Kaon Decays in the Standard Model

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1 Kaon Decays in the Standard Model Helmut Neufeld University of Vienna 13 December 2012, Wien

2 Review: Kaon Decays in the Standard Model V. Cirigliano, G. Ecker, H. N., A. Pich, J. Portoles, Rev. Mod. Phys. 84 (2012) 399 Discussion of all decay modes with BRs > Theoretical framework Estimates of low energy constants Leptonic and semileptonic decays Nonleptonic decays Rare and radiative decays

3 Theoretical Framework Short-distance description SM strangeness-changing transitions via W exchange M K M W integrate out W effective Lagrangian at E = M K W exchange between quark current and leptonic current L semilept. eff = G F 2 S 1/2 EW lγ µ (1 γ 5 )ν l ū i γ µ (1 γ 5 )V ij d j + h.c. W exchange between two quark currents L S=1 eff = G F 2 V ud V us e.g. Q 2 = sγ µ (1 γ 5 )u ūγ µ (1 γ 5 )d gluonic corrections Q 1, Q 3 -Q 6 C i (µ)q i (µ) virtual electromagnetic interactions Q 7 -Q 10 i

4 Theoretical Framework Short-distance description ctd. leptons in final state Q 11 -Q 13 Q 13 Q( νν) = sγ µ (1 γ 5 )d νγ µ (1 γ 5 )ν exchange of two W bosons (box diagram) L S=2 eff = G 2 F M2 W (4π) 2 C S=2 (µ)q S=2 (µ) Q S=2 = sγ µ (1 γ 5 )d sγ µ (1 γ 5 )d

5 Theoretical Framework Chiral Perturbation Theory Low energy effective theory of the SM Construct the most general QFT consistent with the symmetries of the SM in terms of the asymptotic fields relevant at low energies chiral symmetry of QCD in the limit m u = m d = m s = 0 8 Goldstone bosons π, K, η SU(3) L SU(3) R SSB SU(3) V interaction of the pseudoscalars vanishes for p 0 m u, m d, m s 0 M 2 Bm q chiral counting: orders of p 2 M 2 LECs add addition degrees of freedom: photon, light leptons LECs not restricted by chiral symmetry

6 Chiral Lagranian L chiral order (# of LECs) loop order L p 2(2) + L odd p 4 (0) + LG S=1 8 p (1) + L S=1 2 G 27 p (1) L = L emweak G 8 e 2 p 0 (1) + L em e 2 p 0 (1) + L lepton kin (0) + L p 4(10) + L odd p 6 (23) + L S=1 G 8 p (22) + L S=1 4 G 27 p (28) L L emweak G 8 e 2 p 2 (14) + L em e 2 p 2 (13) + L lepton e 2 p 2 (5) + L p 6(90) L 2

7 Estimates of Low-Energy Constants Matching between CHPT and the underlying SM LECs from low-energy data Resonance saturation Lattice determinations

8 Leptonic and Semileptonic Decays K l2 (and π l2 ) decays Electromagnetic corrections Extraction of V us /V ud (K,π) The ratio R e/µ K l2γ K l3 Electromagnetic effects in Kl3(γ) decays Quark mass ratios and Kl3 decays Form factors and phase space integrals SU(3) breaking effects in f K 0 π + (0) Determination of V us and CKM unitarity tests T violation in K µ3 decays K l3γ K l4 K e5

9 Leptonic and Semileptonic Decays K l2 (and π l2 ) decays Electromagnetic corrections Extraction of V us /V ud (K,π) The ratio R e/µ K l2γ K l3 Electromagnetic effects in Kl3(γ) decays Quark mass ratios and Kl3 decays Form factors and phase space integrals SU(3) breaking effects in f K 0 π + (0) Determination of V us and CKM unitarity tests T violation in K µ3 decays K l3γ K l4 ππ scattering lengths from K e4 data K e5

10 Nonleptonic decays K ππ ππ phase shifts from K ππ decays ɛ /ɛ CP violation in K 0 -K 0 mixing K 3π CP violation in K 3π decays ππ scattering lengths from K 3π decays near threshold

