Dispersive analysis of D K Franz Niecknig · 2015. 4. 17. · Dispersive analysis of D → Kππ Franz Niecknig with B. Kubis Helmholtz-Institut für Strahlen- und Kernphysik University

Dispersive analysis of D → Kππ

Franz Niecknig

with B. Kubis

Helmholtz-Institut für Strahlen- und KernphysikUniversity Bonn

PWA 8 / ATHOS 3Ashburn VA, April 13-17, 2015

F. Niecknig Dispersive analysis of D → Kππ 1 / 17

Outline

Motivation

• Heavy-meson Dalitz plots: hunting for CP violation

• How Dalitz plots are commonly analysed

Dispersive analysis of three-body decays

Dispersive analysis of D → Kππ FN, Kubis in progress

• challenges of higher decaying masses

• experimental comparison

Summary & Outlook


Heavy-meson Dalitz plots: hunting for CP violation

CP violation in partial widths Γ(P → f ) 6= Γ(P → f )

• at least two interfering decay amplitudes

• different weak (CKM) phases

• different strong (final-state-interaction) phases







two-body decays: D → ππ, K K

• decay at fixed total energy −→ fixedstrong phase







two-body decays: D → ππ, K K

• decay at fixed total energy −→ fixedstrong phase

three-body decays: D → 3π, ππK

• Dalitz plot = density distribution intwo kinematical variables

• resonances −→ rapid phasevariation enhances CP violation inparts of the decay region


How Dalitz plots are commonly analyzed

Dalitz plots typically analysed by the isobar model

D

σ

π

π

π

D+

π+

κ,K ∗...

π+

K−




D

σ

π

π

π

D+

π+

κ,K ∗...

π+

K−

Challenges

• some resonances do not look like Breit-Wigners! κ, f0(500) · · ·→ use scattering phase-shifts

• three-particle rescattering effects

+++ · · ·

• unitarity fixes Im/Re relation ⇒ additional contact term?




D

σ

π

π

π

D+

π+

κ,K ∗...

π+

K−

Dispersion theory

⊲ DR fulfill unitarity and analyticity by construction

⊲ phase-shifts with good accuracy availableBüttiker et al. 2004, Garcia-Martin et al. 2011 and Ananthanaryan et al. 2001

⊲ includes three-particle rescattering effects

⊲ applied to η → 3π Lanz, PhD 2011 talk by Peng Guoη′ → ηππ Schneider, PhD 2013

ω/φ → 3π FN, Schneider, Kubis 2012 talk by Igor Danilkin


Dispersion theory

Unitarity relation: discMfi(s) = 2i∑

n ρn(s)M∗fn(s)Mni(s) Θ(s − sn)

• analytic structure:cut on real axis

• Cauchy’s integral formula

• extend contour

• introduce subtractions(if necessary)

i n f


Dispersion theory





• extend contour


Re

Im

s0


Dispersion theory





• extend contour


Re

Im

s0

Css

Mfi(s) =1

2πi

∮

Cs

Mfi(s′)

s′ − sds′


Dispersion theory





• extend contour


Re

Im

s0

Css

Mfi(s) =1

2πi

∮

Cs

Mfi(s′)

s′ − sds′


Dispersion theory





• extend contour


Re

Im

s0

s

Mfi(s) =1

2πi

∮

Cs

Mfi(s′)

s′ − sds′ =

12πi

∫ ∞

s0

disc Mfi (s′)

s′ − sds′


Dispersion theory





• extend contour


Re

Im

s0

s

Mfi(s) =n−1∑

i=0

ci si +

sn

2πi

∫ ∞

s0

discMfi(s′)

s′n(s′ − s)ds′


Three-particle decay

Ansatz:

Mdecay < 3Mfinal Mdecay > 3Mfinal

s ∈ [(Mdecay +Mfinal)2,∞] s ∈ [4M2

final, (Mdecay −Mfinal)2]

continuation inMdecay and s

scatteringregion

decayregion



Ansatz:





scatteringregion

decayregion

• set up a dispersive treatment for:

