Precision measurement of ππ scattering lengths in Ke4decays by NA48/2
Brigitte Bloch-Devaux
CEA Saclay, IRFU-SPP
KAON 09, EPOCHAL Tsukuba, 9-12 June 2009

On behalf of the NA48/2 collaboration:
Cambridge, CERN, Chicago, Dubna, Edinburgh, Ferrara, Firenze,Mainz, Northwestern, Perugia, Pisa, Saclay, Siegen, Torino, Wien
June 10, 2009

B.Bloch-Devaux @ KAON09

1outline• Brief introduction to NA48/2• QCD tests from study of Kaon decays @ NA48/2• Ke4 decays (K± Æ π +π − e± ν ) :Form Factors and π π scattering lengths• Combining cusp and Ke4 results• SummaryJune 10, 2009

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The NA48/2 experiment at the CERN-SPS : primarily designed for
CP violating charge asymmetries studies in K3π decays
2003 run: ~ 50 days +Simultaneous K+ and K- beams:
large charge symmetrization ofexperimental conditions
June 10, 20092004 run: ~ 60 days

PK spectra,
60±3 GeV/c

B.Bloch-Devaux @ KAON09

Beams coincide within ~1mm
all along the 114m decay volume
flux ratio K+/K- ~1.8
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The NA48/2 experiment: detector and performances
LKr electromagnetic calorimeter :quasi-homogenous and high granularity
ΔE/E = (3.2/√E ⊕ 9.0/E ⊕ 0.42)% (E in GeV)
σx = σy ~ 1.5 mm for E=10 GeV
Very good resolution for neutrals (π0 Æ γγ)
σ (Mππ0π0) = 1.4 MeV/c2+Magnetic spectrometer :4 high-resolution DCH’s + dipole magnet
Δp/p = (1.0 ⊕ 0.044 p)% (p in GeV/c)Very good resolution for charged invariant
masses: σ (M3π± ) = 1.7 MeV/c2

E/p ratio used for e /π discrimination
Hodoscope for charged fast trigger
σt = 150 ps
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Kaon decays : what can be learned on QCD @ Low Energy ?
Hadronic decay modes into 3 pions: ( D.Madigojine ‘s talk)• large Br’s : K± Ø π0π0 π± (1.7 %) and K± Ø π+ π − π± (5.6 %),60 Millions events now analyzed(PRL B633 (2006) partial sample)• three pions : π0π0 system + nearby hadron• accessible Μππ range from π0π0 threshold to (MK-Mπ )Semileptonic decay mode Ke4: (this talk)• small Br’s : K± Ø π+ π− e± ν ( 4.1 10-5),
1.1 Million events now analyzed(EPJC 54 (2008) partial sample)• only two π+ π− pions, very clean environment• accessible Μππ range from π+ π− threshold to (MK-Me) ≅ MK
Two different but complementary approaches to ππ scattering near thresholdto extract s-wave scattering lengths ( a0, a2) for Isospin I = 0 and I = 2
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Ke4 decays : formalism
Five kinematic variables (Ca.Ma. variables)(Cabibbo-Maksymowicz 1965)
Sπ (M2ππ), Se (M2eν),dipioncosθπ, cosθe and φ.

dilepton

Partial Wave expansion of the amplitude
into s and p waves (Pais-Treiman 1968)+ Watson theorem (T-invariance) for δlI
δ00 ≡ δs and δ11 ≡ δp

Map the distributions of the Ca.Ma.
variables in the five-dimensional space with
4 Form factors and one phase shift ,assuming identical phases for the p-wave
Form Factors Fp, Gp, Hp:

F, G = 2 Axial Form Factors
F = Fs eiδs + Fp eiδp cosθπ
G = Gp eiδg
H = 1 Vector Form Factor
H = Hp eiδhThe fit parameters are :FsFpGp

Hp and δ = δs- δp(Fs , Fp , Gp, , Hp are real )F, G, H are complex and dimensionlessJune 10, 2009

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Ke4 decays: event selection and background rejection
Signal (π +π − e ± ν) topology:- 3 charged tracks and a good vertex- two opposite sign pions,- 1 electron (E(LKr) /p ~ 1),- some missing energy and pT (ν)- reconstruct PK (missing ν hypothesis)

π+π-

K+e+

ν

π2+K+π-Background main sources :π2+
π 1+ (dominant)K+ Æ π ν or mis-identeս e- ν orսemis-ident
սe+ νeor
mis-ident e++K+ Æ π 0 (π 0 ) π +ս(e+e–γ)
սս
mis-ident
π+πmis-ident

e+π1++ν(π20)and γ (s) undetected

π+K+(e- ) π –e+π10(γ) νJune 10, 2009

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Ke4 decays: background rejection

