Chieh-Jen (Jerry) Yang Recent progress towards a chiral effective field theory for the NN system Chieh-Jen (Jerry) Yang 楊傑仁 The Seventh International Symposium on Chiral Symmetry in Hadrons and Nuclei 10/27/2013 Collaborators: Bingwei Long, Bruce Barrett, D. Phillips, U. van Kolck, J. Rotureau

Answer: Yes, almost! But,… Chiral EFT potential Weinberg, van Kolck Kaiser, Brockmann, Weise. Epelbaum, Glockle, Meissner. Entem, Machleidt. Krebs, Epelbaum, Meissner. and more… NN Amplitude Conventional way： Weinberg counting + iterate-to-all order (solve it in Schrodinger or Lippmann-Schwinger eq.) But,…

Problems of Weinberg’s counting Singular attractive potentials demand contact terms. (Nogga, Timmermans, van Kolck (2005)) Beyond LO: Has RG problem at Λ>1 GeV. Yang, Elster, Phillips (2009) N3LO(Q4) Ch. Zeoli R. Machleidt D. R. Entem (2012)

Why is that a problem?

(in field theory => contact/counter terms) Main idea of EFT Absorbs the detail of high-E (more generally, un-relevant) physics into simpler form. (in field theory => contact/counter terms) Need to perform renormalization properly (a.k.a., power counting), so that the amplitude can be improved order by order (ideally, expand in (Q/Mhi)n).

How to check ? (that you are doing the sensible thing)

Renormalization group (RG) : included Cutoff Cutoff Cutoff Un-important detail Un-important detail Un-important detail More Un-important detail + Physics relevant (after proper PC) Un-important detail + (after proper PC) (after renormalization) (after renorm.) = = ~ ~ Un-important detail ~ Physics relevant Physics relevant 600 MeV *Only source of error: given by the high order terms. If not so, the power counting isn’t completely correct! (unimportant are not really unimportant)

Origin of the RG problem (in the conventional treatment)

O.k., as long as pcm is small enough, so that All O(Q2) + + + + O.k., as long as pcm is small enough, so that Λ Conventional power counting ….. Has problem, as Λ-dependence enter here; contact term aren’t enough. The expansion parameter is no longer ! Problems at Λ>1 GeV also imply that WPC might not give you (to make use of) all the counter terms which are legitimate (according to a truly RG-invariant theory) to be used.

New power counting Long & Yang, (2010-present) Also, Valderrama (2010-present) LO: Still iterate to all order (at least for l<2). Start at NLO, do perturbation. Reason: van Kolck, Bedaque,… etc. Thus, O(Q0): T(0) + +… ≡ (T = T(0)+T(1)+T(2)+T(3)+…) If V(1) is absent: T(2) = V(2) + 2V(2)GT(0) + T(0)GV(2)GT(0). V(2) V(2) T(0) T(0) V(2) T(0) V(2) T(0) T(3) = V(3) + 2V(3)GT(0) + T(0)GV(3)GT(0).

Results (All RG-invariant)

Tlab=30 MeV 3P0 Tlab=50 MeV Tlab=100 MeV Tlab=40 MeV

Road Map for the future RG-invariant Check/arrange the power counting in detail (Lepage plot). Optimize the NN fit. Further details in Bingwei’s talk (6pm today)

Chiral EFT in discrete space For ab-initio nuclear structure/reaction calculation

Nuclear Structure: Two ways of approach #1. Take the bare interaction (in the continuum), then manipulates it (unitary trans) to speed up the convergence with (truncated) model space. #2. Take only the symmetry and power counting, then establishes interaction directly in the given space (where we want to perform the calculation). (the conventional one) (this work)

From QCD to nuclear structure 4+ nucleons Strong short-range interaction Difficulty: Model space grow combinatorial. Many-body Reasonable truncation of model space + Unitary transformation of NN, NNN forces Establish the (short-range) EFT in truncated model space No-core shell model (NCSM) EFT approach to NCSM ☺

Why want alternatives to unitary transformation (V-lowk, SRG, etc…) ? I. Whenever a model space is truncated, (artificial) higher body forces arise II. If using EFT interaction, power counting may not be preserved O(1) O(Q1) O(Q2) O(Q3) … Well-organized power counting in EFT could be destroyed! Especially when Vsubleading need to be treated perturbatively. Unitary trans. EFT: separation of scale

Directly perform renormalization of EFT in the truncated model space

Good idea. But, not enough bound-state to decide LECs.

Solution: Use trapped space HO basis in free space HO basis in trapped space Energy shift due to different b.c. … … En Busch’s formula: through the E-shift, relates En to phase shift. Analogy: Lattice QCD (Luscher formula)

Generalization of Busch formula Allow fixing (ren. ) L. E. C Generalization of Busch formula Allow fixing (ren.) L.E.C.’s to NN phase shift Uncoupled channels: Exact only for Nmax→∞, l=0 and zero-range interaction case. Otherwise has error ~ Coupled channels:

LO results: 1S0 CS renormalized by E0(∞) Fix ω(=0.5 MeV), increase Nmax Fix Λ, decrease ω Converge to continuum limit !

LO results: 3S1-3D1 CT renormalized by E0(∞) Fix ω(0.5 MeV), increase Nmax Fix Λ, decrease ω

Results up to O(Q2) Preliminary

ω=2 MeV ω=2 MeV ω=2 MeV

Thank you!

Once <V(n)>trap fixed, do perturbative calculation (without modifying existing code) credit: D. Lee & N. Barnea

An overview: From QCD to nuclear structure 1011 eV Spontaneous symmetry breaking 1010 eV Effective field theory for low energy ππ, πN, NN interaction. Some open issues in NN sector 1. Propose a new power counting scheme 109 eV 108 eV 107 eV Nuclear structure 4+ nucleons Strong short-range interaction Difficulty: Model space grow combinatorial. 106 eV Many-body Low energy Reasonable truncation of model space + Unitary transformation of NN, NNN forces Other bypass ? 2. Build NN force in discrete space Ab-initio methods (e.g., no-core shell model)

Λ=1.5 GeV O(Q3) O(1) O(Q2)

Error analysis: 1S0 Relative error scales as O(ω)!

From VNN to nuclear structure: No-core shell model (NCSM) Unlike traditional shell model: NCSM: (all nucleons are active) Need sufficiently large basis to reach convergence! Add Hcm for faster convergence Has core and valence. (P. Navratil, et al 2009) Defines a (H.O.) basis to solve VNN,ij, VNNN,ijk