[PPT] - Effects of Spin 3/2 Nucleon Resonances in Kaon Photoproduction A. J. PowerPoint Presentation

SLIDE 1

Effects of Spin 3/2 Nucleon Resonances in Kaon Photoproduction

A. J. Arifi1 & T. Mart2

RCNP, Osaka University1 Physics Department, University of Indonesia2 J-Park Hadron Workshop March 2nd, 2016

A. J. Arifi (RCNP)

Kaon photoproduction 1 / 28

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Publication

T. Mart, S. Clymton, and A.J. Arifi, Phys. Rev. D 92, 094019 (2015)

Nucleon resonances with spin 3/2 and 5/2 in the isobar model for kaon photoproduction

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Overview

1

Introduction Motivation Objective

2

Formalism Isobar Model Fitting Procedure

3

Result and Discussion Fitting Result Plots of Observables

4

Summary

A. J. Arifi (RCNP)

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Motivation

Kaon photoproduction provides an essential tool for the studies of strange hadron.

Figure: Kaon photoproduction on proton (a) and nucleus (b)

A. J. Arifi (RCNP)

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Motivation

Experimental side — abundant data are available on kaon photoproduction and we have included 7433 data points in this

research. The experimental data are from

CLAS Collaboration, GRAAL Collaboration, LEPS Collaboration, MAMI Collaboration.

Theoretical side — we have used Isobar model to describe kaon

photoproduction. Some formulations have been well-established but

the formulation of nucleon resonances spin 3/2 or higher is still plagued with the problem of consistency.

A. J. Arifi (RCNP)

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Objective

we aim to study the effect of inclusion of spin 3/2 nucleon resonances with two different formulations using isobar model. Model A R.A. Adelseck, C. Bennhold, and L.E. Wright, Phys. Rev. C32, 1681(1985)

J. C. David, C. Fayard, G. H. Lamot, and B. Saghai, Phys. Rev. C53,

2613 (1996) Model B

V. Pascalutsa, Phys. Lett. B503, 85 (2001)
A. J. Arifi (RCNP)

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Channels of Kaon Photoproduction

Figure: (a) Baryon Octet (b) Pseudoscalar meson nonet

based on conservation of strangeness and isospin, there are six possible channels of kaon photoproduction

1. γ + p → K + + Λ
4. γ + n → K 0 + Σ0
2. γ + p → K + + Σ0
5. γ + n → K 0 + Λ
3. γ + p → K 0 + Σ+
6. γ + n → K + + Σ−
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Kaon Photo- & Electroproduction

Figure: (a) Kaon Photoproduction (b) Kaon Electroproduction

Elementary process of kaon electroproduction e(ki) + p(p) → e′(kf ) + K +(q) + Λ(pΛ), (1) it’s equivalent to virtual photoproduction γν(k) + p(p) → K +(q) + Λ(pΛ), (2) with k = kf − ki. but, we use real photon which has properties k2 = 0 and k · ǫ = 0 for the case of kaon photoproduction.

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Feynman Diagram of Kaon Photoproduction

Figure: (a) s-channel (b) u-channel (c) t-channel of kaon photoproduction

Contribution of mediator particles are categorized into background and

resonances. So, the amplitude reads

M = Mbackground + Mresonances (3)

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Hadronic Form Factor

We include the hadronic form factors in the hadronic vertices by adopting the method developed by Haberzettl. ˜ F = F(Λ, s) sin2 θhad cos2 φhad + F(Λ, u) sin2 θhad sin2 φhad +F(Λ, t) cos2 θhad, (4) where, F(Λ, x) = Λ4 Λ4 + (x − m2)2 (5) and x = s, t, u (mandelstam variables)

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Nukleon Resonances

Table: Status, mass and width of nucleon resonances used in our model

Resonances Status Mass (MeV) Width (MeV) N(1440)P11 1430 ± 20 350 ± 100 N(1520)D13 1515 ± 5 115+10

−15

N(1535)S11 **** 1535+20

−10

150 ± 25 N(1650)S11 **** 1655+15

−10

140 ± 30 N(1700)D13 *** 1700 ± 50 150+100

−50

N(1710)P11 *** 1710 ± 30 100+150

−50

N(1720)P13 **** 1720+30

−20

250+150

−100

N(1875)D13 *** 1875+45

−55

200 ± 25 N(1880)P11 1870 ± 35 235 ± 65 N(1895)S11 1895 ± 15 90+30

−15

N(1900)P13 * 1900 250 N(2120)D13 2140 330 ± 45

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Model A

Propagator P3/2

µν

= / p + / k + √s 3(s − m2

N∗ + imN∗ΓN∗)

gµν + γνγµ − 2

s (p + k)µ(p + k)ν − 1 √s {γµ(p + k)ν − γν(p + k)µ}

,

(6) Electromagnetic vertex factor Γν(±)

