K π form factors from a dispersive approach Emilie Passemar* Indiana University/Jefferson Laboratory « Current and Future Status of the First-Row CKM Unitarity » Workshop UMass Amherst, Amherst, USA., May 17 2019 * Supported by NSF Emilie Passemar
Outline : 1. Introduction and Motivation 2. Dispersive representation of the K π form factors 3. Improvement: Combination of τ → K πν τ and K l3 decays 4. Applications 5. Conclusion and outlook 2
1. Introduction and Motivation Emilie Passemar
1.1 Precise test of the Standard Model Studying � and K l3 decays indirect searches of new physics, • several possible high-precision tests: � Extraction of V us � � l , e K � � W � � � K l l � � � � e � l , � V � us � � � � 1 2 � 3 l � � with I dt F t f , ( ), t f ( ) t l 2 N f (0)V I � � / 0 0 8 K � � � m / 0 us / 0 K l 3 K � / 0 K From Lattice QCD f � (0) ? 2 2 2 � � � Test of unitarity f � (0) V us V us V V V 1 ud us ub Negligible 0 + � 0 + K l3 decays (B decays) � decays 4 Emilie Passemar 4
1.1 Precise test of the Standard Model Studying � and K l3 decays indirect searches of new physics, • several possible high-precision tests: � Extraction of V us � � l , e K � � W � � � K l l � � � � e � l , � V � us � � � � 1 2 � 3 l � � with I dt F t f , ( ), t f ( ) t l 2 N f (0)V I � � / 0 0 8 K � � � m / 0 us / 0 K l 3 K � / 0 K Knowledge of the two form factors: � � � � � � � � � � � � � � � � � � � K K ( p ) s u K(p ) = p p p p f ( ) t p p ( ) f t � � � � � � � � K K � K � K � 0 � � t t scalar vector f ( ) t � � � � � 2 2 2 � � 0, t q ( p p ) ( p p ) , f ( ) t � � � � K 0, � f (0) � 5 Emilie Passemar 5
1.2 Callan-Treiman Low Energy Theorem � Bernard, Oertel, E.P., Stern’06 Bernard, Oertel, E.P., Stern’06, ‘08 • Callan-Treiman theorem: us F V F 1 � � � � � � K � � ud K C f ( ) V r � 0 K CT CT ud us F f (0) F V f (0) V � � � � � 2 2 m m � B exp = 1.2446(41) K Very precisely known from Br(Kl2/ � l2), � (Ke3) and V ud ( ) r = = 1 ln C 0.2141(73) − Δ = − ± 3 ( 3.5 8).10 • In the Standard Model : SM CT � � � � – NLO value + large error bars in � r ≠ agreement with – 1 • In presence of new physics, new couplings : Bijnens&Ghorbani’07 u ℓ Kastner & Neufeld’08 • Ex: H + + " s,d ν s 6 Emilie Passemar
1.2 Callan-Treiman Low Energy Theorem CT point ⎡ ⎡ ⎤ ⎤ Δ = 2 − 2 m m ⎡ ⎡ = − ⎤ ⎤ 2 t ( m m π ) ⎣ ⎣ ⎦ ⎦ ⎣ ⎣ ⎦ ⎦ π π π π 0 K K K Physical Region ? 7 Emilie Passemar
1.3 Test of New Physics Test of New Physics : � u , e SUSY loops g W g � Z’, Charged Higgs, � Right-Handed s � , e Currents,…. V us [E.g. Bernard et al’06,’07, Deschamps et al’09, Cirigliano et al’10, Jung et al’10, Buras et al’10…] 8 Emilie Passemar
1.4 K π form factors � • accessible in K e3 and K � 3 decays f ( ) t � • only accessible in K � 3 (suppressed by m l2 /M K2 ) + correlations f t 0 ( ) difficult to measure Data from Belle and BaBar on � � K �� � decays ( Belle II , SuperB soon!) • Use them to constrain the form factors and especially f 0 � � K K W W • � � K �� � decays � � � � V � � us Hadronic matrix element: Crossed channel � � � � � � � � � � � � � � � � � � � K K K s u 0 = p p p p f ( ) s p p ( ) f s � � � � � � � K � K � K � 0 � � s s vector scalar � � � 2 2 with s q ( p p � ) K 8 9 Emilie Passemar
1.5 Parametrization of the K π form factors p p g g + + + + - • ( p ) s u K(p ) = f ( )( t p p ) f ( )( t p p ) p µ + p µ - p µ K K K f ( ) t t f t ( ) = = = + + + f ( ) t f t ( ) f ( ) t f ( ) t = = = = f ( ) t s , Normalisation: and + - + s - 2 2 m m p f (0) 0 0 f + (0) + K t 1 t = = + + l l ' + l l " + f ( ) t 1 ... • Taylor Expansion: + + + + + ,0 ,0 ,0 2 2 m 2 m p p p p Ok for K l3 but can not combine with tau data and large correlations [Franzini, Kaon’08] Only slope accessible for the scalar FF 10 Emilie Passemar
