Impact of Kaon Physics in determining the CKM matrix Gino Isidori CERN −Theory Division The Cabibbo angle The ε K hyperbola Rare decays
Impact of Kaon Physics in determining the CKM matrix Gino Isidori CERN −Theory Division & CERN −Choir A concert in three movements for soli [key experimental data], choir [secondary exp. info] and orchestra [th. instruments] The Cabibbo angle [ adagio maestoso, con moto ] The ε K hyperbola [ allegro ma non troppo ] Rare decays [ allegro con brio, quasi scherzoso ]
The Cabibbo angle adagio maestoso, con moto Soloist singers: K → π l ν l decays ( K l3 ) Choir: other semileptonic K decays [minor role] Th. instruments: Chiral Perturbation Theory (CHPT)
The rates of the four K l3 decays [ K=K + ,K L l=e, µ ] can be written as 2 M K 5 × | V us | 2 × | f + (0) | 2 × I ( df / dt ) Γ= G F kinematical integral: mild sensitivity to df + / dt vector form factor at zero (and f − / f + for l= µ) momentum transfer [ t =( p’ − p ) 2 =0] and e.m. corrections u γ µ s π p = C p’ � p µ f � t � p’ � p µ f � t K p’ ¯ CVC ⇒ f + (0) = 1 in the SU (3) limit m s = m u = m d Three main issues to address in order to extract | V us | : estimates of the SU (3) breaking term f + (0) − 1 e.m. corrections kinematical dependence of the form factors [ mainly an exp. issue ]
E.m. corrections: s u I. short−distance corrections to the s → u l ν l eff. Hamiltonian sizable [ ~ α log( µ had / M W ) ⇒ δΓ ~ 1% ] W well known Marciano & Sirlin, ’70− ’80 II. pure long−distance corrections (IR div. & bremss.) K π sizable [ ~ α log( M K / m e ) ⇒ δΓ ~ 1% ] Ginsberg, ’66− ’69 partially known (Coulomb corrections) ν III. structure−dependent (intermediate−scale) terms e small [ no large logs ⇒ δΓ ~ 0.1% ] model dependent Coherent analysis of the 3 effects (particularly II. + III.) possible in the framework of CHPT [non−trivial results at O ( e 2 p 2 )] Cirigliano et al. ’01
A crucial point in the analysis of e.m. corrections is the identification of I.R. safe observables. most convenient choice: Γ ( K l3 ) incl. = ∑ Γ K →π l ν l � n γ The recent work by Cirigliano et al. provides a clear prescription to separate, in this observable, known QED corrections (which modify spectra and norm.) from the local counterterms of O ( e 2 p 2 ) + 2 × I ( df / dt ) e.g.: Γ ( K e3 ) incl. ∝ | f + (0) | δ QED = − 1.27% δ CT = (+ 0.36 ± 0.16) % Are we sure that the (old) PDG data on Γ ( K l3 ) are completely inclusive ? ⇒ important exp. issue (together with the kinem. dependence of the f.f.) especially in view of new precise measurements
