Exploring novel scenarios of NNEFT

1. Exploring novel scenarios of NNEFT Ji-Feng Yang East China Normal University

2. OUTLINES • EFT approach to nuclear forces • Contact potential and LSE • Rigorous solutions • Implications for renormalization • Various power counting scenarios • Summary KITPC program: EFTs in P&NP

3. EFT approach to nuclear forces • Pre-EFT era: Various phenomenological models • EFT era (1990-): (a) model independent (guided by chiral symmetry of QCD) and (b) systematic (organized by EFT power counting) • Status of EFT approach: Successful as a field-theoretical basis for nuclear forces and a bridge between nuclear physics and QCD (A recent review: EHM arXiv: 0811.1338) • An interesting issue: an understanding of power counting and renormalization of EFT in nonperturbative regime. (Recent efforts: NTvK 2005; LvK 2007; YEP 2007, 2009; BKV 2008; EG 2009; etc.) KITPC program: EFTs in P&NP

4. Contact potential and LSE • Weinberg’s proposal for NNEFT in nonperturbative regime (due to IR enhancement from “large” nucleon mass): • 1) Potentials from CHPT • 2) T-matrices from Lippmann-Schwinger Equation (LSE) or Schrödinger Equation • 3) EFT power counting (PC) applied to potentials • In pionless EFT (contact potentials): • 1) The framework is simple • 2) LSE allow for rigorous solutions • 3) Nonperturbative essentials become transparent KITPC program: EFTs in P&NP

5. Setups Contact potential in an uncoupled channel (L): external momenta contact couplings EFT expansion order Introducing In the same fashion, Then, LSE reduces to with assuming all divergences KITPC program: EFTs in P&NP

6. General form of Ĩ(E) reads with prescription-dependent ! NOTE: The above formalism does not depend on the specifics of power counting for couplings. For example, in 1S0 channel, Ĩ(E) looks like KITPC program: EFTs in P&NP

7. Rigorous Solutions • Rigorous solutions (uncoupled channels) : • General form of on-shell T-matrix with L: • For coupled channels with J: “J-”= (J-1,J-1); “J+”= (J+1,J+1); “Jx”= (J-1,J+1) and (J+1,J-1). N..., D..., N... and D…: polynomials again in terms of p squared, [λ] and [J…]. • Unitarity: prescription-independent. For coupled channels: KITPC program: EFTs in P&NP

8. Examples KITPC program: EFTs in P&NP

9. Actually, KITPC program: EFTs in P&NP

10. More involved expressions in terms of couplings and [J…]. Remarks • The T-matrices obtained above are intrinsically nonperturbative due to their closed form. • At a finite Δ, only finite many parameters [J…] (or divergences) are present at all, i.e., Rank(Ĩ(E))‹∞. • 0 is universally present and ‘disentangled’ from all contact couplings in the inverse on-shell T-matrices. • N… and D… or N… and D… are 0-independent and ‘perturbative’ in terms of p squared and couplings. KITPC program: EFTs in P&NP

11. Implications for renormalization • The divergences in Ĩ(E) or [J…] must be so removed that the closed form of the T-matrices be preserved. • The conventional subtraction algorithm could not work for the nonperturbativedivergences in the compact T-matrices. The subtraction must be done otherwise. • The functional dependence of T-matrix upon momenta is physical and renormalization group (RG) invariant. For closed form T-matrices, RG invariance is consequential. Let us elaborate on these points below. KITPC program: EFTs in P&NP

12. Failure of ‘exogenous’ counter-terms Endogenous/exogenous counter-terms: introduced before/after nonperturbative summation is finished. It suffices to examine 1S0 and 1P1 channels at Δ=2. Evidently, no exogenous counter-terms could remove the divergences in 0, J3 and J5 in the two compact T’s. Thus, for compact T-matrices, exogenous counter-terms could not succeed. A mismatch between λand Ĩ(E) Since KITPC program: EFTs in P&NP

