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```							Crossing symmetry implemented in ππ D- and
F -waves amplitudes

´
Robert Kaminski

Institute of Nuclear Physics Polish Academy of Sciences, Kraków, Poland

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 1
What?

2
s = mππ
crossing symmetry:

T (s, t) = Cst T (t, s) where Cst is crossing matrix

we ( + J. Pelaez, F. Yndurain from Madrid) did it already for
the ππ amplitudes for the S0, S2 and P1 waves
(notation: JI)
(see e.g. proceedings of QCD08 conference)

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 2
What?

2
s = mππ
crossing symmetry:

T (s, t) = Cst T (t, s) where Cst is crossing matrix

we ( + J. Pelaez, F. Yndurain from Madrid) did it already for
the ππ amplitudes for the S0, S2 and P1 waves
(notation: JI)
(see e.g. proceedings of QCD08 conference)

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 2
What?

2
s = mππ
crossing symmetry:

T (s, t) = Cst T (t, s) where Cst is crossing matrix

we ( + J. Pelaez, F. Yndurain from Madrid) did it already for
the ππ amplitudes for the S0, S2 and P1 waves
(notation: JI)
(see e.g. proceedings of QCD08 conference)

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 2
How?

Roy’s equations (Roy 1971) - twice subtracted dispersion
relations
Re fℓI (s) = a0 δI0 δℓ0 + a2 δI2 δℓ0
0            0
s − 4mπ 2                          1          1
+           2
(2a0 − 5a2 ) (δI0 δℓ0 + δI1 δℓ1 −
0      0                       δI2 δℓ0 )
12mπ                              6          2
2       1    smax
′                  ′
+                             II                           I
− ds′ Kℓℓ′ (s, s′ )Im fℓI′ (s′ ) + dℓ (s, smax )
I ′ =0 ℓ′ =0     2
4mπ
(1)
s − 4mπ         2
a0 + a2 +
0    0                     (2a0 − 5a2 ) : subtracting term (ST)
0      0
12mπ 2
′              ′                                      ′
II
Kℓℓ′ (s, s′ )Im fℓI′ (s′ ): kernel term (KT(s)) (∼ ImfℓI′ (s′ )/s′3 )
I
dℓ (s, smax ): driving term (DT(s))
′
Im fℓI′ (s′ ): INPUT

´
Robert Kaminski, IFJ PAN, Kraków, Poland    QCD 2010, Montpellier, page 3
How?

Roy’s equations (Roy 1971) - twice subtracted dispersion
relations
Re fℓI (s) = a0 δI0 δℓ0 + a2 δI2 δℓ0
0            0
s − 4mπ 2                          1          1
+           2
(2a0 − 5a2 ) (δI0 δℓ0 + δI1 δℓ1 −
0      0                       δI2 δℓ0 )
12mπ                              6          2
2       1    smax
′                  ′
+                             II                           I
− ds′ Kℓℓ′ (s, s′ )Im fℓI′ (s′ ) + dℓ (s, smax )
I ′ =0 ℓ′ =0     2
4mπ
(1)
s − 4mπ         2
a0 + a2 +
0    0                     (2a0 − 5a2 ) : subtracting term (ST)
0      0
12mπ 2
′              ′                                      ′
II
Kℓℓ′ (s, s′ )Im fℓI′ (s′ ): kernel term (KT(s)) (∼ ImfℓI′ (s′ )/s′3 )
I
dℓ (s, smax ): driving term (DT(s))
′
Im fℓI′ (s′ ): INPUT

´
Robert Kaminski, IFJ PAN, Kraków, Poland    QCD 2010, Montpellier, page 3
How?

