Pole position of the a 1 ( 1260 ) Misha Mikhasenko Joint Physics Analysis Center, COMPASS @ CERN, Universit¨ at Bonn, HISKP , Bonn, Germany CHARM 2018 Akademgorodok, Novosibirsk 23/05/2018 J P C A M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 1 / 14
Overview Introduction 1 Hadrons in QCD Analytical structure of the scattering amplitude Three pions dynamics 2 Constrains Data Extraction of the resonance parameters 3 Fit Analytical continuation Remarks 4 COMPASS analysis CLEO analysis a 1 ( 1420 ) phenomenon M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 2 / 14
Introduction Hadrons in QCD Flavor and excitation Quark model color-binding, L s s 1 2 q q ◮ Radial excitation ( n ) , ◮ Orbital excitation ( L ) , many states ( J PC = 0 ++ , 1 −− . . . ) are coupled to ππ . Some other to 3 π system [Amsler et al., Phys. Rept. 389, 61 (2004)] M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 3 / 14
Introduction Hadrons in QCD Lattice QCD Lattice QCD spectrum matches experimental observations well but predicts more a 1 ρ π [Dudek et. al, Phys.Rev. D82 (2010) 034508] M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 4 / 14
Introduction Hadrons in QCD Resonances on the Lattice ππ system in the box [EPJ Web Conf. 175 (2018)] 180 150 Tracking pole position 120 90 For high m π the ρ -becomes stable. 60 Pole of ρ approaches real axis. 30 0 400 500 600 700 800 900 1000 [Wilson, D. et al.,PRD 92,(2015)] M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 5 / 14
Introduction Analytical structure of the scattering amplitude Resonances = Poles at the Complex plane Breit-Wigner amplitude Features of the complex s plane: s = E 2 – the total inv.mass squared The Real axis → physical world The Imaginary axis → analytical continuation M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 6 / 14
Introduction Analytical structure of the scattering amplitude Resonances = Poles at the Complex plane Breit-Wigner amplitude Features of the complex s plane: s = E 2 – the total inv.mass squared The Real axis → physical world The Imaginary axis → analytical continuation M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 6 / 14
Introduction Analytical structure of the scattering amplitude Resonances = Poles at the Complex plane Breit-Wigner amplitude Features of the complex s plane: s = E 2 – the total inv.mass squared The Real axis → physical world The Imaginary axis → analytical continuation Unitarity constaints for two-body scattering S † ˆ T † = i ˆ T † ˆ ˆ ˆ I + i ˆ T − ˆ ˆ S = ˆ S = ˆ I T T . � � � 2 | ˆ p ′ 1 p ′ d Φ 2 T ∗ ( s , t ′ ) T ( s , t ′′ ) T ( s , t ) = T | p 1 p 2 2 Im T ( s , t ) = Partial wave expansion − → 2Im t l ( s ) = t ∗ l ( s ) ρ ( s ) t l ( s ) The final form M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 6 / 14
Introduction Analytical structure of the scattering amplitude Resonances = Poles at the Complex plane Breit-Wigner amplitude Features of the complex s plane: s = E 2 – the total inv.mass squared The Real axis → physical world The Imaginary axis → analytical continuation Unitarity constaints for two-body scattering S † ˆ T † = i ˆ T † ˆ ˆ ˆ I + i ˆ T − ˆ ˆ S = ˆ S = ˆ I T T . � � � 2 | ˆ p ′ 1 p ′ d Φ 2 T ∗ ( s , t ′ ) T ( s , t ′′ ) T ( s , t ) = T | p 1 p 2 2 Im T ( s , t ) = Partial wave expansion − → 2Im t l ( s ) = t ∗ l ( s ) ρ ( s ) t l ( s ) The final form M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 6 / 14
Three pions dynamics Constrains Quasi-two-body unitarity [MM (JPAC) in preparation] Three-body unitarity [Eden, Landshoff et al.(2002)] Disconnected Connected � �� � � �� � ξ ξ } σ ′ Singularity splitting: { � σ { = + + T � s Final state interaction: = + + · · · + + . . . K M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 7 / 14
