U. van Kolck. Institut de Physique Nucléaire d Orsay. and University of Arizona. Supported in part by CNRS, Université Paris Sud, and US DOE

Transcription

1 THE NON-PC STORY A SERMON OF ABOUT CHIRAL PC EFT U. van Kolck Institut de Physique Nucléaire d Orsay and University of Arizona Supported in part by CNRS, Université Paris Sud, and US DOE 1

2 Why? Chiral EFT potentials based on Weinberg s power counting widely used in nuclear physics because of their supposed link to QCD Problem: Weinberg s power counting inconsistent with renormalization Solution: Certain counterterms appear at lower order than expected; subleading terms should be treated in perturbation theory Kaplan, Savage + Wise 96,, Nogga, Timmermans + Nogga 05, S. The problem doesn t exist: Renormalization not important Epelbaum + Meissner 06,, Epelbaum + Gegelia 09, Anyway, there is a solution for the problem that doesn t exist: Relativity essential in a non-relativistic problem Epelbaum + Gegelia 1 (Oh, yeah, this solution doesn t completely solve the problem that doesn t exist ---counterterms still need to be promoted--- but that is a detail which barely needs acknowledgement) the talk yesterday

3 Outline Effective field theory & model spaces Pre-story: ChiPT The story Conclusion & Outlook 3

4 E EFT Weinberg, Wilson,... M Λ Z ( 4 i d x ) und = Φ exp ( Φ) ϕδ ϕ f ( Φ) ( ) Λ m = ( 4 i d x ) EFT ϕ ϕ exp ( ) = EFT i i d= 0 idn (, ) underlying dynamics renormalization-group invariance Z Λ = 0 c ( m ϕ ) ( M, Λ) O (, ) d n local most general underlying symmetries 4

5 ν ( ) Q Q T T ( Q)~ N( M ) c, i( ), i ; i M F Λ = ν Λ ν ν= ν m m normalization min T 0 Λ = ν =ν ( d, n, ) power counting e.g. # loops L non-analytic, from loops T For Q ~ m, truncate Q M Q Λ ( ν ) = T 1 +, controlled consistently with RG invariance: Λ T ( v ) T Q = 1 ( v ) Λ Λ model independent If so want Λ > M realistic estimate of errors comes from variation Λ [ M, ) 5

6 Cutoffs define model spaces E M Λ m λ To limit the number of one-particle states, introduce IR cutoff in addition to U cutoff momentum T λ Λ ( ) Q Q λ = T ν 1 +,, M Λ Q To minimize model space error (to converge ), want Λ > M λ Q 6

7 Popular examples Lattice Box Lattice Field Theory Harmonic-Oscillator Box No-Core Shell Model energy ml L = Na N ml Λ m λ m N mb L max N 1 mb max b b = mω nuclear matter Müller et al. 99 finite nuclei Stetcu et al. 06 few nucleons Lee et al. 05 few atoms Stetcu et al. 07 atomic matter Bulgac et al. 06 few atoms Kaplan et al. 10 7

8 Extrapolations in a HO basis Coon, Avetian, Kruse, Maris, ary + v.k., 1 scaling λ = Λ for much more see Furnstahl, Hagen + Papenbrock 1 More et al L 8

9 Chiral EFT Q 1 Ge m M QCD d.o.f.s: pions, nucleons, deltas ( m mn ~ m ) ++ + ( + ) + 1 p + = N = ( ) = i = 0 n 0 3 symmetries: Lorentz, P, T, chiral ( ) f 9 Me = M QCD 4 spontaneously broken: non-linear realization Weinberg 68 Callan, Coleman, Wess + Zumino 69 (chiral) covariant derivatives chiral invariants pion µ Dµ 1 + f 4f baryon, isospin T µ µ i T Dµ f + chiral breaking as in quark mass terms non-derivative interactions proportional to masses (( u d) QCD ) m = m + m M 9

10 Pre-story: ChiPT Example: pion sector (similar in one-nucleon sector) Weinberg 79 Gasser + Leutwyler 84 Manohar + Georgi 84 µ 1 f = 0 = f Dµ D m 1 + 4f 4 m + c1f µ + c f µ ν + c3m µ f + µ µ ν µ 4 ( D D ) D D D D D D ( 1 ) c ( 1 ) T = Weinberg 66 current algebra quantum corrections 10

11 + 1 f Λ (,, ) (,, ) i ( l + ) 4 d l lkm lkm ( ) l m ε k m iε 4 4 f 1 ( 4 f ) forbidden by chiral sym { Λ Λ ( k m ) ( k mk m ) 6 Λ k Q # + # + # + # + # + # ln + # ln + m m Λ absorbed in non-analytic 4 1 Q Q ( # k + # m ) f f 4 f f ( ) c c i i Q ( # c1,k + # c3mk + # c4m ) c ( Λ ) f i f # Λ = ln + c (4 f ) m ( Λ) c ( R) i # Λ #lnα = + + c ( 4 f) m ( 4 f) ( αλ) ln (( 4 f ) ) ( 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 M QCD 11

