Precision tests of the Standard
Model with kaon decays
Matteo Palutan
INFN/Frascati
FlaviaNet working group

KAON ‘09, Tsukuba, 9 June 2009

Outline• Introduction• Vud from nuclear β decays
Hardy-Towner, Marciano-Sirlin• Vus, Vus/Vud from kaon (and pion) decays
FlaviaNet working group• CKM unitarity and GF universality

6/9/09Matteo Palutan, SM test with kaons2http://www.lnf.infn.it/wg/vus

6/9/09Matteo Palutan, SM test with kaons

3

CKM unitarity: GF universalityg
Wα+ U LVCKM γ α DL + e L γ α v e L + µ L γ α v µ L + τ L γ α vτ L
2(

SM coupling to W:

one gauge coupling

€

2

2

unitary matrix
2?

Vud + Vus + Vub = 1

Most precise test of CKM
unitarity from 1st row

negligibleµ decay€K, π and nuclear β decaysµνµ(gµ ge )6/9/09

W
2

4
W

M =G

e

u

νe

s, d

2
F

?

=

l
W
2

(

2

(gq g ) Vud + Vus

Matteo Palutan, SM test with kaons

ν
2

)

2
MW4 = GCKM

4)

CKM unitarity: GF universality
New Physics extensions of the SM can break gauge universality
u
W

s, d

+

ν

2
F

Z’

u

l

2

s, d

(

W

SM + NP ∝ G Vuq 1+ a( MW M X )

2 2

)

l

u

ν

s, d

H+

+…
ν

naively atree ≈ 1, aloop ≈ g2

µ lifetime:

GF= 1.166371(6)×10-5 GeV-2

τ decay:Gτ= 1.1678(26)ew precision test:

Gew = 1.1655(12)Vus at 0.5%:

GCKM = 1.16xx(4)6/9/09l⇒ Mtree ≈ 10 TeV, Mloop ≈ 1 TeV

Matteo Palutan, SM test with kaons

5Vud from nuclear β decays

Experimental survey + radiative corrections (nucleus-dep.)
Towner, Hardy arXiv:0812.1202
Radiative corrections (nucleus-indep.)
Marciano, Sirlin PRL 96 (2006)

6/9/09Matteo Palutan, SM test with kaons

6Vud from Fermi transitions (0+→0+)

nucleus independent• Measured on 13 nuclei:t = t1/2/BR partial half lifef = statistical rate function f(Z,QEC)• Radiative corrections:
ΔR = 2.361(38)%, nucleus-independent (Marciano-Sirlin 2006)
δ́’R , δNS nucleus-dependent• δC nucleus dependent isospin breaking
6/9/09Matteo Palutan, SM test with kaons7World data for 0+→0+ decay, 2008

10 cases with ft-values measured
to ∼0.1% precision; 3 morecases with <0.3% precision

6/9/09Matteo Palutan, SM test with kaons

8Vud from 0+→ 0+ decay in 2008
error budget

Vud = 0.97424 ± 0.00022
Hardy-Towner, CIPANP ’09• Nuclear-structure dependent
corrections, δC and δNS, can betested by experiment• Add new transistions

nuclear 0+→0+
still the mostprecise(by a factor ∼10)

6/9/09Matteo Palutan, SM test with kaons

9Vus and Vus/Vud from kaon
decaysby FlaviaNet working group- “official” analysis: arXiv:0801.1817- partial update presented today

6/9/09Matteo Palutan, SM test with kaons

10Vus from Kl3 decay rates
Γ(Kl3(γ)) =CK2 GF2MK5
192π3

K0π-

SEW |Vus|2 | f+ (0)|2 IKl(λ) (1+2ΔKSU(2) +2ΔKlEM )with K = K+, K0; l = e, µandInputs from theory:

Inputs from experiment:

Universal short distance
EW correction (1.0232)

SEW

CK2 = 1/2 for K+, 1 for K0

Γ(Kl3(γ)) Branching ratios withwell determinedtreatment of radiative
decays; lifetimes

0 -f+K π(0) Hadronic matrix element

at zero momentum transfer (t=0)ΔKSU(2) Form factor correction for SU(2)IKl(λ)

breaking

ΔKlEM

Long distance EM effects

6/9/09Matteo Palutan, SM test with kaons

Phase space integral: λs
parameterize form factor
dependence on t :Ke3 : only λ+ (or λ+′ λ+′′ )Kµ3 : need λ+ and λ0
11Vus from kaon decays: recent history→ 2002
Old Kl3 data give 1 - Vud2 - Vus2 = 0.0035(15)(2004 PDG)
A 2.3σ hint of unitarity violation?
2003

