The Two Nucleon System in Chiral Effective Field Theory: Searching for the Power Counting M. Pav´ on Valderrama Instituto de F´ ısica Corpuscular (IFIC), Valencia Hadron 2011, Munich, June 2011 Perturbative Two Pion Exchange – p. 1
Contents • The NN Potential in ChPT (Weinberg Counting): • Power Counting in the Chiral NN Potentials. • However, breakdown of counting in NN Observables. • Building a Power Counting for the Two-Nucleon System: • Perturbative Treatment of NLO and N 2 LO • Cut-off independence: modifications to W counting. • Results for S- and P-waves. • Conclusions Based on: PRC83, 024003 (2011), arXiv:0912.0699 Perturbative Two Pion Exchange – p. 2
The Nucleon-Nucleon Chiral Potential (I) • The nuclear force is a fundamental problem in nuclear physics • Many phenomenological descriptions available which are, however, not grounded in QCD. • Chiral Perturbation Theory (Weinberg counting): • Problem: NN interaction is non-perturbative • Weinberg’s solution: • apply ChPT to construct the nuclear potential (instead of the scattering amplitude) • insert the potential into the Schrödinger equation, as traditionally done in nuclear physics. Perturbative Two Pion Exchange – p. 3
The Nucleon-Nucleon Chiral Potential (I) • The nuclear force is a fundamental problem in nuclear physics • Many phenomenological descriptions available which are, however, not grounded in QCD. • Chiral Perturbation Theory (Weinberg counting): O ( Q 0 ) + V NN = O ( Q 2 ) + + + + + + . . . Weinberg (90); Ray, Ordoñez, van Kolck (93,94); etc. Perturbative Two Pion Exchange – p. 3
The Nucleon-Nucleon Chiral Potential (II) The two essential ingredients: • Chiral Symmetry provides the connection with QCD. It constraints the nature of pion exchanges (specially TPE). • Power counting allows to express the NN potential as a low energy expansion in terms of a ratio of scales Q/ Λ 0 : q ) + O ( Q 4 q ) = V (0) q ) + V (2) q ) + V (3) V χ ( � χ ( � χ ( � χ ( � ) Λ 4 0 Q ∼ | � q | ∼ p ∼ m π ∼ 100 − 200 MeV (low energy scale) Λ 0 ∼ m ρ ∼ M N ∼ 4 πf π ∼ 0 . 5 − 1GeV (high energy scale) The resulting potential should convergence quickly at low energies / large distances (and diverge at high energies). Power counting is essential for having a systematic scheme! Perturbative Two Pion Exchange – p. 4
The Nucleon-Nucleon Chiral Potential (III) The NN chiral potential in coordinate space: 0 −0.5 −1 1 S 0 V(r) [MeV] −1.5 −2 −2.5 LO −3 NLO NNLO −3.5 1.5 2 2.5 3 3.5 4 4.5 5 r [fm] At long distances power counting implies: Perturbative Two Pion Exchange – p. 5
The Nucleon-Nucleon Chiral Potential (IV) However, at short distances the situation is just the opposite: ... as can be checked in coordinate space: 0 −50 V(r) [MeV] −100 1 S 0 −150 −200 LO NLO NNLO −250 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 r [fm] Perturbative Two Pion Exchange – p. 6
The Nucleon-Nucleon Chiral Potential (IV) However, at short distances the situation is just the opposite: In fact, on dimensional grounds we expect the following behaviour: q | ν q ) ∼ | � f ( | � q | 1 V ( ν ) V ( ν ) χ, pions ( � ) or χ, pions ( � r ) ∼ Λ ν Λ ν 0 r 3+ ν m π 0 This problem is usually dealt with by a renormalization procedure: • including a cut-off r c or Λ ( ≃ π/ 2 r c ) in the computations • the counterterms, which partly absorb the bad behaviour of the potential at scales of the order of the cut-off Perturbative Two Pion Exchange – p. 6
Weinberg Counting: Description • Potential expanded according to counting: V = V (0) + V (2) + V (3) + O ( Q 4 / Λ 4 0 ) • The potential is conveniently regularized and iterated: V R V → Λ V R Λ + V R T = Λ G 0 T • Counterterms are fitted to reproduce scattering observables. • Great phenomenological success at N 3 LO ! ( χ 2 /d.o.f. ≃ 1 ) Entem, Machleidt (03); Epelbaum, Glöckle, Meißner (05) But there are problems, like the cut-off issue, the power counting issue or the sistematicity issue ( Nogga, Timmermans, van Kolck (05); Birse (05); Epelbaum, Meißner (06); Epelbaum, Gegelia (09); Entem, Machleidt (10); etc. ). Perturbative Two Pion Exchange – p. 7
Weinberg Counting: Problems (I) However... Do observables follow a power counting? • The Weinberg prescription prodives a counting for the potential, which is not an observable. • There has not been any systematic effort to determine whether the resulting scattering observables follow the power counting. • Without this ingredient, the Weinberg prescription would merely be a (useful) recipe for constructing nuclear potentials. • Iteration can play very ugly tricks with us. Perturbative Two Pion Exchange – p. 8
