Well-known unknown:new measurements of the Cabibbo angle

A. Glazov

DESY / University of Chicago

Outline

• Vus and unitarity of CKM matrix

• Ways to extract Vus

• Situation before 2004: did we have self-consistent picture ?

• Hints of a problem – K+ case

• Breakthrough – KL, KS case

• Support – hyperons.

• Conclusions

Vus and Unitarity check

CKM matrix describes the quark mixing:

V =

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

(1)

Vus is the oldest known mixing element (Cabibbo angle). Yetmany exciting developments have happened just this year !

Unitarity of CKM matrix requires:

1− (|Vud|2 + |Vus|2 + |Vub|2) = δ = 0 (2)

Largest contribution comes from |Vud|, next from |Vus|,negligible from |Vub|.According PDG-02, δ = 0.0043± 0.0019, about 2.2σ deviationfrom unitarity, with uncertainty from Vus of 0.0010.

Methods to extract VusThe most accurate approach to extract Vus is to use rate ofsemileptonic kaon decays:

KL

π

l

ν

W

ΓK`3 =G2FM

5K

192π3SEW (1 + δ`K)C2 |Vus|2 f2

+(0)I`K , (3)

Here:

• SEW , δ`K – universal short- and mode dependent long-distanceradiative corrections.

• C = 1 for KL and C = 1/2 for K±.

• f2+(0) is calculated in theory form factor value for t = 0

• I`K are mode and form factor (f + (t) for Ke3 and f+(t), f0(t)for Kµ3) dependent decay phase space integrals.

Situation before 2004Apart from unitarity problem, Vus seemed to be wellunderstood before the new data has arrived:

• Measured with KLe3 (0.2182± 0.0012exp), K±e3(0.2208± 0.0016exp) and Hyperon decays (0.2176± 0.0026).The most precise measurement came from KLe3 decays.

• KLe3 branching fraction is extracted from variousmeasurements of 36 different experiments performedbetween 1967-1995, they show good internal agreement

• f+(t) form factor is measured by ∼ 10 experiments, welldescribed by linear λ+ term. The value of λ+ is consistentbetween K± (0.028± 0.003) and KL (0.030± 0.002) as wellas with theory (chiral QCD) expectations (∼ 0.028).

• f+(0) is calculated by Leutweyler and Roos in 1984, theiranalysis shows that K±e3 and KLe3 data are consistent.

The only problem in this picture was BNL E865 determinationof Vus based on K±e3 data (PRL 91 261802, published on 31Dec 2003) which triggered a lot of new experimental activity.

Consistency check: Ke3 vs Kµ3Vus measured with Ke3 should be equal to Vus measured with Kµ3(“lepton universality”). Also, fKe3+ (t) = fKµ3

+ (t). For a linearparameterization of f0(t) this allows to extract λ0 fromBr(Kµ3)/Br(Ke3):.

0

2

4

6

8

10

12

14

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06λ0 (for λ+=0.030)

µ-polarization

Dalitz plot

ΓKµ3/ΓKe3

TheoryBijnens-94

PDG BR

Birulev 81

Cho 80

Hill 79

Buchanan 75

Donaldson 74

Clark 77

Sandweiss 73

λ0 (for λ+=0.030)λ0 (for λ+=0.030)λ0 (for λ+=0.030)

• unsatisfactory experimental situation.

• theory (which is used for f+(0)) largely disagree with BR result

K+ result from BNL E865

Measurement of Br(K+e3) based on 70.000 decays normalizedto Br(K+ → π+π0), Br(K+ → π0µ+ν) andBr(K+ → π+π0π0).

π0 is detected using Dalitz π0 → e+e−γ decay.

Assuming PDG-02 values for the branching fractions of thenormalization modes, using also new calculations of f + (0) andlong distance radiative corrections, E865 experiment extracts:

Vus = 0.2272± 0.0023rate ± 0.0007λ+ ± 0.0018f+(0) (4)

With this value, CKM unitarity is satisfied within 1σ.

New results in 2004

After the BNL result, we got much more experimental andtheoretical attention to Vus.

