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Dalitz plot analysis in

Laura Edera

Universita’ degli Studi di Milano

DAFNE 2004

Physics at meson factories

June 7-11, 2004 INFN Laboratory, Frascati, Italy

lifetime measurements @ better than 1%

CPV, mixing and rare&forbidden decays

investigation of 3-body decay dynamics: Dalitz plot analysis

Phases and Quantum Mechanics interference: FSI

CP violation probe Focus D+ K+K-p+ (ICHEP 2002), Cleo D0 Ksp+p-

The high statistics and excellentquality of charm data now available allow for unprecedented sensitivity & sophisticated studies:

but decay amplitude parametrization problems arise

Laura Edera

Complication for charm Dalitz plot analysis

Focus had to face the problem of dealing

with light scalar particles populating charm meson hadronic decays, such as Dppp,D Kppincluding s(600) and k(900),

(i.e, pp and Kp states produced close to threshold), whose existenceandnatureis still controversial

Laura Edera

D r + 3

2

1 + 2

3

r

r

1

Amplitude parametrization

The problem is to write the propagator for theresonance r

For a well-defined wave with specific isospin and spin (IJ)

characterized by narrow and well-isolated resonances, we know how:

the propagator is of the simple BW type

Laura Edera

Spin 0

Where

Spin 1

Spin 2

and

The isobar model

Dalitz

total amplitude

fit fraction

fit parameters

traditionally applied to charm decays

Laura Edera

In contrast

when the specific IJ–wave is characterized by large and heavily

overlapping resonances (just as the scalars!), the problem is not

that simple.

Indeed, it is very easy to realize that the propagation is no longer dominated by a single resonance but is the result of a complicated interplay among resonances.

In this case, it can be demonstrated on very general grounds that the

propagator may be written in the context of the K-matrix approach as

where K is the matrix for the scattering of particles 1 and 2.

i.e., to write down the propagator we need the scattering matrix

Laura Edera

E.P.Wigner,

Phys. Rev. 70 (1946) 15

K-matrix formalism

T transition matrix

K-matrix is defined as:

i. e.

real & symmetric

r = phase space diagonal matrix

Add two BW ala Isobar model

Adding BW violates unitarity

Add two K matrices

Adding K matrices respects unitarity

The Unitarity circle

Laura Edera

pioneering work by Focus

Dalitz plot analysis of D+ and D+sp +p -p +

Phys. Lett. B 585 (2004) 200

first attempt to fit charm data with the K-matrix formalism

Laura Edera

p+

p+

D

P

(I-iK·r)-1

p-

“K-matrix analysis of the 00++-wave in the mass region below 1900 MeV”

V.V Anisovich and A.V.Sarantsev Eur. Phys.J.A16 (2003) 229

fixed to

BNL

p-p K K n

:

p+ p- p+ p-

CERN-Munich

Crystal Barrel

p p p0 p0 p0, p0 p0 h

p p p+ p- p0, K+ K- p0, Ks Ks p0, K+ Ks p-

Crystal Barrel

etc...

carries the production information

COMPLEX

I.J.R. Aitchison

Nucl. Phys. A189 (1972) 514

D decay picture in K-matrix

Laura Edera

D+ p + p - p +

Yield D+ = 1527 51

S/N D+ = 3.64

Laura Edera

Low mass projection

High mass projection

K-matrix fit results

C.L. ~ 7.7%

Decay fractions Phases

(S-wave) p+ (56.00 3.24 2.08) %0 (fixed)

f2(1275) p+ (11.74 1.90 0.23) %(-47.5 18.7 11.7) °

r(770) p+ (30.82 3.14 2.29) %(-139.4 16.5 9.9) °

No new ingredient (resonance) required not present in the scattering!

Laura Edera

C.L. ~ 10-6

Isobar analysis of D+ p+p +p - would instead require

an ad hoc scalar meson: s(600)

preliminary

m = 442.6 ± 27.0 MeV/c

G = 340.4 ± 65.5 MeV/c

Withs

C.L. ~ 7.5%

Withouts

Laura Edera

Ds+ p + p - p +

Yield Ds+ = 1475 50

S/N Ds+ = 3.41

- Observe:
- f0(980)
- f0(1500)
- f2(1270)

Laura Edera

High mass projection

Low mass projection

K-matrix fit results

C.L. ~ 3%

Decay fractions Phases

(S-wave) p+ (87.04 5.60 4.17) % 0 (fixed)

f2(1275) p+ (9.74 4.49 2.63) % (168.0 18.7 2.5) °

r(1450) p+ (6.56 3.43 3.31) % (234.9 19.5 13.3) °

Laura Edera

low

high

from D+p + p - p +to D+K- p + p +

fromppwave toKpwave

froms(600)tok(900)

from1500events tomore than 50000!!!

