mixing properties of a 1 (1260) axial vector meson ~ composite and elementary natures ~ hideko...

Mixing properties of Mixing properties of aa11(1260) axial vector meson(1260) axial vector meson

~ Composite and elementary natures ~~ Composite and elementary natures ~

Hideko NAGAHIRO (Nara Women’s University)Hideko NAGAHIRO (Nara Women’s University)

H. NagahiroH. Nagahiro, K. Nawa, S. Ozaki, D. Jido and A. Hosaka, arXiv:1101.3623 [hep-ph], K. Nawa, S. Ozaki, D. Jido and A. Hosaka, arXiv:1101.3623 [hep-ph]

Collaborate with :Collaborate with :

K. Nawa, S. Ozaki, D. Jido, A. HosakaK. Nawa, S. Ozaki, D. Jido, A. Hosaka

““Hadron Nuclear Physics 2011”Hadron Nuclear Physics 2011”““Quarks in hadrons, nuclei, and hadronic matter” Quarks in hadrons, nuclei, and hadronic matter”

Feb. 21(Mon), 2011, Feb. 21(Mon), 2011, POSCO International Center, Pohang, Republic of KoreaPOSCO International Center, Pohang, Republic of Korea

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IntroductionIntroduction

Exotic hadrons :Exotic hadrons : not not simple simple qq qq (or (or qqqqqq)) statestate› , , ff00(980), (980), aa00(980), … , (980), … , aa11(1260), (1260), KK11(1270), …, N*(1535), (1270), …, N*(1535), (1405), … etc…(1405), … etc…

molecule, molecule, qqqq qqqq state, state, oror mixed states among themmixed states among them ??

as a as a dynamically generated resonancedynamically generated resonance [as [as composite particle composite particle]]

[coupled-channel BS][coupled-channel BS] Lutz-Kolomeitsev, NPA730(04)392, … Lutz-Kolomeitsev, NPA730(04)392, …

aa11

as an as an elementary fieldelementary field (or (or qqqq) : candidate for the chiral partner of ) : candidate for the chiral partner of

[Hidden local sym.][Hidden local sym.] Bando-Kugo-Yamawaki; PR164(88)217; Bando-Kugo-Yamawaki; PR164(88)217; Kaiser-Meissner, NPA519(90)671, … Kaiser-Meissner, NPA519(90)671, …

[[qqqq-NJL]-NJL] M.Wakamatsu M.Wakamatsu et al,et al,. ZPA311(88)173, A.Hosaka, PL. ZPA311(88)173, A.Hosaka, PLB244B244(90)(90)363-367 363-367 , …, …

[Lattice QCD][Lattice QCD] M. Wingate M. Wingate et al.et al., PRL74(95)4596, ..., PRL74(95)4596, ...

[Holographic QCD][Holographic QCD] T. Sakai, S. Sugimoto, PTP113 (05) 843; T. Sakai, S. Sugimoto, PTP113 (05) 843; ibid.ibid.114(05)1083, …114(05)1083, …

[Chiral Unitary model][Chiral Unitary model] Roca-Oset-Singh, PRD72(05)014002, … Roca-Oset-Singh, PRD72(05)014002, …

in coupled-channel approach based on the chiral effective theoryin coupled-channel approach based on the chiral effective theory

Nature ofNature of axial vector mesonaxial vector meson a a11(1260) : (1260) : m m = 1230 = 1230 40 MeV, 40 MeV, =260 to 600 MeV [PDG]=260 to 600 MeV [PDG]

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aa11(1260) axial vector meson … a good example(1260) axial vector meson … a good example

we have some models that include the we have some models that include the elementary aelementary a1 1 mesonmeson (dominated by q (dominated by q

qqbarbar) as well as the ) as well as the and and mesons mesons

the same model gives an attractive potential between the the same model gives an attractive potential between the and and mesons, mesons,that would generate the that would generate the composite acomposite a11 meson meson

We can treat these two pictures in a unified way We can treat these two pictures in a unified way

a good example to study the mixing nature of elementary and composite comp.a good example to study the mixing nature of elementary and composite comp.

