Chiral dynamics and NN potential

Transcription

1 Chiral dynamics and NN potential Department of Physics University of Bologna Bologna (Italy) Otranto Summer School

2 1935 Yukawa: Meson Theory The Pion Theories 1950 s One-Pion Exchange: o.k. Multi-Pion Exchange: disaster Many pions multi-pion resonances: 1960 s σ, ρ, ω,... The One-Boson-Exchange Model: success Refined meson models, including 1970 s sophisticated 2π exchange contributions (Stony Brook, Paris, Bonn) Nuclear physicists discover 1980 s QCD (Quark Cluster Models) Nuclear physicists discover EFT 1990 s Weinberg, van Kolck and beyond Back to Pion Theory! but constrained by Chiral Symmetry: success! [From R. Machleidt arxiv: v1 nucl-th]

3 Definition of Effective Field Theories (EFT) EFTs are low-energy approximations to (more) fundamental theories. Instead of solving the underlying theory, the low-energy physics is described with a set of variables (effective degrees of freedom) that is suited for the particular energy region you are interested in.

4 Definition of Effective Field Theories (EFT) EFTs are low-energy approximations to (more) fundamental theories. Instead of solving the underlying theory, the low-energy physics is described with a set of variables (effective degrees of freedom) that is suited for the particular energy region you are interested in.

5 Definition of Effective Field Theories (EFT) EFTs are low-energy approximations to (more) fundamental theories. Instead of solving the underlying theory, the low-energy physics is described with a set of variables (effective degrees of freedom) that is suited for the particular energy region you are interested in. [Adapted from R. J. Furnstahl]

6 Definition of Effective Field Theories (EFT) EFTs are low-energy approximations to (more) fundamental theories. Instead of solving the underlying theory, the low-energy physics is described with a set of variables (effective degrees of freedom) that is suited for the particular energy region you are interested in. [Adapted from R. J. Furnstahl]

7 Definition of Effective Field Theories (EFT) There exists a regime where both fundamental and effective theories yeld the same results. How to make sure EFT is not just a phenomenology? EFT must observe all relevant symmetries of the underlying theory. EFTs are based on the the most general Lagrangian which includes all terms that are compatible with the symmetries of the underlying theory. Infinite number of terms and each term is accompanied by a low-energy coupling constant (LEC).

8 Definition of Effective Field Theories (EFT) There exists a regime where both fundamental and effective theories yeld the same results. How to make sure EFT is not just a phenomenology? EFT must observe all relevant symmetries of the underlying theory. EFTs are based on the the most general Lagrangian which includes all terms that are compatible with the symmetries of the underlying theory. Infinite number of terms and each term is accompanied by a low-energy coupling constant (LEC).

9 Definition of Effective Field Theories (EFT) There exists a regime where both fundamental and effective theories yeld the same results. How to make sure EFT is not just a phenomenology? EFT must observe all relevant symmetries of the underlying theory. EFTs are based on the the most general Lagrangian which includes all terms that are compatible with the symmetries of the underlying theory. Infinite number of terms and each term is accompanied by a low-energy coupling constant (LEC).

10 Definition of Effective Field Theories (EFT) There exists a regime where both fundamental and effective theories yeld the same results. How to make sure EFT is not just a phenomenology? EFT must observe all relevant symmetries of the underlying theory. EFTs are based on the the most general Lagrangian which includes all terms that are compatible with the symmetries of the underlying theory. Infinite number of terms and each term is accompanied by a low-energy coupling constant (LEC).

11 Definition of Effective Field Theories (EFT) With an infinite number of terms I need a computational scheme: Weinberg s power counting. Physical quantities are calculated in terms of an expansion in p/λ where p stands for momenta or masses that are smaller than a certain momentum scale Λ. In practical calculations only a finite number of terms in the expansion in p/λ has to be considered Predictions. Effective field theories are non-renormalizable in the traditional sense. However, as long as one considers all terms that are allowed by the symmetries, divergencies that occur in calculations up to any given order of p/λ can be renormalized by redefining fields and parameters (LEC) of the EFT Lagrangian.

12 Definition of Effective Field Theories (EFT) With an infinite number of terms I need a computational scheme: Weinberg s power counting. Physical quantities are calculated in terms of an expansion in p/λ where p stands for momenta or masses that are smaller than a certain momentum scale Λ. In practical calculations only a finite number of terms in the expansion in p/λ has to be considered Predictions. Effective field theories are non-renormalizable in the traditional sense. However, as long as one considers all terms that are allowed by the symmetries, divergencies that occur in calculations up to any given order of p/λ can be renormalized by redefining fields and parameters (LEC) of the EFT Lagrangian.

13 Definition of Effective Field Theories (EFT) With an infinite number of terms I need a computational scheme: Weinberg s power counting. Physical quantities are calculated in terms of an expansion in p/λ where p stands for momenta or masses that are smaller than a certain momentum scale Λ. In practical calculations only a finite number of terms in the expansion in p/λ has to be considered Predictions. Effective field theories are non-renormalizable in the traditional sense. However, as long as one considers all terms that are allowed by the symmetries, divergencies that occur in calculations up to any given order of p/λ can be renormalized by redefining fields and parameters (LEC) of the EFT Lagrangian.

