# Presentation: 466754

## Dalitz plot analysis in

- Laura Edera

**DA********NE 2004**
- Physics at meson factories

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

- Universita’ degli Studi di Milano

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

- but decay amplitude parametrization problems arise

- Laura Edera

## Focus had to face the problem of  dealing 

- with **light scalar particles** populating  charm meson hadronic decays, such as  **D******,** ****D ********K**********including ******(600)** and ******(900)**,
-   (i.e,  and K states  produced close to threshold), whose **existence**** **and** ****nature**** **is still **controversial**

**Complication** for charm **Dalitz** plot **analysis**

- Laura Edera

## Amplitude parametrization

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

- The problem is to write the propagator for the resonance r

- the propagator is of the simple BW type

- Laura Edera

## The  isobar model

- Dalitz
- total amplitude

- fit fraction

- fit parameters

- traditionally applied to charm decays

- Laura Edera

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

- overlapping resonances (just as the scalars!), the problem is not
- that simple.

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

- 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

- 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.

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

**In contrast**

- Laura Edera

## Slide 7

- Laura Edera

## Dalitz plot analysis of D+ and D+s  

- Phys. Lett. B 585 (2004) 200

- pioneering work by Focus

- Dalitz plot analysis of D+ and D+s ** **

- Laura Edera

## D decay picture in K-matrix

- I.J.R. Aitchison
- Nucl. Phys. A189 (1972) 514

- Laura Edera

## D+ 

**Yield D****+**** = 1527 ******** 51**
**S/N D****+**** = 3.64**

**D****+******** **************************

- Laura Edera

## K-matrix fit results

*K*-matrix fit results

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

**       ****	****           ****Decay fractions****	****	****     Phases**
- (S-wave)        (56.00   3.24  2.08) %		0 (fixed)
- f2(1275)         (11.74   1.90   0.23) %	(-47.5  18.7  11.7) °
- (770)       (30.82   3.14  2.29) %		(-139.4  16.5  9.9) °

- C.L.  ~ 7.7%

- Laura Edera

## Isobar analysis of D+   would instead require 

- With 

- Without 

- C.L.  ~ 7.5%

- Isobar analysis of D+   would instead require
- an ad hoc scalar meson: (600)

-  m = 442.6 ± 27.0 MeV/c
-   = 340.4 ± 65.5 MeV/c

**preliminary**

- Laura Edera

## Ds+ 

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

**D****s****+******** **************************

**Yield D****s****+**** = 1475 ******** 50**
**S/N D****s****+**** = 3.41**

- Laura Edera

## K-matrix fit results

**       ****	****           ****Decay fractions                      Phases**
- (S-wave)        (87.04   5.60  4.17) %	          0 (fixed)
- f2(1275)         (9.74   4.49   2.63) % 	          (168.0  18.7  2.5) °
- (1450)     (6.56   3.43  3.31) % 	          (234.9  19.5  13.3) °

*K*-matrix fit results

- C.L.  ~ 3%

- Laura Edera

## from D+  to D+  

- from wave to wave

- from to 

- from 1500events to more than 50000!!!

- Laura Edera

## Isobar analysis of D+ K -  + + would require an ad hoc scalar meson: 

- Without **

-  m = (797  19) MeV/c
-   = (410  43) MeV/c

- Phys.Rev.Lett.89:121801,2002

- With **

**preliminary FOCUS analysis**

- E791

- Laura Edera

## First attempt to fit the D+  Kin the K-matrix approach

- First attempt to fit the D+  Kin the K-matrix approach

- very preliminary

- K scattering data available from LASS experiment

- in progress...

- a lot of work to be performed!!

- Laura Edera

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

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

**Yield D****s****+**** = 567 ******** 31**
**S/N D****s****+**** = 2.4**

**Yield D****+**** = 189 ******** 24**
**S/N D****+**** = 1.0**

-  & K s-waves are necessary...

