Predictions on the second-class currents

Predictions on the second-class currents τ η ( ) ν τ decays Sergi Gonzàlez-Solís Institut de Física d Altes Energies Universitat Autònoma de Barcelona in collaboration with Rafel Escribano and Pablo Roig, PRD 94 (26) 348 4th International Workshop on Tau lepton physics Beijing, 9th september 26

Outline Introduction 2 η ( ) Form Factors Vector Form Factor Scalar Form Factor Breit-Wigner Dispersion relation: Omnès integral Coupled channels 3 Branching ratio predictions τ ην τ τ η ν τ η ( ) π + l ν l (l = e, µ) 4 Conclusions

Introduction Hadronic τ-decays Inclusive decays: τ (ūd, ūs)ν τ Fundamental SM parameters: α s(m τ ), m s, V us (see Tau properties (III) and QCD (I)) Exclusive decays: τ (PP, PPP,...)ν τ, (P = π, K, η ) τ Hadronization of QCD currents, Form Factors, resonance parameters (M R, Γ R ) ν τ W d = V ud d + V us s ū,k h5 Tau properties (II) (Escribano, Gonzàlez-Solís and Roig JHEP 3 (23) 39) η, η (Escribano, Gonzàlez-Solís, Jamin and Roig JHEP 49 (24) 42) this work (Escribano, Gonzàlez-Solís and Roig PRD 94 (26) 348) S.Gonzàlez-Solís TAU 6 9 september 26 3 / 24 hadronization

Introduction τ η ( ) ν τ decays Motivations τ η ( ) ν τ belong to the second-class current processes unobserved in Nature so far (Weinberg 58) G Parity G X = e iπiy C X = ( ) I C X G dγ µ u = + dγ µ u G η = η It is an isospin violating process (m u m d, e ) Sensitive to the intermediate vector and scalar resonances (ρ, ρ, a, a...) coupled to the ūd operator Purposes To describe the participating hadronic form factors To predict the decay spectra and to estimate the branching ratios To stimulate people from B-factories (Belle-II) to measure these decays S.Gonzàlez-Solís TAU 6 9 september 26 4 / 24

Introduction τ K η ( ) ν τ : Amplitude and decay width τ W ν τ η ( ) q 2 M 2 W = η ( ) dγ µ u = [(p η ( ) p π) µ + η ( ) q µ ] C V πη( ) πη ( )F s G F 2 V ud ū(p ντ )γ µ ( γ 5 )u(p τ ) η ( ) dγ µ ( γ 5 )u + (s)+ QCD K K + s, + +, PQ = M 2 P M 2 Q q µ C S η ( ) F π η ( ) (s) dγ (τ η ( ) ν τ ) d = G2 F M3 τ s 24π 3 s S EW V ud 2 π η( ) F+ () 2 ( s Mτ 2 ( + 2s M τ 2 ) q 3 (s) F π η ( ) η ( ) + (s) 2 + 3 2 η ( ) q 4s η ( )(s) F π η ( ) (s) 2 F π η ( ) +, (s) = F π η( ) +, (s), F π η ( ) F π η ( ) + () = CS η ( ) QCD K K + π η( ) +, () C V F () η ( ) η ( ) S.Gonzàlez-Solís TAU 6 9 september 26 5 / 24 2 )

η ( ) Form Factors Framework and mixing Chiral Perturbation Theory: valid up to the first resonance ρ mass Large-N C : to include η singlet Resonance Chiral Theory: to access resonance regime π -η-η mixing (P. Kroll, Mod. Phys. Lett. A2, 2667 (25)) π η η = ε πη cθ ηη + ε πη sθ ηη ε πη cθ ηη ε πη sθ ηη ε πη cθ ηη sθ ηη ε πη sθ ηη cθ ηη where ε πη ( ) and θ ηη are the π -η ( ) and η-η mixing angles π 3 η 8 η S.Gonzàlez-Solís TAU 6 9 september 26 6 / 24

η ( ) Form Factors Vector Form Factor Vector Form Factor: RChT The vector contribution current occurs via π η η mixing, so it is O(ε πη ( )) and hence suppressed η ( ) η ( ) = F π η + (s) F π η + (s) = ( τ π ν τ + + P ρ (77) ρ (45) P η ( ) + επη ) F V G V s ε + πη V =ρ,ρ,ρ F 2 MV 2 s suppresion Fujikawa et. al. PRD78 726 (Belle) s 2 F ΠΠ.. η ( ) + cancel each other Belle data η ( )..5..5 2. 2.5 3. s GeV 2 S.Gonzàlez-Solís TAU 6 9 september 26 7 / 24

