pp scattering lengths measurement from Ke4 and K3p
decays at CERN SPS experiment NA48/2
Sergio Giudici
University of Pisa and INFN
On behalf of the NA48/2 collaboration:
Cambridge, CERN, Chicago, Dubna, Edimburgh,
Ferrara, Firenze, Mainz, Northwestern, Perugia,
Pisa, Saclay, Siegen, Torino , Wien
Heavy Quarks & Lepton
University of Melbourne, June 5-9 2008
pp scattering lengths : why interesting ?
p
R
p
2mE
K
h
At low energy KR << 1 S-wave dominates total cross section
Isospin I = 0,2 only allowed by Bose statistics
Scattering matrix S|pp> = exp(2id) |pp>
may be parametrized with 2 phases: d0,2 = a0,2 k
related to scattering lengths a0 , a2
2 clean measurements can be done
cusp-effect in K3p decay:
a0 , a2
phase shift in Ke4 decays: ds- dp  a0 , a2
At low energy the S-wave scattering lengths are essential
parameters of Chiral Pertubation Theory (CHPT)
Cusp effect in K  pp0p0
Initial curiosity was to observe p+p- bound
states ( “pionium” ) annihilation in p0p0
K p (p+p-)Bound = pionium
“Pionium” would produce an excess of
events at the m00 = 2m+= 2x139.57 MeV
p0p0
s = 0.56 MeV at pionium
p0 masses constraint
optimizes Resolution at low
m00 values
Good Resolution due to excellent NA48 LKR calorimeter
performances and small Q-value
Cusp effect in

K


0
0
ppp
Zoom in Cusp region
Statistics:
2003 data : 16.0 M
2004 data : 43.6 M
Total : ~ 60 M events
4m+2
4m+2
Clear visible discontinuity
in the first derivative = CUSP

Unexpected Kaon Gothic behaviour

Cusp in Notre Dame, Paris ...
Theory: pp rescattering.
Decay Amplitude :
M(K pp0p0) = M0 + M1 + M2 + ...
p+
Dalitz plot variable
u = 2mK (mk/3 – Eodd)/mp2
v = 2mK (E1 – E2) /mp2
K+
p+
p+
K+
p0
p0
p0
Direct emission
M0 = A0 (1 + g0 u/2 + h’u2/2 + k’v2/2)
M1 = -2/3 (a0-a2) m+ A+
0
p
p1 loop Rescattering
(N. Cabibbo , PRL 93 , 2004, 121801)
2 loop Rescattering:
(N. Cabibbo & G. Isidori
JHEP 0503:021 , 2005)
Combination of S-wave
pp scattering length
 M00 
1- 

2m
+


2
K 3p Amplitude
Theory describes the data ... Fitting procedure
Combined 2003+2004 samples
2loops Cabibbo-Isidori fit
One dimensional fit to M002 distribution
MINUIT minimization of 2
of data/MC spectra shapes
Fitting up to half spectrum
0.097 (GeV/c2) since
Cabibbo Theory is an expansion
around 2m+ threshold
Fit to 5 parameters:
Norm, g, h’ , (a0- a2) and a2 (k’ fixed)
For final result 7 bins around cusp
excluded from the fit : EM corrections
Not yet included in the model
The excess of events in this
region is interpreted as pionium
combined with E.M. corrections


Γ K   π A2π
 1.82  0.21 10 -5
ΓK  3π 
Th. Prediction = 0.8 x 10-5
(JTEP lett. 60, 1994, 689)
Scattering length from CUSP
a0- a2 = 0.261 ± 0.006stat ± 0.003syst ± 0.0013ext ± 0.013th
a2
= -0.037 ± 0.013stat ± 0.009syst ± 0.002ext
External uncertainty: from the uncertainty on the ratio of
K+ → p+p+p- and K+ → p+pp decay widths A+ /A0 = 1.97  0.015
Theoretical uncertainty on (a0 – a2) ± 5% DOMINATES !!!
(Cabibbo-Isidori Theory uncertainty from neglecting higher order diagrams
and radiative corrections)
From (a0 – a2) and a2 can be extracted a0 (taken into account the statistical
error correlation coefficient ≈ -0.92)
a0 = 0.224 ± 0.008stat ± 0.006syst ± 0.003ext ± 0.013th
Uncertainties - CUSP method
Systematic effect
(a0-a2) x 102
a2 x 102
Analysis technique
 0.10
 0.20
Trigger inefficiency
 negl.
 0.50
Description of
resolution
 0.06
 0.11
LKR non linearity
 0.06
 0.26
Geometric Acceptance
 0.02
 0.01
MC sample
 0.03
 0.21
Simulation of LKR
shower
 0.17
 0.38
V – dependence on
amplitude
 0.17
 0.38
TOTAL Systematic
 0.28
 0.90
Comparison: NA48 vs DIRAC
DIRAC experiment measured
pionium 1S state lifetime to be
0.49
1S  2.91 -+0.62
fs
Corresponding to
0.033
| a0 - a2 |  0.264 -+0.020
(PLB 619, 50, 2005)
Black Ellipse = NA48 CUSP measurement
(Statistical systematic error)
Yellow area = theoretical uncertainty
in Cabibbo-Isidori Model (assumed
Gaussian)
K  p+p-e±ν : Theory
5 kinematic variables (Cabibbo – Maksymowicz)
N.B. Kaon and electron with
same sign DS  DQ rule
Sp = M2pp , Se = M2eν, cosθp, cosθe and Φ
pp direction
in the K+ rest frame
qp
p*(p+)
p-
p*(e+)
f
K+
ne
en direction
in the K+ rest frame
Expansion in power of q2 = Sp/4m2p
Fs  fs + fs'q2 + fs''q4 + fe Se /4mπ2 + ...
Partial wave (S,P) expansion of the Amplitude:
F,G = Axial Form Factors
F = FS e id s + FP e id p cosθp + d-wave term
G = GP e idg + d-wave term
H = Vector Form Factor
H = HP eidh + d-wave term