14 Definition of Effective Field Theories (EFT) With an infinite number of terms I need a computational scheme: Weinberg s power counting. Physical quantities are calculated in terms of an expansion in p/λ where p stands for momenta or masses that are smaller than a certain momentum scale Λ. In practical calculations only a finite number of terms in the expansion in p/λ has to be considered Predictions. Effective field theories are non-renormalizable in the traditional sense. However, as long as one considers all terms that are allowed by the symmetries, divergencies that occur in calculations up to any given order of p/λ can be renormalized by redefining fields and parameters (LEC) of the EFT Lagrangian.

15 EFT: the most important constrains Symmetry exploited in EFT all the symmetries of QCD (not just the Lorentz one) are accounted for Renormalizability relaxed completely in EFT infinite # of parameters not an issue, if only finite # relevant in the range of validity

16 EFT: non-renormalizable non-feasible Effective Field Theory needs some organizing principle L eff = n L (n) for any n the higher is the n non-renormalization L (n) should contain finite # of terms the less important should L (n) be may require higher and higher n...but they will be less and less important

17 Weinberg theorem If one writes down the most general possible Lagrangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition and the assumed symmetry principles. [S. Weinberg, Physica A 96 (1979) 327]

18 Weinberg theorem If one writes down the most general possible Lagrangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition and the assumed symmetry principles. [S. Weinberg, Physica A 96 (1979) 327] causality [Zuber and Itzykson, Quantum Field Theory, sec ]

19 Weinberg theorem If one writes down the most general possible Lagrangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition and the assumed symmetry principles. [S. Weinberg, Physica A 96 (1979) 327] the sum over the probabilities of the final states must yeld: f S i 2 = 1 f

20 Weinberg theorem If one writes down the most general possible Lagrangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition and the assumed symmetry principles. [S. Weinberg, Physica A 96 (1979) 327] distant experiments must yeld uncorrelated results: S γ+δ α+β S δ β S γ α

21 Weinberg theorem If one writes down the most general possible Lagrangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition and the assumed symmetry principles. [S. Weinberg, Physica A 96 (1979) 327] Everything: Poincaré, discrete (CPT) and internal (isospin and chiral) symmetries. The possibility of a spontaneous symmetry breakdown must be included.

22 EFT step by step for nuclear physicists Identify the degrees of freedom relevant at the resolution scale of (low-energy) nuclear physics: nucleons and pions.

23 EFT step by step for nuclear physicists Identify the degrees of freedom relevant at the resolution scale of (low-energy) nuclear physics: nucleons and pions.

24 EFT step by step for nuclear physicists Identify the degrees of freedom relevant at the resolution scale of (low-energy) nuclear physics: nucleons and pions. Identify the relevant symmetries of low-energy QCD and investigate if and how they are broken (chiral symmetry). Construct the most general Lagrangian consistent with those symmetries and the symmetry breaking (L χ ). Design an organizational scheme that can distinguish between more and less important contributions: a low-momentum expansion (Weinberg s power counting). Guided by the expansion, calculate Feynman diagrams to the the desired accuracy for the problem under consideration. [From R. Machleidt arxiv: v1 nucl-th]

25 EFT step by step for nuclear physicists Identify the degrees of freedom relevant at the resolution scale of (low-energy) nuclear physics: nucleons and pions. Identify the relevant symmetries of low-energy QCD and investigate if and how they are broken (chiral symmetry). Construct the most general Lagrangian consistent with those symmetries and the symmetry breaking (L χ ). Design an organizational scheme that can distinguish between more and less important contributions: a low-momentum expansion (Weinberg s power counting). Guided by the expansion, calculate Feynman diagrams to the the desired accuracy for the problem under consideration. [From R. Machleidt arxiv: v1 nucl-th]

26 EFT step by step for nuclear physicists Identify the degrees of freedom relevant at the resolution scale of (low-energy) nuclear physics: nucleons and pions. Identify the relevant symmetries of low-energy QCD and investigate if and how they are broken (chiral symmetry). Construct the most general Lagrangian consistent with those symmetries and the symmetry breaking (L χ ). Design an organizational scheme that can distinguish between more and less important contributions: a low-momentum expansion (Weinberg s power counting). Guided by the expansion, calculate Feynman diagrams to the the desired accuracy for the problem under consideration. [From R. Machleidt arxiv: v1 nucl-th]

27 EFT step by step for nuclear physicists Identify the degrees of freedom relevant at the resolution scale of (low-energy) nuclear physics: nucleons and pions. Identify the relevant symmetries of low-energy QCD and investigate if and how they are broken (chiral symmetry). Construct the most general Lagrangian consistent with those symmetries and the symmetry breaking (L χ ). Design an organizational scheme that can distinguish between more and less important contributions: a low-momentum expansion (Weinberg s power counting). Guided by the expansion, calculate Feynman diagrams to the the desired accuracy for the problem under consideration. [From R. Machleidt arxiv: v1 nucl-th]

28 Bibliography S. Weinberg, The quantum theory of fields, Vol. II. V. Bernard, N. Kaiser and U.-G. Meissner [hep-ph/ ] E. Epelbaum, [nucl-th/ ] S. Scherer and M. R. Schindler [hep-ph/ ]

29 Chiral Symmetry What is chirality? The structural characteristic of a finite system (molecule, atom, particle) that makes it impossible to superimpose it on its mirror image. Any object that is different from its reflection is said to be chiral. Your left hand and your right hand are distinct chiral objects.