- Laura Edera

## D+ K+ - +

- (770)

- K*(892)

- D+ K+ ****- ****+

- Laura Edera

## Doubly Cabibbo Suppressed Decay

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

- K+  - projection

- +  -  projection

- D+ K+ ****- ****+

- isobar *effective* model

- C.L.  ~ 9.2%

- Laura Edera

## Ds+ K+ - +

- visible contributions: (770), K*(892)

- (770)

- K*(892)

- Ds+ K+ ****- ****+

- Laura Edera

## Singly Cabibbo Suppressed Decay

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

- K+  - projection

- +  -  projection

- Ds+ K+ ****- ****+

- isobar *effective* model

- C.L.  ~ 5.5%

- Laura Edera

## Conclusions

-  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  resonances coming from 	light-quark experiments works for charm decays too

-  Dalitz analysis      interesting and promising results

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

- What we have just learnt will be crucial at higher charm statistics and for future beauty studies, such as B  

- Laura Edera

## slides for possible questions...

- Laura Edera

## K-matrix formalism

- T transition matrix

-  = phase space diagonal matrix

- real & symmetric

- K-matrix is defined as:

- i. e.

- from scattering to production (from T to F):

- carries the production information
- COMPLEX

**production vector**

- Resonances are associated with poles of the S-matrix

- Laura Edera

## from scattering to production (from T to F):

- carries the production information
- COMPLEX

**production vector**

- I.J.R. Aitchison
- Nucl. Phys. A189 (1972) 514

- vector and tensor contributions

- Dalitz
- total amplitude

- Laura Edera

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

- If m0 = ma = mb

- The results is a single BW form where  = a + b

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

- The observed width is the sum of the two individual widths

- 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 ( and KK)

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

**total amplitude**

- the unitarity is not respected!

- no ‘mixing’ terms!

- Laura Edera

## Scattering amplitude: 

- suppresses false kinematical singularity at s=0 near  threshold

- describe a smooth part of the K-matrix elements

- and

- is the coupling constant of the bare state  to the meson channel

- Production of resonances:

- fit parameters

- Laura Edera

## *

*****

******p****************0****n,********n, ********’n, |t|********0.2 (GeV/c****2****)**

**GAMS**

*****

******p********************n, 0.30********|t|********1.0 (GeV/c****2****)**

**GAMS**

*****

**BNL**

*****

********************************

*****

**Crystal Barrel**

*****

**Crystal Barrel**

*****

**Crystal Barrel**

*****

**Crystal Barrel**

******-****p****************0****n, 0********|t|********1.5 (GeV/c****2****)**

**E852**

*****

- At rest, from liquid

-         At rest, from gaseous

- At rest, from liquid

- At rest, from liquid

**“****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

** ****A description of the scattering ...**

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

- 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 ********as measured in the isobar fit **

- Laura Edera

## Ds production coupling constants

- 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)}

- Ds production coupling constants

- 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

- ’and 4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 (770), that alone accounts for about 30 % of the D+ decay.
- This interpretation also bears on the role of the annihilation diagram in the Ds+   decay:
-  the S-wave annihilation contribution is negligible over much   of the dipion mass spectrum. It might be interesting to search for annihilation contributions in higher spin channels, such as 0(1450) and f2(1270)

- Laura Edera

## CP violation on the Dalitz plot 

- CP violation on the Dalitz plot** **

******i** = strong phase

**CP asymmetry:**

**2 different amplitudes**

**strong phase-shift**

- Laura Edera

## CP violation & Dalitz analysis

**COEFFICIENTS** and **PHASES** for each amplitude

- Measured phase:

******=****************+**********

- CP conserving

- CP violating

- CP conjugate

**E831**

**a****CP****=0.006±0.011±0.005**

- Measure of direct CP violation:
- asymmetrys in decay rates of  **D************K********K **********

****

- CP violation & Dalitz analysis

- Laura Edera

## Marginal role of annihilation in charm hadronic decays 

**No significant direct three-body-decay component**

**No significant  ********(770) ******** contribution**

- Marginal role of annihilation in charm hadronic decays

**But need more data!**

- Laura Edera