Electromagnetic Form Factors of Electromagnetic Form Factors of - - PowerPoint PPT Presentation

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Electromagnetic Form Factors of Electromagnetic Form Factors of Electromagnetic Form Factors of Electromagnetic Form Factors of Λ(1405) Λ(1405) in Chiral Dynamics in Chiral Dynamics in Chiral Dynamics in Chiral Dynamics

arXiv:0803.4068 [ arXiv:0803.4068 [nucl nucl-

• th

th] ]

Takayasu Sekihara Takayasu Sekihara Takayasu Sekihara Takayasu Sekihara （ （Kyoto Univ. Kyoto Univ. Kyoto Univ. Kyoto Univ.） ）

in Collaboration with in Collaboration with in Collaboration with in Collaboration with

Tetsuo Hyodo Tetsuo Hyodo Tetsuo Hyodo Tetsuo Hyodo （ （YITP and Tech. Univ. YITP and Tech. Univ. YITP and Tech. Univ. YITP and Tech. Univ. M Mü ünchen nchen nchen nchen） ） and Daisuke Jido and Daisuke Jido and Daisuke Jido and Daisuke Jido （ （YITP YITP YITP YITP） ） 1. 1. 1.

• 1. Introduction:

Introduction: Introduction: Introduction: Λ(1405) Λ(1405) and and and and Chiral Chiral Chiral Chiral Unitary Model ( Unitary Model ( Unitary Model ( Unitary Model (ChUM ChUM ChUM ChUM) ) ) )

• 2. Electromagn
• 2. Electromagnetic In

etic Interaction in teraction in

• 2. Electromagn
• 2. Electromagnetic In

etic Interaction in teraction in Ch ChUM UM Ch ChUM UM

• 3. Electromagnetic Form Factors and Mean Square
• 3. Electromagnetic Form Factors and Mean Square
• 3. Electromagnetic Form Factors and Mean Square
• 3. Electromagnetic Form Factors and Mean Squared Radii

d Radii d Radii d Radii

• 4. Conclusions
• 4. Conclusions
• 4. Conclusions
• 4. Conclusions

夏の学校 夏の学校 '08 @ '08 @ '08 @ '08 @ 代々木オリンピックセンター 代々木オリンピックセンター (Aug. 19 (Aug. 19 (Aug. 19 (Aug. 19 -

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• 1. Introduction
• 1. Introduction
• 1. Introduction
• 1. Introduction

++ ++ Excited baryon Excited baryon Λ Λ(1405) (1405) ++ ++

・ ・ Λ Λ(1405) (1405) ---

• -- mass

mass = 1406 = 1406± ±4 MeV, width = 50 4 MeV, width = 50± ±2 2 MeV MeV (PDG), (PDG), the lightest excited baryon with J the lightest excited baryon with JP

P = 1/2

= 1/2－

－.

. ー ー> > quasi bound state of quasi bound state of KN KN ??? ???

？ ？ < <ー ーー ー> >

< <ー ー It is difficult to observe It is difficult to observe Λ Λ(1405) (1405) directly, directly, because because Λ Λ(1405) (1405) stands below the stands below the KN KN threshold threshold. . Therefore, Therefore, theoretical studies theoretical studies are important are important for for Λ Λ(1405) (1405). .

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• 1. Introduction
• 1. Introduction
• 1. Introduction
• 1. Introduction

++ ++ Λ Λ(1405) (1405) and and kaonic kaonic nuclei ++ nuclei ++

・ ・ It is considered that It is considered that Λ Λ(1405) (1405) plays an important role for plays an important role for the the kaonic kaonic nuclei nuclei (cf. (cf. KNN KNN state state). ). ー ー> > The structure of the The structure of the KN KN system system, , which strongly couples to which strongly couples to Λ Λ(1405), (1405), might be might be essential to essential to the properties of the properties of kaonic kaonic nuclei nuclei. . ・ ・ Also in this point of view, Also in this point of view, we want to know we want to know the structure of the structure of Λ Λ(1405) (1405) as dynamically generated as dynamically generated KN KN quasi bound state quasi bound state. .

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• 1. Introduction
• 1. Introduction
• 1. Introduction
• 1. Introduction

++ Dynamics based on ++ Dynamics based on chiral chiral symmetry symmetry ++ ++

・ ・ Low energy Low energy KN KN interaction interaction in coupled channel approach in coupled channel approach: :

• -- Tree

Tree-

• level interaction

level interaction based on based on chiral chiral symmetry symmetry. .

• -- Satisfying

Satisfying elastic elastic unitarity unitarity ( (N/D N/D method). method). ー ー> > chiral chiral unitary model unitary model. .