Roy’s equations (Roy 1971) - twice subtracted dispersion
relations
Re fℓI (s) = a0 δI0 δℓ0 + a2 δI2 δℓ0
0            0
s − 4mπ 2                          1          1
+           2
(2a0 − 5a2 ) (δI0 δℓ0 + δI1 δℓ1 −
0      0                       δI2 δℓ0 )
12mπ                              6          2
2       1    smax
′                  ′
+                             II                           I
− ds′ Kℓℓ′ (s, s′ )Im fℓI′ (s′ ) + dℓ (s, smax )
I ′ =0 ℓ′ =0     2
4mπ
(1)
s − 4mπ         2
a0 + a2 +
0    0                     (2a0 − 5a2 ) : subtracting term (ST)
0      0
12mπ 2
′              ′                                      ′
II
Kℓℓ′ (s, s′ )Im fℓI′ (s′ ): kernel term (KT(s)) (∼ ImfℓI′ (s′ )/s′3 )
I
dℓ (s, smax ): driving term (DT(s))
′
Im fℓI′ (s′ ): INPUT

´
Robert Kaminski, IFJ PAN, Kraków, Poland    QCD 2010, Montpellier, page 3
How?

Roy’s equations (Roy 1971) - twice subtracted dispersion
relations
Re fℓI (s) = a0 δI0 δℓ0 + a2 δI2 δℓ0
0            0
s − 4mπ 2                          1          1
+           2
(2a0 − 5a2 ) (δI0 δℓ0 + δI1 δℓ1 −
0      0                       δI2 δℓ0 )
12mπ                              6          2
2       1    smax
′                  ′
+                             II                           I
− ds′ Kℓℓ′ (s, s′ )Im fℓI′ (s′ ) + dℓ (s, smax )
I ′ =0 ℓ′ =0     2
4mπ
(1)
s − 4mπ         2
a0 + a2 +
0    0                     (2a0 − 5a2 ) : subtracting term (ST)
0      0
12mπ 2
′              ′                                      ′
II
Kℓℓ′ (s, s′ )Im fℓI′ (s′ ): kernel term (KT(s)) (∼ ImfℓI′ (s′ )/s′3 )
I
dℓ (s, smax ): driving term (DT(s))
′
Im fℓI′ (s′ ): INPUT

´
Robert Kaminski, IFJ PAN, Kraków, Poland    QCD 2010, Montpellier, page 3
How?

Roy’s equations (Roy 1971) - twice subtracted dispersion
relations
Re fℓI (s) = a0 δI0 δℓ0 + a2 δI2 δℓ0
0            0
s − 4mπ 2                          1          1
+           2
(2a0 − 5a2 ) (δI0 δℓ0 + δI1 δℓ1 −
0      0                       δI2 δℓ0 )
12mπ                              6          2
2       1    smax
′                  ′
+                             II                           I
− ds′ Kℓℓ′ (s, s′ )Im fℓI′ (s′ ) + dℓ (s, smax )
I ′ =0 ℓ′ =0     2
4mπ
(1)
s − 4mπ         2
a0 + a2 +
0    0                     (2a0 − 5a2 ) : subtracting term (ST)
0      0
12mπ 2
′              ′                                      ′
II
Kℓℓ′ (s, s′ )Im fℓI′ (s′ ): kernel term (KT(s)) (∼ ImfℓI′ (s′ )/s′3 )
I
dℓ (s, smax ): driving term (DT(s))
′
Im fℓI′ (s′ ): INPUT

´
Robert Kaminski, IFJ PAN, Kraków, Poland    QCD 2010, Montpellier, page 3
How?

´
once subtracted dispersion relations (Kaminski, Pelaez,
Yndurain 2008)
′
Re fℓI (s) =                 I′
st
CII ′ aI
0
2         1     smax

+                              ˜ II ′             ′
˜I
− ds′ Kℓℓ′ (s, s′ )Im fℓI′ (s′ ) + dℓ (s, smax )
I ′ =0 ℓ′ =0      2
4mπ
(2)
st I ′
I ′ CII ′ a0 : subtracting term (ST)
˜ II ′              ′
Kℓℓ′ (s, s′ )Im fℓI′ (s′ ): kernel term (KT(s))
′
(∼ ImfℓI′ (s′ )/s′2 )
˜I
dℓ (s, smax ): driving term (DT(s))
′
Im fℓI′ (s′ ): INPUT

´
Robert Kaminski, IFJ PAN, Kraków, Poland       QCD 2010, Montpellier, page 4
How?