BNL 865 measures higher BR(K+e3) = 5.13(10)%Vus consistent with unitarity

2004present

Many new measurements from KTeV, KLOE, ISTRA+, NA48
• BRs, lifetimes, form-factor slopes• Much higher statistics than older measurements• Importance of radiative corrections• Proper reporting of correlations between measurements

2008beyond
6/9/09

Value of Vus used in precision tests of Standard Model
Matteo Palutan, SM test with kaons

12Extraction of Vus from Kl3 data
1) BR and lifetime averages: KL and KS
2) BR and lifetime averages: K±
3) Form factor slopes and phase space
4) Extraction of |Vus| f+ (0) (and Vus )6/9/09Matteo Palutan, SM test with kaons13Fit to KL BR and lifetime measurements
FlaviaNet fit

21 input measurements:
5 KTeV ratios
NA48 BR(Ke3 /2 track)
4 KLOE BRs
KLOE, NA48 BR(π+π-/Kl3)
KLOE, NA48 BR(γγ/3π0)
PDG ETAFIT BR(2π0/π+π-)
KLOE τL from 3π0
Vosburgh ’72 τL
KTeV BR(π+π-γ/π+π-(γ))
E731, 2KTeV BR(π+π-γDE/π+π-γ)
all publishedresults

1 constraint: ΣBR=1
6/9/09Parameter

Value

S

Ke3
Kµ3
3π0
π+π-π0π+π-(γIB)π+π-γπ+π-γDE
2π0γγ
τL0.4056(9)0.2704(10)0.1952(9)0.1254(6)1.967(7)×10-3
4.15(9)×10-5
2.84(8)×10-58.65(4)×10-4
5.47(4)×10-4
51.16(21) ns

1.3
1.5
1.2
1.1
1.6
1.3
1.1
1.4
1.1
1.1

χ2/ ndf = 19.8/12 (Prob = 7.1%)
compare PDG’08: 35.7/17 (0.5%)
Matteo Palutan, SM test with kaons

14

Evolution of KL BRs
BR(Ke3) %BR(Kµ3) %BR(3π0) %BR(π+π-) 10-3

PDG 04
PDG 06This fit

Differences between FlaviaNet and PDG are minor

Since PDG ‘04:
• Proper radiative corrections, especially for Ke3• Exclusion of old measurementsfrom KL BRs:

6/9/09

PDG ‘04

This fit

Kµ3/Ke30.701(9)0.6666(29)ok with lepton universality|η+−|×103

2.284(14)2.223(6)>3σ difference

Matteo Palutan, SM test with kaons

15

BR(KS→ πeν) and KS lifetime
BRs from KLOE tagged KS beam, 1.2×108 events
KLOE 2006

BR(KS → πeν)/BR(KS → π+π-) = 10.19(13) × 10-4σ∼1.3%

BR(KS → π+π-) /BR(KS → π0π0) = 2.2459(54)Add NA48 result (fit KS and KL components in decay-length distribution )NA48 2007

Γ(KS → πeν) /Γ(KL → πeν) = 0.993(26)(22)σ∼3.4%

we obtain BR(KS → πeν) = 7.05(8) × 10-4

PDG

τS = 0.08958(5) ns
From fit to CP parameters, does not assume CPT
Dominated by NA48 ‘02 and KTeV ‘03 values

6/9/09Matteo Palutan, SM test with kaons

16Extraction of Vus from Kl3 data
1) BR and lifetime averages: KL and KS
2) BR and lifetime averages: K±
3) Form factor slopes and phase space
4) Extraction of |Vus| f+ (0) (and Vus )6/9/09Matteo Palutan, SM test with kaons17Recent results on K± BRs
BNL 865
2003
NA48/2 2007

BR(K+→π0De+ν)/BR(K+→π0DX+) = 0.1962(8)(35)
the start of the Vus revolution

BR(K±e3)/BR(π± π0) = 0.2470(9)(4)
BR(K±µ3)/BR(π± π0) = 0.1637(6)(3)
K+ K- simultaneous beams

