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Impact of Kaon Physics in determining the CKM matrix Gino Isidori - - PowerPoint PPT Presentation
Impact of Kaon Physics in determining the CKM matrix Gino Isidori - - PowerPoint PPT Presentation
Impact of Kaon Physics in determining the CKM matrix Gino Isidori CERN Theory Division The Cabibbo angle The K hyperbola Rare decays Impact of Kaon Physics in determining the CKM matrix Gino Isidori CERN Theory Division & CERN
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The Cabibbo angle
Soloist singers: K → π l νl decays (Kl3) Choir: other semileptonic K decays [minor role]
- Th. instruments: Chiral Perturbation Theory (CHPT)
adagio maestoso, con moto
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vector form factor at zero momentum transfer [ t=(p’−p)2=0]
The rates of the four Kl3 decays [ K=K+,KL l=e,µ ] can be written as
Γ=GF
2 M K 5 ×|Vus| 2×|f+(0)| 2 × I(df/dt)
kinematical integral: mild sensitivity to df+/dt (and f−/f+ for l=µ) and e.m. corrections
K p’ ¯ u γ µ s π p = C p’ p µ f t p’ p µ f t
CVC ⇒ f+(0) = 1 in the SU(3) limit ms = mu = md
Three main issues to address in order to extract |Vus|: estimates of the SU(3) breaking term f+(0)−1 e.m. corrections kinematical dependence of the form factors [mainly an exp. issue]
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E.m. corrections:
s u
- I. short−distance corrections to the s → u l νl eff. Hamiltonian
sizable [ ~ α log(µhad/MW) ⇒ δΓ ~ 1% ] well known
Marciano & Sirlin, ’70− ’80
- II. pure long−distance corrections (IR div. & bremss.)
sizable [ ~ α log(MK/me) ⇒ δΓ ~ 1% ] partially known
Cirigliano et al. ’01
- III. structure−dependent (intermediate−scale) terms
small [ no large logs ⇒ δΓ ~ 0.1% ] model dependent
Ginsberg, ’66− ’69
(Coulomb corrections)
Coherent analysis of the 3 effects (particularly II. + III.) possible in the framework
- f CHPT [non−trivial results at O(e2p2)]
K π ν e W
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A crucial point in the analysis of e.m. corrections is the identification of I.R. safe observables. most convenient choice: Γ(Kl3) incl. = ∑ Γ K →πl νln 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(e2p2) δQED = −1.27% δCT = (+0.36 ± 0.16)% Are we sure that the (old) PDG data on Γ(Kl3) are completely inclusive ?
⇒ important exp. issue (together with the kinem. dependence
- f the f.f.) especially in view of new precise measurements
e.g.: Γ(Ke3) incl. ∝ |f+(0)|
2 × I(df/dt)
+
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- Th. estimates of f+(0) − 1
At O(p2) [LO in CHPT]: At O(p4): finite (unambiguous) non−polynomial corr. induced by meson loops no linear corrections in (ms−mu ) [Ademollo−Gatto theorem, ’64] [Furlan et al. ’65]
K 0π+
| f+ (0)| < 1
[ ~ mP log mP ⇒ ~ (ms−mu )2/ms ] numerically small: δ(4) = −2.2%
K 0π+ K
+π0
f+ (0)/f+ (0)
K 0π+
f+ (0) − 1 δ = = 0
At O(p6): appearance of B2 (ms−mu )2/Λ4χ local terms conservative estimate: δ(6) = −1.6 ± 0.8 % − precise determination of the ratio [ SU(2) ] [Leutwyler−Roos, ’84]
Is it really conservative ? Can we do better by means of recent improvements in CHPT and/or by means of Lattice ?
⇒ discussion in the subgroup on Vus
Leutwyler−Roos sum−rule: K 0π+
| f+ (0)|
2 = 1− ∑
K
0 Q us n
n ≠ π+
2
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Ke3 Ke3 Kµ3 Kµ3
+ 0 + 0
Extraction of |Vus| e.g. Γ(Ke3) [Cirigliano et al. ’01] :
+
∆ V us V us = 1 2 ∆Γ Γ 0.05 ∆λ λ
- ∆ f 0
f 0
± 0.6% ± 0.4% ± 0.85%
substantial reduction possible at KLOE within ~ 1 yr
|Vus| = 0.2207 ± 0.0024
Ke3
+
combined analysis of all Kl3 modes: |Vus| = 0.2187 ± 0.0020
Kl3
[Calderon−Lopez Castro, ’01]
At which precision we would like to know |Vus|? Reference figures provided by δ|Vcb| ~ 5% ⇒ almost negligible impact of δ|Vus| in the usual UT plane δ|Vud| ~ 0.08% (realistic ?) ⇒ |Vus|unitarity = 0.2287 ± 0.0034 [ 2.5σ discrepancy !]
