Dalitz plot analysis in

Dalitz plot analysis in Laura Edera Universita degli Studi di Milano DAΦNE 004 Physics at meson factories June 7-11, 004 INFN Laboratory, Frascati, Italy 1

The high statistics and excellent quality of charm data now available allow for unprecedented sensitivity & sophisticated studies: 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 π + (ICHEP 00), Cleo D 0 K s π + π but decay amplitude parametrization problems arise Daphne004, June 7-11 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 D πππ, D Kππ including σ(600) and κ(900), (i.e, ππ and Kπ states produced close to threshold), whose existence and nature is still controversial Daphne004, June 7-11 Laura Edera 3

D r + 3 1 + Amplitude parametrization 3 r r 1 The problem is to write the propagator for the resonance 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 A= F F p p P J J r D r 1 3 J(cos ϑ13) mr m1 imrγr Daphne004, June 7-11 Laura Edera 4 1

The isobar model A = FF p p P(cos ) BWm ( ) J J r D r 1 3 J ϑ13 1 Where F F F = 1 = (1 + R = (9 + 3R p ) p 1 + 3R 4 p 4 ) 1 Spin 0 Spin 1 Spin PJ = 1 P = ( p p ) J 3 1 P = ( p p ) (3cos ϑ 1) J 3 1 13 and BW (1 r) = 1 M m iγm r 1 r Γ = Γ r j + 1 M r Fr 0 m1 Fr p0 p p ( p) ( ) Dalitz total amplitude M i j = ae δ A j j j fit fraction f = iδ r r A r dm 1dm 13 ae r iδ j ae j j A j dm 1dm 13 fit parameters traditionally applied to charm decays Daphne004, June 7-11 Laura Edera 5

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 ( I ik ρ) where K is the matrix for the scattering of particles 1 and. 1 i.e., to write down the propagator we need the scattering matrix Daphne004, June 7-11 Laura Edera 6

K-matrix formalism E.P.Wigner, Phys. Rev. 70 (1946) 15 K-matrix is defined as: real & symmetric S= I+ it 1 1 K T iρ ρ = phase space diagonal matrix T transition matrix = + i. e. ( ) 1 T = I ik K ρ Add two BW ala Isobar model Adding BW violates unitarity Add two K matrices Adding K matrices respects unitarity The Unitarity circle Daphne004, June 7-11 Laura Edera 7

pioneering work by Focus Dalitz plot analysis of D + and D + s π + π π + Phys. Lett. B 585 (004) 00 first attempt to fit charm data with the K-matrix formalism Daphne004, June 7-11 Laura Edera 8

D decay picture in K-matrix π + I.J.R. Aitchison Nucl. Phys. A189 (197) 514 P 1 = ππ = KK D P 3 = ππππ (I-iK ρ) -1 P F = I ik ρ i β γ m Γ = + α 4 = ηη 5 =ηη ' fixed to α iα α α d ( ) i m mα m carries the production information COMPLEX π + π - 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 (003) 9 BNL CERN-Munich : Crystal Barrel π p K K n π + π π + π p p π 0 π 0 π 0, π 0 π 0 η Crystal Barrel p p π + π π 0, K + K π 0, etc... K s K s π 0, K + Κ s π Daphne004, June 7-11 Laura Edera 9

D + π + π π + Yield D + = 157 ± 51 S/N D + = 3.64 Daphne004, June 7-11 Laura Edera 10

K-matrix fit results C.L. ~ 7.7% Low m π π mass projection High m π π mass projection Decay fractions Phases (S-wave) π + (56.00 ± 3.4 ±.08) % 0 (fixed) f (175) π + (11.74 ± 1.90 ± 0.3) % (-47.5 ± 18.7 ± 11.7) ρ(770) π + (30.8 ± 3.14 ±.9) % (-139.4 ± 16.5 ± 9.9) No new ingredient (resonance) required not present in the scattering! Daphne004, June 7-11 Laura Edera 11

Isobar analysis of D + π + π + π would instead require an ad hoc scalar meson: σ(600) m = 44.6 ± 7.0 MeV/c With σ Γ = 340.4 ± 65.5 MeV/c preliminary C.L. ~ 7.5% Without σ C.L. ~ 10-6 Daphne004, June 7-11 Laura Edera 1

D s+ π + π π + Yield D s+ = 1475 ± 50 S/N D s+ = 3.41 Observe: f 0 (980) f 0 (1500) f (170) Daphne004, June 7-11 Laura Edera 13

