Resonances and their N C fates in U (3) chiral perturbation theory - PowerPoint PPT Presentation

Outline Preface Analytical calculation Phenomenological discussion Conclusions Resonances and their N C fates in U (3) chiral perturbation theory Zhi-Hui Guo Universidad de Murcia & Hebei Normal University Hadron 2011, 13 June -17 June 2011, Munich, Germany in collaboration with Jose Oller, based on arXiv:1104.2849[hep-ph] A work dedicated to Joaquim Prades Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions Outline 1. Preface 2. Analytical calculation ◮ Chiral Lagrangian & perturbative amplitudes ◮ Resummation of s -channel loops : a variant N/D method 3. Phenomenological discussion ◮ Fit quality ◮ Poles in the complex energy plane & their residues ◮ N C trajectories 4. Conclusions Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions Preface In the chiral limit m u = m d = m s = 0 the QCD Lagrangian is invariant under U L (3) ⊗ U R (3) symmetry at the classical level. U A (1) ≡ U L − R : violated at the quantum level, i.e. U A (1) anomaly, which is also responsible for the massive η 1 . U V (1) ≡ U L + R : conserved baryon number. SU L (3) ⊗ SU R (3) → SU V (3) is spontaneously broken. Goldstone bosons appear π , K , η 8 : SU (3) χ PT [Gasser, Leutwyler, NPB’85]. In large N C limit, U A (1) anomaly disappears and the η 1 mass vanishes: M 2 η 1 ∼ O (1 / N C ). So η 1 together with π , K , η 8 constitute the nonet of pesudo Goldstone bosons. [t’Hooft, NPB’74] [Witten, NPB’79] [Coleman & Witten, PRL’80] Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions U (3) χ PT takes π , K , η 8 and η 1 as its dynamical degrees of freedom and employs the triple expansion scheme: momentum, quark masses and 1 / N C , i.e. δ ∼ p 2 ∼ m q ∼ 1 / N C . ◮ Set up in: [ Witten, PRL’80] [ Di Vecchia & Veneziano,’80 ] [ Rosenzweig, Schechter & Trahern, ’80 ] ◮ Chiral Lagrangian to O ( p 4 ) completed in: [Herrera-Siklody, Latorre, Pascual, Taron, NPB’97 ] . See also [Kaiser, Leutwyler, EPJC’00 ] . ◮ Applications Light quark masses: [Leutwyler, PLB’96 ] η − η ′ mixing: [Herrera-Siklody, Latorre, Pascual, Taron, PLB’98] [Leutwyler, NPB(Proc.Suppl)’98 ] η ′ → ηππ decay: [Escribano,Masjuan, Sanz-Cillero, JHEP’11] Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions ◮ Our current work offers the complete one-loop amplitudes of the meson-meson scattering within U (3) χ PT. And then we study the properties of various resonances, such as their pole positions, residues and N C behaviour, by unitarizing the U (3) χ PT amplitudes. Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions There are variant methods to treat η ′ in the market ◮ Matter filed: M 2 η ′ ∼ O (1) and Infrared Regularization method used to handle the loops. [Beisert, Borasoy, NPA’02, PRD’03] ◮ Non-relativistic field [Kubis, Schneider, EPJC’09] Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions Relevant Chiral Lagrangian Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions L ( δ 0 ) = F 2 4 ⟨ u µ u µ ⟩ + F 2 4 ⟨ χ + ⟩ + F 2 0 ln 2 det u , 3 M 2 (1) where Φ 2 F , U = u 2 , u = e i √ u µ = iu † D µ Uu † = u † µ , χ ± = u † χ u † ± u χ † u ,   √ √ 3 π 0 + η 8 + 2 η 1 π + K + √ 6  √ √  3 π 0 + η 8 +  2 η 1  − K 0 π − Φ =  . (2) √  6 √ ¯ − 2 η 8 + 2 η 1 K − K 0 √ 6 Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions L i s correspond to the higher order local operators. At O ( δ ) one has O ( N C p 4 ) and O ( N 0 C p 2 ) operators: L ( δ ) = L 2 ⟨ u µ u ν u µ u ν ⟩ + (2 L 2 + L 3 ) ⟨ u µ u µ u ν u ν ⟩ + L 5 ⟨ u µ u µ χ + ⟩ + L 8 / 2 ⟨ χ + χ + + χ − χ − ⟩ + . . . + F 