Weinberg Counting: Problems (II) The interesting question is whether power counting is preserved in observables: T = T (0) + T (2) + T (3) + O ( Q 3 / Λ 3 0 ) ? So what can fail? The contribution of subleading pieces can eventually grow larger than the leading ones, spoiling the counting. Why? Chiral potentials are increasingly singular! (a) Λ small enough: T (0) > T (2) > T (3) > . . . (b) Λ large enough: T (0) < T (2) < T (3) < . . . (or whatever) In Weinberg Λ ∼ 0 . 5 GeV : is that within (a) or (b)? Not everyone agrees on this view: see Epelbaum, Meißner (06) for an example. Perturbative Two Pion Exchange – p. 9
Weinberg Counting: an Example (I) The previous question can be answered by doing some computations: Weinberg at N 2 LO with a gaussian cut-off Λ = 400 MeV 70 Nijm 2 NNLO 60 50 δ [deg] 40 30 20 10 0 0 50 100 150 200 250 300 k cm [MeV] Which piece of the chiral long range interaction dominates? Perturbative Two Pion Exchange – p. 10
Weinberg Counting: an Example (II) Answer: if the subleading contributions to the scattering amplitude are small, we should be able to approximate them in perturbation theory. The scattering amplitude should behave as: Perturbative Two Pion Exchange – p. 11
Weinberg Counting: an Example (II) Answer: if the subleading contributions to the scattering amplitude are small, we should be able to approximate them in perturbation theory. The previous scheme leads to the following approximations: 70 L(non-pert) SL(pert) 60 50 δ [deg] 40 30 20 10 0 0 50 100 150 200 250 300 k cm [MeV] (a) Power counting is already lost at k ∼ 100 MeV !!!. Perturbative Two Pion Exchange – p. 11
Weinberg Counting: an Example (III) However, the situation is even more paradoxical than we can expect. We can try a different approximation... (different choices are possible depending on the regulator, the cut-off, the value of the chiral couplings, etc.) Perturbative Two Pion Exchange – p. 12
Weinberg Counting: an Example (III) However, the situation is even more paradoxical than we can expect. ... which gives us the following phase shifts 70 L(non-pert) SL(pert) 60 50 δ [deg] 40 30 20 10 0 0 50 100 150 200 250 300 k cm [MeV] (b) The original assumptions made by the power counting are completely broken by the results, which obey a different counting instead. See related comments in Lepage (97). Perturbative Two Pion Exchange – p. 12
Overcoming the Inconsistencies Lesson: don’t iterate unless you are sure what you are doing! Power counting inconsistencies avoided by enforcing the counting, that is, treating the subleading pieces of the potential as perturbations: V (0) + V (0) G 0 T (0) T (0) = V (2) + T (0) G 0 V (2) + V (2) G 0 T (0) T (2) = . . . = . . . and now (i) T (2) ∝ V (2) , (ii) T = T (0) + T (2) + O ( Q 3 / Λ 3 0 ) . Recent examples are given by Shukla, Phillips, Mortenson (07) and the EFT lattice computations by Epelbaum, Krebs, Lee, Meißner. Perturbative Two Pion Exchange – p. 13
Perturbative Weinberg (I) However, there is still a problem with cut-off dependence: 70 60 50 δ [deg] 40 1.0 fm 0.8 fm 30 0.6 fm 0.5 fm 20 0.4 fm 10 0.3 fm Nijm2 0 0 50 100 150 200 250 300 k c.m. [MeV] (a) Perturbative Two Pion Exchange – p. 14
Perturbative Weinberg (II) By analyzing the cut-off dependence of the T-matrix in the singlet channel we find the following T (Λ) = T (0) (Λ) + T (2) (Λ) + T (3) (Λ) + O ( Q 4 / Λ 4 0 ) � �� � � �� � ∼ log Λ ∼ Λ • Problem: the Weinberg counting counterterms χ, contact = C 0 + C 2 ( p 2 + p ′ 2 ) + O ( Q 4 / Λ 4 V (2 , 3) 0 ) are not enough to render the amplitudes cut-off independent. • Solution: promote the C 4 counterterm (which is Q 4 in Weinberg) to order Q 2 to achieve cut-off independence (Birse 05/10). χ, contact = C 0 + C 2 ( p 2 + p ′ 2 ) + C 4 ( p 4 + p ′ 4 ) + O ( Q 4 / Λ 4 V (2 , 3) 0 ) Perturbative Two Pion Exchange – p. 15
Perturbative Weinberg (III) Can be illustrated by the following N 2 LO results in the singlet: 1.0 fm 70 70 0.8 fm 60 60 0.6 fm 0.5 fm 50 50 0.4 fm δ [deg] δ [deg] 0.3 fm 40 40 1.0 fm Nijm2 0.8 fm 30 30 0.6 fm 0.5 fm 20 20 0.4 fm 0.3 fm 10 10 Nijm2 0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 k c.m. [MeV] (a) k c.m. [MeV] (b) (a) with the Weinberg counterterms C 0 and C 2 ( ∆ δ ∼ k 4 /r c ) (b) with the additional counterterm C 4 ( ∆ δ ∼ k 6 r c ) Perturbative Two Pion Exchange – p. 16
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