D D D1 2 3

3 1 22 3 1



Ansatz:





scatteringregion

decayregion

• set up a dispersive treatment for:

D D D1 2 3

3 1 22 3 1

• Reconstruction theorem: M(s, t , u) ⇒ MIJ(s), N

IJ(t), F

IJ(u)

• analytic continuation into the decay region

• coupled integral equations for MIJ(s), N

IJ(t), F

IJ (u)


Unitarity relation

Unitarity relation:

disc MIJ(s) = 2iMI

J(s) + MIJ(s)e−iδI

J (s) sin δIJ(s)


Unitarity relation

Unitarity relation:

︸︷︷︸

left-hand cut

︸︷︷︸

right-hand cut

disc MIJ(s) = 2iMI


J (s) sin δIJ(s)


Unitarity relation

Unitarity relation:

︸︷︷︸

right-hand cut

disc MIJ(s) = 2iMI

J(s) e−iδIJ (s) sin δI

J(s)

Homogeneous solution:

MIJ(s) = P(s) Ω(s) = P(s) exp

[sπ

∫ ∞

th

δIJ (s

′)

s′(s′ − s − iǫ)ds′

]

Omnès 1958

M(s, t, u) =

2, 3-pair

1, 3-pair 1, 2-pair1

2 3


Unitarity relation

Unitarity relation:

︸︷︷︸

left-hand cut

︸︷︷︸

right-hand cut

disc MIJ(s) = 2iMI


J (s) sin δIJ(s)

Integral equation:

MIJ(s) = ΩI

J (s)[

P(n)(s) +sn

π

∫ ∞

th

MIJ(s

′) sin δIJ(s

′)

|Ω(s′)|s′n(s′ − s − iǫ)ds′

]

Anisovich & Leutwyler 1998

MIJ(s) =

2J + 12

∫ 1

−1PJ(zs)M(s, t(s, zs)) |Is dzs −MI

J(s)

MIJ(s) = +++ ...

Dispersion integral generates crossed-channel rescattering contributions


D+→ K−π+π+

D+

π+

π+

K−

u

u

sc

u

d

d

dW+

• Cabibbo favored decay,good statistics

E761 2006, CLEO 2008, FOCUS 2009

• coupled to D+ → K0π0π+

BESIII 2014

Partial waves :


D+→ K−π+π+

D+

π+

π0

K 0

u

d

sc

d

d

d

d

W+


E761 2006, CLEO 2008, FOCUS 2009


BESIII 2014

Partial waves :


D+→ K−π+π+

D+

π+

π+

K−

u

u

sc

u

d

d

dW+


E761 2006, CLEO 2008, FOCUS 2009


BESIII 2014

Partial waves :

pion-pion : S2ππ P1

ππ

pion-kaon : S1/2Kπ P1/2

Kπ

S3/2Kπ P3/2

Kπ(

D1/2Kπ

)


D+→ K−π+π+

D+

π+

π+

K−

u

u

sc

u

d

d

dW+


E761 2006, CLEO 2008, FOCUS 2009


BESIII 2014

Partial waves :


ππ


Kπ

S3/2Kπ P3/2

Kπ(

D1/2Kπ

)


D+→ K−π+π+

D+

π+

π+

K−

u

u

sc

u

d

d

dW+


E761 2006, CLEO 2008, FOCUS 2009


BESIII 2014

Partial waves :


ππ


Kπ

S3/2Kπ P3/2

Kπ(

D1/2Kπ

)

0 0,5 1 1,5 2 2,5 30

10∣ ∣ ∣Ω

1/2

0

∣ ∣ ∣

√s [GeV]

κ

K∗

0 (1430)


D+→ K−π+π+

D+

π+

π+

K−

u

u

sc

u

d

d

dW+


E761 2006, CLEO 2008, FOCUS 2009


BESIII 2014

Partial waves :


ππ


Kπ

S3/2Kπ P3/2

Kπ(

D1/2Kπ

)

0 0,5 1 1,5 2 2,5 30

20

∣ ∣ ∣Ω

1/2

1

∣ ∣ ∣

√s [GeV]

K∗(892)


