On the two-pole interpretation of HADES data

J-PARC collaboration September 3-5, 03 On the two-pole interpretation of HADES data Yoshinori AKAISHI

nucl U K MeV 0-50 Σ+ Λ+ -00 K - + p 3r fm Λ(405) E Γ K = -7 MeV = 40 MeV nucl U K MeV 0-50 Σ+ Λ+ -00 K - + pp 3r fm E K Γ KH = -48 MeV = 6MeV 0-50 Σ+ Λ+ nucl U K MeV -00 K - + 3 He 3r fm 3 KH E = -08 MeV K Γ = 0 MeV Shrinkage! -300-300 -300-400N.V. Shevchenko, A. Gal -400 & J. Mares, Phys. Rev. Lett. 98-400 (007) 0830 E = -55~-70 MeV, Γ = 90~0 MeV Y. Ikeda & T. Sato, Phys. Rev. C 76 (007) 03503 E = -80 MeV, Γ = 73 MeV -500-500 -500 DAΦNE Conf. (999) Y. Akaishi & T. Yamazaki, Phys. Rev. C 65 (00) 044005 T. Yamazaki & Y. Akaishi, Phys. Lett. B 535 (00) 70

Last year! J. Esmaili, Y. Akaishi & T. Yamazaki, Phys. Lett. B 686 (00) 3

K bar N scattering amplitude T. Hyodo and W. Weise, Phys. Rev. C 77 (008) 03504 Akaishi-Yamazaki Chiral s 405 40 Which is the Λ(405) mass?

Double pole structure of Λ(405) D. Jido, J.A. Oller, E. Oset, A. Ramos & U.G. Meissner, Nucl. Phys. A 75 (003) 8 46 Channel = K - + p 0 Λ + η E = -6 MeV, Γ = 3 MeV 3 MeV D. Jido et al., Nucl. Phys. A 835 (00) 59 E = -4 MeV, Γ = 3 MeV Σ Σ 390 ~60 MeV -00 MeV Channel Σ + + - g W M R Σ R g R Σ iγ R R R gσ gσ + / W MR iγr / Breit-Wigner amplitudes

High Acceptance Di-Electron Spectrometer Λ(405) from HADES@GSI G. Agakishiev et al., Phys. Rev. C 87 (03) 050 3.5 GeV The peak structure of Λ(405) appears below 400 MeV/c!

Λ(405) shift in HADES data J. Siebenson & L. Fabbietti, arxiv:306.583v [nucl-ex] T iφ Λ( 405) = Cp.s. BW m) e + BW( m) BW ( m) i ( q = Ai m mi Fitting constraint + im Γ i i c.m. Assume a maximal interference between the Λ(405) and non-resonant contributions. (3.3 μb / total 9 μb) (Main part is non-resonant.)

Interpretation of the Λ(405) shift in HADES data J. Siebenson & L. Fabbietti, arxiv:306.583v [nucl-ex] T iφ Λ( 405) = Cp.s. BW m) e + BW( m) BW ( m) i ( q = Ai m mi + im Γ i i c.m. Best fit

T iφ Λ( 405) = Cp.s. BW m) e + BW( m) ( q c.m. Geng-Oset z =46-i4 MeV z =375-i73 MeV φ =05 deg. Best fit z =48-i9 MeV z =375-i73 MeV φ =78 deg. 30 340 360 380 400 40 440 460 480 30 340 360 380 400 40 440 460 480 Mai-Meissner z =47-i0 MeV z =490-i84 MeV φ =64 deg. Not compatible with HADES "z dominance" Physical property of poles (Ikeda-Hyodo-Weise) z =44-i6 MeV z =38-i8 MeV φ =78 deg. "z dominance" 30 340 360 380 400 40 440 460 480 30 340 360 380 400 40 440 460 480

Author's conclusions Possible explanations of the Λ(405) mass shift in HADES : Peak structure below 400 MeV/c Interference between resonant and non-resonant contributions Coherent sum of two pole contributions with a rather large contribution from the low-mass broad pole Questions Origins of the resonance poles : Feshbach resonance ; pole on [+,-] sheet > K bar N threshold effect Strongly energy-dependent chiral interaction ; decaying-state moving poles > non Breit-Wigner amplitude What physics is obtained from the "chiral HADES" analysis?

Peak position of the st pole in M Σ spectrum T k Hyodo-Weise s two-channel model 43-7i, 398-73i st pole nd pole ch.-cut Peak Observation on [+, +] sheet KN threshold 400 40 40 430 440 s / MeV ch.- & ch.-cuts The peak position differs from the pole position. K bar N threshold effect st pole on [ +, - ] sheet physical, unphysical L405/Zychor.f m B c =4000 MeV

Chiral SU(3) dynamics and the nd pole Weinberg-Tomozawa term I = 0 c ij c ij 3 = ω + ω 3 i 4f 4 j 3 KN Σ ω s i M i Energy dependent interaction with a large positive imaginary part (source term) Im ω 0-00 MeV Σ channel 00 MeV Re ω nd pole T = V VG VG = 0 V = ; WT { V ij } Total 0 channels G T = V + V T T = V G Resonance (bound) states nd pole st pole Full ch. 400-i76 48-i7 ch. 398-i73 43-i7 ch. 388-i96 Hyodo-Weise

