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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
F. Niecknig Dispersive analysis of D → Kππ 2 / 17
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
F. Niecknig Dispersive analysis of D → Kππ 3 / 17
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
F. Niecknig Dispersive analysis of D → Kππ 3 / 17
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
three-body decays: D → 3π, ππK
• Dalitz plot = density distribution intwo kinematical variables
• resonances −→ rapid phasevariation enhances CP violation inparts of the decay region
F. Niecknig Dispersive analysis of D → Kππ 3 / 17
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−
F. Niecknig Dispersive analysis of D → Kππ 4 / 17
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−
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?
F. Niecknig Dispersive analysis of D → Kππ 4 / 17
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−
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
F. Niecknig Dispersive analysis of D → Kππ 4 / 17
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
F. Niecknig Dispersive analysis of D → Kππ 5 / 17
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)
Re
Im
s0
F. Niecknig Dispersive analysis of D → Kππ 5 / 17
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)
Re
Im
s0
Css
Mfi(s) =1
2πi
∮
Cs
Mfi(s′)
s′ − sds′
F. Niecknig Dispersive analysis of D → Kππ 5 / 17
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)
Re
Im
s0
Css
Mfi(s) =1
2πi
∮
Cs
Mfi(s′)
s′ − sds′
F. Niecknig Dispersive analysis of D → Kππ 5 / 17
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)
Re
Im
s0
s
Mfi(s) =1
2πi
∮
Cs
Mfi(s′)
s′ − sds′ =
12πi
∫ ∞
s0
disc Mfi (s′)
s′ − sds′
F. Niecknig Dispersive analysis of D → Kππ 5 / 17
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)
Re
Im
s0
s
Mfi(s) =n−1∑
i=0
ci si +
sn
2πi
∫ ∞
s0
discMfi(s′)
s′n(s′ − s)ds′
F. Niecknig Dispersive analysis of D → Kππ 5 / 17
Three-particle decay
Ansatz:
Mdecay < 3Mfinal Mdecay > 3Mfinal
s ∈ [(Mdecay +Mfinal)2,∞] s ∈ [4M2
final, (Mdecay −Mfinal)2]
continuation inMdecay and s
scatteringregion
decayregion
F. Niecknig Dispersive analysis of D → Kππ 6 / 17
Three-particle decay
Ansatz:
Mdecay < 3Mfinal Mdecay > 3Mfinal
s ∈ [(Mdecay +Mfinal)2,∞] s ∈ [4M2
final, (Mdecay −Mfinal)2]
continuation inMdecay and s
scatteringregion
decayregion
• set up a dispersive treatment for:
D D D1 2 3
3 1 22 3 1
F. Niecknig Dispersive analysis of D → Kππ 6 / 17
Three-particle decay
Ansatz:
Mdecay < 3Mfinal Mdecay > 3Mfinal
s ∈ [(Mdecay +Mfinal)2,∞] s ∈ [4M2
final, (Mdecay −Mfinal)2]
continuation inMdecay and s
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)
F. Niecknig Dispersive analysis of D → Kππ 6 / 17
Unitarity relation
Unitarity relation:
disc MIJ(s) = 2iMI
J(s) + MIJ(s)e−iδI
J (s) sin δIJ(s)
F. Niecknig Dispersive analysis of D → Kππ 7 / 17
Unitarity relation
Unitarity relation:
︸ ︷︷ ︸
left-hand cut
︸ ︷︷ ︸
right-hand cut
disc MIJ(s) = 2iMI
J(s) + MIJ(s)e−iδI
J (s) sin δIJ(s)
F. Niecknig Dispersive analysis of D → Kππ 7 / 17
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
F. Niecknig Dispersive analysis of D → Kππ 7 / 17
Unitarity relation
Unitarity relation:
︸ ︷︷ ︸
left-hand cut
︸ ︷︷ ︸
right-hand cut
disc MIJ(s) = 2iMI
J(s) + MIJ(s)e−iδI
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
F. Niecknig Dispersive analysis of D → Kππ 7 / 17
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 :
F. Niecknig Dispersive analysis of D → Kππ 8 / 17
D+→ K−π+π+
D+
π+
π0
K 0
u
d
sc
d
d
d
d
W+
• Cabibbo favored decay,good statistics
E761 2006, CLEO 2008, FOCUS 2009
• coupled to D+ → K0π0π+
BESIII 2014
Partial waves :
F. Niecknig Dispersive analysis of D → Kππ 8 / 17
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 :
pion-pion : S2ππ P1
ππ
pion-kaon : S1/2Kπ P1/2
Kπ
S3/2Kπ P3/2
Kπ(
D1/2Kπ
)
F. Niecknig Dispersive analysis of D → Kππ 8 / 17
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 :
pion-pion : S2ππ P1
ππ
pion-kaon : S1/2Kπ P1/2
Kπ
S3/2Kπ P3/2
Kπ(
D1/2Kπ
)
F. Niecknig Dispersive analysis of D → Kππ 8 / 17
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 :
pion-pion : S2ππ P1
ππ
pion-kaon : S1/2Kπ P1/2
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)
F. Niecknig Dispersive analysis of D → Kππ 8 / 17
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 :
pion-pion : S2ππ P1
ππ
pion-kaon : S1/2Kπ P1/2
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)
F. Niecknig Dispersive analysis of D → Kππ 8 / 17
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 :
pion-pion : S2ππ P1
ππ
pion-kaon : S1/2Kπ P1/2
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)
F. Niecknig Dispersive analysis of D → Kππ 8 / 17
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)
F. Niecknig Dispersive analysis of D → Kππ 9 / 17
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
F. Niecknig Dispersive analysis of D → Kππ 10 / 17
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
F. Niecknig Dispersive analysis of D → Kππ 11 / 17
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
• insert F(s) into F(s)
F(s) = A(s) +1π
∫ ∞
thF(s′)K (s, s′)ds′
⊲ A(s) subtraction polynomial part ⇒ basis functions
F. Niecknig Dispersive analysis of D → Kππ 11 / 17
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
• 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)
F. Niecknig Dispersive analysis of D → Kππ 11 / 17
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
F. Niecknig Dispersive analysis of D → Kππ 12 / 17
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
• decay symmetric under interchanging the two pions (s ↔ t)
• fit in terms of 7 complex subtraction constants (-1 phase)
F. Niecknig Dispersive analysis of D → Kππ 13 / 17
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
• 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)
F. Niecknig Dispersive analysis of D → Kππ 13 / 17
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
F. Niecknig Dispersive analysis of D → Kππ 14 / 17
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
F. Niecknig Dispersive analysis of D → Kππ 14 / 17
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
F. Niecknig Dispersive analysis of D → Kππ 15 / 17
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
⇒ remove dispersive integrals over inhomogeneities
F. Niecknig Dispersive analysis of D → Kππ 15 / 17
Two-particle rescattering
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
F. Niecknig Dispersive analysis of D → Kππ 15 / 17
Two-particle rescattering
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
F. Niecknig Dispersive analysis of D → Kππ 15 / 17
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
F. Niecknig Dispersive analysis of D → Kππ 16 / 17
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
F. Niecknig Dispersive analysis of D → Kππ 17 / 17
Spares
F. Niecknig Dispersive analysis of D → Kππ 1 / 4
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
F. Niecknig Dispersive analysis of D → Kππ 2 / 4
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
F. Niecknig Dispersive analysis of D → Kππ 2 / 4
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]
F. Niecknig Dispersive analysis of D → Kππ 3 / 4
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]
F. Niecknig Dispersive analysis of D → Kππ 4 / 4
Recommended