D+→ K−π+π+

D+

π+

π+

K−

u

u

sc

u

d

d

dW+


E761 2006, CLEO 2008, FOCUS 2009


BESIII 2014

Partial waves :


ππ


Kπ

S3/2Kπ P3/2

Kπ(

D1/2Kπ

)

0 0.5 1 1.5 2 2.5 30

10∣ ∣ ∣Ω

1/2

2

∣ ∣ ∣

√s [GeV]

K∗

2 (1430)


Full system

S2ππ(u) = Ω

20(u)

u2∫

∞

4M2π

S2ππ (u′)

u′2(u′ − u)dµ2

0

P1ππ(u) = Ω

11(u)

c0 + c1u + u2∫

∞

4M2π

P1ππ (u′)

u′2(u′ − u)dµ1

1

S1/2πK (s) = Ω

1/20 (s)

c2 + c3s + c4s2+ c5s3

+ s4∫

∞

(MK +Mπ )2

S1/2πK (s′)

s′4(s′ − s)dµ1/2

0

S3/2πK (s) = Ω

3/20 (s)

s2∫

∞

(MK +Mπ )2

S3/2πK (s′)

s′2(s′ − s)dµ3/2

0

P1/2πK (s) = Ω

1/21 (s)

c6 + s∫

∞

(MK +Mπ )2

P1/2πK (s′)

s′(s′ − s)dµ1/2

1

D1/2πK (s) = Ω

1/22 (s)

∫

∞

(MK +Mπ )2

D1/2πK (s′)

(s′ − s)dµ1/2

2

System linear in subtraction constants ⇒ basis functions

M(s, t , u) =6∑

i=0

ciMi(s, t , u)


Solution strategy

Classic solution strategy

Input MIJ

Calculate MIJ Calculate MI

J

Accuracy?Result Yes

No

• input Omnès function

• iteration builds up rescattering effects

• fails for D → Kππ ⇒ different solution strategy is needed


Solution strategy

Matrix inversion

• hypothetical equation:

F(s) ≡κ2L+1(s)

2

∫ 1

−1zmF(t(s, z))dz.

F(s) = Ω(s)

P(s) + sn∫ ∞

th

F(s′)

κ2L+1(s′)s′n(s′ − s)dµ


Solution strategy

Matrix inversion


F(s) ≡κ2L+1(s)

2

∫ 1

−1zmF(t(s, z))dz.

F(s) = Ω(s)

P(s) + sn∫ ∞

th

F(s′)


• insert F(s) into F(s)

F(s) = A(s) +1π

∫ ∞

thF(s′)K (s, s′)ds′

⊲ A(s) subtraction polynomial part ⇒ basis functions


Solution strategy

Matrix inversion


F(s) ≡κ2L+1(s)

2

∫ 1

−1zmF(t(s, z))dz.

F(s) = Ω(s)

P(s) + sn∫ ∞

th

F(s′)


• insert F(s) into F(s)

F(s) = A(s) +1π

∫ ∞

thF(s′)K (s, s′)ds′

⊲ A(s) subtraction polynomial part ⇒ basis functions

• discretisation yields

A(si) = (δij − K (si , sj)) F(sj)


Experimental comparison

0.5

1

1.5

2

2.5

3

0.5 1 1.5 2 2.5 3

s [GeV2]

t[G

eV2]

CLEO 2008

Validity range:

• elastic approximation required in our framework

• major inset of inelasticities: ≥ Mη′ + MK ≈ 1.45 GeV

→ beyond 1.45 GeV treatment of inelastic channels needed



0.5

1

1.5

2

2.5

3

0.5 1 1.5 2 2.5 3

s [GeV2]

t[G

eV2]

CLEO 2008

• decay symmetric under interchanging the two pions (s ↔ t)

• fit in terms of 7 complex subtraction constants (-1 phase)



0.5

1

1.5

2

2.5

3

0.5 1 1.5 2 2.5 3

s [GeV2]

t[G

eV2]

CLEO 2008

• decay symmetric under interchanging the two pions (s ↔ t)

• fit in terms of 7 complex subtraction constants (-1 phase)