1.5 Parametrization of the K π form factors p p g g + + + + - • ( p ) s u K(p ) = f ( )( t p p ) f ( )( t p p ) p µ + p µ - p µ K K K t f ( ) t f t ( ) = = + + f t ( ) f ( ) t f ( ) t = = = = = + Normalisation: and f ( ) t f ( ) t s , f + - s - 2 2 m m p + 0 0 f + (0) f (0) K + t 1 t = = + + l l ' + l l " + f ( ) t 1 ... • Taylor Expansion: + + + + + ,0 ,0 ,0 2 2 m 2 m p p p p Ok for K l3 but can not combine with tau data and large correlations • Pole parametrization: 2 m = V S , f ( ) t with m V,S to be fitted from data + ,0 - 2 m t V S , Ok for K l3 but can not combine with tau data: will explode at the resonance mass! Ok for vector but not so obvious for scalar 11 Emilie Passemar
2. Dispersive representation of the K π form factors Emilie Passemar
2.1 Introduction Parametrization to analyse both K l3 and � • – Vector form factor: Dominance of K*(892) resonance � � f s � � � �� K decays � K 3 decays � � l � f ( ) s BW s ( ) � i 2 � � i s 1 s � � � � � � ' '' � � f ( ) s 1 ... � � � 2 2 m 2 m � � � � � � � m m � K � � � m � m � m m � s [GeV] K 13 Emilie Passemar
2.1 Introduction Parametrization to analyse both K l3 and � • – Scalar form factor: No obvious dominance of a resonance � � f s 0 � � �� K decays K 3 decays � l 2 � � � s s � � � � � ' '' � � f ( ) s 1 ... 0 0 0 2 2 m � m � � � s � f 0 ( ) ? � � � m m � K CT � � � m � m m � s [GeV] K 14 Emilie Passemar
2.2 Dispersive representation for the form factors • Parametrization to analyse both K l3 and Unitarity : τ K πν τ Use dispersion relations ⎡ ⎤ ⎦ ∝ t ℓ I ∗ ( s ) f 0, + ( s ) disc f 0, + ( s ) ⎣ ⎡ φ + ,0 ( s ') ⎤ s ds ' ∞ ∫ f + ,0 ( s ) = exp ⎢ ⎥ • Omnès representation: π s ' − s − i ε s ' s th ⎣ ⎦ ϕ +, 0 (s) : phase of the form factor < < φ φ = = δ δ s s : ( ) s ( ) s - + π in ,0 K K π scattering phase ≥ φ + s s : ( ) s ( ) - unknown 2 ≡ + s m m π in ,0 th K ( ) φ φ = = φ φ = = π ± π ( ) s ( ) s → f ,0 ( ) s 1/ s + + + + ,0 ,0 as + Brodsky & Lepage • Subtract dispersion relation to weaken the high energy contribution of the phase. Improve the convergence but sum rules to be satisfied! Emilie Passemar 15
2.2 Dispersive representation Bernard, Oertel, E.P., Stern’06,’09 s • Dispersion relation with n subtractions in : ( ) ⎡ ⎡ ⎤ ⎤ n φ − ( ') s s s ds ' ∞ ∫ + = = ⎢ ⎢ + + ,0 ⎥ ⎥ f ( ) s exp P ( ) s ( ) + − ,0 n 1 π π − − − − ε ε n s ' s i − ⎢ ⎢ ⎥ ⎥ s s ' s ⎣ ⎣ ⎦ ⎦ th f 0 ( ) s Ø dispersion relation with 2 subtractions: 1 in s=0 and 1 in s= Δ K π [Callan-Treiman] ( ) ⎡ ⎤ ⎛ ⎞ ln C + s − Δ K π φ 0 ( s ') s ds ' ∞ ∫ f 0 ( s ) = exp ⎢ ⎥ ⎜ ( ) s ' − s − i ε ⎟ ( ) Δ K π π ( ) s ' − Δ K π 2 s ' ⎝ ⎠ m K + m π ⎢ ⎥ ⎣ ⎦ For s<s in :K � scattering phase For s < s in : K π scattering phase extracted from the data extracted from the data Buettiker, Descotes-Genon, & Moussallam ’ 02 = Δ ln C ln f ( ) λ ' 1 parameter to fit to the data: 2 parameters to fit to the data and π K 0 Emilie Passemar 16
2.2 Dispersive representation Bernard, Oertel, E.P., Stern’06,’09 s • Dispersion relation with n subtractions in : ( ) ⎡ ⎡ ⎤ ⎤ n φ − ( ') s s s ds ' ∞ ∫ + = = ⎢ ⎢ + + ,0 ⎥ ⎥ f ( ) s exp P ( ) s ( ) + − ,0 n 1 π π − − − − ε ε n s ' s i − ⎢ ⎢ ⎥ ⎥ s s ' s ⎣ ⎣ ⎦ ⎦ th f ( ) s Ø dispersion relation with 2 subtractions in s=0 + ⎡ ⎤ φ + ( s ') 2 + s 2 s ds ' ∞ ∫ f + ( s ) = exp λ + ' ⎢ ⎥ ( ) ( ) π s ' − s − i ε 2 s ' 2 m π m K + m π ⎣ ⎦ Extracted from a model including 1 resonances K*(892) a la Gounaris-Sakourai λ + ' 1 parameter to fit to the data: Emilie Passemar 17
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