´
once subtracted dispersion relations (Kaminski, Pelaez,
Yndurain 2008)
′
Re fℓI (s) =                 I′
st
CII ′ aI
0
2         1     smax

+                              ˜ II ′             ′
˜I
− ds′ Kℓℓ′ (s, s′ )Im fℓI′ (s′ ) + dℓ (s, smax )
I ′ =0 ℓ′ =0      2
4mπ
(2)
st I ′
I ′ CII ′ a0 : subtracting term (ST)
˜ II ′              ′
Kℓℓ′ (s, s′ )Im fℓI′ (s′ ): kernel term (KT(s))
′
(∼ ImfℓI′ (s′ )/s′2 )
˜I
dℓ (s, smax ): driving term (DT(s))
′
Im fℓI′ (s′ ): INPUT

´
Robert Kaminski, IFJ PAN, Kraków, Poland       QCD 2010, Montpellier, page 4
How?

´
once subtracted dispersion relations (Kaminski, Pelaez,
Yndurain 2008)
′
Re fℓI (s) =                 I′
st
CII ′ aI
0
2         1     smax

+                              ˜ II ′             ′
˜I
− ds′ Kℓℓ′ (s, s′ )Im fℓI′ (s′ ) + dℓ (s, smax )
I ′ =0 ℓ′ =0      2
4mπ
(2)
st I ′
I ′ CII ′ a0 : subtracting term (ST)
˜ II ′              ′
Kℓℓ′ (s, s′ )Im fℓI′ (s′ ): kernel term (KT(s))
′
(∼ ImfℓI′ (s′ )/s′2 )
˜I
dℓ (s, smax ): driving term (DT(s))
′
Im fℓI′ (s′ ): INPUT

´
Robert Kaminski, IFJ PAN, Kraków, Poland       QCD 2010, Montpellier, page 4
How?

´
once subtracted dispersion relations (Kaminski, Pelaez,
Yndurain 2008)
′
Re fℓI (s) =                 I′
st
CII ′ aI
0
2         1     smax

+                              ˜ II ′             ′
˜I
− ds′ Kℓℓ′ (s, s′ )Im fℓI′ (s′ ) + dℓ (s, smax )
I ′ =0 ℓ′ =0      2
4mπ
(2)
st I ′
I ′ CII ′ a0 : subtracting term (ST)
˜ II ′              ′
Kℓℓ′ (s, s′ )Im fℓI′ (s′ ): kernel term (KT(s))
′
(∼ ImfℓI′ (s′ )/s′2 )
˜I
dℓ (s, smax ): driving term (DT(s))
′
Im fℓI′ (s′ ): INPUT

´
Robert Kaminski, IFJ PAN, Kraków, Poland       QCD 2010, Montpellier, page 4
How?

´
once subtracted dispersion relations (Kaminski, Pelaez,
Yndurain 2008)
′
Re fℓI (s) =                 I′
st
CII ′ aI
0
2         1     smax

+                              ˜ II ′             ′
˜I
− ds′ Kℓℓ′ (s, s′ )Im fℓI′ (s′ ) + dℓ (s, smax )
I ′ =0 ℓ′ =0      2
4mπ
(2)
st I ′
I ′ CII ′ a0 : subtracting term (ST)
˜ II ′              ′
Kℓℓ′ (s, s′ )Im fℓI′ (s′ ): kernel term (KT(s))
′
(∼ ImfℓI′ (s′ )/s′2 )
˜I
dℓ (s, smax ): driving term (DT(s))
′
Im fℓI′ (s′ ): INPUT

´
Robert Kaminski, IFJ PAN, Kraków, Poland       QCD 2010, Montpellier, page 4
INPUT and OUTPUT

fℓI (s)Input = RefℓI (s)Input + iImfℓI (s)Input
RefℓI (s)Output = ST + KT (s) + DT (s)
ImfℓI (s)Input are inside of ST , KT (s) and DT (s)
How do we proceed?
2
χ2 = RefℓI (s)Output − RefℓI (s)Input                    /∆2 output
INPUT:
Set of ImfℓI (s)Input for S0, S2, P1, D0, D2, F 1 and G0
waves ﬁtted to:
a) forward dispersion relations,
b) Omnes sum rules,
c) Roy’s eqs and once subtracted dispersion relations
for the S0, S2 and P1 waves.
!!!see talk of Jacobo Ruiz de Elvira Carrascal!!!
BUT NOT for the D0, D2 and F 1 waves!