η ( ) Form Factors Scalar Form Factor Scalar Form Factor: Breit-Wigner F π η ( ) (s) = c π η ( ) 8cm(cm c d ) 2mK 2 m2 π F 2 M 2 S + 4cm F 2 (c m c d )2m2 π + c d (s + mπ 2 m 2 ) η ( ) M 2 S s Imposing F π η ( ) (s) to vanish for s c d c m = and 4c d c m = F 2 Resummation of self-energy insertions in propagator F π η ( ) (s) = c π η ( ) c π η = cos θ P 2 sin θ P c π η = cos θ P + sin θ 2 P i + Σ(s) + Σ(s) Σ(s) +...= s MK 2 +Σ(s) resonance: M S = 98(2), Γ S = 75(25) M 2 S + η ( ) M 2 S s im SΓ S (s) s ΠΗ, ΠΗ' F 25 2 5 5..5..5 2. s GeV S.Gonzàlez-Solís TAU 6 9 september 26 8 / 24

η ( ) Form Factors Scalar Form Factor Scalar Form Factor: Breit-Wigner 5 5 η η mixing: Next-to-leading order prediction in Res. ChPT F π η ( ) + () = CS η ( ) QCD K K + π η( ) C V F () η ( ) η ( ) F π η ( ) + () = ε πη ( ) ε F π η ( ) () = c π η ( ) M 2 S + πη = 9.8(3) 3 η ( ) ε πη M 2 = 2.5(.5) S 4 Breit-Wigner with 2 resonances: a (98) and a (45) B. Wigner a 98 B. Wigner a 98 a 45..5..5 s ΠΗ' F 4 2 8 6 4 2 B. Wigner a 98 B. Wigner a 98 a 45..5..5 S.Gonzàlez-Solís TAU 6 9 september 26 9 / 24

η ( ) Form Factors Scalar Form Factor Scalar Form Factor: Dispersion relation Analyticity and elastic unitarity through a dispersion relation F π η ( ) + (s) = π η ( ) s th ds ImF π η( ) + (s ) s s iε η ( ) η ( ) Im = T ImF π η ( ) + (s) = σ π η ( )(s)f π η ( ) + T (s) = F π η ( ) + sin δ π η ( ) η ( ) (s)e iδπ (s) Watson theorem: phase of F π η ( ) + (s) is δ π η ( ) (s) in the elastic approx. Omnès solution (Omnès 58) F π η ( ) + (s) = P(s) exp s s π ds δ π η( ) (s ) s th (s s )(s s iε) S.Gonzàlez-Solís TAU 6 9 september 26 / 24

η ( ) Form Factors Scalar Form Factor Scalar Form Factor: Omnès integral Analyticity and elastic unitarity through the Omnès solution π s s δ η ( ) (s) = P(s) exp ds, (s ) = P(s)Ω(s) π s th (s s )(s s iε) F π η ( ) Elastic unitarity: Form factor phase= δ η ( ) 2 2 elastic scattering δ π η ( ), (s) = arctan Imt,(s) Ret, (s), t N, (s),(s) = + g(s)n, (s) = N(s) D(s) N, : U(3) U(3) amplitudes in RχT (Guo-Oller: Phys.Rev. D84 (2) 345) c d = c d / 3 c m = c m/ 3 c d = 9.8 +2. 5.2 MeV c m = 4.9 +3.9 9.2 MeV M a,s 8 = 397 +73 6 MeV M S = +3 63 MeV S.Gonzàlez-Solís TAU 6 9 september 26 / 24

η ( ) Form Factors Scalar Form Factor Scalar Form Factor: Omnès Assuming F π η ( ) (s) to behave as s F πη( ) (s) = P(s)Ω(s), P(s) constant. Our choice: P(s) = F Breit Wigner () F ΠΗ s 8 6 4 2..5..5 2. 2.5 3. s GeV s F ΠΗ' 5 2 5 2 Omnès integral..5..5 2. s GeV S.Gonzàlez-Solís TAU 6 9 september 26 2 / 24

η ( ) Form Factors Scalar Form Factor Scalar Form Factor: Closed expression ( s/s F πη( ) pi ) (s) = i,j ( s/s zj ) D(s) D(s )F (s ) s p and s z: poles and zeros of D(s) = ( + g(s)n(s)) Iwamura, Kurihara, Takahashi 77 Kamal 79, Kamal, Cooper 8 Jamin, Oller, Pich s = F (s ) = F π η,bw () =.5 s z =.397 GeV N(s) = N πη πη s F ΠΗ' 5.. 5...5 Omnès Closed expression..5..5 2. 2.5 3. s GeV S.Gonzàlez-Solís TAU 6 9 september 26 3 / 24

