RoySteiner equations for Martin Hoferichter 1 , 2 Daniel R. - PowerPoint PPT Presentation

Roy–Steiner equations for γγ → ππ Martin Hoferichter 1 , 2 Daniel R. Phillips 2 Carlos Schat 2 , 3 1 Helmholtz-Institut f¨ ur Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, Universit¨ at Bonn 2 Institute of Nuclear and Particle Physics and Department of Physics and Astronomy, Ohio University 3 CONICET - Departamento de F´ ısica, FCEyN, Universidad de Buenos Aires Munich, June 14, 2011 Roy–Steiner equations for γγ → ππ M. Hoferichter (HISKP & BCTP , Uni Bonn) Munich, June 14, 2011 1

Outline Roy equations for ππ scattering 1 Roy–Steiner equations for γγ → ππ 2 es solution for γγ → ππ Muskhelishvili–Omn` 3 Results 4 Roy–Steiner equations for γγ → ππ M. Hoferichter (HISKP & BCTP , Uni Bonn) Munich, June 14, 2011 2

Motivation Roy equations = coupled system of partial wave dispersion relations + crossing symmetry + unitarity Roy equations respect analyticity , unitarity , and crossing symmetry Partial wave dispersion relations in combination with unitarity (and chiral symmetry) allow for high-precision studies of low-energy processes ππ scattering: Roy (1971), Ananthanarayan et al. (2001), Garc´ ıa-Mart´ ın et al. (2011) π K scattering: B¨ uttiker et al. (2004) Roy–Steiner equations for γγ → ππ M. Hoferichter (HISKP & BCTP , Uni Bonn) Munich, June 14, 2011 3

Motivation Roy equations = coupled system of partial wave dispersion relations + crossing symmetry + unitarity Roy equations respect analyticity , unitarity , and crossing symmetry Partial wave dispersion relations in combination with unitarity (and chiral symmetry) allow for high-precision studies of low-energy processes ππ scattering: Roy (1971), Ananthanarayan et al. (2001), Garc´ ıa-Mart´ ın et al. (2011) π K scattering: B¨ uttiker et al. (2004) Application: determination of the pole position of the σ -meson ππ Roy equations + Chiral Perturbation Theory (ChPT) Caprini et al. (2006) M σ = 441 + 16 Γ σ = 544 + 18 − 8 MeV − 25 MeV γγ → ππ provides alternative access to the σ ⇒ two-photon width Γ σγγ Aim: constrain Γ σγγ at a similar level of rigor as M σ and Γ σ Roy–Steiner equations for γγ → ππ M. Hoferichter (HISKP & BCTP , Uni Bonn) Munich, June 14, 2011 3

Roy equations for ππ scattering Start from twice-subtracted dispersion relation at fixed Mandelstam t ∞ s 2 u 2 T ( s , t ) = c ( t )+ 1 � � � d s ′ Im T ( s ′ , t ) s ′ 2 ( s ′ − s ) + π s ′ 2 ( s ′ − u ) 4 M 2 π Determine subtraction functions c ( t ) from crossing symmetry Partial wave projection (angular momentum J and isospin I ) ⇒ coupled system of integral equations for partial waves t I J ( s ) ∞ ∞ 2 t I J ( s ) = k I ∑ ∑ � d s ′ K II ′ JJ ′ ( s , s ′ ) Im t I ′ J ( s )+ J ′ ( s ′ ) I ′ = 0 J ′ = 0 4 M 2 π Kernel functions K II ′ JJ ′ known analytically δ JJ ′ δ ll ′ K II ′ K II ′ JJ ′ ( s , s ′ ) = JJ ′ ( s , s ′ ) s ′ − s − i ε + ¯ Roy–Steiner equations for γγ → ππ M. Hoferichter (HISKP & BCTP , Uni Bonn) Munich, June 14, 2011 4

