Nuclear forces from chiral effective field theory: Achievements and challenges R. Machleidt...

Nuclear forces from chiraleffective field theory:

Achievements and challenges

R. Machleidt R. Machleidt

University of IdahoUniversity of Idaho

R. MachleidtR. Machleidt 11Nuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011

OutlineOutline

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

Basic ideas and resultsBasic ideas and results

• Are we done? Are we done? No!No!

• Proper renormalization of chiral forcesProper renormalization of chiral forces

• Sub-leading many-body forcesSub-leading many-body forces

• Is THE END near?Is THE END near?

R. MachleidtR. Machleidt 22Nuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011

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. MachleidtNuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011 33

• 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

Nuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011


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

NNLON3LO


N3LO Potential by Entem & Machleidt, PRC 68, 041001 (2003).NNLO and NLO Potentials by Epelbaum et al., Eur. Phys. J. A19, 401 (2004).


This is, of course, all very nice;However there is a “hidden” issue

here that needs our attention:Renormalization


“I about got this one renormalized”


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


Now, what’s so important with this renormalization?


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”).


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.


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)

R. MachleidtR. MachleidtNuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011 1515R. MachleidtR. Machleidt 1515

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.


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):

16161616

In diagrams: T = + + + …




• 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:

1717

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:

1818

Non-perturbative renormalization 1818

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.


“Modified Weinberg counting” for LO

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


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?


Renormalization beyond leading Renormalization beyond leading order –order –

IssuesIssues





IssuesIssues

• In lower partial waves (In lower partial waves (≅≅ short distances), in general, no short distances), in general, no order by order convergence; data are not reproduced.order by order convergence; 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?


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

Observations and problemsObservations and problems


• In lower partial waves (In lower partial waves (≅≅ short distances), in short distances), in general, no order by order convergence; data are not general, no order by order convergence; 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!


Option 2: Perturbative, using DWBA Option 2: Perturbative, using DWBA and finite cutoff (Valderrama ‘11)and finite cutoff (Valderrama ‘11)

• Renormalize LO non-perturbatively with finite Renormalize LO non-perturbatively with finite 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!


• Renormalize LO non-perturbatively with finite Renormalize LO non-perturbatively with finite 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.

Option 2: Perturbative, using DWBA Option 2: Perturbative, using DWBA and finite cutoff (Valderrama ‘11)and finite cutoff (Valderrama ‘11)

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.






IssuesIssues

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

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


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).


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

∑


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

There is a range of cutoff independence!

InvestigationsAt NNLO and N3LOAre under way.


Chiral three-nucleon forces (3NF)

R. MachleidtR. MachleidtNuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011 3333R. MachleidtR. MachleidtNuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011 3333


Leading Order





Forces



Leading Order





Forces



Leading Order





Forces

The 3NFat NNLO;

used so far.



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


2N (N3LO) force only


2N (N3LO) +3N (N2LO)

forces


LLNL


AnalyzingPower

Ay

p-d

p-3He

2NF only

2NF+3NFCalculations bythe Pisa Group

2NF only

2NF+3NF

The Ay puzzle is NOT solved

by the 3NF at NNLO.


And so,And so,we need 3NFs beyond NNLO,we need 3NFs beyond NNLO,because …because …

• The 2NF is N3LO;The 2NF is N3LO;

consistency requires that all consistency requires that all contributions are included up to the contributions are included up to the same order.same order.

• There are unresolved problems in 3N There are unresolved problems in 3N and 4N scattering, and nuclear and 4N scattering, and nuclear structure.structure.


The 3NFat NNLO;

used so far.


Ishikawa & Robilotta,PRC 76, 014006 (2007)

Bernard,Epelbaum,Krebs,Meissner,PRC 77, 064004 (2008)

Bernard et al., arXiv:1108.3816

The 3NF at N3LO explicitlyOne-loop, leading vertices

R. MachleidtR. MachleidtNuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011 4444Nuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011

The 3NFat NNLO;

used so far.

Small?!


✗

1-looplead. vert.

1-loopOne ci vert.

Corresponding 2NF contributions

2π


✗

1-looplead. vert.

1-loopOne ci vert.

Corresponding 2NF contributions

2π 3π

Kaiser (2000)

Kaiser (2001)

Small

Large!!


The 3NFat NNLO;

used so far.

Small?!

Large?!


The N4LO 3NF 2PE-1PE diagrams

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

Still a lot to come in the 3NF

business.

Stay tuned!

R. MachleidtR. MachleidtNuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011 4949R. MachleidtR. MachleidtNuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011 4949R. MachleidtR. MachleidtNuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011 4949

The 3NFat NNLO;

used so far.

Small?!

Large?!

And there is alsothe Δ-full version

R. MachleidtR. MachleidtNuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011 5050R. MachleidtR. MachleidtNuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011 5050R. MachleidtR. MachleidtNuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011 5050R. MachleidtR. MachleidtNuclear Forces from ChEFT Nuclear Forces from ChEFT APFB2011, Seoul, 08/22/2011 APFB2011, Seoul, 08/22/2011 5050

Small?!

The 3NFat NNLO;

used so far.

Large?!

And there is alsothe Δ-full version


What’s the better approach is a matter of taste

•Bochum (E. Epelbaum & H. Krebs) pursues Δ-full

• Idaho (R. M. & D. R. Entem) Δ-less

ConclusionsConclusions


• The chiral EFT approach has substantially The chiral EFT approach has substantially advanced our understanding of nuclear advanced our understanding of nuclear forces. forces.

• One major milestone of the past decade: One major milestone of the past decade: “high precision” NN pots. at N3LO.“high precision” NN pots. at N3LO.

• But there are still some open issues:But there are still some open issues:

• The renormalization of the chiral 2NFThe renormalization of the chiral 2NF

• Sub-leading 3NFsSub-leading 3NFs

• 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; short distances), should not be a surprise; no clear power counting scheme, no no clear power counting scheme, no systematic improvements order by order.systematic improvements order by order.

• Perturbative beyond LO: academically Perturbative beyond LO: academically interesting (cf. good work by Valderrama); interesting (cf. good work by Valderrama); but impractical in nuclear structure but impractical in nuclear structure applications, tensor force (wound integral) applications, tensor force (wound integral) too large.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).


Concerning the Concerning the renormalization renormalization issue,issue,we take the view:we take the view:

3NF issue3NF issue

• The 3NF at NNLO is insufficient.The 3NF at NNLO is insufficient.

• The 3NF at N3LO (in the The 3NF at N3LO (in the ΔΔ-less theory) is -less theory) is probably weak.probably weak.

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

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



And so, we are not yet completely done with the

nuclear force problem, but …

Date post:	16-Dec-2015
Category:	Documents
Upload:	nora-shadwick
View:	215 times
Download:	1 times

Nuclear forces from chiral effective field theory: Achievements and challenges R. Machleidt...

Documents