Bastian Kubis - indico.ph.tum.de · Final-state interactions and Dalitz plots Various reasons why...

Precision tools in hadron physics for Dalitz plot studies

Bastian Kubis

Helmholtz-Institut fur Strahlen- und Kernphysik (Theorie)

Bethe Center for Theoretical Physics

Universitat Bonn, Germany

Quark Confinement and the Hadron Spectrum XMunich, October 12th 2012

B. Kubis, Precision tools in hadron physics for Dalitz plot studies – p. 1

Precision tools in hadronic physics for Dalitz plot studies

Introduction

• final-state interactions and Dalitz plots

• CP violation in three-body decays

Final-state interactions and dispersion relations

• scattering consistent with analyticity and unitarity: Roy equations

• decays linked to scattering: form factors and Omnès solution

• three-particle dynamics for the example of ω/φ → 3π

Niecknig, BK, Schneider, EPJC72 (2012) 2014

• application: transition form factors in ω/φ → π0ℓ+ℓ−

Schneider, BK, Niecknig, PRD86 (2012) 054013

Outlook


Final-state interactions and Dalitz plots

Various reasons why final-state interactions are important

• if rescattering strong, significantly enhances decay probabilities





• if rescattering strong, significantly shapes decay probabilities(resonances!)






• introduce phases/imaginary parts (prerequisite for CPV!)






• introduce phases/imaginary parts (prerequisite for CPV!)

• new analytic features in the Dalitz plot (cusp effect in K → 3π)

0

200

400

600

800

1000

1200

1400

1600

x 10 2

0.08 0.09 0.1 0.11 0.12 0.13

25000

30000

35000

40000

45000

50000

0.076 0.077 0.078 0.079 0.08


CP violation in three-body decays

Advantage of 3-body decays:

• resonance-rich environment

• larger branching fractions

here: B± → K±π∓π±

e.g. 3.7σ signal in KρBELLE 2006, BABAR 2008

How to analyse CP violation inDalitz plots?

1. strictly model-independentextraction from data directly

Gardner et al. 2003, 2004

Bediaga et al. 2009

2. theoretical information onstrong amplitudes as input!

Sensitive Observables

)2

Even

ts /

(0.0

15 G

eV/c

20

40

60

80

100

120

)2

Even

ts /

(0.0

3 G

eV/c

20

40

60

80

100

)2 (GeV/cππm

0.6 0.7 0.8 0.9 1 1.1

)2

Ev

ents

/ (

0.0

3 G

eV/c

20

40

60

80

100

120

)2 (GeV/cππm

0.7 0.8 0.9 1 1.1 1.2

B− → K−π+π− B+→ K+π−π+

cos(θH) > 0cos(θH) > 0

cos(θH) < 0cos(θH) < 0

ρρ f0f0

Here:

Consistent with SMImportant next steps:


Direct data analysis: significance

Significance in Dalitz plot distributions: Bediaga et al. 2009

DpSCP(i).=

N(i)− N(i)√

N(i) + N(i)

N , N : CP-conjugate decays; i: label of a specific Dalitz plot bin

• allows to study local asymmetries

• no theoretical input required at all — strictly model-independent

• B decays: some evidence, in particular in B± → K±ρ0(→ π±π∓)consistent with Standard Model BELLE 2006, BABAR 2008

• D decays: only upper limits (at few-percent level)Standard Model prediction tiny BABAR 2008

−→ certainly more restrictive / potentially more useful to interpret if⊲ reliable hadronic input amplitudes available⊲ errors of that input quantifiable

see also poster by P. C. Magalhães on Tuesday


ππ scattering constrained by analyticity and unitarity

see also talk by R. Kaminski on Thursday

Roy equations = coupled system of partial-wave dispersion relations+ crossing symmetry + unitarity

• twice-subtracted fixed-t dispersion relation:

T (s, t) = c(t) +1

π

∫ ∞

4M2π

ds′

s2

s′2(s′ − s)+

u2

s′2(s′ − u)