» We focus on the nature of hadrons consisting of multiple components,We focus on the nature of hadrons consisting of multiple components,and study how to and study how to disentangle their mixturedisentangle their mixture appearing in the physical states. appearing in the physical states.

1. large 1. large NNCC classification classification

2. wave functions by residues2. wave functions by residues

aa11 physphys == CC11 ++ CC22 + …+ …

aa11

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Large-NLarge-NCC, large-, large-(=(=YMYM22 N NCC) QCD ) QCD can be analyzed by the classical supergravitycan be analyzed by the classical supergravity

Lagrangians : holographic QCD (HQCD)Lagrangians : holographic QCD (HQCD)

Important concept to avoid the double-counting of molecule and bare comp.Important concept to avoid the double-counting of molecule and bare comp.

[Sakai, Sugimoto, PTP113(05)843; PTP114(05)1083, [Sakai, Sugimoto, PTP113(05)843; PTP114(05)1083, Nawa, SuganumaNawa, Suganuma, , Kojo, PRD75Kojo, PRD75(07)(07)086003086003 etc.] etc.]

aa11 meson in HQCD meson in HQCD

[E. Witten, Nucl. Phys. B 160, 57 (1979)][E. Witten, Nucl. Phys. B 160, 57 (1979)]

elementary elementary aa11 meson does meson does notnot havehave molecular component molecular component [large Nc limit][large Nc limit]

» essentially the same as essentially the same as those of the those of the hidden-local symmetryhidden-local symmetry

aa11

gg44 = 1, = 1, ggaa11= 0.26= 0.26 ff = 92.4MeV, = 92.4MeV, mm= 776MeV= 776MeV

mmaa11= 1189 MeV and couplings are determined automatically= 1189 MeV and couplings are determined automatically

» aa11 meson appears through the mode expansion of meson appears through the mode expansion of AA((xx,, zz) [5D gauge field] ) [5D gauge field]

ii

wherewhere

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Dynamically generated resonances : composite Dynamically generated resonances : composite aa11 meson meson

aa11 meson as a meson as a -composite : chiral unitary model -composite : chiral unitary model

generated in generated in ss-wave -wave scattering scattering trough non-perturbative dynamics trough non-perturbative dynamics

ttWTWT((ss)) = = vvWTWT((ss))+ + vvWTWT (s)(s)GG(s)(s) t tWTWT(s)(s)

Bethe-Salpeter equation for unitarized Bethe-Salpeter equation for unitarized ss-wave amplitude-wave amplitude

= = vvWTWT + + vvWTWT G vG vWTWT + + vvWTWT G v G vWT WT G vG vWTWT+ …+ …

L.Roca, E.Oset and J.Singh, PRD72(05)014002L.Roca, E.Oset and J.Singh, PRD72(05)014002

++++ ++++ ++++ …………==

tttt tttt== ++

vv GG

vvWTWTttWTWT((ss))==11− − vvWTWTGG

formal solutionformal solutionIf If vvWTWT is attractive,is attractive,

a pole appears in the amp. a pole appears in the amp.

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WT potential WT potential vvWT WT obtained from obtained from ss-wave projection of WT int.-wave projection of WT int.

aa11 meson as a meson as a -composite : chiral unitary model -composite : chiral unitary model

L.Roca, E.Oset and J.Singh, PRD72(05)014002L.Roca, E.Oset and J.Singh, PRD72(05)014002

KK**

−−++

coupled-channel calculation coupled-channel calculation

aa11 pole as mainly pole as mainly composite composite at 1011-84 at 1011-84ii MeV MeV

Loop function GLoop function G

with on-shell with on-shell energies (low energy amplitude ~ threshold) energies (low energy amplitude ~ threshold)

==

== ……

aapheno.pheno. = = −−1.85 (1.85 (=900MeV)=900MeV)

a a parameter : subtraction constant … parameter : subtraction constant … arbitrary.arbitrary. Suitably chosen to reproduce data : Suitably chosen to reproduce data : aaphenopheno

vvWTWT

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== ==

== ……

a a parameter : subtraction constant … parameter : subtraction constant … arbitrary.arbitrary. Suitably chosen to reproduce data : Suitably chosen to reproduce data : aaphenopheno