Laura Edera

Isobar analysis of D+ K - p+p+ would require an ad hoc scalar meson: k(900)

m = (797 19) MeV/c

G = (410 43) MeV/c

Withk

E791

Phys.Rev.Lett.89:121801,2002

preliminary FOCUS analysis

Withoutk

Laura Edera

First attempt to fit the D+ K- p + p + in the K-matrix approach

very preliminary

Kp scattering data available from LASS experiment

a lot of work to be performed!!

a “real” test of the method (high statistics)…

in progress...

Laura Edera

Doubly Cabibbo Suppressed

Singly Cabibbo Suppressed

D+K+p+p-

Ds+K+p+p-

The excellent statistics allow for investigation of suppressed and even heavily suppressed modes

Yield D+ = 189 24

S/N D+ = 1.0

Yield Ds+ = 567 31

S/N Ds+ = 2.4

pp & Kp s-waves are necessary...

Laura Edera

D+K+p -p+

r(770)

K*(892)

Laura Edera

isobar effective model

D+K+p -p+

Doubly Cabibbo Suppressed Decay

C.L. ~ 9.2%

K+p- projection

p+p- projection

Decay fractions Coefficients Phases

K*(892) = (52.2 6.8 6.4) % 1.15 0.17 0.16 (-167 14 23) °

r (770) = (39.4 7.9 8.2) % 1 fixed (0 fixed)

K2(1430) = (8.0 3.7 3.9) % 0.45 0.13 0.13 (54 38 21) °

f0(980) = (8.9 3.3 4.1) % 0.48 0.11 0.14 (-135 31 42) °

Laura Edera

Ds+K+p -p+

visible contributions: r(770), K*(892)

r(770)

K*(892)

Laura Edera

First Dalitz

analysis

isobar effective model

Ds+K+p -p+

C.L. ~ 5.5%

Singly Cabibbo Suppressed Decay

p+p- projection

K+p- projection

Decay fractions coefficients Phases

r (770) = (38.8 5.3 2.6) % 1 fixed (0 fixed)

K*(892) = (21.6 3.2 1.1) % 0.75 0.08 0.03 (162 9 2) °

NR = (15.9 4.9 1.5) % 0.64 0.12 0.03 (43 10 4) °

K*(1410) = (18.8 4.0 1.2) % 0.70 0.10 0.03 (-35 12 4) °

K*0(1430) = (7.7 5.0 1.7) % 0.44 0.14 0.06 (59 20 13) °

r(1450) = (10.6 3.5 1.0) % 0.52 0.09 0.02 (-152 11 4) °

Laura Edera

Conclusions

Dalitz analysis interesting and promising results

Focus has carried out a pioneering work!

The K-matrix approach has been applied to charm decay for the first time

The results are extremaly encouraging since the same parametrization of two-body pp resonances coming from light-quark experiments works for charm decays too

Cabibbo suppressed channels started to be analyzed now easy (isobar model), complications for the future (pp and Kp waves)

What we have just learnt will be crucial at higher charmstatistics and for future beauty studies, such as B rp

Laura Edera

slides for possible questions...

Laura Edera

gia=coupling constant to channel i

ma= K-matrix mass

Ga= K-matrix width

decay channels

sum over all poles

K-matrix formalism

Resonances are associated with poles of the S-matrix

T transition matrix

K-matrix is defined as:

i. e.

r = phase space diagonal matrix

real & symmetric

from scattering to production (from T to F):

carries the production information

COMPLEX

production vector

Laura Edera

I.J.R. Aitchison

Nucl. Phys. A189 (1972) 514

from scattering to production (from T to F):

carries the production information

COMPLEX

production vector

Dalitz

total amplitude

vector and tensor contributions

Laura Edera

Only in a few cases the description through a simple BW is satisfactory.

If m0 = ma = mb

The observed width is the sum of the two individual widths

The results is a single BW form where G = Ga + Gb

If ma and mb are far apart relative to the widths (no overlapping)

The transition amplitude is given merely by the sum of 2 BW

Laura Edera

The K-matrix formalism gives us the correct tool to deal with the nearby resonances

e.g. 2 poles (f0(1370) - f0(1500)) coupled to 2 channels (pp and KK)

total amplitude

if you treat the 2 f0 scalars as 2 independent BW:

no ‘mixing’ terms!

the unitarity is not respected!