ImT (s′, t)

• subtraction function c(t) determined from crossing symmetry



see also talk by R. Kaminski on Thursday

Roy equations = coupled system of partial-wave dispersion relations+ crossing symmetry + unitarity

• twice-subtracted fixed-t dispersion relation:

T (s, t) = c(t) +1

π

∫ ∞

4M2π

ds′

s2

s′2(s′ − s)+

u2

s′2(s′ − u)

ImT (s′, t)

• subtraction function c(t) determined from crossing symmetry

• project onto partial waves tIJ(s) (angular momentum J , isospin I)−→ coupled system of partial-wave integral equations

tIJ(s) = kIJ(s) +

2∑

I′=0

∞∑

J ′=0

∫ ∞

4M2π

ds′KII′

JJ ′(s, s′)ImtI′

J ′(s′)

Roy 1971

• subtraction polynomial kIJ (s): ππ scattering lengthscan be matched to chiral perturbation theory Colangelo et al. 2001

• kernel functions KII′

JJ ′(s, s′) known analytically



• elastic unitarity −→ coupled integral equations for phase shifts

• modern precision analyses:⊲ ππ scattering Ananthanarayan et al. 2001, García-Martín et al. 2011

⊲ πK scattering Büttiker et al. 2004

• example: ππ I = 0 S-wave phase shift & inelasticity

400 600 800 1000 1200 1400

s1/2

(MeV)

0

50

100

150

200

250

300

CFDOld K decay dataNa48/2K->2 π decayKaminski et al.Grayer et al. Sol.BGrayer et al. Sol. CGrayer et al. Sol. DHyams et al. 73

δ0

(0)

1000 1100 1200 1300 1400

s1/2

(MeV)

0

0.5

1

η00(s)

Cohen et al.Etkin et al.Wetzel et al.Hyams et al. 75Kaminski et al.Hyams et al. 73Protopopescu et al.CFD .

ππ KK

ππ ππ

García-Martín et al. 2011

• strong constraints on data from analyticity and unitarity!


Form factors constrained by analyticity and unitarity

• just two particles in final state: form factor; from unitarity:

=disc

1

2idiscFI(s) = ImFI(s) = FI(s)×θ(s−4M2

π)× sin δI(s) e−iδI(s)

−→ final-state theorem: phase of FI(s) is just δI(s) Watson 1954


Form factors constrained by analyticity and unitarity

• just two particles in final state: form factor; from unitarity:

=disc

1

2idiscFI(s) = ImFI(s) = FI(s)×θ(s−4M2

π)× sin δI(s) e−iδI(s)

−→ final-state theorem: phase of FI(s) is just δI(s) Watson 1954

• solution to this homogeneous integral equation known:

FI(s) = PI(s)ΩI(s) , ΩI(s) = exp

s

π

∫ ∞

4M2π

ds′δI(s

′)

s′(s′ − s)

PI(s) polynomial, ΩI(s) Omnès function Omnès 1958

completely given in terms of phase shift δI(s)


Application: limits on SUSY couplings (1)

• constrain new-physics operators (e.g. SUSY) from low-energydecays; example: lepton-flavour violation τ− → µ−π+π−

• problem: huge model uncertainties e.g. in τ− → µ−f0(980):strength of scalars coupling to quark currents??depends on (controversial) nature of scalar resonances??

Herrero et al. 2009



• constrain new-physics operators (e.g. SUSY) from low-energydecays; example: lepton-flavour violation τ− → µ−π+π−

• problem: huge model uncertainties e.g. in τ− → µ−f0(980):strength of scalars coupling to quark currents??depends on (controversial) nature of scalar resonances??