Choice of regularization constant : Choice of regularization constant : naturalnatural renormalization renormalization

A choice of the subtraction constant A choice of the subtraction constant aapheno pheno ≠ a≠ anaturalnatural is equivalent to the is equivalent to the

introduction of the CDD (or elementary) pole that is not included in the modelintroduction of the CDD (or elementary) pole that is not included in the modelspace of the scattering problem. space of the scattering problem. [T. Hyodo, D.Jido, A.Hosaka, PRC78(08)025203][T. Hyodo, D.Jido, A.Hosaka, PRC78(08)025203]

aa(() = ) = 0.2 [natural]0.2 [natural]aa(()= )= 1.85 (1.85 (=900MeV) [Roca05]=900MeV) [Roca05]

to ensure that the resulting state is a purely hadronic composite without quark-core to ensure that the resulting state is a purely hadronic composite without quark-core

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Formalism : Formalism : elementary elementary aa11 field field through through additional interactionadditional interaction

(coupled-channel) Bethe-Salpeter eq.(coupled-channel) Bethe-Salpeter eq.

++++

++++ ++++ ++++ ++++ …………++++

potentialpotential

WT interactionWT interaction

aa11 pole term pole term

wherewhere

bare bare aa11

++++ ++++ ++++ …………==

×x

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Numerical result 1 : pole-flow of T-matrix Numerical result 1 : pole-flow of T-matrix

00

100100

200200

300300

400400

500500

600600900900 10001000 11001100 12001200 13001300 14001400 15001500 16001600 17001700 18001800

[Roca [Roca et al.et al., PRD72], PRD72]aa==

channel onlychannel onlyaa==

ssp p = = 101110118484ii MeV MeV

ssp p = 1012= 10129898i i MeVMeV


[natural][natural] ssp p = 1012= 1012221221i i MeVMeV

single channelsingle channel

naturalnatural

++

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00

100100

200200

300300

400400

500500

600600900900 10001000 11001100 12001200 13001300 14001400 15001500 16001600 17001700 18001800

elementary elementary aa11 pole pole

mma a = 1189 MeV (HQCD)= 1189 MeV (HQCD)

ssp p = 1012= 1012221221i i MeVMeV

where

V = +bare a1

×x


[natural][natural]

a1

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00

100100

200200

300300

400400

500500

600600900900 10001000 11001100 12001200 13001300 14001400 15001500 16001600 17001700 18001800

ssp p = 1012= 1012221221i i MeVMeV

xx 1 1xx

1 1

x x = 1= 1

x x = 1= 1

x x = 0= 0

x x = 0= 0


[natural][natural]

elementary elementary aa11 pole pole

mma a = 1189 MeV (HQCD)= 1189 MeV (HQCD)

a1

where

V = +

×x

bare a1

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Numerical result 1 : pole-flow of T-matrixNumerical result 1 : pole-flow of T-matrix

00

100100

200200

300300

400400

500500

60060010001000 11001100 12001200 13001300 14001400 15001500 16001600 17001700 18001800

x x = 1= 1

x x = 1= 1

x x = 0= 0

x x = 0= 0

Pole from “Pole from “-composite”-composite” is closer to the real axis is closer to the real axis

than that of “elementary” pole than that of “elementary” pole

pole-apole-a pole-bpole-b

bare bare aa11

xx

a1

?

??

Q. What is the nature of these physical (Q. What is the nature of these physical (xx=1) poles ?=1) poles ?

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Resonant structure tested by behaviors in large NResonant structure tested by behaviors in large NCC limit limit

» To see the nature of the physical resonant states ( To see the nature of the physical resonant states ( x x = 1 ), we first check the = 1 ), we first check the NNCC (color number) dependence of the pole positions. (color number) dependence of the pole positions.

