Peng Guo !

Indiana Univ.-Bloomington &

JLab Physics Analysis Center

Three-body final state interaction and its applications

!1

1

3

2

Extracting properties of resonances in multi-hadron final state (PWA): spin, mass, decay width and coupling constant

Mission: Hadron Spectroscopy

!2

1

3

2

Extracting properties of resonances in multi-hadron final state (PWA): spin, mass, decay width and coupling constant

Mission: Hadron Spectroscopy

!2

1

3

2

Extracting properties of resonances in multi-hadron final state (PWA): spin, mass, decay width and coupling constant

Mission: Hadron Spectroscopy

!2

• Isobar Model: quasi two-body decays

1

3

2

Extracting properties of resonances in multi-hadron final state (PWA): spin, mass, decay width and coupling constant

Mission: Hadron Spectroscopy

!2

• Isobar Model: quasi two-body decays

1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)

Extracting properties of resonances in multi-hadron final state (PWA): spin, mass, decay width and coupling constant

Mission: Hadron Spectroscopy

!2

• Isobar Model: quasi two-body decays

1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)

Extracting properties of resonances in multi-hadron final state (PWA): spin, mass, decay width and coupling constant

Mission: Hadron Spectroscopy

!2

• Isobar Model: quasi two-body decays

1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)

mpippi020.08 0.1 0.12 0.14 0.16 0.18

mpi

ppim

2

0.08

0.1

0.12

0.14

0.16

0.18

0.2

mpippim2:mpippi02 {mpippim2>0&&mpippi02>0.06}

Extracting properties of resonances in multi-hadron final state (PWA): spin, mass, decay width and coupling constant

Mission: Hadron Spectroscopy

!2

• Isobar Model: quasi two-body decays

1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)

mpippi020.08 0.1 0.12 0.14 0.16 0.18

mpi

ppim

2

0.08

0.1

0.12

0.14

0.16

0.18

0.2

mpippim2:mpippi02 {mpippim2>0&&mpippi02>0.06}

?

Extracting properties of resonances in multi-hadron final state (PWA): spin, mass, decay width and coupling constant

Mission: Hadron Spectroscopy

!2

• Isobar Model: quasi two-body decays

1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)

• What are we missing?

mpippi020.08 0.1 0.12 0.14 0.16 0.18

mpi

ppim

2

0.08

0.1

0.12

0.14

0.16

0.18

0.2

mpippim2:mpippi02 {mpippim2>0&&mpippi02>0.06}

?

Extracting properties of resonances in multi-hadron final state (PWA): spin, mass, decay width and coupling constant

Mission: Hadron Spectroscopy

!2

• Isobar Model: quasi two-body decays

1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)

• What are we missing?

Isobar Model: quasi two-body decays

How to extract coupling to resonancesHow one establishes new resonances

Answer?It’s important to construct amplitudes which contain all the known physics

1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)

Tuesday, January 26, 2010

mpippi020.08 0.1 0.12 0.14 0.16 0.18

mpi

ppim

2

0.08

0.1

0.12

0.14

0.16

0.18

0.2

mpippim2:mpippi02 {mpippim2>0&&mpippi02>0.06}

?

Extracting properties of resonances in multi-hadron final state (PWA): spin, mass, decay width and coupling constant

Mission: Hadron Spectroscopy

!2

• Isobar Model: quasi two-body decays

1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)

• What are we missing?

Isobar Model: quasi two-body decays

How to extract coupling to resonancesHow one establishes new resonances

Answer?It’s important to construct amplitudes which contain all the known physics

1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)

Tuesday, January 26, 2010

mpippi020.08 0.1 0.12 0.14 0.16 0.18

mpi

ppim

2

0.08

0.1

0.12

0.14

0.16

0.18

0.2

mpippim2:mpippi02 {mpippim2>0&&mpippi02>0.06}

?

Extracting properties of resonances in multi-hadron final state (PWA): spin, mass, decay width and coupling constant

Mission: Hadron Spectroscopy

!2

• Isobar Model: quasi two-body decays

1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)

• What are we missing?

Isobar Model: quasi two-body decays

How to extract coupling to resonancesHow one establishes new resonances

Answer?It’s important to construct amplitudes which contain all the known physics

1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)

Tuesday, January 26, 2010

mpippi020.08 0.1 0.12 0.14 0.16 0.18

mpi

ppim

2

0.08

0.1

0.12

0.14

0.16

0.18

0.2

mpippim2:mpippi02 {mpippim2>0&&mpippi02>0.06}

?

Extracting properties of resonances in multi-hadron final state (PWA): spin, mass, decay width and coupling constant

Unitarity &

Analyticity

Mission: Hadron Spectroscopy

!2

Is three-body interaction important?

!3

Is three-body interaction important?

!3

Dalitz plot can be described by distribution function:

Is three-body interaction important?

!3

Dalitz plot can be described by distribution function:

Is three-body interaction important?

!3

Dalitz plot can be described by distribution function:

Is three-body interaction important?

!3

Dalitz plot can be described by distribution function:

Is three-body interaction important?

!3

Theory

Experiment

S.P.Schneider, B.Kubis and C. Ditsche, JHEP02(2011)028

Dalitz plot can be described by distribution function:

Is three-body interaction important?

