Dispersive analysis of K 3 and cusps ahal 1 , 2 , Karol Kampf 1 , - PowerPoint PPT Presentation

Dispersive analysis of K → 3 π and cusps ∗ ahal 1 , 2 , Karol Kampf 1 , 3 Martin Zdr´ 1 IPNP, Charles University, Czech Republic 2 Faculty of Physics, University of Vienna, Austria 3 PSI, Switzerland martin.zdrahal ∈ univie.ac.at, karol.kampf ∈ psi.ch Durham, September 2008 ∗ Work in progress in collaboration with M. Knecht and J. Novotn´ y Outline: What is the cusp? Theoretical approaches - overview of existing approaches Dispersive approach introduction to dispersive analysis first iteration - results at O ( p 4 ) sketch of second iteration ( O ( p 6 )) Conclusions 1 /20 M. Zdr´ ahal, K. Kampf Dispersive analysis of cusps

Motivation - What is the cusp? Beach cusps - Kootenay lake 2 /20 M. Zdr´ ahal, K. Kampf Dispersive analysis of cusps

What is the cusp? Decay K + → π + π 0 π 0 - 6 · 10 8 reconstructed events at NA48/2 Pictures taken from L. DiLella, Kaon 07 3 /20 M. Zdr´ ahal, K. Kampf Dispersive analysis of cusps

What is the cusp? Decay K + → π + π 0 π 0 - 6 · 10 8 reconstructed events at NA48/2 Pictures taken from L. DiLella, Kaon 07 3 /20 M. Zdr´ ahal, K. Kampf Dispersive analysis of cusps

Theory – Why is the cusp? Cabibbo ’04 Amplitude for K + → π + π 0 π 0 : M 0 + M 1 , schematically: π + π + π 0 K + + π − π 0 4 /20 M. Zdr´ ahal, K. Kampf Dispersive analysis of cusps

Theory – Why is the cusp? Cabibbo ’04 Amplitude for K + → π + π 0 π 0 : M 0 + M 1 , schematically: π + π + π 0 K + + π − π 0 ¯ J m π + ( s ) √ 1 − 4 m 2 /s Im ր ¯ log v − 1 J m ( s ) ∼ 2 + v v +1 = ≈ iπv + regul. = ⇒ M 1 ∼ iπv 3.0 3.5 4.0 4.5 5.0 Re thus we have square root singularity at 4 m 2 + above physical threshold 4 m 2 0 and � ( M 0 ) 2 + ( M 1 ) 2 + 2 M 0 M 1 s < 4 m 2 : |M| 2 = + ( M 0 ) 2 + ( i M 1 ) 2 s > 4 m 2 : + depends on the scattering length of ππ Meißner, M¨ uller, Steininger ’97 4 /20 M. Zdr´ ahal, K. Kampf Dispersive analysis of cusps

Where is the cusp? The same should appear for the K L → π 0 π 0 π 0 π + π 0 π + π 0 π + π 0 K + K L π − π − π 0 π 0 This second cusp is much weaker – roughly: (Cabibbo, Isidori ’05, DiLella – Kaon07 ) Decay K + → π + π 0 π 0 � “size of amplitude” ∼ A +;+ − A +;00 + A +;+ − A +;00 “cusp effect” � ≈ 6 � | A +;00 | 2 � “branch. point” Decay K L → π 0 π 0 π 0 “size of amplitude” ∼ A L ;+ − A L ;00 “cusp effect” � � ≈ 0 . 5 � | A L ;00 | 2 � “branch. point” 5 /20 M. Zdr´ ahal, K. Kampf Dispersive analysis of cusps

Where is the cusp? The same should appear for the K L → π 0 π 0 π 0 π + π 0 π + π 0 π + π 0 K + K L π − π − π 0 π 0 This second cusp is much weaker Decay K L → π 0 π 0 π 0 - 9 · 10 8 reconstructed events at NA48/2 Pictures taken from L. DiLella, Kaon07 5 /20 M. Zdr´ ahal, K. Kampf Dispersive analysis of cusps

Where is the cusp? The same should appear for the K L → π 0 π 0 π 0 π + π 0 π + π 0 π + π 0 K + K L π − π − π 0 π 0 This second cusp is much weaker Decay K L → π 0 π 0 π 0 - 9 · 10 8 reconstructed events at NA48/2 However, it is seen both at KTeV and NA48/2!! — NA48/2: not published yet — KTeV: arXiv:0806.3535 Pictures taken from L. DiLella, Kaon07 5 /20 M. Zdr´ ahal, K. Kampf Dispersive analysis of cusps

