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Dalitz analysesDalitz analyses IntroductionBelle Analysis School
October 1-2 2009
Hi, could you talk b t D lit
Sure...A couple of months ago, somwehere on
about Dalitzat the BAS?
g ,the net....
A couple of weeks ago, somwehere on
Yes...?
WHO did you saythe net.... WHO did you say I should talk about...?
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 1 BAS, KEK, October 2009
Dalitz analysesDalitz analyses IntroductionBelle Analysis School
Boštjan GolobBelle & Belle II
University of Ljubljana Jožef Stefan Institute
October 1-2 2009
University of Ljubljana, Jožef Stefan Institute
1. Introduction2. Kinematics3. Physics4. Parametrization4. Parametrization5. Experimental issues6. Specifics, Outlook
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 2 BAS, KEK, October 2009
IntroductionHistoryHistory
Richard Henry Dalitz (28 February 1925 – 13 January 2006);Australian physicist;Australian physicist;
@ Cornell introduced “phase space plotsphase space plots”, i.e. Dalitz technique (as called today), e a t tec que (as ca ed today),to study 3π “tau” (kaon) decays;
On the analysis of τ meson data and the nature of the τ mesonOn the analysis of τ-meson data and the nature of the τ-meson Author: R. H. Dalitz a
Affiliation: a Department of Mathematical Physics, University of Birmingham, y gDOI: 10.1080/14786441008520365 Published in: Philosophical Magazine Series 7, Volume 44, Issue 357 October 1953 , pages 1068 - 1080
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 3 BAS, KEK, October 2009
Kinematicsn-body decayJ.D. Jackson, D.R. Tovey, Kinematics, in RPP
( )Φ=Γ nn
d
ppPdM
d
3
12
4
),;(22π
KM
∏∑==
−=Φni i
i
niinn E
pdpPppPd,1
3
3
,1
41 2)2(
)(),;(π
δK LISP:Lorent InvariantPhase Sace
independent variables: 4-vectors: 4 n +
ti l 4 3 7n=3 2n=3 2
conservation laws: 4 = 3 n - 7final state masses: n -arbitrary rotations: 3
n=4 5n=5 8
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 4 BAS, KEK, October 2009
Kinematics3 body decay
J.D. Jackson, D.R. Tovey, Kinematics, in RPP,
3-body decay
take two inv. masses asindependent variables decaying particle:independent variables
213
212
233 32
1)2(
1 dmdmM
d M=Γ
decaying particle:scalar or averagingover spin states33 32)2( Mπ p
mij: inv. mass of part. i,j.2
322
21
2223
213
212 constmmmMmmm =+++=++
211M
dΓ332
13212 32)2(
MMdmdm π
= standard form of Dalitz plot
if |M|2 const ⇒ dΓ/dm 2dm 2 const
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 5 BAS, KEK, October 2009
if |M|2 const. ⇒ dΓ/dm122dm13
2 const.
