The light Scalar Mesons: σ and κ

Transcription

1 7th HEP Annual Conference Guilin, Oct 29, 2006 The light Scalar Mesons: σ and κ Zhou Zhi-Yong Southeast University

2 1 Background Linear σ model Nonlinear σ model σ particle: appear disappear reappear in P DG Pi Pi s0-channel f 0 (980) πk s c a tte rin g s 1 /2 c h a n n e l phase 300 δ f 0 (600)? s 1/ s 1 /2

3 1 0.8 Im t 0 0 unitarity limit Energy (GeV) There is the broad object seen in ππ scattering, often called background, which extends from about 400 MeV up to about 1700 MeV. This object we consider as a single broad resonance 2 which we identify as the lightest glueball with quantum numbers J P C = we refer to it as red dragon P. Minkowski and W. Ochs, Eur. Phys. J. C9 (1999) 283 H. Leutwyler Bern The lowest resonance of QCD p.8/39

4 The Red Dragon the red dragon I. Caprini, G. Colangelo and H. Leutwyler, Phys. Rev. Lett. 96 (2006) Does QCD have a resonance near threshold? Why care? Concerns the nonperturbative domain of QCD Quark and gluon degrees of freedom useless there Understanding very poor, pattern of energy levels? Lowest resonance: σ? ρ? Resonance pole on second sheet Poles are universal Pole position is unambiguous, even if width is large Where is the pole closest to the origin? as pointed by H.Leutwyler in QCHS VII H. Leutwyler Bern The lowest resonance of QCD p.9/39

5 Comparison with compilation of PDG

6 Different methods in studying ππ physics 1. Chiral Perturbation Theory(without the information of resonance) 2. Phenomenological Analysis(model-dependent, hardly to be trusted in studying broad resonance) 3. Dispersive technique(including Roy equation method, S matrix theory...), modelindependent

7 2 Analytical Continuation pole on second sheet zero on first sheet S 0 0 (s) = η0 0 (s) exp 2iδ0 0 (s) s-plane S 0 0 (s) is analytic in the cut plane 0 4 M π 2 physical region For 0 < s < 4Mπ 2, S0 0 (s) is real S 0 0 (s ) = S 0 0 (s) x in elastic interval: S 0 0 (x ± iɛ) = exp ±2iδ0 0 (x) Second sheet is reached by continuation across the elastic interval of the right hand cut S0 0(x iɛ)ii = S0 0(x + iɛ)i = 1/S0 0 (x iɛ)i Analyticity S 0 0 (s)ii = 1/S 0 0 (s)i valid s Pole in S 0 0 (s)ii zero in S 0 0 (s)i H. Leutwyler Bern The lowest resonance of QCD p.12/39

8 3 PKU Parametrization Form The most important characters in studying low energy physics 1. Unitarity 2. Crossing symmetry 3. Analyticity S phy = S cut S p i, (1) S cut is expressed based on dispersion relation: S cut = exp[2iρ(s)f(s)], (2) i f(s) = f(s 0 ) + (s s 0) disc L f(z) 2πi L(z s)(z s 0 ) dz + (s s 0) Im R f(z) dz. (3) π (z s)(z s 0 ) R

9 It conserves unitarity, The Simplest S Matrices 1 discf = disc{ 2iρ(s) log [ S phy (s) ] }. (4) Im R f(s) = 1 1 log η(s) = 2ρ(s) 4ρ(s) log(s 11S 22 dets ), (5) 1. A virtual state pole at s 0, with s 0 real. S(s) = 1 + iρ(s) s s0 s L s s L s R s 0. (6) 1 iρ(s) s s0 s L s s L s R s 0 As for a bound state at s 0, change sign. s L < s 0 < s R. 2. A resonance poles at z 0 and z 0 on second sheet: S(s) = M 2 (z 0 ) s + iρ(s)sg M 2 (z 0 ) s iρ(s)sg, (7)

10 Some examples of the second sheet poles;in different Im[z 0 ], the relations of M 2 (z 0 ) and Re[z 0 ].

11 Recast Eq.(1) into 1 2iρ(s) log(1 + 2iρ(s)T phy (s)) = 1 log(s R i (s)) + f(s). (8) 2iρ(s) i its behavious when s 0 + : f(0) = 0, (9) u =0 s channel t =0 s =0 u channel t channel In the case of equal-mass, TJ I 1 0 2t (s) = 32π(s 4m 2 dtp J (1 + π) 4m s 4m 2 π s 2 )T I (s, t, u), π u = 4m 2 π s t (10) T I J (0 +) is finite based on Mandelstam analyticity.