12 Generalizing, EFT p f n + D,, m m N m ψψ c{ n, p, f } { npf,, } M QCD MQ CD f f M QCD = f M QCD calculated from QCD: lattice, fitted to data ( ) = f f n+ p+ d + 0 = 0 = (1) α = ε, 4 isospin conserving (NDA) isospin breaking chiral symmetry chiral index Q Q Λ = ( )~ ( M QCD ) c ν, i( Λ) Fν, i ; ν= νmin i M QCD m m ( ) T T Q N ν = A+ L+ ν = A i i i min ν # nucleons = 0,1 # loops # vertices of type i 1

13 The era of the scriptures E = k m N m N 3 d l = + 3 ( ) l k The story* 4 1 d l 1 1 i + iε + m iε ( ) l k m N l mn l k mn l N infrared enhancement: no ChiPT expansion for T for A Weinberg 90, 91, 9 Ordonez + v.k. 9 v.k. 94 Ordonez, Ray + v.k. 94, 96 Brockmann, Kaiser + Weise 96 Gerstendoerfer, Kaiser + Weise 97 potential = sum of subdiagrams without IR enhancement: amenable to ChiPT expansion, cutoff absorbed in counterterms of NDA size 1 Friar 99 Kaiser 99 0 l Weinberg s recipe ( W PC ): truncate potential, solve dynamical equation exactly [and, as always, check assumptions] * Not a history, not even Whiggish 13

14 Q Q Λ ( Λ)~ N( M ) c ν, i( Λ) fν, i ; ν= ν i M m m min ν = A+ L+ ν = A i i i ν min not an observable: in general depends on cutoff, form of dynamical equation, choice of nucleon fields, etc. 14

15 -body 3-body 4-body LO in German 1 NLO f NNLO 1 f 1 f Q M Q M QCD QCD (parity violating) NNNLO 1 f Q M 3 3 QCD NNNNLO 1 f Q M 4 4 QCD etc. 15

16 Q Q Λ ( Λ)~ N( M ) c ν, i( Λ) fν, i ; ν= ν i M m m min ν = A+ L+ ν = A i i i ν min not an observable: in general depends on cutoff, form of dynamical equation, choice of nucleon fields, etc. Potential to O(Q^3) with and to O(Q^4) without delta isobar derived Fit of NN phase shifts to O(Q^3) with delta encouraging; similar accuracy (or lack thereof) for three cutoffs from 500 to 1000 Me TPE potential to O(Q^3) without delta improves Nijmegen PWA Pions perturbative in F waves and higher 16

17 Weinberg 9 Rho 93 Park, Min + Rho 94 Beane, Lee + v.k. 95 Also, many processes with external probes: o pion elastic scattering o electroweak currents o pion photoproduction o pion production o Compton scattering o 17

18 The Reformation Amplitude in 1S0 solved in semi-analytic form for W LO: T Y = + χ = + I = + T Y χ TY Kaplan, Savage + Weise 96 Cohen + Phillips 97 Kaplan v.k. 97 Kaplan, Savage + Weise 98 Gegelia 98 Bedaque, Hammer + v.k. 98,... (0) χ( p ; k) χ( pk ; ) T ( p, pk ; ) = TY ( p, pk ; ) + 1 Ik ( ) c 4 mn m Λ k I( k) = # Λ + # ln + mn 4 f f m Λ c ( m) W PC: = C + D m + NDA fails for chiral symmetry-breaking operators: W PC not entirely correct 0 LO NNLO

19 Detailed study of renormalization, validity of NDA, perturbativity of sublos, power counting, etc. in simpler pionless EFT for Q< m Some lessons: 1) fine-tuning necessary for large scattering lengths can be incorporated into PC for amplitude ) non-perturbative renormalization intrinsically different from renormalization of corresponding perturbative series 3) one gains no understanding of the renormalization of the A-body system by just monkeying around with higher-order terms in the A-1-body system 4) NDA has very limited usefulness; e.g., three-body force of very high order by NDA, but renormalization requires it at LO 5) subleading interactions must be treated in perturbation theory 6) fully consistent theory works well for very low-energy processes involving (at least) light nuclei and cold atoms, incorporating universal properties such as the Efimov effect, Phillips and Tjon lines, Wigner SU(4), ; yet, mostly ignored by nuclear physics community Moral: faced with W PC vs RG, choose RG 19

20 Proposal for perturbation approach to pion exchange in chiral EFT ( KSW PC ) 1) manifestly consistent PC Some Results ) rescues NDA for chiral symmetry-breaking operators 3) converges only for Q < Me; at that point pion tensor force no longer perturbative 0

21 -body 3-body 4-body LO 4 m N Q NLO 4 m M N NN l = 0 NNLO 4Q m M N NN? NNNLO 4 Q m M N 3 NN l = 1 l = 0 etc. M NN 4 f m N 1

22 NNLO Fleming, Mehen + Stewart 00 LO Nijmegen PWA NNLO + h.o. ct. NLO NNLO NLO LO Nijmegen PWA

23 E k = m N m N 3 d l = + 3 ( ) l k mnq 4 m Q N 4 Weinberg s IR enhancement Q instead of 4 4pi enhancement compared to ChPT ( ) (0) = +? 1 4 f M m N NN M NN 4 f m N f f Resum when Q> M NN (0) T = (0) + T (0) (0) b.s. at MNN f B 10 Me m 4 N But, since Weinberg s PC inconsistent, then what? 3

