POWER COUNTING WHAT? WHERE?

Transcription

1 POWER COUNTING WHAT? WHERE? U. van Kolck Institut de Physique Nucléaire d Orsay and University of Arizona Supported by CNRS and US DOE 1

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4 Outline Meeting the elephant What? Where? Walking out of McDonald s Conclusion with Y.-H. Song & R. Lazauskas

5 The elephant

6 1996: Chiral-symmetry breaking interaction, absent according to Weinberg s power counting, needed in 1 S 0 channel at LO π π = χ EFT Dmπ NP1 NNP1 N 1 S 4 0 S 0 Fπ Fπ 1 NDA D = fπ M QCD vs nonperturbative 1 D = 4 renormalization f π D.B. Kaplan, M.J. Savage, M.B. Wise, Nucl. Phys. B 478 (1996) 69

7 S.R. Beane, P.F. Bedaque, M.J. Savage, U. van Kolck, Nucl. Phys. A 700 (00) 377 B. Long, C.-J. Yang, Phys. Rev. C 86 (01) 04001

8 1996: Chiral-symmetry breaking interaction, absent according to Weinberg s power counting, needed in 1 S 0 channel at LO π π = χ EFT Dmπ NP1 NNP1 N 1 S 4 0 S 0 Fπ Fπ 1 NDA D = fπ M QCD vs nonperturbative 1 D = 4 renormalization f π D.B. Kaplan, M.J. Savage, M.B. Wise, Nucl. Phys. B 478 (1996) 69

9 1997: Nonperturbative treatment of higher-order interactions beset by RG problems e.g. π = + + EFT CNP NNP N S S Wigner bound T.D. Cohen, D.R. Phillips, Phys. Lett. B 390 (1997) 7 K.A. Scaldeferri, D.R. Phillips, C.W. Kao, T.D. Cohen, Phys. Rev. C 56 (1997) 679

10 005: Derivative contact interactions, down two or more orders according to Weinberg s power counting, needed at LO in (triplet) channels with attractive tensor force where pions are iterated to all orders e.g. χ = + CNP N( N) P N+ EFT P P NDA 1 C = f π M QCD vs nonperturbative renormalization 1 C = 4 f π A. Nogga, R.G.E. Timmermans, U. van Kolck, Phys. Rev. C 7 (005) M. Pavón Valderrama, E. Ruiz Arriola, Phys. Rev. C 74 (006) C.-J. Yang, C. Elster, D.R. Phillips, Phys. Rev. C 77 (008) 01400

11 01: Two-derivative contact interaction, down two orders according to Weinberg s power counting, needed in 1 S 0 channel at NLO (i.e. ONE, ONE, ONE, ONE power down) χ = + + EFT CNP NNP N S S nonperturbative 1 NDA C C = = vs 3 f M renormalization f π M π QCD QCD B. Long, C.-J. Yang, Phys. Rev. C 86 (01) Y.-H. Song, R. Lazauskas, U. van Kolck, Phys. Rev. C 96 (017) 0400

12 Reactions Pionless EFT Power counting has VERY LITTLE to do with Weinberg s e.g. 4π, S S 1+ n n mn M lo Mhi 4π Cn =, S S n mn M lom hi 4π, S S 1+ n mn M hi M m B A lo N A Mhi m π P.F. Bedaque, U. van Kolck, Phys. Lett. B 48 (1998) 1 U. van Kolck, Lect. Notes Phys. 513 (1998) 6 D.B. Kaplan, M.J. Savage, M.B. Wise, Phys. Lett. B 44 (1998) 390 D.B. Kaplan, M.J. Savage, M.B. Wise, Nucl. Phys. B 534 (1998) 39 For more, see talk by König

13 Chiral EFT with perturbative pions Mlo π fπ Mhi MNN fπ f m Same with 4 D.B. Kaplan, M.J. Savage, M.B. Wise, Phys. Lett. B 44 (1998) 390 D.B. Kaplan, M.J. Savage, M.B. Wise, Nucl. Phys. B 534 (1998) 39 m π N π Chiral EFT with partly perturbative pions M, lo MNN m π M M 4 π f, m, hi QCD π N NDA except when running dominated by pions, i.e. low waves where OPE tensor force is attractive A. Nogga, R.G.E. Timmermans, U. van Kolck, Phys. Rev. C 7 (005) For more, see talks by Long, Pavón, Yang (?)

14 Walking out of McDonald s

15 Mid-term quizz 1) We should power counting where? a) The potential b) The scattering amplitude c) Both d) None of the above ) We should count powers of what? a) b) Q f π Q (4 π ) f π c) Both d) It doesn t matter which

16 + 1 f Λ (,, π ) (,, π ) i ( l + ) 4 d l lkm lkm ( π) l m ε k m iε 4 4 π π π f π 1 ( 4π fπ ) forbidden by chiral sym { Λ Λ ( k mπ ) ( k mπk mπ ) 6 Λ k Q # + # + # + # + # + # ln + # ln + m m Λ absorbed in non-analytic 4 1 Q Q ( # k + # mπ ) fπ f π 4π f π π f π ( ) π c c i i Q ( # c1,k + # c3mk π + # c4mπ ) c ( Λ ) f i π fπ # Λ = ln + c (4 π fπ ) mπ ( Λ) c ( R) i # Λ #lnα = + + c ( 4π fπ) mπ ( 4π fπ) ( αλ) ln (( 4π fπ ) ) ( M QCD ) = = ( R ) i four parameters; if omitted: cutoff becomes physical only one parameter = model ( R) i 4 NDA: naïve dimensional analysis cf. error not dominant as long as Λ > Q M QCD 6 4 f π M QCD

