Meson baryon components in the states of the baryon decuplet

Transcription

1 Meson baryon components n the states of the baryon decuplet Insttuto de Físca Corpuscular, Unversdad de Valenca - CSIC Oct 1st, 2013 Collaborators: L. R. Da, L. S. Geng, E. Oset and Y. Zhang

2 Outlne ntroducton bref overvew of the formalsm nterpretaton of the sumrule applcaton to the J P = 3/2 + baryon decuplet results and conclusons

3 Resonances: composte states of hadrons or genune states? S. Wenberg, Phys. Rev. 137, B672-B678 (1965): - s-waves and small bndng energes - appled to the deuteron C. Hanhart, Y..S. Kalashnkova and A. V. Nefedev, Phys. Rev. D 81, (2010) - s-waves D. Gamermann, J. Neves, E. Oset and E. Ruz Arrola, Phys. Rev. D 81, (2010): - bound states n coupled channels (stll n s-waves) - bgger bndng energes J. Yamagata-Sekhara, J. Neves and E. Oset, Phys. Rev. D 83, (2011): - also resonances are consdered F. A. and E. Oset, Phys. Rev. D 86, (2012): - generalzaton to any partal waves, for both bound states and resonances - appled to the ρ and K mesons

4 Overvew of the formalsm We start from the potental: p V p = (2l + 1) v Θ(Λ p)θ(λ p ) p l p l P l (cos θ) Λ: cutoff n the momentum space v: N N matrx N: number of channels l-wave character of the process p l, p l, P l(cos θ) v s a constant matrx From the Lppmann-Schwnger equaton: T = (2l + 1)P l (ˆp, ˆp )Θ(Λ p)θ(λ p ) p l p l t where: t = v[1 vg] 1 does not contan the factors p l

5 Overvew of the formalsm G = p<λ d 3 p p 2l E m 1 m 2 p2 2µ modfed to contan the p 2l factor ths choce s necessary to get the generalzed sumrule g 2 [ ] dg = 1 de E=E P - g : couplng to the channel g g j = lm E E P (E E P)t j - E P: poston of the pole t holds for any partal waves t holds for both bound states and resonances [F. A. and E. Oset, Phys. Rev. D 86, (2012)] the nterpretaton of the sumrule s dfferent for bound states and resonances

6 Interpretaton of the sumrule the case of bound states the sumrule follows drectly from the normalzaton of the wave functon of the state, snce [ ] g 2 dg de = d 3 p Ψ (p) 2 d 3 p Ψ (p) 2 = 1 Z wth Z = d 3 p Ψ β (p) 2 E=E P each term s nterpreted as the probablty to fnd a certan channel n the wave functon Ψ β (p) : genune component the case of resonances - they show themselves as complex poles of the T -matrx, n the 2nd Remann sheet for open channels ( [ ] ) Re g 2 dg II de = 1 Z wth Z = Re d 3 p(ψ β(p)) 2 E=E P - nterpretaton n terms of probablty no more possble

7 Interpretaton of the sumrule related to the fact that the egenstates of a complex H are not generally orthogonal we need a borthogonal bass The relatonshp we have to use to derve the sumrule s d 3 p( Ψ (p)) Ψ (p) = d 3 pψ 2 (p) = 1 Ψ Ψ = [n the case of bound states we had Ψ Ψ = d 3 p Ψ (p) 2 ]

8 Interpretaton of the sumrule Snce the wave functon s gven by Ψ (p) = g Θ(Λ p)p E m 1 m 2 p 2 /2µ [F. A. and E. Oset, Phys. Rev. D 86, (2012)] d 3 p(ψ (p)) 2 = g 2 dg II de whch means g 2 dg II de = 1 g 2 dg II de extrapolaton of the probablty to the complex plane but t s not a probablty we can thnk of t as the weght of one gven channel n the wave functon

9 Sumrule and wave functons d 3 p(ψ (p)) 2 : can gve us nformaton about the wave functon n coordnate space snce Ψ ( r) = d 3 p (2π) 3/2 e p r Ψ ( p) and d 3 pψ ( p) 2 = d 3 rψ ( r) 2 for an open channel n the lmt r Ψ (r) 1 r e 2µ E R r e 2µ E Γ R r 4E R - E R = Re(E P ) - Γ = 2Im(E P ) and n the 2nd Remann sheet Ψ II (r) 1 r e 2µ E R r 2µ E e R Γ r 4E R = the concept of probablty s useless when we have open channels However, (Ψ II (r)) 2 oscllatory behavor d 3 r(ψ II (r)) 2 0 when r