• New measurements/determinations of semileptonicbranching fractions: KTeV (also preliminary by NA48,KLOE)

• New measurements of semileptonic form factors: ISTRA+,KTeV (also preliminary by NA48).

• New (preliminary) measurement of KL lifetime: KLOE

• New calculations of f+(0) – chiral QCD, lattice QCD.

• New results for Ξ0 beta decay

KTeV measurement of KL branching fractions

KTeV is (was) a fixed target experiment to measure <(ε′/ε)with 10−4 precision. Since there is no way to tag the kaon,measure all six largest decay modes in terms of five branchingfraction ratios and use the constraint that the remaining widthis just 0.03%. Use external τL to convert branching fractionsinto partial widths.

The five measured ratios are:

ΓKµ3/ΓKe3 ≡ Γ(KL → π±µ∓ν)/Γ(KL → π±e∓ν) (5)

Γ+−0/ΓKe3 ≡ Γ(KL → π+π−π0)/Γ(KL → π±e∓ν) (6)

Γ000/ΓKe3 ≡ Γ(KL → π0π0π0)/Γ(KL → π±e∓ν) (7)

Γ+−/ΓKe3 ≡ Γ(KL → π+π−)/Γ(KL → π±e∓ν) (8)

Γ00/Γ000 ≡ Γ(KL → π0π0)/Γ(KL → π0π0π0), (9)

KTeV: Acceptance vs Z

0

20

40

60

130 140 150

0.96

0.98

1

1.02

1.04

130 140 150

0

10

20

130 140 150

Thou

sands

of eve

nts / m

eter

0.96

0.98

1

1.02

1.04

130 140 150

Data/

MC ra

tio

0

5

10

130 140 150

0.96

0.98

1

1.02

1.04

130 140 150

0

10

20

30

130 140 150z vertex (m)

0.96

0.98

1

1.02

1.04

130 140 150z vertex (m)

π0π0π0

(a) (b)

Slope:(-0.87 ± 1.20) × 10−4/m

πeνSlope:(-2.30 ± 1.30) × 10−4/m

πµνSlope:(2.04 ± 2.04) × 10−4/m

π+π−π0 Slope:(2.37 ± 1.33) × 10−4/m

• Acceptance is different for different modes but well described byMC

• Special effort to minimize effects from different particle types(e.g. µ vs π). For example, µ system is not used in the mainKµ3 analysis and π0 decay products are ignored for π+π−π0.

KTeV: Reconstruction of the charged modes

electron E/p

11010 210 310 410 510 6

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Ent

ries

/ 0.0

1DataMC

pion E/p

11010 210 310 410 510 6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ent

ries

/ 0.0

4

DataMC

min Kµ3 track-energy deposit in CsI (GeV)

1010 210 310 410 5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Ent

ries

/ 80

MeV Data

MC

0

500

1000

1500

2000

x 10 2

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6mππ (GeV/c2)

Data

mππ (GeV/c2)mππ (GeV/c2)mππ (GeV/c2)mππ (GeV/c2)mππ (GeV/c2)mππ (GeV/c2)

(a)

mππ (GeV/c2)

Ke3 MC

Kµ3 MC

3π MCMC Sum

2π MC

Ent

ries/

10 M

eV/c

2

1010 210 310 410 510 6

0 0.01 0.02 0.03 0.04 0.05 0.06p2

t (GeV2/c2)p2t (GeV2/c2)p2t (GeV2/c2)p2t (GeV2/c2)p2t (GeV2/c2)p2t (GeV2/c2)p2t (GeV2/c2)

(b)Data

p2t (GeV2/c2)

3π MC2π MC Kµ3 MC Ke3 MC

MC Sum

Ent

ries/

500

MeV

2 /c2

0

1000

2000

3000

4000

x 10 2

-0.06 -0.04 -0.02 0 0.02k+-0 (GeV2/c2)k+-0 (GeV2/c2)k+-0 (GeV2/c2)k+-0 (GeV2/c2)k+-0 (GeV2/c2)k+-0 (GeV2/c2)k+-0 (GeV2/c2)k+-0 (GeV2/c2)k+-0 (GeV2/c2)

(c)

Ent

ries/

0.01

GeV

2 /c2

Data

k+-0 (GeV2/c2)

3π MC

MC SumKe3 MCKe3 MC

2π MCKµ3 MC

MC Sum

Different charged modes are distinguished from each otherusing CsI calorimeter energy response (left) and kinematicrequirements (right).