17 d d 1 d 4 Q (4 π ) 1 Λ (4 π ) d + d d momenta ~ cutoff = short-range physics 1 4 d Q Q NDA, more generally + d d 1 1 ( Λ) C ( Λ) d C ( Λ) C ( Λ ) Q + d C d C ( Λ) d d d1 = d = d C d (4 π ) ( Λ) d Λ strong naturalness C d ( M ) (4 π ) d M arbitrary diagram fermions more loops, vertices other interactions Moral: NDA comes from the renormalization of perturbative amplitudes number of fields in operator (4 π ) N red ci = c D 4 i dimension of operator c M red i = (( ) # ) red g reduced underlying theory parameter Georgi + Manohar 86 reduced coupling insertions

18 E V V m N 3 d l = V V + 3 ( π ) l k mnq VV 4π Weinberg s IR enhancement mnq Q = k instead of 4π ( 4π ) m N Are nuclear amplitudes perturbative? 4π enhancement compared to ChPT Does it make sense? (0) V = e Q 4π mα mq Q expansion in m α Q Q < mα (0) T = (0) V + T V (0) (0) B b.s. at ( αm) α m m

19 E V V m N 3 d l = V V + 3 ( π ) l k mnq VV 4π Weinberg s IR enhancement mnq Q = k instead of 4π ( 4π ) m N Are nuclear amplitudes perturbative? 4π enhancement compared to ChPT (0) V = +? Resum in S wave when Q M NN > (0) T = 1 4π f M π (0) V + but still keep perturbative expansion in Q M QCD m N NN T V (0) (0) expansion in M NN N 4π f m N π b.s. at f π Q M NN MNN fπ B 10 MeV m 4π CANNOT JUST COUNT POWERS OF Q f π

20 + + + expansion in Q ( ) η f π nonperturbative for where η ( ) cr f π < M QCD cr singlets η ( ) ( M QC π ) s D f M. Pavón Valderrama et al., Phys. Rev. C 95 (017) but regular pot, so can be resummed without RG penalty triplets M.C. Birse, Phys. Rev. C 74 (006) For more, see Long s talk

21 When OPE s tensor force is attractive and resummed, V() r 3 r ψ( r) r cos + δ + MNNr if no counterterm, will depend on cutoff V p, p f ( p Λ) V p, p f ( p Λ) ( ) ( ) n n ( ) f ( x) = exp x, n=, 4, 6 n Y.-H. Song, R. Lazauskas, U. van Kolck, Phys. Rev. C 96 (017) 0400 n n = 4 (similar for others) E lab = 10, 50,100, 00 MeV

22 Following Birse LO (similar for all)

23 E lab = 10, 50,100, 00 MeV

24 Nijmegen PWA Λ = 06., 1, 4,10 GeV Extension of similar results from A. Nogga, R.G.E. Timmermans, U. van Kolck, Phys. Rev. C 7 (005) E. Epelbaum, U.-G. Meißner, Few-Body Syst. 54 (013) 175

25 1 S 0 is special blue magenta black

26 NLO Nijmegen PWA E lab = 10, 50,100, 00 MeV Λ = 06., 1, 4,10 GeV

27 blue magenta black

28 -body 3-body LO NLO 4π m Q N 4π mn M QCD S = 0 l = 0 S =1 l < S = 0 l = 0 N LO 4πQ m M N QCD N 3 LO N 4 LO 4πQ m M N 4πQ m M N 3 QCD 3 4 QCD S = 0 l = 0 S =1 l < S = 0 l = 0 etc. (Details still being worked out!)

29 -body 3-body LO NLO 4π m Q N 4π mn M QCD S = 0 l = 0 S =1 l < S = 0 l = 0 N LO 4πQ m M N QCD N 3 LO N 4 LO etc. 4πQ m M N 4πQ m M N 3 QCD 3 4 QCD S = 0 l = 0 S =1 l < S = 0 l = 0 Friar counting (includes factors of 4π) J. Friar, Few-Body Syst. (1997) 161

30 Deltaless -body 3-body LO NLO 4π m Q N 4π mn M QCD S = 0 l = 0 S =1 l < S = 0 l = 0? N LO 4πQ m M N QCD N 3 LO N 4 LO etc. 4πQ m M N 4πQ m M N 3 QCD 3 4 QCD S = 0 l = 0 S =1 l < S = 0 l = 0 e.g. A. Kievsky et al., Phys. Rev. C 95 (017) 04001

31 LO wavefunctions of deep B bound states large and positive V.I. Kukulin, V.N. Pomerantsev, Ann. Phys. 111 (1978) 330 expt j max = 1,,3, 4

32 blue magenta black finite but small NO RG NEED FOR LO 3BF because triton energy too low, ie, amenable to Pionless EFT? or motivation to promote 3B contact? cf. Kievsky et al. (017)

33 NLO expt green blue magenta black Corrections in the right direction but small

34 blue magenta black finite NO RG NEED FOR NLO 3BF but small somewhat larger than before indication of relatively larger N LO corrections?

35 Conclusion Power counting: beyond Weinberg (?) Pionless EFT Chiral EFT with perturbative pions Chiral EFT with partly perturbative pions No RG need for LO and NLO 3BFs with partly perturbative pions --- but more work needed to understand three- (and more-) nucleon systems