Natures of mesons in large Nc limitNatures of mesons in large Nc limit [’t Hooft, NPB72(74)461][’t Hooft, NPB72(74)461]

qqqqbarbar (elementary) meson (elementary) meson

becomes becomes stablestable as N as NCC ∞ ∞

M M M M11MM2 2 O(N O(NCC−−1/21/2))

mmqqqq-meson-meson O(1)O(1)−−

qqqq-meson-meson O(1/O(1/NNCC))−−TTMMMMMM MM O(1/NO(1/NCC))

hadronic composite mesonhadronic composite meson

is is vanishedvanished as N as NCC ∞∞

scaling law of the pion decay constant scaling law of the pion decay constant ff O(N O(NCC1/21/2))

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Numerical result 1 : large NNumerical result 1 : large NCC flow flow

00

100100

200200

300300

400400

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600600900900 10001000 11001100 12001200 13001300 14001400 15001500 16001600 17001700 18001800

bare bare aa11

xxx x = 0= 0

x x = 1= 1

x x = 1= 1

x x = 0= 0

NNCC

large large

NNCC=5=5

NNCC=10=10

x x = 0.7= 0.7x x = 0.65= 0.65

NNCC

large large

NNCC

large large

To see the nature of poles :To see the nature of poles :

and and NNCC large largepole-apole-a

L.Geng L.Geng et al.et al.,,EPJA39(09)81EPJA39(09)81

pole-bpole-b

?

??

flip!flip!

a1

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Alternative expression for the full Alternative expression for the full scattering amplitude T scattering amplitude T

= (= ( ggRR, , gg ) ) sssspp

ssmma1a122 gGggGgRR gGggGg

ggRRGgGg11

ggRR

gg

++++ ++++ ++++ ………… ==

aa11 pole term pole term

bare bare aa11

-composite -composite aa11 pole pole

,,== (( )) 11 11

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Alternative expression for the full Alternative expression for the full scattering amplitude T scattering amplitude T

,,== (( )) 11

,,== (( ))++ ++ ……

== ++ ++++ ++++

11

= (= ( ggRR, , gg ) ) sssspp

ssmma1a122 gGggGgRR gGggGg

ggRRGgGg11

ggRR

gg

== ++ ++++ggRR ggRR gg gg gg ggggRR ggRR

DD1111 DD2121 DD1212 DD2222

In this form, we can analyze the mixing nature of the physical In this form, we can analyze the mixing nature of the physical aa11

in terms of the in terms of the original two basesoriginal two bases: and : and composite composite aa11 elementary elementary aa11

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Features of the full propagatorsFeatures of the full propagators

The propagators have two poles exactly at the same position as The propagators have two poles exactly at the same position as TTfullfull

the residues the residues zzaaiiii have the information on the mixing rate have the information on the mixing rate

+ regular term+ regular term

Probability of finding the original composite aProbability of finding the original composite a11 in pole-a in pole-a

DD1111

In this form, we can analyze the mixing nature of the physical In this form, we can analyze the mixing nature of the physical aa11

in terms of the in terms of the original two basesoriginal two bases: and : and composite composite aa11 elementary elementary aa11

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-0.1-0.1

-0.2-0.2

-0.3-0.3

-0.4-0.4

00

11 1.11.1 1.21.2 1.31.3 1.41.4 1.51.5 1.61.6 1.71.7 1.81.80.90.9

xx=0=0

xx=0=0

xx=1=1

pole

-apo

le-a pole-b

pole-b

mixing parameter mixing parameter xx00 110.80.80.60.60.40.40.20.2

22

33

44

11

00

|z|

|z|

zzbb1111

zzaa2222

pole flowpole flow

residueresidue

xx=1=1

zzbb2222

zzaa1111

Large residueLarge residue due to the ene-dep. due to the ene-dep.