30 Notes on chiral symmetry Starting point L QCD = q(iγ µ D µ M)q 1 4 G µν,ag µν a considering m u = m d = 0 L QCD = L heavy (s,c,b,t) + L u,d defining right- and left-handed quark fields q R = P R q, q L = P L q, with P R = 1 2 (1 + γ 5), P L = 1 2 (1 γ 5), right- and left-handed components are decoupled L u,d = L ur,d R + L ul,d L

31 Notes on chiral symmetry defining right- and left-handed quark fields q R = P R q, q L = P L q, with P R = 1 2 (1 + γ 5), P L = 1 2 (1 γ 5), right- and left-handed components are decoupled L u,d = L ur,d R + L ul,d L

32 Notes on chiral symmetry (continues..) in fact L QCD = q R iγ µ D µ q R + q L iγ µ D µ q L 1 4 G µν,ag µν a. L QCD is now invariant under q L q L = exp( i τ j 2 Θj L )q L {j = 1, 3} q R q R = exp( i τ j 2 Θj R )q R where τ j are Pauli matrices (generators of the SU(2) group) and Θ j R,L arbitrary parameters. SU(2) R SU(2) L symmetry is called Chiral Symmetry. [see Volker Koch, nucl-th/ , pags. 8-9]

33 Noether theorem To each element j of a continuos symmetry a current J µ j is conserved µ J µ j = 0 and a charge Q j = d 3 x J 0 j (t, x) is a constant of motion, i.e. is time independent in case of SU(2) R SU(2) L we have 6 conserved 4-currents: 6 charges R 0 i, L0 i R µ j = q R γ µ τ j 2 q R j = 1, 3 L µ j = q L γ µ τ j 2 q L constants of motion. [see Scherer, hep-ph/ , pags.11-18, Gell-Mann and Lévy, Nuovo Cim. 16 (1960) 705]

34 ...some manipulations 3 vector currents V µ i 3 axial vector currents = R µ i + Lµ i = qγµ τ i 2 q µv µ i = 0 A µ i = Rµ i Lµ i = qγµ γ 5 τ i 2 q µa µ i = 0, now the symmetry group is SU(2) V SU(2) A. q q = exp( i τ j q q = exp( i τ j 2 Θj V 2 γ 5Θ j A )q )q {j = 1, 3} SU(2) V mixes states with different isospin (u and d flavor) SU(2) A mixes states with different parity [see Scherer, hep-ph/ , pags.19,20]

35 with an explicit chiral symmetry breaking term... the mass term qm q breaks chiral symmetry M = ( mu 0 0 m d = 1 2 (m u + m d ) ) ( = 1 2 (m u + m d ) I (m u m d ) τ 3. ) + 1 ( (m u m d ) 0 1 )

36 with an explicit chiral symmetry breaking term... the mass term qmq breaks chiral symmetry M = ( mu 0 0 m d = 1 2 (m u + m d ) ) ( = 1 2 (m u + m d ) I (m u m d ) τ 3. ) + 1 ( (m u m d ) 0 1 )

37 with an explicit chiral symmetry breaking term... the mass term qmq breaks chiral symmetry M = ( mu 0 0 m d = 1 2 (m u + m d ) ) ( = 1 2 (m u + m d ) I (m u m d ) τ 3. ) + 1 ( (m u m d ) 0 1 ) The first term in the last equation in invariant under SU(2) V (isospin symmetry)

38 with an explicit chiral symmetry breaking term... the mass term qmq breaks chiral symmetry M = ( mu 0 0 m d = 1 2 (m u + m d ) ) ( = 1 2 (m u + m d ) I (m u m d ) τ 3. ) + 1 ( (m u m d ) 0 1 ) The first term in the last equation in invariant under SU(2) V (isospin symmetry) and the second term vanishes for m u = m d. Thus, isospin is an exact symmetry if m u = m d.

39 with an explicit chiral symmetry breaking term... the mass term qmq breaks chiral symmetry M = ( mu 0 0 m d = 1 2 (m u + m d ) ) ( = 1 2 (m u + m d ) I (m u m d ) τ 3. ) + 1 ( (m u m d ) 0 1 ) The first term in the last equation in invariant under SU(2) V (isospin symmetry) and the second term vanishes for m u = m d. Thus, isospin is an exact symmetry if m u = m d. However, both terms break SU(2) A. Since the up and down quark masses are small the explicit chiral symmetry breaking is very small.

40 ...but [H, Q] = 0 H 0 = 0 Q V 0 = Q A 0 = 0 isospin-multiplets and parity-multiplets are degenerate Å Å Î ¾¼¼¼ ½ ¼¼ ½ ¼¼ ½ ¼¼ ½ ¼¼ ½ ¼¼ ½ ¼¼ ½ ¼¼ ½¾¼¼ ½½¼¼ ½¼¼¼ ¼¼ ½ ¾ ½ ¾ ¾ Æ ¾ ¾ ¾ ½ ¾ ½ ¾ ¾ ¾ ¾ ¾ [From Cohen and Glozman: the low lying N and experimental spectra] ¾ ¾ chiral symmetry breaking is not small SU(2) R SU(2) L is broken at the level of the ground-state.

