Nuclear Forces - Lecture 2 - R. Machleidt University of Idaho

Transcription

1 CNS Summer School, Univ. of Tokyo, at Wako campus of RIKEN, Aug. 18-3, 005 Nuclear Forces - Lecture - R. Machleidt University of Idaho 1

2 Lecture : The Meson Theory of Nuclear Forces Yukawa s historic idea The mesons How do those mesons couple to the nucleon? The One-Boson-Exchange Potential Closing remarks

3 Yukawa and his idea S. Tomonaga, H. Yukawa, and S. Sakata in the 1950s. 3

4 From: H. Yukawa, Proc. Phys. Math. Soc. Japan 17, 48 (1935). 4

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7 Repulsive force due to the exchange of a basket ball between two people standing on ice 7

8 pseudo scalar vector scalar STOP 8

9 What do those mesons do to the NN interaction? To find out, we have to do some calculations. Proper calculations are done in the framework of Quantum Field Theory. That means, we have to take the following steps: Write down appropriate Lagrangians for the interaction of the mesons with nucleons. Using those interaction Lagrangians, calculate Feynman diagrams that contribute to NN scattering. 9

10 Feynman diagram for NN scattering p 1 ' p ' α Γ1 Γ q = ( p ' p ) 1 1 p 1 Amplitude: Fα ( p', p) = p u ' Γ u P u ' Γ u α q + mα χs χs E + M χs with Dirac spinor ups (, ) = σ p σ p M χ 0 s χ s E M E M + + where E = p + M and χ is a two-component Pauli spinor. s 10

11 Pseudo-vector coupling of a pseudo-scalar meson Lagrangian: L f = πnn µ ( π ) π NN ψγ γ5τψ µ ϕ mπ Vertex: i times the Lagrangian stripped off the fields f () NN NN i π µ π γ γ5τ qµ mπ Γ = f πnn mπ ( σ q) τ Potential: i times the amplitude ( Pπ = i, q q ) f ( )( q) Vπ = ifπ τ τ σ1 q σ NN π mπ q + mπ 1 11

12 Pseudo-vector coupling of a pseudo-scalar meson, cont d Using the operator identity with q ( σ1 q)( σ q) = σ1 σ S1( qˆ ) 3 + S () qˆ 3( σ qˆ)( σ qˆ) σ σ ( Tensor operator ), the one-pion exchange potential (OPEP) can be written as f Vπ = σ1 σ S 1() qˆ τ1 τ πnn q 3mπ q + mπ 1

13 Scalar coupling Lagrangian: L σnn = gσ ψψ ϕ ( σ ) Vertex: Potential: ( σ p')( σ p) igσ 1 ( E ' + M)( E + M) 1 k q σ L p' p + iσ ( p' p) ig 1 ig 1 4 = σ σ ( E ' M)( E M) + + 4M up ( ') Γ σnn up ( ) = igσ up ( ') up ( ) 1 i keeping all terms up to Q / M P σ i, k ( p' p), L S ( 1 ) ( q k ) [ = + = σ + σ ] g k q L S V if σ σ = σ 1 + q + mσ M 8M M 13

14 Vector coupling of a vector meson Lagrangian: L µ ( ω) ωnn = gω ψγ ψ ϕµ Vertex: 0 0 ( σ p')( σ p) µ = 0: up ( ') Γ ωnnup ( ) = igω up ( ') γ up ( ) igω 1 + ( E ' M)( E M) + + σ L igω 1, keeping only the σ L term. 4M Potential, including also the γ terms: Pω = igµν + g L S V if ω ω = ω 1 3 q + mω M 14

15 Tensor coupling of a vector meson Lagrangian: Vertex: L f ( ) (tensor) ρ µν ( ρ) ( ρ) ρnn = ψσ τψ µ ϕν ν ϕµ 4M (tensor) fρ µν fρ µν fρ Γ ρnn = σ ( qµ qν ) τ = σ qµ τ ( σ q) τ 4M M M Potential: V Pρ = igµν + (tensor) (tensor) fρ ( σ1 q)( σ q) ρ = ifρ = τ 1 τ 4M q + mρ fρ σ1 σ q ( σ1 q )( σ q ) = τ 1 τ 4M q + mρ ρ q M q + mρ f = σ1 σ + S1( qˆ ) τ1 τ 1 15