Kinematics Belle, PRL 99, 131803 (2007)
)
3-body decay
example: D0 → Ksπ -π+ m2 (
KSπ
+ )
cos2
θ
non-uniformity of Dalitz plot ⇒contribution of intermediate states
Ki ti li itKinematic limitsD0 → π+K*- m2(KSπ-)J.D. Jackson, D.R. Tovey, Kinematics, in RPP
θ23*=π
( ) ( ) 222 2mmm +θ23*=0
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 6 BAS, KEK, October 2009
( ) ( )( ) ( )min
223max
223
23max23min23*23
2cos
mmmmm
−−+
=θ 34
KinematicsM1
3-body decay
various intermediate states Mk
contributing to same final state ⇒ interfereinterferenncece
Mn
|M|2 is not incoherent sum, (|M|2 ≠ |M1|2 + |M2|2 + ... ) ,
n
but a coherent sum, ( |M|2 = |M1+ M2 + ...|2 )
example: pp → 3π0
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 7 BAS, KEK, October 2009
PhysicsCleo-c, arXiv:0903.1301
New states and propertiesof known states
,
Dalitz analysis usually not needed for narrow, non-overlapping
K*0
resonances(negligible interference)example: Ds → K+K-π+
(but interf. φπ/KKπ, φπ/f0πimportant in precise Br(Ds → φπ)determination)
φ
determination)
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 8 BAS, KEK, October 2009
another example: 37
PhysicsB → KZ+(4430)
Belle, PRD 80, 031104(R) (2009)
New states and propertiesof known states
KZ (4430)
In most cases intermediate states strongly interfere
Z+(4430)example: B → Kπ+ψ’
by fitting fitting DalitzDalitz distributiondistributionobtain evidence of newobtain evidence of new states, measure properties(mass width spin)
B → K*(892)ψ’
B → K0*(1430)ψ’
(mass, width, spin)
projection of Dalitz disribution and fit to
24
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 9 BAS, KEK, October 2009
disribution and fit to M(π+ψ’)
Physics20 )( tAyixqAefDd t +
+=→Γ −
D0 Mixing and CPV
x, y: mixing parameters;
2tA
pAe
dt ff +=
due to mixing, D0 → D0 → fx, y << 1
ΓΓ−Γ
=Γ−
==2
;;' 2121 ymmxttτ
t-dependent pDalitz analyses
different types of interm states;different types of interm. states;example: D0 → π+π- KS
CF: D0 → K*-π+
DCS: D0 → K*+π-
CP: D0 → ρ0 KS
if f = f ⇒ populate same Dalitz plot; l ti h d t i dl ti h d t i d
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 10 BAS, KEK, October 2009
relative phases determined relative phases determined 28
Physics Belle, PRL 99, 131803 (2007)
t dependentt
t-dependent Dalitz analyses
t-dependence:t-dependence:regions of Dalitz plane →specific t dependence F(x, y);
time evolution of Dalitz distribution⇒⇒ x, y x, y
[ ]titi
S
eemm
tDKtmm
21)(1
)(),,(
22
022
λλ
ππ
−−
−++−
++=
=≡
A
MD0→f
28
[ ]
[ ]titi eemmpq
eemm
21),(21
),(2
22 λλ −−+−
+−
−+
++=
A
AD0→f
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 11 BAS, KEK, October 2009
p2λ1,2=F(x,y); m±
2 = m2(KSπ±),
PhysicsBelle, arXiv:0803.3375
φ measurementφ3 measurement
B- →K- D0(→ f)B- →K- D0(→ f)B →K D0(→ f)interference ⇒ |M|2 = F(φ3) f= π+π- KS
),(),())(( 22220)(
3±
+±±
−±± +=→→ mmAremmAfDKB ii
mmδϕM(
A: D0 decays; r: ratio of two B amplitudes; δ: strong phase diff. of two B amplitudes
A from fit to Dalitz fit to Dalitz distribution of D0 decays;
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 12 BAS, KEK, October 2009
r, φ3, δ (in principle) from fit to Dalitz fit to Dalitz distribution of B± decays30
sin 2φ eff in b → sqq
Physics SM:
sin 2φ1eff in b → sqq
NP contrib. ⇒ sin 2φ1eff ≠ sin 2φ1
effVtbVts* : no weak phase
quasi twoquasi two--bodybody, B → ρ0(770)KS, f0(980)KSBelle, PRD76, 091103(R) (2007)BaBar, PRL99, 161802 (2007)B B PRL98 051803 (2007)
B→BaBar, PRL98, 051803 (2007)
Vtd* 2: (mixing): sin2φ1
