolck Kolck PRC72, 054006 (2005):PRC72, 054006 (2005):



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Quantitative chiral NN potentials are at N3LO. So, we need to go substantially beyond LO.

• Nonperturbative or perturbative?Nonperturbative or perturbative?

• Infinite cutoff or finite cutoff?Infinite cutoff or finite cutoff?



20112011 3232

Renormalization beyond leading Renormalization beyond leading order –order –

IssuesIssues

Renormalization beyond leading Renormalization beyond leading order –order –

OptionsOptions

11 Continue with the nonperturbative Continue with the nonperturbative infinite-cutoff renormalization.infinite-cutoff renormalization.

22 Perturbative using DWBA.Perturbative using DWBA.

33 Nonperturbative using finite Nonperturbative using finite cutoffs ≤ Λχ ≈ 1 GeV.cutoffs ≤ Λχ ≈ 1 GeV.



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Option 1: Nonperturbative infinite-Option 1: Nonperturbative infinite-cutoff renormalization up to N3LOcutoff renormalization up to N3LO



20112011 3535

S=1 T=1 NLO

NNLO

LO

N3LO

Different partialwaves are windowson differentranges of the force.

• In lower partial waves (In lower partial waves (≅≅ short distances), in some cases short distances), in some cases convergence, in some not; data are not reproduced.convergence, in some not; data are not reproduced.

• In peripheral partial waves (In peripheral partial waves (≅≅ long distances), always long distances), always good convergence and reproduction of the data.good convergence and reproduction of the data.

• Thus, long-range interaction o.k., short-range not (should Thus, long-range interaction o.k., short-range not (should not be a surprise: the EFT is designed for Q < Λχ).not be a surprise: the EFT is designed for Q < Λχ).

• At all orders, either one (if pot. attractive) or no (if pot. At all orders, either one (if pot. attractive) or no (if pot. repulsive) counterterm, per partial wave: What kind of repulsive) counterterm, per partial wave: What kind of power counting scheme is this? power counting scheme is this?

• Where are the systematic order by order improvements?Where are the systematic order by order improvements?



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Option 1: Nonperturbative infinite-cutoff Option 1: Nonperturbative infinite-cutoff renormalization up to N3LOrenormalization up to N3LO

Observations and problemsObservations and problems



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• In lower partial waves (In lower partial waves (≅≅ short distances), in some short distances), in some cases convergence, in some not; data are not cases convergence, in some not; data are not reproduced.reproduced.

• In peripheral partial waves (In peripheral partial waves (≅≅ long distances), long distances), always good convergence and reproduction of the always good convergence and reproduction of the data.data.

• Thus, long-range interaction o.k., short-range not Thus, long-range interaction o.k., short-range not (should not be a surprise: the EFT is designed for Q (should not be a surprise: the EFT is designed for Q < Λχ).< Λχ).

• At all orders, either one (if pot. attractive) or no (if At all orders, either one (if pot. attractive) or no (if pot. repulsive) counterterm, per partial wave: What pot. repulsive) counterterm, per partial wave: What kind of power counting scheme is this? kind of power counting scheme is this?

• Where are the systematic order by order Where are the systematic order by order improvements?improvements?


Option 1: Nonperturbative infinite-cutoff Option 1: Nonperturbative infinite-cutoff renormalization up to N3LOrenormalization up to N3LO

Observations and problemsObservations and problems

No good!



20112011 3838

Option 2: Perturbative, using DWBAOption 2: Perturbative, using DWBA(Valderrama ‘09)(Valderrama ‘09)

• Renormalize LO non-perturbatively with infinite Renormalize LO non-perturbatively with infinite cutoff using modified Weinberg counting.cutoff using modified Weinberg counting.

• Use the distorted LO wave to calculate higher Use the distorted LO wave to calculate higher orders in perturbation theory.orders in perturbation theory.

• At NLO, 3 counterterms for 1S0 and 6 for 3S1: a At NLO, 3 counterterms for 1S0 and 6 for 3S1: a power-counting scheme that allows for power-counting scheme that allows for systematic improvements order by order systematic improvements order by order emerges.emerges.

• Results for NN scattering o.k., so, in principal, Results for NN scattering o.k., so, in principal, the scheme works.the scheme works.

• But how practical is this scheme in nuclear But how practical is this scheme in nuclear structure?structure?

• LO interaction has huge tensor force, huge LO interaction has huge tensor force, huge wound integral; wound integral; bad convergence of the bad convergence of the many-body problem. Impractical!many-body problem. Impractical!




Option 2: Perturbative, using DWBAOption 2: Perturbative, using DWBA(Valderrama ‘09)(Valderrama ‘09)

• Renormalize LO non-perturbatively with infinite Renormalize LO non-perturbatively with infinite cutoff using modified Weinberg counting.cutoff using modified Weinberg counting.