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 5
INPUT and OUTPUT

fℓI (s)Input = RefℓI (s)Input + iImfℓI (s)Input
RefℓI (s)Output = ST + KT (s) + DT (s)
ImfℓI (s)Input are inside of ST , KT (s) and DT (s)
How do we proceed?
2
χ2 = RefℓI (s)Output − RefℓI (s)Input                    /∆2 output
INPUT:
Set of ImfℓI (s)Input for S0, S2, P1, D0, D2, F 1 and G0
waves ﬁtted to:
a) forward dispersion relations,
b) Omnes sum rules,
c) Roy’s eqs and once subtracted dispersion relations
for the S0, S2 and P1 waves.
!!!see talk of Jacobo Ruiz de Elvira Carrascal!!!
BUT NOT for the D0, D2 and F 1 waves!

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 5
INPUT and OUTPUT

fℓI (s)Input = RefℓI (s)Input + iImfℓI (s)Input
RefℓI (s)Output = ST + KT (s) + DT (s)
ImfℓI (s)Input are inside of ST , KT (s) and DT (s)
How do we proceed?
2
χ2 = RefℓI (s)Output − RefℓI (s)Input                    /∆2 output
INPUT:
Set of ImfℓI (s)Input for S0, S2, P1, D0, D2, F 1 and G0
waves ﬁtted to:
a) forward dispersion relations,
b) Omnes sum rules,
c) Roy’s eqs and once subtracted dispersion relations
for the S0, S2 and P1 waves.
!!!see talk of Jacobo Ruiz de Elvira Carrascal!!!
BUT NOT for the D0, D2 and F 1 waves!

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 5
INPUT and OUTPUT

fℓI (s)Input = RefℓI (s)Input + iImfℓI (s)Input
RefℓI (s)Output = ST + KT (s) + DT (s)
ImfℓI (s)Input are inside of ST , KT (s) and DT (s)
How do we proceed?
2
χ2 = RefℓI (s)Output − RefℓI (s)Input                    /∆2 output
INPUT:
Set of ImfℓI (s)Input for S0, S2, P1, D0, D2, F 1 and G0
waves ﬁtted to:
a) forward dispersion relations,
b) Omnes sum rules,
c) Roy’s eqs and once subtracted dispersion relations
for the S0, S2 and P1 waves.
!!!see talk of Jacobo Ruiz de Elvira Carrascal!!!
BUT NOT for the D0, D2 and F 1 waves!

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 5
INPUT and OUTPUT

fℓI (s)Input = RefℓI (s)Input + iImfℓI (s)Input
RefℓI (s)Output = ST + KT (s) + DT (s)
ImfℓI (s)Input are inside of ST , KT (s) and DT (s)
How do we proceed?
2
χ2 = RefℓI (s)Output − RefℓI (s)Input                    /∆2 output
INPUT:
Set of ImfℓI (s)Input for S0, S2, P1, D0, D2, F 1 and G0
waves ﬁtted to:
a) forward dispersion relations,
b) Omnes sum rules,
c) Roy’s eqs and once subtracted dispersion relations
for the S0, S2 and P1 waves.
!!!see talk of Jacobo Ruiz de Elvira Carrascal!!!
BUT NOT for the D0, D2 and F 1 waves!

