3-BODY QUANTIZATION CONDITION IN UNITARY FORMALISM [Eur.Phys.J. A53 - - PowerPoint PPT Presentation

SLIDE 1

3-BODY QUANTIZATION CONDITION IN UNITARY FORMALISM

M a x i m M a i T h e G e

• r

g e W a s h i n g t

• n

U n i v e r s i t y

[Eur.Phys.J. A53 (2017) no.9] [Eur.Phys.J. A53 (2017) no.12] [Phys.Rev. D97 (2018) no.11] [arXiv:1807.04746]

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• Many unsolved questions of QCD involve 3-body channels
• Roper-puzzle & π

π N channel

• a

1

( 1 2 6 ) ↔ π ρ / π σ channels ↔ spectroscopy spin-exotics

• X

( 3 8 7 2 ) etc..

π π π π

ρ

π π π π

σ

a1

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Ma x i m Ma i ( G WU )

3

• Many unsolved questions of QCD involve 3-body channels
• Roper-puzzle & π

π N channel

• a

1

( 1 2 6 ) ↔ π ρ / π σ channels ↔ spectroscopy spin-exotics

• X

( 3 8 7 2 ) etc..

• Best theoretical tool: Lattice QCD → some (preliminary) studies:
• π

π N & a

1

( 1 2 6 )

• π

ρ I = 2

• more is under way...

π π π π

ρ

π π π π

σ

a1

Lang et al.(2014)Lang et al.(2016) [I=2,πρ] Woss et al. (2017)

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this talk: Q U A N T I Z A T I O N C O N D I T I O N F O R 3

• B

O D Y S Y S T E M S

• Many unsolved questions of QCD involve 3-body channels
• Roper-puzzle & π

π N channel

• a

1

( 1 2 6 ) ↔ π ρ / π σ channels ↔ spectroscopy spin-exotics

• X

( 3 8 7 2 ) etc..

• Best theoretical tool: Lattice QCD → some (preliminary) studies:
• π

π N & a

1

( 1 2 6 )

• π

ρ I = 2

• more is under way...
• However,

Lattice spectrum is discretized → m a p p i n g t

• i

n fi n i t e v

• l

u m e spectrum

π π π π

ρ

π π π π

σ

a1

Lang et al.(2014)Lang et al.(2016) [I=2,πρ] Woss et al. (2017)

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2-body case

• o

n e

• t
• n

e m a p p i n g

• Various extensions: multi-channels, spin, ...

Lüscher(1986) Gottlieb,Rummukainen,Feng,Meißner, Li,Liu,Doring,Briceno,Rusetsky,Bernard…

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2-body case

• o

n e

• t
• n

e m a p p i n g

• Various extensions: multi-channels, spin, ...

3-body case

• p

r e s u m a b l y no o n e

• t
• n

e m a p p i n g → complex kinematics (8 variables) → sub-channel dynamics

• important theoretical developments and p

i l

• t

numerical investigation

• First data driven study of the volume spectrum

→ (π

+

π

+

) and (π

+

π

+

π

+

) systems → comparison with Lattice QCD results

Lüscher(1986) Gottlieb,Rummukainen,Feng,Meißner, Li,Liu,Doring,Briceno,Rusetsky,Bernard… Sharpe,Hansen,Briceno,Hammer,Rusetsky,Polejaeva,Griesshammer,Davoudi,Guo… MM/Doring(2017) Pang/Hammer/Rusetsky/Wu(2017) Hansen/Briceno/Sharpe(2018) Doring/Hammer/MM/Pang/Rusetsky/Wu (2018)

MM/Doring (2018) > this talk <

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1) T is a sum of a dis/connected parts

UNITARY ISOBAR INF.-VOL. AMPLITUDE

Eur.Phys.J. A53 MM et al. (2017)

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1) T is a sum of a dis/connected parts 2) Disconnected part = spectator + tower of “i s

• b

a r s ”