ISTRA+ 2007

BR(K-e3)/BR(π- π0) = 0.2449(4)(14)still preliminary, not used in the updated fit

KLOE
2004-2008

BR(K+→µ+ν) = 63.66(9)(15)%BR(K+→π+π0) = 20.65(5)(8)%
BR (K±e3) = 4.965(53)(38)%BR (K±µ3) = 3.233(29)(26)%BR(K+→π+π0π0) = 1.763(13)(22)%absolute BR measurements with tagged K± beams
Matteo Palutan, SM test with kaons

18

KLOE: measurement of K± lifetime
Poor consistency (CL=0.2%) between old meas., need confirmation
Tag events with K± → µ± ν decay
Identify a kaon decay vertex on the opposite side

Tag(Kµ2)K-• 1st method: obtain τ± from the K decay lengthK+take into account the energy loss: τK = ∑i ΔLi/(βiγic)• 2nd method: obtain τ± from the K decay time
use photons from K± → π±π0 decay

Compatible results,
average with correlations:

6/9/09γγ

τ±= 12.347(30) ns
τ+/τ-= 1.004(4)Matteo Palutan, SM test with kaons

JHEP (01)2008
19

K± lifetime: older data
PDG
2008 average

τ±= 12.379(21) ns
S = 1.9

Serious doubts about accuracy oferror estimates for older
experiments

Ott & Pritchard ‘71

Statistical errors incorrect by x2.3 (∼0.56/√N)
Uncorrelated combination of systematic errors

Koptev et al. ‘95∼2σ difference begtween results for same experiment,
different stoppers → use average with scaled errors

Lobkowicz et al. ‘69

Focused on τ+/τ- , precision absolute scale notemphasized, fit not compatible with exponential
behaviour → not used in our fit

6/9/09Matteo Palutan, SM test with kaons20Fit to K± BR and lifetime measurements
Compared to PDG, our fit:
We exclude all six Chiang ’72 measurements• No radiative corrections• 6 BRs constrained by ΣBR = 1, correlations unavailable
After a carefull review, we exclude also
• 2 Ke3/2-body (Ecshstruth ‘68, Cester ‘66)
• 2 Kµ3/Ke3 (Botterill ’68, Heintze ‘77)
new measurement• Γ(K+→π+π+π-) from Ford ’67
needed!
Lifetime measurements• No Lobkowicz ‘69• Koptev ’95 measurements combined with scaled errors

Compared to previous Flavianet fit (KAON 07):
• Ke3/ππ0 from ISTRA+ excluded (never published)• new KLOE BR(K+→π+π0), important as normalization channel
6/9/09Matteo Palutan, SM test with kaons21Fit results to K± BR and lifetime
17 input measurements:
3 older τ values in PDG
1 KLOE τ
KLOE BR(µν), BR(ππ0)
KLOE Ke3, Kµ3 BRs
NA48/2 Ke3/π π0, Kµ3/π π0E865 Ke3/Kdal
KEK246 Kµ3/Ke3
3 old ππ0/µν
1 old + 1 KLOE results on 3π
all publishedresults

ParameterBR(µν)BR(ππ0)BR(πππ)BR(Ke3)BR(Kµ3)BR(ππ0π0)

τ±

Value

S

63.47(18)%
20.61(8)%5.73(16)%
5.078(31)%3.359(32)%1.757(24)%
12.384(15) ns

1.3
1.1
1.2
1.3
1.9
1.0
1.2

χ2/ ndf = 25.8/11 (Prob = 0.7%)

1 constraint: ΣBR=1
6/9/09FlaviaNet fit

compare PDG’08: 52/24 (0.1%)

Matteo Palutan, SM test with kaons

22

K± BR fit vs. data

Pulls from
6/9/093π

τ±Fitch65

Ott71
Kop95

π+π0π0π+π0π0/π+π+π-Kµ3

Kµ3/ππ0Kµ3/Ke3

Ke3/ππ0Kµ2/ππ0

Kµ2
ππ0

Ke3

Ke3/KDalKe3

Kµ3

2 bodyKLOE E865 NA48/2 others
Matteo Palutan, SM test with kaons23

Evolution of K± BRs
BR(Κ± → π0e±ν)

BR(Κ± → π0µ±ν)

PDG ’04

PDG ’04

PDG ’07

PDG ’07

PDG ’08

PDG ’08

This fit

This fit

BR(Κ± → µ±ν)BR(Κ± → π±π0)