NP:
[ ]tmqStmqAetP CPCP
t
∆∆−∆∆+=∆∆−
sincos14
)(/||
τ
τ
interf. between various states and non-resonant contrib. ⇒ Dalitz analysis
sη’
B0 g
g~b s
+(δ 23
dRR)b
~R
s~
NP:
d d
s
s Ks
B0 gs~R
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 13 BAS, KEK, October 2009
∆S=sin2φ1eff -sin2φ1
Physicssin 2φ eff in b → sqqsin 2φ1
eff in b → sqq
[ ]tmqStmqAetP CPCP
t
∆∆−∆∆+=∆∆−
sincos14
)(/||
τ
τ
4τ
),( 22mmmAA ±=
each point in Dalitz space has a specific time evolution depending onevolution, depending on |A|2-|A|2 (direct CPV)(direct CPV)and ℑ(AA*) (indirect CPVindirect CPV corresponding to
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 14 BAS, KEK, October 2009
ℑ(AA ) (indirect CPVindirect CPV, corresponding to specific two-body contribution sin 2sin 2φφ11
effeff (i)(i))
Parametrizationdescription of Dalitz distributiondescription of Dalitz distribution
a matter of statistics102 pp → 3π0 events10 pp → 3π events
Adopted from K. Peters, talk at Charm 2006, Beijing
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 15 BAS, KEK, October 2009
Parametrizationdescription of Dalitz distributiondescription of Dalitz distribution
a matter of statistics103 pp → 3π0 events10 pp → 3π events
Adopted from K. Peters, talk at Charm 2006, Beijing
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 16 BAS, KEK, October 2009
Parametrizationdescription of Dalitz distributiondescription of Dalitz distribution
a matter of statistics104 pp → 3π0 events104 pp → 3π0 events
Adopted from K. Peters, talk at Charm 2006, Beijing
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 17 BAS, KEK, October 2009
Parametrizationdescription of Dalitz distributiondescription of Dalitz distribution
a matter of statistics105 pp → 3π0 events
larger stat. ⇒ larger sensitivity to model details
Breit-Wigner resonances
10 pp → 3π events
Breit Wigner resonancessimple consideration of spin 0 elastic scatteringleads to the Breit-Wigner amplitude for a b → r → a b
Adopted from K. Peters, talk at Charm 2006, Beijing
// 2
4/)(4/
2/2/
22
22
Γ+−Γ
=Γ−−
Γ=
EmT
iEmT
rr
|T|2 |T|2
mr=1.0Γ=0.3
mr=0.5Γ=0.2
| |
Γ
mR
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 18 BAS, KEK, October 2009
E E
ParametrizationBreit Wigner resonances
|T1+ T2 |2
Breit-Wigner resonances
several intermediate statesb ba b → r1 → a b
r2model amplitude as sum of BW amplitudes;
E
sum of BW’s amplitudes; - the approach violates unitarity for wide overlapping resonances;
- the BW shape is distorted close
(“isobar model”)most commonly used most commonly used to model Dalitz distributionsthe BW shape is distorted close
to the thresholds;
parametrization of Dalitz distribution is
distributions
22
D0, B0 → ABCpa model, phenomenological object;
it should provide adequate description adequate description ∑ Φ
=
rACAB
BWr
ir
ACAB
mmAea
mmr ),(
),(22
22A
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 19 BAS, KEK, October 2009
of dataof data; caution needed in interpretation
r
ParametrizationBW resonancesBW resonances
parametrization usually follows Cleo PRD63 092001 (2001) J 0Cleo, PRD63, 092001 (2001) JA,B,C=0
resonance spin
J=0
J=1
J=2
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 20 BAS, KEK, October 2009
ParametrizationBW resonancesBW resonances
parametrization usually follows Cleo PRD63 092001 (2001)Cleo, PRD63, 092001 (2001)
JA,B,C=0
mass dependent width;pAB: momentum of A/B in AB rest frame;pr : momentum of A/B in rest frame of r;
Fi
Ji=0
pr ;(note: MAB may not equal Mr)