+ (regular)+ (regular)


Residues : probabilities of finding two Residues : probabilities of finding two aa11’s in pole-a and -b’s in pole-a and -b

interchangeinterchange

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-0.1-0.1

-0.2-0.2

-0.3-0.3

-0.4-0.4

00

11 1.11.1 1.21.2 1.31.3 1.41.4 1.51.5 1.61.6 1.71.7 1.81.80.90.9

xx=0=0

xx=0=0

xx=1=1

pole

-apo

le-a pole-b

pole-b


22

33

44

11

00

|z|

|z|

zzbb1111

zzaa2222

non-zero comp. of non-zero comp. of

pole flowpole flow

residueresidue

xx=1=1

zzbb2222

zzaa1111

at physical point (at physical point (xx=1)=1)

• pole-apole-a has a component of the elementary has a component of the elementaryaa1 1 meson meson comparable tocomparable to that of that of

composite composite aa11..

(pole-a at x=1 is possibly observed one) (pole-a at x=1 is possibly observed one)



~~

~~

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Last question : large NLast question : large NCC procedure vs. nature of poles procedure vs. nature of poles

-0.1-0.1

-0.2-0.2

-0.3-0.3

-0.4-0.4

00

11 1.11.1 1.21.2 1.31.3 1.41.4 1.51.5 1.61.6 1.71.7 1.81.80.90.9

xx=0=0xx=1=1

xx=1=1po

le-a

pole

-a pole-bpole-b

pole flowpole flow• for pole-b, for pole-b, large Nlarge NCC flip point ~ residue-interchange flip point ~ residue-interchange

• for pole-a, for pole-a, large Nlarge NCC flip point flip point residue-interchange residue-interchange

22

33

44

11

00

|z|

|z|

NNCC

33 44 55 66 77 88 99 1010

zzbb1111

zzaa2222

zzaa1111

zzbb2222 Residues interchangeResidues interchangeby large Nby large NCC procedure. procedure.

xx=0.8=0.8

xx=0=0

1/ NC

Large NC itself changesthe nature of resonances

1/NC

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22

33

44

11

00

|z|

|z|

NNCC

33 44 55 66 77 88 99 1010

zzbb1111

zzaa2222

zzaa1111


xx=0.8=0.8

1/ NC

Large NC limit doesn’t alwaysreflect the world at NC=3.

1/NC

two sources of Ntwo sources of NCC dependence: dependence:

aa11 vertex controlling the mixing strength vertex controlling the mixing strength

WT interaction determining the pole position WT interaction determining the pole position sspp

of the of the basis statebasis state for composite for composite aa11

sssspp((NNCC))

ssmma1a122 gg((NNCC))GgGgRR((NNCC))

11ggRR((NNCC))GgGg((NNCC))

gg((NNCC))GgGg((NNCC))DD = =^̂

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ConclusionsConclusions

» We discussed the We discussed the mixing propertiesmixing properties of of aa11(1260) meson as the superposition (1260) meson as the superposition

of the hadronic of the hadronic composite and elementary composite and elementary aa11 based on the holographic based on the holographic

QCD Lagrangian.QCD Lagrangian.› bare bare aa11 … doesn’t have molecule nature … doesn’t have molecule nature

› molecule … “natural” regularizationmolecule … “natural” regularization

» We analyzed the pole nature by residuesWe analyzed the pole nature by residues

Important to avoid the double-countingImportant to avoid the double-counting

the pole expected to be observed is the pole expected to be observed is pole-apole-a: having finite comp.: having finite comp.

Non-trivial NNon-trivial NCC dependence pole-nature dependence pole-nature ?? large N large NCC

Future worksFuture works

phenomenological interestsphenomenological interests» -decay spectrum with our model parameter-decay spectrum with our model parameter

[Wagner and Leupold, PRD78(08)053001, …][Wagner and Leupold, PRD78(08)053001, …]

» radiative decay widthradiative decay width [H. Nagahiro, L. Roca, A. Hosaka , E. Oset, PRD79(09)014015, …] [H. Nagahiro, L. Roca, A. Hosaka , E. Oset, PRD79(09)014015, …]

» etc…etc…

to seeto see how the nature of poles affects how the nature of poles affects observablesobservables

backupbackup

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L.Roca, L.Roca, et al.et al., PRD72(05)014002, PRD72(05)014002

WT interactionWT interaction

(coupled-channel) Bethe-Salpeter eq.(coupled-channel) Bethe-Salpeter eq.

aa(() = ) = 0.2 [natural]0.2 [natural]

L. Roca L. Roca et al.et al., PRD72(05)014002, PRD72(05)014002

[T. Hyodo, D.Jido, A.Hosaka, PRC78(08)025203][T. Hyodo, D.Jido, A.Hosaka, PRC78(08)025203]

aa11 pole as pole as composite at 1011-84 composite at 1011-84ii MeV MeV

aa(()= )= 1.85 (1.85 (=900MeV) [Roca =900MeV) [Roca et al.et al.]]

regularization constantregularization constant

to avoid the double-counting.to avoid the double-counting.