Roy equations for ππ scattering ∞ ∞ 2 � t I J ( s ) = k I ∑ ∑ d s ′ K II ′ JJ ′ ( s , s ′ ) Im t I ′ J ( s )+ J ′ ( s ′ ) I ′ = 0 J ′ = 0 4 M 2 π Free parameters: ππ scattering lengths in k I J ( s ) (“subtraction constants”) ⇒ Matching to ChPT Colangelo et al. (2001) Use elastic unitarity to obtain a coupled integral equation for the phase shifts Im t I J ( s ) = σ ( s ) | t I J ( s ) | 2 t I t I J J J ( s ) = e 2 i δ I J ( s ) − 1 t I � 1 − 4 M 2 σ ( s ) = π 2 i σ ( s ) s Roy–Steiner equations for γγ → ππ M. Hoferichter (HISKP & BCTP , Uni Bonn) Munich, June 14, 2011 5

Roy–Steiner equations for γγ → ππ q 1 , λ 1 q 2 , λ 2 Kinematics: s = ( p 1 + q 1 ) 2 , t = ( q 1 − q 2 ) 2 , u = ( q 1 − p 2 ) 2 Amplitude for γπ → γπ : F λ 1 λ 2 ( s , t ) = ε µ ( q 1 , λ 1 ) ε ∗ ν ( q 2 , λ 2 ) W µν ( s , t ) ∆ µ = p 1 µ + p 2 µ p 1 p 2 � t W µν ( s , t ) = A ( s , t ) 2 g µν + q 2 µ q 1 ν � + B ( s , t ) � 2 t ∆ µ ∆ ν − ( s − u ) 2 g µν + 2 ( s − u )(∆ µ q 1 ν +∆ ν q 2 µ ) � Use dispersion relations for A ( s , t ) and B ( s , t ) ⇒ constraints from gauge invariance automatically fulfilled Crossing symmetry couples γγ → ππ and γπ → γπ ( s − a )( u − a ) = ( s ′ − a )( u ′ − a ) ⇒ use hyperbolic dispersion relations Hite, Steiner (1973) ∞ d t ′ Im A ( t ′ , z ′ t ) 1 1 π − a + 1 1 � A ( s , t ) = π − s + π − u − M 2 M 2 M 2 π t ′ − t 4 M 2 π ∞ + 1 � 1 1 1 � � d s ′ Im A ( s ′ , t ′ ) s ′ − s + s ′ − u − π s ′ − a M 2 π Roy–Steiner equations for γγ → ππ M. Hoferichter (HISKP & BCTP , Uni Bonn) Munich, June 14, 2011 6

Roy–Steiner equations for γγ → ππ Coupled system for γγ → ππ partial waves h I J , ± ( t ) and γπ → γπ partial waves f I J , ± ( s ) (photon helicities ± ), e.g. ∞ ∞ ∞ J ( t ) + 1 J ′ , + ( s ′ )+ 1 � � h I ∑ JJ ′ ( t , s ′ ) Im f I d t ′ ∑ JJ ′ ( t , t ′ ) Im h I J , − ( t ) = ˜ N − d s ′ G − + ˜ K −− ˜ J ′ , − ( t ′ ) π π J ′ = 1 J ′ M 2 4 M 2 π π Subtraction constants ⇔ pion polarizabilities π , t ) = α 1 ± β 1 + t ± 2 α F + ± ( s = M 2 12 ( α 2 ± β 2 )+ O ( t 2 ) ˆ M π t Transition between isospin and particle basis       h π ± h 0 1 1 √ √ J , ± J , ± 3 6  = etc.         �    h π 0 h 2 1 2 √ − J , ± J , ± 3 3 Roy–Steiner equations for γγ → ππ M. Hoferichter (HISKP & BCTP , Uni Bonn) Munich, June 14, 2011 7