Kaiser, Siegel, Kaiser, Siegel, Weise Weise, , Nucl. Phys

• Nucl. Phys.

. A594 A594 (1995) 325, Oset, Ramos, (1995) 325, Oset, Ramos, Nucl. Phys.

• Nucl. Phys. A635

A635 (1998) 99, (1998) 99, Oller, Oset, Oller, Oset, Phys. Rev.

• Phys. Rev. D60

D60 (1999) 074023, Oller, Mei (1999) 074023, Oller, Meiβ βner, ner, Phys. Lett.

• Phys. Lett. B500

B500 (2001) 263. (2001) 263.

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• 1. Introduction
• 1. Introduction
• 1. Introduction
• 1. Introduction

++ Dynamics based on ++ Dynamics based on chiral chiral symmetry symmetry ++ ++

・ ・ Scatt

• Scatt. Amp.

. Amp. T Tij

ij is written as following

is written as following argeblaic argeblaic form: form:

• -- Weinberg

Weinberg-

• Tomozawa

Tomozawa term term (from (from chiral chiral symmetry) symmetry)

• -- Loop integral

Loop integral, which contains , which contains parameter parameter of

• f chiral

chiral unitary model, unitary model,

. .

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• 1. Introduction
• 1. Introduction
• 1. Introduction
• 1. Introduction

++ Chiral unitary model and ++ Chiral unitary model and Λ Λ(1405) (1405) ++ ++

・ ・ The model reproduce The model reproduce total cross sections of total cross sections of K K-

• p

p ー ー> several channels > several channels. .

Kaiser, Siegel, Kaiser, Siegel, Weise Weise, , Nucl. Phys

• Nucl. Phys.

. A594 A594 (1995) 325, (1995) 325, Oset Oset, Ramos, , Ramos, Nucl. Phys.

• Nucl. Phys. A635

A635 (1998) 99. (1998) 99.

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• 1. Introduction
• 1. Introduction
• 1. Introduction
• 1. Introduction

++ Chiral unitary model and ++ Chiral unitary model and Λ Λ(1405) (1405) ++ ++

・ ・ The model reproduce The model reproduce mass distribution of mass distribution of Λ Λ(1405) (1405). .

Kaiser, Siegel, Kaiser, Siegel, Weise Weise, , Nucl. Phys

• Nucl. Phys.

. A594 A594 (1995) 325, (1995) 325, Oset Oset, Ramos, , Ramos, Nucl. Phys.

• Nucl. Phys. A635

A635 (1998) 99. (1998) 99.

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• 1. Introduction
• 1. Introduction
• 1. Introduction
• 1. Introduction

++ Chiral unitary model and ++ Chiral unitary model and Λ Λ(1405) (1405) ++ ++

・ ・ The model The model dynamically generates poles dynamically generates poles, which , which correspond to excited baryons correspond to excited baryons, in , in Scatt

• Scatt. Amp.

. Amp.

Oller Oller, , Mei Meiβ βner ner, , Phys.

• Phys. Lett

Lett. . B500 B500 (2001) 263, (2001) 263, Oset Oset, Ramos and , Ramos and Bennhold Bennhold, , Phys.

• Phys. Lett

Lett. . B527 B527 (2002) 99, (2002) 99, Jido Jido, , Oller Oller, , Oset Oset, Ramos, , Ramos, Mei Meiβ βner ner, , Nucl. Phys.

• Nucl. Phys. A725

A725 (2003) 181. (2003) 181.

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• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in ChUM

ChUM ChUM ChUM

++ EM interactions of ++ EM interactions of Λ Λ(1405) (1405) ++ ++

・ ・ We want to know We want to know electromagnetic interactions of electromagnetic interactions of Λ Λ(1405) (1405). . < <ー ー Photo Photo-

• couplings to

couplings to Scatt

• Scatt. Amp.

. Amp. T Tij

ij: .

: .

ー ー> >

• -- Photo

Photo-

• couplings to

couplings to meson meson, , baryon baryon and and WT term WT term. . ・ ・ Following Following three diagrams three diagrams contribute to the form factors. contribute to the form factors.

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• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii

++ Form factors of excited baryon ++ ++ Form factors of excited baryon ++

・ ・ Definition of form factors of excited baryon Definition of form factors of excited baryon ( (Breit Breit frame): frame): . . ・ ・ Behavior of Behavior of Scatt

• Scatt. Amp. close to

. Amp. close to the pole the pole: : . .

• -- Scatt
• Scatt. Amp.

. Amp. with double pole with double pole contribute to form factors. contribute to form factors.

