# Eta and kaon production in a chiral quark model - IJS · 2015-07-13 ·...

### Transcript of Eta and kaon production in a chiral quark model - IJS · 2015-07-13 ·...

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Eta and kaon production in a chiral quark model

Bojan Golli

Faculty of Education, University of Ljubljana andJ. Stefan Institute, Ljubljana, Slovenia

Exploring hadron resonances, Bled, 10 July 2015

in collaboration with

Simon Sirca (Ljubljana) and Manuel Fiolhais, Pedro Alberto(Coimbra)

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Outline

I Motivation

I A short review of the coupled channel formalism

I Low and intermediate energy resonances in the CloudyBag Model

I Preliminary results for γp→ ηN, K Λ, K Σphotoproduction involvingE0+, E1+, E2− and M1−, M1+, M2− multipoles

I Conclusions

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Motivation

I What is the principal mechanism that explains the structure ofbaryon resonances in low and intermediate energy regime: excitationof the quark core or dynamical generation?

I The pion scattering amplitudes can be well reproduced in variousmodels by a suitable adjustment of parameters; a better criterion toasses models is the ability to reproduce the meson photo- andelectro-production amplitudes.

I An independent test is pion and photon production of strangemesons (baryons), in particularly in those channels in which thebackground contribution is expected to be small.

I Introducing strange degrees of freedom it is desirable that no newadjustable parameters are introduced in the model; a SU(3)extension of the Cloudy Bag Model is such an example.

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Some general features of the method

I Baryons are treated as composite particles from the very beginning;the strong and electro-weak form-factors are derived from baryoninternal structure and not inserted a posteriori ; as a consequence themethod introduces a much smaller number of free parameters.

I The physical resonances appear as linear superpositions of bareresonances.

I The bare quark-meson and quark-photon vertices are modifiedthrough meson loops as well as through mixing of resonances andcoupling to the background.

I The meson cloud around baryons is included in a consistent way alsoin the asymptotic states.

I The method yields a symmetric K matrix and hence respects theunitarity of the S matrix.

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Constructing the K-matrix

Aim: to include many-body states of quarks in the scattering formalism(Chew-Low type approach)

Construct K-matrix in the spin-isospin (JI) basis:

KJIM′B′MB = −π

√ωMEBkMW

〈ΨMBJI (W)||VM′(k)||ΨB′〉

by using principal-value (PV) states

|ΨMBJI (W)〉 =

√ωMEBkMW

{[a†(kM)|ΨB〉

]JI− P

H−W[V(kM)|ΨB〉]JI

}normalized as

〈ΨMB(W)|ΨM′B′(W′)〉 = δ(W−W′)δMB,M′B′(1 + K2)MB,MB

dressed states

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Ansatz for the channel PV states

|ΨMBJI 〉 =

√ωMEBkMW

{[a†(kM)|ΨB〉]JI

+∑R

cMBR |ΦR〉

+ ∑M′B′

∫ dk χM′B′MB(k, kM)

ωk + EB′(k)−W[a†(k)|ΨB′〉]JI

}

Above the meson-baryon (MB) threshold:

KM′B′MB(k, kM) = π√

ωMEBkMW

√ωM′EB′kM′W

χM′B′MB(k, kM)

free meson(defines the channel)

bare (genuine) baryons (3q)

meson “clouds”with amplitudes χ

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Assumption about the two- and three-meson channels

2π, πη and 3π decays through intermediate baryons (∆(1232),N(1535)S11, . . .) or mesons (σ, ρ, . . .)

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Intermediate baryon state

For µ ∼ MR, the intermediate baryon state can be written as

|Ψα(µ)〉 = ∑β

(1 + K2

)− 12

βα|Ψβ(µ)〉

=1√2π

√ΓMB(µ)√

(MR − µ)2 + 14 Γ2(µ)

|ΨR〉+ . . . ,

µ is the invariant mass of the baryon-meson system into which theintermediate baryon decays.

ΨR is the three-quark state surrounded by the meson cloud:

|ΨR〉 = Z−12R

[|ΦR〉 −∑

MB

∫ dk VMBR(k)

ωk + EB −W[a†(k)|ΨB〉]JI

].

