What can we learn from hadronic and radiative decays of light ...B. Kubis, Hadronic and radiative decays of light mesons – p. 8 From unitarity to integral equation Decay amplitude

What can we learn from hadronic and radiative decaysof light mesons?

Bastian Kubis

Helmholtz-Institut fur Strahlen- und Kernphysik (Theorie)

Bethe Center for Theoretical Physics

Universitat Bonn, Germany

Light Meson Decays Workshop, Jefferson Lab, August 5th 2012

B. Kubis, Hadronic and radiative decays of light mesons – p. 1

Hadronic and radiative decays of light mesons

Chiral perturbation theory and dispersion relations

Final-state interactions of two pions

• Pion vector form factor

• Decays η, η′ → π+π−γ

Final-state interactions of three pions

• 3-particle dynamics for ω/φ → 3π talk by S. Schneider on Wednesday

• η → 3π: quark mass ratios and Dalitz plot parameterstalk by S. Lanz on Tuesday

Meson transition form factors

• η, η′ → γℓ+ℓ− thanks to A. Wirzba

• ω/φ → π0ℓ+ℓ− talk by S. Schneider on Wednesday

Outlook


Light mesons without models

Chiral perturbation theory (ChPT) . . . talk by J. Bijnens on Monday

• Effective field theory: simultaneous expansion inquark masses + small momenta

⊲ systematically improvable⊲ well-established link to QCD⊲ so what’s wrong with it?!



Chiral perturbation theory (ChPT) . . . talk by J. Bijnens on Monday

• Effective field theory: simultaneous expansion inquark masses + small momenta

⊲ systematically improvable⊲ well-established link to QCD⊲ so what’s wrong with it?!

. . . and its limitations

• strong final-state interactions render corrections large

• physics of light pseudoscalars (π, K, η) only⊲ (energy) range limited by resonances

e.g. pion–pion scattering: σ(500), ρ . . .⊲ not applicable to decays of (e.g.) vector mesons at all

−→ marry ChPT and dispersion relations in order to

apply ChPT where it works best!



Dispersion relations on one page

Re(z)

Im(z)

s

4M2π

analyticity & Cauchy’s theorem:

T (s) =1

2πi

∮

∂Ω

T (z)dz

z − s




Re(z)

Im(z)

s

4M2π


T (s) =1

2πi

∮

∂Ω

T (z)dz

z − s




Re(z)

Im(z)

s

4M2π


T (s) =1

2πi

∮

∂Ω

T (z)dz

z − s




Re(z)

Im(z)

s

4M2π


T (s) =1

2πi

∮

∂Ω

T (z)dz

z − s




Re(z)

Im(z)

s

4M2π


T (s) =1

2πi

∮

∂Ω

T (z)dz

z − s

−→ 1

2πi

∫ ∞

4M2π

discT (z)dz

z − s

=1

π

∫ ∞

4M2π

ImT (z)dz

z − s




Re(z)

Im(z)

s

4M2π


T (s) =1

2πi

∮

∂Ω

T (z)dz

z − s

−→ 1

2πi

∫ ∞

4M2π

discT (z)dz

z − s

=1

π

∫ ∞

4M2π

ImT (z)dz

z − s

• discT (s) = 2i ImT (s) calculable by “cutting rules”:

T (s) T (s)

e.g. if T (s) is a ππ partial wave −→

discT (s)

2i= ImT (s) =

2qππ√sθ(s−4M2

π)|T (s)|2




Re(z)

Im(z)

s

4M2π


T (s) =1

2πi

∮

∂Ω

T (z)dz

z − s

−→ 1

2πi

∫ ∞

4M2π

discT (z)dz

z − s

=1

π

∫ ∞

4M2π

ImT (z)dz

z − s

• discT (s) = 2i ImT (s) calculable by “cutting rules”:

T (s) T (s)inelastic intermediate states (KK, 4π)suppressed at low energies

−→ will be neglected in the following


Pion form factor 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


Pion form factor 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

• today: high-accuracy ππ (and πK) phase shifts availableAnanthanarayan et al. 2001, García-Martín et al. 2011, Caprini et al. 2012

(Büttiker et al. 2004)


Pion vector form factor

• pion vector form factor clearly non-perturbative: ρ resonance

-0,2 0 0,2 0,4 0,6 0,8 1

sππ [GeV2]

