Nature of the sigma meson as revealed by its softening process Tetsuo Hyodo a, Daisuke Jido b, and Teiji Kunihiro c Tokyo Institute of Technology a YITP, Kyoto b Kyoto Univ. c supported by Global Center of Excellence Program Nanoscience and Quantum Physics 202, Sep. 5th
Introduction The sigma meson f0() f0() or σ J P = 0 +, I = 0 Mass : - -550 MeV Width : -000-700 MeV The sigma meson - is the lowest resonance in QCD - plays an important role in hadron mass generation due to spontaneous chiral symmetry breaking - provides attraction in phenomenological nuclear force Developments in scattering theory + accurate data --> determination of pole position is now possible. I. Caprini, G. Colangelo, H. Leutwyler, Phys. Rev. Lett. 96, 3 (6),... 2
Introduction Structure of the sigma meson Sigma meson in naive constituent quark model (qq ) has some difficulties: light mass (v.s. p-wave excitation), mass ordering of scalar nonet (v.s. σ > κ > f0 ~ a0) Alternative descriptions of the sigma meson - Chiral sigma (e.g. linear sigma model) σ chiral π M. Gell-Mann, M. Levy, Nuovo Cim. 6, 705 (960),... - Dynamical sigma (e.g. mesonic molecule generated by ππ attraction) J.I. Basdevant, B.W. Lee, Phys. Rev. D2, 680 (970),... π π - CDD pole contribution (pre-formed state) (e.g. diquark-antidiquark model, glueball,...) We want to clarify the structure <-- softening 3
Introduction Softening of chiral sigma T. Hatsuda, T. Kunihiro, H. Shimizu Phys. Rev. Lett. 82, 2840 (999) Softening of the sigma meson Partial restoration of chiral sym. --> Spectral enhancement in I=J=0 channel near threshold Threshold enhancement of ππ cross section, also for the dynamical sigma meson D. Jido, T. Hatsuda, T. Kunihiro, Phys. Rev. D63, 090 () 4
Introduction Mechanism of the softening (chiral sigma) In the previous studies, it seems that the softening takes place, irrespective to the structure of the sigma meson. Mechanism of the softening? Softening of the chiral sigma (linear sigma model) Sigma meson: bare sigma pole acquires finite width through the coupling to ππ ππ bare ππ Chiral symmetry restoration: --> lowering bare sigma mass --> reduction of the phase space --> narrow peak in spectrum im im im ππ re re re 5
Introduction Softening of the dynamical sigma (ChPT + unitarization) Sigma meson: dynamically generated by ππ attraction Chiral symmetry restoration: --> fπ ~ <σ> decreases --> (attractive) interaction (fπ) -2 increases --> resonance turns into bound state, peak gets narrow Special nature of the s-wave resonance: attractive Mechanism of the softening (dynamical sigma) resonance virtual state pole on the 2nd Riemann sheet below threshold ex.) spin singlet NN bound state --> novel softening pattern? im virtual state (II) bound state (I) ππ re resonance (II) 6
Model setup Tree level interaction Lagrangian of 2-flavor linear sigma model L = 4 Tr M M µ 2 MM 2 4! (MM ) 2 + h(m + M ), M = + i 3 parameters <-- mπ, mσ, <σ> at mean field level. ππ scattering amplitude in general (crossing symmetry) T tree (s, t, u) =A(s, t, u) ab cd + A(t, s, u) ac bd + A(u, t, s) ad bc Tree-level ππ scattering amplitude A(s, t, u) = m2 m 2 2 (m 2 m 2 ) 2 2 s m 2 + = s m2 2 (s m 2 ) 2 2 s m 2 leading order term of ChPT: 7
Model setup Unitarization Unitarity of S-matrix: conservation of probability. Tree-level amplitude violates unitarity at certain energy. Optical theorem: Im T (s) = (s) 2 for s>4m 2, (s) = 6 4m 2 Scattering amplitude (N/D method + matching with Ttree) J. A. Oller, E. Oset, Phys. Rev. D60, 074023 (999) T (s; x) = Ttree(s; x) +G(s) s G(s) = 2 (4 ) 2 a(µ)+ln m2 µ 2 + 4m 2 s ln 4m 2 s + 4m 2 s Subtraction constant: to exclude the CDD pole G(s) = 0 at s = m 2, a(m )= 3 T. Hyodo, D. Jido, A. Hosaka, Phys. Rev. C78, 025203 (8) 8
Model setup Amplitude in vacuum Numerical result in vacuum Input: mπ = 40 MeV, mσ = 550 MeV, <σ> = 93 MeV sigma origin I=0 scattering length (mπ) - pole position [MeV] chiral 44 423-26 i dynamical 74 364-356 i (experiment) 22 [] 44-272 i [2] [] J.R. Batley et al., Eur. Phys. J. C54, 4 (8) [2] I. Caprini, G. Colangelo, H. Leutwyler, Phys. Rev. Lett. 96, 3 (6) (we do not aim at fine-tuning of the parameters) 9
Chiral symmetry restoration Prescription for symmetry restoration We introduce the effect of chiral symmetry restoration from the outside of the model, by modifying mπ, mσ, <σ>. ) chiral condensate (pion decay constant): decreases = 0, 0 2) mass of pion: no change m = const. 3) mass of chiral sigma for : decreases m = m, m = 2 0 3 with λ and mπ being fixed. + m 2 Symmetry restoration is modeled by the change of Φ. 0
Numerical analysis spectrum Softening in 0-00 Re 0 Im - - - 00 Re = pole position - Linear sigma model + unitarization : chiral sigma - Softening takes place, as expected. - peak at threshold : Φ ~ 0.6 <=> bare sigma pole moves below the threshold
Numerical analysis Softening in spectrum Re 0 Im 0-00 - - - 00 Re pole position = - ChPT + unitarization : dynamical sigma - Softening takes place, but virtual state appears. - at Re[Mpole] = 2mπ (Φ ~ 0.6), due to finite width, spectrum does not show the peak structure - peak at threshold : Φ ~ <=> formation of bound state 2
Numerical analysis Comparison of and = = 0.9 = 0.8 = 0.7 = 0.6 = = = = = - Strong threshold enhancement : different from each other. - Shape of the spectrum? T 2 /s chiral sigma s /2 T 2 /s dynamical sigma s /2 3
Summary Summary We study the structure of the sigma meson using chiral symmetry restoration. Dynamical scattering models with (i) chiral sigma (ii) dynamical sigma Dynamical sigma softens qualitatively differently from chiral sigma. <-- virtual state (s-wave resonance) T. Hyodo, D. Jido, T. Kunihiro, Nucl. Phys. A848, 34-365 (200). 4