Elementarity of composite systems

Transcription

1 Workshop Bound states in QCD and beyond 4 th 7 th of March, 05 Schlosshotel Rheinfels, St. Goar, Germany Elementarity of composite systems Hideko Nagahiro,, Atsushi Hosaka Nara Women s University, Japan RCNP, Osaka University, Japan References: H. Nagahiro, and A. Hosaka, PRC90(04)0650 H. Nagahiro, and A. Hosaka, PRC88(03)05503 (Editors Suggestion)

2 Introduction & motivation Many candidates for exotic hadrons : not simple q q (or qqq) state PP VP BP VP σ, f 0 (980)/a 0 (980),, a (60),, Λ(405), N (535),, X(387), vs. Dynamically generated resonance Nature of possible exotic hadrons elementary particle ( quasi-particle ) (~ can be q q or qq q q ) The question : Physical state must be a mixture of possible quantum states physical state C +C + How much they contain elementary components?

3 compositeness or elementarity the wave function renormalization Z compositeness condition Z 0 for a bound state Z : probability of finding the elementary particle Bound state Weinberg, PR30(63)776 Lurie-Macfarlane, PR36(65)B86 Hyodo-Jido-Hosaka, PRC() bare, un-renormalized, field φ Zφ R vanishes for a bound state φ + + quasi-particle of infinite mass with Z 0 Weinberg, PR30(63)776 a bound state can be represented by introducing a quasi-particle with infinite bare mass and hence Z 0 Lurie-Macfarlane, PR36(65)B86 equivalence between a four-fermi theory and a Yukawa theory the renormalization constant Z for a Yukawa particle is equal to zero Weinberg, PR36(65)B86 Z < 0. for deuteron system for a resonant state?? Hyodo-Jido-Hosaka, PRC85()050 3

4 Elementary particle model of s-wave composite states the wave function renormalization Z compositeness condition Z 0 for a bound state Z : probability of finding the elementary particle Bound state + + quasi-particle of infinite mass with Z 0 bare, un-renormalized, field φ Zφ R vanishes for a bound state φ energy-dependent + + +? φ + + Dynamically generated s-wave resonance equivalent quasi-particle model? What do we obtain for an s-wave resonant state? 4

5 A brief review of compositeness condition D. Lurie et al., PR36(64)B86 Bound state (four-point) model T F v G(s) Yukawa theory with constant g 0 T Y g 0 s m 0 g 0 G(s) G μ s μ g 0 g 0 G μ s μ g R s μ T F T Y g R s μ Z 0 g R G (μ ) Weinberg also uses this eq. by estimating g R from low energy p-n scattering. wave function renormalization Z g 0 G μ + g R G (μ ) 5

6 Equivalent Yukawa model to a resonant model? Bound Resonance state case model (composite model) T F v vs G(s) Yukawa theory with constant g 0 T Y g 0 s m 0 g 0 G(s) G μ s μ g 0 g 0 G μ s μ g R s μ T F T Y? g R s μ Z 0? g R G (μ ) energy dep. leads to Z 0? wave function renormalization Z g 0 G μ + g R G (μ ) 6

7 model set-up for s-wave composite states Interaction kernel : Weinberg-Tomozawa type v s f (s m ) energy-dependent [] Olle-Oset, NPA60(97)438 scattering amplitude with on-shell factorization [] T s v s + v s G s v s v s G(s) g R s s μ composite pole loop function G s i d 4 q π 4 q m + iε P q m + iε regularize appropriately by dim. regularization / 3dim cut-off physical coupling g R μ f μ m G μ 7

8 fictitious elementary model of the composite state shifted amplitude δ 0 T +δ T δ T T δ v s μ +δ μ + δ G s δ scattering amplitude with on-shell factorization [] T s v s + v s G s v s v s G(s) g R s s μ composite pole 8

9 Equivalent Yukawa model shifted amplitude T T δ Yukawa term v s + δ G s V Y (s) V Y s v s + δ g 0 (s) (s m fic )/g 0 G(s) g 0 (s) s m fic g 0 (s)g(s) s m fic bare mass of the fictitious elementary particle m fic m + f δ (cf. Hyodo08) Yukawa term g 0 m fic g 0 fictitious particle Energy-dependent Yukawa coupling g 0 s f s m fic m fic m 9

10 Equivalent Yukawa model Composite model T T δ v s + δ G s Yukawa model (s m fic )/g 0 G(s) g 0 (s) s m fic g 0 (s)g(s) full propagator of the fictitious elementary particle Δ s s m fic Π(s) self-energy Π s g 0 s G(s) wave function renormalization constant Z Z Π μ δ g 0 μ δ G s g0 μ δ G μ δ due to energy-dependence of g 0 bound state case Z g 0 G μ (Weinberg s formula) g R μ δ g 0 μ δ 0