Three pions dynamics Constrains Quasi-two-body unitarity [MM (JPAC) in preparation] Three-body unitarity [Eden, Landshoff et al.(2002)] Disconnected Connected � �� � � �� � ξ ξ } σ ′ Singularity splitting: { � σ { = + + T � s Final state interaction: = + + · · · + + . . . K T ( σ ′ , s , σ ) = K ξ ( s , σ ′ ) t ( s ) K ξ ( s , σ ) M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 7 / 14
Three pions dynamics Constrains Quasi-two-body unitarity [MM (JPAC) in preparation] Three-body unitarity [Eden, Landshoff et al.(2002)] Disconnected Connected � �� � � �� � ξ ξ } σ ′ Singularity splitting: { � σ { = + + T � s Final state interaction: = + + · · · + + . . . K T ( σ ′ , s , σ ) = K ξ ( s , σ ′ ) t ( s ) K ξ ( s , σ ) 2 Im t ( s ) = t ∗ ( s ) ρ ( s ) t ( s ) , Symmetrized quasi-two-body phase space factor − m 1318 MeV/ c 2 < 100 MeV/ c 2 3 π ] 2 ) 800 c 2 ρ (770) 1.5 (GeV/ 700 [ 600 + � � 2 π − 2 π � � � � m 1 500 � � ρ ( s ) = 1 � � � � 400 d Φ 3 − = d Φ 3 − � � � � 300 2 � � � � 0.5 200 � � 100 � �� � 0 0 0 0.5 1 1.5 interference [ ] m 2 (GeV/ c 2 ) 2 π π − + M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 7 / 14
Three pions dynamics Constrains Quasi-two-body unitarity [MM (JPAC) in preparation] Three-body unitarity [Eden, Landshoff et al.(2002)] Disconnected Connected � �� � � �� � ξ ξ } σ ′ Singularity splitting: { � σ { = + + T � s Final state interaction: = + + · · · + + . . . K T ( σ ′ , s , σ ) = K ξ ( s , σ ′ ) t ( s ) K ξ ( s , σ ) 2 Im t ( s ) = t ∗ ( s ) ρ ( s ) t ( s ) , Symmetrized quasi-two-body phase space factor − m 1318 MeV/ c 2 < 100 MeV/ c 2 3 π ] 2 ) 800 2 c ρ (770) 1.5 (GeV/ 700 [ 600 + � � 2 π − 2 π � � � � m 1 500 � � ρ ( s ) = 1 � � � � 400 d Φ 3 − = d Φ 3 − � � � � 300 2 � � � � 0.5 200 � � 100 � �� � 0 0 0 0.5 1 1.5 interference [ ] m 2 (GeV/ c 2 ) 2 π − π + The model: symmetrized [Bowler, Phys.Lett.B182 (1986)] g 2 t ( s ) = m 2 − s − ig 2 ρ ( s ) / 2 M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 7 / 14
Three pions dynamics Constrains Quasi-two-body unitarity [MM (JPAC) in preparation] Three-body unitarity [Eden, Landshoff et al.(2002)] Disconnected Connected � �� � � �� � ξ ξ } σ ′ Singularity splitting: { � σ { = + + T � s Final state interaction: = + + · · · + + . . . K T ( σ ′ , s , σ ) = K ξ ( s , σ ′ ) t ( s ) K ξ ( s , σ ) 2 Im t ( s ) = t ∗ ( s ) ρ ( s ) t ( s ) , Symmetrized quasi-two-body phase space factor − m 1318 MeV/ c 2 < 100 MeV/ c 2 π 3 ] 2 ) 800 2 c ρ (770) 1.5 (GeV/ 700 [ 600 + � � 2 π − 2 π � � � � m 1 500 � � ρ ( s ) = 1 � � � � 400 d Φ 3 − = d Φ 3 − � � � � 300 2 � � � � 0.5 200 � � 100 � �� � 0 0 0 0.5 1 1.5 interference [ ] m 2 (GeV/ c 2 ) 2 π − π + The model: symmetrized, dispersive � ∞ g 2 ρ ( s ′ ) ρ ( s ) = s s ′ ( s ′ − s ) d s ′ , t ( s ) = ρ ( s ) / 2 , ˜ Im i ˜ ρ = i ρ. m 2 − s − ig 2 ˜ π i 9 m 2 π M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 7 / 14
Three pions dynamics Data 1 ++ light meson spectrum Axial vector states below 2 GeV Dominated by 3 π scattering ◮ ρπ ∼ 60 % − 80 % ◮ σπ ∼ 5 % − 10 % ◮ f 2 π ∼ < 5 % K ¯ K π < 3 % M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 8 / 14
Three pions dynamics Data 1 ++ light meson spectrum Axial vector states below 2 GeV Dominated by 3 π scattering ◮ ρπ ∼ 60 % − 80 % ◮ σπ ∼ 5 % − 10 % ◮ f 2 π ∼ < 5 % K ¯ K π < 3 % τ − → π − π + π − ν V-A: Vector (1 −− ) or Axial (1 ++ ) π + V-A Isospin 1 due to the charge W − π − π − Negative G -parity ⇒ positive C -parity τ − ν ⇒ J PC = 1 ++ M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 8 / 14
Extraction of the resonance parameters Fit Fit to ALEPH data [data from ALEPH, Phys.Rept.421 (2005)] χ 2 function ALEPH(3 ) χ 2 ( c , m , g ) = 2.5 (a. u. / 0.025 GeV 2 ) M ( c , m , g )) T C − 1 ( � D − � stat ( � D − � M ( c , m , g )) , 2.0 Stat. errors ∼ × 5 Systematic errors 1.5 Stat. cov. matrix is used in the fit 1.0 Syst. cov. matrix – in the bootstrap d /d s 0.5 0.0 0 1 2 3 M 2 3 (GeV 2 ) s Stat. cov. matrix Syst. cov. matrix M. Mikhasenko (HISKP , Uni Bonn) Ground axial vector 25/05/2018 9 / 14
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