➢ f

u n c t i

• n

s w i t h c

• r

r e c t r i g h t

• h

a n d

• s

i n g u l a r i t i e s f

• r

e a c h Q N τ ( M

i n v

)

➢ c

• u

p l i n g t

• a

s y m p t

• t

i c s t a t e s : c u t

• f

r e e

• f

u n c t i

• n

v ( q , p )

UNITARY ISOBAR INF.-VOL. AMPLITUDE

Eur.Phys.J. A53 MM et al. (2017)

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1) T is a sum of a dis/connected parts 2) Disconnected part = spectator + tower of “i s

• b

a r s ”

➢ f

u n c t i

• n

s w i t h c

• r

r e c t r i g h t

• h

a n d

• s

i n g u l a r i t i e s f

• r

e a c h Q N τ ( M

i n v

)

➢ c

• u

p l i n g t

• a

s y m p t

• t

i c s t a t e s : c u t

• f

r e e

• f

u n c t i

• n

v ( q , p ) 3) Connected part = general 4d BSE-like equation w.r.t kernel B ( p , q ; s )

UNITARY ISOBAR INF.-VOL. AMPLITUDE

Eur.Phys.J. A53 MM et al. (2017)

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→ relativistic 3d integral-equation → useful for phenomenological applications → unknowns: v , C , m 1) T is a sum of a dis/connected parts 2) Disconnected part = spectator + tower of “i s

• b

a r s ”

➢ f

u n c t i

• n

s w i t h c

• r

r e c t r i g h t

• h

a n d

• s

i n g u l a r i t i e s f

• r

e a c h Q N τ ( M

i n v

)

➢ c

• u

p l i n g t

• a

s y m p t

• t

i c s t a t e s : c u t

• f

r e e

• f

u n c t i

• n

v ( q , p ) 3) Connected part = general 4d BSE-like equation w.r.t kernel B ( p , q ; s ) 4) 2- and 3-body unitarity constrains B , τ

UNITARY ISOBAR INF.-VOL. AMPLITUDE

Eur.Phys.J. A53 MM et al. (2017)

+ + +...

τ

• 1

= B = C

1 / m v v v v

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• Power-law fjnite-volume efgects

↔ on-shell confjgurations in T ↔ I m T ↔ Unitarity is crucial

• Replace integrals by sums:

{ E * | T

• 1

( E * ) = } = { E n . E i g e n v a l u e s i n a b

• x

} ⚠ B is NOT regular → projection to irreps essential

3-BODY QUANTIZATION CONDITION

Eur.Phys.J. A53 MM/Doring(2017)

some useful techniques: Doring/Hammer/MM/… (2018)

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• Power-law fjnite-volume efgects

↔ on-shell confjgurations in T ↔ I m T ↔ Unitarity is crucial

• Replace integrals by sums:

{ E * | T

• 1

( E * ) = } = { E n . E i g e n v a l u e s i n a b

• x

} ⚠ B is NOT regular → projection to irreps essential

➢ Final result in terms of shells s(/) and basis vector index u(/)

3-BODY QUANTIZATION CONDITION

Eur.Phys.J. A53 MM/Doring(2017)

some useful techniques: Doring/Hammer/MM/… (2018)

W – total energy ϑ – multiplicity L – lattice size Es – 1p. energy

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• Power-law fjnite-volume efgects

↔ on-shell confjgurations in T ↔ I m T ↔ Unitarity is crucial

• Replace integrals by sums:

{ E * | T

• 1

( E * ) = } = { E n . E i g e n v a l u e s i n a b

• x

} ⚠ B is NOT regular → projection to irreps essential

➢ Final result in terms of shells s(/) and basis vector index u(/)

• Possible work-fmow:

1)Fix & to 2-body channel (Lattice or Exp. data) 2)Fix C

in to 3-body data (Lattice or Exp. data)

3-BODY QUANTIZATION CONDITION

Eur.Phys.J. A53 MM/Doring(2017)

+

+ +... τ

• 1

= C

B =

some useful techniques: Doring/Hammer/MM/… (2018)

W – total energy ϑ – multiplicity L – lattice size Es – 1p. energy

v =

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PHYSICAL APPLICATION

• Interesting system: π

+

π

+

π

+

➢ LatticeQCD results for ground level available for π

+

π

+

& π

+

π

+

π

+

➢ Repulsive channel →

Q : d

• e

s t h e “ i s

• b

a r ” p i c t u r e h

• l

d ?