Σ-Σ invariant-mass spectrum Hyodo-Weise's chiral SU(3) dynamics U = [ T = U + UGT ] U = T + G T ( Ξ ) U( Ξ ) G( Ξ ) T Shift due to U(E pole ) - U(E) nd pole peak = T ( Ξ pole) U( Ξ pole) G( Ξ pole) = 0 = T ( E) U( Ξ pole) G( E) 09 + i 73 f = Breit-Wigner T ( E) U( E) G( E) MeV no longer BW Moving pole of decaying state, PJA B84 (008) 64 nd pole position 300 350 400 450 500 550 600 st pole position M Σ [MeV/c ]

Σ invariant mass spectrum.5.0.5 Scattering amplitude TΣ, Σ( s ) q -Imω source 96.4 [MeV].0 90.0 0.5 80.0 0.0 70.0 Born 50.0 30.0 0.0 0.0-0.5 -.0 300 30 340 360 380 400 40 440 Hyodo-Weise Fig. 300 30 340 360 380 400 40 440 460 s / [MeV]

Moving pole Σ single channel Re E -50 MeV -00-50 0 50 00 Virtual states Observation T ( E ; V ( E)) Reference spectra T ( z ; V ( Z T ( z ; V ( E ref )) ω = m + Z Observed spectrum obs )) at z = E obs ref Dispersion relation x' Im f ( x') Re f ( x) = P dx' x' x Cross section Optical theorem Im f(e) 0 Re f(e) -00 + i 96.4 MeV source T ( Z ; V ( Z)) Difference in dynamics Weinberg-Tomozawa Reω i Imω F f(z) Analytic continuation + i 0-00 MeV Pole + i 40 + i 60 + i 80 Im E Wpole/poleScut5.f

Σ mass spectrum ; KN ; Σ U opt = U + U.0 G U G f U U Σ + Σ + _ K + N (KN)* Σ + QBS. Decaying-st. pole. Threshold cusp 3. Interference 0.8 0.6 f = 0.5 Escape U U G Uf G -.0 -.0 0.0 0. 0.4 0.5 370 400 430 MeV Re S / f =.0 0.9 0.8-50 MeV 330 350 370 390 40 430 450 470 490 M Σ [MeV/c ] Σ threshold KN threshold 0.9.0 0.8 Im E Superposition of two B-W amplitudes is not a proper explanation.

Im F KN [fm] 4 0 Missing mass spectrum Im T = 4 μ h T KN >Σ invariant mass spectrum q Chiral SU(3) dynamics Two poles 4 05 Σ >Σ invariant mass spectrum

+ + Σ + (660) Σ Λ* T Σ,Σ Σ Σ + (660) Λ* T Σ,NK Σ + T 405 MeV A-Y R.J. Hemingway, Nucl. Phys. B53 (985) 74 Σ + (660) T K - + p C Σ T Σ, q + Σ Interference C' T Σ,NK q Wpole/SpiInv.f 330 380 430 480 530 M Σ [MeV/c ]

Hassanvand et al.'s analysis of HADES data Feshbach resonance on Riemann's [ +, - ] sheet T spectrum and T spectrum K bar N threshold effect on T spectrum Moving pole effect on T spectrum Resonance-continuum interference effect

_ Observables of KN-Σ coupled system Hyodo-Weise's chiral SU(3) dynamics Im T KN missing mass spectrum C T k C T k Arbitrary unit : Σ : KN 40 KN >Σ invariant mass spectrum Two kinds of M inv spectra Σ "Conversion Σ" : T "Scattering Σ" : T 405 Σ >Σ invariant mass spectrum Escape of K Conversion 330 350 370 390 40 430 450 470 490 s / [MeV]

Pole position vs. peak position T k without interference Pole position K - p threshold Γ [MeV] 70 60 50 40 30 0 0 Peak position shifts from the pole position due to K - p threshold effect, when width becomes wide. 30 340 360 380 400 40 440 Carla/HEMING.f M(Σ) [MeV/c ]

Pole position vs. peak position T k with interference Pole position K - p threshold Γ [MeV] 70 60 50 40 30 0 0 See Fig.8 of Hassanvand. Further (small) change of spectrum comes from interference with I=0 L=0 Σ continuum. 30 340 360 380 400 40 440 Carla/HEMING.f M(Σ) [MeV/c ]

Pole position vs. peak position T k with interference Pole position K - p threshold Γ [MeV] 70 60 50 40 30 0 0 Further (large) change of spectrum comes from interference with I=0 L=0 Σ continuum. 30 340 360 380 400 40 440 Carla/HEMING.f M(Σ) [MeV/c ]

Λ(405) from HADES G. Agakishiev et al., Phys. Rev. C 87 (03) 050 M. Hassanvand et al., Phys. Rev. C 87 (03) 0550 PDG 4

Pole position vs. peak position Pole position from pp collision @ 3.5 GeV T k Our best fit M = 405 + -9 MeV/c Γ = 6 +0-0 MeV Γ [MeV] 70 60 50 40 30 0 0 Peak position at Γ =50 MeV The peak position shifts from the pole position. K bar N threshold effect K - p threshold 30 340 360 380 400 40 440 M(Σ) [MeV/c ] Carla/HADES.f

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