41 Vector and axial vector mesons Isospin symmetry SU(2) V is respected (π +, π, π 0 have approximately the same mass) If axial symmetry is respected ρ ρ + Θ a 1 isovector dipole and axial dipole excitations must have the same mass [M.Barate et al. (ALEPH): Eur. Phys. J. C4 (1998) 409; ) K. Ackerstaff et al. (OPAL): Eur. Phys. J. C7 (1999) 571; sum rule analysis: E. Marco et al. : Phys. Lett. B 482 (2000) 87 ]

42 Spontaneous Symmetry Breaking (SSB) O(3) Model Having in mind the smallness of the u- and d- quark masses, one naturally expects that an approximate chiral symmetry manifests itself in the observable properties of hadrons...but it does not happen! (...only the isospin symmetry SU(2) V is respected) Any solution? When a Lagrangian density L is invariant with respect to a set of transformations (SU(2) V SU(2) A in our case), two situations can occur: Wigner-Weyl mode, where ground-state has the same symmetry of L Nambu-Goldstone mode, where the ground-state does not respect the symmetry of the system

43 Goldstone Theorem To each generator that does not annihilate the ground state exists a massless boson degree of freedom. [J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127 (1962) 965] H 0 QCD is invariant under SU(2) L SU(2) R φ 0 is invariant under SU(2) V 3 massless Goldstone bosons. If Q A 0 0, then there must be a set of physical states φ A = Q A 0 which are energetically degenerate H φ A = HQ A 0 = Q A H 0 since Q A represents axial charges, the states φ A must represent a triplet of pseudoscalar mesons Guess: Pions (π +, π 0, π )?

44 Pions Pions are degenerate but not massless (m π 135 MeV) because the up and down quark masses are not exactly zero either (explicit symmetry breaking) δl = ψmψ ψψ = ψ R ψ L + ψ L ψ R Pions reflect spontaneous as well as explicit symmetry breaking

45 Chiral Lagrangian The relevant degrees of freedom are pions (Goldstone bosons) and nucleons. Interactions of Goldstone bosons must vanish at zero momentum transfer and in the chiral limit (m π 0) the low-energy expansion of the Lagrangian is arranged in powers of derivatives and pion masses. with L eff = L ππ + L πn + L NN L ππ = L (2) ππ + L (4) ππ +... L πn = L (1) πn + L(2) πn + L(3) πn +... L NN = L (2) NN + L(4) NN +... [S. Coleman, J. Wess, and B. Zumino, Phys. Rev. 177 (1969) 2239 and 2247]

46 Chiral Lagrangian L ππ Pions are represented by a 2 2 SU(2) matrix field U(x) U(x) = exp[iτ a φ a (x)] φ a = π a /f with the pion decay constant f in the chiral limit as normalisation. Write an effective Lagrangian for the field U(x) and its derivatives L QCD L eff (U, U, 2 U,...) Goldstone bosons can only interact when they carry momentum low-energy expansion in powers of µ U Lorentz invariance only an even numbers of derivatives

47 Chiral Lagrangian L ππ Transformation behaviour under SU(2) L SU(2) R for the U field U RUL U LU R and for its derivatives µ U µ U µ (RUL ) = µ RUL +R µ UL +RU µ L = R µ UL µ U L µ U R The most general chirally invariant L with the minimal number of derivatives is L (2) ππ = f 2 4 T r[ µu µ U ] [see Scherer, hep-ph/ , pags.63...and also V. Koch, nucl-th/ ]

48 Chiral Lagrangian L ππ in fact L (2) ππ f 2 4 T r[r µul L µ U R ] = f 2 4 T r[r R µ U µ U ] = L (2) ππ The symmetry breaking mass term is small, so that it can be handled perturbatively. The leading contribution in the quark mass matrix M is: where L (2) = f 2 4 T r[ µu µ U] + f 2 ( ) mu 0 M = 0 m d ; 2 B 0 T r[mu + M U]. { M RML M LM R

49 Chiral Lagrangian L ππ Expanding L (2) ππ L (2) ππ = (m u +m d )f 2 B µπ a µ π a 1 2 (m u +m d )B 0 π 2 a +...

50 Chiral Lagrangian L ππ Expanding L (2) ππ L (2) ππ = (m u +m d )f 2 B µπ a µ π a 1 2 (m u +m d )B 0 π 2 a +... The first (constant) term corresponds to the shift of the vacuum energy density by the non-zero quark masses (m u ūu + m d dd ) = (m u + m d )f 2 B 0 where ūu = dd = f 2 B 0 in the chiral limit V(σ, π=0) V(σ, π=0) f π σ fπ σ [From V. Koch (Potential of linear sigma model without (left) and with (right) explicit symmetry breaking)]

51 Chiral Lagrangian L ππ Expanding L (2) ππ L (2) ππ = (m u +m d )f 2 B µπ a µ π a 1 2 (m u +m d )B 0 π 2 a +... The first (constant) term corresponds to the shift of the vacuum energy density by the non-zero quark masses (m u ūu + m d dd ) = (m u + m d )f 2 B 0 where ūu = dd = f 2 B 0 in the chiral limit The pion mass term is m 2 π = (m u + m d )B 0 from which we have the GOR relation m 2 π f 2 π = (m u + m d ) qq + O(m 2 u,d ) [M. Gell-Mann, R. Oakes and B. Renner, Phys. Rev. 122 (1968) 2195]

52 Chiral Lagrangian L πn At leading order where L (1) πn = ψ ( ) i /D M (0) N + g 2 γµ γ 5 u µ ψ }{{} axial coupling /D = µ + Γ µ Γ µ = (u µ u + u µ u )/2 = iτ (π µ π)/4f 2 }{{} +O(π4 ) W einberg T omozawa u µ = i(u µ u u µ u ) = τ µ π/f + O(π 3 ) u = U = 1 + iτ π/2 + O(π 2 ) with g πnn = (M N /f)g = 13.1