Th. estimates of f + (0) − 1 no linear corrections in ( m s − m u ) [Ademollo−Gatto theorem, ’64] K 0 π + | f + (0) | < 1 [Furlan et al. ’65] 2 0 Q us n K 0 π + 2 = 1− ∑ K | f + (0) | Leutwyler−Roos sum−rule: n ≠ π + K 0 π + δ = = 0 At O ( p 2 ) [LO in CHPT]: f + (0) − 1 At O ( p 4 ): finite (unambiguous) non−polynomial corr. induced by meson loops [ ~ m P log m P ⇒ ~ ( m s − m u ) 2 / m s ] numerically small: δ (4) = − 2.2% + π 0 K 0 π + K − precise determination of the ratio [ SU (2) ] f + (0) / f + (0) At O ( p 6 ): appearance of B 2 ( m s − m u ) 2 / Λ 4 χ local terms conservative estimate: δ (6) = −1 .6 ± 0.8 % [Leutwyler−Roos, ’84] ⇒ discussion in the Is it really conservative ? Can we do better by means of recent improvements in CHPT and/or by means of Lattice ? subgroup on V us
Extraction of | V us | ∆ V us ∆ f � 0 ∆Γ � 0.05 ∆λ + = 1 e.g. Γ ( K e3 ) [Cirigliano et al. ’01] : � Γ λ V us 2 f � 0 | V us | = 0.2207 ± 0.0024 ± 0.6% ± 0.4% ± 0.85% + K e3 substantial reduction possible f + (0) | V us | at KLOE within ~ 1 yr combined analysis of all K l3 modes: | V us | = 0.2187 ± 0.0020 K l3 [Calderon−Lopez Castro, ’01] + 0 + 0 K e3 K e3 K µ 3 K µ 3 At which precision we would like to know | V us | ? Reference figures provided by δ | V cb | ~ 5% ⇒ almost negligible impact of δ | V us | in the usual UT plane δ | V ud | ~ 0.08% (realistic ?) ⇒ | V us | unitarity = 0.2287 ± 0.0034 [ 2.5 σ discrepancy !]
The ε K hyperbola allegro ma non troppo Soloist singer: ε K Choir: CP−conserving data on K → 2 π [minor role] Th. instruments: Perturbative QCD, Lattice, CHPT, 1/ N C expansion, etc.
Master formula for ε K s s only one i π⁄ 4 ℑ M 12 � 2 ℑ A 0 e q u = u,c,t dim−6 ε K = ℜ M 12 W ℜ A 0 2 ∆ M K operator d d ∆ S = 2 ∼ ¯ ∗ = 1 0 H eff ∆ S = 2 K s d V � A ≡ Q L L H eff s d V � A ¯ 0 ¯ M 12 K 2 M K = 8 2 B K × 2 M K Q L L µ 3 f K 2 η tt F tt � 2 λ c λ t η ct F ct �λ c 2 η cc F cc ε K ∝ ℑ λ t B K × α(µ) −2/9 [1+ O ( α )] 1 ∗ V qd η ij = NLO QCD corrections λ q = V qs η [Buras et al. ’90, Herrlich & Nierste, ’95−’96] _ _ η [ (1 − ρ ) A 2 η tt F tt + P charm ] A 2 B K = 0.204 1.36 ± 0.07 0.30 ± 0.05 ρ - 1 1 0 ~ 4% error from pert. QCD ⇒ the key problem is B K
Non−Lattice estimates of B K Chiral limit [ CHPT at O ( p 2 ) ]: B K = 0.3 [ extracted from Γ ( K + → π + π 0 ) ] [ limit of light m s ] Factorization: B K = 1 [ → 3 / 4 for N C → ∞ ] [ limit of heavy m s ] Corrections to the chiral limit are potentially large and cannot be computed in a model − independent way [ ( m K / Λ χ ) 2 ~ 25% ⇒ O (1) effects are not so unlikely ] Subleading 1 / N C corrections decrease the leading 1 / N C result NLO 1 / N C + chiral limit ⇒ B K = 0.4 ± 0.1 [Pich & de Rafael, ’00] we are still far from a precise estimate of B K at the physical point low values are certainly more favoured the chiral limit result is an important test for Lattice approaches
Lattice estimates of B K NDR B K ( 2 GeV ) B K [ RGE−inv. ] 0.86 ± 0.06 JLQCD ’98 Kogut−Susskind 0.628(42) ± 0.14 [ quench. ] [Reference figure @ Lattice 2000] 0.91 ± 0.17 SPQ CD R ’01 Wilson [with subtr.] 0.71(13) 0.90 ± 0.13 SPQ CD R ’01 Wilson [Ward id.] 0.70(10) 0.79 ± 0.03 CP−PACS ’01 DWF [pert. ren.] 0.575(20) 0.70 ± 0.02 RBC ’01 DWF [non pert. ren.] 0.513(11) reasonable agreement with the chiral symmetry at ∆ I=3/2 amplitude in all cases finite lattice spacing study of quenching effects still both DWF analysis very preliminary shows a significance decrease of B K My conclusion: (δ B K ) tot ≥ 25% in the chiral limit CP−PACS
