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A study for “elementarity” of composite systems

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### A study for “elementarity” of composite systems

“Structure and production of charmed baryons II”

2014, Aug. 7-9, J-PARC, Tokai

Hideko Nagahiro1,2, Atsushi Hosaka2

1 Nara Women’s University, Japan

2RCNP, Osaka University, Japan

References:

H. Nagahiro, and A. Hosaka, e-print arXiv:1406.3684 [hep-ph]

H. Nagahiro, and A. Hosaka, PRC88(2013)055203 (published as an Editors’ Suggestion)

Introduction & motivation

Many candidates for exotic hadrons : not simple (or )state

, (980)/(980), …, (1260), …, (1405), (1535), …, (3872), …

vs.

“elementary” particle (“quasi-particle”)

Dynamically generated resonance

(~ can be or …)

Nature of possible exotic hadrons

Physical state must be a mixture of possible quantum states

+ …

physical state

The question :

How much they contain “elementary” components ?

“compositeness” or “elementarity”

bare, un-renormalized, field vanishesfor a bound state

Weinberg, PR130(63)776

Lurie-Macfarlane, PR136(65)B816

Weinberg, PR137(65)B682

Hyodo-Jido-Hosaka, PRC(12)015201

...

the wave function renormalization Z

“compositeness condition for a bound state

probability of finding the elementary particle

:

…

…

+

+

+

+

+

“quasi-particle” of infinite masswith

Bound state

- Weinberg, PR130(63)776
- a bound state can be represented by introducing a “quasi-particle”
- with infinite bare mass and hence Z = 0

- Lurie-Macfarlane, PR136(65)B816
- equivalence between a four-Fermi theory and a Yukawa theory the renormalization constant Z for a Yukawa particle is equal to zero

- Weinberg, PR136(65)B816
- Z < 0.2 for deuteron system

for a resonant state ??

Discussions given for bound states

Hyodo-Jido-Hosaka, PRC85(12)015201

Contents of this talk

- Wave function renormalization constant ( “elementarity”) is zero for any resonant or bound statedynamically generated by WT type interaction
- How in a Yukawa model we can introduce the “fictitious” elementary particle which is equivalent to the s-wave dynamical state by
- wave function renormalization constant for the fictitious particle
- Underlying mechanism of

- Model (cut-off & representation) dependence of
- Choice of “elementary particle” as a measure
- How we should employ the constant to understand the hadron natures.

- A special case of zero-energy bound state
- Underlying mechanism for can be different from others

D. Lurie, A.J.Macfarlane, PR136(64)B816

D. Lurie, Particle and Fields, 1968

A brief review of “compositeness condition”Yukawa theory with constant

Bound state (four-point) model

…

+

=

+

+ …

+

+

=

=

=

wave function renormalization

Weinberg also uses this eq. by estimating from low energy p-n scattering.

Equivalent Yukawa model to a resonant model?

Yukawa theory with constant

Resonance case (composite model)

Bound state model

…

+

=

+

+ …

+

+

=

?

=

=

wave function renormalization

For an -wave composite states

Interaction kernel : Weinberg-Tomozawa type

energy-dependent

[1] Olle-Oset, NPA620(97)438

scattering amplitude with on-shell factorization[1]

composite pole

+

=

+

+ …

regularize appropriately

loop function

by dim. regularization / 3dim cut-off

bound state case

(constant )

(physical) coupling

Equivalent Yukawa model

shifted amplitude

scattering amplitude with on-shell factorization[1]

composite pole

+

=

+

+ …

Equivalent Yukawa model

shifted amplitude

Yukawa term

Yukawa term

bare mass of the fictitious elementary particle

(cf. Hyodo08)

“fictitious” particle

Energy-dependent Yukawa coupling

Equivalent Yukawa model

Composite model

Yukawa model

…

+

+

+ …

+

+

self-energy

full propagator of the fictitious elementary particle

How about ?

bound state case

wave function renormalization constant

due to energy-dependence of

Wave function renormalization constant

wave function renormalization constant

due to energy-dependence of

Wave function renormalization constant

zero !

renormalized coupling

finite

bare mass of the fictitious elementary particle

wave function renormalization constant

infinite

bare coupling

infinite

due to energy-dependence of

The condition

Composite model

Yukawa model

- The composite states can be equivalently represented by a “quasi-particle” with infinite bare massand hence with [Weinberg(63)]
- The “elementarity” is zero for any composite state by WT term

…

+

+

+

+

+ …

hadronic scale

[1] Jido-Oller-Oset-Ramos-Meissner, NPA725(03)181.