Herrero et al. 2009

• R-parity-violating SUSY operators:

τ−

ui

µ−

dj

uk

τ−

dj

µ−

ui

dℓ

τ−

µ−

dj

νi

dk

τ−

µ−

dj

νi

dk

−→ effective operators generated by heavy SUSY exchanges:

Leff =λ′

31jλ′∗

21j

2m2

dj︸︷︷︸

eff. coupling

(µγαPLτ)(uγαPLu)

︸︷︷︸

vector×vector

+ . . .+λ3i2λ

′∗

i11

2m2νi

︸︷︷︸

eff. coupling

(µPLτ)(dPRd)︸︷︷︸

scalar×scalar

+ . . .



• but hadronisation into ππ given by vector / scalar form factors:⟨π+π−∣∣ 1

2 (uγαu− dγαd)

∣∣0⟩= FV

π (s)(pπ+ − pπ−)α

⟨π+π−∣∣ 1

2 (uu+ dd)∣∣0⟩= BnΓn(s)

⟨π+π−∣∣ss

∣∣0⟩= BsΓs(s)

−→ both normalisation and line shape known frome+e− → π+π− / dispersive analyses

0.4 0.6 0.8 1.00

1

2

3

4

5

6

0.4 0.6 0.8 1.00

2

4

6

8

0.4 0.6 0.8 1.00

5

10

15

20(a) (b) (c)

√s [GeV]

√s [GeV]

√s [GeV]

dΓ/d√s×

103

ρ

“σ”f0(980)→

• integrate spectra within experimental cuts for ρ0(770)/f0(980)Belle 2009–2011

−→ upper limits on effective SUSY couplings with very littlehadronic model uncertainty Dreiner, Hanhart, BK, Meißner, in progress


Dispersion relations for three-body decays

Example: ω/φ → 3π

• beyond chiral energy range; half way to D → 3 light mesons

• vector mesons highly important for (virtual) photon processes


Dispersion relations for three-body decays

Example: ω/φ → 3π

• beyond chiral energy range; half way to D → 3 light mesons

• vector mesons highly important for (virtual) photon processes

• typically used for 3π decays: improved tree-level models(vector-meson dominance, hidden local symmetry. . . )

+ crossed +ω

ρ

π

π

π

ω

π

π

π

• similarly in experimental analyses φ → 3π: KLOE 2003, CMD-2 2006

sum of 3 Breit–Wigners (ρ+, ρ−, ρ0)+ constant background term

−→ obviously, unitarity relation cannot be fulfilled!

• particularly simple system: restricted to odd partial waves

−→ P-wave interactions only (neglecting F- and higher)


From unitarity to integral equation

Decay amplitude can be decomposed into single-variable functions

M(s, t, u) = iǫµναβnµpνπ+pαπ−p

βπ0 F(s, t, u)

F(s, t, u) = F(s) + F(t) + F(u)

Unitarity relation for F(s):

discF(s) = 2iF(s)︸︷︷︸

right-hand cut

+ F(s)︸︷︷︸

left-hand cut

× θ(s− 4M2

π)× sin δ11(s) e−iδ11(s)





right-hand cut

+ F(s)︸︷︷︸

left-hand cut

× θ(s− 4M2






right-hand cut

× θ(s− 4M2


=disc

• right-hand cut only −→ Omnès problem

F(s) = aΩ(s) , Ω(s) = exp

s

π

∫ ∞

4M2π

ds′

s′δ11(s

′)

s′ − s− iǫ

−→ amplitude given in terms of pion vector form factor

++pair0V +0pair+V +0pairVF(s, t, u) =





right-hand cut

+ F(s)︸︷︷︸

left-hand cut

× θ(s− 4M2


• inhomogeneities F(s): angular averages over the F(s)

F(s) = aΩ(s)

1 +s

π

∫ ∞

4M2π

ds′

s′sin δ11(s

′)F(s′)

|Ω(s′)|(s′ − s− iǫ)

F(s) =3

2

∫ 1

−1

dz (1− z2)F(t(s, z)

)Khuri, Treiman 1960

Aitchison 1977

Anisovich, Leutwyler 1998

F(s) = +++ ...