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• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii

++ Form factors of excited baryon ++ ++ Form factors of excited baryon ++

・ ・ There are There are three diagrams three diagrams for for T Tγ

γ with double pole:

with double pole: ・ ・ Form factors are obtained from Form factors are obtained from residues of residues of T Tγ

γ/T

/T: :

. .

• -- These form factors are

These form factors are free from non free from non-

• resonant background

resonant background. .

・ ・ Mean squared radii Mean squared radii: . : .

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• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii

++ Electromagnetic FF and MSR ++ ++ Electromagnetic FF and MSR ++

Experimental data fitting of neutron: Experimental data fitting of neutron: Platchkov Platchkov et al. et al. Nucl. Phys.

• Nucl. Phys. A510

A510 740. 740.

・ ・ EM form factors of EM form factors of higher higher Λ(1405) Λ(1405) state (1426 state (1426ー ー17 i 17 i MeV MeV) ), , which is considered to be which is considered to be originated from

• riginated from KN

KN bound state bound state. .

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• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii

++ Electromagnetic FF and MSR ++ ++ Electromagnetic FF and MSR ++

・ ・ EM mean squared radii EM mean squared radii are obtained from form factors: are obtained from form factors: using formulae: using formulae: . .

• -- Note that electric MSR of

Note that electric MSR of Λ(1405) Λ(1405) is not affected by common FF. is not affected by common FF.

・ ・ Note that Note that no clear relations are known between no clear relations are known between (complex) mean squared radii and its size for resonance (complex) mean squared radii and its size for resonance. .

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• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii

++ Electromagnetic FF and MSR ++ ++ Electromagnetic FF and MSR ++

・ ・ The absolute electric form factor of The absolute electric form factor of Λ(1405) Λ(1405) has has large energy dependence large energy dependence than that of neutron. than that of neutron. ー ー> The electric form factor is > The electric form factor is “ “softer softer” ”, and , and the production the production cross section of the cross section of the Λ(1405) Λ(1405) may have large energy Dep. may have large energy Dep. ・ ・ K K-

• inside the

inside the Λ Λ(1405) (1405) has less has less “ “virtuality virtuality” ” than than pions pions in in N N. .

• -- 140MeV

140MeV are needed for the neutron, whereas are needed for the neutron, whereas several several MeV MeV for for Λ(1405) Λ(1405). .

ー ー> > With the negative and large real part of electric FF With the negative and large real part of electric FF, , the the K K-

• inside the

inside the Λ Λ(1405) (1405) can be widely distributed can be widely distributed (electric FF = Fourier Trans. of charge distribution). (electric FF = Fourier Trans. of charge distribution).

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• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii

++ Electromagnetic FF and MSR ++ ++ Electromagnetic FF and MSR ++

・ ・ The absolute electric form factor of The absolute electric form factor of Λ(1405) Λ(1405) has has large energy dependence large energy dependence than that of neutron. than that of neutron. ー ー> The electric form factor is > The electric form factor is “ “softer softer” ”, and , and the production the production cross section of the cross section of the Λ(1405) Λ(1405) may have large energy Dep. may have large energy Dep. ・ ・ K K-

• inside the

inside the Λ Λ(1405) (1405) has less has less “ “virtuality virtuality” ” than than pions pions in in N N. .

• -- 140MeV

140MeV are needed for the neutron, whereas are needed for the neutron, whereas several several MeV MeV for for Λ(1405) Λ(1405). .

ー ー> > With the negative and large real part of electric FF With the negative and large real part of electric FF, , the the K K-

• inside the

inside the Λ Λ(1405) (1405) can be widely distributed can be widely distributed (electric FF = Fourier Trans. of charge distribution). (electric FF = Fourier Trans. of charge distribution).

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• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii

++ Form factors (without decay width) ++ ++ Form factors (without decay width) ++

・ ・ Also it is interesting to consider Also it is interesting to consider KN KN bound state bound state. .

• -- Obtained by

Obtained by switching off couplings of switching off couplings of KN KN to others to others. . ー ー> > Λ Λ(1405) (1405) as as KN KN bound state without width bound state without width appears. appears.

• -- mass=1429

mass=1429 MeV MeV, considered as , considered as origin of higher state of

• rigin of higher state of Λ

Λ(1405) (1405). .

• for

for proton proton, , for for neutron neutron (from PDG). (from PDG).

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• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii

++ Form factors (without decay width) ++ ++ Form factors (without decay width) ++

・ ・ Also it is interesting to consider Also it is interesting to consider KN KN bound state bound state. .

