Фундаментальные параметры рассеяния в распадах заряженных каонов эксперимент NA48/2

15 мая 200915 мая 2009 В.Кекелидзе, "Марковские чтения"В.Кекелидзе, "Марковские чтения"11

ФундаментальныеФундаментальные параметрыпараметры рассеяниярассеяния вв распадахраспадах

заряженныхзаряженных каоновкаоновэксперимент NA48/2


Эксперимент NA48/2

«cusp» эффект и параметры киральной теории

сравнение с Ke4 результатами

содержание


Эксперимент Эксперимент NA48/2NA48/2

2003 run: ~ 50 days

2004 run: ~ 60 days

Total statistics in 2 years:

K +: ~4·109

K 00: ~1·108

Rare K± decays:

BR’s down to 10–9

can be measured

> 200 TB of data recorded

Основная задача - Поиск зарядовой асимметрии в распадах K(3)


Установка Установка NA48NA48

High resolution

• magnetic spectrometer (+):

2 + 2 drift chambers; magnet with momentum kick 265 MeV/c

• electromagnetic cal. (00): liquid krypton (~10m3 at 120 K) 13212 cells granularity 2x2 cm2

Rate ~106 KL decays / spill


LKr - э-м жидко-криптоновый калориметр


NA48/2 beam line

1cm

50 100

Front-end

achromat

• Momentum

selection

Quadrupole

quadruplet

• Focusing• sweeping

Second

achromat

• Cleaning• Beam spectrometer

(resolution 0.7%)

~71011 ppp,

400 GeV

K+

K

Beams coincide within ~1mm

all along 114m decay

volume

focusing beamsfocusing beams

BM

z

magnet

K+

K

beam pipebeam pipe

Simultaneous K+ and K beams:

large charge symmetrization of

experimental conditions

Be target

2-3M K/spill (/K~10), decay products stay in

pipe.

Flux ratio: K+/K– 1.8

PK spectra,

603 GeV/c

54 60 66

10 cm

200

vacuum

tank

not to scale

250 m

He tank

+ spectrometer


An anomaly was observed at invariant mass M(00) = 2m+in K 00 decaysnever observed in previous experiments «Сusp» - эффект

Cusp - «остриё», особенность в спектре, связанная с изломом; в отличие от пика, достоверное наблюдение такой особенности требует • большой статистики и • высокого разрешения

2003 + 2004: 59.6 mln

events

+– threshold


длины пионного рассеянияважнейшим свободным параметром киральной теории (ChPT) является кварковый конденсат <qq>, который определяет относительные размеры масс и импульсов в степенном разложении

a0 и a

2 - длины рассеяния (S-волны) соответственно с

изоспином I=0 и I=2, также входят в амплитуды -рассеяния

связь <qq> с a0 и a

2 известна из теории с высокой точностью;

поэтому измерения a0 и a

2 позволяют получить важные ограничения

для параметров ChPT

Pion scattering lengths can be measured in the study of the cusp-effect in K decays

2003 result: PLB 633 (2006) 173, Cabibbo-Isidori theoretical framework]

«Сusp» - эффект и параметры киральной теории

Анализ был проведен независимо 2-мя группами из Пизы и Дубны

N. Cabibbo, PRL 93 (2004) 121801


m0 – mass of

Ei, Ek – energy of ik

Dik – distance between iandk on LKrzik – distance from decay vertex to LKr

KK selection selection

m02 = 2EiEk(1-cos) ≈ EiEk2 = EiEk

(Dik)2

(zik)2

M2(), (GeV/c2)2

, MeV/c2

2 mass resolution

… excellent at low M00!

MK-3 mass spectrum(2004 data)

103

Background isnegligible

Events

for each photon pair (i,k) a decay vertex is reconstructedalong the beam axis under the assumption of decay

accidental photon

selected photon

LKrDik

zzzik

zlm

K-decay vertexK-decay vertex

El

Ei

Emz

Ek


No M1 amplitude

M1 amplitude present:

13% depletion

under the threshold

Arbitrary scale

N. Cabibbo, PRL 93 (2004) 121801

M(K00) = M0 + M1

M0 =

A0(1+g0u/2+h0u2/2+k0v

2/2)

+

0

0K+Kaon rest frame:

u = 2mK∙(mK/3Eodd)/m2

v = 2mK∙(E1E2)/m2

Direct emission:

Rescattering amplitude:

M1 = –2/3(a0–a2)m+M+

Combination of S-wave scattering

lengths

Negative

interference

under threshold

K3

amplitude

at threshold

M2(00), (GeV/c2)2

(isospin symmetry

assumed here)

Объяснение «Cusp» - рассеяние в конечном состоянии

1–

( )2

M00

2m+


• S-wave scattering lengths (ax, a++, a+–, a+0,

a00) expressed as linear combinations of a0

and a2

• isospin symmetry breaking - following J.

Gasser

• for example, ax = (1+/3)(a0–a2)/3, where

=(m+2–m0

2)/m+2=0.065 - isospin breaking

parameter• all rescattering processes at one- & two-

loop level

• radiative corrections missing: (a0– a2)

precision ~5%

b) irreducible 3 scattering

Two-loop diagrams:

c) reducible 3 scattering

a) 2 scattering

N. Cabibbo and G. Isidori (“CI”),

JHEP 503 (2005) 21

Arbitrary scale

No rescattering

amplitudeSubleading

effect

M2(00), (GeV/c2)2

Cusp point

Leading effect

Prediction of

the

two-loop

theory

One-loop diagrams:

Теория: двух-петлевые диаграммы


Theory: effective fields

• Non-relativistic Lagrangian for effective fields; expanding in another small parameters.• Valid in the whole decay region.• Another (in comparison with CI) part of amplitude is absorbed in the polinomial terms (so another correlations).• At two loops, algebraically different formulae for amplitude• FORTRAN code written by authors

G. Colangelo, J. Gasser, B. Kubis, A. Rusetsky (Bern-Bonn group: “BB”)Phys.Lett. B638 (2006) 187-194


процедура фитирования

2(g0,h0,m+(a0–a2),m+a2,N)= (FDATA–NFMC)2

FDATA2+N2FMC

2s3 bins

Detector response matrix Rij obtained with a GEANT-based MC simulation

5 free parameters

1-dimensional fit of the M00 projection

MINUIT minimization of 2

of data/MC spectra shapes

Generated distribution

G(M00) = G(g0,h’,a0,a2,M00)

Generated s3=M200, (GeV/c2)2

Reconstructed s3

(GeV/c2)2

420x420 bins Reconstructed

distribution:

FjMC = RijGifit region

Log(Rij)


Fit quality & pionium signature

Combined 2003+2004 sample

Points excluded from the fitdue to absence of EM corrections

in the used model

7 data bins skipped aroundthe M(+–) threshold

Excess of events in the excluded interval,

if interpreted as due to pionium decaying as A200,

gives R=(K+A2)/(K+–) = (1.820.21)10–5.Prediction [Z.K. Silagadze, JETP Lett. 60 (1994) 689]: R=0.810–5.


Электромагнитные поправки в конечном состоянии (Gevorkian, Tarasov, Voskresenskaya, Phys.Lett. B649 159 (2007)) Two contributions from K±→ ± decay to the K±→ ± cusp region: Pionium formation : atom → (negligible width) Additional unbound states with resonance structure →

resonant structure(no experimental resolution)

resonant structurewith experimental resolution

atoms and resonant structure with experimental resolution

In our fit we use a freeparameter fatom (relative excess in the bin of width 0.00015 GeV2) to represent Pionium + resonant structure

measured value% (6 1) % is in good agreeemen with the prediction 5.8%


Effect of radiative corrections

• Electromagnetic part is included into Lagrangian and allthe first-order diagrams are taken into account

• But the bound states and resonant peak at the very thresholdcan not be calculated perturbatively (a free parameter fatom is used to represent the peak-like contribution)

• Could be extracted as a relative effect for amplitude andImplemented also to Cabibbo-Isidori case

Bern — Bonn radiative correction to pion rescattering amplitude for 3pi decays.

(Bissegger M., Fuhrer A., Gasser J., Kubis B., Rusetsky A.Nucl. Phys., B806, 2009, 178-223.)


Uncorrected

Corrected

68% - probability ellipsesfor statistical errors onlyto illustrate the shift.

Effect of radiative corrections


Example of correlation coefficients (CI)Example of correlation coefficients (CI)


(a0–a2)m+= 0.2571 0.0048stat. 0.0025syst. 0.0014ext.

a2m+= –0.024 0.013stat. 0.009syst. 0.002ext.

Результат по данным 2003+2004

Additional theoretical uncertainty (higher order terms neglected)

is estimated from the difference between results

obtained using Bern-Bonn and Cabibbo-Isidori formulae:

(a0–a2)m+= 0.0088; a2m+= 0.015.

External uncertainty: due to (A++–/A+00)|threshold = (1.93-1.94)±0.015;

(central value depends on fitted matrix element,

error comes from PDG data)

(окончательный)

The relative errors for a2

are essentially larger than (a0-a

2) ones,

as a2 alone leads to the second-order contribution to the cusp shape


Systematical errors for Systematical errors for independent aindependent a

0,0, a a22in units of 10-4


(a0–a2)m+= 0.2633 0.0024stat. 0.0014syst.

0.0019ext.

[Colangelo et al., PRL 86 (2001) 5008]:

Theory precision uncertainty for this case is estimated by theorists:

(a0–a2)m+= 0.0053 (2%).

Результат по данным 2003+2004

учитывающий ограничение из киральной теории(окончательный)

ChPT prediction : 0.2650 +/- 0.0040

Experimental precision is compatible with the theoretical one, the only problem is the theoretical uncertainty of measurement !


Systematical errors with ChPT constraint (a2 is a function of a0) In units of 10-4


scattering in K± → e± decay

15 мая 200915 мая 2009 В.Кекелидзе, "Марковские чтения"В.Кекелидзе, "Марковские чтения"

K

p*(e)

e

e

Direction of the total e+emomentum

in the K+

rest frame

Direction of the total

momentumin the K+

rest frame

p*()

a rare decay [ B.R. = (4.09 ± 0.09) x 105 ] described by five variables

Cabibbo – Maksymowicz variables : s ≡ M2

se ≡ Me2

e

For K K

e e

Ke4 formalism


In the partial wave expansion (only S and P waves) the amplitude can be written using form factors:

F=Fseis+Fpeipcos

G=Gpeip

H=Hpeip

The form factors can be expanded as a function of M

2 and Me

2:

F (Fp,Fs), G, H and =p-s will be used as fit parameters

Fs=fs+fs’q2+fs’’q4+fe’(Me2/4m

2)+...

Fp=fp+fp’q2+...

Gp=gp+gp’q2+...

Hp=hp+hp’q2+...

q2=(M2/4m

2)-1

Ke4 formalism


Ke4: selection & background Selection:Selection:

• 3 tracks

• Missing energy & missing Pt

• LKr/DCH to electron PID

739500 K+ ; 411000 K-

The background is studied using the electron “wrong” sign events (we assume DQ=DS & total charge ±1) and cross check with MC.

The total bkg is at level of 0.5%.

Kaon

momentum

GeV/c

Main background sources:Main background sources:

•

(→e

e

K

• with

misidentified

• or +(Dalitz) +e misidentified and s outside the LKr


Ke4: Fitting procedure

• The form factors (F,G,H and ) are extracted minimizing a log-likehood estimator in each of 10(M)x5(Me)x5(cose)x5(cos)x12()=15000 equi-populated bins. In each bin the correlation between the 4+1 parameters is taken into account.

Evts/

bin

evts

Evts/

bin

evts

MCData

9.8 M

650

17.7 M

1180

411000

27

739500

49

K-

K+• the form factors structure is studied in 10 bins of M, assuming constant form factors in each bins

• a 2D fit (M, Me) is used to study the Fs expansion

• all the results are given wrt to Fs(q=0) constant term, due to theunspecified overall normalization (BR is not measured)

M

● Data

▬ MC


KKe4e4: form factors result: form factors result

f’s/fs = 0.158±0.007±0.006

f’’s/fs= -0.078±0.007±0.007

f’e/fs = 0.067±0.006±0.009

fp/fs = -0.049±0.003±0.004

gp/fs = 0.869±0.010±0.012

g’p/fs = 0.087±0.017±0.015

hp/fs = -0.402±0.014±0.008

f’s/fs = 0.158±0.007±0.006

f’’s/fs= -0.078±0.007±0.007

f’e/fs = 0.067±0.006±0.009

fp/fs = -0.049±0.003±0.004

gp/fs = 0.869±0.010±0.012

g’p/fs = 0.087±0.017±0.015

hp/fs = -0.402±0.014±0.008• All the Form factors are measured relatively to fs

• first evidence of fp≠0 and fe’≠0

• The f.f. are measured at level of <5% of precision while the slopes at ~15% (factor 2 or 3 improvement wrt previous measurements)

Separately measured on K+ and K- - then combined (different statistical error)

Systematics checks:

• Acceptance

• Background

• PID

• Radiative corrections

• Evaluation of the sensitivity of the form factors on the Medependence of the normalization

Preliminay

(2003+2004

data)


а0 = 0.218 ± 0.013(stat) ± 0.007(syst) ± 0.002(theor)

a2 = -0.0457 ± 0.0084(stat) ± 0.0041(syst) ± 0.0030(theor)

(correlation coefficient 97%)

Using ChPT link:

a0 = 0.220 ± 0.005(stat) ± 0.002(syst) ± 0.006(theor)

Ke4 results:


Измерение длин пионного рассеяния- фундаментальных параметров киральной теории

68% - probability ellipses,full uncertainties


QuickTime™ Ë a ‰ÂÍÓÏÔ ÂÒÒÓ

Ú Â·ÛÂÚÒfl, ˜ÚÓ·˚ ‚Ë‰ÂÚ¸ ˝ÚÛ Í‡ ÚËÌÍÛ.

a0=0.2186 0.0065

2% th. uncert.


физика каонов, как и 50 лет назад остается актуальной и является потенциальным источником новых открытий

заключение


SPARE


The Ke4 decay amplitude depends on two complex phasesidentified with

0 : the scattering phase shift in the I = 0 , l = 0 state (s – wave) 1 : the scattering phase shift in the I = 1 , l = 1 state (p – wave)

The Ke4 decay rate depends on the phase shift difference = 0 – 1

is an increasing function of M ; → 0 for M → 2m

( from scattering theory : (k) ≈ ak at very low centre-of-mass momentum k ; a is the scattering length ; a ≠ 0 for s – waves only )

≠ 0 in the Ke4 decay amplitude asymmetric distribution ofthe charged lepton direction with respect to the plane definedby the two pions ( distribution)(Shabalin 1963)

The asymmetry of the distribution increases with M


NA48 / 2 (Preliminary)

distributions for four M bins

2m+ < M < 0.291 GeV

0.309 < M < 0.318 GeV

0.335 < M < 0.345 GeV

0.373 GeV < M < mK


versus M(no isospin – breaking corrections)

Geneva – Saclay : ~ 30,000 events , p K+ = 2.8 GeV/cBNL E865 : 406,103 events (with ~ 4.4% background), p K+ = 6 GeV/c NA48/2 : 677,510 events (with ~0.5% background), p K± = 60 GeV/c

NOTE: the isospin – breaking corrections reduce by 0.01 – 0.012(J. Gasser)

OLD PICTURE !!!


To extract scattering lengths from some external data (I=2 scatteringg at higher energies) and theoretical input are needed, for example:

Solution of Roy equations - ACGL, Phys.Rep.353(2001), DFGS, EPJ C24 (2002) that relates and (a

0,a

2) .

The Universal Band centre line parametrisation corresponds to 1-parameter fit with a

2=f(a

0)

ChPT constrain gives another relation between (a0,a

2) inside

Universal Band.

Isospin symmetry breaking is taken into account (new)

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Фундаментальные параметры рассеяния в распадах заряженных каонов эксперимент NA48/2