K-matrix fit results C.L. ~ 3% Low m π π mass projection High m π π mass projection Decay fractions Phases (S-wave) π + (87.04 ± 5.60 ± 4.17) % 0 (fixed) f (175) π + (9.74 ± 4.49 ±.63) % (168.0 ± 18.7 ±.5) ρ(1450) π + (6.56 ± 3.43 ± 3.31) % (34.9 ± 19.5 ± 13.3) Daphne004, June 7-11 Laura Edera 14

from D + π + π π + to D + Κ π + π + from ππ wave to Κπ wave from σ(600) to κ(900) from 1500 events to more than 50000!!! high m Kπ low Daphne004, June 7-11 Laura Edera 15 m Kπ

Isobar analysis of D + K - π + π + would require an ad hoc scalar meson: κ(900) m = (797 ± 19) MeV/c With κ Γ = (410 ± 43) MeV/c E791 Phys.Rev.Lett.89:11801,00 preliminary FOCUS analysis Without κ Daphne004, June 7-11 Laura Edera 16

First attempt to fit the D + K π + π + in the K-matrix approach very preliminary Kπ scattering data available from LASS experiment a lot of work to be performed!! a real test of the method (high statistics) in progress... Daphne004, June 7-11 Laura Edera 17

The excellent statistics allow for investigation of suppressed and even heavily suppressed modes Doubly Cabibbo Suppressed D + K + π + π - Singly Cabibbo Suppressed D s+ K + π + π - Yield D + = 189 ± 4 S/N D + = 1.0 Yield D s+ = 567 ± 31 S/N D s+ =.4 mk ππ ππ & Kπ s-waves are necessary... Daphne004, June 7-11 Laura Edera 18

D + K + π - π + m π π ρ(770) K * (89) m K π Daphne004, June 7-11 Laura Edera 19

D + K + π - π + Doubly Cabibbo Suppressed Decay isobar effective model C.L. ~ 9.% K + π - projection π + π - projection Decay fractions Coefficients Phases K*(89) = (5. ± 6.8 ± 6.4) % 1.15 ± 0.17 ± 0.16 (-167 ± 14 ± 3) ρ (770) = (39.4 ± 7.9 ± 8.) % 1 fixed (0 fixed) K (1430) = (8.0 ± 3.7 ± 3.9) % 0.45 ± 0.13 ± 0.13 (54 ± 38 ± 1) f 0 (980) = (8.9 ± 3.3 ± 4.1) % 0.48 ± 0.11 ± 0.14 (-135 ± 31 ± 4) Daphne004, June 7-11 Laura Edera 0

D s+ K + π - π + m + - π π visible contributions: ρ(770), K * (89) ρ(770) m + - K π K * (89) Daphne004, June 7-11 Laura Edera 1

D s+ K + π - π + Singly Cabibbo Suppressed Decay First Dalitz analysis isobar effective model C.L. ~ 5.5% K + π - projection π + π - projection Decay fractions coefficients Phases ρ (770) = (38.8 ± 5.3 ±.6) % 1 fixed (0 fixed) K*(89) = (1.6 ± 3. ± 1.1) % 0.75 ± 0.08 ± 0.03 (16 ± 9 ± ) NR = (15.9 ± 4.9 ± 1.5) % 0.64 ± 0.1 ± 0.03 (43 ± 10 ± 4) K*(1410) = (18.8 ± 4.0 ± 1.) % 0.70 ± 0.10 ± 0.03 (-35 ± 1 ± 4) K* 0 (1430) = (7.7 ± 5.0 ± 1.7) % 0.44 ± 0.14 ± 0.06 (59 ± 0 ± 13) ρ (1450) = (10.6 ± 3.5 ± 1.0) % 0.5 ± 0.09 ± 0.0 (-15 ± 11 ± 4) Daphne004, June 7-11 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 ππ 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 (ππ and Kπ waves) What we have just learnt will be crucial at higher charm statistics and for future beauty studies, such as B ρπ Daphne004, June 7-11 Laura Edera 3

