Effective operators from wave function factorization

Transcription

1 Effective operators from wave function factorization Scott Bogner NSCL/FRIB Laboratory & Department of Physics and Astronomy Michigan State University

2 Some motivating questions Can we understand the general form of effective operators independent of detailed implementation (SRG, OLS, Vlowk, UCOM, )? Does it buy us anything? What state/system independent aspects of NME in A-body systems that can be informed/extracted by few-body calculations? Is there a way to identify/understand correlations between different observables?

3 Some motivating questions Can we understand the general form of effective operators independent of detailed implementation (SRG, OLS, Vlowk, UCOM, )? Does it buy us anything? What state/system independent aspects of NME in A-body systems that can be informed/extracted by few-body calculations? Is there a way to identify/understand correlations between different observables? Disclaimer: Anderson, SKB et al., PRC 82 (21) SKB and Roscher, PRC 86 (212) But see recent generalizations of Barnea, Bazak, Weiss, et al. PRL 114 (215) PRC 92 (215) arxiv: arxiv:

4 Progress in Ab Initio Calculations quasi-exact methods (QMC, NCSM) limited to p-shell

5 Progress in Ab Initio Calculations Explosion of methods with polynomial scaling (CC, IMSRG, SCGF, MBPT) Enabled by soft NN and NNN interactions (chiral EFT, RG transformations)

6 Renormalization Group Methods k k k k Λ 2 Λ 1 Λ λ λ 1 λ 2 Bogner, Furnstahl, Schwenk, Prog. Part. Nucl. Phys. 65 (21) Folklore low Λ simple Ψ(Λ) ==> complicated O(Λ)? What about large q >> Λ operators? How do interpretations change with Λ?

7 Basic Problem Goal: Extract nuclear properties from experiments and predict them from theory d d / h f O(q) b i i 2 d e.g., nucleon knockout reaction s h f {z} structure reaction z } { bo(q) i i {z} structure Factorization to isolate components and extract process-independent properties hard scale factorization structure reaction

8 Analogy with DIS in QCD High-E QCD Low-E Nuclear hard scale factorization Observable: cross section Structure model: spectroscopic factor Reaction model: single-particle cross section long-distance parton density short-distance Wilson coefficient Open Questions Separation not unique, depends on the scale μf Form factor F2 independent of μf but pieces not fa(x, μf) runs with μf 2 = Q 2, but is process independent When does factorization hold? What is the scale/scheme dependence of extracted props? Can we extract at one scale and evolve to another? Scale/scheme dependence of interpretations? Structure of evolved operators?

9 Ground rules Want to understand form of effective operators without getting bogged down in a particular scheme (OLS, SRG, etc.) Λ I only assume for low-energy states q 1) P n i Z() n i Λ 1I) Q n i p Met by all softening transformations I know of

10 Wave function factorization Consider low-k components of low-e wf s for A=2. Λ q Λ p (k) [fm 3/2 ] RG doesn t change long distance/ir structure =1 3 S 1 =2 3 S 1 =1 3 D 1 =2 3 D k [fm 1 ] (p) Z (p)

11 Wave function factorization Consider high-k components of low-e wf s for A=2. Λ Scale separation (E α << Λ 2 << q 2 ) q (q) (q; ) Z d 3 pz (p) + (q; ) Z d 3 p p 2 Z (p) Λ p Anderson et al., PRC 82 (21) SKB and Roscher, PRC 86 (212)

12 Wave function factorization Consider high-k components of low-e wf s for A=2. Λ Scale separation (E α << Λ 2 << q 2 ) q (q) (q; ) Z d 3 pz (p) + (q; ) Z d 3 p p 2 Z (p) Λ Operator Product Expansion of wave function a-la Lepage p (q; ) = (q; ) = Z Z dq 1 hq QH q iv (q, ) Q dq 1 hq QH q 2 V (q, p) p= State-independent Wilson Coefficients Anderson et al., PRC 82 (21) SKB and Roscher, PRC 86 (212)

13 Wave function factorization LO: (q) (q; ) Z d 3 pz (p) state-independent ratio for well-separated scales (q) (r = ) (q; ) E. 2 q &

14 Wave function factorization LO: (q) (q; ) Z d 3 pz (p) state-independent ratio for well-separated scales Ψ n (q)/ψ λ n (r=) Argonne v 18 λ = 2.1 fm -1 k = ik D k =.5 k =.2 k =.43 k =.72 k = 1.5 k = 1.37 k = 1.64 k = 1.87 k = 2.2 k = 2.8 k = 2.73 k = 4.43 k = 7.2 k = 1.2 (q) (r = ) (q; ) E. 2 q & 1e q [fm -1 ]

15 Effective operators from w.f. factorization h b O i = Z dp Z dp (p)o(p, p ) (p )+ Z dp Z dq (p)o(p, q) (q) + Z dq Z dp (q)o(q, p) (p) + Z dq Z dq (q)o(q, q ) (q )