➢ L

= 2 . 5 f m , m

π

= 2 9 1 / 3 5 2 / 4 9 1 / 5 9 1 M e V → B

• n

u s Q : c h i r a l e x t r a p

• l

a t i

• n

i n 3 b

• d

y s y s t e m ?

Detmold et al.(2008) arXiv:1807.04746 MM/Doring(2018)

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PHYSICAL APPLICATION

• I. 2-body subchannel:

➢ one-channel problem: π

π

• system in S-wave, I=2

➢ 2-body amplitude consistent with 3-body one

• Interesting system: π

+

π

+

π

+

➢ LatticeQCD results for ground level available for π

+

π

+

& π

+

π

+

π

+

➢ Repulsive channel →

Q : d

• e

s t h e “ i s

• b

a r ” p i c t u r e h

• l

d ?

➢ L

= 2 . 5 f m , m

π

= 2 9 1 / 3 5 2 / 4 9 1 / 5 9 1 M e V → B

• n

u s Q : c h i r a l e x t r a p

• l

a t i

• n

i n 3 b

• d

y s y s t e m ?

Detmold et al.(2008) arXiv:1807.04746 MM/Doring(2018)

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PHYSICAL APPLICATION

• I. 2-body subchannel:

➢ one-channel problem: π

π

• system in S-wave, I=2

➢ 2-body amplitude consistent with 3-body one

1) Fix λ , M to exp. data

☹ incoorrect m

π

behavior!

ChPT @ NLO K-mat @ LO IAM Isobar: λ=const. Isobar: IAM

400 600 800 1000

• 100
• 80
• 60
• 40
• 20

σ [MeV] δ [°]

• Interesting system: π

+

π

+

π

+

➢ LatticeQCD results for ground level available for π

+

π

+

& π

+

π

+

π

+

➢ Repulsive channel →

Q : d

• e

s t h e “ i s

• b

a r ” p i c t u r e h

• l

d ?

➢ L

= 2 . 5 f m , m

π

= 2 9 1 / 3 5 2 / 4 9 1 / 5 9 1 M e V → B

• n

u s Q : c h i r a l e x t r a p

• l

a t i

• n

i n 3 b

• d

y s y s t e m ?

Detmold et al.(2008) arXiv:1807.04746 MM/Doring(2018)

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PHYSICAL APPLICATION

• I. 2-body subchannel:

➢ one-channel problem: π

π

• system in S-wave, I=2

➢ 2-body amplitude consistent with 3-body one

1) Fix λ , M to exp. data

☹ incoorrect m

π

behavior! 2) Chiral NLO & K-matrix

☹ works badly for high energies

Gasser/Leutwyler(1984)

ChPT @ NLO K-mat @ LO IAM Isobar: λ=const. Isobar: IAM

400 600 800 1000

• 100
• 80
• 60
• 40
• 20

σ [MeV] δ [°]

• Interesting system: π

+

π

+

π

+

➢ LatticeQCD results for ground level available for π

+

π

+

& π

+

π

+

π

+

➢ Repulsive channel →

Q : d

• e

s t h e “ i s

• b

a r ” p i c t u r e h

• l

d ?

➢ L

= 2 . 5 f m , m

π

= 2 9 1 / 3 5 2 / 4 9 1 / 5 9 1 M e V → B

• n

u s Q : c h i r a l e x t r a p

• l

a t i

• n

i n 3 b

• d

y s y s t e m ?