i
Ji=1 form factors (Blatt-Weisskopf)
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 21 BAS, KEK, October 2009
Ji=2 R: phenomenological factor(0 GeV-1 – 10 GeV-1 for D0, 0 GeV-1 – 3 GeV-1 for r)
ParametrizationBW resonancesBW resonances
Refinements - FlattFlattéé parametrizationFlattFlattéé parametrization
if 2nd channel for r decays opens close to mr;example f0(980) → ππ, +−−
=0
220
0 )(1
KKKKfff ggimmm
Aρρ ππππππ
S.M. Flatté, PLB63, 224 (1976)
p 0( ) ,KK threshold close to mf0
- parametrization of nonnon resonantresonant contribution
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−⎟
⎟⎠
⎞⎜⎜⎝
⎛ +−= 2
2
2
2 )(1
)(1
ij
ji
ij
jiij
ff
mmm
mmm
ρ
nonnon--resonantresonant contribution in D0 decays usually a constantterm; in B0 charmless decays phase
gij: couplings r/ij(note: mismatch in several papers on Flatté parameters of f0(980) and a (980); contact A Zupanc Zhao Li orin B charmless decays phase
space larger, some variations of ANR; example B0 →KSπ+π- ANR ∝ exp(-αmij
2) Belle, PRD79, 072004 (2009)
a0(980); contact A. Zupanc, Zhao Li orB.G.)
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 22 BAS, KEK, October 2009
p S NR p( ij )B → Dππ ANR = const.
, , ( )
Belle, PRD76, 012006 (2007)
ParametrizationK matrix
E.P. Wigner, Phys. Rev. 70, 15 (1946);S.U. Chung et al., Ann. Phys. 4, 404 (1995)
K-matrix
- sum of BW’s violates unitaritybroad S wave ππ Kπ states are
ρρ TiIS
iSfS fi
+=
=2/12/12
- broad S-wave ππ, Kπ states are not simple BW’s
scattering operator Sρ
ρ
iTKKiKIT
+=
−=−−
−
11
1)(
scattering operator Stransition operator TK matrix
++ =⇔= KKISS
iijij = ρδρoriginally developed for description of scattering
can be adopted to production processproduction process, smm ii
i
221 )(1 +
−=ρ
e.g. D, B decays, using scattering data scattering data
kfkf KiKIT 1)( −−= ρ D
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 23 BAS, KEK, October 2009
I.J.R. Aitchison, Nucl. Phys. A189, 514 (1972); kfkf PiKIF 1)( −−= ρ P (I-iKρ)-1
ParametrizationK matrix PiKIF 1)( −−= ρK-matrix
assume we are parametrizing a 2 body
Drescattering of k→ ππ
production of k K
kk PiKIF ,)(= ππππ ρ
parametrizing a 2-body intermediate state of a 3-body final state(example: ππ in KSπ+π-):
P (I-iKρ)-1of k→ ππ of k KS
k=ππ, KK, ηη, ...
(example: ππ in KSπ π ):Belle, PRL99, 131803 (2007)BaBar, hep-ex/0507101 scatt
scattprod
kr r
rkr
k sssf
smgP
0
0,2
1−−
+−
= ∑ ππβ
s=m2ππ
scattrr 1 ⎪⎫⎪⎧
resonances - poles;(may not coincide withphysical states)
slow-varying part;
factorkinemsssf
smgg
K scatt
scattscatt
ijr r
rj
ri
ij .1
0
02 ⋅
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−
+−
= ∑
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 24 BAS, KEK, October 2009
gir, mr, fijscatt, s0
scatt : from compilation of ππ scattering datascattering data
ParametrizationK matrix gg ρρ D. Asner, Dalitz plot analysis K-matrix
for single channel (e g ππ)
)()1(2
1
2
gKiKT
smggK
ρππ
ρ
ππππ
==
−=
−
formalism, in RPP
(e.g. ππ) and single pole(e.g. ρ0): )(
)()1( 22
smg
gismKiKT
ρρρππ
ρππρ
Γ=
−−=−=
KK--matrix = single BWmatrix = single BW
for multiple channels )(
)(2 simsm
smT
ρρρ
ρρ
Γ−−Γ
=
pand/or multiple poles:
KK--matrix matrix ≠≠ multiple BW multiple BW
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 25 BAS, KEK, October 2009
S. Malvezzi, talk at Charm07, Cornell
Experimental issues
Inv. mass resolutiontypically σ << Γ;e g B → Dππ: σ ~3 MeV;e.g. B → Dππ: σππ 3 MeV;non-negligible for φ(K+K-), ω(π+π-)→ ABW⊗G(σ) numerically (or included in syst.)