53 Chiral Lagrangian L πn At leading order where L (1) πn = ψ ( ) i /D M (0) N + g 2 γµ γ 5 u µ ψ }{{} axial coupling /D = µ + Γ µ Γ µ = (u µ u + u µ u )/2 = iτ (π µ π)/4f 2 }{{} +O(π4 ) W einberg T omozawa u µ = i(u µ u u µ u ) = τ µ π/f + O(π 3 ) u = U = 1 + iτ π/2 + O(π 2 ) with g πnn = (M N /f)g = 13.1

54 Chiral Lagrangian L πn Invariance under SU(2) L SU(2) R very lengthy [see J. Gasser, M. E. Sainio and A. Švarc, Nucl. Phys. B 307 (1988) 779] The nucleon mass M N in the chiral limit does not vanish a dimensional scale to be taken in account. A solution (not the only one): Heavy Baryon expansion where the Lagrangian is expanded in powers of 1/M N [ L (1) πn = N i 0 1 4fπ 2 τ (π 0 π) g ] A τ ( σ 2f )π N+... π [E. Jenkins and A. V. Manohar, Phys. Lett. B 255 (1991) 558] At next-to-leading order the symmetry breaking term enters M N = M (0) N + σ N L (2) πn =... + σ N ψψπ 2 2fπ 2

55 Chiral Lagrangian L πn In general the Lagrangians must contain all terms consistent with chiral symmetry and Lorentz invariance. The parameters c i are the low-energy constants (LECs). 2 nd order L (2) πn = 4 i=1 (2) c i ΨO i Ψ 4 c i parameters to be determined from fits to πn data 3 rd order L (3) 23 πn = (3) d i ΨO i Ψ i=1 23 d i parameters to be determined (not all relevant for the NN potential) [Operators: Fettes et al., Ann. Phys. (N.Y.) 283 (2000) 283; 288 (2001) 249; Nucl. Phys. A 640 (1998) 199]

56 Chiral Lagrangian L NN Nucleon contact interactions consist of four nucleon fields to renormalize loop integrals to make results reasonably independent of regulators to parametrize the unresolved short-distance contributions Because of parity only even numbers of derivatives L NN = L (0) NN + L(2) NN + L(4) NN +... L (0) NN = 1 2 C S NN NN 1 2 C T L (2) 14 NN = C i NÕ (2) i N i=1 N σn N σn where LECs are determined by a fit to the NN data.

57 Power counting contributions in terms of powers of (Q/Λ χ ) ν, Power counting: the power ν at which a given diagram contributes for an irreducible NN diagram (A = 2) ν = 2L + i i where L is the number of loops and i = d i + n i 2 2 ν is bounded from below; e.g., for A = 2, ν 0. [see for the general case R. Machleidt arxiv: v1 nucl-th]

58 Power counting for an irreducible NN diagram (A = 2) ν = 2L + i i where L is the number of loops and i = d i + n i 2 2 At leading order [LO, O(Q 0 ), ν = 0]: 2 contact terms the static 1π exchange no contributions at order O(Q 1 )

59 Power counting for an irreducible NN diagram (A = 2) ν = 2L + i i where L is the number of loops and i = d i + n i 2 2 At leading order [LO, O(Q 0 ), ν = 0]: 2 contact terms the static 1π exchange no contributions at order O(Q 1 )

60 Power counting for an irreducible NN diagram (A = 2) ν = 2L + i i where L is the number of loops and i = d i + n i 2 2 The first graph with four nucleon legs and a solid square represent the seven contact terms of O(Q 2 ). All 2π exchange diagrams.

61 Nuclear Force hyerarchy Q 0 LO Q 2 NLO 2N Force 3N Force 4N Force generation of many body forces estimates of missing terms Q 3 NNLO Q 4 3 N LO For high precision potentials (χ 2 1) go to N 3 LO

62 Nuclear Force hyerarchy Q 0 LO Q 2 NLO 2N Force 3N Force 4N Force generation of many body forces estimates of missing terms Q 3 NNLO Q 4 3 N LO For high precision potentials (χ 2 1) go to N 3 LO

63 Nuclear Force hyerarchy Q 0 LO Q 2 NLO 2N Force 3N Force 4N Force generation of many body forces estimates of missing terms Q 3 NNLO Q 4 3 N LO For high precision potentials (χ 2 1) go to N 3 LO

64 Nuclear Force hyerarchy Q 0 LO Q 2 NLO 2N Force 3N Force 4N Force generation of many body forces estimates of missing terms Q 3 NNLO Q 4 3 N LO For high precision potentials (χ 2 1) go to N 3 LO

65 Strategy Following Weinberg: use perturbation theory to calculate the N N potential and to apply this potential in a scattering equation to obtain the NN amplitude [S. Weinberg, Phys. Lett. B 251 (1990) 288; 295 (1992) 114; Nucl. Phys. B 363 (1991) 3] Derive the N N potential perturbatively pion-exchange contribution contact terms contribution T-matrix derivation Partial wave analysis

66 π-exchange contributions Momentum-space N N potential in the center-of-mass system V ( p, p) = V C + τ 1 τ 2 W C + [ V S + τ 1 τ 2 W S ] σ 1 σ 2 ( + [ V LS + τ 1 τ 2 W LS ] is ( q ) k) + [ V T + τ 1 τ 2 W T ] σ 1 q σ 2 q + [ V σl + τ 1 τ 2 W σl ] σ 1 ( q k ) σ 2 ( q k ),