!3

Theory

Experiment

S.P.Schneider, B.Kubis and C. Ditsche, JHEP02(2011)028

Dalitz plot can be described by distribution function:

Is three-body interaction important?

!3

Theory

Experiment

S.P.Schneider, B.Kubis and C. Ditsche, JHEP02(2011)028

Dalitz plot can be described by distribution function:

Is three-body interaction important? Yes

!3

One of Missions at JPAC

• Build in three-body interaction (unitarity&analyticity) for future experimental analysis tools

• Ongoing projects at JPAC: eta, omega to 3 pions

Isobar Model: quasi two-body decays

How to extract coupling to resonancesHow one establishes new resonances

Answer?It’s important to construct amplitudes which contain all the known physics

1

3

2

1

2

3

+1

2

3

=(23)

(31)

1

2

3

+

(12)

Tuesday, January 26, 2010

!4

A concrete example: assuming only isoscalar S-wave contribution

Building integral equations for three-body

!5

A concrete example: assuming only isoscalar S-wave contribution

Building integral equations for three-body

!5

A concrete example: assuming only isoscalar S-wave contribution

Building integral equations for three-body

+

−

0

g0f(s1 2)

!5

A concrete example: assuming only isoscalar S-wave contribution

Building integral equations for three-body

+

−

0

g0f(s1 2)

!5

A concrete example: assuming only isoscalar S-wave contribution

Building integral equations for three-body

+

−

0

g0f(s1 2)

+

−

0

T(s1 2)f *(s1 2)

+

+

−

0

0

0

f*(s1 2)

T(s2 3)+T(s3 1)

Disc1 2T(s1 2) =

Unitarity & Analyticity

!5

A concrete example: assuming only isoscalar S-wave contribution

Building integral equations for three-body

+

−

0

g0f(s1 2)

+

−

0

T(s1 2)f *(s1 2)

+

+

−

0

0

0

f*(s1 2)

T(s2 3)+T(s3 1)

Disc1 2T(s1 2) =

Unitarity & Analyticity

!5

Building integral equations for three-body

!6

Building integral equations for three-body

sth

s

!6

Building integral equations for three-body

sth

s

!6

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1sqrt(s) (GeV)

ReT3bisobar

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1sqrt(s) (GeV)

ImT3bisobar

!7

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1sqrt(s) (GeV)

ReT3bisobar

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1sqrt(s) (GeV)

ImT3bisobar

!7

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1sqrt(s) (GeV)

ReT3bisobar

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1sqrt(s) (GeV)

ImT3bisobar

+

−

0

g0f(s1 2)

!7

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1sqrt(s) (GeV)

ReT3bisobar

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1sqrt(s) (GeV)

ImT3bisobar

+

−

0

g0f(s1 2)

!7

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1sqrt(s) (GeV)

ReT3bisobar

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1sqrt(s) (GeV)

ImT3bisobar

+

−

0

g0f(s1 2) +

−

0

T(s1 2)f *(s1 2)

+

+

−

0

0

0

f*(s1 2)

T(s2 3)+T(s3 1)

Disc1 2T(s1 2) =

!7

Current status and outlook• A test version of eta to 3 pions package (with 3b, Isospin-I and

partial wave-L up to 2) has been passed onto experimental colleagues (Dennis Weygand, Diane Schott)

• Omega to 3 pions is still on working progress (Igor Danilkin, Peng)

• Coupled channel effects will be incorporated into our framework

B. Discontinuity for P-Wave

if we only consider P-wave, we have

Disc12T3�11 (s, s12) (39)

=1

128⇥2

⇥

1� 4m2�

s12M�11

2�⇥2�(s12)T3�11 (s, s12)

+ 21

128⇥2

⇥

1� 4m2�

s12

�d cos ��

3

3

2d1

1,0(��3 )d1

1,0(��1 )M�11

2�⇥2�(s12)T3�11 (s, s⇤23)

+1

64⇥2

⇥

1� 4m2K

s12M�11

KK̄⇥��(s12)T(K+K�)�11 (s, s12)

� 21

64⇥2

⇥

1� 4m2K

s12

�d cos �K

3

3

2d1

1,0(�K3 )d1

1,0(�K1 )M�11

KK̄⇥��(s12)TK+(K��)11 (s, s⇤23)

and

0

−

++

− +

0

−

+

+

−

0

−

+

−

0

K +

K −

+

+

0

−

+

0

K −

K +

K −

Disc(12)T(K+K�)�11 (s, s12) (40)

=1

64⇥2

⇥

1� 4m2K

s12M�11

KK̄⇥KK̄(s12)T(K+K�)�11 (s, s12)

� 21

64⇥2

⇥

1� 4m2K

s12

�d cos �K

3

3

2d1

1,0(�K3 )d1

1,0(�K1 )M�11

KK̄⇥KK̄(s12)TK+(K��)11 (s, s⇤23)

+1

128⇥2

⇥

1� 4m2�

s12M�11

2�⇥KK̄T 3�11 (s, s12)

+ 21

128⇥2

⇥

1� 4m2�

s12

�d cos ��

3

3

2d1

1,0(��3 )d1

1,0(��1 )M�11

2�⇥KK̄(s12)T3�11 (s, s⇤23)

31

!8