Toward a new effective interaction in energy density functional theory

Transcription

1 Toward a new effective interaction in energy density functional theory Chieh-Jen (Jerry) Yang With M. Grasso, D. Lacroix, U. van Kolck INT workshop 26/4/2016

2 Outline Beyond mean field approach. (C.J. Yang, M. Grasso, X. Roca-Maza, G. Colo, and K. Moghrabi, arxiv: ) Renormalizability. (C.J. Yang, M. Grasso, K. Moghrabi, U van Kolck, coming soon!) Low density limit. (C.J. Yang, M. Grasso, D Lacroix, arxiv: ) Conclusion and future work.

3 Motivation Skyrme-type interaction works o.k. (able to do the fitting in EDF framework) No way to get with ab-initio! UNEDF collaboration Need to think about other expansion (than on NN d.o.f.).

4 Present status of EDF Energy density functional (EDF) framework gives reasonable results at mean field, when sufficient amount of parameters (~10) are included. Include more parameters won t necessarily help. Limited predictive power. Maybe the correct theory has a structure where different terms appears at different order. Need to go beyond mean field to perform the test.

5 Interaction & mean field EoS Interaction: Skyrme without spin-orbit v= t0(1 + xp 0 σ) + t1(1 + xp 1 σ)( k' + k ) + t2(1 + xp 2 σ) k' k 2 S wave S wave p wave O(0) 2 2 O(q ) O(q ) 1 α 1 + t3(1 + xp 3 σ) ρ. P σ = (1 + σ1 σ2) 6 2 s wave, higher body No pion! Like pionless EFT, except for the density-dependent term. k -k =, k -k ' = k k q kf1 kf2 3 3 k1 0 0 E 1 EoS : d d k2v A ρ

6 2 nd order correction (symmetric & neutron matter) (0) (2) E E E = A A A E mean field nd 2 order (2) * sym 3m A G = 1, = (2T + 1)(2S + 1) d k1 d k2 d q[ vgv] 64 πk (2 π) q F 1 + q (k - k ) 1,2 ST, Contour of integral (C ): k [0, k ] F k +q > k, k -q > k 1 F I F C I 1 st order 2 nd order

7 Previous attempts 1. t 0 -t 3 model, done by: K. Moghrabi, M. Grasso, G. Colo, and N.V. Giai, Phys. Rev. Lett. 105, (2010). 2. Full Skyrme (no spin-orbit): K. Moghrabi, M. Grasso, G. Colo, X. Roco-Maza, Phys. Rev. C 85, (2012). K. Moghrabi, M. Grasso, Phys. Rev. C 86, (2012). Some mistakes 3. Full Skyrme (no density-dep.): N.Kaiser, J. Phys. G 42,095111(2015)

8 Details: partial-waves mixing In general, need to sum over contribution from nn, pp and np. If k k, G no longer symmetric under the contour C! F F I Thus, d qd k1d k2 [ vlgvl' ] 0. => In general any partial - waves can mix! C I p = medium Symmetric or pure n: k F1 =k F2 n Reduce to p (no even-old l mixing) Through 2 nd order effect, medium (G) act as 3 rd particle, but does not destroy the original parity of the interaction. n

9 Details: partial-waves mixing In asymmetric matter, need to sum over contribution from nn, pp and np. For np part, k k, G no longer symmetric under the contour C! F F I Thus, d qd k1d k2 [ vlgvl' ] 0. => any partial - waves can mix! C I p Asymmetric case: k F1 k F2 n, diff. fermi surface. medium Reduce to p n medium acts as 3 rd body (by G), and mixes even-odd partial-wave.

10 Results for nuclear matter In agreement with N.Kaiser, J. Phys. G 42,095111(2015) Diverge as Λ 5 Diverge as Λ 5

11 Renormalization I Pick a Λ, and simply re-adjust the 9 (skyrme) parameters.

12 Symmetric matter before after

13 Pure neutron matter Asymmetric matter (δ=0.5)

14 Keep only the finite part (Dimensional regularization) Without density-dep.(t 3 ) term, Kaiser 2015 With density-dep., without rearrangement term

15 Before renormalization

16 Pure neutron matter (DR)

17 PART II: RENORMALIZABILITY

18 When Λ, how the 2 nd order terms behaves? Converge terms Diverge, k F -dep appears in MF Diverge, k F -dep not in MF

19 Idea: Absorb the Λ-divergence in 2 nd order into mean field terms with the same k F -dependence. converge Diverge, k F -dep appears in MF Diverge, k F -dep not in MF eliminate by setting α=1/3 and t 1 =t 2 =0, or seting t 1 =t 2 =t 3 =0.

