s 2 ( 2573 ) and the prediction of novel exotic charmed mesons R. - PowerPoint PPT Presentation

Introduction Formalism: The VV interaction Results Conclusions A new interpretation for the D ∗ s 2 ( 2573 ) and the prediction of novel exotic charmed mesons R. Molina 1 , T. Branz 2 , and E. Oset 1 1 Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Institutos de Investigación de Paterna, Aptdo. 22085, 46071 Valencia, Spain 2 Institut für Theoretische Physik, Universität Tübingen, Kepler Center for Astro and Particle Physics, Auf der Morgenstelle 14, D-72076 Tübingen, Germany

Introduction Formalism: The VV interaction Results Conclusions Outline Introduction Formalism: The VV interaction Convolution The PP decay mode Results Conclusions

Introduction Formalism: The VV interaction Results Conclusions Introduction • Heavy quark symmetry framework (HQS): with l = 1 two doublets of D s states are generated: - light quark → j l = 3 / 2, total angular momentum: J P = 1 + , 2 + - light quark → j l = 1 / 2, total angular momentum: J P = 0 + , 1 + • The doublet with J P = 1 + , 2 + is identified with the D s 1 ( 2536 ) and D s 2 ( 2573 ) in HQS • However, the doublet with J P = 0 + , 1 + and very broad states cannot be identified with the narrow states discovered: the D ∗ s 0 ( 2317 ) and the D s 1 ( 2460 ) (100 MeV lower in mass than the predictions) • D ∗ s 0 ( 2317 ) : strong s-wave Coupling to DK , E. van Beveren and G. Rupp, PRL (2003) ; Couple Channels: D ∗ s 0 ( 2317 ) ∼ DK , D. Gamermann, E. Oset, D. Strottmann, M. J. Vicente Vacas , PRD (2007) ; D s 1 ( 2460 ) ∼ KD ∗ ( η D ∗ s ) , D s 1 ( 2536 ) ∼ DK ∗ ( D s ω )) D. Gamermann and E. Oset, EPJA (2007)

Introduction Formalism: The VV interaction Results Conclusions The VV interaction Bando, Kugo, Yamawaki L ( 3 V ) 4 � V µν V µν � = ig � ( ∂ µ V ν − ∂ ν V µ ) V µ V ν � L III = − 1 III L ( c ) III = g 2 2 � V µ V ν V µ V ν − V ν V µ V µ V ν � V µν , g V µ ρ 0 K ∗ +  ω  ρ + 2 + √ √ V µν = ∂ µ V ν − ∂ ν V µ − ig [ V µ , V ν ] 2 − ρ 0 K ∗ 0 ρ − ω  2 +  √ √ g = M V   2 K ∗− K ∗ 0 ¯ 2 f φ µ

Introduction Formalism: The VV interaction Results Conclusions The VV interaction V V V V + − → V V V a ) b ) c ) d ) • The VV interaction comes from 1. a) and c) • 1. d): - p-wave repulsive for equal masses (R. Molina, 2008) - minor component of s-wave for different masses (L. S. Geng, 2009)

Introduction Formalism: The VV interaction Results Conclusions Formalism: The VV interaction Approximation ǫ µ k µ = ( k 0 , 0 , 0 , | � k | ) 1 = ( 0 , 1 , 0 , 0 ) ǫ µ 1 = ( 0 , 1 , 0 , 0 ) ǫ µ 2 = ( 0 , 0 , 1 , 0 ) ǫ µ k / m ≃ 0, � 2 = ( 0 , 0 , 1 , 0 ) 3 = ( | � k | , 0 , 0 , k 0 ) / m ǫ µ ǫ µ j ǫ ( l ) k µ 3 = ( 0 , 0 , 0 , 1 ) µ ≃ 0 Spin projectors 1 P ( 0 ) 3 ǫ µ ǫ µ ǫ ν ǫ ν = 1 2 ( ǫ µ ǫ ν ǫ µ ǫ ν − ǫ µ ǫ ν ǫ ν ǫ µ ) P ( 1 ) = { 1 2 ( ǫ µ ǫ ν ǫ µ ǫ ν + ǫ µ ǫ ν ǫ ν ǫ µ ) − 1 P ( 2 ) 3 ǫ α ǫ α ǫ β ǫ β } =

Introduction Formalism: The VV interaction Results Conclusions Formalism: The VV interaction (a) + (b) • (a) and (b) → Pole + mass and width (c) • (c) → p-wave + repulsive (not included) (d) • (d) → Pole width Bethe equation � q max q 2 dq T = [ I − VG ] − 1 V G = ω 1 + ω 2 0 ( 2 π ) 2 ω 1 ω 2 [( P 0 ) 2 − ( ω 1 + ω 2 ) 2 + i ǫ ]