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 5
INPUT and OUTPUT

fℓI (s)Input = RefℓI (s)Input + iImfℓI (s)Input
RefℓI (s)Output = ST + KT (s) + DT (s)
ImfℓI (s)Input are inside of ST , KT (s) and DT (s)
How do we proceed?
2
χ2 = RefℓI (s)Output − RefℓI (s)Input                    /∆2 output
INPUT:
Set of ImfℓI (s)Input for S0, S2, P1, D0, D2, F 1 and G0
waves ﬁtted to:
a) forward dispersion relations,
b) Omnes sum rules,
c) Roy’s eqs and once subtracted dispersion relations
for the S0, S2 and P1 waves.
!!!see talk of Jacobo Ruiz de Elvira Carrascal!!!
BUT NOT for the D0, D2 and F 1 waves!

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 5
limits and errors of the outputs

limit for integration of lower waves (S, P and D):
smax ≈ 1.4 − 1.6 GeV - consequence of quality of the data
(we take smax = 1.42 GeV)
RefℓI (s) can be calculated up to s ≈ 1.1 GeV - convergence
at s′ = smax one has to match smoothly left amplitudes
(from phase shifts and inelasticities) and right ones (regge)

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 6
limits and errors of the outputs

limit for integration of lower waves (S, P and D):
smax ≈ 1.4 − 1.6 GeV - consequence of quality of the data
(we take smax = 1.42 GeV)
RefℓI (s) can be calculated up to s ≈ 1.1 GeV - convergence
at s′ = smax one has to match smoothly left amplitudes
(from phase shifts and inelasticities) and right ones (regge)

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 6
limits and errors of the outputs

limit for integration of lower waves (S, P and D):
smax ≈ 1.4 − 1.6 GeV - consequence of quality of the data
(we take smax = 1.42 GeV)
RefℓI (s) can be calculated up to s ≈ 1.1 GeV - convergence
at s′ = smax one has to match smoothly left amplitudes
(from phase shifts and inelasticities) and right ones (regge)

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 6
limits and errors of the outputs

limit for integration of lower waves (S, P and D):
smax ≈ 1.4 − 1.6 GeV - consequence of quality of the data
(we take smax = 1.42 GeV)
RefℓI (s) can be calculated up to s ≈ 1.1 GeV - convergence
at s′ = smax one has to match smoothly left amplitudes
(from phase shifts and inelasticities) and right ones (regge)
Errors of outputs:
are composed of errors of all inputs in the ST , KT (s) and
DT (s)
Monte Carlo with Gaussian distributions of all 52
parameters within their 3σ
analysis of Gaussian distributions of outputs for each s
(we do not take extreme values of outputs but their σ)

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 7
limits and errors of the outputs

limit for integration of lower waves (S, P and D):
smax ≈ 1.4 − 1.6 GeV - consequence of quality of the data
(we take smax = 1.42 GeV)
RefℓI (s) can be calculated up to s ≈ 1.1 GeV - convergence
at s′ = smax one has to match smoothly left amplitudes
(from phase shifts and inelasticities) and right ones (regge)
Errors of outputs:
are composed of errors of all inputs in the ST , KT (s) and
DT (s)
Monte Carlo with Gaussian distributions of all 52
parameters within their 3σ
analysis of Gaussian distributions of outputs for each s
(we do not take extreme values of outputs but their σ)

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 7
limits and errors of the outputs

limit for integration of lower waves (S, P and D):
smax ≈ 1.4 − 1.6 GeV - consequence of quality of the data
(we take smax = 1.42 GeV)
RefℓI (s) can be calculated up to s ≈ 1.1 GeV - convergence
at s′ = smax one has to match smoothly left amplitudes
(from phase shifts and inelasticities) and right ones (regge)
Errors of outputs:
are composed of errors of all inputs in the ST , KT (s) and
DT (s)
Monte Carlo with Gaussian distributions of all 52
parameters within their 3σ
analysis of Gaussian distributions of outputs for each s
(we do not take extreme values of outputs but their σ)