13. Arising of endogenous counter-terms Then the issue boils down to renormalization of Ĩ(E). Consequently: The divergences in Ĩ(E) could not be completely absorbed by couplings in a sensible and consistent manner.Thus, beyond the leading order, we must seek for other sources for the endogenous counter-termsbeyond couplings. In underlying theory viewpoint, EFT’s are built from low-energy projections. Then, divergences arise as projection and loop integration do not commute: C.T.T. : short-hand notation for commutator KITPC program: EFTs in P&NP

14. LE projection from UT onto EFT Rearranging the commutator: well-defined loop integral ill-defined or divergent loop integral Therefore, the underlying-theory perspective shows that counter-terms naturally arise at the level of loop integrals, hence, ‘endogenous’. So, subtractions should be performed at the level of loop integrals to render [J…] finite, which could not be fully accomplished with counter-terms from couplings within EFT framework due to the reasons given above. The closed form T-matrices will develop nontrivial dependence upon renormalization prescriptions. To remove the nontrivial prescription dependence, it is crucial to impose physical boundary conditions. KITPC program: EFTs in P&NP

15. Consequences of RG invariance From now on, [J...] denote renormalized parameters, which should depend upon scales in the range: (0, Λ], with Λ being the EFT upper scale. In principle, J… may depend on Λ and other physical scales besides a running scale μ, i.e.: Now, for on-shell T-matrices, RG invariance reads, For the closed form T-matrices, R…: appropriate ratios in terms of [N…,D…] or [N…,D…]. So, RG inv beyond LO: [I0, R…([N…,D…] or [N…,D…])] 1) J0 (=Re(0)) depends on physical scales only, or it is a physical quantity: KITPC program: EFTs in P&NP

16. The concrete form or value of ρ0 could in principle be determined from physical boundary conditions. 2) A possible form of the ratios R…([N…,D…]) in uncoupled channels: 3) An example of nonperturbative running couplings: KITPC program: EFTs in P&NP

17. 4) More physical scales or constraints on couplings and parameters [J...] from RG invariance of [R…]: At leading order, Ĩ(E)=0, T≠0 only for L=0 : but J 0is not! At this order, we can put then KSW scaling could be reproduced: KITPC program: EFTs in P&NP

18. Various power counting scenarios Due to the nontrivial prescription dependence, the EFT power counting for coupling should be supplemented with that for the prescription paramters [J…]. Below, we explore three different scenarios for comparisons. • PC A: • PC B: • PC C: (Below we setμ=Λε) KITPC program: EFTs in P&NP

19. Qualitative behaviors of T 1S0 (Δ=4) • PC A with a fine-tuning of C0 (does not matter for PC A) • PC B • PC C Motivations: preserving the conventional EFT power counting while yielding unnatural scattering length. As J0 is RG invariant, we simply set [J…] in the way shown. KITPC program: EFTs in P&NP

20. with a finer-tuning of C0 KITPC program: EFTs in P&NP

21. Results for coupled channel 3S1-3D1 (Δ=4) KITPC program: EFTs in P&NP

22. Through qualitative analysis, a nonperturbative realization of unnatural scattering lengths, etc., is shown to be possible within EFT approach. In contrast to PC A and B, the simple scenario of PC C yields unnatural scattering lengths only, without ruining the naturalness of the rest ERE parameters. All these result from the nontrivial ‘entanglement’ of the EFT power counting and the nonperturbative prescription structures. KITPC program: EFTs in P&NP

23. Summary • Within a finite order of EFT expansion, only finitemany nonperturbative divergences appear • EFT expansion of potential is incompatible with the nonperturbative structures of divergences in LSE • Subtractions should be performed at the level of integrals, and the resulting nontrivial prescription dependence needs to be removed through physical boundary conditions. • RG invariance is consequential, which constrains some of the prescription-dependent parameters to be physical or RG invariant • EFT power counting is ‘entangled’ with prescriptions • Nonperturbative description of origin of unnatural scattering length is possible within EFT approach KITPC program: EFTs in P&NP

24. Thank you! KITPC program: EFTs in P&NP