Laura Edera

- wave has been reconstructed on the basis of a complete available data set

1 = pp 2 = KK

3 = hh 4 = hh’

5 = multimeson states (4p)

KIJab is a 5x5 matrix (a,b = 1,2,3,4,5)

Scattering amplitude:

is the coupling constant of the bare state a to the meson channel

and

describe a smooth part of the K-matrix elements

suppresses false kinematical singularity at s=0 near pp threshold

Production of resonances:

fit parameters

Laura Edera

pp- KKn

:

CERN-Munich

pp p0p0p0, p0p0h , p0hh

pp p0p0p0, p0p0h

pp p+p-p0, K+K-p0, KsKsp0, K+Ksp-

np p0p0p-, p-p-p+, KsK-p0, KsKsp-

A description of the scattering ...

A global fit to all the available data has been performed!

“K-matrix analysis of the 00++-wave in the mass region below 1900 MeV’’

V.V Anisovich and A.V.Sarantsev Eur.Phys.J.A16 (2003) 229

*

GAMS

ppp0p0n,hhn, hh’n, |t|0.2 (GeV/c2)

*

GAMS

ppp0p0n, 0.30|t|1.0 (GeV/c2)

*

BNL

*

p+p- p+p-

At rest, from liquid

*

Crystal Barrel

*

Crystal Barrel

At rest, from gaseous

*

Crystal Barrel

At rest, from liquid

*

Crystal Barrel

At rest, from liquid

E852

*

p-pp0p0n, 0|t|1.5 (GeV/c2)

Laura Edera

A&S K-matrix poles, couplings etc.

Laura Edera

A&S T-matrix poles and couplings

A&S fit does not need a s as measured in the isobar fit

Laura Edera

Ds production coupling constants

f_0(980)(1.019,0.038) 1 e^{i 0} (fixed)

f_0(1300)(1.306,0.170) (0.43 \pm 0.04) e^{i(-163.8 \pm 4.9)}

f_0(1200-1600)(1.470,0.960) (4.90 \pm 0.08) e^{i(80.9 \pm1.06)}

f_0(1500)(1.488,0.058) (0.51 \pm 0.02) e^{i(83.1 \pm 3.03)}

f_0(1750)(1.746,0.160) (0.82 \pm0.02) e^{i(-127.9 \pm 2.25)}

D+ production coupling constants

f_0(980) (1.019,0.038) 1 e^{i0} (fixed)

f_0(1300)(1.306,0.170)(0.67 \pm 0.03) e^{i(-67.9 \pm 3.0)}

f_0(1200-1600)(1.470,0.960)(1.70 \pm 0.17) e^{i(-125.5\pm 1.7)}

f_0(1500)(1.489,0.058) (0.63 \pm 0.02) e^{i(-142.2\pm 2.2)}

f_0(1750)(1.746,0.160) (0.36 \pm 0.02) e^{i(-135.0 \pm 2.9)}

Laura Edera

The Q-vector approach

- We can view the decay as consisting of an initial production of the five virtual states pp, KK,
hh, hh’and 4p, which then scatter via the physical T-matrix into the final state.

The Q-vector contains the production amplitude of each virtual channel in the decay

Laura Edera

The resulting picture

- The S-wave decay amplitude primarily arises from a ss contribution.
- For the D+ the ss contribution competes with a dd contribution.
- Rather than coupling to an S-wave dipion, the dd piece prefers to couple to a vector state like r(770), that alone accounts for about 30 % of the D+ decay.
- This interpretation also bears on the role of the annihilation diagram in theDs+p+p-p+ decay:
- the S-wave annihilation contribution is negligible over much of thedipion mass spectrum. It might be interesting to search for annihilationcontributions in higher spin channels, such as r0(1450)p andf2(1270)p.

Laura Edera

Atot= g1M1eid1 + g2M2eid2

CP conjugate

Atot= g1M1eid1 + g2M2eid2

*

*

2Im(g2 g1*) sin(d1-d2)M1M2

|Atot|2- |Atot|2

aCP=

=

|Atot|2+ |Atot|2

|g1|2M12+|g2|2M22+2Re(g2 g1*)cos(d1-d2)M1M2

CP violation on the Dalitz plot

- For a two-body decay

di= strong phase

CP asymmetry:

strong phase-shift

2 different amplitudes

Laura Edera

f =-f

d = d

q =d-f

CP violation & Dalitz analysis

Dalitz plot = FULL OBSERVATION of the decay

COEFFICIENTS and PHASES for each amplitude

q =d+f

Measured phase:

CP conserving

CP violating

CP conjugate

aCP=0.006±0.011±0.005

Measure of direct CP violation:

asymmetrys in decay rates ofDKK p

E831

Laura Edera

p+

p-

r

p-

p+

p+

p+

- No significant direct three-body-decay component

- No significant r(770) p contribution

Marginal role of annihilation in charm hadronic decays

But need more data!

Laura Edera