1. 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

2. 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,…

3. 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)

4. Why is that a problem?

5. (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).

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

7. 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)

8. Origin of the RG problem (in the conventional treatment)

9. 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.

10. 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) V(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) V(3)GT(0) T(0)GV(3)GT(0).

11. Results (All RG-invariant)

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

13. 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)

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

15. 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)

16. 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 ☺

17. 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

18. Directly perform renormalization of EFT in the truncated model space

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

20. 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)

21. 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:

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

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

24. Results up to O(Q2)
Preliminary

25. ω=2 MeV
ω=2 MeV
ω=2 MeV

26. Thank you!

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

28. 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)

29. Λ=1.5 GeV
O(Q3)
O(1)
O(Q2)

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

31. 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