11 Equivalent Yukawa model Composite model T T δ v s + δ G s Yukawa model (s m fic )/g 0 G(s) g 0 (s) s m fic g 0 (s)g(s) full propagator of the fictitious elementary particle Δ s s m fic Π(s) wave function renormalization constant Z Z Π μ δ g 0 μ δ T Y T δ δ 0 G s g0 μ δ G μ δ due to energy-dependence of g 0 T How about Z δ 0? self-energy Π s g 0 s G(s) bound state case Z g 0 G μ (Weinberg s formula) g R μ δ g 0 μ δ

12 Wave function renormalization constant wave function renormalization constant Z Z Π μ δ g 0 μ δ G s g0 μ δ G μ δ due to energy-dependence of g 0 g R μ δ g 0 μ δ δ 0 zero! 0 renormalized coupling g R μ δ f μ δ m G μ δ δ 0 g R (μ ) finite bare mass of the fictitious elementary particle m fic m + f δ δ 0 infinite bare coupling g 0 μ δ f μ δ m fic m fic m δ 0 infinite

13 The condition Z 0 Composite model T v WT G T Y Yukawa model g 0 s m fic g 0 G + +» The composite states can be equivalently represented by a quasi-particle with infinite bare mass and hence with Z 0 [Weinberg(63)]» The elementarity is zero for any composite state by WT term chiral unitary approach [] Jido-Oller-Oset-Ramos-Meissner, NPA75(03)8. [] Inoue-Oset-Vicente Vacas, PRC65(0) [3] Hyodo-Jido-Hosaka, PRC78(08)0503. Λ(405) KN bound state [] Z 0 N(535) KΛ bound state [] (but large qqq? [3] ) Z 0!? 3

14 The condition Z 0 Composite model T v WT G T Y Yukawa model g 0 s m fic g 0 G + +» The composite states can be equivalently represented by a quasi-particle with infinite bare mass and hence with Z 0 [Weinberg(63)]» The elementarity is zero for any composite state by WT term Λ Λ 405 hadronic scale chiral unitary approach [] Jido-Oller-Oset-Ramos-Meissner, NPA75(03)8. [] Inoue-Oset-Vicente Vacas, PRC65(0) [3] Hyodo-Jido-Hosaka, PRC78(08)0503. Λ(405) KN bound state [] Z 0 N(535) KΛ bound state [] (but large qqq? [3] ) Z 0!? Λ N (535): un-natural equivalent to introduce an explicit pole term [Hyodo et al.,prc(08)0503] 4

15 With an explicit pole term Interaction kernel : Weinberg-Tomozawa type + explicit pole term bare mass of fictitious particle m fic v s f (s m ) + f s m s M 0 m M 0 m δ + f M 0 M 0 m δ + f δ 0 M 0 Introduced by by an an un-natural un- natural cut-off cut-off [Hyodo (08)] (08)] Equivalent Yukawa term V Y δ 0 v f M 0 m s m s M 0 renormalized coupling g R s δ 0 finite bare coupling g 0 s δ 0 finite Wave function renormalization constant Z g R /g 0 δ 0 finite If there is an explicit pole term, elementarity Z is finite. 5

16 With an explicit pole term Interaction kernel : Weinberg-Tomozawa type + explicit pole term v s f (s m ) Scattering amplitude T + f s m s M 0 (s) G(s; Λ un nat ) v WT + v pole G s; Λnat v WT Introduced by by an an un-natural un- natural cut-off cut-off [Hyodo (08)] (08)] Z 0 Z 0 Arbitrariness of elementarity Z Physical observables are invariant under the simultaneous change in G and v. Multiple interpretations for a physical state Z can be any value and cannot be determined in a model-independent manner. Necessary to specify a model (cut-off scale to be used as a measure ) 6

17 Representation dependence of Z cf.) ππ scattering in sigma model Yukawa model (fictitious particle or quasi-particle ) V Y s (s m ) m fic m f s m fic quasi-particle dominates Nonlinear model V NL s f s m + s m f s m fic practically zero in m fic + Linear model V L s f m fic m + m fic m f s m fic + V Y s V NL s V L (s) They all have the same s m fic, but Z is different 7

18 Re z Representation dependence of Z.4 Linear model..0 f 9.4 MeV m 38 MeV Re Z L m fic 0!? nonlinear Re Yukawa (quasi-particle) 0. Re z bare mass m fic [MeV] Z Y m fic 0!? Arbitrariness of the value of Z multiple interpretation Each Z indicates the elementarity measured by different elementary particle : the elementary particle in different models are different We need to specify a model used as a measure 8

19 Summary» Wave function renormalization constant Z can be zero for any resonant state dynamically generated by WT type interaction v(s) We have shown that the amplitude can be equivalently represented by a Yukawa model with a quasi-particle having infinite bare mass and hence with Z 0 (with minimal quantum correction) The underlying mechanism is the same as a bound state (constant interaction) case» Model (cut-off & representation) dependence of Z The arbitrariness leads to multiple interpretations for a physical state Among a number of possible models, we have a model with Z 0. Z cannot be determined from experiments in a model-independent manner. Specify firstly : What is an elementary particle to be used as a measure? choice of a model : problem of economization» (A special case of zero-energy bound state) Z 0 does not exclude an elementary state near the physical state Nagahiro-Hosaka, PRC90(4)0659 9