f+(0)|Vus|
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The εK hyperbola
Soloist singer: εK Choir: CP−conserving data on K → 2π [minor role]
- Th. instruments: Perturbative QCD, Lattice, CHPT,
1/NC expansion, etc. allegro ma non troppo
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1
- 1
1
η ρ
s
d
W d s qu = u,c,t
Master formula for εK
εK = e
iπ⁄4
2∆ M K ℑ M 122 ℑ A0 ℜ A0 ℜM 12 M 12
∗ =
1 2 M K ¯ K
0 H eff ∆ S=2 K
Heff
∆ S=2 ∼ ¯
s d VA ¯ s d VA≡QL L
- nly one
dim−6
- perator
εK ∝ ℑ λt
2ηtt F tt2λcλt ηct F ctλc 2ηccF cc
BK
NLO QCD corrections
[Buras et al. ’90, Herrlich & Nierste, ’95−’96]
ηij= λq= V qs
∗ V qd
1.36 ± 0.07 0.30 ± 0.05
η [ (1− ρ) A2 ηtt Ftt + Pcharm ] A2 BK = 0.204
_ _
~ 4% error from pert. QCD ⇒ the key problem is BK
QL L µ = 8 3 f K
2 M K 2 BK ×
× α(µ)−2/9 [1+O(α)]
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Non−Lattice estimates of BK
Chiral limit [CHPT at O(p2)]: BK = 0.3 [extracted from Γ(K+ → π+π0
)]
[limit of light ms ]
Factorization: BK = 1 [→ 3/4 for NC → ∞]
[limit of heavy ms]
Corrections to the chiral limit are potentially large and cannot be computed in a model−independent way
[ (mK/Λχ) 2 ~ 25% ⇒ O(1) effects are not so unlikely ]
Subleading 1/NC corrections decrease the leading 1/NC result
NLO 1/NC + chiral limit ⇒ BK = 0.4 ± 0.1
[Pich & de Rafael, ’00]
we are still far from a precise estimate of BK at the physical point low values are certainly more favoured the chiral limit result is an important test for Lattice approaches
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Lattice estimates of BK
JLQCD ’98 Kogut−Susskind 0.628(42) 0.86 ± 0.06 [Reference figure @ Lattice 2000] ± 0.14 [quench.] SPQCDR ’01 Wilson [with subtr.] 0.71(13) 0.91 ± 0.17 SPQCDR ’01 Wilson [Ward id.] 0.70(10) 0.90 ± 0.13 CP−PACS ’01 DWF [pert. ren.] 0.575(20) 0.79 ± 0.03 RBC ’01 DWF [non pert. ren.] 0.513(11) 0.70 ± 0.02 BK (2 GeV) BK [RGE−inv.]
NDR
chiral symmetry at finite lattice spacing
CP−PACS
both DWF analysis shows a significance decrease of BK in the chiral limit
reasonable agreement with the ∆I=3/2 amplitude in all cases study of quenching effects still very preliminary My conclusion: (δBK )tot ≥ 25%
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At which level can we shrink the εK hyperbola ?
NNLO perturbative corrections to Heff (d=6) ⇒ ~ 4% (scale with BK) d=8 operators [ O(GF
2), no hard GIM & large logs enhancement, indep. of BK]
~ O( mK
2/[mc2 ln(mc/MW)] ) of the d=6 charm contribution ⇒ < 2%
Genuine long−distance effects (∆S=1 × ∆S=1)
- Th. errors besides BK :
K0 K0
_
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 ∆mK (where l.d. terms are CKM enhanced) As long as (δBK )tot ≥ 10% we can forget about terms not included in Heff (d=6)
~
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Rare decays
Soloist singers: K+
→ π+νν & KL → π0νν
[a beautiful but difficult part, which require expensive singers...] Choir: Dalitz decays (KL → µ+µ−, KL,S → π0e+e− , KL → γl+l−,...) [the most interesting choir part]
- Th. instruments: mainly Perturbative QCD & CHPT
allegro con brio, quasi scherzoso
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Xi @ NLO:
Buchalla & Buras ’94 Misiak & Urban ’99
Thanks to the "hard" GIM mechanism these decays are largely dominated by short−distance dynamics:
Heff = GF α 2 2πsW
2
λc X cλt X t ¯ sd V A ¯ νν V A
2 2 2 2
K → π νν
−
N.B.: the hadronic matrix element 〈 π | (sd)V−A | K 〉 is known from Kl3
Marciano & Parsa, ’96
with excellent accuracy
Z q=u,c,t
+ box ⇒ Aq ~ mq
VqsVqd ∼ 2 *
λq
2
ΛQCD λ
(u) mc λ
+ i mc λ 5
(c) mt λ
5
+ i mt λ
5
(t)
s d
Genuine ∆S=1 O(GF) transition
2
W
[ λ = sin θc]
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Theoretical predictions for BR(K → π νν) within the SM: K
+
- Th. error dominated by the charm contribution
[NNLO perturbative corr. (+ d=8 terms)]
Lu & Wise ’94 Buchalla & Buras ’97 Falk et al. ’00
[error dominated by the uncertainty on CKM param.]