11 Nonleptonic decays K ππ ππ phase shifts from K ππ decays ɛ /ɛ CP violation in K 0 -K 0 mixing K 3π CP violation in K 3π decays ππ scattering lengths from K 3π decays near threshold

12 Rare and radiative decays K πν ν K ππν ν K γ ( ) γ ( ) KS γγ K L γγ K S γl + l KL γl + l KL γl + 1 l 1 l+ 2 l 2 K l + l K S l + l KL l + l

13 Rare and radiative decays K πν ν K ππν ν K γ ( ) γ ( ) KS γγ K L γγ K S γl + l KL γl + l KL γl + 1 l 1 l+ 2 l 2 K l + l K S l + l KL l + l

14 Rare and radiative decays ctd. K ππγ ( ) K + π + π 0 γ K L π + π γ KS π + π γ K ππl + l Other decays K 0 γγγ KL γγl + l KL γν ν K S l + 1 l 1 l+ 2 l 2 K L, K S π 0 π 0 γ KL, K S π 0 π 0 γγ K 3πγ

15 K l2 (and π l2 ) decays V us/v ud from decay widths and lattice data, V. Cirigliano, H. N., PLB 700 (2011) 7 Γ Kl2(γ) Γ πl2(γ) = Vus 2 V ud 2 F 2 K F 2 π M K ± M π ± ( ) 1 m 2 l /M 2 2 ( ) K ( ± ) 1 m 2 l /M δem + δ SU(2) π ± F K /F π in the isospin limit (m u = m d, e = 0) δ EM = α π ( F (m l /M K ) F (m l /M π) + 3 Z ) ln M2 K 4 Mπ 2 = (17), M 2 π ± M2 π 0 = 8παZF 2 0 δ SU(2) = [ 3 ε 4 3 (F K /F π 1) + 1 3(4π) 2 F 2 0 ( )] MK 2 Mπ 2 Mπ 2 ln M2 K Mπ 2 = (5)(11), ε = 3 m d m u = (13), F 4 m s m K /F π = 1.193(6) from FLAG

16 Update of V us /V ud V us F K = V ud F π Experimental input: ( ΓKl2(γ) M ) 1/2 π ± 1 m 2 l /M 2 ( π ± 1 δem Γ πl2(γ) M K ± 1 ml 2 /2 δ SU(2) /2 ) /M2 K ± = (25) = (5) ( ) 1/2 ΓKl2(γ) Γ πl2(γ) Γ πµ2(γ) = (7) (µs) 1, Γ Kµ2(γ) = 51.33(13) (µs) 1 PDG 2012 Lattice input: F K /F π = 1.193(6) (FLAG) V us /V ud = (12)

17 Isospin Breaking on the Lattice G. M. de Divitiis et al. (RM123 Collaboration), JHEP (2012) 1204 Method: expand path integral in powers of m d m u kaon masses kaon decay constant K l3 form factor neutron-proton mass splitting δsu(2) lattice = (7) to be compared with δchpt SU(2) = (12)

18 The ratio R K e/µ V. Cirigliano, I. Rosell, JHEP 10 (2007) 005; Phys. Rev. Lett. 99 (2007) Re/µ K = Γ ( ) ( ) 2 K e2(γ) me MK 2 = 2 m2 e Γ Kµ2(γ) m µ MK 2 (1 + δ EM) m2 µ = 2.477(1) 10 5 Two-loop calculation in CHPT with virtual photons and leptons Large-N C calculation of an associated counterterm Summation of leading logarithms α n ln n (m µ/m e) (Marciano, Sirlin 1993) Most recent NA62 result: Re/µ K = 2.488(10) 10 5 Full 2007 data set: article to be submitted in 2012, G. Ruggiero, CKM 2012