2 Λ 1 / 12 D µ ψ D µ ψ − i F 2 Λ 2 / 12 ψ ⟨ U † χ − χ † U ⟩ + . . . At O ( δ 2 ) (same order as the one-loop contribution), one then has O ( N − 2 C p 0 ), O ( N − 1 C p 2 ), O ( N 0 C p 4 ) and O ( N C p 6 ) operators: 0 X 4 + ˜ L ( δ 2 ) v (4) v (2) 1 X 2 ⟨ u µ u µ ⟩ + L 4 ⟨ u µ u µ ⟩⟨ χ + ⟩ = ˜ + C 1 ⟨ u ρ u ρ h µν h µν ⟩ + . . . , with ψ = − i ln det U , X = log det( U ) and h µν = ∇ µ u ν + ∇ ν u µ . [Herrera-Siklody, Latorre, Pascual, Taron, NPB’97 ] [Bijnens, Colangelo, Ecker, JHEP’99] Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions Alternatively, one could use resonances to estimate the higher order low energy constants: c d ⟨ S 8 u µ u µ ⟩ + c m ⟨ S 8 χ + ⟩ L S = c d S 1 ⟨ u µ u µ ⟩ + � + � c m S 1 ⟨ χ + ⟩ + ... (3) L V = iG V √ ⟨ V µν [ u µ , u ν ] ⟩ + ... , (4) 2 2 [Ecker, Gasser, Pich, de Rafael, NPB’89] In the current discussion, we assume the resonance saturation and exploit the above resonance operators to calculate the meson-meson scattering. The monomials proportional to Λ 1 and Λ 2 are not generated through resonance exchange. No double counting. Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions Perturbative calculation of the scattering amplitudes Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions P S Figure: Relevant Feynman diagrams for mass, wave function renormalization and η − η ′ mixing The leading order η - η ′ mixing has to be solved exactly Figure: The dot denotes the mixing of η 8 and η 1 at leading order, which is proportional to m 2 K − m 2 π . Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions Scattering amplitudes consist of + + + crossed (c) (b) (a) S + + crossed S , V (d) (e) Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions S Figure: Relevant Feynman diagrams for the pseudo Goldstone decay constant. The wiggly line corresponds to the axial-vector external source. We expressed all the amplitudes in terms of physical masses and F π , i.e. reshuffling the leading order contributions. Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions Partial wave amplitude and its unitarization Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions Partial wave projection: ∫ 1 1 T I dx P J ( x ) T I [ s , t ( x ) , u ( x )] , √ J ( s ) = (5) 2) N 2( − 1 √ 2) N is a where P J ( x ) denote the Legendre polynomials and ( symmetry factor to account for the identical particles, such as ππ, ηη, η ′ η ′ . Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions The essential of the N / D method is to construct the unitarized T J : [Chew, Mandelstam, PR’60] T J = N D , (6) where for s > 4 m 2 , Im D = N Im T J = − ρ N , for s < 4 m 2 , Im D = 0 , Im N = D Im T J , for s < 0 , Im N = 0 , for s > 0 , (7) due to the fact that the unitarity condition for the elastic channel is Im T − 1 s > 4 m 2 = − ρ , (8) J √ 1 − 4 m 2 / s / 16 π . where ρ = Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory

Outline Preface Analytical calculation Phenomenological discussion Conclusions One can now write the dispersion relations for N and D : ∫ ∞ N ( s ′ ) ρ ( s ′ ) a SL ( s 0 ) − s − s 0 ( s ′ − s )( s ′ − s 0 ) ds ′ + ... , D ( s ) = � (9) π 4 m 2 ∫ 0 D ( s ′ ) Im T J ( s ′ ) ds ′ . N ( s ) = (10) s ′ − s −∞ It can be greatly simplified if one imposes the perturbative solution for N ( s ) instead of the left hand discontinuity [Oller, Oset, PRD’99] , N ( s ) T J ( s ) = 1 + g ( s ) N ( s ) , (11) where ∫ ∞ g ( s ) = a SL ( s 0 ) ρ ( s ′ ) − s − s 0 ( s ′ − s )( s ′ − s 0 ) ds ′ . (12) 16 π 2 π 4 m 2 Zhi-Hui Guo UM&HEBNU Resonances and their N C fates in U (3) chiral perturbation theory