The background for each charged mode is ≤ 0.1%.

KTeV results for KL Branching Fractions

00.511.522.5

0.38 0.39 0.4 0.41B(KL→πeν)

KTEVPDG 02

B(KL→πeν)

00.511.522.5

0.27 0.2725 0.275B(KL→πµν)

KTEVPDG 02

B(KL→πµν)

00.511.522.5

0.19 0.2 0.21B(KL→3π0)

KTEVPDG 02

B(KL→3π0)

00.511.522.5

0.124 0.126 0.128B(KL→π+π−π0)

KTEVPDG 02

B(KL→π+π−π0)

00.511.522.5

1.9 2 2.11000 × B(KL→π+π−)

KTEVPDG 02

1000 × B(KL→π+π−)

00.511.522.5

0.85 0.9 0.95 11000 × B(KL→π0π0)

KTEVPDG 02

1000 × B(KL→π0π0)

Large change compared to PDG for 4 out of 6 decay modes. Inparticular, Ke3 is about 5% higher. But Kµ3 is consistent witholder values.

KTeV vs old experiments

05101520253035

-6 -4 -2 0 2 4(MKTeV - MExp)/σ∆M

Baldo-Ceolin 75James 72

Cho 71Meisner 71

Webber 70Behr 66

Chan 71Aubert 65

Burgun 72Webber 71

Cho 70Franzini 65

Cho 80Williams 74

Brandenburg 73Evans 73

Budagov 68NA31 95

Budagov 68Budagov 68

Kulyukina 68Anikina 64

Budagov 68Aleksanyan 64

NA31 95Cho 77

Alexander 73Brandenburg 73

Evans 73Kulyukina 68Hopkins 67

Hawkins 66Astbury 65

Guidoni 65

(MKTeV - MExp)/σ∆M

Γ+−0

ΓKe3

ΓKe3+ΓKµ3

ΓKµ3/ΓKe3

Γ000/Γtotal

Γ000/(ΓKe3+ΓKµ3+Γ+−0)

Γ000/Γ+−0

Γ+−0/ΓKe3

Γ+−0/(ΓKe3+ΓKµ3+Γ+−0)

• For all experiments: χ2/dof = 83/34

• Excluding Cho80, NA31: χ2/dof = 42/31

Another problem: η+−Using the measured KL → ππ branching fractions, external values ofτS (KTeV, NA48) and τL = 5.15± 0.04 (PDG02), and correcting forsmall effects of <(ε′/ε) and KS semileptonic branching fraction oneobtains η+−

η+− = τSτL

BL(π+π−)+BL(π0π0)[1+6<(ε′/ε)]1−BS(Kl3)

= (2.228± 0.010)× 10−3(10)

0.511.522.533.544.5

2.2 2.22 2.24 2.26 2.28 2.3 2.32 2.341000 x Iη+−I

KTeV

Semileptonic charge asymmetry

CPLEAR 99

Geweniger 74

• Most of the error for KTeV is from external τL uncertainty

• Geweniger-74 and CPLEAR-99 are KL −KS interference basedmeasurements, depend on τS , corrected to new τS .

From another talk: status of <(ε′/ε)

NA48 + KTeV-97 → <(ε′/ε) = (16.7± 2.3)× 10−4.KTeV analysis of 99 data is still in progress:

χ2/dof = 64.9 / 47

• π+π− analysis is essentially finalized

• π0π0 analysis: tuning CsI MC.

KTeV measurement of semileptonic form factorsSince kaon energy is unknown (2-fold ambiguity) use boost invarianttransverse-t determined using p⊥ of the particles.

