ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES MODEL INDEPENDENT - PowerPoint PPT Presentation

INTRODUCTION PION SCALAR FORM FACTOR ANALYSIS OF S 0 0 ππ PHASE SHIFT DATA PION SCALAR FF PHASE REPRESENTATION PARAMETERS OF f 0 (500) AND f 0 (980) CONCLUSIONS ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES MODEL INDEPENDENT PARAMETERS OF f 0 (500) and f 0 (980) MESONS. S.Dubnicka,Anna Z. Dubnickova,A.Liptaj Institute of Physics, Slovak Academy of Sciences, Bratislava and Department of Theoretical Physics, Comenius University, Bratislava, Slovak Republic May 26, 2014 MESON’14, Krakow, 29 May - 3 June, 2014 S.Dubnicka,Anna Z. Dubnickova,A.Liptaj ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES

INTRODUCTION PION SCALAR FORM FACTOR ANALYSIS OF S 0 0 ππ PHASE SHIFT DATA PION SCALAR FF PHASE REPRESENTATION PARAMETERS OF f 0 (500) AND f 0 (980) CONCLUSIONS Outline INTRODUCTION 1 PION SCALAR FORM FACTOR 2 ANALYSIS OF S 0 0 ππ PHASE SHIFT DATA 3 PION SCALAR FF PHASE REPRESENTATION 4 PARAMETERS OF f 0 (500) AND f 0 (980) 5 CONCLUSIONS 6 S.Dubnicka,Anna Z. Dubnickova,A.Liptaj ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES

INTRODUCTION PION SCALAR FORM FACTOR ANALYSIS OF S 0 0 ππ PHASE SHIFT DATA PION SCALAR FF PHASE REPRESENTATION PARAMETERS OF f 0 (500) AND f 0 (980) CONCLUSIONS In contrast to other SU (3) known multiplets of hadrons, the identification of the scalar mesons - long-standing puzzle. Despite of this fact, all experimentally established scalar mesons are now classified: into light scalar nonet comprising the f 0 (500) , K ∗ 0 (800) , f 0 (980) and a 0 (980) mesons - not necessarily to be q ¯ q states into regular nonet consisting of the f 0 (1370) , K ∗ 0 (1430) , a 0 (1450) and f 0 (1500) (or f 0 (1700)) mesons. The f 0 (500) or sigma-meson is the lightest hadronic resonance with vacuum quantum numbers 0 ++ - to be identical with glueballs . S.Dubnicka,Anna Z. Dubnickova,A.Liptaj ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES

INTRODUCTION PION SCALAR FORM FACTOR ANALYSIS OF S 0 0 ππ PHASE SHIFT DATA PION SCALAR FF PHASE REPRESENTATION PARAMETERS OF f 0 (500) AND f 0 (980) CONCLUSIONS In the past σ − meson has been: - listed in PDG as ”not well established” until 1974 - removed from PDG in 1976 - listed back in 1996 , after missing more than two decades, although still with an obscure denotation f 0 (400 − 1200) - from 2002 as ”well established” f 0 (600), but with conservative estimate of the mass: 400 − 1200 MeV and width : 600 − 1000 MeV A clarification of this controversial situation has been achieved in the papers I.Caprini, G.Colangelo, H.Leutwyler: Phys. Rev. Lett. 96 (2006) 132001 R.Garcia-Martin, R.Kaminski, J.R.Pelaez, J.Ruiz de Elvira: Phys. Rev. Lett 107 (2011) 072001 S.Dubnicka,Anna Z. Dubnickova,A.Liptaj ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES

INTRODUCTION PION SCALAR FORM FACTOR ANALYSIS OF S 0 0 ππ PHASE SHIFT DATA PION SCALAR FF PHASE REPRESENTATION PARAMETERS OF f 0 (500) AND f 0 (980) CONCLUSIONS In this presentation we confirm an existence of f 0 (500) by the pion scalar FF analysis. As for the latter a representation of the pion scalar form factor is valid in the whole elastic region up to 1 GeV 2 , one can determine also the f 0 (980) meson parameters as well. S.Dubnicka,Anna Z. Dubnickova,A.Liptaj ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES

INTRODUCTION PION SCALAR FORM FACTOR ANALYSIS OF S 0 0 ππ PHASE SHIFT DATA PION SCALAR FF PHASE REPRESENTATION PARAMETERS OF f 0 (500) AND f 0 (980) CONCLUSIONS The pion scalar FF Γ π ( t ) is defined by the matrix element of the quark density uu + ¯ < π i ( p 2 ) | � dd ) | π j ( p 1 ) > = δ ij Γ π ( t ) m (¯ t = ( p 2 − p 1 ) 2 and m = 1 where � 2 ( m u + m d ). NOTE : Γ π ( t ) is not directly measurable quantity and it enters e.g. in the matrix element for the decay of the Higgs particle into two pions. S.Dubnicka,Anna Z. Dubnickova,A.Liptaj ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES

INTRODUCTION PION SCALAR FORM FACTOR ANALYSIS OF S 0 0 ππ PHASE SHIFT DATA PION SCALAR FF PHASE REPRESENTATION PARAMETERS OF f 0 (500) AND f 0 (980) CONCLUSIONS Properties of Γ π ( t ): it is analytic in the whole complex t -plane besides for a cut along the positive real axis starting at t = 4 m 2 π for real values t < 4 m 2 Γ π ( t ) is real π ⇒ it implies the so-called reality condition Γ ∗ π ( t ) = Γ π ( t ∗ ) at t = 0 Γ π ( t ) coincides with the pion sigma term Γ π (0) = (0 . 99 ± 0 . 02) m 2 π , however, in our considerations we normalize it exactly to m 2 π , i.e. Γ π (0) = 1 S.Dubnicka,Anna Z. Dubnickova,A.Liptaj ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES

INTRODUCTION PION SCALAR FORM FACTOR ANALYSIS OF S 0 0 ππ PHASE SHIFT DATA PION SCALAR FF PHASE REPRESENTATION PARAMETERS OF f 0 (500) AND f 0 (980) CONCLUSIONS if Γ π ( t ) is evaluated on the upper boundary of the cut ⇒ the unitarity condition is obeyed � uu + ¯ < π ( p ′ ) π ( p ) | T | n >< n | � Im Γ π ( t ) = m (¯ dd ) | 0 > n where the sum runs over a complete set of allowed states like 2 π, 4 π, ... K ¯ K , etc., which create additional branch cuts on the positive real axis of the t -plane between 4 m 2 π and ∞ . in the elastic region 4 m 2 π ≤ t ≤ 16 m 2 π only the first term in the unitarity condition contributes , then ∗ Im Γ π ( t ) = Γ π M 0 0 where M 0 0 is I = J = 0 ππ scattering amplitude 0 = e i δ 0 M 0 0 sin δ 0 0 ; with δ 0 0 the S -wave isoscalar ππ phase shift . S.Dubnicka,Anna Z. Dubnickova,A.Liptaj ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES

INTRODUCTION PION SCALAR FORM FACTOR ANALYSIS OF S 0 0 ππ PHASE SHIFT DATA PION SCALAR FF PHASE REPRESENTATION PARAMETERS OF f 0 (500) AND f 0 (980) CONCLUSIONS ⇒ the elastic unitarity condition is Im Γ π ( t ) = Γ π e − i δ 0 0 sin δ 0 0 from where the identity δ Γ ≡ δ 0 0 follows, where δ Γ is phase of the pion scalar FF . NOTE : However, the phenomenological analysis of the ππ interactions reveals that this identity is valid well above t = 4 m 2 K ≈ 1 GeV 2 , where the inelastic two-body channel ππ → K ¯ K is opened The asymptotic behavior of Γ π ( t ) is predicted to be Γ π ( t ) | t |→∞ ∼ 1 / t . S.Dubnicka,Anna Z. Dubnickova,A.Liptaj ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES

INTRODUCTION PION SCALAR FORM FACTOR ANALYSIS OF S 0 0 ππ PHASE SHIFT DATA PION SCALAR FF PHASE REPRESENTATION PARAMETERS OF f 0 (500) AND f 0 (980) CONCLUSIONS Starting from S -wave isoscalar ππ scattering amplitude unitarity condition ImM 0 0 = − | M 0 0 | 2 one can do analytic continuation of M 0 0 through the upper and lower boundaries of the elastic unitary cut and to come to M 0 I M 0 II = 0 0 1 − 2 iM 0 I 0 which reveals the singularity at t = 4 m 2 π to be a square root branch point the same is valid for Γ π ( t ) Γ I Γ II π = π 1 − 2 iM 0 I 0 Moreover, they have identical denominators! S.Dubnicka,Anna Z. Dubnickova,A.Liptaj ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES

INTRODUCTION PION SCALAR FORM FACTOR ANALYSIS OF S 0 0 ππ PHASE SHIFT DATA PION SCALAR FF PHASE REPRESENTATION PARAMETERS OF f 0 (500) AND f 0 (980) CONCLUSIONS NOTE: From this identity of denominators it follows: If f 0 (500) and f 0 (980) resonances appear as poles on the II. Riemann sheet of M 0 0 ⇒ they have to appear also as the poles on the II. Riemann sheet of Γ π ( t ) . For the model independent identification of these poles we use phase representation of Γ π ( t ) . Now, by an application of the conformal mapping q = [( t − 4) / 4] 1 / 2 , m π = 1 (1) two-sheeted Riemann surface of Γ π ( t ) is mapped into one absolute valued pion c.m. three-momentum q -plane and the elastic cut disappears. S.Dubnicka,Anna Z. Dubnickova,A.Liptaj ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES

INTRODUCTION PION SCALAR FORM FACTOR ANALYSIS OF S 0 0 ππ PHASE SHIFT DATA PION SCALAR FF PHASE REPRESENTATION PARAMETERS OF f 0 (500) AND f 0 (980) CONCLUSIONS Neglecting all higher branch points , there are only poles and zeros of Γ π ( t ) in q -plane ⇒ Γ π ( t ) can be represented by a Pad‘e-type approximation � M n =0 a n q n Γ π ( t ) = . � N i =1 ( q − q i ) Because Γ π ( t ) is a real analytic function , ⇒ coefficients a n with M even (odd) real (pure imaginary). The poles q i can appear on the imaginary axis or they are placed always two of them symmetrically according to it. S.Dubnicka,Anna Z. Dubnickova,A.Liptaj ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES

INTRODUCTION PION SCALAR FORM FACTOR ANALYSIS OF S 0 0 ππ PHASE SHIFT DATA PION SCALAR FF PHASE REPRESENTATION PARAMETERS OF f 0 (500) AND f 0 (980) CONCLUSIONS If one multiplies both, the numerator and the denominator by the complex conjugate factor � N i =1 ( q − q i ) ∗ ⇒ new denominator is a polynomial with real coefficients and tan δ Γ ( t ) = Im [ � N i =1 ( q − q i ) ∗ � M n =1 a n q n ] . Re [ � N i =1 ( q − q i ) ∗ � M n =1 a n q n ] By using the identity δ Γ = δ 0 0 and the threshold behavior of δ 0 0 , the following parametrization 0 ( t ) = A 1 q + A 3 q 3 + A 5 q 5 + A 7 q 7 + ... tan δ 0 1 + A 2 q 2 + A 4 q 4 + A 6 q 6 + ... S.Dubnicka,Anna Z. Dubnickova,A.Liptaj ANALYSIS OF THE PION SCALAR FORM FACTOR PROVIDES