right-hand cut

+ F(s)︸︷︷︸

left-hand cut

× θ(s− 4M2


• inhomogeneities F(s): angular averages over the F(s)

F(s) = aΩ(s)

1 +s

π

∫ ∞

4M2π

ds′

s′sin δ11(s

′)F(s′)

|Ω(s′)|(s′ − s− iǫ)

F(s) =3

2

∫ 1

−1

dz (1− z2)F(t(s, z)

)Khuri, Treiman 1960

Aitchison 1977

Anisovich, Leutwyler 1998

−→ crossed-channel scattering between s-, t-, and u-channel


ω/φ → 3π Dalitz plots

• only one subtraction constant a −→ fix to partial width

• normalised Dalitz plot in y = 3(s0−s)2MV (MV −3Mπ)

, x =√3(t−u)

2MV (MV −3Mπ):

ω → 3π : φ → 3π :

• ω Dalitz plot is relatively smooth

• φ Dalitz plot clearly shows ρ resonance bands


ω/φ → 3π Dalitz plots

• only one subtraction constant a −→ fix to partial width

• normalised Dalitz plot in y = 3(s0−s)2MV (MV −3Mπ)

, x =√3(t−u)

2MV (MV −3Mπ):

ω → 3π : φ → 3π :

ω → 3π

• ω Dalitz plot is relatively smooth

• φ Dalitz plot clearly shows ρ resonance bands


Experimental comparison to φ → 3π

• fit to Dalitz plot: 1.98× 106 events in 1834 bins KLOE 2003

0 100 200 300 400 500 600 700 800 9000

2000

4000

6000

8000

Bin number

#eve

nts

(effi

cien

cyco

rrec

ted)

Omnès χ2 = 1.71 ... 2.06

KLOE (2003)

F(s) = aΩ(s) = exp

[

s

π

∫ ∞

4M2π

ds′

s′δ11(s

′)

s′ − s

]




0 100 200 300 400 500 600 700 800 9000

2000

4000

6000

8000

Bin number

#eve

nts

(effi

cien

cyco

rrec

ted)

Omnès χ2 = 1.71 ... 2.06

Disp1 χ2 = 1.17 ... 1.50

KLOE (2003)

F(s) = aΩ(s)

[

1 +s

π

∫ ∞

4M2π

ds′

s′F(s′) sin δ11(s

′)

|Ω(s′)|(s′ − s− iǫ)

]




0 100 200 300 400 500 600 700 800 9000

2000

4000

6000

8000

Bin number

#eve

nts

(effi

cien

cyco

rrec

ted)

Omnès χ2 = 1.71 ... 2.06

Disp1 χ2 = 1.17 ... 1.50

Disp2 χ2 = 1.02 ... 1.03

KLOE (2003)

F(s) = aΩ(s)

[

1 + b s+s2

π

∫ ∞

4M2π

ds′

s′2F(s′) sin δ11(s

′)

|Ω(s′)|(s′ − s− iǫ)

]




0 100 200 300 400 500 600 700 800 9000

2000

4000

6000

8000

Bin number

#eve

nts

(effi

cien

cyco

rrec

ted)

Omnès χ2 = 1.71 ... 2.06

Disp1 χ2 = 1.17 ... 1.50

Disp2 χ2 = 1.02 ... 1.03

KLOE (2003)

• once-subtracted DR: highly predictive

• twice-subtracted DR: one additional complex parameter fitted

• phenomen. contact term seems to emulate rescattering effects!