1

10

|FV

(sπ

π)|2

ChPT at one loop

data on e+e− → π+π−

Omnès representation

Stollenwerk et al. 2012

−→ Omnès representation vastly extends range of applicability


Application: η, η′→ π+π−γ

• η(′) → π+π−γ driven by the chiral anomaly, π+π− in P-wave

−→ final-state interactions the same as for vector form factor

• ansatz: Aη(′)

ππγ = A× P (sππ)× FVπ (sππ), P (sππ) = 1 + α(′)sππ







• spectra with fitted normalisation and slope(s) α(′)

0 0.05 0.1 0.15 0.2

Eγ[GeV]

0

1

2

3

4

5

6

7

8

dΓ/dEγ[arb. units]

0 0.1 0.2 0.3 0.4

Eγ[GeV]

0

5

10

15

20

dΓ/dEγ[arb. units]








• divide data by pion form factor −→ P (sππ)

0 0.05 0.1 0.15 0.2 0.25 0.3

sππ [GeV2]

1

1.2

1.4

1.6

P(s π

π)

0 0.2 0.4 0.6 0.8

sππ [GeV2]

1

1.5

2

2.5

3

P(s π

π)


−→ exp.: αWASA = (1.89± 0.64)GeV−2, αKLOE = (1.31± 0.08)GeV−2

−→ interpret α(′) by matching to chiral perturbation theory


Dispersion relations for three-body decays

Example: ω/φ → 3π

• beyond ChPT: copious efforts to develop EFT for vector mesonsBijnens et al.; Bruns, Meißner; Lutz, Leupold; Gegelia et al.; Kampf et al.. . .

• vector mesons highly important for (virtual) photon processes


Dispersion relations for three-body decays

Example: ω/φ → 3π

• beyond ChPT: copious efforts to develop EFT for vector mesonsBijnens et al.; Bruns, Meißner; Lutz, Leupold; Gegelia et al.; Kampf et al.. . .

• 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)B. Kubis, Hadronic and radiative decays of light mesons – p. 8

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!


Predictions for experiment: ω → 3π Dalitz plot parameters

• ω → 3π Dalitz plot smooth −→ polynomial parameterisation

|Fpol(z, φ)|2 = |N |2

1 + 2αz + 2βz3/2 sin 3φ+ 2γz2 + 2δz5/2 sin 3φ+O(z3)




|Fpol(z, φ)|2 = |N |2

1 + 2αz+2βz3/2 sin 3φ+ 2γz2 + 2δz5/2 sin 3φ

(

|Fpol(z, φ)|2/|F(z, φ)|2 − 1

)

[%]

x [MeV]

y[M

eV]

α× 103 β × 103 γ × 103 δ × 103

84 . . . 96 — — —

74 . . . 84 24 . . . 28 — —

73 . . . 81 24 . . . 28 3 . . . 6 —

74 . . . 83 21 . . . 24 0 . . . 2 7 . . . 8




|Fpol(z, φ)|2 = |N |2

1 + 2αz + 2βz3/2 sin 3φ+2γz2 + 2δz5/2 sin 3φ

(

|Fpol(z, φ)|2/|F(z, φ)|2 − 1

)

[%]

x [MeV]

y[M

eV]

α× 103 β × 103 γ × 103 δ × 103

84 . . . 96 — — —

74 . . . 84 24 . . . 28 — —

73 . . . 81 24 . . . 28 3 . . . 6 —

74 . . . 83 21 . . . 24 0 . . . 2 7 . . . 8




|Fpol(z, φ)|2 = |N |2

1 + 2αz + 2βz3/2 sin 3φ+ 2γz2+2δz5/2 sin 3φ

(

|Fpol(z, φ)|2/|F(z, φ)|2 − 1

)

[%]

x [MeV]

y[M

eV]

α× 103 β × 103 γ × 103 δ × 103

84 . . . 96 — — —

74 . . . 84 24 . . . 28 — —

73 . . . 81 24 . . . 28 3 . . . 6 —

74 . . . 83 21 . . . 24 0 . . . 2 7 . . . 8




|Fpol(z, φ)|2 = |N |2

1 + 2αz + 2βz3/2 sin 3φ+ 2γz2 + 2δz5/2 sin 3φ

(

|Fpol(z, φ)|2/|F(z, φ)|2 − 1

)

[%]

x [MeV]

y[M

eV]