KL
Charm contribution suppressed by the CP structure [ The state produced by is a CP eigenstate ]
¯ νν Heff
BR K L = 4.30×10
10
mt mt 170 GeV
2.3
ℑ V ts
∗V td
λ
5 2
= 2.3±1.3 ×10
11 (SM)
Littenberg, ’89 Buchalla & Buras ’97 Buchalla & G.I. ’98
- th. error ~ 2% !
− The best way to directly measure the area of the (full) UT
(SM)
BR K
- = C |Vcb|
4 [( ρ−ρc )2 + (σ η)2]
_ _
= 7.2±2.1 ×10
11
⇒ 0.04 error on ρ around the origin of the UT plane
_
ρc = 1.40 ± 0.06
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B K
→π ν¯
ν = 1.57 0.82
1.75 ×10 10
Littenberg ’02
2 events observed at BNL−Ε787 (0.15 bkg) [hep−ex/0111091] central value 2×SM ! δB ~ 30% (assuming BSM) expected before 2005 from BNL−E949
Experimental status of K
+ → π+ νν
−
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A) KL
→ π0 e+e−
- 2. indirect CPV: BCPV−ind < 5×10
−10
- 3. CPC amplitude: BCPC ~ 10
−12
strong constraints from KL→ π0γγ [NA48 ’00] B(KL→ π0e+e−)CPV−dir ~ 4×10
−12
- 1. direct CPV amplitude
short−distance dominated,
proportional to Im(λt)
KL,S → π0e+e−
B) KS
→ π0 e+e−
long−distance dominated
(γ−exchange amplitude)
π π- π+ K γ
B(KS→ π0e+e−) ‹ 1.4×10
−7
[NA48 ’00]
Using the exp. bound B(KL→ π0e+e−) ‹ 5.6×10
−10 [KTeV ’00]
we can derive the following solid (but weak) constraint: Im(λt) < 1.7×10
−3
minor improvement possible with a better bound on B(KS→ π0e+e−), new generation of experiments needed for as substantial improvement (SM level)
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KL known @ NLO
|A(KL→ µ+µ−)|
2 = |ℑAγγ| 2 + |ℜAγγ + ℜAshort | 2
fixed by Γ( KL
→ γγ )
+
clean s.d. terms (as in K +→ π+νν )
KL → µ+µ−
BNL−E871 ’98, ’00
B(KL
→ µ+µ− )=(7.18±0.17)×10 −9
Buchalla & Buras, ’94
depends on the KL
→ γ∗ γ∗
form factor (at all energies) Γabs= (7.07±0.18)×10
−9
The dispersive integral does not contains large logs [contrary to the KL
→ e+ e− case]
and is naturally of the same order of the s.d. contrib.
theoretical constraints @ high q2 low−energy constraints from KL
→ γ l +l − [good data]
−
Γs.d.= 0.9×10
−9 ×(1.2−ρ) 2
⇒ large negative values of ρ [ρ < −0.5] are certainly disfavoured, but at present is
difficult to extract a reliable probabilistic information
~
Bergstrom, Masso, Singer ,’90 D’Ambrosio, G.I., Portoles, ’98 Dumm & Pich, ’98 Geng & Hwang, ’01
& KL
→ e+ e− µ+ µ− [poor data]
_
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(rare) K decays & the unitarity triangle
Good consistency Negligible impact on precision comparison with B−physics data except for the εK hyperbola the lower bound on B(K
+ → π+ νν)
η − ρ −
KL
→ π0e+e−
5 5
KL
→ µ+µ−
K +→ π+νν εK
Disclaimer:
I have not discussed role of ε’/ε in this game because of time ε’/ε could in principle provide a bound on η but I think we are still very far from being able to extract this info in a systematic way
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central value (no exp. error)
∆mBd
full 1σ range
εK ΑψKs
More about the impact of B(K
+ → π+ νν) on the UT:
excellent agreement with Vub & εK slight disagreement with ΑψKs & ∆mBd [∆B=2]
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The scherzoso part of the movement: UT fit ignoring any ∆B=2 information [i.e. allowing non−standard effects in ∆B=2]
[D’Ambrosio & G.I. ’01]
εK
68% & 90% intervals at present error on B(K
+ → π+ νν)
would decrease by a factor 2 68% if the exp.
values of γ > 90o are (slightly) favoured a similar (still weak) indication comes also from B→Kπ ... ... we should not give up the hope to observe non− standard effects !
[wide literature]
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