Introduction Formalism: The VV interaction Results Conclusions The VV interaction 1. f 0 ( 1370 ) , f 2 ( 1270 ) ∼ ρρ R. Molina, D. Nicmorus and E. Oset, Phys. Rev. D 78 , 114018 (2008) 2 ( 1525 ) ∼ ρρ , K ∗ ¯ 2. f 0 ( 1370 ) , f 0 ( 1710 ) , f 2 ( 1270 ) , f ′ K ∗ ... K ∗ 2 ( 1430 ) ∼ ρ K ∗ , ω K ∗ ... L. S. Geng and E. Oset, Phys. Rev. D 79 , 074009 (2009) 3. D ∗ ( 2640 ) , D ∗ 2 ( 2460 ) ∼ ρ ( ω ) D ∗ R. Molina, H. Nagahiro, A. Hosaka and E. Oset, Phys. Rev. D 80 , 014025 (2009) 4. Y(3940), Z(3930), X(4160) ∼ D ∗ ¯ D ∗ , D ∗ D ∗ s ¯ s R. Molina and E. Oset, Phys. Rev. D 80 , 114013 (2009)

Introduction Formalism: The VV interaction Results Conclusions The VV interaction • C = 0 ; S = 1 ; I = 1 / 2 • C = 1 ; S = 2 ; I = 1 / 2: (hidden charm): D ∗ s K ∗ D ∗ D ∗ , J /ψ K ∗ s ¯ • C = 2 ; S = 0 ; I = 0, 1: • C = 1 ; S = − 1 ; I = 0, 1: D ∗ D ∗ D ∗ ¯ K ∗ • C = 2 ; S = 1 ; I = 1 / 2: • C = 1 ; S = 1 ; I = 0: D ∗ s D ∗ D ∗ K ∗ , D ∗ s ω , D ∗ s φ • C = 2 ; S = 2 ; I = 0: • C = 1 ; S = 1 ; I = 1: D ∗ s D ∗ s D ∗ K ∗ , D ∗ s ρ

Introduction Formalism: The VV interaction Results Conclusions Convolution Convolution due to the width of the ρ meson ( D ∗ s ρ channel) � ( m ρ + 2 Γ ρ ) 2 1 1 ( − 1 1 G ( s ) ˜ ( m ρ − 2 Γ ρ ) 2 d ˜ m 2 π ) I m G ( s , ˜ m 2 1 , m 2 = s ) D ∗ N m 2 1 − m 2 ρ + i Γ ˜ m 1 ˜ m 2 − 4 m 2 m ) = Γ ρ ( ˜ Γ D ∗ < 2 . 1 MeV m − 2 m π ) π ) 3 / 2 θ ( ˜ Γ( ˜ m 2 ρ − 4 m 2 Γ ρ = 146 . 2 MeV π Γ K ∗ = 48 MeV • The ρ ∗ -mass convolution gives Γ ≃ 8 MeV ( D ∗ s ππ ) • The K ∗ -mass convolution gives Γ ≃ 3 MeV (or less)( D ∗ π K )

Introduction Formalism: The VV interaction Results Conclusions The PP decay mode • The PP box diagram has only J P = 0 + and J P = 2 + quantum numbers • We only find atractive interaction in the sectors: - C = 1 ; S = − 1 ; I = 0: D ∗ ¯ K ∗ - C = 2 ; S = 0 ; I = 0 ; J = 1: D ∗ D ∗ - C = 1 ; S = 1 ; I = 0: D ∗ K ∗ , D ∗ s φ , - C = 2 ; S = 1 ; I = 1 / 2 ; J = 1: D ∗ s ω D ∗ s D ∗ - C = 1 ; S = 1 ; I = 1: D ∗ K ∗ , D ∗ s ρ D ∗ ( k 1 ) D ∗ ( k 3 ) D ∗ D ∗ D ∗ D ∗ s s s D ( q ) D D π ( k 1 − q ) π ( k 3 − q ) π K K K K ( P − q ) K K K ∗ ( k 2 ) K ∗ ( k 4 ) K ∗ φ φ φ D ∗ D ∗ D π π ¯ K ¯ ¯ K ∗ K ∗

Introduction Formalism: The VV interaction Results Conclusions The VV interaction • Model A: b − m 2 Λ 2 F 1 ( q 2 ) = 1 q | 2 , b − ( k 0 1 − q 0 ) 2 + | � Λ 2 b − m 2 Λ 2 F 3 ( q 2 ) = 3 q | 2 , b − ( k 0 3 − q 0 ) 2 + | � Λ 2 with q 0 = s + m 2 2 − m 2 q running variable, Λ b = 1 . 4 , 1 . 5 GeV and , � 2 √ s 4 g = M ρ / 2 f π • Model B: q | 2 ) / Λ 2 , F ( q 2 ) = e (( q 0 ) 2 −| � with Λ = 1 , 1 . 2 GeV, q 0 = s + m 2 2 − m 2 , g = M ρ / 2 f π , 2 √ s 4 s / 2 f D s = 5 . 47 and g D = g exp g D s = M D ∗ D ∗ D π = 8 . 95 (experimental value)