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Equations for meson amplitudes (Lippmann-Schwinger)

χM′B′MB(k, kM) = −∑R

cMBR VM′

B′R(k) +KM′B′MB(k, kM)

+ ∑M′′B′′

∫dk′KM′B′M′′B′′(k, k′)χM′′B′′MB(k′, kM)

ω′k + EB′′(k′)−W

with kernels

KM′B′MB(k, k′) = ∑B′′

f B′′BB′

VM′B′′B′(k

′)VMB′′B(k)

ωk + ω′k + EB′′(k)−W

(f B′′BB′ are spin-isospin coefficients)

The solution assumes the form

χM′B′MB(k, kM) = −∑R

cMBR V

M′B′R(k) +D

M′B′MB(k, kM)

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Solving the coupled equations

Dressed vertices then satisfy:

VMBR(k) = VM

BR(k) + ∑M′B′

∫dk′KMB M′B′(k, k′)VM′

B′R(k′)

ω′k + EB′(k′)−W

and similarly the background part of the amplitude:

DM′B′MB(k, kM) = KM′B′MB(k, kM)

+ ∑M′′B′′

∫dk′KM′B′M′′B′′(k, k′)DM′′B′′MB(k′, kM)

ω′k + EB′′(k′)−W

The coefficients cMBR′ in front of the quasi-bound states satisfy a set of

equations:

∑R′

ARR′(W) cMBR′ (W) = VM

BR(kM)

ARR′ = (W−M0R)δRR′ + ∑

B′

∫dkVM′

B′R(k)VM′B′R′(k)

ωk + EB′(k)−W

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Mixing of bare resonances

To solve the set of equations, diagonalize A to obtain U, along with thepoles of the K matrix, and wave-function normalization Z:

UAUT =

ZR(W)(W−MR) 0 00 ZR′(W)(W−MR′) 00 0 ZR′′(W)(W−MR′′)

As a consequence, ΦR mix:

|ΦR〉 = ∑R′

URR′ |ΦR〉 VBR =1√

ZR(W)∑R′

URR′VBR′

Solution for the K matrix

KMB,M′B′ = π√

ωMEBkMW

√ωM′EB′kM′W

[∑R

VMBRV

M′B′R

(MR −W)+DMB,M′B′

]and for the T matrix

TMB,M′B′ = KMB,M′B′ + i ∑M′′K′′

TMB,M′′B′′KM′′B′′ ,M′B′

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Including the γN channel

Only the strong TMB,M′B′ appears on the RHS:

TMB,γN = KMB,γN + i ∑M′B′

TMB,M′B′KM′B′ ,γN

where

KM′B′ ,γN = −π

√ωγENkγW 〈Ψ

M′B′JI ||Vγ||ΨN〉

The meson production amplitudes, proportional to KMB,γN, are obtainedby solving the Heitler equation:

MMB γN =MKMB γN + i ∑

M′B′TMB M′B′MK

M′B′ γN .

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

The EM interaction

The current appearing in Vγ is expanded as

jEM(r) · εMeikz = ∑Llmn

il√

4π(2l + 1) jl(kr)CLMl01M CLM

lm1n Ylm(r)jEM,n(r)

Quark contribution to the current

j qEM µ =

3

∑i=1

[ψ†

j= 32αµ(i)ψS + ψ†

j= 12αµ(i)ψS

]12 τ0(i)

Pion contribution

j πEM µ = i ∑

ttπt(r)∇µπ−t(r) .

Kaon contribution

j KEM µ = i

[K−(r)∇µK+(r)− K+(r)∇µK−(r)

].

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Underlying quark model

The Cloudy Bag Model extended to pseudo-scalar SU(3) octet and the ρmeson:

Hint = −∫

dr[

i2f

qλa(γ5φa + γ ·Aa)q δS +1

4f 2 qλaγµq(φ× ∂µφ)aθV

],

a = 1, 2, . . . , 8

provides a consistent parameterization of the baryon-meson andbaryon-photon coupling constants and form factors in terms of ”fπ” andthe bag radius Rbag.

Parameters:Rbag = 0.83 fm (from the ground state calculations)f π = 76 MeV (reproducing the experimental value of gπNN)f K = 1.2 f π, f η = f π or 1.2 f π, f ρ = f π

similar results for 0.75 fm < Rbag < 1.0 fm

Free parameters: bare masses of the resonances

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

P-wave resonances

P11 and P33 resonances:

Single-quark excitation 1s→ 2s1/2

N∗ = cN|(1s)3〉+ cR|(1s)2(2s)1〉

P13 and P31 resonances: single-quark excitation 1s→ 1d1/2 and 1d3/2

N(1720) = −sin ϑs|483/2〉+ cos ϑs|283/2〉= cl

D|(1s)21d5/2〉+ clA|(1s)21d3/2〉MA + cl

S|(1s)21d3/2〉MS ,

N(1900) = cos ϑs|483/2〉+ sin ϑs|283/2〉= cu

D|(1s)21d5/2〉+ cuA|(1s)21d3/2〉MA + cu

S|(1s)21d3/2〉MS ,

ϑsis a free parameter.