slides for possible questions... Daphne004, June 7-11 Laura Edera 4

K-matrix formalism Resonances are associated with poles of the S-matrix T transition S= I+ it matrix 1 1 K T iρ K-matrix is defined as: i. e. = + ( ) 1 T = I ik K ρ ij decay channels real & symmetric γ γ m Γ K = + c m α iα jα α α ( ) ij mα m sum over all poles ρ = phase space diagonal matrix γ ia =coupling constant to channel i m a = K-matrix mass Γ a = K-matrix width from scattering to production (from T to F): P production vector i β γ m Γ = + α α iα α α d ( ) i m mα m carries the production information COMPLEX ( ρ) 1 F = I ik P Daphne004, June 7-11 Laura Edera 5

from scattering to production (from T to F): I.J.R. Aitchison Nucl. Phys. A189 (197) 514 ( ρ) 1 T = I ik K ( ρ) 1 F = I ik P P i β γ m Γ = + α α iα α α d ( ) i m mα m carries the production information COMPLEX production vector Dalitz total amplitude M = + + iϑ0 j ae F ae BW 0 j j iϑ vector and tensor contributions Daphne004, June 7-11 Laura Edera 6

Only in a few cases the description through a simple BW is satisfactory. If m 0 = m a = m b γ m Γ γ m Γ [ ] K = + m m m m a a a b b b a b T = m Γ ( m) +Γ ( m) 0 a 0 0 b [ ( ) ( )] m m im Γ m +Γ m a b The results is a single BW form where Γ = Γ a + Γ b The observed width is the sum of the two individual widths If m a and m b are far apart relative to the widths (no overlapping) T 0 0 maγ a ma q mbγ b mb q ma m ima a( m) m + qa mb m imb b( m) m Γ Γ qb The transition amplitude is given merely by the sum of BW Daphne004, June 7-11 Laura Edera 7

The K-matrix formalism gives us the correct tool to deal with the nearby resonances e.g. poles (f 0 (1370) - f 0 (1500)) coupled to channels (ππ and KK) K ij γ γ m Γ γ γ m Γ = + m m m m ai aj a a bi bj b b a b P i β γ m Γ β γ m Γ = + m m m m a ai a a b bi b b a b total amplitude F i = βamaγaγa 1( mb m ) + βbmbγbγb 1( ma m ) imaγambγbρ( γaβb βγ a b)( γaγb 1 γa1γb) ( ma m )( mb m ) imaγ a( γa 1ρ1+ γaρ)( mb m ) imbγ b( γb 1ρ1+ γbρ)( ma m ) maγambγbρρ 1 ( γaγb 1 γa1γb) if you treat the f 0 scalars as independent BW: F i β γ m Γ β γ m Γ = + m m im m m im a ai a a b bi b b a aγ a a + a b bγ b b + b ( γ 1ρ1 γ ρ) ( γ 1ρ1 γ ρ) the unitarity is not respected! no mixing terms! Daphne004, June 7-11 Laura Edera 8

IJ PC = 00 ++ - wave has been reconstructed on the basis of a complete available data set Scattering amplitude: K IJ ab is a 5x5 matrix (a,b = 1,,3,4,5) 1 = ππ = KK 3 = ηη 4 = ηη 5 = multimeson states (4π) ( α) ( α) scatt g 00 i g j scatt 1GeV s 0 s sa mπ Kij () s = + f ij scatt M s s s α α 0 ( s sa0)(1 sa0) g α ( α) ( α) is the coupling constant of the bare state α to the meson channel g ( m) = m Γ ( m) ( ) i scatt f ij and scatt s0 describe a smooth part of the K-matrix elements ( s samπ ) ( s sa0)(1 sa0 P ) Production of resonances: suppresses false kinematical singularity at s=0 near ππ threshold β g GeV s s s m ( )(1 ) ( α ) prod α j 1 0 A π j = + f bck prod α M α s s s 0 s sa0 sa0 i α i fit parameters Daphne004, June 7-11 Laura Edera 9

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 (003) 9 * ** * ** * * GAMS GAMS BNL CERN-Munich Crystal Barrel Crystal Barrel Crystal Barrel Crystal Barrel E85 : πp π 0 π 0 n,ηηn, ηη n, t <0. (GeV/c ) πp π 0 π 0 n, 0.30< t <1.0 (GeV/c ) πp - KKn π + π π + π pp π 0 π 0 π 0, π 0 π 0 η, π 0 ηη pp π 0 π 0 π 0, π 0 π 0 η pp π + π π 0, K + K - π 0, K s K s π 0, K + Κ s π np π 0 π 0 π -, π π π +, K s K - π 0, K s K s π - π - p π 0 π 0 n, 0< t <1.5 (GeV/c ) H At rest, from liquid At rest, from gaseous At rest, from liquid At rest, from liquid H H D Daphne004, June 7-11 Laura Edera 30

