474 th International Heraeus Seminar Bad Honnef Germany

474 -th International Heraeus Seminar Bad Honnef, Germany, February 12 – 16, 2011 The nuclear force problem: Have we finally reached the end of the tunnel? R. Machleidt Collaborators: E. Marji, Ch. Zeoli University of Idaho

Outline • Historical perspective • Nuclear forces from chiral EFT: Overview & achievements • Are we done? No! • Sub-leading many-body forces • Proper renormalization of chiral forces • The end of the tunnel? R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 2

e l c r i s c i e y r h T isto ! g h n f i o los c R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 3

From QCD to nuclear physics via chiral EFT (in a nutshell) • QCD at low energy is strong. • Quarks and gluons are confined into colorless • • hadrons. Nuclear forces are residual forces (similar to van der Waals forces) Separation of scales R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 4

• Calls for an EFT soft scale: Q ≈ mπ , hard scale: Λχ ≈ mρ ; pions and nucleon relevant d. o. f. • Low-energy expansion: (Q/Λχ)ν with ν bounded from below. • Most general Lagrangian consistent with all symmetries of low-energy QCD. • π-π and π-N perturbatively • NN has bound states: (i) NN potential perturbatively (ii) apply nonpert. in LS equation. (Weinberg) R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 5

2 N forces 3 N forces 4 N forces Leading Order Next-to Leading Order The Hierarchy of Nuclear Forces Next-to Leading Order R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 6

NN phase shifts up to 300 Me. V Red Line: N 3 LO 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. A 19, 401 (2004). LO NNLO N 3 LO R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 7

N 3 LO Potential by Entem & Machleidt, PRC 68, 041001 (2003). NNLO and NLO Potentials by Epelbaum et al. , Eur. Phys. J. A 19, 401 (2004). R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 8

Applications of the chiral NN potential at N 3 LO R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 9

Chiral NN potential at N 3 LO underbinds by ~1 Me. V/nucleon. (Size extensivity at its best. ) Nucleus DE / A [Me. V] 4 He 1. 08 (0. 73 FY) 16 O 1. 25 40 Ca 0. 84 48 Ca R. Machleidt 1. 27 The Nuclear Force Problem 48 Ni Bad Honnef, 141. 21 February 2011 10

… including the chiral 3 NF at N 2 LO R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 11

Calculating the properties of light nuclei using chiral 2 N and 3 N forces “No-Core Shell Model “ Calculations by P. Navratil et al. , LLNL R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 12

Calculating the properties of light nuclei using chiral 2 N and 3 N forces “No-Core Shell Model “ Calculations by P. Navratil et al. , LLNL R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 2 N (N 3 LO) force only 13

Calculating the properties of light nuclei using chiral 2 N and 3 N forces “No-Core Shell Model “ Calculations by P. Navratil et al. , LLNL R. Machleidt 2 N (N 3 LO) +3 N (N 2 LO) The Nuclear Force Problem forces Bad Honnef, 14 February 2011 2 N (N 3 LO) force only 14

Analyzing Power Ay p-3 He R. Machleidt p-d 2 NF+3 NF 2 NF only Calculations by the Pisa Group d e v l o s T O N. s i O L e l N z z N u t p a y F A N The y the 3 b The Nuclear Force Problem Bad Honnef, 14 February 2011 15

Why do we need 3 NFs beyond NNLO? • The 2 NF is N 3 LO; consistency requires that all contributions are at the same order. • There are unresolved problems in 3 N, 4 N scattering and nuclear structure. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 16

The 3 NF at NNLO; used so far. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 17

The 3 NF at NNLO; used so far. Small? Large!! R. Machleidt Nuclear forces. Force from chiral EFT The Nuclear Problem EFB 21, Salamanca, 08 -31 -2010 Bad Honnef, 14 February 2011 See contribution to This Seminar By H. Krebs. 18