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 7
limits and errors of the outputs

limit for integration of lower waves (S, P and D):
smax ≈ 1.4 − 1.6 GeV - consequence of quality of the data
(we take smax = 1.42 GeV)
RefℓI (s) can be calculated up to s ≈ 1.1 GeV - convergence
at s′ = smax one has to match smoothly left amplitudes
(from phase shifts and inelasticities) and right ones (regge)
Errors of outputs:
are composed of errors of all inputs in the ST , KT (s) and
DT (s)
Monte Carlo with Gaussian distributions of all 52
parameters within their 3σ
analysis of Gaussian distributions of outputs for each s
(we do not take extreme values of outputs but their σ)

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 7
threshold behavior of output amplitudes

Threshold expansion: (here let’s keep mπ = 1)
RefℓI (s ≈ 4) = (s − 4)ℓ aI + bℓ (s − 4) + ...
ℓ
I

once subtracted dispersion relations near threshold:
Wave       Thr. exp.           ST                 KT (s)               DT (s)
D0            0                0                   0                    0
F1            0          −(a0 − 5 a2 )/8
0    2 0
A + B(s − 4)         C + D(s − 4)
D2            0                0                   0                    0

so, necessary are partial cancellations of constant terms in
the F 1-wave.

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 8
threshold behavior of output amplitudes

Threshold expansion: (here let’s keep mπ = 1)
RefℓI (s ≈ 4) = (s − 4)ℓ aI + bℓ (s − 4) + ...
ℓ
I

once subtracted dispersion relations near threshold:
Wave       Thr. exp.           ST                 KT (s)               DT (s)
D0            0                0                   0                    0
F1            0          −(a0 − 5 a2 )/8
0    2 0
A + B(s − 4)         C + D(s − 4)
D2            0                0                   0                    0

so, necessary are partial cancellations of constant terms in
the F 1-wave.

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 8
threshold behavior of output amplitudes

Threshold expansion: (here let’s keep mπ = 1)
RefℓI (s ≈ 4) = (s − 4)ℓ aI + bℓ (s − 4) + ...
ℓ
I

once subtracted dispersion relations near threshold:
Wave       Thr. exp.           ST                 KT (s)               DT (s)
D0            0                0                   0                    0
F1            0          −(a0 − 5 a2 )/8
0    2 0
A + B(s − 4)         C + D(s − 4)
D2            0                0                   0                    0

so, necessary are partial cancellations of constant terms in
the F 1-wave.

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 8
ππ D0 wave

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 9
ππ F 1 wave

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 10
ππ D2 wave

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 11
ﬁrst results for the D0, D2 and F 1 waves

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 12
Conclusions

in our hands we have promising tool for testing the ππ
amplitudes in the D0, D2 and F 1 waves,

together with similar relations with two subtractions
(G. Colangelo and H. Leutwyller) it will create similar set or
equations as is for the S0, S2 and P1 waves,

let’s try to ﬁt the D and F wave amplitudes to these new
equations

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 13
Conclusions

in our hands we have promising tool for testing the ππ
amplitudes in the D0, D2 and F 1 waves,

together with similar relations with two subtractions
(G. Colangelo and H. Leutwyller) it will create similar set or
equations as is for the S0, S2 and P1 waves,

let’s try to ﬁt the D and F wave amplitudes to these new
equations

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 13
Conclusions

in our hands we have promising tool for testing the ππ
amplitudes in the D0, D2 and F 1 waves,

together with similar relations with two subtractions
(G. Colangelo and H. Leutwyller) it will create similar set or
equations as is for the S0, S2 and P1 waves,

let’s try to ﬁt the D and F wave amplitudes to these new
equations

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 13
Conclusions

in our hands we have promising tool for testing the ππ
amplitudes in the D0, D2 and F 1 waves,

together with similar relations with two subtractions
(G. Colangelo and H. Leutwyller) it will create similar set or
equations as is for the S0, S2 and P1 waves,

let’s try to ﬁt the D and F wave amplitudes to these new
equations

Thanks to Stephan for organizing of this conference
and for all
Merci et a bientot!

´
Robert Kaminski, IFJ PAN, Kraków, Poland   QCD 2010, Montpellier, page 14

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