η ( ) Form Factors Scalar Form Factor Scalar Form Factor: Coupled channels case F i (s) = π 2 j= s i Other cuts (K K, πη...) ds Σ j(s )F j (s )T i j (s ) (s s iε) F πη (s) = π ds σπη(s πη )F (s )T πη πη(s ) s th s s iε F πη (s) = π ds σπη(s πη )F (s )T πη πη (s ) s th s s iε Im Im η = + π η = η, η T ds σ πη (s πη )F (s )T πη πη (s ) s th2 s s iε + π η, η T η ds σ πη (s πη )F (s )T πη πη (s ) s th2 s s iε S.Gonzàlez-Solís TAU 6 9 september 26 4 / 24 η

η ( ) Form Factors Scalar Form Factor Scalar Form Factor: Coupled channels case (closed expression) ( s/s F πη( ) pi ) (s) = i,j ( s/s zj ) D(s) D(s )F (s ) s p and s z : poles and zeros of detd(s) Iwamura, Kurihara, Takahashi PTF 58 (977) Kamal 79, Kamal, Cooper 8 F (s) = ( F πη (s) F πη (s) ), D(s) = + g(s)n(s), s z =.397 GeV F (s ) = F BW () 5 2 Elastic ΠΗ coupled to ΠΗ' g(s) = ( g πη ), g πη F ΠΗ s 5 2 N(s) = ( N πη πη N πη πη N πη πη N πη πη ),..5..5 2. s GeV S.Gonzàlez-Solís TAU 6 9 september 26 5 / 24

η ( ) Form Factors Scalar Form Factor Scalar Form Factor: Coupled channels case (closed expression) ( s/s F πη( ) pi ) (s) = i,j ( s/s zj ) D(s) D(s )F (s ) s p and s z : poles and zeros of detd(s) Iwamura, Kurihara, Takahashi PTF 58 (977) Kamal 79, Kamal, Cooper 8 F (s) = ( F πη (s) F πη (s) ), s z =.397 GeV F (s ) = F BW () 5 Elastic ΠΗ' coupled to ΠΗ D(s) = + g(s)n(s), g(s) = ( g πη ), g πη s F ΠΗ' 5 5 N(s) = ( N πη πη N πη πη N πη πη N πη πη ),..5..5 2. s GeV S.Gonzàlez-Solís TAU 6 9 september 26 6 / 24

η ( ) Form Factors Scalar Form Factor η ( ) Form Factors: recapitulate 5 2 5 2 Elastic ΠΗ coupled to ΠΗ' ΠΗ coupled to KK ΠΗ coupled to ΠΗ' and KK Elastic 5 ΠΗ' coupled to ΠΗ ΠΗ' coupled to KK ΠΗ' coupled to ΠΗ and KK 5 5..5..5..5..5 Vector Form Factor: Driven by the π vector form factor Scalar Form Factor Breit-Wigner: with a (98) and a (45) resonances 2 Omnès solution: analyticity+elastic final state interactions 3 Closed Form: coupled-channels S.Gonzàlez-Solís TAU 6 9 september 26 7 / 24

Branching ratio predictions τ ην τ T H IS W O R K τ ην τ : Invariant mass distribution and Branching Ratio 6 ÿ dgêd s... -4 Full: vector + elastic Full: vector + 3 coupled channels Full: vector + B. Wigner H2 resl Vector BR V 5 BR S 5 BR 5 Reference.25.6.85 Tisserant, Truong 82.2.38.5 Bramón, Narison, Pich 87.5.6.2 Neufeld, Rupertsberger 94.36..36 Nussinov, Soffer 8 [.2,.6] [.2, 2.3] [.4, 2.9] Paver, Riazuddin.44.4.48 Volkov, Kostunin 2.3.2.33 Descotes-Genon, Moussallam 4.26(2).72 +.46.22.98(5) Breit-Wigner [a (98)].26(2).48 +.29.4.74(32) Breit-Wigner [a (98) + a (45)].26(2). +.2.3.36(4) Elastic Omnès solution.26(2).5(9).4(9) 2 coupled channels ( η to η ).26(2).86() 2.2() 2 coupled channels ( η to K K ).26(2).4(9).67(9) 3 coupled channels -5.6.8..2.4.6.8 s HGeVL BRexp BaBar < 9.9 5 95%CL (PRD 83 (2) 322) BR Belle exp < 7.3 5 9%CL (PoS EPS -HEP29, 374 (29)) S.Gonzàlez-Solís TAU 6 9 september 26 8 / 24