So, we are obviously not done! Some of the more crucial open issues: • Subleading few-nucleon forces: N 4 LO in Δ-less or N 3 LO in Δ-full. • Renormalization of chiral nuclear forces I will focus now on this one. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 19

“I about got this one renormalized” R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 20

The issue has produced lots and lots of papers; this is just a small sub-selection. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 21

So, what’s the problem with this renormalization? R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 22

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”). R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 23

For calculating pi-pi and pi-N reactions no problem. However, the NN case is tougher, because it involves two kinds of (divergent) loop integrals. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 24

The first kind: • “NN Potential”: irreducible diagrams calculated perturbatively. Example: Counter terms Ø perturbative renormalization (order by order) R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 25

The first kind: • “NN Potential”: . e n i f irreducible diagrams calculated perturbatively. Example: s i h o T N ms. e l b o r p Counter terms Ø perturbative renormalization (order by order) R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 26

The second kind: • Application of the NN Pot. in the Schrodinger or Lippmann-Schwinger (LS) equation: non-perturbative summation of ladder diagrams (infinite sum): In diagrams: R. Machleidt + The Nuclear Force Problem Bad Honnef, 14 February 2011 + +… 27

The second kind: • Application of the NN Pot. in the Schrodinger or Lippmann-Schwinger (LS) equation: non-perturbative summation of ladder diagrams (infinite sum): • Divergent integral. • Regularize it: • Cutoff dependent results. • Renormalize to get rid of the cutoff dependence: ØNon-perturbative renormalization R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 28

The second kind: • Application of the NN Pot. in the Schrodinger or Lippmann-Schwinger (LS) equation: non-perturbative summation of ladderto diagrams (infinite sum): this With what renormalize time? Weinberg’s • Divergent integral. silent assumption: The same counter terms as • Regularize it: before. (“Weinberg counting”) • Cutoff dependent results. • Renormalize to get rid of the cutoff dependence: ØNon-perturbative renormalization R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 29

Weinberg counting fails already in Leading Order (for Λ ∞ renormalization) • • 3 S 1 and 1 S 0 (with a caveat) renormalizable with LO • counter terms. However, where OPE tensor force attractive: 3 P 0, 3 P 2, 3 D 2, … Nogga, Timmermans, v. Kolck a counter term PRC 72, 054006 (2005): must be added. “Modified Weinberg counting” for LO R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 30

Quantitative chiral NN potentials are at N 3 LO. So, we need to go substantially beyond LO. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 31

Renormalization beyond leading order – Issues • Nonperturbative or perturbative? • Infinite cutoff or finite cutoff? R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 32

Renormalization beyond leading order – Options 1 Continue with the nonperturbative 2 3 infinite-cutoff renormalization. Perturbative using DWBA. Nonperturbative using finite cutoffs ≤ Λχ ≈ 1 Ge. V. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 33

Option 1: Nonperturbative infinite-cutoff renormalization up to N 3 LO R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 34

NNLO S=1 T=1 NLO LO N 3 LO R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 Different partial waves are windows on different ranges of the force. 35

Option 1: Nonperturbative infinite-cutoff renormalization up to N 3 LO Observations and problems • In lower partial waves (≅ short distances), in some cases • • convergence, in some not; data are not reproduced. In peripheral partial waves (≅ long distances), always good convergence and reproduction of the data. Thus, long-range interaction o. k. , short-range not (should not be a surprise: the EFT is designed for Q < Λχ). At all orders, either one (if pot. attractive) or no (if pot. repulsive) counterterm, per partial wave: What kind of power counting scheme is this? Where are the systematic order by order improvements? R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 36