Roy–Steiner equations for γγ → ππ ∞ ∞ ∞ J ( t ) + 1 J ′ , + ( s ′ )+ 1 h I � ∑ JJ ′ ( t , s ′ ) Im f I � d t ′ ∑ JJ ′ ( t , t ′ ) Im h I J , − ( t ) = ˜ N − d s ′ ˜ G − + K −− ˜ J ′ , − ( t ′ ) π π J ′ = 1 J ′ M 2 4 M 2 π π Unitarity relation is linear in h I J , ± ( t ) Im h I J , ± ( t ) = σ ( t ) h I J , ± ( t ) t I J ( t ) ∗ t I h I J, ± J ⇒ less restrictive than for ππ scattering “Watson’s theorem” : phase of h I J , ± ( t ) equals δ I J ( t ) Watson (1954) es problem for h I J , ± ( t ) Muskhelishvili (1953), Omn` ⇒ Muskhelishvili–Omn` es (1958) Equations are valid up to t max = ( 1 GeV ) 2 (assuming Mandelstam analyticity) Roy–Steiner equations for γγ → ππ M. Hoferichter (HISKP & BCTP , Uni Bonn) Munich, June 14, 2011 8

es solution for γγ → ππ Muskhelishvili–Omn` Truncate the system at J = 2 Input for Im f I J , ± ( s ) : approximate multi-pion states by sum of resonances Garc´ ıa-Mart´ ın, Moussallam (2010) Assume h I J , ± ( t ) to be known above t m = ( 0 . 98 GeV ) 2 ⇒ Muskhelishvili–Omn` es problem with finite matching point B¨ uttiker et al. (2004) es functions, e.g. for h I 0 , + ( t ) (one subtraction) Solution in terms of Omn` 0 , + ( t )+ M π h I 0 , + ( t ) = ∆ I 2 α ( α 1 − β 1 ) I t Ω I 0 ( t ) t m ∞ d t ′ sin δ I 0 ( t ′ )∆ I Im h I + t 2 Ω I 0 , + ( t ′ ) 0 , + ( t ′ ) 0 ( t ) � � � � d t ′ 0 ( t ′ ) | + t ′ 2 ( t ′ − t ) | Ω I t ′ 2 ( t ′ − t ) | Ω I π 0 ( t ′ ) | t m 4 M 2 π with the Omn` es function t m d t ′ δ I � t J ( t ′ ) � � Ω I J ( t ) = exp t ′ ( t ′ − t ) π 4 M 2 π Roy–Steiner equations for γγ → ππ M. Hoferichter (HISKP & BCTP , Uni Bonn) Munich, June 14, 2011 9

es solution for γγ → ππ Muskhelishvili–Omn` t m ∞ d t ′ sin δ I 0 ( t ′ )∆ I Im h I 0 ( t )+ t 2 Ω I 0 , + ( t ′ ) 0 , + ( t ′ ) 0 , + ( t )+ M π 0 ( t ) � � � � h I 0 , + ( t ) = ∆ I 2 α ( α 1 − β 1 ) I t Ω I d t ′ 0 ( t ′ ) | + t ′ 2 ( t ′ − t ) | Ω I t ′ 2 ( t ′ − t ) | Ω I π 0 ( t ′ ) | t m 4 M 2 π ∆ I 0 , + ( t ) describes left-hand cut ∞ 0 , + ( t ) + 1 � ∆ I 0 , + ( t ) = N I d t ′ � 02 ( t , t ′ ) Im h I 02 ( t , t ′ ) Im h I � K ++ ˜ 2 , + ( t ′ ) + ˜ K + − 2 , − ( t ′ ) π 4 M 2 π ∞ d s ′ ∑ + 1 � � G ++ ˜ 0 J ′ ( t , s ′ ) Im f I J ′ , + ( s ′ )+ ˜ G + − 0 J ′ ( t , s ′ ) Im f I J ′ , − ( s ′ ) � π J ′ = 1 , 2 M 2 π Input Above t m use Breit–Wigner description of f 2 ( 1270 ) ππ phases: Caprini et al. (in preparation), Garc´ ıa-Mart´ ın et al. (2011) Roy–Steiner equations for γγ → ππ M. Hoferichter (HISKP & BCTP , Uni Bonn) Munich, June 14, 2011 10