0

50

100

150

200

0 0.05p⊥

2,ν (GeV2/c2)p⊥

2,ν (GeV2/c2)

(a)

dataMC

χ2/dof = 21.2/22

πeν

0

50

100

150

200

0 0.05p⊥

2,ν (GeV2/c2)p⊥

2,ν (GeV2/c2)

(b)

dataMC

χ2/dof = 16.1/18

πµν

0

50

100

0 0.05p⊥

2,e (GeV2/c2)p⊥

2,e (GeV2/c2)

(c)

dataMC

χ2/dof = 12.7/25

πeν

Thou

sand

s of e

ntries

per 0

.002 G

eV2 /c2

0

50

100

0 0.05p⊥

2,µ (GeV2/c2)p⊥

2,µ (GeV2/c2)

(d)

dataMC

χ2/dof = 14.8/21

πµν

0

25

50

75

100

0 0.05p⊥

2,π (GeV2/c2)p⊥

2,π (GeV2/c2)

(e)dataMC

χ2/dof = 24.7/26

πeν

0

25

50

75

100

0 0.05p⊥

2,π (GeV2/c2)p⊥

2,π (GeV2/c2)

(f)

dataMC

χ2/dof = 25.1/22

πµν

→ good agreement btw data and MC. MC study shows that

t⊥-method to extract FF is only about 15% less precise statistically

compared to ideal t-based extraction.

Form factors: non-linear termParameterization of the form factors:

f+(t) = f+(0)×[1 + λ′+

tM2π

+ 12λ′′+

t2

M4π

]

f0(t) = f+(0)×[1 + λ′0

tM2π

] (11)

KTeV sees improvement in the fit to t⊥ distribution using thequadratic parameterization for f+(t) :.

0

50

100

150

200

0 2 4 6

χ2/dof = 33.5/ 16

Data1st order fit

0.95

0.975

1

1.025

1.05

0 2 4 6

14.69 / 14A0 1.008 0.2503E-02A1 -0.7856E-02 0.1910E-02A2 0.1338E-02 0.3095E-03

0

50

100

150

200

0 2 4 6t⊥π/Mπ

2et⊥

π/Mπ2e

χ2/dof = 13.7/ 16

Data2nd order fit

Thou

sand

s of e

ntries

per 0

.4

0.95

0.975

1

1.025

1.05

0 2 4 6

13.73 / 14A0 1.000 0.2494E-02A1 -0.1351E-03 0.1905E-02A2 0.1872E-04 0.3084E-03

t⊥π/Mπ

2e

→ the second order fit changes IK integrals by about −1%

Form factor results

1011121314151617181920

25 26 27 28 29 30103 × λ+103 × λ+103 × λ+103 × λ+103 × λ+

Averageπµνπeν

103 ×

λ 0

(a)

1011121314151617181920

10 15 20 25103 × λ,

+103 × λ,+103 × λ,+103 × λ,+103 × λ,+

(b)

103 ×

λ 0

1011121314151617181920

0 2 4 6103 × λ,,

+103 × λ,,+103 × λ,,+103 × λ,,+103 × λ,,+

(c)

103 ×

λ 0

0

1

2

3

4

5

6

15 20 25103 × λ,

+103 × λ,+103 × λ,+103 × λ,+

MK*

Mv

103 × λ,+

(d)

103 ×

λ,, +

KTeV result is consistent with ISTRA+ result for K+

λ′+ λ′′+ λ0 (for λ+ = 0.0277)

×10−3

KTeV 20.64± 1.75 3.20± 0.69 16.5± 1.1

ISTRA+ 23.24± 1.55 1.68± 0.82 18.3± 1.1

KTeV check: lepton universalityVus measured with Ke3 and Kµ3 should be the same – leptonuniversality. More directly, the ratio of the Fermi coupling constantsfor electrons and muons must be the same:

(GµFGeF

)2

=[

Γ(KL → π±µ∓ν)

Γ(KL → π±e∓ν)

]/(1 + δµK1 + δeK

· IµK

IeK

)(12)

• Theoretical uncertainties in f+(0) cancel for this ratio

• “Matching scale” uncertainties for δ`K are reduced:(1 + δµK)/(1 + δeK) = 1.0058± 0.0010

• Uncertainties for the “rate” measurement ofΓ(KL → π±µ∓ν)/Γ(KL → π±e∓ν) = 0.6640± 0.0026differ vs the “shape” measurement of the form factors.