16 Recall: We found the mesons below in PDG Table and asked: What do they do? Now, we have the answer. Let s summarize. pseudo scalar vector scalar STOP 16

17 Summary π (138) fπ NN q π 3m q + m V = σ1 σ S1() qˆ τ1 τ π π Long-ranged tensor force σ (600) ω(78) g Vω ρ(770) g L S V σ σ 1 q + mσ M ω L S q + mω M f ρ q Vρ = σ1 σ + S1( qˆ ) τ1 τ 1M q + mρ intermediate-ranged, attractive central force plus LS force short-ranged, repulsive central force plus strong LS force short-ranged tensor force, opposite to pion It s EVERYTHING we need to describe the nuclear force! 17

18 Central force Summary: Most important parts of the nuclear force Short ω ρ Intermediate σ Long range π Tensor force π ρ ω σ R. Spin-orbit Machleidt force Nuclear Forces - Lecture 18

19 The One-Boson Exchange Potential (OBEP) V OBEP = α= π, σ, ρ, ωη,, a, 0 V α η(548) is a pseudo-scalar meson with I = 0, therefore, Vη is given by the same expression as Vπ, except that Vη carries no ( τ τ ) factor. 1 a0(980) is a scalar meson with I = 1, therefore, Va is given by 0 the same expression as Vσ, except that V a carries a ( τ τ ) factor

20 Some comments Note that the mathematical expressions for the various V α given on previous slides are simplified (many approximations) --- for pedagogical reasons. For a serious OBEP, one should make few approximations. In fact, it is quite possible to apply essentially no approximations. This is known as the relativistic (momentum-space) OBEP. Examples are the OBEPs constructed by the Bonn Group, the latest one being the CD-Bonn potential (R. M., PRC 63, (001)). If one wants to represent the OBE potential in r-space, then the momentum-space OBE amplitudes must be Fourier transformed into r-space. The complete, relativistic momentum-space expressions do not yield analytic expressions in r- space after Fourier transform, i.e., it can be done only numerically. However, it is desirable to have analytic expressions. For this, the momentum-space expressions have to be approximated first, e.g., expanded up to Q / M, after which an analytic Fourier transform is possible. The expressions one gets by such a procedure are shown on the next slide. Traditionally, the Nijmegen group has taken this approach; their latest r-space OBEPs are published in: V. G. J. Stoks, PRC 49, 950 (1994). 0

21 OBEP expressions in r-space (All terms up to Q / M are included.) 1

22 Does the OBE model contain everything? NO! It contains only the so-called iterative diagrams. 1 Lippmann-Schwinger eqn: T = V + V T e T = V + V V + V V V + e e e In diagrams: i-t-e-r-a-t-i-v-e However: There are also non-iterative diagrams which contribute to the nuclear force (see next slide).

23 Some examples for non-iterative meson-exchange contributions not included in the OBE model (or OBEP). The Bonn Full Model (or Bonn Potential ) contains these and other non-iterative contributions. It is the most comprehensive meson-model ever developed (R. M., Phys. Reports 149, 1 (1987)). The Paris Potential is based upon dispersion theory and not on field theory. However, one may claim that, implicitly, the Paris Potential also includes these diagrams; M. Lacombe et al., Phys. Rev. C 1, 861 (1980). 3

24 Reviews on Meson Theory Pedagogical introduction which also includes a lot of history: R. M., Advances in Nuclear Physics 19, (1989). The derivation of the meson-exchange potentials in all mathematical details is contained in: R. M., The Meson Theory of Nuclear Forces and Nuclear Matter, in: Relativistic Dynamics and Quark-Nuclear Physics, M. B. Johnson and A. Picklesimer, eds. (Wiley, New York, 1986) pp Computer codes for relativistic OBEPs and phase-shift calculations in momentum-space are published in: R. M., One-Boson Exchange Potentials and Nucleon-Nucleon Scattering, in: Computational Nuclear Physics Nuclear Reactions, K. Langanke, J.A. Maruhn, and S.E. Koonin, eds. (Springer, New York, 1993) pp