N∗pγ

= ga

N∗pγ

k2

(k2ǫν − k · ǫkν) −

1 √s ± mp (k2/ ǫ − k · ǫ/ k)kν +gb

N∗pγ

p · ǫkν − p · kǫν (√s ± mp)2 + gc

N∗pγ

k · ǫkν − k2ǫν (√s ± mp)2

Γ±,

(7) Hadronic vertex factor Γµ(±)

KΛN∗ = gKΛN∗

mN∗ pµ

ΛΓ∓,

(8)

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Model B

Propagator P3/2

µν

= / p + / k + mN∗ (s − m2

N∗ + imN∗ΓN∗)

−gµν + 1

3γµγν + 2 3s (p + k)µ(p + k)ν + 1 3√s {γµ(p + k)ν − γν(p + k)µ}

,

(9) Electromagnetic vertex factor Γν(±)

N∗pγ

= − i m2

N∗

g(1)(ǫν/

k − kν/ ǫ)/ p + g(2)(kνp · ǫ − ǫνp · k) +g(3)pν(/ ǫ/ k − / k/ ǫ) + g(4)γν(/ k/ ǫ − / ǫ/ k)/ p +g(5)γν(p · k/ ǫ − p · ǫ/ k)

Γ±,

(10) Hadronic vertex factor Γµ(±)

KΛN∗

= gKYN∗ m2

N∗

Γ∓

(pΛ · q − /

pΛ/ q)γµ + / pΛqµ − / qpµ

Λ

,

(11)

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where we define g(1) = −2iga

N∗pγ + 3igc N∗pγ + gd N∗pγ,

g(2) = −2iga

N∗pγ − gb N∗pγ + 2igc N∗pγ − 2gd N∗pγ,

g(3) = −iga

N∗pγ + igc N∗pγ,

(12) g(4) = −iga

N∗pγ + igc N∗pγ,

g(5) = −2iga

N∗pγ + igc N∗pγ − gd N∗pγ,

and parities are also defined by Γ+ = iγ5 and Γ− = 1.

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Scattering Amplitude

The corresponding scattering amplitude is written as M(±)

res

= ¯ uΛΓµ(±)

KΛN∗P3/2 µν Γν(±) N∗pγup,

(13) = ¯ uΛ

6

i=1

Ai(s, t, u, k2)Miup, (14) Then, we decompose them into invariant matrices as below M1 = 1 2γ5(/ ǫ/ k − / k/ ǫ), M2 = γ5 [(2q − k) · ǫP · k − (2q − k) · kP · ǫ] , M3 = γ5(q · k/ ǫ − q · ǫ/ k), (15) M4 = iǫµνρσγµqνǫρkσ, M5 = γ5(q · ǫk2 − q · kk · ǫ), M6 = γ5(k · ǫ/ k − k2/ ǫ)

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Amplitude of Model B

A1 = mp

1

2(mp + mΛ) (cΛ − mΛmp − 3scs) + mΛ (2scs − ck)

±mN∗
1

2(mp + mΛ)(mΛ − 3mpcs − 1 s cΛmp)

+2cΛ − 2m2

Λ − 3c1 − 1 s cΛck

G1

+

1

2(mp + mΛ)

mpcΛ + mΛ(k2 − s)
+ bpcΛ
± mN∗
mΛbp

+ 1

2(mp + mΛ)

3(c1 − bpcs) + 1

s cΛ(k2 − s) + mΛmp

G2

+2

bpcΛ − ckm2

Λ − 3sc1

± mN∗mΛ
ckcs − 3c1 − k2

G3, (16)

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A2 = mp t − m2

k

mΛk2 ± mN∗
3(k2 − 2bq) − 2

s cΛk2

G1 +

1 t−m2

k

3s(c1 − bpcs) − mΛmpk2

± mN∗mp ×

3(bpcs − c1) − 1

s cΛk2

G2 + 4k2 t − m2

k

cΛ ± mN∗mΛ
G3,

(17) A3 =

1 2mp

3s − mΛmp
± mN∗
3(mp − mΛ) + 2

s cΛmp

G1

+ 1

2

mpcΛ + mΛ(k2 − s)
± mN∗
mΛmp + 1

s cΛ(k2 − s)

+3

c1 + bp(1 + 1

s cΛ)

G2

−2

mΛck ± mN∗
3c1 + 1

s cΛck

G3,

(18)