• Ratio of IµK/IeK = 0.6622± 0.0018 has reduced dependence on

the form factor parameterization.

(GµF /GeF )2 = 0.9969± 0.0048

NA48

NA48 presents new preliminary results for

• Measure B(KL → 3π0) = 0.1966± 0.033 (normalized toKS → 2π0) — consistent with KTeV

• B(KLe3)/B(KL → all 2 track) = 0.498± 0.004. UsingB(KL → 3π0) NA48 determines B(Ke3) = 0.4010± 0.0045— again consistent with KTeV.

• B(K±e3) = (5.14± 0.06)% (using K± → π±π0) asnormalization mode — consistent with E865.

• New results for Ξ0 beta decay (see later)

• Measurement of KLe3 form factor (linear parameterizationonly) λ+ = 0.0288± 0.0012, also in agreement with KTeV(0.0283± 0.0006).

KLOE: KS, KL, K+

KLOE — ability to tag KS , KL and K± makes KLOE an idealexperiment to measure branching fractions. A number of new(preliminary) results:

• Precision measurement ofBr(KS → πeν) = (7.09± 0.11)× 10−3

• Measurement of the four largest KL branching fractions:

Br(KL → πeν) = 0.3985± 0.0035

Br(KL → πµν) = 0.2702± 0.0025

Br(KL → π0π0π0) = 0.2010± 0.0022

Br(KL → π+π−π0) = 0.1268± 0.0011

(13)

• Measurement of the KL lifetime.

KTeV vs KLOE vs NA48

00.511.522.533.544.5

0.38 0.39 0.4 0.41B(KL→πeν)

KTEVPDG 02

KLOE 04 (prelim)

NA48 04 (prelim)

B(KL→πeν)

00.511.522.533.5

0.27 0.275B(KL→πµν)

KTEV

PDG 02

KLOE 04 (prelim)

B(KL→πµν)

00.511.522.533.544.5

0.19 0.2 0.21B(KL→3π0)

KTEVPDG 02

KLOE 04 (prelim)

NA48 04 (prelim)

B(KL→3π0)

00.511.522.533.5

0.124 0.126 0.128B(KL→π+π−π0)

KTEV

PDG 02

KLOE 04 (prelim)

B(KL→π+π−π0)

Taking into account correlation between KTeV measurements,χ2/dof = 13.1/6→ 4% consistency probability between thenew experimental results.

KLOE — lifetimeAnother very important development from KLOE: newmeasurements of KL lifetime (K± in progress !)

Two methods:

• Unitarity condition that the four measured modes sum to99.7% of the total width

• Use KL → 3π0 decays.

5 10 15 20 25 30 35

t* (ns)

10

counts

/0.5

ns

350

100

150

200

KL lifetime

KLOE 04 3π0

KLOE 04 Br

CPLEAR+Geweniger-KTeV

PDG-02 Ave.

49 49.5 50 50.5 51 51.5 52ns

Difference between interference and KTeV determination ofη+− can be used to determine τL.

τL = (51.18± 0.19) ns (χ2/dof = 5.3/3) (14)

New Hyperon data from NA48

Ξ0 beta decay:

Ξ0 → Σ+e−νe (15)

→ for the exact SU(3) symmetry identical to neutron β-decay.

NA48 sample from 2002-KS run – about 6000 events (with2.4% background) → much larger compared to the publishedKTeV sample — 176 events.

Br(Ξ0 → Σ+e−νe) = (2.51± 0.11)× 10−4 (16)

Assuming no SU(3) symmetry breaking, Vus = 0.214± 0.030,consistent with unitarity but the errors are large.

Radiative corrections for K`3 decays

Two parts of radiative corrections:

δtot = SEW (1 + δK) (17)

Universal short distance radiative corrections, SEW = 1.022,calculated by Sirlin in 1981.

Mode dependent radiative corrections δK :

• Originally calculated by Ginsberg in the late 1960s.