At which level can we shrink the ε K hyperbola ? Th. errors besides B K : NNLO perturbative corrections to H eff ( d =6) ⇒ ~ 4% (scale with B K ) 2 ), no hard GIM & large logs enhancement, indep. of B K ] d=8 operators [ O( G F ~ O ( m K 2 /[ m c 2 ln( m c / M W ) ] ) of the d=6 charm contribution ⇒ < 2% ~ Genuine long−distance effects ( ∆ S=1 × ∆ S=1) _ K 0 K 0 same parametrical suppression as d=8 terms but with a potential ∆ I=1/2 enhancement leading chiral contribution vanish by construction ⇒ small effects (~1%) suggested by explicit model calculations [ e.g. Donoghue & Holstein, ’84 ] and by ∆ m K (where l.d. terms are CKM enhanced) As long as (δ B K ) tot ≥ 10% we can forget about terms not included in H eff ( d =6)
Rare decays allegro con brio, quasi scherzoso → π + νν & K L → π 0 νν Soloist singers: K + [a beautiful but difficult part, which require expensive singers...] Choir: Dalitz decays ( K L → µ + µ − , K L,S → π 0 e + e − , K L → γ l + l − ,. .. ) [the most interesting choir part] Th. instruments: mainly Perturbative QCD & CHPT
− K → π νν Thanks to the "hard" GIM mechanism these decays are largely dominated by short−distance dynamics: s Λ QCD λ 2 (u) q=u,c,t + i m c λ 5 + box ⇒ A q ~ m q V qs V qd ∼ m c λ 2 * (c) 2 2 W Z λ q 5 5 m t λ + i m t λ 2 2 d (t) 2 Genuine ∆ S=1 O ( G F ) transition [ λ = sin θ c ] G F α X i @ NLO: H eff = λ c X c �λ t X t νν V � A sd V � A ¯ ¯ Buchalla & Buras ’94 2 2 2 π s W Misiak & Urban ’99 N.B.: the hadronic matrix element 〈 π | ( sd ) V−A | K 〉 is known from K l3 with excellent accuracy Marciano & Parsa, ’96
− Theoretical predictions for BR ( K → π νν ) within the SM: + Th. error dominated by the charm contribution Lu & Wise ’94 K Buchalla & Buras ’97 [ NNLO perturbative corr. ( + d =8 terms ) ] Falk et al . ’00 _ _ (SM) � 4 [ ( ρ−ρ c ) 2 + ( σ η) 2 ] � 11 = C | V cb | = 7.2 ± 2.1 × 10 BR K [error dominated by the ρ c = 1.40 ± 0.06 uncertainty on CKM param.] _ ⇒ 0.0 4 error on ρ around the origin of the UT plane Charm contribution suppressed by the CP structure Littenberg, ’89 K L Buchalla & Buras ’97 νν [ The state produced by is a CP eigenstate ] H eff ¯ Buchalla & G.I. ’98 2 2.3 ∗ V td ℑ V ts m t m t (SM) � 10 � 11 = 4.30 × 10 = 2.3 ± 1.3 × 10 BR K L 5 170 GeV λ th. error ~ 2% ! The best way to directly measure the area of the (full) UT
− + → π + νν Experimental status of K � →π � ν¯ ν B K � 1.75 × 10 � 10 = 1.57 � 0.82 2 events observed at BNL −Ε 787 (0.15 bkg) [hep−ex/0111091] central value 2 × SM ! δ B ~ 30% (assuming B SM ) expected before 2005 from BNL−E949 Littenberg ’02
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