==++++ ++++ ++++ …………== ==

dynamics in dynamics in aa11 channel : chiral unitary approach channel : chiral unitary approach

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effectiveeffective aa11 field that is equivalent to composite field that is equivalent to composite aa11

++++ ++++ ++++ …………== ++++ ++++== …………

Or considering the quantum effects as..Or considering the quantum effects as..

CRITERIONCRITERION==

Ref. D. Lurie, “Particles and fields”, Interscience,Ref. D. Lurie, “Particles and fields”, Interscience, D. Lurie and A.J. Macfarlane, PR136(64)B816.D. Lurie and A.J. Macfarlane, PR136(64)B816.

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G function : G function :

700700 800800 900900 10001000 11001100 12001200 13001300 14001400

ImGImG

0.0180.018

0.0160.016

0.0140.014

0.0120.012

0.0100.010

0.0060.006

0.0080.008

0.0040.004

0.0020.002

00

0.0020.002

ReG(ReG(aa(()=)=0.2 : natural)0.2 : natural)

ReG(ReG(aa(()=)=1.85)1.85)mm

““natural” renormalizationnatural” renormalization[T. Hyodo, D.Jido, A.Hosaka, PRC78(08)025203][T. Hyodo, D.Jido, A.Hosaka, PRC78(08)025203]

G(E=G(E=mmTT)=0 so that T(E=)=0 so that T(E=mmTT)=V)=VWTWT


L.Roca L.Roca et alet al, PRD72, PRD72

T. Hyodo, D. Jido, A. Hosaka, Phys.Rev.C78:025203,2008;T. Hyodo, D. Jido, A. Hosaka, Phys.Rev.C78:025203,2008;

http://maru.bonyari.jp/texclip/texclip.php?s=%24%24%0D%0AG(%5Csqrt%7Bs%7D%2Ca_%7B%5Crm%20pheno%7D%7D)%3DG(%5Csqrt%7Bs%7D%2Ca_%7B%5Crm%20natural%7D)%2B%5CDelta%20a%0D%0A%24%24

http://maru.bonyari.jp/texclip/texclip.php?s=%24%24%0D%0AT%3D%5Cfrac%7B1%7D%7Bv_%7BWT%7D%5E%7B-1%7D-G(%5Csqrt%7Bs%7D%2Ca_%7B%5Crm%20pheno%7D)%7D%0D%0A%24%24

http://maru.bonyari.jp/texclip/texclip.php?s=%24%24%0D%0A%3D%5Cfrac%7B1%7D%7Bv_%7BWT%7D%5E%7B-1%7D-%5CDelta%20a%20-G(%5Csqrt%7Bs%7D%2Ca_%7B%5Crm%20natural%7D)%7D%0D%0A%24%24

http://maru.bonyari.jp/texclip/texclip.php?s=%24%24%0D%0A%3D%5Cfrac%7B1%7D%7B(v_%7BWT%7D%2Bv_%7B%5Crm%20pole%7D)%5E%7B-1%7D%20-G(%5Csqrt%7Bs%7D%2Ca_%7B%5Crm%20natural%7D)%7D%0D%0A%24%24

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0.80.8 0.90.9 1.01.0 1.11.1 1.21.2 1.31.3 1.41.4 1.51.5

GeVGeV

aa((GeV) = GeV) = 0.2 & energy-indep. 3 point vertex.0.2 & energy-indep. 3 point vertex.