Option 1: Nonperturbative infinite-cutoff renormalization up to N 3 LO Observations and problems ! d • In lower partial waves (≅ short distances), in some cases • • convergence, in some not; data are not reproduced. In peripheral partial waves (≅ long distances), always good convergence and reproduction of the data. Thus, long-range interaction o. k. , short-range not (should not be a surprise: the EFT is designed for Q < Λχ). At all orders, either one (if pot. attractive) or no (if pot. repulsive) counterterm, per partial wave: What kind of power counting scheme is this? Where are the systematic order by order improvements? R. Machleidt g o N o o The Nuclear Force Problem Bad Honnef, 14 February 2011 37

Option 2: Perturbative, using DWBA (Valderrama ‘ 09) • Renormalize LO non-perturbatively with infinite cutoff • • • using modified Weinberg counting. Use the distorted LO wave to calculate higher orders in perturbation theory. At NLO, 3 counterterms for 1 S 0 and 6 for 3 S 1: a power -counting scheme that allows for systematic improvements order by order emerges. Results for NN scattering o. k. , so, in principal, the scheme works. But how practical is this scheme in nuclear structure? LO interaction has huge tensor force, huge wound integral; bad convergence of the many-body problem. Impractical! R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 38

Option 2: Perturbative, using DWBA (Valderrama ‘ 09) • Renormalize LO non-perturbatively with s infinite cutoff • • • n o i t e a d r u e t i d l i s p n am o c r r N o f Fo he N. k. l a e o c t i r t u c of t a c r u p r. t m s i s n t r o u a i B cle cat nu ppli a using modified Weinberg counting. Use the distorted LO wave to calculate higher orders in perturbation theory. At NLO, 3 counterterms for 1 S 0 and 6 for 3 S 1: a power -counting scheme that allows for systematic improvements order by order emerges. Results for NN scattering o. k. , so, in principal, the scheme works. But how practical is this scheme in nuclear structure? LO interaction has huge tensor force, huge wound integral; bad convergence of the many-body problem. Impractical! R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 39

What now? R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 40

Option 3: Rethink the problem from scratch • EFFECTIVE field theory for Q ≤ Λχ ≈ 1 Ge. V. • So, you have to expect garbage above Λχ. • The garbage may even converge, but that • doesn’t convert the garbage into the good stuff (Epelbaum & Gegelia ‘ 09). So, stay away from territory that isn’t covered by the EFT. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 41

Option 3: Nonperturbative using finite cutoffs ≤ Λχ ≈ 1 Ge. V. Goal: Find “cutoff indepence” for a certain finite range below 1 Ge. V. 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). R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 42

Cutoff dependence of NN Phase shifts at NLO 1000 400 Where is the range of cutoff independence? ? ? R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 43

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 NN Data below 350 Me. V). Therefore, it makes no sense to look at single data sets (observables). Instead, one should calculate with N the number of NN data in a certain energy range. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 44

Χ 2/datum for the neutron-proton data as function of cutoff in energy intervals as denoted There is a range of cutoff independence! R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 45

Conclusions • Chiral effective field theory is a useful tool to deal • • with the nuclear force problem. Substantial advances in chiral nuclear forces during the past decade. The major milestone of the decade: “high precision” NN pots. at N 3 LO, good for nuclear structure. But there are still issues: Subleading 3 NFs: additional and stronger 3 NFs are needed (see next talk by H. Krebs). Renormalization: more subtle, more controversial. R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 46

Our views on reno • Forget about non-perturbative infinite-cutoff • • reno: not convergent (in low partial waves ≅ short distances), should not be a surprise; no clear power counting scheme, no systematic improvements order by order. Perturbative beyond LO: may be o. k. for the NN amplitude (cf. work of Valderrama); but impractical in nuclear structure applications, tensor force (wound integral) too large. Identify “Cutoff Independence” within a range ≤ Λχ ≈1 Ge. V. Most realistic approach (Lepage!). I have demonstrated this at NLO (NNLO and N 3 LO to come, stay tuned). R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 47

And so, Have we finally reached the end of the tunnel? Not quite, But certainly we see the light at the end of the tunnel! R. Machleidt The Nuclear Force Problem Bad Honnef, 14 February 2011 48