POWER COUNTING WHAT? WHERE? - PDF Free Download

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

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

The elephant

1996: Chiral-symmetry breaking interaction, absent according to Weinberg s power counting, needed in 1 S 0 channel at LO π π = + + + 4 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

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

1997: Nonperturbative treatment of higher-order interactions beset by RG problems e.g. π = + + EFT CNP NNP N S S 1 1 0 0 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

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 3 3 0 0 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) 054006 M. Pavón Valderrama, E. Ruiz Arriola, Phys. Rev. C 74 (006) 064004 C.-J. Yang, C. Elster, D.R. Phillips, Phys. Rev. C 77 (008) 01400

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 1 1 0 0 1 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) 04001 Y.-H. Song, R. Lazauskas, U. van Kolck, Phys. Rev. C 96 (017) 0400

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

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) 054006 For more, see talks by Long, Pavón, Yang (?)

Walking out of McDonald s

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

+ 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 4 4 4 { 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 1 4 4 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

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

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

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

+ + + 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) 054001 but regular pot, so can be resummed without RG penalty triplets M.C. Birse, Phys. Rev. C 74 (006) 014003 For more, see Long s talk

When OPE s tensor force is attractive and resummed, V() r 3 r 1 1 4 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

Following Birse LO (similar for all)

E lab = 10, 50,100, 00 MeV

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) 054006 E. Epelbaum, U.-G. Meißner, Few-Body Syst. 54 (013) 175

1 S 0 is special blue magenta black

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

blue magenta black

-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!)

-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

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

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

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)

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

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

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