α× 103 β × 103 γ × 103 δ × 103

84 . . . 96 — — —

74 . . . 84 24 . . . 28 — —

73 . . . 81 24 . . . 28 3 . . . 6 —

74 . . . 83 21 . . . 24 0 . . . 2 7 . . . 8




|Fpol(z, φ)|2 = |N |2

1 + 2αz + 2βz3/2 sin 3φ+ 2γz2 + 2δz5/2 sin 3φ

(

|Fpol(z, φ)|2/|F(z, φ)|2 − 1

)

[%]

x [MeV]

y[M

eV]

α× 103 β × 103 γ × 103 δ × 103

84 . . . 96 — — —

74 . . . 84 24 . . . 28 — —

73 . . . 81 24 . . . 28 3 . . . 6 —

74 . . . 83 21 . . . 24 0 . . . 2 7 . . . 8

−→ 2 Dalitz plot parameters sufficient at 1% accuracy

−→ compare η → 3π0 (same 3-fold symmetry):α = (−31.7± 1.6)× 10−3 PDG average

β ≈ −4× 10−3 γ ≈ +1× 10−3 Schneider, BK, Ditsche 2011


Quark masses and η → 3π decays

• η → 3π isospin violating; two sources in the Standard Model:

mu 6= md e2 6= 0

• electromagnetic contribution small Sutherland 1967

Baur, Kambor, Wyler 1996; Ditsche, BK, Meißner 2009

η → π+π−π0 : ALOc (s, t, u) =

B(mu −md)

3√3F 2

π

1 +3(s− s0)

M2η −M2

π

s = (pπ+ + pπ−)2 , 3s0.= M2

η + 3M2π

• ∆I = 1 relation between charged and neutral decay amplitudes:

η → 3π0 : An(s, t, u) = Ac(s, t, u) +Ac(t, u, s) +Ac(u, s, t)


Quark masses and η → 3π decays

• η → 3π isospin violating; two sources in the Standard Model:

mu 6= md e2 6= 0

• electromagnetic contribution small Sutherland 1967

Baur, Kambor, Wyler 1996; Ditsche, BK, Meißner 2009

η → π+π−π0 : ALOc (s, t, u) =

B(mu −md)

3√3F 2

π

1 +3(s− s0)

M2η −M2

π

s = (pπ+ + pπ−)2 , 3s0.= M2

η + 3M2π

• ∆I = 1 relation between charged and neutral decay amplitudes:

η → 3π0 : An(s, t, u) = Ac(s, t, u) +Ac(t, u, s) +Ac(u, s, t)

• relevance: (potentially) clean access to mu −md

but : large higher-order / final-state interactions

−→ require good theoretical Dalitz-plot description

to extract normalisation


η → 3π: final-state interactions

• strong final-state interactions among pions

⊲ tree level: Γ(η → π+π−π0) = 66 eV Cronin 1967

⊲ one-loop: Γ(η → π+π−π0) = 160± 50 eV Gasser, Leutwyler 1985

⊲ experimental: Γ(η → π+π−π0) = 296± 16 eV PDG

• major source: large S-wave final-state rescattering −→use dispersion relations to resum those beyond loop expansion

• similar formalism to ω/φ → 3π, but more partial waves

(S waves I = 0, 2, P wave I = 1)

match subtraction constants to ChPT and/or to datatalk by S. Lanz on Tuesday


η → 3π: final-state interactions

• strong final-state interactions among pions

⊲ tree level: Γ(η → π+π−π0) = 66 eV Cronin 1967

⊲ one-loop: Γ(η → π+π−π0) = 160± 50 eV Gasser, Leutwyler 1985

⊲ experimental: Γ(η → π+π−π0) = 296± 16 eV PDG

• major source: large S-wave final-state rescattering −→use dispersion relations to resum those beyond loop expansion

• similar formalism to ω/φ → 3π, but more partial waves

(S waves I = 0, 2, P wave I = 1)

match subtraction constants to ChPT and/or to datatalk by S. Lanz on Tuesday

• on the other hand: consider r =Γ(η → 3π0)

Γ(η → π+π−π0)

ChPT: rtree = 1.54 , r1-loop = 1.46 , r2-loop = 1.47

PDG: r = 1.432± 0.026 (fit) , r = 1.48± 0.05 (average)

−→ agrees rather well Bijnens, Ghorbani 2007


η → 3π0 Dalitz plot parameter α

|An(x, y)|2 = |Nn|21 + 2α z + . . .

z ∝ (s− s0)

2 + (t− s0)2 + (u− s0)

2

-0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10

ChPT O(p4)

ChPT O(p6)

Dispersive (KWW)