also non-negligible at Dalitz boundaries→ mass constraint fit mass constraint fit of ABC to D0, B0;also same Dalitz region signal/sideband (background description)
EfficiencyABW(mAB
2,mAC2) → ABW (mAB
2,MAC2) • ε (mAB
2,mAC2);
ε (mAB2,mAC
2) from MC generated uniformly over phase space;parametrization (factorize mAB
2,mAC2 dependence)
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 26 BAS, KEK, October 2009
Experimental issues
BackgroundD0, B0 mass sidebands 2222
,
22,
22 //
ACAB
side
ACABside
MCbkg
ACABsig
MCbkg
ACAB
bkg
dmdmdN
dmdmdNdmdmdN
dmdmdN
⋅=
Fitmany issues (talk by E. White)(talk by E. White);- unbinned sometimes binned;
Dalitz model testunbinned, sometimes binned;
- fit quality bin Dalitz plot, χ2;model appropriateness
test
pp p(hypothesis testing)→ compare lnL for various models;significances of ind. contributions
Dalitz + decay tfit test(talk by B. (talk by B. YabsleyYabsley););
- biastest using MC with generated Dalitz model,
l D0 K +
fit test
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 27 BAS, KEK, October 2009
example D0 →KSπ+π- Belle, PRL99, 131803 (2007)Belle Note 9799
Experimental issuesFit
Belle, PRD79, 072004 (2009)
Fit- multiple solutions,
example B0 →KSπ+π-
test for multiple solutions by performing numerous fits with varying starting values of solution 1 solution 2y g gparameters;
toy MC to checks if some solutions due to stat fluctuations;
K0*(1430)+π- fit fraction:~ 65% ~ 17%
due to stat. fluctuations;
- individual contribution∑ Φ= 2222 ),(),(
rACAB
BWr
irACAB mmAeamm rA
- individual contribution → fit fraction, fi
Σfi ≠ 100%
∫ ∑
∫ Φ
= 2
22222 ),( ACABACABBWi
ii
i
r
dmdmmmAeaf
i
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 28 BAS, KEK, October 2009
∫ ∑ Φ 2222 ),( ACABr
ACABBWr
ir dmdmmmAea r
Experimental issuesFitFit
- systematic uncertainty due to Dalitz model
variation of resonance parameters (width mass );variation of resonance parameters (width, mass, ...);variation of form factors, dependence of width on mass; inclusion/exclusion of intermediate states with low signif.; difference isobar / K-matrix model;;possible biases (MC)
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 29 BAS, KEK, October 2009
Specifics, outlookZ+(4430)
Belle, PRD 80, 031104(R) (2009)
Z (4430)New states and propertiesof known states
Larger statistics important for significance;
ibilit t di ti i hpossibility to distinguish JZ=1 and JZ=0
(hypothesis testing)(hypothesis testing)
Outlook:Outlook:
more surprises
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 30 BAS, KEK, October 2009
BaBar, PRD 79, 112001 (2009)