Introduction Formalism: The VV interaction Results Conclusions C = 1 ; S = − 1 ; I = 0 (exotic) • Channels: D ∗ ¯ K ∗ ( α = − 1 . 6) • V ∼ − 10 g 2 for I = 0 ; J = 0 , 1 √ s pole (MeV) I [ J P ] Γ (MeV) Model • V ∼ − 16 g 2 for 0 [ 0 + ] 2848 A, Λ = 1400 MeV 23 I = 0 ; J = 2 A, Λ = 1500 MeV 30 B, Λ = 1000 MeV 25 B, Λ = 1200 MeV 59 √ s pole I [ J P ] g D ∗ ¯ 0 [ 1 + ] K ∗ ) 2839 Convolution 3 0 [ 0 + ] 0 [ 2 + ] 2848 12227 2733 A, Λ = 1400 MeV 11 0 [ 1 + ] 2839 13184 A, Λ = 1500 MeV 14 0 [ 2 + ] 2733 17379 B, Λ = 1000 MeV 22 B, Λ = 1200 MeV 36

Introduction Formalism: The VV interaction Results Conclusions C = 1 ; S = 1 ; I = 0 • V ∼ − 18 g 2 for √ s (MeV) I [ J P ] Model Γ (MeV) I = 0 ; J = 0 , 1 0 [ 0 + ] 2683 A, Λ = 1400 MeV 20 • V ∼ − 26 g 2 for A, Λ = 1500 MeV 25 I = 0 ; J = 2 B, Λ = 1000 MeV 44 B, Λ = 1200 MeV 71 0 [ 1 + ] 4 × 10 − 3 2707 Convolution • α = − 1 . 6 0 [ 2 + ] 2572 A, Λ = 1400 MeV 7 Γ exp = 20 ± 5 MeV A, Λ = 1500 MeV 8 B, Λ = 1000 MeV 18 B, Λ = 1200 MeV 23 √ s I [ J P ] g D ∗ K ∗ g D ∗ g D ∗ s ω s φ 0 [ 0 + ] 2683 15635 − 4035 6074 0 [ 1 + ] Channels: 2707 14902 − 5047 4788 C = 1 ; S = 1 ; I = 0: D ∗ K ∗ , D ∗ s φ , D ∗ 0 [ 2 + ] s ω 2572 18252 − 7597 7257

Introduction Formalism: The VV interaction Results Conclusions C = 1 ; S = 1 ; I = 1 • V ∼ − 7 g 2 for • Channels: D ∗ K ∗ , D ∗ s ρ ( α = − 1 . 6) I = 0 ; J = 0 , 1 • V ∼ − 13 g 2 for √ s pole (MeV) I [ J P ] Model Γ (MeV) I = 0 ; J = 2 1 [ 2 + ] 2786 A, Λ = 1400 MeV 8 A, Λ = 1500 MeV 9 B, Λ = 1000 MeV 9 √ s pole I G [ J PC ] g D ∗ K ∗ g D ∗ s ρ B, Λ = 1200 MeV 11 1 [ 2 + ] 2786 11041 11092

Introduction Formalism: The VV interaction Results Conclusions C = 2 ; S = 0 ; I = 0 and C = 2 ; S = 1 ; I = 1 / 2 (exotics) Channels: D ∗ D ∗ ( α = − 1 . 4) Channels: D ∗ D ∗ s ( α = − 1 . 4) • V ∼ 0 for I = 0 ; J = 0 , 2 • V ∼ 20 for I = 1 / 2 ; J = 0 , 2 • V ∼ − 25 g 2 for I = 0 ; J = 1 • V ∼ − 20 g 2 for I = 0 ; J = 1 √ s pole √ s pole I [ J P ] I [ J P ] g D ∗ D ∗ g D ∗ s D ∗ 0 [ 1 + ] 1 / 2 [ 1 + ] 3969 16825 4101 13429

Introduction Formalism: The VV interaction Results Conclusions Summary √ s √ s exp I [ J P ] C , S Γ A (Λ = 1400 ) Γ B (Λ = 1000 ) State Γ exp 0 [ 0 + ] 1 , − 1 2848 23 25 0 [ 1 + ] 2839 3 3 0 [ 2 + ] 2733 11 22 0 [ 0 + ] 1 , 1 2683 20 44 0 [ 1 + ] 4 × 10 − 3 4 × 10 − 3 2707 0 [ 2 + ] D s 2 ( 2573 ) 2572 . 6 ± 0 . 9 2572 7 18 20 ± 5 1 [ 2 + ] 2786 8 9 0 [ 1 + ] 2 , 0 , 3969 0 0 1 / 2 [ 1 + ] 2 , 1 4101 0 0 Table: Summary of the nine states obtained. The width is given for the model A, Γ A , and B, Γ B . All the quantities here are in MeV. See the talk of A. Valcarce 16/06 (16.30h quarkonia session)