20 Idea: Absorb the Λ-divergence in 2 nd order into mean field terms with the same k F -dependence. converge Diverge, k F -dep appears in MF Diverge, k F -dep not in MF Treatment 1: Absorb divergence into redefinition of parameters. Treatment 2: Add counter terms correspond to each divergence. eliminate by setting α=1/3 and t 1 =t 2 =0, or seting t 1 =t 2 =t 3 =0.

21 Results Treatment 1 doesn t work (cannot obtain reasonable fit). Treatment 2 works.

22 Lesson The leading order quite possible just contains only t 0 -t 3 terms. However, the regulator dependence tells us the power counting cannot be established in this way.

23 Part III: Matching the low density limit

24 Neutron matter at very low-ρ Lee & Yang formula (1957) describes the dilute system. 2 2 ENM k N 2 3 = + ( kna) + (11 2ln 2)( kna) + O( kn) A 2m 5 3π 35 higher order KE.. fixed to t nd 0 term automatically recover in 2 of t0 The 2nd order EoS automatically recover the (11-2ln2)(k N a) 3 term. If take physical value of a=-18.9 fm, then impossible to fit pure neutron matter EoS outside region k N a<<1 (adding t1, t2, t3 terms doesn t help). (k N a) needs to be re-summed. (Steele (2000), Schafer (2005), Kaiser (2011))

25

26 This work: Resumed-inspired functional V B β Bβ 2/3 = + D 1/3 2/3 β ρ + Fβ ρ 1 R C β ρ + β ρ + velocity dep term* 3 body, R β higher order in L&Y to be resumed* are fixed to reproduce first two term in Lee & Yang. 2 2 v 1 6 6π => Bβ = 2 π aβ, Rβ = (11 2 ln 2) aβ. m v 35π v (degeneracy: v = 2(4) for β = 0 ( 1 )) a 0 1 avg. of a, a, a in pure n = 18.9fm, a = 20 fm. 1 nn pp np S0 sym E B ρ = KE + + D ρ + F A R C 1/3 β 5/3 α + 1 β 1/3 2/3 β β 1 βρ + βρ ρ α

27 Results Able to describe pure neutron matter from very-dilute-limit to twice saturation density.

28 Very dilute limit FP: B. Friedman and V. Pandharipande, Nucl. Phys. A361,502 (1981). Akmal: A. Akmal, V. R. Pandharipande, and D. G. Ravenhall, Phys. Rev. C 58, 1804 (1998). HS: K. Hebeler and A. Schwenk Phys. Rev. C 82, (2010).

29 Up to ρ=0.3 fm -3 Able to describe both sym and pure neutron matter EoS up to 2ρ 0 very well with only 4 free parameters each.

30 Asymmetric case Before: Lots of models fail Parabolic approximation E E A A ( δ = ( ρ ρ ) /( ρ + ρ )) δ sym 2 ( ρ) = ( ρ) + S( ρδ ), L = 3 ρ ( ds / d ρ ) ρ = 0 N p N p ρ 0 X. Roca-Maza, et al., (2015).

31 Asymmetric case Our result (prediction) Satisfies the experimental constraint.

32 Summary and conclusion 2 nd order contribution is considered, and with the rearrangement term included, results can be refitted to standard data very well in both cutoffand dimensional-regularization. Renormalizability suggests that the leading order(mf) likely to be a t 0 -t 3 model (to avoid a rapid grows of counter terms). Resuming the t 0 part of interaction is necessary to describe low ρ limit correctly.

33 Future prospects Try to bridge EFT ideas/techniques to mean field (and beyond) within EDF framework. Mean field with potential models (effective interaction). (e.g., Skyrme-type) 2nd order corrections Higher order corrections Add new effective interactions? What is the proper form of it? Is the improvement systematic? Goal: Systematic treatment of the interactions. Renormalization-group analysis + power counting check

34 Thank you!

35 Simplification for symmetric and pure neutron matter kf k 1 F2 3 3 Mean field: E d k 1 d k2v( k', k) k k q * -m sym, neut 2 2. kf k 1 F2 Λ (2) 3 3 k 1 k nd 2 order correction: E d d vgv. k -k k -k Define =, ' = G k' -k