19 K πlν (K l3 ) decays Γ Kl3(γ) = G F 2 V us 2 MK 5 C K 2 ( ) 128 π 3 S EW f K 0 π + (0) 2 I (0) Kl (λ i) 1 + δem Kl + δkπ SU(2) C K = 1 for K 0 l3, C K = 1/ 2 for K + l3 I (0) Kl phase space integral δ Kπ SU(2) = ( f Kπ + (0)/f K 0 π + (0) )2 1 Long-distance EM corrections δ Kl EM = δ Kl EM(D 3) + δ Kl EM(D 4 3) Short-distance factor S EW = 1 + 2α π A. Sirlin, 1977; 1981; W.J. Marciano, A. Sirlin, 1993 ( 1 α s ) 4π ln M Z M ρ + O ( αα s ) π = (5) 2

20 Electromagnetic effects in K l3(γ) decays V. Cirigliano, M. Giannotti, H. N., JHEP 11 (2008) 006 δ Kl EM (D 3)(%) δ Kl EM (D 4 3)(%) δ Kl EM (%) Ke ± 0.22 K e3 ± ± 0.25 Kµ ± 0.22 K µ3 ± ± 0.25

21 Electromagnetic effects in K l3(γ) decays ctd. Differential decay distribution d 2 Γ dy dz = G F 2 V us 2 MK 5 C K 2 [ ρ ] 128 π 3 S EW f+ Kπ (0) 2 (0) (y, z) + δ ρ EM (y, z) y = 2p K p l /M 2 K = 2E l/m K, z = 2p π p K /M 2 K = 2E π/m K

22 Ratio δ ρ EM (y, z)/ ρ (0) (y, z) for K 0 e3

23 Ratio δ ρ EM (y, z)/ ρ (0) (y, z) for K + e3

24 Ratio δ ρ EM (y, z)/ ρ (0) (y, z) for K 0 µ3

25 Ratio δ ρ EM (y, z)/ ρ (0) (y, z) for K + µ3

26 Isospin Violation in K l3 decays CHPT (quark mass ratios & deviation from Dashen s limit): A. Kastner, H. N., 2008 δ K ± π 0 SU(2) = 0.058(8) Combine recent K l3 data with δ K ± π 0 SU(2) = 2Γ K + l3 Γ K 0 l3 I K 0 l I K + l ( MK 0 M K + ) 5 1 ( δ K + l EM ) δk 0 l EM δ K ± π 0 SU(2) = 0.054(8)

27 Determination of V us and CKM unitarity tests Experimental average: V us f + (0) = (5) Recent update M. Moulson, CKM 2012, Cincinnati Lattice (RBC/UKQCD 2010): f + (0) = (34)( )(14) V us = (5) exp (12) th

28 Fit of V us, V ud V. Cirigliano, H. N., PLB 700 (2011) 7 Input: V ud = (22) from , J.C. Hardy, I.S. Towner, 2009 V us/v ud from K l2 /π l2 V us Kl3 = (5) exp(12) th from K l3 decays and f +(0) from lattice V ud = (22), V us = (9) CKM unitarity test CKM = V ud 2 + V us 2 + V ub 2 1 = (6)

29 Vus unitarity Vus/Vud (K!2/"!2) fit Vus (Kl3) Vud (0 + # 0 + ) Vud

30 K ππlν (K l4 ) decays ππ scattering lengths from K e4 data ππ s-wave scattering lengths: a 0 = 7M2 π 32πFπ 2 = }{{} tree a 2 = M2 π 16πFπ 2 = }{{} tree }{{} one loop }{{} one loop }{{} 0.220(5) }{{} two loop disp. th. + CHPT } {{} (10) }{{} two loop disp. th. + CHPT most precise prediction: G. Colangelo, J. Gasser, H. Leutwyler, 2001

31 Results from NA48/2 K e4 data J. R. Batley et al., 2010 two-parameter fit: a 0 = (128) stat (50) syst (37) th a 2 = (86)(34)(28) alternative: using additional theoretical constraint one-parameter fit: a 0 = (49)(18)(64) (a 2 = ) from constraint to be compared with theory: a 0 = 0.220(5), a 2 = (10)