Theoretical approaches - Overview a direct computation from χ PT (with weak part) Bijnens, Borg ’04, ’04, ’05 - up to now just NLO - includes isospin breaking and elmag. corrections b use of analyticity and unitarity Cabibbo ’04 Cabibbo, Isidori ’05 G´ amiz, Prades, Scimemi ’06 full dispersive approach ← − this talk c Nonrelativistic QFT Colangelo, Gasser, Kubis, Rusetsky, Bissegger, Fuhrer ’06, ’07, ’08 - nonrelativistic approach - double expansion in velocities and scattering lengths - possible to add photons (cf. Gevorkyan, (Madigozhin), Tarasov, Voskresenskaya ’06, ’07) - interesting alternative approach 6 /20 M. Zdr´ ahal, K. Kampf Dispersive analysis of cusps

Theoretical approaches - Overview a direct computation from χ PT (with weak part) Bijnens, Borg ’04, ’04, ’05 - up to now just NLO - includes isospin breaking and elmag. corrections b use of analyticity and unitarity Cabibbo ’04 Cabibbo, Isidori ’05 G´ amiz, Prades, Scimemi ’06 full dispersive approach ← − this talk c Nonrelativistic QFT Colangelo, Gasser, Kubis, Rusetsky, Bissegger, Fuhrer ’06, ’07, ’08 - nonrelativistic approach - double expansion in velocities and scattering lengths - possible to add photons (cf. Gevorkyan, (Madigozhin), Tarasov, Voskresenskaya ’06, ’07) - interesting alternative approach 6 /20 M. Zdr´ ahal, K. Kampf Dispersive analysis of cusps

Theoretical approaches - Use of analyticity and unitarity Cabibbo 04 Cabibbo, Isidori 05 G´ amiz, Prades, Scimemi 06 Dispersive approach 08 1 2 NLO NNLO NNLO NNLO Order Ass’tions (inputs) ( p 2 ) K → 3 π = ( p 2 ) amplitudes = first O real part of A +;00 ≈ real part of A +;00 ≈ O first order polynomial second order polynomial isospin symm. result of χ PT order polynomial Parametri- s > 4 m 2 � | s − 4 m 2 zation � A + Bv ( s ) + , + | A +;00 = v ( s ) = A +;00 = Re + Im s < 4 m 2 s A + iBv ( s ) + , direct computation: imaginary part of the amplitude - given by unitarity discontinuity ( ∼ im.part) ππ - scattering relations - given by (generalised) computation → toy model unitarity relations Method of Lagrangian the full amplitude - given by reconstruc- tion theorem from discontinuity omits diagram assumes simple analytic structure of the amplitude the correct analytic struc- of method Problem π 0 ⇒ some contributions are missed out ture of amplitudes and π 0 π 0 the correct integration K L π 0 π 0 contours taken into ac- count → complicated 7 /20 M. Zdr´ ahal, K. Kampf Dispersive analysis of cusps

Theoretical approaches - Use of analyticity and unitarity Our dispersive approach discontinuity ( ∼ imaginary part) of A +;00 given by (generalised) unitarity relations the full amplitude given by reconstruction theorem from this discontinuity includes second order rescattering the correct analytic structure of the amplitudes and correct integration contours taken into account does not take explicitly into account photons ⇒ full-featured approach based just on the unitarity, analyticity (subtracted dispersion relations), crossing symmetry and chiral power-counting 8 /20 M. Zdr´ ahal, K. Kampf Dispersive analysis of cusps

Dispersive approach for now we ignore photons, CP violation instead of computing K → πππ amplitudes directly, we use crossing symmetry (analytic continuation) of the Kπ → ππ amplitudes to the decay region partial wave decomposition: A ( s, t, u ) = 16 π ( f 0 ( s ) + 3 f 1 ( s ) cos θ ) + A ℓ ≥ 2 , Re A ℓ ≥ 2 ∼ O ( p 4 ) , Im A ℓ ≥ 2 ∼ O ( p 8 ) , Re f 0 , 1 ( s ) ∼ O ( p 2 ) , Im f 0 , 1 ( s ) ∼ O ( p 4 ) . Reconstruction theorem Assuming validity of (subtracted) DR’s (and further conditions), we can reconstruct the amplitude of the process AB → CD : Stern, Sazdjian, Fuchs ’93 M.Z., Novotn´ y ’08 S ( s, t ; u )= R + Φ 0 ( s ) + [ s ( t − u ) + ( m 2 A − m 2 B )( m 2 C − m 2 D )]Φ 1 ( s ) + crossed channels + O ( p 8 ) , R - third order polynomial in s, t, u with same symmetries as S ( s, t ; u ) , ∞ Φ 0 ( s )= 16 s 3 dx Im f 0 ( x ) � , x 3 x − s Σ ∞ Φ 1 ( s )= 48 s 3 � dx Im f 1 ( x ) CD ( x ) , x 3 ( x − s ) λ 1 / 2 AB ( x ) λ 1 / 2 Σ and similar for the t − and u − crossed channel � s − ( m X + m Y ) 2 �� s − ( m X − m Y ) 2 �� � λ XY ( s ) = 9 /20 M. Zdr´ ahal, K. Kampf Dispersive analysis of cusps