Specifics, outlookD0 Mi i d CPV
Belle, PRL 99, 131803 (2007), 540fb-1
D0 Mixing and CPVK*X(1400)+
)%290800( 130±±x
K*(892)+
)%24.033.0(
)%29.080.0(
14.010.0
16.013.0
±±=
±±=
y
x
(89 )
K*(892)- ρ/ωpq 09.010.0
29.030.086.0/ ±±=
rad)09.030.028.024.0( ±±−=ϕ
relative sign of x, y; K-matrix used for systematics
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 31 BAS, KEK, October 2009
1038
Specifics, outlookBelle PRL 96 151801 (2006)D0 Mixing and CPV
direct determination of x, y0
∑ +−Φ
+− =r
BWr
ir mmAeamm r ),(),( 2222A
Belle, PRL 96, 151801 (2006)
instead of x’, y’ as in D0 →K+π-
(rotated for unknown δ between CF and DCS)
r
a1=1, Φ1=0relative phases determined from the fit to Dalitz distribution
2⎤⎡
Outlook:
fit to Dalitz distribution
[ ] 2
2
2 08030.0⎥⎤
⎢⎡
[ ] 2
0
2 06.0/
30.0 radrad+
⎥⎥⎦
⎤
⎢⎢⎣
⎡=
LLϕσ need to reduce Dalitzmodel systematics;
[ ] 2
0
2|/| 08.0
/30.0
+⎥⎥⎦⎢
⎢⎣
=LLpqσ other modes: KSKK, πππ0,...
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 32 BAS, KEK, October 2009
Specifics, outlookφ3 measurement
B- →K- D0(→ f) ))(( 0)(−
±± =→→ fDKBM(
Belle, arXiv:0803.3375
( )B- →K- D0(→ f)interference ⇒ |M|2 = F(φ3) f= π+π- KS
),(),(
))((2222 3±
+±± +
→→
mmAremmA
fDKBii
mmδϕ
M(
S
use x±, y± instead of r, φ3, δ;→ more free parameters; )sin(
)cos(
3
3
δφδφ
+±=+±=
±±
±±
ryrx
lower bias, simple distributions
φ3=76o (+12o -13o) ± 4o ± 9o
Outlook:model independent determination;
22222
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 33 BAS, KEK, October 2009from Cleo-c
)),(),((|| 22222±± −= mmmm DD mm δδFM
12
Specifics, outlooksin 2φ eff in b → sqq BaBar PRD 79 112001 (2009)
example B →J/ψKπ
Belle, PRD79, 072004 (2009)
sin 2φ1eff in b → sqq
change of variables →rectangular Dalitz plot
22
BaBar, PRD 79, 112001 (2009)
22
122arccos1'
''det θ
πππ ⎞⎜⎜⎛
−−
=
→±
mmm
ddmJdmdm m
01'
2
θθ
π π
=
⎠⎜⎝ −− mmm KsB
K
π-
θ
K*0
π
B
KSθ0
a oid concentration of e ents
π+
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 34 BAS, KEK, October 2009
avoid concentration of events at Dalitz boundaries 39
Specifics, outlooksin 2φ eff in b → sqq
Belle, PRD79, 072004 (2009)
sin 2φ1eff in b → sqq
φ1eff(f0)=(12.7 ±6.9
6.5±2.8±3.3)o
Outlook:
⎨⎧≤∆ sKfS 03.0)( 0
⎩⎨⎧≤∆
s
sK
fS ϕσ 05.003.0)( 0
@ 50ab-1 for ∆S=0
b D lit ln.b.: Dalitz analyses measuring φ1
eff
⇒σ(S) = 2 cosφ1eff σ(φ1
eff)∆S
SuperBelle, 50 ab-1
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 35 BAS, KEK, October 2009
Summaryi t d f D lit l i flinstead of summary.... Dalitz analysis flow....
improved methodsimproved methodsgive you larger luminosity!
is Belle note on t-dependent Dalitzready?
NP!NP!