67 π-exchange contributions Momentum-space N N potential in the center-of-mass system V ( p, p) = V C + τ 1 τ 2 W C + [ V S + τ 1 τ 2 W S ] σ 1 σ 2 ( + [ V LS + τ 1 τ 2 W LS ] is ( q ) k) + [ V T + τ 1 τ 2 W T ] σ 1 q σ 2 q + [ V σl + τ 1 τ 2 W σl ] σ 1 ( q k ) σ 2 ( q k ), V i represents isoscalar contributions

68 π-exchange contributions Momentum-space N N potential in the center-of-mass system V ( p, p) = V C + τ 1 τ 2 W C + [ V S + τ 1 τ 2 W S ] σ 1 σ 2 ( + [ V LS + τ 1 τ 2 W LS ] is ( q ) k) + [ V T + τ 1 τ 2 W T ] σ 1 q σ 2 q + [ V σl + τ 1 τ 2 W σl ] σ 1 ( q k ) σ 2 ( q k ), V i represents isoscalar contributions W i represents isovector contributions

69 π-exchange contributions Momentum-space N N potential in the center-of-mass system V ( p, p) = V C + τ 1 τ 2 W C + [ V S + τ 1 τ 2 W S ] σ 1 σ 2 ( + [ V LS + τ 1 τ 2 W LS ] is ( q ) k) + [ V T + τ 1 τ 2 W T ] σ 1 q σ 2 q + [ V σl + τ 1 τ 2 W σl ] σ 1 ( q k ) σ 2 ( q k ), q p p is the momentum transfer k 1 2 ( p + p) is the average momentum S 1 2 ( σ 1 + σ 2 ) is the total spin

70 π-exchange contributions Momentum-space N N potential in the center-of-mass system V ( p, p) = V C + τ 1 τ 2 W C + [ V S + τ 1 τ 2 W S ] σ 1 σ 2 ( + [ V LS + τ 1 τ 2 W LS ] is ( q ) k) + [ V T + τ 1 τ 2 W T ] σ 1 q σ 2 q + [ V σl + τ 1 τ 2 W σl ] σ 1 ( q k ) σ 2 ( q k ), Dimensional regularization typically generates also polynomial and nonpolynomial terms. The polynomials will be absorbed by the contact interactions.

71 π-exchange contributions At leading order [O(Q 0 )] the static one-pion exchange V 1π ( p, p) = g2 A σ 1 q σ 2 q 4fπ 2 τ 1 τ 2 q 2 + m 2. 1π exchange π At second order (ν = 2, NLO) 1π iterated (no iterative contributions) W C = L(q) [ 4m 2 384π 2 fπ 4 π (5gA 4 4gA 2 1) + q 2 (23gA 4 10gA 2 1) ] + 48g4 A m4 π w 2, V T = 1 q 2 V S = 3g4 A L(q) 64π 2 f 4 π where L(q) w q ln w+q 2m π and w 4m 2 π + q 2,

72 π-exchange contributions insertions from L (2) πn rel. 1/M N corrections [D. R. Entem and R. Machleidt, Phys. Rev. C 68 (2003) ] 3-pion exchange rel. 1/M N corrections rel. (1/M N ) 2 corrections

73 N N contact contributions Short distance part of the nuclear force Contact terms pick up infinities from regularization and remove scale dependence I (ν) +1 L = Q ν P L (cos θ)d cos θ = 1 +1 where P L is a Legendre polynomial. Due to the orthogonality of the P L, 1 I (ν) L = 0 for L > ν 2. order zero contribute only in S-waves order-two contribute up to P -waves order-four contribute up to D-waves... f ν 2 (cos θ)p L(cos θ)d cos θ,

74 N N contact contributions Short distance part of the nuclear force Contact terms pick up infinities from regularization and remove scale dependence I (ν) +1 L = Q ν P L (cos θ)d cos θ = 1 +1 where P L is a Legendre polynomial. Due to the orthogonality of the P L, 1 I (ν) L = 0 for L > ν 2. order zero contribute only in S-waves order-two contribute up to P -waves order-four contribute up to D-waves... f ν 2 (cos θ)p L(cos θ)d cos θ,

75 N N contact contributions Short distance part of the nuclear force Contact terms pick up infinities from regularization and remove scale dependence I (ν) +1 L = Q ν P L (cos θ)d cos θ = 1 +1 where P L is a Legendre polynomial. Due to the orthogonality of the P L, 1 I (ν) L = 0 for L > ν 2. order zero contribute only in S-waves order-two contribute up to P -waves order-four contribute up to D-waves... f ν 2 (cos θ)p L(cos θ)d cos θ,

76 N N contact contributions Order zero Order two Order four V (0) ( p, p) = C S + C T σ 1 σ 2 V (2) ( p, p) = C 1 q 2 + C 2 k 2 + ( C 3 q 2 + C 4 k 2) σ 1 σ 2 + C 5 ( i S ( q ) k) + C 6 ( σ 1 q) ( σ 2 q) + C 7 ( σ 1 k) ( σ 2 k) V (4) ( p, p) = D 1 q 4 + D 2 k 4 + D 3 q 2 k 2 + D 4 ( q k) 2 + D 5 q 4 + D 6 k 4 + D 7 q 2 k 2 + D 8 ( q k) 2 σ 1 σ 2 + D 9 q 2 + D 10 k 2 is ( q k) + D 11 q 2 + D 12 k 2 ( σ 1 q) ( σ 2 q) + D 13 q 2 + D 14 k 2 ( σ 1 k) ( σ 2 k) + D 15 σ 1 ( q k) σ 2 ( q k)