∆(1910) = |2101/2〉 = |(1s)21d3/2〉 .

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Meson scattering in P11 and P13 wave

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Pion photoproduction in P11 and P13 wave

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Meson scattering in P33 and P31 wave

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Pion photoproduction in P33 wave

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

S-wave resonances

Single-quark excitation 1s→ 1p1/2 and 1s→ 1p3/2S11 resonances:

N(1535) = −sin ϑs|481/2〉+ cos ϑs|281/2〉= cl

P|(1s)21p3/2〉+ clA|(1s)21p1/2〉MA + cl

S|(1s)21p1/2〉MS ,

N(1650) = cos ϑs|481/2〉+ sin ϑs|281/2〉= cu

P|(1s)21p3/2〉+ cuA|(1s)21p1/2〉MA + cu

S|(1s)21p1/2〉MS .

ϑs = ϑs(W) determined by resonance mixing through pion loops

S31 resonance:

∆(1650) = |2101/2〉 = −√

83 |(1s)21p3/2〉+ 1

3 |(1s)21p1/2〉 ,

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Meson scattering in S11 and S31 wave

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Pion photoproduction in S11 and S31 wave

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

D-wave resonances in quark models

Single-quark excitation 1s→ 1p1/2 and 1s→ 1p3/2

D13 resonances:

N(1520) = − sin ϑd|483/2〉+ cos ϑd|283/2〉= cl

S|(1s)21p3/2〉MS + clA|(1s)21p3/2〉MA + cl

P|(1s)21p1/2〉 ,

N(1700) = cos ϑd|483/2〉+ sin ϑd|283/2〉= cu

S|(1s)21p3/2〉MS + cuA|(1s)21p3/2〉MA + cu

P|(1s)21p1/2〉 .

D33 resonance: ∆(1700) = |2103/2〉 =√

53 |(1s)21p3/2〉 − 2

3 |(1s)21p1/2〉 ,

Modification of quark-meson coupling constants with respect to thecorresponding quark-model values:

D13: gπNN∗ = 1.43 gπNN∗ (QM), gs−waveπ∆N∗ = 0.58 gs−wave

π∆N∗ (QM).

D33: gπNN∗ = 2.4 gπNN∗ (QM),

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Meson scattering in D13 and D33 wave

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Pion photoproduction in D13 and D33 wave

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Meson photoproduction amplitudes

MKMB,γN = fMJT NMB

[〈[ΨBaM]|Vγ|ΨN〉 −

VMBR(kM)〈ΨR|Vγ|ΨN〉

WR −W

],

MMB γN =MKMB γN + i ∑

M′B′TMB M′B′MK

M′B′ γN .

Isospin decomposition in the case of KΣ channels:

A(γ + p→ K+Σ0) = A(1/2)p +

23

A(3/2)

A(γ + p→ K0Σ+) =√

2 A(1/2)p − 1

3A(3/2)

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Eta photoproduction

Re MAID

Im MAID

Re BonnGachina 2014

Im BonnGachina 2014

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Eta photoproduction

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

p + γ→ K+Λ photoproduction

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

p + γ→ K+Λ photoproduction

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

Isospin decomposition in the case of KΣ channels:

A(γ + p→ K+Σ0) = A(1/2)p +

23

A(3/2)

A(γ + p→ K0Σ+) =√

2 A(1/2)p − 1

3A(3/2)

Only Kaon-pole background included (in the case of K+Σ0)

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

p + γ→ KΣ photoproduction, E multipoles, J = 32

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

p + γ→ KΣ photoproduction, M multipoles, J = 12

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

p + γ→ KΣ photoproduction, E multipoles, J = 12

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

p + γ→ KΣ photoproduction, M multipoles, J = 12

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

p + γ→ KΣ photoproduction, M multipoles, J = 32

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

p + γ→ KΣ photoproduction, E multipoles, J = 32

Introduction Formalism Electro-production Model P wave S wave D wave ηN and KΛ KΣ Conclusions

(Very preliminary) conclusions

I ηN channels: dominating E0+ multipole well reproduced. Othermultipoles small, large experimental uncertainty; our model predictsorder of magnitude.

I KΛ channels: similar conclusions as in the case of ηN channel

I KΣ channels: the model predicts E0+ to be dominated by the S11partial wave (rather than S31)

I M1− underestimated in our approach probably due to non-inclusionof background processes; similar conclusion for E2− and M2−

I Surprisingly good agreement for M1+, predicted to be dominated bythe P33 resonances (rather than P13) in agreement with thequark-model scenario.

I Reasonably good agreement for E1+ with the main contributioncoming from the P13 resonances (again in agreement with thequark-model scenario.