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A study for “elementarity” of composite systems

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Mini workshop on

“Structure and production of charmed baryons II”

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

A study for “elementarity” of composite systems

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)

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 ?

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

- 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

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.

Yukawa theory with constant

Resonance case (composite model)

Bound state model

…

+

=

+

+ …

+

+

=

?

=

=

wave function renormalization

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

shifted amplitude

scattering amplitude with on-shell factorization[1]

composite pole

+

=

+

+ …

shifted amplitude

Yukawa term

Yukawa term

bare mass of the fictitious elementary particle

(cf. Hyodo08)

“fictitious” particle

Energy-dependent Yukawa coupling

Composite model

Yukawa model

…

+

+

+ …

+

+

Energy-dependent Yukawa coupling

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

due to energy-dependence of

zero !

renormalized coupling

finite

bare mass of the fictitious elementary particle

wave function renormalization constant

infinite

bare coupling

infinite

due to energy-dependence of

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

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

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”)

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

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)

for

finite

finite

0

finite

renormalized coupling

wave function renormalization constant or “elementarity”

Derivative of the loop function

for with

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]

- 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

backup

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]

Yukawa model (“quasi-particle”)

“quasi-particle” dominates

practically zero in

Nonlinear model

+

linear model

+

tree amplitude in the linear model

+

Scattering amplitude

+

+…

+

,

,

+…

+

+

Wave function renormalization constant

,

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

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)