Crystal Ball@BNL

Crystal Barrel@LEAR

GAMS-2000

KLOE

MAMI-B

MAMI-C

SND

WASA@CELSIUS

WASA@COSY

O(p4) + NREFT (full)

103 × α

+13

+13± 32

−7 . . .− 14

−25± 5

Schneider, BK,Ditsche 2011

PDG average:−31.7± 1.6


η → π+π−π0: Dalitz plot parameters

• |Ac(x, y)|2 = |Nc|21 + ay + by2 + dx2 + . . .

x =

√3

2MηQη(u− t )

y = 32MηQη

(s0 − s)

a b d

KLOE −1.09 +−

0.010.02 0.12±0.01 0.057+

−0.0090.017

ChPT O(p6) −1.27 ± 0.08 0.39±0.10 0.055±0.057

ChPT O(p4)+ NREFT −1.21 ± 0.01 0.31±0.02 0.050±0.003

dispersive −1.16 0.24 . . . 0.26 0.09 . . . 0.10

Bijnens, Ghorbani 2007

Schneider, BK, Ditsche 2011

Kambor, Wiesendanger, Wyler 1995

• only one modern precision experiment so far KLOE 2008

−→ WASA-at-COSY coming up soon

• note large discrepancy theory vs. experiment for b !


η → 3π Dalitz parameters, charged vs. neutral

• Dalitz plot vs. amplitude expansion: x ∝ t− u , y ∝ s− s0

|Ac|2 = |Nc|21 + ay + by2 + dx2 + . . .

|An|2 = |Nn|2

1 + 2αz + . . .

Ac = Nc

1 + ay + by2 + dx2 + . . .

An = Nn

1 + αz + . . .

a = 2Re a , b = |a|2 + 2Re b , d = 2Re d , α = Re α




|Ac|2 = |Nc|21 + ay + by2 + dx2 + . . .

|An|2 = |Nn|2

1 + 2αz + . . .

Ac = Nc

1 + ay + by2 + dx2 + . . .

An = Nn

1 + αz + . . .


• isospin relation between neutral and charged parameters:

α =1

2

(b+ d

)−→ α =

1

4

(

b+ d− a2

4− (Im a)2

)

<1

4

(

b+ d− a2

4

)

Bijnens, Ghorbani 2007




|Ac|2 = |Nc|21 + ay + by2 + dx2 + . . .

|An|2 = |Nn|2

1 + 2αz + . . .

Ac = Nc

1 + ay + by2 + dx2 + . . .

An = Nn

1 + αz + . . .



α =1

4

(

b+ d− a2

4

)

− ζ1(1 + ζ2a)2 , ζ1 = 0.050± 0.005 , ζ2 = 0.225± 0.003

ζ1/2 determined by ππ phases Schneider, BK, Ditsche 2011




|Ac|2 = |Nc|21 + ay + by2 + dx2 + . . .

|An|2 = |Nn|2

1 + 2αz + . . .

Ac = Nc

1 + ay + by2 + dx2 + . . .

An = Nn

1 + αz + . . .



α =1

4

(

b+ d− a2

4

)

− ζ1(1 + ζ2a)2 , ζ1 = 0.050± 0.005 , ζ2 = 0.225± 0.003

ζ1/2 determined by ππ phases Schneider, BK, Ditsche 2011

• use precise KLOE data on a, b, d as input KLOE 2008

αtheoKLOE = −0.062± 0.003stat

+0.004−0.006syst ± 0.003ππ

αexpKLOE = −0.030± 0.004stat

+0.002−0.004syst

significanttension!




|Ac|2 = |Nc|21 + ay + by2 + dx2 + . . .

|An|2 = |Nn|2

1 + 2αz + . . .

Ac = Nc

1 + ay + by2 + dx2 + . . .

An = Nn

1 + αz + . . .



α =1

4

(

b+ d− a2

4

)

− ζ1(1 + ζ2a)2 , ζ1 = 0.050± 0.005 , ζ2 = 0.225± 0.003

ζ1/2 determined by ππ phases

• displayed as constraint in a− b plane:

-1.40 -1.35 -1.30 -1.25 -1.20 -1.15 -1.10 -1.05 -1.00a

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

b

O(p4) + NREFT

KLOE

NREFT imaginary part

NO imaginary part


Hadronic decays of the η′

• large final-state interactions expected in η′ → ηππ and η′ → 3π

−→ unitarised U(3) ChPT e.g. Borasoy, Nißler 2005

• only includes two-particle unitarity:

−→ dispersive study in progress Schneider, BK


Hadronic decays of the η′

• large final-state interactions expected in η′ → ηππ and η′ → 3π

−→ unitarised U(3) ChPT e.g. Borasoy, Nißler 2005

• only includes two-particle unitarity:

−→ dispersive study in progress Schneider, BK

• claim: Gross, Treiman, Wilczek 1979

Γ(η′ → π0π+π−)

Γ(η′ → ηπ+π−)∝

(md −mu

ms

)2

assumptions: (a) A(η′ → π0π+π−) = ǫπ0η ×A(η′ → ηπ+π−)

(b) amplitudes "essentially flat" in phase space

• refuted in unitarised U(3) ChPT: assumptions too simplisticBorasoy, Meißner, Nißler 2006


(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 η → γℓ+ℓ−

• 2-pion contribution to Fηγ∗γ(s, 0) intimately linked to Aηππγ :

η π+

π−

γ∗

γ

discFηγ∗γ(s, 0) ∝ Aηππγ(s, 0)× FV ∗

π (s) = A× P (s)× |FVπ (s)|2

F(I=1)ηγ∗γ (s, 0) = 1 +

B(η→π+π−γ)︷︸︸︷

Aηππγ

Aηγγ

︸︷︷︸

B(η→γγ)

e s

12π2

∫ ∞

4M2π

ds′q3ππ(s

′)

s′3/2P (s′)︸︷︷︸

1+α s′

|FVπ (s′)|2s′ − s

• corrections from isoscalar contributions −→ here small

• in particular: form factor slope bη function of α(η → π+π−γ)

−→ indications for significant deviation from VMD pictureHanhart, Stollenwerk, Wirzba, work in progress


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 → π0γ∗ 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−

6G

eV−1]

0 0.1 0.2 0.3 0.4 0.5 0.610-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0.2 0.3 0.4 0.5 0.6 0.70

3

6

9

10em

6√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−/d

s[10−

8G

eV−1]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

0.2 0.3 0.4 0.5 0.6 0.7 0.801

2

34

5

6

7

10em

8

√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

Dispersion relations for meson decays

• based on fundamental principles ofunitarity, analyticity, crossing symmetry

• rigorous treatment of two- and three-hadron final states

• extends range of applicability (at least) to full elastic regime

• matching to ChPT where it works best:

(sub)threshold, normalisation, slopes. . .

• relates hadronic to radiative decays / transition form factors


Summary

Dispersion relations for meson decays

• based on fundamental principles ofunitarity, analyticity, crossing symmetry

• rigorous treatment of two- and three-hadron final states

• extends range of applicability (at least) to full elastic regime

• matching to ChPT where it works best:

(sub)threshold, normalisation, slopes. . .

• relates hadronic to radiative decays / transition form factors

Omissions

• not-extremely-rare decays not yet seen experimentally: η′ → 4πGuo, BK, Wirzba 2011

• impact of transition form factors on rare leptonic decays

π0 → e+e−, η → ℓ+ℓ− KTeV 2007


Spares


ππ scattering constrained by analyticity and unitarity

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



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!


Pion vector form factor and ahvpµ

• more refined representation: taken from talk by G. Colangelo 2008

F πV (s) = Ω1(s)×Gω(s)× Ωinel(s)

Gω(s): ρ− ω mixingΩinel(s): inelastic for

√s & (Mπ +Mω), parametrized using

conformal mapping techniques Trocóniz, Ynduráin 2002Comparison CMD2(04)–KLOE

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

E (GeV)

0

10

20

30

40

50

|Fπ|2

CMD2 dataKLOE dataFit to both sets

• achieve amazing precisionfor hadronic contribution toaµ below 1 GeV:

ahvpµ (√s ≤ 2MK)

= (493.7± 1.0)× 10−10

Colangelo et al. (preliminary)

• check of data compatibilitywith analyticity / unitarity

• also extension to higher energies Hanhart 2012


Transition form factor beyond the πω threshold

0 0.2 0.4 0.6 0.8 1 1.2 1.40.1

1

10

100

√s [GeV]

|Fφπ0(s)|2

NA60 ’09NA60 ’11Lepton-G

VMDCMD-2

Terschlüsen et al.f1(s) = aΩ(s)

once subtracted f1(s)

• full solution above naive VMD, but still too low

• higher intermediate states (4π / πω) more important?


Documents

What can we learn from hadronic and radiative decays of light ...B. Kubis, Hadronic and radiative decays of light mesons – p. 8 From unitarity to integral equation Decay amplitude