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A4 = − 1

2mp

3(cΛ + scs) + mΛmp
± mN∗
3(mp + mΛ) − 2

s mpcΛ

G1

+ 1

2

mpcΛ + mΛ(k2 − s)
± mN∗
mΛmp + 1

s cΛ(k2 − s)

+3 (c1 − bpcs)

G2 − 2
mΛck ± mN∗
3c1 + 1

s cΛck

G3,

(19) A5 =

1 2 mp t−m2

k

−mΛc5 ± mN∗
3(k2 − 2bq) + 2

s cΛc5

G1

+ 1

2 1 t−m2

k

mΛmpc5 − 3s(c1 + 3bpcs − 2bp)
± mN∗mp

×

3(bp + bΛ) − 2

s cΛc5

G2 −

2c5 t−m2

k

cΛ ± mN∗mΛ
G3,

(20)

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A6 =

1 2mp

mΛmp + 3scs − cΛ
± mN∗
3mpcs + 1

s mpcΛ − mΛ

G1

+ 1

2

mp
cΛ − mpmΛ
± mN∗
3(c1 + bpcs) − 1

s cΛm2 p + mpmΛ

G2

+2

mpcΛ − mΛs
± mN∗
mpmΛ − cΛ
G3,

(21) where G (i) = g(i)gKΛN∗ 3m4

N∗(s − m2 N∗ + imN∗ΓN∗),

(22) which g(i) are as in equation (12) for i = 1, ..., 5.

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we also define some variables in our calculation as below bp = p · k = 1 2(s − k2 − m2

p),

(23) bΛ = pΛ · k = 1 2(k2 + m2

Λ − u),

(24) bq = q · k = 1 2(k2 + m2

K − t),

(25) cΛ = (p + k) · pΛ, ck = (p + k) · k, cs = 1 − 1 s cΛ, (26) c± = 1 √s ± mp , d± = 1 s (mΛ √s ± cΛ), (27) c1 = bΛ − 1 s cΛck, c2 = m2

Λ − 1

s c2

Λ,

c3 = 1 s c2

k − k2,

(28) c4 = 2bp + k2, c5 = 4bp + k2, (29)

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Fitting Procedure

We have included all the latest K +Λ photoproduction data which consist of differential cross section, recoil polarization, photon and target asymmetries, beam-recoil double polarization Cx and Cz, as well as Ox′ and Oz′. (7433 data points) We have fitted a number of resonance coupling constants, background parameters, mass and width of resonaces. In total, model A (B) contains 54 (66) parameters. The minimization process of χ2/N is performed by using the CERN- MINUIT code. χ2 N = 1 Nmax − Npar

Nmax

i=1

σi(exp) − σi(th) δσi(exp) 2 (30)

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Fitting Result

Table: Fitting result with N=7433 data points

Model A Model B gKΛN/ √ 4π

3.00
4.40

gKΣN/ √ 4π 0.90 0.90 ΛB (GeV) 0.71 0.70 ΛR (GeV) 1.87 1.89 θhad. (deg) 144 90 φhad. (deg) 180 46 χ2 25548 20719 Npar 54 66 χ2/N 3.46 2.81

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Plots of Observables

0.5 1 1.5 2 2.5 3 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

σtot (µb) W (GeV)

p ( γ , K + ) Λ CLAS 2006 Kaon-Maid Model A Model B

Figure: Total cross section of kaon photoproduction

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0.05 0.1 0.15 0.2 0.25 0.3 1.6 1.8 2 2.2 2.4 2.6 2.8

dσ / dΩ (µb/sr) W (GeV) cos θ = 0.30

Figure: Differential cross section of kaon photoproduction

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1
0.5

0.5 1 1.6 1.7 1.8 1.9 2 2.1 2.2

W (GeV) cos θ = 0.50

P

1
0.5

0.5 1 1.6 1.7 1.8 1.9 2 2.1 2.2

W (GeV) cos θ = 0.50

T

0.4
0.2

0.2 0.4 0.6 0.8 1 1.6 1.7 1.8 1.9 2 2.1 2.2

W (GeV) cos θ = 0.50

Σ

Figure: Recoil polarization (P), target (T) & photon (Σ) asymmetry.

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1
0.5

0.5 1 1.6 1.7 1.8 1.9 2 2.1 2.2

W (GeV) cos θ = 0.50

Ox

1
0.5

0.5 1 1.6 1.7 1.8 1.9 2 2.1 2.2

W (GeV) cos θ = 0.50

Oz

Figure: Beam-recoil double polarization (Ox′ & Oz′)

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Summary

We have analyzed kaon photoproduction p(γ, K +)Λ by using isobar model with two different formulation of spin 3/2 nucleon resonances. We have derived the corresponding amplitudes and decomposed them into invariant matrices. All available nucleon with spin up to 3/2 listed by PDG are taken into account. The corresponding coupling constants are extracted by fitting all available kaon photoproduction data. It is found that Pascalutsa’s prescription leads to a better agreement with the experimental data.