• -- Obtained by

Obtained by switching off couplings of switching off couplings of KN KN to others to others. . ー ー> > Λ Λ(1405) (1405) as as KN KN bound state without width bound state without width appears. appears.

• -- mass=1429

mass=1429 MeV MeV, considered as , considered as origin of higher state of

• rigin of higher state of Λ

Λ(1405) (1405). . , , . .

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• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii

++ Form factors (without decay width) ++ ++ Form factors (without decay width) ++

・ ・ Also it is interesting to consider Also it is interesting to consider KN KN bound state bound state. .

• -- Obtained by

Obtained by switching off couplings of switching off couplings of KN KN to others to others. . ー ー> > Λ Λ(1405) (1405) as as KN KN bound state without width bound state without width appears. appears.

• -- mass=1429

mass=1429 MeV MeV, considered as , considered as origin of higher state of

• rigin of higher state of Λ

Λ(1405) (1405). .

• for

for proton proton, , for for neutron neutron (from PDG). (from PDG).

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• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii
• 3. Form Factors and Mean Squared Radii

++ Mean squared radii (without width) ++ ++ Mean squared radii (without width) ++

・ ・ MSR vs. binding energy of MSR vs. binding energy of KN KN bound state bound state is shown. is shown.

• -- By changing

By changing parameter parameter (subtraction constant) (subtraction constant). . ・ ・ Λ Λ(1405) (1405) has has small binging small binging energy (about a few energy (about a few MeV MeV) ) in in chiral chiral unitary model. unitary model. ー ー> Our result implies that > Our result implies that

• verlap between
• verlap between K

K-

• and

and p p is small inside is small inside Λ Λ(1405) (1405). .

• -- Consistent with meson

Consistent with meson-

• baron picture

baron picture in our approach. in our approach. However, knowledge of precise pole position is needed. However, knowledge of precise pole position is needed.

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• 4. Conclusion
• 4. Conclusion
• 4. Conclusion
• 4. Conclusion

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• 4. Conclusion
• 4. Conclusion
• 4. Conclusion
• 4. Conclusion

++ Conclusion ++ ++ Conclusion ++

・ ・ We formulate We formulate electromagnetic form factors of electromagnetic form factors of Λ Λ(1405) (1405), , based on based on meson meson-

• baryon interaction picture

baryon interaction picture, , ChUM ChUM. . － －> The electric mean squared radii of higher > The electric mean squared radii of higher Λ Λ(1405) (1405) is is “ “soft soft” ”, which may affect , which may affect the energy dependence of the energy dependence of the production cross section the production cross section. . ・ ・ As a consequence of As a consequence of the small binding energy the small binding energy in the in the model, our result implies that model, our result implies that the the K K-

• inside the

inside the Λ Λ(1405) (1405) can be widely distributed around can be widely distributed around p p. . ・ ・ Our results for the Our results for the Λ Λ(1405) (1405) as as KN KN bound state are bound state are consistent with meson consistent with meson-

• baryon interaction picture

baryon interaction picture. .

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• 4. Conclusion
• 4. Conclusion
• 4. Conclusion
• 4. Conclusion

++ Future plan ++ ++ Future plan ++

・ ・ Relations between our results and Relations between our results and the observable which is obtained from Exp the observable which is obtained from Exp.? .? ・ ・ Comparison with other approach Comparison with other approach. . ・ ・ Consideration Consideration about complex mean squared radii about complex mean squared radii. . ・ ・ Applications to Applications to other excited hadrons

• ther excited hadrons.

.

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御清聴有難うございます 御清聴有難うございます. .

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• A. Appendix
• A. Appendix
• A. Appendix
• A. Appendix

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Introduction to ChUM Introduction to ChUM Introduction to ChUM Introduction to ChUM

++ ++ Λ(1405) Λ(1405) in ChUM ++ in ChUM ++

・ ・ Coefficients Coefficients C Cij

ij are as follows:

are as follows:

• -- for seagull interactions:

for seagull interactions: . .

Kaiser, Siegel and Weise, Kaiser, Siegel and Weise, Nucl. Phys

• Nucl. Phys.

. A594 A594 (1995) 325, (1995) 325, Oset and Ramos, Oset and Ramos, Nucl. Phys.

• Nucl. Phys. A635

A635 (1998) 99. (1998) 99.

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• 1. Introduction
• 1. Introduction
• 1. Introduction
• 1. Introduction

++ Dynamics based on ++ Dynamics based on chiral chiral symmetry symmetry ++ ++

・ ・ Loop integral Loop integral is calculated in following form: is calculated in following form:

• -- Cut

Cut-

• off
• ff μ

μ and and parameter parameter (subtraction constant) (subtraction constant) a ak

k

are determined from Exp. data. are determined from Exp. data.