Fit scenarios:

• full fit

• only two particle rescattering (Omnès)


Experimental comparison I

Dalitz plot slices

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

1

1.5

2

2.5

3

0 100 200 300 400 500 600 700 800

s [GeV2]

t[G

eV2]

50 100 150

200

400

600

800

400 500

200

400

bin number

• full fit: χ2/ndof ≈ 1.1


Experimental comparison I

fit fractions slices

Full fit

S2ππ (8 ± 3)%

S1/2πK (72 ± 12)%

P1/2πK (10 ± 2)%

S3/2πK (16 ± 3)%

D1/2πK (0.15 ± 0.1)%

Σ (106 ± 20)%

50 100 150

200

400

600

800

400 500

200

400

bin number

• full fit: χ2/ndof ≈ 1.1

• fit fractions: hierachy of partial-wave amplitudes compare to previousanalyses


Two-particle rescattering

Full set of equations

S2ππ(u) = Ω

20(u)

u2∫

∞

4M2π

S2ππ (u′)

u′2(u′ − u)dµ2

0

P1ππ(u) = Ω

11(u)

c0 + c1u + u2∫

∞

4M2π

P1ππ (u′)

u′2(u′ − u)dµ1

1

S1/2πK (s) = Ω

1/20 (s)

c2 + c3s + c4s2+ c5s3

+ s4∫

∞

(MK +Mπ )2

S1/2πK (s′)

s′4(s′ − s)dµ1/2

0

S3/2πK (s) = Ω

3/20 (s)

s2∫

∞

(MK +Mπ )2

S3/2πK (s′)

s′2(s′ − s)dµ3/2

0

P1/2πK (s) = Ω

1/21 (s)

c6 + s∫

∞

(MK +Mπ )2

P1/2πK (s′)

s′(s′ − s)dµ1/2

1

D1/2πK (s) = Ω

1/22 (s)

∫

∞

(MK +Mπ )2

D1/2πK (s′)

(s′ − s)dµ1/2

2

inhomogeneities build up crossed-channel rescattering



Full set of equations

S2ππ(u) = Ω

20(u)

u2∫

∞

4M2π

S2ππ (u′)

u′2(u′ − u)dµ2

0

P1ππ(u) = Ω

11(u)

c0 + c1u + u2∫

∞

4M2π

P1ππ (u′)

u′2(u′ − u)dµ1

1

S1/2πK (s) = Ω

1/20 (s)

c2 + c3s + c4s2+ c5s3

+ s4∫

∞

(MK +Mπ )2

S1/2πK (s′)

s′4(s′ − s)dµ1/2

0

S3/2πK (s) = Ω

3/20 (s)

s2∫

∞

(MK +Mπ )2

S3/2πK (s′)

s′2(s′ − s)dµ3/2

0

P1/2πK (s) = Ω

1/21 (s)

c6 + s∫

∞

(MK +Mπ )2

P1/2πK (s′)

s′(s′ − s)dµ1/2

1

D1/2πK (s) = Ω

1/22 (s)

∫

∞

(MK +Mπ )2

D1/2πK (s′)

(s′ − s)dµ1/2

2

inhomogeneities build up crossed-channel rescattering

⇒ remove dispersive integrals over inhomogeneities



Omnès representation

S2ππ(u) = Ω2

0(u) P1ππ(u) = c0 + c1u Ω1

1(u)

S3/2πK (s) = Ω

3/20 (s) S1/2

πK (s) =

c2 + c3s + c4s2 + c5s3

Ω1/20 (s)

P1/2πK (s) = c6 Ω

1/21 (s) D1/2

πK (s) = Ω1/22 (s)

Direct comparison with ⇔ without inhomogeneities not possible

⇒ rearrange fit constants



Omnès representation

S2ππ(u) = c0 Ω

20(u)

S3/2πK (s) = c1 Ω

3/20 (s) S1/2

πK (s) =

c2 + c3s + c4s2 + c5s3

Ω1/20 (s)

P1/2πK (s) = c6 Ω

1/21 (s) D1/2

πK (s) = c7 Ω1/22 (s)