• Use the distorted LO wave to calculate higher Use the distorted LO wave to calculate higher orders in perturbation theory.orders in perturbation theory.

• At NLO, 3 counterterms for 1S0 and 6 for 3S1: a At NLO, 3 counterterms for 1S0 and 6 for 3S1: a power-counting scheme that allows for power-counting scheme that allows for systematic improvements order by order systematic improvements order by order emerges.emerges.

• Results for NN scattering o.k., so, in principal, Results for NN scattering o.k., so, in principal, the scheme works.the scheme works.

• But how practical is this scheme in nuclear But how practical is this scheme in nuclear structure?structure?

• LO interaction has huge tensor force, huge LO interaction has huge tensor force, huge wound integral; wound integral; bad convergence of the bad convergence of the many-body problem. Impractical!many-body problem. Impractical!

For considerations

of the NN

amplitude o.k.

But impractical fo

r

nuclear stru

cture

applications.



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What now?

Option 3: Rethink the problem Option 3: Rethink the problem from scratchfrom scratch

• EFFECTIVE EFFECTIVE field theory for Q ≤ Λχ ≈ 1 GeV.field theory for Q ≤ Λχ ≈ 1 GeV.

• So, you have to expect garbage above Λχ.So, you have to expect garbage above Λχ.

• The garbage may even converge, but that The garbage may even converge, but that doesn’t convert the garbage into the good doesn’t convert the garbage into the good stuff (Epelbaum & Gegelia ‘09).stuff (Epelbaum & Gegelia ‘09).

• So, stay away from territory that isn’t So, stay away from territory that isn’t covered by the EFT.covered by the EFT.



20112011 4141

Option 3: Nonperturbative using finite Option 3: Nonperturbative using finite cutoffs ≤ Λχ ≈ 1 GeV.cutoffs ≤ Λχ ≈ 1 GeV.

Goal: Find “cutoff indepence” for a Goal: Find “cutoff indepence” for a certain finite range below 1 GeV.certain finite range below 1 GeV.



20112011 4242

Very recently, a systematic investigation of this kind has been conducted by us at NLO using Weinberg Counting, i.e.

2 contacts in each S-wave(used to adjust scatt. length and eff. range),

1 contact in each P-wave(used to adjust phase shift at low energy).



20112011 4343

Cutoff dependence ofNN Phase shifts at NLO

Where is the range of cutoff independence???

400

1000 1000

400



20112011 4444

Note that the real thing are DATA (not phase shifts), e.g., NN cross sections, etc. Therefore better: Look for cutoff independence in the description of the data.

Notice, however, that there are many data (about 6000 NNData below 350 MeV). Therefore, it makes no senseto look at single data sets (observables). Instead, one shouldcalculate

with N the number of NN data in a certain energy range.

χ 2 =zitheory − zi

exp( )2

Δziexp( )

2i=1

i=N

∑



20112011 4545

Χ2/datum for the neutron-proton data as functionof cutoff in energy intervals as denoted

There is a range of cutoff independence!

ConclusionsConclusions



20112011 4646

• Chiral effective field theory is a useful tool Chiral effective field theory is a useful tool to deal with the nuclear force problem.to deal with the nuclear force problem.

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

• But there are still issues:But there are still issues:

• Subleading 3NFs: additional and stronger Subleading 3NFs: additional and stronger 3NFs are needed (see next talk by H. Krebs).3NFs are needed (see next talk by H. Krebs).

• Renormalization:Renormalization: more subtle, more more subtle, more controversial.controversial.

• Forget about non-perturbative infinite-cutoff Forget about non-perturbative infinite-cutoff reno: not convergent (in low partial waves reno: not convergent (in low partial waves ≅≅ short distances), should not be a surprise; no short distances), should not be a surprise; no clear power counting scheme, no systematic clear power counting scheme, no systematic improvements order by order.improvements order by order.

• Perturbative beyond LO: may be o.k. for the Perturbative beyond LO: may be o.k. for the NN amplitude (cf. work of Valderrama); but NN amplitude (cf. work of Valderrama); but impractical in nuclear structure applications, impractical in nuclear structure applications, tensor force (wound integral) too large.tensor force (wound integral) too large.

• Identify “Cutoff Independence” within a Identify “Cutoff Independence” within a range ≤ Λχ ≈1 GeV. Most realistic approach range ≤ Λχ ≈1 GeV. Most realistic approach (Lepage!). I have demonstrated this at NLO (Lepage!). I have demonstrated this at NLO (NNLO and N3LO to come, stay tuned).(NNLO and N3LO to come, stay tuned).



20112011 4747

Our views on Our views on renoreno



20112011 4848

Have we finally finally reachedthe end of the tunnel?

Not quite,But certainly we see the light

at the end of the tunnel!

And so,

r. machleidt collaborators: e. marji, ch. zeoli university of idaho

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