00

-100-100

-200-200

-300-300

-50-50

-150-150

-250-250

xx=0=0

xx=1=1

xx=0=0

xx=0.8=0.8at physical point (at physical point (xx=1)=1) there is a single pole (there is a single pole (pole-bpole-b))

1.21.2

1.01.0

0.80.8

0.60.6

0.40.4

0.20.2

00

|z|

|z|

zzaa1111

zzaa2222

zzbb1111

zzbb2222

00 0.20.2 0.40.4 0.60.6 0.80.8 11mixing parameter mixing parameter xx

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|T||T|22 spectrum : spectrum : aa==1.85 w/o bare 1.85 w/o bare aa11 vs. vs. aa==0.2 w/ bare 0.2 w/ bare aa11|T

||T

|22

1.41.4

1.21.2

1.01.0

0.80.8

0.60.6

0.40.4

0.20.2

00

×10×1044

11 1.11.1 1.21.2 1.31.3 1.41.4 1.51.5 1.61.6 1.71.7 1.81.8

aa==1.851.85

aa==0.20.2 ++

GeVGeV

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Numerical result 1 : pole flowNumerical result 1 : pole flow

00

100100

200200

300300

400400

500500

600600900900 10001000 11001100 12001200 13001300 14001400 15001500 16001600 17001700 18001800

x x = 1= 1

x x = 1= 1

x x = 0= 0

x x = 0= 0

To see the nature of poles :To see the nature of poles :

and and NNCC large largePole from “WT-molecule”Pole from “WT-molecule” is closer to the real axis is closer to the real axis

than that of “bare” pole than that of “bare” pole


bare bare aa11

xx

|T|

|T|22

ssp p = 1012= 10129898i i MeVMeV

• WT-WT- molecule ( molecule (aa==1.851.85) only) only

pole-bpole-b17281728314314i i MeVMeV

0.50.5 1.01.0 1.51.5 22 2.52.5 3.03.0

1.41.4

1.21.2

1.01.0

0.80.8

0.60.6

0.40.4

0.20.2

00

×10×1044

The present caseThe present case

pole-apole-a10331033107 107 i i MeVMeV

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nature of two poles analyzed by residuenature of two poles analyzed by residue

» To see the nature of two poles, one can see the residues of To see the nature of two poles, one can see the residues of T-matrixT-matrix at poles. at poles.

c.f. c.f. (1405) with double-pole structure in chiral unitary approach(1405) with double-pole structure in chiral unitary approach[[c.f.c.f. Jido, Oller, Oset, Ramos, Meißner, NPA725(03)181] Jido, Oller, Oset, Ramos, Meißner, NPA725(03)181]

KK

NN NN

» This case :This case :

KK

To know To know the mixing strength of “elementary the mixing strength of “elementary aa11” and “composite ” and “composite aa11”,”,

we define the appropriate we define the appropriate base base as elementary as elementary aa11 and and composite composite aa11

bare bare aa11 bare bare aa11

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2D-matrix propagator 2D-matrix propagator

expand expand fullfull T-matrix elementT-matrix element

bare bare aa11

11stst term term

++++ ++++ ++++ …………

22ndnd term term

++++

==

++++

++++

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,,,,

2D-matrix propagator2D-matrix propagator

== ++++

++++

++++

++++++++ ++++ …………

==== (((( ))))++++ ++++ …………

~ 2D-matrix propagator ~~ 2D-matrix propagator ~

Residues of Residues of GGfullfull = comp. of wave function = comp. of wave function These residues are These residues are normalized to benormalized to be 11 at at xx=0.=0.

expand expand fullfull T-matrix elementT-matrix element

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Numerical result 1 : pole flowNumerical result 1 : pole flow

00

100100

200200

300300

400400

500500

600600900900 10001000 11001100 12001200 13001300 14001400 15001500 16001600 17001700 18001800

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C. Amsler et al. (Particle Data Group), PL B667, 1 (2008)C. Amsler et al. (Particle Data Group), PL B667, 1 (2008)

aa11(1260) meson(1260) meson

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mixing properties analyzed by Residue mixing properties analyzed by Residue