PDG ’04

PDG ’04

PDG ’07

PDG ’07

PDG ’08

PDG ’08

This fit

This fit

6/9/09Matteo Palutan, SM test with kaons24Extraction of Vus from Kl3 data
1) BR and lifetime averages: KL and KS
2) BR and lifetime averages: K±
3) Form factor slopes and phase space
4) Extraction of |Vus| f+ (0) (and Vus )6/9/09Matteo Palutan, SM test with kaons

25€

Kl3 form-factors
Hadronic matrix element:

π JαV K = f + (0) × f˜+ (t)( P + p)α + f˜− (t)( P − p)α()tf-(t) term only important f˜ (t) = f˜ (t) + f˜ (t)0+−for Kµ3 → scalar ff
mK2 − mπ2t = ( P − p)2

To evaluate phase space integrals (for Vus) need to parameterize (and€measure) ff-dependence on t
€
2

 t 
 t 
˜
′  2  + λ+,0′′  2 
Quadratic: f +,0 (t) = 1+ λ+,0 mπ +  mπ + 

Fits exhibit high correlation for λ ′, λ ′′€Polar:

6/9/092
V ,SM
f˜+,0 (t) = 2
MV €,S − t€λ′ = (mπ + M )λ ′′ = 2( λ ′)22Matteo Palutan, SM test with kaons

26

Current data on Ke3 form-factor slopes
λ′+ × 103
ISTRA+
PLB 589 (2004)
KTeV
PRD 70 (2004)
NA48
PLB 604 (2004)
KLOE
PLB 632 (2006)

24.9 ± 1.7
21.7 ± 2.0
28.0 ± 2.4
25.5 ± 1.8λ′′+ × 103

analysis

1.9 ± 0.9

0.9×106 K-e3
ρ(Ee,Eπ)

2.9 ± 0.8

1.9×106 KLe3tπ⊥0.4 ± 0.9

5.6×106 KLe3
ρ(t low , t high)

1.4 ± 0.8

2.0×106 KLe3t from KS* ISTRA+ results are rescaled by (mπ+/mπ0)2
6/9/09Matteo Palutan, SM test with kaons27

Ke3 slopes: quadratic fits
Slopes from

KTeV KLOE ISTRA+ NA48

FlaviaNet fit

Slope parameters × 103
λ′+ = 25.15 ± 0.87λ′′+ = 1.57 ± 0.38ρ(λ′+, λ′′+) = - 0.941
χ2/ ndf = 5.3/6 (51%)
Excellent compatibility
Significance of λ′′+ ~ 4σI(K0e3)= 0.15463(21)I(K+e3)= 0.15900(22)6/9/09Matteo Palutan, SM test with kaons

28

λ″+ × 103

Ke3 slopes: Quadratic vs. pole fits
KLe3 only

KLe3 only

KTeV
Pole model
KLOE

Average: 875.3 ± 5.4 MeV

MV frompole fit

KTeV
881 ± 7 MeV

KLOE

870 ± 9 MeV

NA48

NA48

Pole fits
λ′′ = 2λ′2859 ± 18 MeV
MV [MeV]λ′+ × 103

KLe3 dataP(χ2) fit
Quad

KLe3 integral
QuadP(χ2) fit
Pole

KLe3 integral
Pole

Difference

KTeV
KLOE

54%
92%

0.15378(51)0.15472(42)0.15463(21)

43%
92%

0.15449(25)0.15489(33)0.15481(18)+0.46%+0.11%+0.12%

KLe3 avg
6/9/09Matteo Palutan, SM test with kaons

29Current data on Kµ3 form-factor slopes
λ′+ × 103
ISTRA+
PLB 581 (2004)
KTeV
PRD 70 (2004)
NA48
PLB 647 (2007)
KLOE
JHEP 12 (2007)

23.0 ± 6.4

17.0 ± 3.7

20.5 ± 3.3

25.6 ± 1.8

λ0 × 103

analysis17.1 ± 2.2

0.9×106 K-µ3
ρ(Eµ,Eπ)

12.8 ± 1.8

1.5×106 KLµ3
ρ(tµ⊥,Mπµ)2.6 ± 1.4

9.5 ± 1.4

2.3×106 KLµ3
ρ(Eµ,Eπ)low

1.5 ± 0.815.4± 2.2

1.8×106 KLµ3
Eν spectrum + KLe3λ′′+ × 103
2.3 ± 2.3

4.4 ± 1.5

* KLOE quotes Ke3+Kµ3 average* ISTRA+ don’t quote systematic error for quadratic fit: we exclude from global fit
6/9/09Matteo Palutan, SM test with kaons30Fit to Kl3 form factor slopes
e3-µ3 averages from