77 Constructing the T matrix Nonrelativistic Lippmann-Schwinger (LS) equation T ( p, p) = V ( p, p)+ d 3 p V ( p, p M ) T ( p, p) p 2 p 2 + iɛ V and T satisfy the Blankenbecler and Sugar equation [Excellent introduction: M. Hjorth-Jensen, ] No manifest covariance. Relativity is accounted for in terms of a Q/Λ expansion up to given order Iteration of V requires cutting V off for high momenta to avoid infinities cutoff V ( p, p) V ( p, p) e (p /Λ) 2n e (p/λ)2n { [ (p ) ] } V ( p 2n ( p ) 2n, p) Λ Λ

78 Constructing the T matrix For large n, the regulator introduces contributions that are beyond the given order or removed by contact terms What cutoff? A too small Λ will remove the truly long-distance physics and reduce any predictive power. On the other hand with large cutoffs high-momentum states are included that are sensitive to short-distance dynamics Entem and Machleidt: 500 MeV [R. Machleidt arxiv: v1 nucl-th]] Epelbaum, Meissner,...: MeV (systematic investigations) [Epelbaum, Prog. Part. Nucl. Phys. 57 (2006) 654] More on cutoffs

79 Phase shifts with L 3 and T lab 300 MeV At order four in small momenta no contributions from contact interactions f π = 92.4 and g A = 1.29 LEC NN potential πn at N 3 LO empirical c ± 0.15 a c ± 0.23 b c ± 1.34 a c ± 0.04 a d 1 + d ± 0.21 b d ± 0.73 b d ± 0.42 b d 14 d ± 0.41 b Phase Shift (deg) F 3 N2LO LO N3LO NLO Phase Shift (deg) F 2 N3LO N2LO LO NLO Phase Shift (deg) F 3 Bonn OPE N3LO Phase Shift (deg) F 2 N3LO Bonn OPE Lab. Energy (MeV) Lab. Energy (MeV) Lab. Energy (MeV) Lab. Energy (MeV) 3 F F 4 3 F F 4 Phase Shift (deg) N3LO N2LO NLO LO Phase Shift (deg) N3LO N2LO NLO LO Phase Shift (deg) N3LO Bonn OPE Phase Shift (deg) N3LO Bonn OPE Lab. Energy (MeV) Lab. Energy (MeV) Lab. Energy (MeV) Lab. Energy (MeV)

80 Phase shifts Nijmegen CD-Bonn Contact Potentials partial-wave high-precision Q 0 Q 2 Q 4 analysis potential LO NLO/NNLO N 3 LO 1 S S S1-3 D P P P P P2-3 F D D D D D3-3 G F F F F Total

81 Phase shifts (np scattering with Λ = 500 MeV) NLO: dotted line NNLO: dashed line N 3 LO: solid line [From R. Machleidt arxiv: v1 nucl-th]

82 Phase shifts np database T lab bin Idaho Argonne (MeV) N 3 LO V 18 ( ) pp database T lab bin Idaho Argonne (MeV) N 3 LO V 18 ( ) Deuteron properties Idaho N 3 LO CD-Bonn AV18 Empirical (500) B d (MeV) (9) A S (fm 1/2 ) (9) η (4) r d (fm) (85) Q (fm 2 ) (3) P D (%)

83 Phase shifts by Epelbaum Spectral Function regularization [Epelbaum et al. Eur. Phys. J. A 19 (2004) 125] NLO: dashed bands NNLO: light shaded bands N 3 LO: dark shaded bands Phase Shift [deg] Phase Shift [deg] Phase Shift [deg] Phase Shift [deg] ε 1 3 D2 1 S0 3 P Lab. Energy [MeV] D2 3 D3 3 S1 3 P Lab. Energy [MeV] [Epelbaum, nucl-th/ ] P1 3 P2 3 D1 ε Lab. Energy [MeV]

84 Many-body forces Do we need 3body forces? Some evidence from few-body observables p + d p + d [Witala et al., Phys. Rev. Lett. 81 (1998) 1183; Machleidt, talk at FB18]

85 Many-body forces EFT approaches create two- and many-nucleon forces at the same time Old approaches based on phenomenology [Pieper, Pandharipande, Wiringa and Carlson, Phys. Rev. C 64 (2001) ]

86 Many-body forces [From Epelbaum, Talk (2006)] Calculations performed at NNLO; N 3 LO under construction

87 O(3) Model Considering a system of spins, located on the sites of an infinite dimensional lattice. X H = S i S j near.neighbours Back is invariant under rotations g O(3) in 3-dimensional space S i g S i disordered system (O(3) conserved) ordered system (O(3) broken into O(2)) Ferromagnet Low-energy QCD H symmetry O(3) SU(2) V SU(2) A Vacuum symmetry O(2) SU(2) V Collective modes spin waves (massless) Goldstone bosons Explicit breaking external magnetic field (small) quark masses

88 1π-exchange pseudovector coupling two-body amplitude L pv = f π m π ψγ5 γ µ ψ µ φ π Vpv π = f π 2 ū(p 1 )γ 5γ µ (p 1 p 1 )µ u(p 1 )ū(p 2 )γ 5γ ν (p 2 p 2 )ν u(p 2 ) m 2 π (p 1 p 1 )2 m 2 π the Dirac spinors obey γ µ p µ u(p) = mu(p) ū(p)γ µ p µ = mū(p)