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 36 BAS, KEK, October 2009
Bckups
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 37 BAS, KEK, October 2009
Kinematics more J.D. Jackson, D.R. Tovey, Kinematics, in RPP,
3-body decay
decaying particle:scalar or averaging
i t tover spin states⇒integration over dΩ1*, dΩ3
213
212dmdmd
=Γ
233 32
1)2(
1M
Mπ=
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 38 BAS, KEK, October 2009
6
Kinematics more J.D. Jackson, D.R. Tovey, Kinematics, in RPP,
3-body decay
2/)()( 222*2**2 MEEM + v
2/)(;2/)(),(
1222
21
2*2
*2
*1
*2
*112
1223
212
23
23312
2
mmmMEppEEmmmmMEpEmM
+−=⇒=+=−−=⇒+=
)cos,(cos22
*23
212
223
*23
23
2*3
22
2*2
*3
*2
23
22
223
ϑϑ
mfmmEmEEEmmm
=−−−++=
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 39 BAS, KEK, October 2009
6
Kinematics more J.D. Jackson, D.R. Tovey, Kinematics, in RPP,
3-body decay
( ) ( )( ) ( )22
223max
223min
223*
232
cosmm
mmm−
−+=ϑ ( ) ( )min23max23 mm −
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 40 BAS, KEK, October 2009
6
Physics moreB → K ψ(2S)
Belle, PRL 91, 262001 (2003), 140fb-1
New states
Dalitz analysis not needed B → K X(3872)
B → K ψ(2S)(→ J/ψ π -π+)
for narrow, non-overlaping resonances(negligible interference)example: B → K J/ψ π -π+
( )(→ J/ψ π -π+)
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 41 BAS, KEK, October 2009
8
Physics moret d d t
Belle, PRL 99, 131803 (2007)
t-dependent Dalitz analyses
t d d [ ]titi
S tDKtmm
1
)(),,(
22
022
λλ
ππ −++− =≡M
t-dependence:regions of Dalitz plane →specific t dependence F(x, y);
[ ]
[ ]titi
titi
eemmq
eemm
21
21
),(1
),(21
22
22
λλ
λλ
−−+
−−+−
−+
++=
A
A
time evolution of Dalitz distribution⇒⇒ x, y x, y
[ ]eemmp
),(2 +−+ A
2,12121
Γ−= imλ
22,12,1
)()(
),(),(
2222
2222
Φ
+−Φ
+−
=
=
∑
∑mmAeamm
mmAeamm
BWi
r
BWr
ir
r
r
A
A
,,
),(),( −++−
Φ=Φ=⎯⎯⎯ →⎯
= ∑aa
mmAeamm
rrrrDCPVno
rrrA
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 42 BAS, KEK, October 2009
),(),( 2222−++− = mmmm AA 28
Physics moreBelle, PRD79, 072004 (2009), , ( )
sin 2φ1eff in b → sqq
ci determines amount of DCPVci determines amount of DCPV
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 43 BAS, KEK, October 2009
31
Physics moreBelle, PRD79, 072004 (2009), , ( )
sin 2φ1eff in b → sqq
vtx reconstr (non-scaling)vtx reconstr. (non-scaling)improved with better tracking
bkg model (scaling)CPV in bkg; improved with measurements of contributing modes e g ’Ks
⎩⎨⎧≤
s
sKKfS ϕσ 06.0
04.0)( 0 @ 50 ab-1
of contributing modes – e.g. η Kssig. model (non-scaling)misreconstructed events; comparison with MC with correct reconstr.
D lit d l ( li )
some syst. errors cancel in ∆S = S(sqq)-S(J/ψKs)
⎧ Kf030Dalitz model (scaling)parametr. of NR; inclusion of other resonances;
⎩⎨⎧≤∆
s
sKKfS ϕσ 05.0
03.0)( 0
for ∆S=0
n.b.: Dalitz analyses measuring φ1eff ⇒σ(S) = 2 cosφ1
eff σ(φ1eff)
B. Golob, Ljubljana Univ., IJS Introd. to Dalitz analyses 44 BAS, KEK, October 2009
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