Fp  fp + fp'q2 + ...
Gp  gp + gp' q2 + ...
Assuming same phase for F,G,H
The fit parameter are : FS FP GP HP and
qe
Hp  hp + hp' q2 + ...
d = ds - dp

K  p+p-e±ν : Selection and background
Topology: 3 charged track , Signal: 2p with opposite charge
1 e identified with E/p ~1 , additional Missing v energy and pt cuts
Background main sources:
p+p+p- decays and p  eν (dominant) or p misidentified as e
p+p0p0 decays and p0 dalitz decay , g undetected or e misidentified as p
Background estimated by Montecarlo Simulation ... But....
Wrong sign events Event p+p+e- (violating DS = DQ rule)
provide a check for MC background estimate
Fitting procedure and Statistics
•
•
Define 10x5x5x5x12 iso-populated bins in (Mππ , Meν , cos qπ , cos qe , f )
The form factors are extracted from the data using simulated events by
minimizing a log-likehood estimator in each of the Mpp bins:
– In each Mpp bin the form factors are assumed to be constant
– 10 independent fits (one fit per Mpp bin) of 4 parameters (Fp, Gp , Hp
and d) plus free normalization (related to Fs) in 4D space.
– The correlation between the 4+1 parameters is taken into account.
– K+ and K- fitted separately and combined.
Statistics
K+
evts
Evts/bin
K-
evts
Evts/bin
Data (2003 )
MC
435654
29
10.0 M
667
241856
16
5.6 M
373
f distributions
d = ds – dp of the Ke4 decay
amplitude is extracted from
the measured asymmetry of
the f distribution as function
of Mpp
2m+ < Mpp < 0.291 GeV
0.309 < Mpp < 0.318 GeV
f
f
The asymmetry of the f
distribution increases with Mpp
 Increasing sensitivity to d
0.335 < Mpp < 0.345 GeV
K+ and K- have opposite
f asymmetry
f
0.373 GeV < Mpp <
mK
f
Phase shift VS Mpp
Direct measured points (NO MODEL ASSUMED SO FAR)
From NOW on MODEL assumptions are needed
To extract information from d variation, some theoretical work is needed:
Numerical solution of Roy equation which relates d and a0 , a2
(ACGL Phys. Rep. 352 , 2001 ; DFGS EPJ C24 , 2002)
Phase shift : Comparison
BNL E865 quotes various
values ranging from
a0 = 0.203 to a0 = 0.237
Note the last BNL point !!!
Predictions for a0=0.26 and a0=0.22
(a0, a2) plane Ke4 result
NA48/2 Ke4
Under the assumption of Isospin
symmetry and using Roy Equation
a0 = 0.233  0.016 stat  0.007 syst
a2 = - 0.0471  0.011 stat  0.004 syst
EPJC 54, 2008, 411
CHPT predictions
a0 = 0.220  0.005
a2 = - 0.0444  0.0010
NPB 603, 125 , 2001
Conclusions
The pion pion scattering lengths have been measured by NA48/2.
Two methods based on two different charge Kaon decay processes
Give results in good agreement.
The experimental measured scattering lengths agree with
CHPT predicted values at the per cent level.
This measurement is one of the most stringent test for CHPT
... Final Invitation ...
CUSP effect in KL 3p0
Ratio data / prediction
Change of slope where it has to be....
4m+2
K long sample of ~ 100M events collected in 2000
The CUSP visibility is ~ 13 smaller
CALL TO KTEV : LET THE CUSP BE SEEN IN YOUR HUGE Klong statistics
CUSP VISIBILITY
two possible p+p- pairs
2M+ + -M+ 00 2M+ + R(K ) 

2
(M+ 00 )
M+ 00
M+ + - : K+ → p+ p+ p- matrix element
M+ 0 0 : K+ → p+ p p matrix element
M+ -0M000 M+ -0
R(KL ) 

2
(M000 )
M000
M+ - 0 : KL → p+ p- p matrix element
M0 0 0 : KL → p p p matrix element
+
Calculate matrix elements at cusp point (Mpp = 2m+)
from measured partial width ratios and slope parameters:
R
(K+)
≈ 6.1 ; R(KL) ≈ 0.47
R(K+)
≈ 13
R(KL)
Cusp “visibility” is ~ 13 times higher in K+ → p+pp decays
than in KL → ppp decays