16 Effective operators from w.f. factorization h b O i = Z dp Z dp (p)o(p, p ) (p )+ Z dp Z dq (p)o(p, q) (q) + Z dq Z dp (q)o(q, p) (p) + Z dq Z dq (q)o(q, q ) (q ) Now use: (q) (q; ) Z d 3 pz (p) + OPE for w.f. s (p) Z (p) IR structure unaltered O(q, p) O(q, ) + Scale separation

17 Effective operators from w.f. factorization h b O i Z 2 h b O i + g () () h (3) (r) i + state-independent high-q physics depends on operator state dependent soft m.e. (low-k) same for all high-q operators E.g., g () () 2Z 2 Z + Z 2 d qo(,q) (q; ) Z d q Z d q (q; )O(q, q ) (q ; ) Generically: b O = Z 2 b O + g () () (r) +g (2) () r 2 (r) +

18 Scaling of high momentum operators How does an operator that probes high-momentum w.f. components look in a low-momentum effective theory? h b O i Z 2 h b O i + g () () h (3) (r) i + = since P O P = E.g., momentum distribution for q >> Λ h a qa q i 2 (q; ) Z 2 h (r) i 2 low-e states have the same large-q tails Generalize to arbitrary A-body states?

19 Scaling of high momentum tails SKB and Roscher, PRC 86 (212) Creation/annihilation operators under RG evolution: a () q = a q + X k 1,k 2 C q (k 1, k 2 )a k 1 a k 2 a k1 +k 2 q + a q + a () q fixed from RGE in A=2 system

20 Scaling of high momentum tails SKB and Roscher, PRC 86 (212) Creation/annihilation operators under RG evolution: a () q = a q + X k 1,k 2 C q (k 1, k 2 )a k 1 a k 2 a k1 +k 2 q + a q + a () q Scale separation (Λ << q < Λ): h,a a qa q,a i = h,a a qa q + a qa q + a q a q + a q a q,a i h,a a q a q,a i

21 Scaling of high momentum tails SKB and Roscher, PRC 86 (212) Creation/annihilation operators under RG evolution: a () q = a q + X k 1,k 2 C q (k 1, k 2 )a k 1 a k 2 a k1 +k 2 q + a q + a () q Scale separation (Λ << q < Λ): h,a a qa q,a i = h,a a qa q + a qa q + a q a q + a q a q,a i h,a a q a q,a i 2 (q; ) X k,k,k Z 2 h,a a K 2 +ka K 2 k a K 2 k a K 2 +k,a i - hard (high q) physics - Universal (state-indep) - fixed from A=2 X - soft (low-k) m.e. - same for all high-q probes - A-dependent scale factor

22 Scaling of high momentum tails N(p) / A A=2, 2 body only A=3, 2 body only A=4, 2 body only A=2, PHQ 2 body only, λ=2 A=3, PHQ 2 body only, λ=2 A=4, PHQ 2 body only, λ= p natural explanation why high-q tails scale C(A, 2) n A(q) n D (q) P k,k,k P k,k,k h,a a K h,d a K 2 +ka K 2 +ka K a 2 k K 2 k a K 2 +k,a i a 2 k K 2 k a K 2 +k,d i

23 Scaling of high momentum tails E.g., static structure functions b S(q) =b (q)b (q) h,a b S(q),A i n 2 (q; )+ X P o (P + q; ) (P; ) X K,k,k Z 2 h,a a K 2 +ka K 2 k a K 2 k a K 2 +k,a i Universal (state-indep) q-dependence => connects few-body and A-body State dependence encoded in low-k m.e. => linear correlations between observables with same leading OPE? (Javier s talk)

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26 Scaling of high momentum tails Factorization of generic high-q operators (schematic) h n Ô q n i = h n Ô q ni Ô q = a qa q, X p,p a p+qa p a p a p +q... Expand evolved operator as polynomial in creation/annihilation operators at Λ Ô q = X g q () Â c-number running couplings string of creation/annihilation operators = p 1...p

27 Scaling of high momentum tails Factorization of generic high-q operators (schematic) h n Ô q n i = h n Ô q ni q = X g q () h n  ni 1) Decoupling => only modes p < Λ in α contribute 2) Taylor expand c-# coefficients about p = => q-dependence factorizes => state-dependence from soft matrix elements A α Scaling if leading term dominates

28 Conclusions Simple decoupling + scale separation arguments generically give the form of effective operators softened by OLS, SRG, Vlowk, Can we use scaling of A-body tails w.r.t. few-body systems to constrain the form of short-distance contributions to NME? Can we use factorization/ope-like arguments to identify quantities that correlate w/vbb NME? How do interpretations change as Λ varied by RG transformations (See Sushong More s talk)