-0.1-0.1

-0.2-0.2

-0.3-0.3

-0.4-0.4

00

11 1.11.1 1.21.2 1.31.3 1.41.4 1.51.5 1.61.6 1.71.7 1.81.80.90.9

xx=0=0xx=1=1

pole

-apo

le-a pole-b

pole-b

1616

1414

1212

1010

88

66

44

22

00


zzbb1111

zzbb2222

zzaa2222

zzaa1111|z|

|z|

pole flowpole flow

residueresidue

xx=1=1

xx=0=0


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-0.1-0.1

-0.2-0.2

-0.3-0.3

-0.4-0.4

00

11 1.11.1 1.21.2 1.31.3 1.41.4 1.51.5 1.61.6 1.71.7 1.81.80.90.9

xx=0=0

xx=0=0

xx=1=1

pole

-apo

le-a pole-b

pole-b


22

33

44

11

00

|z|

|z|

zzbb1111

zzaa2222

pole flowpole flow

residueresidue

xx=1=1

zzbb2222

zzaa1111





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Probabilities of finding two Probabilities of finding two aa11’s in pole-a & -b : residues’s in pole-a & -b : residues

-0.1-0.1

-0.2-0.2

-0.3-0.3

-0.4-0.4

00

11 1.11.1 1.21.2 1.31.3 1.41.4 1.51.5 1.61.6 1.71.7 1.81.80.90.9

xx=0=0

xx=0=0

xx=1=1

pole

-apo

le-a pole-b

pole-b


22

33

44

11

00

|z|

|z|

interchangeinterchange

zzbb1111

zzaa2222

non-zero comp. of non-zero comp. of

pole flowpole flow

residueresidue

xx=1=1

zzbb2222

zzaa1111

at physical point (at physical point (xx=1)=1)

• pole-apole-a remains as a “ remains as a “composite-acomposite-a11””

(bare comp. (bare comp. molecule comp.) molecule comp.)

both poles have molecule comp.both poles have molecule comp.• pole-bpole-b changes into a “ changes into a “composite-acomposite-a11” ”

Difference of transition pointsDifference of transition points energy-dependence (mainly) of energy-dependence (mainly) of



~~>>

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-0.1-0.1

-0.2-0.2

-0.3-0.3

-0.4-0.4

00

11 1.11.1 1.21.2 1.31.3 1.41.4 1.51.5 1.61.6 1.71.7 1.81.80.90.9

xx=0=0xx=1=1

xx=1=1

pole

-apo

le-a pole-b

pole-b

pole flowpole flow• for pole-b, for pole-b, large Nlarge NCC flip point ~ residue-exchange flip point ~ residue-exchange

• for pole-a, for pole-a, large Nlarge NCC flip point flip point residue-exchange residue-exchange

22

33

44

11

00

|z|

|z|

NNCC

33 44 55 66 77 88 99 1010

zzbb1111

zzaa2222

zzaa1111


xx=0.8=0.8

xx=0=0

1/ NC

Large NC procedure itself changesstrength of the mixing int.

1/NC

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00

100100

200200

300300

400400

500500

60060010001000 11001100 12001200 13001300 14001400 15001500 16001600 17001700 18001800

x x = 1= 1

x x = 1= 1

x x = 0= 0

x x = 0= 0


bare bare aa11

xxNumerical result 1 : pole-flow of T-matrixNumerical result 1 : pole-flow of T-matrix

Pole from “Pole from “-composite”-composite” is closer to the real axis is closer to the real axis

than that of “elementary” pole than that of “elementary” pole

900900

|T|

|T|22

1.41.4

1.21.2

1.01.0

0.80.8

0.60.6

0.40.4

0.20.2

00 11 1.11.1 1.21.2 1.31.3 1.41.4 1.51.5 1.61.6 1.71.7 1.81.8

GeVGeV

pole-apole-acoming fromcoming from

the composite the composite aa11

pole-bpole-b

[Roca05][Roca05]

amplitude on the real axisamplitude on the real axis

mixing properties of a 1 (1260) axial vector meson ~ composite and elementary natures ~ hideko...

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