6/9/09

KTeV KLOE ISTRA+ NA48

Matteo Palutan, SM test with kaons

31

Kl3 form factor slopes: fit results
Compatibility poor, inconsistency parameterized by scale factors for fit results

Slopes × 103:

Phase space integrals

λ′+ = 24.54 ± 0.87

S = 1.1

I(K0e3)0.15449(20)λ′′+ = 1.81 ± 0.41

S = 1.2

I(K+e3)0.15885(21)λ0 = 11.67 ± 1.38

S = 1.9I(K0µ3)0.10171(32)I(K+µ3)0.10467(33)χ2/ ndf = 29/8 (3×10-4)
Correlation coefficients:
λ′+λ0λ′′+ -0.94 -0.52
λ′+6/9/09+0.44

Without NA48 Kµ3:
• χ2/ ndf = 9.5/7 (22%)• I(Kµ3): +0.7%• e3-µ3 average for |Vus|f+(0): -0.09%
Matteo Palutan, SM test with kaons

32Extraction of Vus from Kl3 data
1) BR and lifetime averages: KL and KS
2) BR and lifetime averages: K±
3) Form factor slopes and phase space
4) Extraction of |Vus| f+ (0) (and Vus )6/9/09Matteo Palutan, SM test with kaons33

SU(2) correction
ΔSU(2) =f+(0)K+π0
f+(0)K0π−−1

Strong isospin breaking
Quark mass differencesη -π0 mixing in K+π0 channel+2.36(22)%Cirigliano ’07 (updating Cirigiliano et al. ’02)
O(p4) chiral correction
Quark mass-ratio analysis based on Leutwyler ‘96+2.9(4)%

Kastner & Neufeld ’08
Quark mass-ratio from Ananthanarayan, Moussallam’04+2.31%

Bijnens, Ghorbani ‘07
O(p6) chiral correction
34

Long-distance EM corrections
ΔEM = mode-dependent integrated correction to the decay rate from Dalitz
plot modifications from EM effects; values depend on acceptance for eventswith additional real photon(s)All recent measurements assumed fully inclusive

ΔEM

Cirigliano et al. ’04
ChPT

Andre ’04
Had. model

Cirigliano et al. ’08
ChPT

K0e3

+0.52(10)%+0.65(15)%+0.50(11)%K+e3+0.03(10)%

K0µ3
K+µ3

+0.05(13)%+0.95(15)%+0.70(11)%+0.01(13)%Cirigliano et al. ’08 perform comprehensive analysis at fixed order e2p2
Fully inclusive prescription to treat real photon emission
Updated values of LECs for structure-dependent terms
Correlations evaluated, e.g.: ρ(K0e3, K0µ3) = +0.69, ρ(K0e3, K+e3) = +0.08
35

|Vus|f+(0) from Kl3 data
Approx. contrib. to % err from:|Vus| f+(0)% err BR

Δ

Int

KLe3

0.21652(56)0.25

0.11 0.20 0.11 0.10KLµ30.21746(69)0.32

0.17 0.19 0.11 0.15

KSe3

0.21572(132)0.61

0.60 0.03 0.11 0.10

K±e30.21624(113)

0.52

0.31 0.06 0.41 0.09

K±µ3

0.21676(141)0.65

0.48 0.06 0.41 0.15

Average: |Vus| f+(0) = 0.21660(47)6/9/09τ

χ2/ndf = 3.03/4 (55%)

Matteo Palutan, SM test with kaons

36

|Vus|f+(0): K± vs KL,S
Fit 5 modes with separate values of |Vus| f+(0) for K± and KL,S modes• Using results of overall fit to form-factor slopes• With SU(2) corrections for K± modes [ΔSU(2)theory = 2.9(4)%]K± modes|Vus| f+(0) = 0.2163(11)KL,S modes|Vus| f+(0) = 0.2167(5)0.3σ differenceχ2/ndf = 2.9/3 (41%)ρ = 0.04

When fit performed without SU(2) corrections for K± modes,
obtain an experimental value for ΔSU(2)K± modes, no SU(2)|Vus| f+(0) = 0.2226(7)6/9/09ΔSU(2)exp = 2.7(4)%Matteo Palutan, SM test with kaons

37

Kℓ3 data and lepton universality
For each state of kaon charge, we evaluate:Modes

2004 BRs*

This fitK±

1.016(12)1.005(9)KL,S1.056(15)1.009(6)*Assuming current values for form-factor parameters and ΔEM; KS not included