A. J. Arifi (RCNP)

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Thank You

A. J. Arifi (RCNP)

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Effects of Spin 3/2 Nucleon Resonances in Kaon Photoproduction

RCNP, Osaka University1 Physics Department, University of Indonesia2 J-Park Hadron Workshop March 2nd, 2016

Publication

Nucleon resonances with spin 3/2 and 5/2 in the isobar model for kaon photoproduction

Overview

1

Introduction Motivation Objective

2

Formalism Isobar Model Fitting Procedure

3

Result and Discussion Fitting Result Plots of Observables

4

Summary

Motivation

Kaon photoproduction provides an essential tool for the studies of strange hadron.

Figure: Kaon photoproduction on proton (a) and nucleus (b)

Motivation

Experimental side — abundant data are available on kaon photoproduction and we have included 7433 data points in this

CLAS Collaboration, GRAAL Collaboration, LEPS Collaboration, MAMI Collaboration.

Theoretical side — we have used Isobar model to describe kaon

the formulation of nucleon resonances spin 3/2 or higher is still plagued with the problem of consistency.

Objective

we aim to study the effect of inclusion of spin 3/2 nucleon resonances with two different formulations using isobar model. Model A R.A. Adelseck, C. Bennhold, and L.E. Wright, Phys. Rev. C32, 1681(1985)

2613 (1996) Model B

Channels of Kaon Photoproduction

Figure: (a) Baryon Octet (b) Pseudoscalar meson nonet

based on conservation of strangeness and isospin, there are six possible channels of kaon photoproduction

Kaon Photo- & Electroproduction

Figure: (a) Kaon Photoproduction (b) Kaon Electroproduction

Feynman Diagram of Kaon Photoproduction

Figure: (a) s-channel (b) u-channel (c) t-channel of kaon photoproduction

Contribution of mediator particles are categorized into background and

M = Mbackground + Mresonances (3)

Hadronic Form Factor

We include the hadronic form factors in the hadronic vertices by adopting the method developed by Haberzettl. ˜ F = F(Λ, s) sin2 θhad cos2 φhad + F(Λ, u) sin2 θhad sin2 φhad +F(Λ, t) cos2 θhad, (4) where, F(Λ, x) = Λ4 Λ4 + (x − m2)2 (5) and x = s, t, u (mandelstam variables)

Nukleon Resonances

Table: Status, mass and width of nucleon resonances used in our model

Resonances Status Mass (MeV) Width (MeV) N(1440)P11 **** 1430 ± 20 350 ± 100 N(1520)D13 **** 1515 ± 5 115+10

−15

N(1535)S11 **** 1535+20

−10

150 ± 25 N(1650)S11 **** 1655+15

−10

140 ± 30 N(1700)D13 *** 1700 ± 50 150+100

−50

N(1710)P11 *** 1710 ± 30 100+150

−50

N(1720)P13 **** 1720+30

−20

250+150

−100

N(1875)D13 *** 1875+45

−55

200 ± 25 N(1880)P11 ** 1870 ± 35 235 ± 65 N(1895)S11 ** 1895 ± 15 90+30

−15

N(1900)P13 *** 1900 250 N(2120)D13 ** 2140 330 ± 45

Model A

Propagator P3/2

µν

= / p + / k + √s 3(s − m2

N∗ + imN∗ΓN∗)

s (p + k)µ(p + k)ν − 1 √s {γµ(p + k)ν − γν(p + k)µ}

(6) Electromagnetic vertex factor Γν(±)

N∗pγ

= ga

N∗pγ

k2

1 √s ± mp (k2/ ǫ − k · ǫ/ k)kν +gb

N∗pγ

p · ǫkν − p · kǫν (√s ± mp)2 + gc

N∗pγ

k · ǫkν − k2ǫν (√s ± mp)2

(7) Hadronic vertex factor Γµ(±)

KΛN∗ = gKΛN∗

mN∗ pµ

ΛΓ∓,

(8)

Model B

Propagator P3/2

µν

Resonances Status Mass (MeV) Width (MeV) N(1440)P11 1430 ± 20 350 ± 100 N(1520)D13 1515 ± 5 115+10

200 ± 25 N(1880)P11 1870 ± 35 235 ± 65 N(1895)S11 1895 ± 15 90+30

N(1900)P13 * 1900 250 N(2120)D13 2140 330 ± 45