(g − 2)µ, light-by-light, and transition form factors

Czerwinski et al., arXiv:1207.6556 [hep-ph]

• leading and next-to-leading hadronic effects in (g − 2)µ:

had

had

−→ hadronic light-by-light soon dominant uncertainty





had

had


• important contribution: pseudoscalar pole terms

singly / doubly virtual form factors

FPγγ∗(M2P , q

2, 0) and FPγ∗γ∗(M2P , q

21 , q

22)

π0, η, η′





had

had


• important contribution: pseudoscalar pole terms

singly / doubly virtual form factors

FPγγ∗(M2P , q

2, 0) and FPγ∗γ∗(M2P , q

21 , q

22)

• for specific virtualities: linked tovector-meson conversion decays

π0, η, η′

−→ e.g. Fπ0γ∗γ∗(M2π0 , q21 ,M

2ω) measurable in ω → π0ℓ+ℓ− etc.


Transition form factors in ω, φ → π0ℓ+ℓ−

• long-standing puzzle: transition form factor ω → π0ℓ+ℓ− far fromvector-meson-dominance picture see e.g. Terschlüsen, Leupold 2010

disc

ω

π0

π0

ωπ

+

π−

=

fωπ0(s) = fωπ0(0) +s

12π2

∫ ∞

4M2π

ds′q3ππ(s

′)FV ∗π (s′)f1(s

′)

s′3/2(s′ − s)Köpp 1974

• f1(s) = fω→3π1 (s) = F(s) + F(s) P-wave projection of F(s, t, u)

• subtracting dispersion relation once yields

⊲ better convergence for ω → π0 transition form factor

⊲ sum rule for ω → π0γ −→ saturated at 90–95%

fωπ0(0) =1

12π2

∫ ∞

4M2π

ds′q3ππ(s

′)

s′3/2FV ∗π (s′)f1(s

′) , Γω→π0γ ∝ |fV π0(0)|2

Schneider, BK, Niecknig 2012


Numerical results: ω → π0γ∗

0 0.1 0.2 0.3 0.4 0.5 0.6

1

10

100

√s [GeV]

|Fωπ0(s)|2

NA60 ’09NA60 ’11Lepton-GVMDTerschlüsen et al.f1(s) = aΩ(s)full dispersive

0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

6

7

8

9

√s [GeV]

dΓω→

π0µ+µ−/d

s[10−6

GeV

−1]

0 0.1 0.2 0.3 0.4 0.5 0.610-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

√s [GeV]

dΓω→

π0e+e−/d

s[G

eV−1]

• clear enhancement vs. pure VMD

• unable to account for steep rise (similar in φ → ηℓ+ℓ−?)


Numerical results: φ → π0γ∗

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1

10

100

√s [GeV]

|Fφπ0(s)|2

VMDf1(s) = aΩ(s)once subtracted f1(s)

twice subtracted f1(s)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6

7

√s [GeV]

dΓφ→

π0µ+µ−/ds

[10−8

GeV

−1]

0 0.2 0.4 0.6 0.810-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

√s [GeV]

dΓφ→

π0e+e−/d

s[G

eV−1]

• measurement would be extremely helpful: ρ in physical region!

• partial-wave amplitude backed up by experiment


Summary / Outlook

Rigorous methods for investigations of hadronic meson decays:

• dispersion relations based on fundamental principles ofunitarity, analyticity and crossing symmetry

• excellent fit to KLOE data for φ → 3π −→ method works

• relation between hadronic decays and transition form factors


Summary / Outlook





Work in progress:

• η′ → ηππ −→ subtractions? πη scattering? Schneider, BK

• D+ → π+π+K− Niecknig, BK


Summary / Outlook





Work in progress:

• η′ → ηππ −→ subtractions? πη scattering? Schneider, BK

• D+ → π+π+K− Niecknig, BK

Challenges for going to heavier meson decays:

• at higher energies: coupled-channel integral equations

• inelasticities certainly not negligible

• perturbative treatment of crossed-channel effects reliable?

• when are higher partial waves non-negligible?


Bastian Kubis - indico.ph.tum.de · Final-state interactions and Dalitz plots Various reasons why...

Documents

Transcript of Bastian Kubis - indico.ph.tum.de · Final-state interactions and Dalitz plots Various reasons why...