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• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in ChUM

ChUM ChUM ChUM

++ Gauge invariance in EM interactions ++ ++ Gauge invariance in EM interactions ++

・ ・ For calculation of EM form factors of For calculation of EM form factors of Λ(1405) Λ(1405), we must , we must keep in mind gauge invariance keep in mind gauge invariance of EM interactions.

• f EM interactions.

ー ー> Photons couple to > Photons couple to everywhere everywhere they can couple. they can couple. ・ ・ We can use We can use “ “Ward identity Ward identity” ” to confirm whether to confirm whether we get correct photon couplings: . we get correct photon couplings: .

夏の学校 夏の学校 '08 @ '08 @ '08 @ '08 @ 代々木オリンピックセンター 代々木オリンピックセンター (Aug. 19 (Aug. 19 (Aug. 19 (Aug. 19 -

• 24, 2008

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• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in ChUM

ChUM ChUM ChUM

++ Gauge invariance in EM interactions ++ ++ Gauge invariance in EM interactions ++

・ ・ We have We have 10 diagrams 10 diagrams for photon couplings: for photon couplings:

• -- We can confirm

We can confirm gauge invariance gauge invariance with them. with them.

Borasoy Borasoy, , Bruns Bruns, , Mei Meiβ βner ner and and Ni Niβ βler ler, , Phys. Rev.

• Phys. Rev. C72

C72 (2005) 065201. (2005) 065201.

夏の学校 夏の学校 '08 @ '08 @ '08 @ '08 @ 代々木オリンピックセンター 代々木オリンピックセンター (Aug. 19 (Aug. 19 (Aug. 19 (Aug. 19 -

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• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in ChUM

ChUM ChUM ChUM

++ EM interactions of mesons and baryons ++ ++ EM interactions of mesons and baryons ++

・ ・ We use following We use following EM interactions of mesons and baryons EM interactions of mesons and baryons. . ・ ・ These are chosen These are chosen for gauge invariance for gauge invariance. .

夏の学校 夏の学校 '08 @ '08 @ '08 @ '08 @ 代々木オリンピックセンター 代々木オリンピックセンター (Aug. 19 (Aug. 19 (Aug. 19 (Aug. 19 -

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• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in
• 2. Electromagnetic Interactions in ChUM

ChUM ChUM ChUM

++ Inclusion of the size effect of hadrons ++ ++ Inclusion of the size effect of hadrons ++

・ ・ For inclusion of For inclusion of the effect of size of mesons and baryons the effect of size of mesons and baryons, , common form factor common form factor can be multiplied in our approach: can be multiplied in our approach:

• -- With this factor,

With this factor, mesons and baryons can have sizes mesons and baryons can have sizes

• f about 0.8 fm.
• f about 0.8 fm.

夏の学校 夏の学校 '08 @ '08 @ '08 @ '08 @ 代々木オリンピックセンター 代々木オリンピックセンター (Aug. 19 (Aug. 19 (Aug. 19 (Aug. 19 -

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Form Factors and Mean Squared Radii Form Factors and Mean Squared Radii Form Factors and Mean Squared Radii Form Factors and Mean Squared Radii

++ Diagram 8 ++ ++ Diagram 8 ++

・ ・ We can easily obtain expression of diagram 8 We can easily obtain expression of diagram 8 (left) in Breit frame: (left) in Breit frame: . .

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Form Factors and Mean Squared Radii Form Factors and Mean Squared Radii Form Factors and Mean Squared Radii Form Factors and Mean Squared Radii

++ Diagram 9 ++ ++ Diagram 9 ++

・ ・ We can easily obtain expression of diagram 9 We can easily obtain expression of diagram 9 (left) in Breit frame: (left) in Breit frame: . .

夏の学校 夏の学校 '08 @ '08 @ '08 @ '08 @ 代々木オリンピックセンター 代々木オリンピックセンター (Aug. 19 (Aug. 19 (Aug. 19 (Aug. 19 -

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Form Factors and Mean Squared Radii Form Factors and Mean Squared Radii Form Factors and Mean Squared Radii Form Factors and Mean Squared Radii

++ Diagram 10 ++ ++ Diagram 10 ++

・ ・ We can easily obtain expression of diagram 10 We can easily obtain expression of diagram 10 (left) in Breit frame: (left) in Breit frame: . . ・ ・ We obtain amplitudes summing up 3 diagrams; We obtain amplitudes summing up 3 diagrams; . .