Control sample from data(assuming ΔS=ΔQ valid)
K ± Æ π ± π ± e∓ ν “Wrong Sign” events- total charge (±1) as “Right Sign” events- electron charge opposite to total charge- same sign pions

P KaonRatio (RS/WS) events:
2/1 if coming from K3π (dominant)
1/1 if coming from K2π (π0)
GeV/cTotal background level can be kept at ~ 2 x 0.3 % relative level
estimated from WS events rate and checked from MC simulation
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Ke4 decays : fitting procedure
Total (2003+2004) 1.13 Million Ke4 decays
Using iso-populated boxes in the 5-dimension space of the Ca.Ma. variables,(Mππ, Meν, cosθπ, cosθe and φ) one defines a grid of
10x5x5x5x12=15000 variable size boxes.
In each Mππ “slice” (1500 boxes), a set of 4 fit parameters is found which minimizes thedifference between the data and predicted populations
The normalisation Fs2 is obtained in each bin/slice by the ratio xslice = Σj in slice Nj/Σj in sliceMCj
K+ sample (726 400 events) 48 events/box

K+ MC (17.4 Million events) 1160 events/box

K- sample (404 400 events) 27 events/box

K- MC ( 9.7 Million events)

650 events/box

MC sample

Data sample

K+ and K- samples fitted separately in 10 independent Mππ bins/slices, then combined ineach slice according to their statistical error.

No assumption is made on the shape of the variation of the phase δ (and FF) fromone Mππ slice to the next ( i.e. “model independent” analysis)
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Ke4 decays : Data/MC comparison after fit
Mππ (GeV/c2)

Meν (GeV/c2)cosθπφ (rad)cosθeK+K-●= Datahistogram= Simulation after fit.Yellow hist= WS Background (x 10 to be visible)

CP symmetry :(K+) φ distribution is opposite of (K-) φ distribution
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Ke4 Form Factors : fit results
Series expansion with q2 (q2=Sπ/4mπ2 -1)
and Se/4mπ2 used to describe the FF
variations, in the isospin symmetry limit(Amoros Bijnens 1999)
Fs2 = fs2(1+fs’/fs q2 + fs’’/fs q4 + fe’/fs Se/4mπ2)2Correlation

fs’’ /fsfe’ /fs

fs’ /fs-0.95

0.08fs” /fs0.02

Gp/fs = gp/fs + gp’/fs q2

Correlation -0.91

valuestatsystfs’/ fs
fs ” / fsfe’/ fs

0.152 ± 0.007 ± 0.005-0.073 ± 0.007 ± 0.006
0.068 ± 0.006 ± 0.007

f p / fs
constant-0.048 ± 0.003 ± 0.004

gp / fs
gp’ /f s

0.868 ± 0.010 ± 0.010
0.089 ± 0.017 ± 0.013

hp / fs
constant-0.398 ±0.015 ± 0.008

systematics• mostly from background + acceptance control

first evidence by NA48

• ~ same size as statistical error or smaller
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Ke4 decays: from phase shifts to scattering lengths (a0,a2)
ππ phases at threshold can bepredicted from data above 0.8 GeV
using Roy equations (unitarity,analyticity and crossing symmetries)
and 2 subtraction constants a0 and a2

Numerical solutions have beendeveloped (ACGL, DFGS) valid in theIsospin symmetry limit (UniversalBand in the [a2,a0] plane), but
broken in the experimental world.

factorization of electromagnetic and mass effects :Gamow factor x PHOTOS generator
Radiative effects (except masseffects) included in the simulation,
Gamow factor : “classical” Coulomb
attraction between the 2 charged pions
PHOTOS generator: real photon(s) areemitted and tracked in the simulation

xIsospin corrections

Mass effects:• recently computed as a correction to themeasurements• even larger than current experimental
precision !
(CGR EPJ C59 (2009) 777,
DK preliminary June 2008 in progress )(-> effect on event selection + possible
bias on reconstructed quantities)
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Ke4 charged decays : isospin corrections to δ
CGR EPJ C59 (2009) 777 formulation developed in close contact with NA48

treeone loop

π0−η mixing

δ00 Æ

Ke4 Fit range [285,390] MeVCorrection is ~ 10-15 mrad
Exp. stat precision (δ) is ~ 7-8 mrad
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Ke4 decays: from phase shifts to scattering lengths (a0,a2)

a tiny effect from theory…. a bigchange in now precise experimental
measurement !