Detmold et al.(2008) arXiv:1807.04746 MM/Doring(2018)

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PHYSICAL APPLICATION

• I. 2-body subchannel:

➢ one-channel problem: π

π

• system in S-wave, I=2

➢ 2-body amplitude consistent with 3-body one

Truong(1988)

1) Fix λ , M to exp. data

☹ incoorrect m

π

behavior! 2) Chiral NLO & K-matrix

☹ works badly for high energies

3) Inverse Amplitude ☺ correct σ & m

π

behavior

☺ parameters known

Gasser/Leutwyler(1984)

ChPT @ NLO K-mat @ LO IAM Isobar: λ=const. Isobar: IAM

400 600 800 1000

• 100
• 80
• 60
• 40
• 20

σ [MeV] δ [°]

• Interesting system: π

+

π

+

π

+

➢ LatticeQCD results for ground level available for π

+

π

+

& π

+

π

+

π

+

➢ Repulsive channel →

Q : d

• e

s t h e “ i s

• b

a r ” p i c t u r e h

• l

d ?

➢ L

= 2 . 5 f m , m

π

= 2 9 1 / 3 5 2 / 4 9 1 / 5 9 1 M e V → B

• n

u s Q : c h i r a l e x t r a p

• l

a t i

• n

i n 3 b

• d

y s y s t e m ?

Detmold et al.(2008) arXiv:1807.04746 MM/Doring(2018)

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PHYSICAL APPLICATION

• I. 2-body subchannel:

➢ one-channel problem: π

π

• system in S-wave, I=2

➢ 2-body amplitude consistent with 3-body one ChPT @ NLO K-mat @ LO IAM Isobar: λ=const. Isobar: IAM

400 600 800 1000

• 100
• 80
• 60
• 40
• 20

σ [MeV] δ [°]

discretize (Lüscher) → predicted fjn-vol. spectrum

• Interesting system: π

+

π

+

π

+

➢ LatticeQCD results for ground level available for π

+

π

+

& π

+

π

+

π

+

➢ Repulsive channel →

Q : d

• e

s t h e “ i s

• b

a r ” p i c t u r e h

• l

d ?

➢ L

= 2 . 5 f m , m

π

= 2 9 1 / 3 5 2 / 4 9 1 / 5 9 1 M e V → B

• n

u s Q : c h i r a l e x t r a p

• l

a t i

• n

i n 3 b

• d

y s y s t e m ?

Detmold et al.(2008) arXiv:1807.04746 MM/Doring(2018)

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PHYSICAL APPLICATION

arXiv:1807.04746 MM/Doring(2018)

• II. 3-body spectrum

Remaining unknown: C

➢ g

e n u i n e ( m

• m

e n t a

• d

e p e n d e n t ) 3

• b
• d

y “ f

• r

c e ”

➢ s

i m p l e s t c a s e :

Detmold et al.(2008)

QUANTIZATION CONDITION

C

• Interesting system: π

+

π

+

π

+

➢ LatticeQCD results for ground level available for π

+

π

+

& π

+

π

+

π

+

➢ Repulsive channel →

Q : d

• e

s t h e “ i s

• b

a r ” p i c t u r e h

• l

d ?

➢ L

= 2 . 5 f m , m

π

= 2 9 1 / 3 5 2 / 4 9 1 / 5 9 1 M e V → B

• n

u s Q : c h i r a l e x t r a p

• l

a t i

• n

i n 3 b

• d

y s y s t e m ?