32 -0.01 NA48/2 Ke4 ( ) UB CHPT with isospin corrections no isospin corrections

33 ππ scattering lengths from K 3π decays near threshold π + -π 0 mass difference cusp in the M π 0 π 0 distribution at M π 0 π 0 = 2M π + K π π 0 + K π π + π 0 π 0 π π 0 seen in: K ± π ± π 0 π 0 J.R. Batley et al. (NA48/2), 2006,2009 K L 3π 0 E. Abouzaid et al. (KTeV), 2008

34 Cusp in K 3π (theory) a 0 a 2 A(π + π π 0 π 0 ) threshold original idea: P. Budini, L. Fonda, 1961; N. Cabibbo, 2004 unitarity analyticity expansion of singular term in M π 0 π 0 powers of scattering lengths: N. Cabibbo, G. Isidori 2005 unitarity, analyticity, CHPT: E. Gámiz, 2007 distribution in two-loop dispersive representation of K 3π amplitudes in the presence of isospin breaking: K. Kampf et al., 2009 most advanced approach: nonrelativistic effective field theory (NRQFT) K 3π amplitudes expanded in powers of the scattering lengths and of the pion momenta in the K rest frame: G. Colangelo et al., 2006; M. Bissegger et al., 2007; J. Gasser et al., 2011 inclusion of electromagnetic corrections: M. Bissegger et al., 2009

35 NA48/2 combined Ke4 + Cusp (stat. + syst.) errors 68% CL contour CHPT DIRAC Cusp Ke

36 Results for the s-wave ππ scattering lengths a 0 a 2 a 0 a 2 K 3π (48)(25)(14) (129)(94)(18) K e (128)(50)(37) (86)(34)(28) theory 0.264(4) 0.220(5) (10) Impressive agreement between theory and experiment!

37 K πν ν d s W d W ν u, c, t u, c, t e Z 0 s W ν Z-penguin and box contributions to K πν ν

38 K πν ν ctd. A(s dν ν) m2 t MW 2 λ t + m2 c MW 2 ln M W λ c + Λ2 QCD λ u, m c M 2 W λ q = V qd V qs GIM mechanism: t contribution (large CP-violating phase): 68% c contribution: 29% u contribution: 3% large suppression of the u-quark (long-distance) contribution and complete dominance of the CP-violating contribution by the t-quark term K πνν decays are short-distance dominated

39 K πν ν ctd. low energies: dominated by Q 13 = sγ µ (1 γ 5 )d νγ µ (1 γ 5 )ν hadronic matrix element of Q 13 can be related to K l3 form factors very accurate theoretical prediction of K πνν rates Contributions of many authors: M. Lu, M.B. Wise, 1994; G. Buchalla, A. Buras, 1994, 1996, 1998, 1999; W.J. Marciano, Z. Parsa, 1996; G. Buchalla, G. Isidori, 1998; M. Misiak, J. Urban, 1999; A. Falk, A. Lewandowski, A. Petrov, 2001; G. Isidori, F. Mescia,, C. Smith, 2005; A. Buras, M. Gorbahn, U. Haisch, U. Nierste, 2005, 2006; J. Brod, M. Gorbahn, 2008; F. Mescia, C. Smith, 2007; J. Brod, M. Gorbahn, E. Stamou, 2011

40 Present status of K πν ν from: V. Cirigliano, Kaon Physics Working Group, Project X Physics Study, June mode SM experiment K + π + ν ν 7.81(75)(29) ( ) E787/949 K L π 0 ν ν 2.43(39)(6) < E391a

41 Summary Interplay of short- and long-distance physics EFT methods: operator product expansion and CHPT Resonance model and lattice calculations to determine LECs Excellent precision for semileptonic decays Electromagnetic contributions Strong isospin breaking CKM unitarity test Strong bound on effective scale Λ of dimension-six operators violating quark-lepton universality of charged current weak interactions: Λ > 11 TeV (90% CL) (V. Cirigliano et al., 2011)

42 Summary ctd. Test of lepton universality in Re/µ K physics) (very sensitive to new Extraction of ππ scattering lengths from K e4 and K 3π Impressive interplay of high precision measurements and refined theoretical methods Short distance dominated decays K πν ν Very accurate theoretical predictions in the SM Sensitive probe for new physics

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