89 1π-exchange using these relations, together with {γ 5, γ µ } = 0, we find ū(p 1)γ 5 γ µ (p 1 p 1) µ u(p 1 ) = 2mū(p 1)γ 5 u(p 1 ) ū(p 2)γ 5 γ µ (p 2 p 2 ) µ u(p 2 ) = 2mū(p 2)γ 5 u(p 2 ) we get Vpv π = f π 2 m 2 4m 2 ū(p 1 )γ 5u(p 1 )ū(p 2 )γ 5u(p 2 ) π (p 1 p 1 )2 m 2 π by inserting expressions for the Dirac spinors ( ) ū(p (E 1 1)γ 5 u(p 1 ) = + m)(e 1 + m) σ 1 p 1 4m 2 E 1 + m σ 1 p 1 E 1 + m and similarly for ū(p 2 )γ 5u(p 2 )

90 1π-exchange In the CM system p 2 = p 1, p 2 = p 1 and so E2 = E 1 and E2 = E 1 Vpv,CM π = f π 2 m 2 4m 2 1 (E 1 + m)(e 1 + m) π (p 1 p 1 )2 m 2 π 4m 2 ( ) ( ) σ 1 p 1 E 1 + m σ 1 p 1 σ 2 p 1 E 1 + m E 1 + m σ 2 p 1 E 1 + m in the non-relativistic limit E 1 = p m2 m E 1 and then (p 1 p 1 )2 = k 2 Vpv,CM π = f π 2 m 2 4m 2 1 σ 1 π k 2 + m 2 2m (p 1 p 1) σ 2 2m (p 1 p 1) = f 2 π m 2 π (σ 1 k)(σ 2 k) k 2 + m 2 τ 1 τ 2 π

91 1π-exchange in coordinate space Back V π (r) = d 3 k 2π 3 eikr V π (k) we obtain V π (r) = f π 2 ( e m πr ) m 2 τ 1 τ 2 σ 1 σ 2 π r carrying out the differentiation V π (r) = 1 f 2 { π e m πr } 3 m 2 4πδ 3 (r) σ 1 σ 2 τ 1 τ 2 π r + 1 f 2 { π m π r + 3 } e m πr m 2 πr 2 S 12 (ˆr)τ 1 τ 2 r m 2 π [T. Ericson and W. Weise, Pions and nuclei, Oxford Press]

92 1π iterated Back The iterative 2PE contribution has to be subtracted from the covariant box diagram, order by order V 2π,it ( p, p) = M 2 N E p d 3 p (2π) 3 V 1π ( p, p ) V 1π ( p, p) p 2 p 2 + iɛ [N. Kaiser, R. Brockmann, and W. Weise, Nucl. Phys. A 625 (1997) 758]

93 More about cutoffs Back Doesn t a finite cutoff introduce a model dependence? A finite Λ reduce the range of applicability to the momentum region Q < Λ. Taking Λ Λ breakdown will not introduce any ambiguity/model dependence [Donoghue, Holstein and Borasoy, Phys. Rev. D 59 (1999)] What about breaking of chiral invariance? Chiral symmetry is preserved. [Dijukanovic et al., hep-ph/ ] What about if I take Λ? Infinite number of counterterms.

94 N N-force: some phenomenology V C (r) [MeV] S0 3 S1 OPEP r [fm] [Lattice calculations from Ishii et al., nucl-th ] The extracted potential has a strong repulsive core of a few hundred MeV at short distances (r 0.5 fm) surrounded by a relatively weak attraction at medium and long distances

95 N N-force: some phenomenology V III II I N N π π π π N N N N π N π N r [µ 1 ] Strong short range repulsion (core) Intermediate attraction Long-range: 1π-exchange [Figure from W. Weise, arxiv: v1 (nucl-th)]

96 N N-force: some phenomenology V III II I N N π π π π N N N N π N π N r [µ 1 ] Strong short range repulsion (core) Intermediate attraction Long-range: 1π-exchange [Figure from W. Weise, arxiv: v1 (nucl-th)]

97 N N-force: some phenomenology V III II I N N π π π π N N N N π N π N r [µ 1 ] Strong short range repulsion (core) Intermediate attraction Long-range: 1π-exchange [Figure from W. Weise, arxiv: v1 (nucl-th)]

98 N N-force: some phenomenology V III II I N N π π π π N N N N π N π N r [µ 1 ] Strong short range repulsion (core) Intermediate attraction Long-range: 1π-exchange [Figure from W. Weise, arxiv: v1 (nucl-th)]

99 N N-force: some phenomenology Relevant aspects Deuteron (J π = 1 + ) as a shallow bound state Spin-dependent force Spin-orbit force Non central force tensor force Charge symmetry breaking (CSB) and Charge indipendence breaking (CIB) app = 17.3 ± 0.4 fm a nn = 18.8 ± 0.5 fm (with some disagreements...) a pn = ± 0.02 fm

100 N N-force: some phenomenology Symmetries Translation invariance Galilean invariance Rotation invariance Space reflection invariance Time reversal invariance Invariance under the interchange of particle 1 and 2 Almost isospin symmetry [For a general two-body non-relativistic model see Okubo and Marshak, Ann. Phys. (NY) 4, 166 (1958); in general see R. Machleidt, Adv. Nucl. Phys. 19 (1989) 189]