Average Kl3

rµe = 1.008(5)Without NA48 Kµ3: rµe = 1.002(5)Compare with:
π → ℓν
(rµe) = 1.0042(33)Ramsey-Musolf, Su, Tulin ’07

τ → ℓνν(rµe) = 1.000(4)Davier, Hoecker, Zhang ’06

6/9/09Matteo Palutan, SM test with kaons

38

Evaluations of f+(0)Compilations: L. Lellouch, Lattice ’08

f+(0)Quark model
ChPT + QM
ChPT + disp
ChPT + 1/NC
Nf = 02 Wilson
2 DWF
2 Stag+Wilson(2+1) DWF

Many evaluations available,
analytic and lattice-based
ChPT results tend to givehigher values for f+(0)Trend is to use lattice
results, but which ones?
FlaviaNet Lattice Averaging
Group (FLAG) has promised
to make a recommendation
For now use:

Provisional coding (Lellouch), similar to FLAG scheme

f+(0) = 0.9644(49)
RBC/UKQCD ’07

39

Vus from Kl3 data and CKM unitarity
Kl3 average: |Vus| f+(0) = 0.21660(47)was 0.2167 at Kaon’07
With f+(0) = 0.9644(49) from lattice QCD:

Kl3 average: |Vus| = 0.2246(12)Using |Vud| = 0.97424(22) from average 0+ → 0+ β decays (Towner-Hardy ‘09)

Vud2 + Vus2 - 1 = -0.0004(7)Compatibility with unitarity -0.6σ
was 0031(15) in PDG04

σ(Vud2)∼0.0004
6/9/09σ(Vus2)∼0.0005

Matteo Palutan, SM test with kaons

40Vus/Vud from K, π → µν decays

Inputs from theory:

Inputs from experiment:

fK/fπ = 1.189(7) from lattice QCD

K± BR fit:
BR(K±µ2(γ)) = 0.6347(18)τK± = 12.384(15) ns(HPQCD/UKQCD ’07)

Cancellation of lattice-scale
uncertainties

1 + α (CK – Cπ) = 0.9930(35)(Marciano ’04)
Uncertainty from SD virtual
corrections (Finkemeier ’96)

PDG:
BR(π±µ2(γ)) = 0.9999
τπ± = 26.033(5) ns|Vus|/|Vud| = 0.2319(15)6/9/09Matteo Palutan, SM test with kaons

41

CKM unitarity
Vus

Vud
V us/V ud

Now can fit:
• Vus from Kl3• Vus/Vud from Kµ2/πµ2• Vud from β decay

fit
Vus

arity

unit

Vud

Vud = 0.97424(22)Vus = 0.2252(9)χ2/ndf = 0.52/1 (47%)
Vud2 + Vus2 - 1 = -0.0001(6)We use f+(0) = 0.9644(49), fK/fπ = 1.189(7)6/9/09Matteo Palutan, SM test with kaons421st-row unitarity and gauge universality
MuLan ’07

Gµ = 1.166371(6) × 10−5 GeV−2

CKM 1st row

1.16626(30) × 10−5 GeV−2

Precision EW tests

1.1655(12) × 10−5 GeV−2

τ decay

1.1678(26) × 10−5 GeV−2

From new physics:
GF → GF [1 + a(MW/ΛNP)2]
Probe mass scales
aloop ~ g2/16π2 ~ 1 TeV
atree ~ 1
~ 10 TeV

Example: Z′ bosons

u

Z′

W+

s,d
6/9/09ℓνZ′ boson from SO(10) GUT:QeL = QuL = −3QdL = 1(Marciano)mZ′ > 700 GeV 95% CL
About same sensitivity as
direct searches
Matteo Palutan, SM test with kaons43

Kµ2: sensitivity to new physics
u

ℓ
H+

K+
s

Scalar currents, e.g. due to Higgs exchange, could
affect K→ µν widthνℓThe observable:
Vus (K µ 2 ) Vud (0 + → 0 + )Rl 23 =
×
Vus (K l 3 )Vud (π µ 2 )mK2 +  mπ2 +  tan 2 β= 1− 2 1− 2 m H +  mK +  1+ ε0 tan β(Hou, Isidori-Paradisi)

we find€ Rl23 = 1.008 ± 0.008(KLOE only)

limited by lattice uncertainty on f+(0) and fK/fπ
6/9/09Matteo Palutan, SM test with kaons44Fit to determine f+(0) and fK/fπAssuming the SM, we can extract the decay costants
Start from Kℓ2 rate expression:

Rewrite as:

Experimental inputs
up to IB and radiative corrections

Vud from 0+ → 0+
Vus f+(0) from Kℓ3
QKℓ2 rate ratio

Parameters from theory
f+(0)
fK/fπ

Constraints from: QKℓ2 rate expression and 1st row CKM unitarity
6/9/09Matteo Palutan, SM test with kaons45Fit to determine f+(0) and fK/fπ
5 parameters:
Vud, Vus, QKℓ2, f+(0), fK/fπ
3 inputs:
Vud, Vus f+(0), QKℓ2
2 constraints:
QKℓ2 rate relation
1st row unitarity

fK/fπ

HPQCD
UKQCD ’07
RBC
UKQCD ‘07

f+(0) = 0.9605(45)fK/fπ = 1.1913(60)ρ = +0.79

6/9/09Matteo Palutan, SM test with kaons

f+(0)
46

SummaryKℓ3 average: |Vus| f+(0) = 0.21660(47)With Vud from 0+ → 0+ β decays and hadronic parameters from lattice
QCD, first-row CKM unitarity is verified:

Vud2 + Vus2 - 1 = -0.0004(7)Compatibility with unitarity -0.6σ
This allows precision tests of the SM and places important limits onuniversality-breaking new physics
Very rich new kaon data set; expect to improveon K± BRs and semileptonic slopes
Impressive results from lattice QCD: very
interesting prospects for precision test of the SM
6/9/09Matteo Palutan, SM test with kaons47

Additional information

6/9/09Matteo Palutan, SM test with kaons48

J.C. Hardy at CIPANP ‘09

6/9/09Matteo Palutan, SM test with kaons49

J.C. Hardy at CIPANP ‘09

6/9/09Matteo Palutan, SM test with kaons50

J.C. Hardy at CIPANP ‘09

6/9/09Matteo Palutan, SM test with kaons

51

J.C. Hardy at CIPANP ‘09

6/9/09Matteo Palutan, SM test with kaons52

J.C. Hardy at CIPANP ‘09

6/9/09Matteo Palutan, SM test with kaons53

KL branching ratios from KTeV
5 ratios of main BRs from independent samples of 105-106
events collected with a single trigger
KTeV, PRD 70 (2004)

2-track ratios

Neutral ratio

BR(Kµ3)/ BR(Ke3) = 0.6640(26)BR(2π0)/BR(3π0) = 0.004446(25)BR(π+π-π0)/ BR(Ke3) = 0.3078(18)Mixed ratioBR(π+π-)/ BR(Ke3) = 0.004856(28)BR(3π0)/ BR(Ke3) = 0.4782(55)• 6 decays = 99.93% of KL width, KTeV combines ratios to
extract BRs• Fit to BRs uses the 5 ratios (correlations available)
6/9/09Matteo Palutan, SM test with kaons54

KL branching ratios from NA48
1) KL beam, 2-track sample, 80×106 events (6×106 signal)
NA48, PLB 602 (2004)

BR(Ke3)/ BR(2 track) = 0.4978(35)• NA48 evaluates BR(Ke3) from BR(2 track) = 1.0048-BR(3π0)KTeV• The measured ratio used in fit to BRs

2) Measurement of Γ(KL → 3π0)/ Γ(KS → 2π0) with same
number of KL and KS produced on target, 2-6×105 signal events;use Γ(KS → 2π0) to extract Γ(KL → 3π0)NA48, 2004 preliminary

BR(KL → 3π0)/ τL = 3.795(58) MHz
6/9/09Matteo Palutan, SM test with kaons

55

KL branching ratios and lifetime from KLOE
1) Absolute BRs: KL decays tagged by KS→π+π-

13×106 tagged, 105 -106 signal

KLOE, PLB 632 (2006)

BR(0)(Ke3) = 0.4049(21)BR(0)(3π0) = 0.2018(24)at τL(0) = 51.54 ns, with
dBR/BR = 0.67 dτL/τL(geometrical acceptance)

BR(0)(π+π-π0) = 0.1276(15)Correlations available

BR(0)(Kµ3) = 0.2726(16)KLOE results: set ΣBR(i)=1 and solve for τL
Fit to BRs: use unconstrained BRs with dependence on τL2) Lifetime: measurement from an independent sample of
15×106 KL→π0π0π0 events
uniform reconstruction eff. over 0.4τL