24 The Counter-Reformation Epelbaum, Gloeckle + Meissner 98 Entem + Machleidt 03 Ekstroem et al., last week Elevate cutoff to physical quantity constrained to M NN <Λ< M QCD Faced with W PC vs RG, choose W s PC Countless improvements under W PC: 1) elimination of redundant operators ) correction of some mistakes 3) smart choice of regulator (cutoff not on transferred momentum, to decouple effects of short-range interactions on various partial waves) 4) careful treatment of relativistic corrections N) fits to NN data at O(Q^4) without delta of similar quality as purely phenomenological pots (But also some steps back, e.g., no deltas until recently, different regulators for different loops) Goes iral Chiral EFT becomes input of choice for a new generation of ab initio methods for light and medium-mass nuclei 4

25 The Reckoning? Conjecture: M Beane, Bedaque, Savage + v.k. 0 Nogga, Timmermans + v.k. 06 Pavon alderrama + Ruiz Arriola 06 Birse 06 Long + v.k. 08 Yang, Elster + Phillips 09 Pavon alderrama 10, 11 so that one can think of T as an expansion around the chiral limit, only necessary resummation being that of the tensor force: o singlet channels ~ KSW (solves the W problem with chiral symmetry breaking) NN m o triplet channels ~ W (solves the KSW problem of convergence) > Long + Yang 11, 1 However, W s PC fails also in triplets! 5

26 W PC at LO Attractive-tensor channels: Nogga, Timmermans + v.k. 05 E (Me) incorrect renormalization That means some counterterms deemed to be sublo because of NDA are actually LO! 6

27 Add needed counterterms at this order, e.g., Nogga, Timmermans + v.k. 05 c1 l= 1, j= 0 = pp 3 ( ) cf. c = t l= 0, j= 1 3 ( ) 7

28 W PC at NNLO Yang, Elster + Phillips 09 incorrect renormalization That means some counterterms deemed to be subnnlo because of NDA are actually NNLO or lower! 8

29 Λ = 5 Ge Λ =1Ge Zeoli, Machleidt + Entem 1 Λ = 0.5 Ge W PC at NNNLO incorrect renormalization That means YOU ARE USING THE WRONG PC 9

30 Root of the problem: pion exchanges (long-ranged, contribute to waves higher than S) are singular (sensitive to short-range physics, require counterterms) This has NOthing to do with relativity (For the opposite opinion, see Epelbaum + Gegelia 1) New, emerging PC: LO: OPE plus needed counterterms (one per wave where OPE is non-perturbative, singular, attractive) sublos: NPE given by ChPT plus counterterms given by NDA with respect to the lowest order they appear at, treated in perturbation theory (contrast with Epelbaum + Gegelia 09, who suggest: if you cannot take a large cutoff when treating certain sublos non-perturbatively, don t take a large cutoff. ) 30

31 -body 3-body LO in German NLO 4 m Q N 4 mn M QCD S = 0 l = 0 S =1 l S = 0 l = 0 NNLO 4Q m M N QCD? NNNLO NNNNLO 4Q mn M 3 QCD 4Q mn M 3 4 QCD S = 0 l = 0 S =1 l S = 0 l = 0 etc. (Details still being worked out, e.g. at ESNT Saclay workshop two weeks ago) 31

32 (0) (0) T = (0) + + = (0) + (0) large enhancement 4 m Q N T (0) (0) ( T + ) = E (0) (0) ψ (0) (0) ψ b.s.s, resonances T (0) (1) T (1) = (1) (0) T T (0) (1) T (1) (0) E = ψ ψ (1) (0) ( 1) (0) smaller 3

33 () T = cf. (0 1) T + = (0 1) + + = (0) T + (0 1) (0 1) T + (1) (1) + + (1) + (1) () + E missed (0) (1) (0) (0) (1) (0) () n n = (0) ( 0) + n E E n ψ ψ ψ ψ ψ T (0) () (0) (1) ψ + sum even smaller + T = T + f Λ M T ( ν ) ( ν ) Λ T T Λ ( ν ) = ( ) 1 ( ν ) uncontrolled model dependent error estimate??? 33

34 e.g. new PC Fits to data Pavon alderrama 10, 11 Long + Yang 11, 1 bands (not error estimates): coordinate-space cutoff variation fm cyan: NNLO in Weinberg s scheme Pavon alderrama 10 34

35 Conclusion & Outlook much has been learned about EFT in a non-perturbative context non-analytic parts of long-range pots derived a chiral EFT NN amplitude consistent with RG being constructed compared to the NN amplitude obtained with W PC: it contains more counterterms (thus parameters) at a given order but sublos require perturbation theory (sorry, but that is what physics asks of you) details still being worked out, but first results suggest possibility of better fits to data than W PC; perhaps a realistic amplitude emerges at NNNLO? few-body forces and currents remain to be studied; effects could be substantial since they are tied to NN amplitude 35