[2] Inoue-Oset-Vicente Vacas, PRC65(02)035204.

[3] Hyodo-Jido-Hosaka, PRC78(08)025203.

chiral unitary approach

un-natural

(1405)… bound state[1]

(1535) … bound state[2](but large ? [3])

in the composite model = in the Yukawa model

our assumption

With an explicit pole term

Interaction kernel : Weinberg-Tomozawa type + explicit pole term

Introduced by an

un-natural cut-off

[Hyodo (08)]

Introduced by an

un-natural cut-off

[Hyodo (08)]

bare mass of “fictitious” particle

Equivalent Yukawa term

renormalized coupling

finite

bare coupling

finite

If there is an explicit pole term,

“elementarity” Z is finite.

Wave function renormalization constant

finite

With an explicit pole term

Interaction kernel : Weinberg-Tomozawa type + explicit pole term

Introduced by an

un-natural cut-off

[Hyodo (08)]

Scattering amplitude

Arbitrariness of “elementarity”

- Physical observables are invariant under the simultaneous change in and .
- Multiple interpretations for a physical state
- Z can be any value and cannot be determined in a model-independent manner.

Necessary to specify a model

(cut-off scale to be used as a “measure”)

Representation dependence of

cf.) scattering in sigma model

Yukawa model (fictitious particle or “quasi-particle”)

“quasi-particle” dominates

practically zeroin

Nonlinear model

+

Linear model

+

- They all have the same , but is different

Representation dependence of

92.4 MeV

= 138 MeV

Linear model

0

1.4

1.2

Yukawa

nonlinear

1.0

0.8

0

Re

0.6

0.4

0.2

- the pole positions are the same for all representations
- Each indicates the “elementarity” measured by different elementary particle: the elementary particle in different models are different

0

0.2

1500

2000

500

1000

2500

3000

bare mass [MeV]

Necessary to specify a model ( = representation)

Another mechanism of for zero energy bound state

for

finite

finite

0

finite

renormalized coupling

wave function renormalization constant or “elementarity”

Derivative of the loop function

for with

Another mechanism of for zero energy bound state

Example :

Interaction kernel : Weinberg-Tomozawa type + explicit pole term

[MeV]

260

255

250

245

240

for

but

( 480 MeV,MeV)

1

0.8

0.6

0.4

- Different mechanism from
- It does not necessarily mean that“infinite bare mass” of quasi-particle
- Z=0 for B=0 does not excludean elementary state near B=0

92.4 [MeV]

550 [MeV]

0.2

0

20

15

10

5

0

binding energy [MeV]

Summary

- Wave function renormalization constant can be zero for any resonant statedynamically generated by WT type interaction
- We have shown that the amplitude can be equivalently represented by a Yukawa model with a “quasi-particle” having infinite bare mass and hencewith .
- Different from the “renormalization” due to a divergence of G(s)
- The underlying mechanism is the same as a bound state (constant interaction) case

- Model (cut-off & representation) dependence of
- The arbitrariness leads to multiple interpretations for a physical state
- Among a number of possible models, we have a model with .
- Z cannot be determined from experiments in a model-independent manner.
- Specify firstly : “What is an “elementary particle” to be used as a measure ? ” … choice of a model : problem of “economization”

- A special case of zero-energybound state
- Underlying mechanism for can be different from other cases
- does not exclude an elementary state near the physical state

Another mechanism of for zero energy bound state

Example :

Interaction kernel : Weinberg-Tomozawa type + explicit pole term

[MeV]

260

255

250

245

240

for

but

( 480 MeV,MeV)

- Different mechanism from
- It does not necessarily mean that“infinite bare mass” of quasi-particle
- Z=0 for B=0 does not excludean elementary state near B=0

92.4 [MeV]

550 [MeV]

20

15

10

5

0

binding energy [MeV]

Model dependence of

Yukawa model (“quasi-particle”)

“quasi-particle” dominates

practically zero in

Nonlinear model

+

linear model

+

Scattering amplitude in the linear model

tree amplitude in the linear model

+

Scattering amplitude

+

+…

+

,

,

+…

+

+

Wave function renormalization constant

,

With an explicit pole term

150

Scattering amplitude

100

50

0

50

Pole position

100

1

Interaction kernel

0.5

0

0.5

1

0

0.2

0.4

1

0.6

0.8

Constant interaction (bound state) case

scattering amplitude with constant

(positive constant)

Yukawa term

fictitious mass and bare coupling

Yukawa model

bare coupling must be proportional to

in the large limit (limit)