10 πn component n the (1232) (model dependent test) We use v = α M 4 ( ) β 1 + sr s CDD pole not dependent on the momenta G(s) = d 3 q M N q 2 (2π) 3 2ω(q)E N (q) s ω(q) EN (q) + ɛ relatvstc modfed to contan the factor q 2 (p-waves) regularzed by a cutoff q max The phaseshft s gven by: T = p 2 t = 4π s M N 1 p cot δ(p) p From the best ft to the πn data we get: α = MeV sr = MeV β = MeV q max = MeV

11 πn component n the (1232) (model dependent test) In order to apply the sumrule we have to extrapolate the ampltude to the complex plane: s a + b go to the second Remann sheet: G(s) G II (s) = G I (s) + M N p3 2π s look for the complex pole s 0 correspondng to the resonance evaluate the couplngs as the resdues at the pole of t II wth Im(p) > 0 We fnd: s0 = ( ) MeV g = ( ) 10 3 MeV 1 [Re( s 0) 1210 MeV, Im( s 0) 50 MeV [PDG]] From where we get: g 2 [ dg II d s ] s0 = ( ) 1 Z = 0.62 szeable amount of πn

12 πn component n the (1232) (phenomenologcal test) We make an estmate of the same quanttes usng a phenomenologcal test g relatvstc ampltude: t = 2 s M + Γon ( ) p 3 wth g 2 = 2πM Γon ponm N 2 pon we have to go to the 2nd Remann sheet: s a + b and p p we evaluate agan the complex pole of t II and the new couplngs as the resdue at the pole We fnd: s0 = ( ) MeV g = ( ) 10 3 MeV 1 [Re( s 0) 1210 MeV, Im( s 0) 50 MeV [PDG]] q max [GeV ] 2 II dg g d s 1 Z rather stable and consstent wth the prevous results

13 The J P = 3/2 + baryon decuplet We apply the phenomenologcal test to the other partcles of the decuplet the πλ and πσ content n the Σ(1385): the branchng ratos to the dfferent channels must be taken nto account: BR πλ = 87% BR πσ = 11.7% [J. Bernger et al. [Partcle Data Group Collaboraton], Phys. Rev. D 86, (2012).] g 2 Σ, = 2πM Σ Γon p 3 () on M BR = πλ, πσ the πξ content n the Ξ(1535): we proceed n complete analogy wth the (1232) snce BR πξ = 100% [J. Bernger et al. [Partcle Data Group Collaboraton], Phys. Rev. D 86, (2012).]

14 The J P = 3/2 + baryon decuplet the KΞ content n the Ω : - the Ω s stable to strong decays and ths prevents us from evaluatng the couplngs as before - the couplng g Ω, KΞ can be related to g,πn usng SU(3) symmetry and we fnd: - the pole of the ampltude s found on the real axs g 2 Ω, KΞ = 2 g 2,πN The results we fnd are: Σ(1385) Channel πλ πσ s0 [MeV ] g [MeV ] 1 g 2 GII E 1 Z ( ) 10 3 ( ) ( ) 10 3 ( ) Ξ(1535) πξ ( ) Ω KΞ ( ) [for dg II /d s we used q max 450 MeV ]

15 The ρ and K mesons the ππ content n the ρ [F. A. and E. Oset, Phys. Rev. D 86, (2012)] Model dependent test s0 = ( ) MeV g ρ = ( ) 1 Z = Phenomenologcal test s0 = ( ) MeV g ρ = ( ) q max [GeV ] 1 Z the Kπ content n the K [C. W. Xao, F. A. and M. Bayar, Eur. Phys. J. A 49, 22 (2013)] Model dependent test s0 = ( ) MeV g ρ = ( ) 1 Z = 0.12 Phenomenologcal test s0 = ( ) MeV g ρ = ( ) q max [GeV ] 1 Z

16 Conclusons we clarfed the meanng of extendng the Wenberg sumrule to the case of resonances - d 3 p( p Ψ ) 2 measures the weght of an open channel n the wave functon - the concept of probablty s no longer useful n the case of resonances - the sum of d 3 p( p Ψ ) 2 for dfferent coupled channels s unty and ths s the generalzed sumrule for the amount of πn n the (1232) we found 60% for the other members of the decuplet we found - decreasng sze of meson baryon component at hgher energy Σ and Ξ better represented by a genune component - an amount of the bound KΞ component n the Ω of 25%

17 Meson baryon components n the states of the baryon decuplet Insttuto de Físca Corpuscular, Unversdad de Valenca - CSIC Oct 1st, 2013 Collaborators: L. R. Da, L. S. Geng, E. Oset and Y. Zhang