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PHYSICAL APPLICATION

arXiv:1807.04746 MM/Doring(2018)

• II. 3-body spectrum

Remaining unknown: C

➢ g

e n u i n e ( m

• m

e n t a

• d

e p e n d e n t ) 3

• b
• d

y “ f

• r

c e ”

➢ s

i m p l e s t c a s e :

Detmold et al.(2008)

QUANTIZATION CONDITION

C

Fit C to NPLQCD ground state level → C = . 2 ± 1 . 5 · 1

− 1

• Interesting system: π

+

π

+

π

+

➢ LatticeQCD results for ground level available for π

+

π

+

& π

+

π

+

π

+

➢ Repulsive channel →

Q : d

• e

s t h e “ i s

• b

a r ” p i c t u r e h

• l

d ?

➢ L

= 2 . 5 f m , m

π

= 2 9 1 / 3 5 2 / 4 9 1 / 5 9 1 M e V → B

• n

u s Q : c h i r a l e x t r a p

• l

a t i

• n

i n 3 b

• d

y s y s t e m ?

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PHYSICAL APPLICATION

arXiv:1807.04746 MM/Doring(2018)

• II. 3-body spectrum

Remaining unknown: C

➢ g

e n u i n e ( m

• m

e n t a

• d

e p e n d e n t ) 3

• b
• d

y “ f

• r

c e ”

➢ s

i m p l e s t c a s e :

Detmold et al.(2008)

QUANTIZATION CONDITION

C

Predict exited spectrum: → novel pattern 1/1 of interacting/non-interacting lvls → all QC-poles are simple → chiral extrapolation to phys point

• Interesting system: π

+

π

+

π

+

➢ LatticeQCD results for ground level available for π

+

π

+

& π

+

π

+

π

+

➢ Repulsive channel →

Q : d

• e

s t h e “ i s

• b

a r ” p i c t u r e h

• l

d ?

➢ L

= 2 . 5 f m , m

π

= 2 9 1 / 3 5 2 / 4 9 1 / 5 9 1 M e V → B

• n

u s Q : c h i r a l e x t r a p

• l

a t i

• n

i n 3 b

• d

y s y s t e m ?

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“Three-body Unitarity with Isobars Revisited”

➢ P

a r a m e t r i z a t i

• n

v i a 2

• b
• d

y s u b

• c

h a n n e l a m p l i t u d e s ( “ i s

• b

a r s ” )

➢ R

e l a t i v i s t i c i n t e g r a l e q u a t i

• n

➢ P

h e n

• m

e n

• l
• g

i c a l a p p l i c a t i

• n

s i n p r

• g

r e s s …

“Three-body Unitarity in the Finite Volume”

➢ D

i s c r e t i z a t i

• n

& P r

• j

e c t i

• n

t

• i

r r e p s

• f

O

h

l e a d s t

• 3

b

• d

y Q C

➢ N

u m e r i c a l t

• y
• e

x a m p l e s e x p l

• r

e d

➢ E

x t e n s i

• n

t

• m

u l t i

• c

h a n n e l s i n p r

• g

r e s s …

“Finite-volume spectrum of π

+

π

+

and π

+

π

+

π

+

systems”

➢ (

e x c i t e d s p e c t r u m )

• f

π

+

π

+

& π

+

π

+

π

+

s y s t e m s p r e d i c t e d f r

• m

3 b Q C

➢ g

r

• u

n d l e v e l c

• m

p a r e d w i t h N P L Q C D r e s u l t s

➢ 3

• b
• d

y fi n

• v
• l

. s p e c t r u m f e a t u r e s e x p l

• r

e d

➢ P

r e d i c t i

• n

s a t p h y s . p i

• n

m a s s

➢ O

u t l

• k

: N * ( 1 4 4 ) , …

[arXiv:1807.04746]

[Eur.Phys.J. A53 (2017) no.9, 177] [Phys.Rev. D97 (2018) no.11] [Eur.Phys.J. A53 (2017) no.9]

“LIBERATION/CAPTURING OF 3 BIRDS (PARTICLES)”= SUMMARY

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BACKUP

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χ@291

2

χ@352

2

χ@491

2

χ@591

2

χall

2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.2 0.4 0.6 0.8 1.0 1.2 C [1] χdof

2

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