KLOE, PLB 626 (2005)
6/9/09

τL = 50.92(30) ns
Matteo Palutan, SM test with kaons

56KL → π+πNew measurements of KL → π+π-(γ) also useful in global fit
KTeV
PRD 70 (2004)

KLOE
PLB 638 (2006)

NA48
PLB 645 (2007)

BR(π+π-/Ke3) = 4.856(29) × 10-3
1 of 5 ratios in KL BR analysis
Contribution from direct emission (DE) negligibleBR(π+π-/Kµ3) = 7.275(68) × 10-3
Fully inclusive of DE componentBR(π+π-/Ke3) = 4.826(27) × 10-3
Residual DE contribution of 0.19% subtracted

For consistency and to better satisfy S BR = 1 in global fit,
DE contribution of 1.52(7)%* added to KTeV and NA48 results* From E731 ’93, KTeV ’01 and KTeV ’06 KL → π+π-γ results
6/9/09Matteo Palutan, SM test with kaons

57

Can we observe λ0′′?
Determination of f+(0) with ChPT calculations at O(p6), would benefit
from the measurement of scalar form factor curvature (Bijnens andTalavera, 2003):

2C12r+ C34r ~ λ0′

C12r ~ λ0′′[Cijr = LEC’s at order O(p6)]

Unfortunately, for Kµ3 there are two FF and 4 parameters → highcorrelations → big errors. If we compute errors on λ0′′ from the (noexperimental resolution) we get:ρ(λ0′, λ0′′) = - 0.999

dλ0′′ ~ 20/√N

dλ0′′ ~ 2.5/√N

Fit to λ+′, λ+′′, λ0′, λ0′′

Fit to λ0′, λ0′′ (λ+′, λ+′′ fixed)

impossible

25M events for dλ0′′ = 0.5×10-3

6/9/09Matteo Palutan, SM test with kaons

58Dispersive treatment of form factors
Ignoring λ0″ changes Κµ3 phase space by −0.1% (0.2σ KLOE)
Better parameterization allows tests of ChPT & low-energy dynamics
Use parameterizations of f+, f0 from twice-subtracted dispersion relations
constrained by fits to Κπ scattering data
Bernard et al. ’06
Expansion gives:

Fit results:

I(KLµ3) phase-space integrals

KLOE e3-µ3
JHEP 12(2007)

λ+C = (25.7 ± 0.6) × 10−3 Quad-lin:
λ0C = (14.0 ± 2.1) × 10−3 Disp:
ρ = −0.26

NA48 µ3 only
PLB 647 (2007)

λ+C = (23.3 ± 0.9) × 10−3 Quad-lin: 0.10039(58)0.1%
λ0C = (8.8 ± 1.2) × 10−3 Disp:
0.10063(31)ρ = −0.44
NB: NA48 value for λ C converted from ln C = 0.1438(138)

0.10271(52)0.10262(47)

0.1%

+

59

Form factors and the Callan-Treiman relation
Callan-Treiman relation:f0(t)f0(tCT)f+(0)tCT = mK2 – mπ2
ΔCT = SU(2)-breaking correction= −(3.5 ± 8.0) × 10−3 in NLO ChPT (mu = md)

Kℓ3
mℓ2

t

(mK−mπ)2 tCT = (mK2−mπ2)

Use dispersive parameterization of f0(t) andstatistical considerations to extrapolate to tCT
Lattice QCD

fK/fπ = 1.190(15)KLOE Ke3-µ3 data:
λ0C = (14.0 ± 2.1) × 10−3

→ f+(0) = 0.968(28)NA48 Kµ3 data:ln C = 0.1438(138)→ f+(0) = 1.027(20)Lattice QCD

f+(0) = 0.964(5)

f+(0)

60

Lattice evaluations of fK/fπ
Compilations: L. Lellouch, Lattice ’08

fK/fπ

Nf = 2
Twisted mass
Nf = 2+1
Staggered

Nf = 2+1
DW or Wilson

Provisional coding

Only published resultwith full evaluation ofsystematic errors is
HPQCD/UKQCD ’07
Nf = 2+1HISQ
fK/fπ = 1.189(7)Results without
staggered quarks havelarge errors (PACS ’08
has stat errs only) or are
preliminary (BMW ’08)

For now use HPQCD/UKQCD’07

61

Vus from τ

6/9/09Matteo Palutan, SM test with kaons

62