• New calculations for K0`3 and K±`3 using chiral QCD(Cirigliano et al, Bytev et al) and effective theory approach(Andre) – for K0e3 about 0.5% lower than Ginsbergestimation

• The radiative corrections are included in MC simulation(e.g. KLOR program in the case of KTeV)

→ δKLe3 = (1.3± 0.3)%, δKLµ3 = (1.9± 0.3)%, the errorsinclude the uncertainty arising from the change of the matchingscale.

New theory developments for f+(0)

Original estimate of f+(0) was made by Leutweyler and Roos(82):

• Complete chiral-QCD calculations up to p4

• Estimate of p6 contribution using quark model(f4 = −0.016± 0.008)

→ f+(0) = 0.961± 0.008 (for KL).

New estimates:

• New chiral-QCD based calculation of p6 terms ( Bijnens etal, Jamin et al.; f4 = −0.002± 0.010 )

• New lattice-QCD (quenched) calculation (Becirevic et al)

The new chiral-QCD calculations tend to return higher values→ f+(0) = 0.980± 0.010

The quenched lattice QCD is close to the original estimate:→ f+(0) = 0.960± 0.009

Check of the theory: form factors vs experiment

The theoretical estimates of f+(0) can be checked comparingf(t) predictions vs experiment.

Lattice calculation, “pole model” fit (from the talk of F. Mesciaat ICHEP04):

Theory KTeV

λ+ 0.025± 0.002 0.0250± 0.0004

λ0 0.012± 0.002 0.0141± 0.0010

→ good agreement between data and theory.

But chiral-QCD also predicts similar values of λ+, λ0.

Putting things togetherExperiment:

KL lifetime is re-measured by KLOE. The new world averagevalue is τL = 51.18± 0.19 ns.

B(KL → πeν) is measured by KTeV, KLOE, NA48. All values are higherthan PDG-02. KTeV and KLOE are different by ∼ 2σ,NA48 agrees with both.Need final results from KLOE/NA48 !

B(KL → πµν) is measured by KTeV, KLOE and agrees well with PDG-02

B(K± → πeν) is measured by NA48, agrees well with E865, significantlyhigher than PDG-02.

f+,0(t) measured by ISTRA+ for K+ and KTeV for KL agree well.

Theory:

f+(0) is re-calculated by chiral-QCD to p6 and by lattice QCD.Differ by ∼ 2%

f+,0(t) for both chiral and lattice QCD are in agreement withISTRA+, KTeV measurements.

Vusf+(0) results

|Vus|f+(0) separates theoretical and experimental errors:

-4

-2

0

2

4

6

8

10

12

0.21 0.215 0.22 0.225IVusI f+(0)

PDG 02E865 Ke3NA48 Ke3 (prelim)

PDGKTEV AveKLOE Ke3 (prelim)KLOE Kµ3 (prelim)NA48 Ke3 (prelim)

KLOE (prelim)

Leutwyler and RoosBijnens and TalaveraBecirevic et al.Jamin et al.

IVusI f+(0)

K+

KL

KS

f+(0)(1-|Vud|2-|Vub|

2)1/2

(All new KL results adjusted for the new average lifetime, NA48 is

adjusted for the new KTeV form factor measurement).

Vus

The final verdict if unitarity problem still exists or not seems tobe on the theory side. Using the lattice QCD calculations forf+(0), and only new experimental data one obtains:

|Vus| =

0.2257± 0.0023 (KL)

0.2261± 0.0029 (KS)

0.2287± 0.0026 (K+)

(18)

Or the average:

|Vus| = 0.2262± 0.0023

With this value for Cabibbo angle, the unitarity:

δ = 1−(|Vud|2 + |Vus|2 + |Vub|2

)= 0.0013± 0.0018

is satisfied at 1σ level.

Conclusions

• New experimental data shows large deviation from oldPDG average values for both KLe3 and K±e3 decay rate

• New results resolve longstanding unitarity issue, (if latticeQCD is selected for f+(0))

Still to do:

• Need final results from KLOE, NA48 → may help toresolve KLe3 difference.

• Need new K± lifetime measurement (from KLOE) → mayhelp to understand ∼ 2.2σ discrepancy between KL andK± Vus determination.

Moral: never leave important measurements un-re-measured