A&S K-matrix poles, couplings etc. Poles g g g g g ππ KK 4 π ηη ηη ' 0.65100 0.4844 0.553 0 0.38878 0.36397 1.070 0.91779 0.5547 0 0.38705 0.9448 1.561 0.3704 0.3591 0.6605 0.18409 0.1893 1.157 0.34501 0.3964 0.97644 0.19746 0.00357 1.81746 0.15770 0.17915 0.90100 0.00931 0.0689 s f f f f f scatt scatt scatt scatt scatt scatt 0 11 1 13 14 15 3.30564 0.6681 0.16583 0.19840 0.3808 0.31193 s A s A0 1.0 0. Daphne004, June 7-11 Laura Edera 31

A&S T-matrix poles and couplings ( m, Γ/) g g g g g ππ KK 4 π ηη ηη ' i13.1 i 96.5 i80.9 i 98.6 i10.1 (1.019, 0.038) 0.415e 0.580e 0.148 e 0.484 e 0.401 e (1.306, 0.167) 0.406 e 0.105 e 0.891 e 0.14 e 0.5 e i116.8 i100. i61.9 i140.0 i 133.0 (1.470, 0.960) 0.758 e 0.844e 1.681 e 0.431 e 0.175 e i 97.8 i 97.4 i 91.1 i115.5 i15.4.115 e i151.5 i149.6 i13.3 i170.6 i 133.9 (1.489, 0.058) 0.46 e 0.134 e 0.4867 e 0.100 e 0 (1.749, 0.165) 0.536 e 0.07 e 0.7334 e 0.160 e 0.313 e i101.6 i134. i13.6 i16.7 i101.1 A&S fit does not need a σ as measured in the isobar fit Daphne004, June 7-11 Laura Edera 3

D s 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(100-1600) (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 0.0) e^{i(83.1 \pm 3.03)} f_0(1750) (1.746,0.160) (0.8 \pm0.0) e^{i(-17.9 \pm.5)} 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(100-1600) (1.470,0.960) (1.70 \pm 0.17) e^{i(-15.5\pm 1.7)} f_0(1500) (1.489,0.058) (0.63 \pm 0.0) e^{i(-14.\pm.)} f_0(1750) (1.746,0.160) (0.36 \pm 0.0) e^{i(-135.0 \pm.9)} Daphne004, June 7-11 Laura Edera 33

The Q-vector approach We can view the decay as consisting of an initial production of the five virtual states ππ, KK, ηη, ηη and 4π, which then scatter via the physical T-matrix into the final state. 1 1 1 1 F = ( I ikρ) P= ( I ikρ) KK P= TK P= TQ The Q-vector contains the production amplitude of each virtual channel in the decay Daphne004, June 7-11 Laura Edera 34

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 ρ(770), that alone accounts for about 30 % of the D + decay. This interpretation also bears on the role of the annihilation diagram in the D s+ π + π π + 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 ρ 0 (1450)π and f (170)π. Daphne004, June 7-11 Laura Edera 35

CP violation on the Dalitz plot For a two-body decay CP asymmetry: A tot = g 1 M 1 e iδ 1 + g M e iδ CP conjugate A tot = g* 1 M 1 e iδ 1 + g * M e iδ δ i = strong phase a CP = A tot - A tot A tot + A tot = Im(g g 1 *)) sin(δ 1 -δ )M 1 M g 1 M 1 + g M +Re(g g 1 *)cos(δ 1 -δ )M 1 M different amplitudes strong phase-shift Daphne004, June 7-11 Laura Edera 36

CP violation & Dalitz analysis Dalitz plot = FULL OBSERVATION of the decay COEFFICIENTS and PHASES for each amplitude Measured phase: CP conjugate θ = δ + φ CP conserving CP violating δ = δ φ =-φ θ = δ - φ E831 Measure of direct CP violation: a CP =0.006±0.011±0.005 asymmetrys in decay rates of D ± K ± K π ± ± Daphne004, June 7-11 Laura Edera 37

No significant direct three-bodydecay component D + s π + π No significant ρ(770) π contribution D + s ρ π + π π + π + Marginal role of annihilation in charm hadronic decays But need more data! Daphne004, June 7-11 Laura Edera 38