Branching ratio predictions τ ην τ τ η ν τ : Invariant mass distribution and Branching Ratio 7 ÿ dgêd s. -4-6 Full: vector + elastic scalar Full: vector + 3 coupled channels Full: vector + B. Wigner H2 resl Vector T H I S W O R K -8 -..2.4.6.8 2. s HGeVL BR V BR S BR Reference < 7 [.2,.3] 6 [.2,.4] 6 Nussinov, Soffer 99 [.4, 3.4] 8 [.6,.8] 7 [.6, 2.] 7 Paver, Riazuddin. 8 2.63 8 3.74 8 Volkov, Kostunin 2 [.3, 5.7] [2, 7 ] [.5,.3 9 ] Breit-Wigner ( res) [.3, 5.7] [5, 2 9 ] [.8, 2.6 9 ] Breit-Wigner (2 res) [.3, 5.7] [2 9, 4 8 ] [2.6 9, 4 8 ] Elastic Omnès solution [.3, 5.7] [2 7, 2 6 ] [2 7, 2 6 ] 2 cc ( η to η) [.3, 5.7] [3 7, 3 6 ] [3 7, 3 6 ] 2 cc ( η to K K ) [.3, 5.7] [ 7, 6 ] [ 7, 6 ] 3 coupled channels BR BaBar exp < 4 6 (9%CL), (PRD 86, 92 (22) S.Gonzàlez-Solís TAU 6 9 september 26 9 / 24

Branching ratio predictions η ( ) π + l ν l (l = e, µ) Branching Ratio estimates: η ( ) π + l ν l (l = e, µ) dγ d s = G2 F V udf + () 2 πη (CV )2 (s ml 2)2 24π 3 Mη 3 s {(2s + m 2 l )q3 πη F + (s) 2 + 3 4s 2 πηm l 2 q πη F (s) 2 }.5..5 Η Π e Νe Η Π Μ ΝΜ Η' Π e Νe 5 Η' Π Μ ΝΜ 4 3 2....2.3.4..2.4.6.8 Decay Descotes-Genon, Moussallam 4 Our estimate η π + e ν e + c.c..4 3.6 3 η π + µ ν µ + c.c..2 3.4 3 η π + e ν µ + c.c..7 7 η π + µ ν µ + c.c..7 7 S.Gonzàlez-Solís TAU 6 9 september 26 2 / 24

Conclusions Outlook τ η ( ) ν τ : second-class currents unseen in Nature Isospin-violating processes and hence suppressed Form Factors: Vector Form Factor: driven by τ π ν τ data Scalar Form Factor: Breit-Wigner, Omnès, coupled-channels We predict τ ην τ.7 5 and τ η ν τ O( 7 6 ) We encourage experimental groups (Belle-II) to pursue this decay mode S.Gonzàlez-Solís TAU 6 9 september 26 2 / 24

Back-up

Back-up Hadronic Matrix Element Taking the divergence we obtain on the L.H.S µ( sγ µ u) K + η ( ) = i(m s m u) su K + η ( ) = i K π C S K η ( ) F K η ( ) (s) () where PQ = M 2 P M2 Q, CS K η = / 6, C S K η = 2/ 3 on the R.H.S (vector current not conserved) iq µ K η ( ) sγ µ u = ic V K η ( ) [(m2 η ( ) m2 K )F K η ( ) + (s) sf K η ( ) (s)] (2) Equating eqs. (,2) allows us to relate F K η ( ) (s) with F K η ( ) (s) as F K η ( ) (s) = K η ( ) s C S K η ( ) C V K η ( ) K π F K η ( ) (s) + F K η ( ) + (s) K η ( ) The hadronic matrix element finally reads (q µ = (p η ( ) + p K ) µ + and q 2 = s) (3) K η ( ) sγ µ u = [(p η ( ) p K ) µ + K η ( ) q µ ] C V K η s ( )F K η ( ) + (s) + K π s qµ C S K η F K η ( ) ( ) (s) (4) S.Gonzàlez-Solís TAU 6 9 september 26 23 / 24

Back-up Scalar Form Factor: Closed expression Once subtracted dispersion relation F(s + iε) = F(s ) + s s ds σ(s )tij (s )F (s ) π s th (s s )(s s iε) = F(s ) + F(s + iε), F(s + iε) F(s iε) = 2iσ(s)t (s + iε)f(s + iε) = 2iσ(s)t (s + iε)[f(s ) + F(s + iε)] t = N/D Imt = σ(s) ImD(s) = Nσ(s) F(s + iε)d(s + iε) F(s iε)d(s iε) = 2iImD(s)F(s ), F(s + iε) = (s s ) D(s + iε) π ds ImD(s )F(s ) s th (s s )(s s) = D(s + iε) [D(s + iε) D(s )] F(s ) S.Gonzàlez-Solís TAU 6 9 september 26 24 / 24