12 u =0 s channel t =0 u =(MK + Mπ) 2 t =4M 2 K s =(MK + Mπ) 2 s =0 u channel t channel In the case of unequal-mass, the behavior of T (s, t) in t is described by ReggeT heory. For simplicity, assume that T (s, t) < t n for a certain s and t TJ I (s) = π 2q 2 s 0 4q 2 s dtp J (1 + t 2q s 2 )T I (s, t, u), qs 2 = (m2 K m2 π) 2, (s 0) (11) 4s So lim T I s (s) < dt t n, 32π 2q 2 s 4qs 2 O(s n ) (12)

13 According to the multi-channel unitarity Im{Tl ii (s)} = ρ i Tl ii (s) 2 + ρ n T in (s + )T ni (s ) n i + (3 and more body channels) (13) so Im R f(s) = 1 2ρ(s) log Sphy (s) = 1 4ρ log [ 1 4ρIm R T + 4ρ 2 T (s) 2] = 1 4ρ log [ 1 4ρ( n ρ n T 1n (s) 2 + ) The behavior of third sheet pole is totally different from that of second sheet pole: Δδ η ] (14) s 0.75 contribution of 3-sheet pole s

14 4 ππ Scattering and σ resonance Z.Y. Zhou et al.,[jhep 0502:043,2005] IJ=20 channel There exist a virtual state in s2 partial wave. s GeV 2, its contribution to a 2 0 is fairly large 0.11m 1 π )

15 IJ=11 channel ReT IJ (s) = q 2J [a I J + b I Jq 2 + O(q 4 )], (q = 1 2 s 4m 2 π ), (15) leads to f(4m 2 π) = i a R i. (16) Thus f is subtracted at s = 0 and s = 4m 2 π, f(s) = f(4m2 π) 4m 2 s + s(s 4m2 π) π π a R s i 4m 2 + s(s 4m2 π) i π π. L+R Imf(s ) s (s 4m 2 π)(s s) L+R Imf(s ) s (s 4m 2 π)(s s) ds, (17)

16 IJ=00 channel two resonance: σ and f 0 (980). Crossing symmetry BNR relations: χ 2 tot = χ χ χ χ 2 BNR = 29.7(36) (39) (23) ; M ρ = ± 0.4MeV, Γ ρ = ± 0.6MeV, M σ = 457 ± 15MeV, Γ σ = 551 ± 28MeV, ( ff' ππ οß,& σ Λ(fi 45 MeV ) e0,3 rmcn]sz^h ( MeV ) CCL, PRL96(2006) table 5.1 ο»lffl xc χpt #5 [68] χ (18) p Roy equation m.#5 [71] 39ffν 3 #5[H#» IJ fl 11.3S?"tL Uc5IFgsfi.»,ο BNR *6T T 11 =μfi>'3 8± Our results χpt [68] Roy Eqs. [71] Exp. [62] Unit a ± ± ± ± 0.05 b ± ± ± ± 0.03 m 2 π a ± ± ± ± b ± ± ± ± m 2 π a ± ± ± ± m 2 π b ± ± ± m 4 π ^ 4.1: χhfiωφffwxyeπx6fixz3± IJ=11 u+fπenr Ref. [17, 64]. 7m.ffiχ#5χpip/B:HΩe=-j35I»>N0, σ l53μ tcvm3ο`ψc-jρ M σ = 470 ± 50MeV, Γ σ = 570 ± 50MeV, (4.28)

17 5 πk Scattering and κ resonance s-plane Kπ singularity a) s-channel unitarity cut s (M K + M π ) 2 b) u-channel unitarity cut u (M K + M π ) 2, leads to s-plane < s (M K M π ) 2 c) t-channel unitarity cut t 4(M π ) 2,leads to i circular cut s = (M 2 K M 2 π)exp(iφ) ii < s 0 d) singularity at s const χ' Kπ οß& κ Λ(fi 53 s plane C o C i Inelastic threshold L 1 L 2 (m K m π) 2 (mk + m π) 2 R m 2 K m2 π Π 5.1: πk 3C@ C»j CCA@ C±

18 By measuring process K π +, LASS Collaboration provide a combined data of a 0 and φ 0 : A 0 = a 0 e iφ 0 = T 1/ T 3/2 0 = 1 2i (η1/2 0 e 2iδ1/2 0 1) + 1 4i (η3/2 0 e 2iδ3/2 0 1), (19)

19 Our result:m σ = 694 ± 53MeV, Γ σ = 606 ± 59MeV [Z. Y. Zhou, H. Q. Zheng, [Nucl. Phy. A775 (2006) ] Roy-steiner relation(an analogue of Roy equation) find the lowest I=1/2, l=0 resonance sits at M σ = 658 ± 13MeV, Γ σ = 557 ± 24MeV [S. Descotes-Genon, B. Moussallam, hep-ph/ ] Calculation confirms our result.

20 QCHS VII, Leutwyler

21 6 Conclusion 1. Including the contributions of left hand cut will stablize the low-lying pole positions in fit. 2. By a model independent approach, the existence of σ and κ is confirmed and determined accurately.

22 Thank you! Address: E_mail: Tel: Institute of Structural Mechanics, CAEP, P.O.Box , Mianyang, Sichuan, China, & (O), (H), (M) Fax: