Probing non perturbative
QCD with the cusp effect inkaon decays by NA48
Dmitry Madigozhin(JINR, Dubna)

on behalf of the NA48/2 Collaboration:
Cambridge, CERN, Chicago, Dubna, Edinburgh, Ferrara,Firenze, Mainz, Northwestern, Perugia, Pisa, Saclay,Siegen, Torino, Vienna

NA48/2 beam line

Simultaneous K+ and K− beams:

2­3M K/spill (π/K~10),5460

large charge symmetrization ofπ decay products stay in pipe.

PK spectra,
60± 3 GeV/cexperimental conditions

Flux ratio: K+/K– ≈ 1.8

magnet66

K+K+Be target

K−~7⋅ 1011 ppp,
400 GeVachromatzfocusing beams

SecondK−Front­end1cmBM

beam pipeachromat• Cleaning

Quadrupole• Beam spectrometer

quadruplet(resolution 0.7%)• Momentum• Focusingselection• µ sweeping

Beams coincide within ~1mm
all along 114m decay volumevacuumtank

He tank+ spectrometer10 cm

not to scale

50

100

200250 m

The NA48 detector
Main detector components:• Magnetic spectrometer (4 DCHs):
4 views/DCH: redundancy ⇒efficiency;
used in trigger logic;
Δp/p = 1.0% + 0.044%*p [GeV/c].
• Hodoscope
fast trigger;
precise time measurement (150ps).
• Liquid Krypton EM calorimeter (LKr)
High granularity, quasi­homogenious;
E/E = 3.2%/E1/2 + 9%/E + 0.42% [GeV];

Beam pipe

x=y=0.42/E1/2 + 0.6mm (1.5mm@10GeV).
• Hadron calorimeter, muon veto counters,
photon vetoes.

Beams

NA48/2 data:

2003 run: ~ 50 days
2004 run: ~ 60 days

A view of the NA48/2 beam line

Total statistics in 2 years:
K: ~4·109K: ~1·108
Rare K± decays:
BR’s down to 10–9
can be measured>200 TB of data recorded

Pion scattering lengths
The important free parameter of ChPT is the quark condensate <qq>,
it determines the relative size of mass and momentum terms in thepower expansion.
a0 and a2 are S-wave  scattering lengths in isospin states I=0 andI=2, correspondingly. They enter into all  scattering amplitudes.
The relation between <qq> and the scattering lengths a0 and a2 isknown from this theory with a high precision, so the experimental
measurement of a0 and a2 provides an important constraints for
ChPT Lagrangian parameters.
Pion scattering lengths can be measured in thestudy of the cusp­effect in K decays[2003 result: PLB 633 (2006) 173, Cabibbo­Isidori theoretical framework]K    selection  Assume  for each  :(Dik)2
m02 = 2EiEk(1-cos) ≈ EiEk2 = EiEk(zik)2
m02 – mass of Ei, Ek – energy of ikDik – distance between
iandk on LKr
zik – distance from  
decay vertex to LKr
Ek

Background isnegligible

K-decay vertex

zik

accidental photonDikEmLKrzzlmEiElzselected photon

3 mass ­ MK

2 mass resolution
, MeV/c2

spectrum… excellent at low M00!
M2(), (GeV/c2)2

Observation of the cusp
2003: ~16 millions events

2004: ~44 millions events

M2(), (GeV/c2)2

M2(), (GeV/c2)2First observation of thecusp was made withthe 2003 data
Inclusion of 2004 data:
statistics increased bya factor of 3.7

0

Uniform selection
conditions andMC/Data statistics, so
we sum the2003 and 2004 datafor the joint analysis

– threshold

M2(), (GeV/c2)2

M2(), (GeV/c2)2

Full data statistics
2003 + 2004
Fit region

60.31 millions
eventsFit region is choosen toreach a minimum total error(systematical contribution
grows with upper M00)
226-th bin instead of 176-th(as was in our first cusp
paper (2006))

Statisticaluncertainties
are shown !

Theory: N.final
state
rescattering
Cabibbo, PRL 93 (2004) 121801
M(K) = M0 + M1
Direct emission (k0,h0 << g):

M0 = A0(1+g0u/2+h0u /2+k0v /2)
22

M+ = A+(1+g+u/2+h+u2/2+k+v2/2)

Rescattering amplitude:

M1 = –2/3(a0–a2)m+M+ 1–((isospin symmetry
assumed here)

Kaon rest frame:u = 2mK∙(mK/3−Eodd)/m

2K+v = 2mK∙(E1−E2)/m2

M00
22m+

Negative interference
under threshold)K amplitude
at threshold

No M1 amplitude
Arbitrary scale

Combinationof S­wave  scatteringlenghts



M1 amplitude present:
13% depletion
under the threshold
M2(), (GeV/c2)2

Theory: two-loop diagrams
N. Cabibbo and G. Isidori (CI),JHEP 503 (2005) 21One­loop diagrams:•Five S­wave scattering lengths (ax, a++, a+–, a+0, a00)
expressed as linear combinations of a0 and a2•Isospin symmetry breaking accounted for following J.
Gasser. For example, ax = (1+/3)(a0–a2)/3, where=(m+2–m02)/m+2=0.065 is isospin breaking parameter•Radiative corrections missing; (a0–a2) precision ~5%•V­dependent terms ~ (k`/2)V2 introduced both intoTwo­loop diagrams:a) 2 scatteringb) irreducible 3 scattering

Arbitrary scale“unperturbed” Kand K amplitudes.

Prediction of thetwo­loop theoryCusp point

No rescattering
amplitude

Subleading
effectc) reducible 3scattering

Leading effectM2(), (GeV/c2)2

Theory: effective fields

G. Colangelo, J. Gasser, B. Kubis, A. Rusetsky (Bern­Bonn group, BB)
Phys.Lett. B638 (2006) 187­194

NA48 technical intervention:polinomial parts of amlitudes are
expressed in terms of (U,V)slopes g,h,k (numerically different
from CI case ones).

• Non­relativistic Lagrangian for effective fields;
expanding in another small parameters.
• Valid in the whole decay region.
• Another (in comparison with CI) part of amplitude isabsorbed in the polinomial terms (so anothercorrelations).
• At two loops, algebraically different formulae foramplitude• FORTRAN code provided by authors

Fitting procedure1­dimensional fit of the M00 projection

Detector response matrix Rij obtained with a GEANT­basedMonte­Carlo simulation

Generated distribution
G(g0,h0,a0,a2,M00)

Reconstructed s3 (GeV/c2)2

Bin width 0.00015 (GeV/c2)2

MINUIT minimization of of data/MC spectra shapesLog(Rij)
420x420 bins

Reconstructed
distribution:
FjMC = RijGiGenerated s3=M2(), (GeV/c2)2

Up to 5 free parameters(FDATA–NFMC)2
2(g,h’,m+(a0–a2),m+a2,N)=s bins F 2+N2F 2DATAMC3

If one fix a2 or link it to a0 by some relation, result for (a0-a2) is more precise

Charged decay (K) amplitude parameters•For the CI fit we take the measured by NA48/2 parameters of the K amplitude

M+ = A+(1+g+u/2+h+u2/2+k+v2/2):• g = ­ 0.21117(15);+h+ = 0.00671(26); k+ = ­ 0.00477(8)• For BB case the similar parameters are not numerically the same due to thepresence of additional rescattering terms in the in M+ amplitude. To define them:simultaneous fit of the NA48/2 K and KDalitz plots by the CGKR
code; central result (slightly depends on other fit conditions) is:• : g = ­ 0.1894(4); h = 0.0003(20); k = ­ 0.0077(19)+++Pionium signature
Points excluded from the fit
due to absence of EM correctionsin the model

7 data bins skipped aroundthe M(–) threshold
Combined 2003+2004 sample

Excess of events in the excluded interval (CI fit),
if interpreted as due to pionium decaying as A2,gives R=(KA)/(K–) = (1.80.3)10–5.
Prediction [Z.K. Silagadze, JETP Lett. 60 (1994) 689]: R=0.810–5.

Electromagnetic corrections to final state interactions in K → decays(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 → Our observable is an additional joint narrow (with respect to our resolution)
contribution into the decay width at the very cusp point.
This measured narrowcontribution N with respect to fullwidth :N / (K→)=(5.6±1.0)10-5 resonant structure(no experimentalresolution) atoms plusresonant structurewith experimental resolution

In our fits we use a freeparameter fatom (relative excess
in threshold bin of width
0.00015 GeV2) to represent
Pionium + resonant structure.
Its predicted value 5.8% is ina good agreement with themeasurement (6 +/- 1 %).

resonant structurewith experimental
resolution

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.
Electromagnetic part is included into Lagrangian and all
the lowest-order diagrams are taken into account — moreconsequent calculation for the wide U region.
But the bound states and resonant peak at the very threshold
can not be calculated perturbatively. That is why we use freeparameter f_atom to represent the peak-like contribution in our fits
of experimental data.
The correction could be extracted as a relative effect for BB
amplitude and implemented also in Cabibbo-Isidori case as amultiplicative factor.

Effect of radiative corrections
Corrected

EM interaction makes cusp
somewhat stronger, so ignoring
it leads to overestimation of thepions scattering constants
68% - probability ellipses
for statistical errors only
to illustrate the shift.
Chiral symmetry constraint[Colangelo et al., PRL 86(2001) 5008]:

Uncorrected

a2 = (–0.0444 +/­ 0.0008)+ 0.236(a0–0.22)– 0.61(a0–0.22)2– 9.9(a0–0.22)3

Example of parameters correlation coefficients(statistical only), BB fit case

Systematics example: decays Z distribution• Good agreement
Between Dataand MC.
• But: huge statistics,
difference is visible.

MC normalized
to Data
Symbols : Data
Histogram : MC

• Define the sources,
measure the result
sensitivity -> syst.err.

Trigger effectDifference inforward decays
acceptance (trackradius cut + beam shape)

Data/MCSystematic uncertainties
for independent a0 and a2

In units of 10-4Correlation
betweena0 and a2 aretaken into
accountFinal result for independent a0 and a2(a0–a2)m+= 0.2571 ± 0.0048stat. ± 0.0025syst. ± 0.0014ext.
a2m+= –0.024 ± 0.013stat. ± 0.009syst. ± 0.002ext.

External uncertainty: due to (A++–/A+00)|threshold = (1.93­1.94)± 0.015;(central value depends on fitted matrix element, error comes frombranching rates PDG data)
Additional theoretical uncertainty (higher order terms neglected) is estimatedfrom the difference between results, obtained using Bern­Bonn andCabibbo­Isidori formulae, that assume different next­order contributions: (a0–a2)m+= 0.0088;

 a2m+= 0.015.

The relative errors of all kinds for a2 are essentially larger than (a0-a2)
ones, as a2 corresponds to second-order contribution to cusp shape.

Systematics with ChPT constraint(a2 is a function of a0).

In units of 10-4

Final result with ChPT constraint[Colangelo et al., PRL 86 (2001) 5008](a0–a2)m+= 0.2633 ± 0.0024stat. ± 0.0014syst. ± 0.0019ext.
Theory precision uncertainty for this case is estimated in theory:(a0–a2)m+= 0.0053 (2%).

S.Gallorini 2008

Theoretical ChPT prediction : 0.265 +/- 0.004(Colangelo, Gasser, Leutwyler 2000)
Experimental precision is compatible with the theoretical one, thelargest is the theoretical uncertainty of (a0-a2) extraction !

Results: other Dalitz plot parameters

M(K) = M0 + M1
Unperturbed amplitude is M0 ~ (1+g0u/2+h u2/2+k v2/2)
00

NB: even without M1not the same parameters as the PDG ones:|M0|2(PDG) ~ (1+gu+hu2+kv2)[g0g, h0h–g2/4, k0k]•Technique:1. k0 is extracted from 2­dimensional CI and BB fits
2. (a0­a2,g0,h0); ChPT a2(a0); fixed k0 (its uncertainty ­> systematics)
CI parameters:

k0CI = –0.0095
g0CI = 0.653
h0CI = –0.043.

BB parameters:

k0BB = –0.0081
g0BB = 0.622
h0BB = –0.052.

uncertainties:± 0.0002stat. ± 0.0005syst.
± 0.001stat.± 0.003syst.
± 0.001stat.± 0.003syst.

For the free a2 the errors are larger.

Final results of pion scattering lengths measurementfrom the cusp and comparison to Ke4 results
68% - probability ellipses,full uncertainties:
combined- Statistical,- Systematical,- External,- Theoretical.
All covariances are
summed together to obtainthe final correlation between
a0-a2 and a2 (-0.879).

Cusp and Ke4 results using
Chiral constraint (shownwith a Gaussian error)

Refer to the next talk for Ke4 analysis details

Conclusions
60.31 millons of fully reconstructed K decays,
cusp anomaly in mass distribution at 2mdue topions rescattering.
Complete analyses are done both for Cabibbo­Isidoriand Bern­Bonn theoretical calculations, taking into
account electromagnetic correction.
Final result with ChPT link between a0 and a2 :(a -a ) =(experiment)(theory)
02[CERN-PH-EP/2009-010, submitted to EPJC],
is in good agreement with Ke4 and with the prediction ofChPT.
Experimental uncertainty is smaller, than the errorof theoretical ChPT prediction 0.265 0.004 (Colangelo,Gasser, Leutwyler 2000).