UBNO
NO Isospin
Isospin effects
effects

This induces a large change on (a0,a2) values
from a 2p fit

from a 1p fit

Δa0 = -0.025, Δa2 = -0.007

Δa0 = -0.022

error statstat

syst

σ(a0): ± 0.0128 ± 0.0050

syst± 0.005 ± 0.002

σ(a2): ± 0.0084 ± 0.0034
Isospin effects ON
June 10, 2009

Ellipses are 68% CL contours in 2p fits
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Ke4 decays: comparison with theoretical predictions
THEORY prediction

Experimental measurement

Using more inputs from ChPT and low energy
constants, the prediction is better constrained(CGL NPB603(2001),PRL86(2001)):

a0 =0.220 ± 0.005

a2 = -0.0444 ± 0.0008

0.2206 ± 0.0049 stat
± 0.0018 syst
± 0.0064 theo *a0free0.2220 ± 0.0128 stat
± 0.0050 syst
± 0.0037 theo*-0.0432 ± 0.0086 stat
± 0.0034 syst
± 0.0028 theo*
Correlation 96.7%

a2free2p fit

Universal BandChPT

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a0ChPT
1p fit*Theory error evaluated fromcontrol of the isospin corrections &inputs to Roy equation numericalsolutions (CGR EPJ C59 (2009)777)

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Comparison of Ke4 phase shift experimental measurements
Apply Isospin corrections (10-15 mrad) to all published points :S118 (Geneva-Saclay): typical error 40-50 mrad

NA48/2 typical error 7-8 mrad

E865: typical error 15-20 mrad

improved precision due to both-larger statistics ~3 x E865-larger acceptance at high ππ mass

Correction small wrt experimental error but coherent
shift downwards for all data points•All Phase points corrected
for isospin mass effects•Independent experiments•Errors = stat + syst
Line from a 2p fit toNA48 data alone
Fit to all data points (21 points) : dominated by NA48/2 measurements
2p fit:

1p fit :a0 = 0.2199 ± 0.0125exp ± 0.0037theoa0= 0.2168 ±0.0048exp ± 0.0064theoa2 = -0.0430 ± 0.0083exp ± 0.0028 theo

(theory error common to all expts)

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Cusp and Ke4 scattering lengths results•Two statistically independent measurements by NA48/2
60 Millions K3π decays ⇔ 1.13 Million Ke4 decays• systematic uncertainties from different origins : mostly independentcalorimeter + trigger ⇔ background + electron identification• different theoretical inputs :(1- + 2-loop) re-scattering models ⇔ Roy equations + isospin mass effects• Large overlap in the (a0,a2) plane or (a0-a2,a2) plane with different
correlations suggests to combine the two results in a single one withimproved uncertainties• similar combination can be done for the results using the same ChPT
constraint

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Combining cusp and Ke4 results : method
Ke4:
ΔstatΔsyst
Δtheoa0 = 0.2220 ± 0.0128 ± 0.0050 ± 0.0037
a2 =-0.0432 ± 0.0086 ± 0.0034 ± 0.0028

x1x2Correlation
ρ12 = 0.967 (stat only)(0.969 with all errors)

Cusp:
Δstat Δsyst+ext Δtheoa0–a2 = 0.2571 ± 0.0048 ± 0.0029 ± 0.0088
a2 = –0.0241 ± 0.0129 ± 0.0096 ± 0.0149x3x4

ρ34 = -0.839 (stat only)
ρ34 = -0.879 (all errors)• Four measurements : (x1, x2,x3,x4)•Two fitted parameters : (a0, a 2) or (a0–a 2, a 2)0
where σii = Δxi2
•Error matrix isσ11 σ12 0
block-diagonal
σ21 σ22 00
and σij= σji =ρij Δxi Δxj
if no correlated
00 σ33 σ34
error cusp-Ke4 4
00 σ43 σ44
Minimize χ 2 =(x i − yi (a ))T Vij −1 x j − y j (a ) where Vij is the covariance matrix∑( )()

i, j=1

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Combined results from cusp and Ke4statsyst

theo

a0 = 0.2210 ± 0.0047 ± 0.0015 ± 0.0049
a2 = - 0.0429 ± 0.0044 ± 0.0016 ± 0.0030
Correlation 0.912
Total errors Δ a0: ± 0.0070 ( 3% rel. precision)
Δ a2: ± 0.0055 (13% rel. precision)
Including the ChPT constraint:

stat

syst

theo

a0 - a2 =0.2639 ± 0.0020 ± 0.0004 ± 0.0021
a2 = - 0.0429 ± 0.0044 ± 0.0016 ± 0.0030
Correlation 0.277
Total errors Δ (a0 - a2): ± 0.0030 ( 1% rel. precision)
Δ a2: ± 0.0055 (13% rel. precision)
stat

syst

theo

a2 = -0.0444 ± 0.007 ± 0.005 ± 0.0012
ora0 - a2 = 0.2640 ± 0.0020 ± 0.0017 ± 0.0035
a0 = 0.2196 ± 0.0027 ± 0.0021 ± 0.0048
Δ a2 : ± 0.0015
Δ (a0 - a2) : ± 0.0044
Total error Δ a0 : ± 0.0059
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Comparison with other experimental measurements
Cusp : (a0-a2) ChPT fit with 2 models,BB is the most complete in terms ofradiative corrections

a0 ChPT band prediction

Events60 106

Ke4 : apply isospin corrections topublished phase points of all
experiments and perform a0 ChPT fit
Note : E865 result dominated by highest energy
data point, otherwise compatible

1.13 106
4 105

ππ atoms DIRAC: |a0-a2 | errors fromPLB619 (2005), use ChPT constraint(only 40% Data analyzed)

3 104
6.5 103

Yellow band: best ChPT prediction

a0 (ChPT) = 0.220 ± 0.005

NA48/2 experimental precision and most precise theory prediction now at the same level!
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Summary
NA48/2 has analyzed ~1.13 M Ke4 events recorded in (2003+2004)• Ke4 Form Factors measured with an improved precision• Scattering lengths, extracted in Ke4 and K3π Cusp analyses, give a consistentexperimental picture and an impressive agreement with predictions from ChPT, both forfree (a0,a2) and constrained fits :ChPT best prediction a0 = 0.220 ± 0.005 , a2 = -0.0444 ± 0.0008 , a0–a2 = 0.264 ± 0.004(ChPT cusp )a0–a2 = 0.2633 ± 0.0024stat ± 0.0014syst ± 0.0019ext ± 0.0053th(ChPT Ke4 )a0= 0.2206 ± 0.0049stat ± 0.0018syst ± 0.0064th(ChPT combined) a0 = 0.2196 ± 0.0027stat ± 0.0021syst ± 0.0048 th
a2 = -0.0444 ± 0.0007stat ± 0.0005syst ± 0.0012 th
a0 - a2 = 0.2640 ± 0.0020stat ± 0.0017syst ± 0.0035 th• The achieved experimental precision (stat + syst) of combined result for a0 and a0–a2 is
now smaller than the theoretical precision .
The collaboration with many theory groups was/still is invaluable in understanding how to
extract scattering lengths from NA48 Data. Thanks to all groups in Bern, Bonn, Dubna,Orsay, Madrid, Marseille, Rome .. !

Two final publications (Cusp and Ke4) in progress
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spares

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Ke4 Form Factors : getting Fs••

variations of the normalization with q2, q4 and Se/4mπ2 are investigatedby a 4-parameter fit in the plane [Mππ,Meν]Introducing a slope with Se improves the fit χ2 (211./83 Æ 94./82)

Fs2 = fs2(1+fs’/fs q2 + fs’’/fs q4 + fe’/fs Se/4mπ2)2q2= (Sπ/4mπ2 -1)
Correlations:f”s /fs f’e /fs
f’s /fsf”s /fs-0.95

0.08
0.02

Other parameterizations could be easily tried
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Ke4 Form Factors: getting Fp, Gp, Hp

fp ≠ 0No q2
dependence

χ2=20.0/9

χ2=16.2/9

Gp/fs = gp/fs + g’p/fs q2
Linear with q2

χ2=16.8/8

June 10, 2009

Correlation
g’p /fs
gp /fsB.Bloch-Devaux @ KAON09

-0.91
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Common errors to Cusp and Ke4 results ?
• Statistical errors from cusp and Ke4 measurements are independent• Systematic errors : if (partly) correlated, contribute to the off-diagonal terms ofthe error matrix but very few common items• Theoretical uncertainties come from different sourcesCusp units 10-4 a0-a2

a2Ke4units 10-4a0a2

Acceptance (z)

42020

13

Acceptance (v)18Acceptance (z) +acceptance (r)+ PK spectrum ( ~ 2/3 )PK spectrum

1332

Trigger eff. (but not the same) 6

4Trigger eff.

1039

Fit method95

LKr resol

929

Bkg shape

3118

LKr nonlinear.

1267

e-ident

1419

LKr shower

1018

Radiative corrections

1610MC (time)

51

Bkg level1910

k0 control

26Se control

137Total

2594

Total

50

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B.Bloch-Devaux @ KAON0925Ke4 analysis : theoretical uncertainties

Work done in collaboration with Colangelo + Gasser
Ingredients to extract (a0,a2) from phase δ• Roy equations : two formulations ACGL(Bern)/ DFGS(Orsay)
Orsay can vary the solutions with phases @ matching point (0.8 GeV)
Reference :Vary : δo by ± 3.4 o
Vary : δ1 by ± 2.0 o• Isospin corrections
Vary R = 37 ± 5
Vary Fπ = (86.2 ± 0.5) MeV
Neglected Higher orders• CHPT constraint
Vary by a2 ± 0.0008

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The cusp in KL Æ 3π0 decays: work in progress

NA48:
S.Giudicci@HQL08

KTeV:
R.Kessler@BEACH08
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