Direct comparison with ⇔ without inhomogeneities not possible

⇒ rearrange fit constants

Fit scenarios

⊲ without D wave: 7 complex fit parameters

⊲ with D wave: additional complex fit constant c7


Experimental comparison II

Dalitz plot Fit fractions

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

1

1.5

2

2.5

3

0 100 200 300 400 500 600 700 800

s [GeV2]

t[G

eV2]

without D wave with D wave

S2ππ (48 ± 16)% (9.5 ± 8)%

S1/2πK (178 ± 22)% (91 ± 22)%

P1/2πK (7 ± 1)% (8 ± 1)%

S3/2πK (395 ± 35)% (240 ± 40)%

D1/2πK - (0.13 ± 0.03)%

• without D wave: χ2/ndof ≈ 1.3

• with D wave: χ2/ndof ≈ 1.1 (+1 complex fit constant)

• huge cancellation between amplitudes

⇒ need crossed-channel rescattering for meaningful fit results


Summary & Outlook

• dispersion relations powerful tool to describe three-body decays

⊲ based on analyticity, unitarity and crossing

• D+ → K−π+π+:

⊲ “elastic region“ well described

⊲ parametrization of inelastic effects needed (η′K , . . . in πK ,)

• D+ → K 0π0π+:

⊲ further constrain on subtraction constants

⊲ data on its way (BESIII 2014)

Outlook

• coupled-channel approaches (e.g. to D → 3π/K Kπ)

• extraction of πK phase-shift information at higher energies


Spares


Analytic continuation

Continuation into the decay region

⇒• M2 → M2 + iδ

Bronzan & Kacser 1963

F(s) =1

κ(s)2L+1

∫ s+(s)

s−(s)h(2s′ − 3s0 + s)F

(s′) ds′

Integration path can cross the branch cut ⇒ path deformation needed

Im s

Re s

s+(s)

s−(s)

M < 3Mf :

Integration does not run over the cut⇒ no path deformation needed


Analytic continuation

Continuation into the decay region

⇒• M2 → M2 + iδ

Bronzan & Kacser 1963

F(s) =1

κ(s)2L+1

∫ s+(s)

s−(s)h(2s′ − 3s0 + s)F

(s′) ds′

Integration path can cross the branch cut ⇒ path deformation needed

Im s

Re s

s+(s)

s−(s)

M > 3Mf :

Integration does run over the cut⇒ path deformation needed


Basis functions

-1

0

-5

0

5

-2

0

2

-1

0

1

-2

-1

01

0

1

-1

0

-3

0

3

-2

0

2

-0.5

0

0.5

-1

0

0

0.5

0

1

-1

0

1

-3

0

3

-1

0

0

2

-0.2

0

0.2

0

1

-1

0

1

-5

0

5

-1

0

0

-0.2

0

0.2

0

1

-0.5

0

0.5

-5

0

5

-0.5

0

-1

0

1

0

0.1

-2

0

2

-2

0

2

-10

0

10

-1

0

1

-4

0

0

0.4

0 0.5 1 1.5

-2

0

0 0.5 1 1.5-1

0

1

0 1 2

-2

0

2

0 1 2

-0.1

0

0.1

0 1 2-15

0

15

0 1 2

-1

0

basisfunctions(a.u.)

F00 F1

1 F1/20 F3/2

0 F1/21 F1/2

2

0

1

2

3

4

5

6

√s [GeV]


Angular amplitudes

0 0.5 1 1.50

100

200

300

0 0.5 1 1.5-8-6-4-202

0 0.5 1 1.50

200

400

600

0 0.5 1 1.5

0

3

0 0.5 1 1.50

200

400

0 0.5 1 1.5

0

2

4

0 0.5 1 1.50

2

4

6

0 0.5 1 1.50

2

4

F00

F1/2

0+

F3/2

0

F1/21

F1/22

√s [GeV]


Documents

Dispersive analysis of D K Franz Niecknig · 2015. 4. 17. · Dispersive analysis of D → Kππ Franz Niecknig with B. Kubis Helmholtz-Institut für Strahlen- und Kernphysik University