Meson cloud e ects in the electromagnetic hadron structure · Constituent quark model, M N = M qqq...

Basics Hadronic level Quark level Outlook

Meson cloud effectsin the electromagnetic hadron structure

Daniel Kupelwieser

Thesis supervisor: Wolfgang Schweiger

Collaborators: Elmar Biernat, Regina Kleinhappel

Universitat Graz

Graz–Jena monitoring workshop, Graz, October 2012

D. Kupelwieser (Graz) Meson cloud effects 1 / 23


Contents

1 Basic Definitions and Motivation

2 Hadronic level

3 Quark level

4 Outlook



Electron–nucleon scattering

Basic setting:

ke //

kγ

k ′e

kN //⊗

k ′N

ke // k ′e

kN //⊗kγ

k ′N

Electron–nucleon scattering via photon exchange

Constituent quark model, MN = Mqqq + Vconf

Point form of relativistic quantum mechanics

Time-ordered diagrams

Photon–nucleon vertex dressed with single-pion exchange



Form factors

Electromagnetic interaction Lagrangian:

Lint = JµN Aµ

To account for inner electromagnetic nucleon structure,modify nucleon current:

JµN(~kN , µN , ~k′N , µ

′N) = e uµN (~kN)

(F1(q2) γµ + F2(q2)

i qν σµν

2mN

)uµ′N (~k ′N)

F1(q2) . . . Dirac form factor, F1(0) = 1

F2(q2) . . . Pauli form factor, F2(0) = 0

σµν := i2 [γµ , γν ]



Point form

Point form of relativistic dynamics:Quantization surface: Spacetime hyperboloid

x2 ≡ t2 − ~x2 = τ2 = const.

(Spacelike hypersurface, invariant under Lorentz group)

Intrinsic Lorentz covariance

Dynamic (interaction-dependent) Poincare generators: only Pµ

Kinematic generators: {~K ,~J} (Lorentz group, nice!)



Bakamjian–Thomas construction

Given: n-particle system with overall 4-momentum Pµ.Bakamjian–Thomas construction:

Pµ = Pµ0 + Pµint = (M0 + Mint)Vµ0 , (P2 = M2)

Mint has to commute with V µ, ~K and ~J (like Pµ does).

Overall velocity conserved at vertices

Advantage: Interactions may be instantaneous,system stays covariant!



Velocity states

Useful basis (V := V0): Velocity states

∣∣{~pi , σi}⟩ −→ ∣∣V ; {~ki , µi}⟩

withn∑

i=1

~ki = 0

Behavior under Lorentz-transformation Λ:

U(Λ)∣∣V ; {~ki , µi}

⟩=∑{µ′i}

∣∣∣ΛV ; {R(ΛV )ki , µ′i}⟩∏

i

D12

µ′iµi(R(ΛV ))

(Spins get transformed with same Wigner rotation!)



Eigenvalue equation

Coupled-channel approach:MNe Kγ Kπ 0

K †γ MNeγ 0 KπK †π 0 MNπe Kγ0 K †π K †γ MNπeγ

∣∣Ne⟩∣∣Neγ⟩∣∣Nπe⟩∣∣Nπeγ⟩ = m

∣∣Ne⟩∣∣Neγ⟩∣∣Nπe⟩∣∣Nπeγ⟩

M...: Relativistic energies of particles in each channel

K (†)... : Particle creation/ annihilation operators (∝ Lint)m: Mass eigenvalue of the whole system.



Feshbach reduction & Optical potential

After Feshbach reduction

neglecting self-energy contributions and double loops,

with P... := (m −M...)−1 :

(m −MNe)∣∣Ne⟩ =

= Kγ PNeγ K†γ

∣∣Ne⟩++ Kγ PNeγ Kπ PNπeγ K

†γ PNπe K

†π

∣∣Ne⟩++Kπ PNπe Kγ PNπeγ K

†π PNeγ K

†γ

∣∣Ne⟩++Kπ PNπe Kγ PNπeγ K

†γ PNπe K

†π

∣∣Ne⟩ =: Vopt

∣∣Ne⟩Vopt . . . optical potential



Contents


2 Hadronic level

3 Quark level

4 Outlook



Hadronic unity operators

Insert hadronic unity operators, e.g.

INeγ =∑∫

DV DkN Dkγ(ωN + ωe + ωγ)3

2ωe(−gµγµγ )

∣∣VNeγ⟩⟨VNeγ∣∣For example, framed line becomes

Kγ PNeγ INeγ Kπ PNπeγ INπeγ K †γ PNπe INπe K †π∣∣Ne⟩

Propagators P... assume eigenvalues

Get matrix elements of vertex operators K (†)...

(sum over emitting/absorbing particles)



Spectator conditions & Nucleon current

Spectator conditions:When two particles interact, others stay unaffected, e.g.:

⟨V ′N ′e ′γ′

∣∣K †Nγ∣∣VNe⟩ = ∆VV ′∆ee′(−1)√

M′3NeγM

3Ne

⟨VN ′γ′

∣∣K †Nγ∣∣VN⟩with ⟨

VN ′γ′∣∣K †Nγ∣∣VN⟩ = JNν(~kN , µN , ~k

′N , µ

′N) ενµ′γ (~k ′γ)

(JµN . . . nucleon current, contains form factors!)



Photon coupling to bare nucleon

Finally, we get the following 10 time-ordered diagrams:⟨V ′N ′e ′

∣∣Vopt

∣∣VNe⟩ = . . .

ke //

kγ

k ′e

kN //⊗ k ′N

+

ke // k ′e

kN //⊗

kγ

k ′N

+ . . .

− 1

m3∆VV ′ JνN(~kN , µN , ~k

′N , µ

′N)

gνλq2

Jλe (~ke , µe , ~k′e , µ′e) + . . .



Photon coupling to dressed nucleon

. . .+

ke //

kγk ′e

kN //⊗

kπ

⊗ ⊗ k ′N+

ke // k ′e

kN //⊗

kπ

⊗

kγ

⊗ k ′N+

+

ke //

kγk ′e

kN //⊗

kπ

⊗ ⊗ k ′N+

ke // k ′e

kN //⊗

kπ

⊗kγ

⊗ k ′N+ . . .

. . .+

∆VV ′

4m3

gνλq2

∑α′′Nα

′′′N απ

∫Dkπ

1

ω′′N ω′′′N

(m −MN′′πe)−1×

(m −MN′′′πe)−1 Q5π(N ′′′,N ′) JνN(N ′′,N ′′′) Jλe (e, e ′)Q5

π(N,N ′′)

+ . . .



Photon coupling to pion

. . .+

ke //kγ

k ′e

kN //⊗kπ

⊗ k ′N

+

ke // k ′ekγ

kN //⊗kπ

⊗ k ′N

+

+

ke //

kγ

k ′e

kN //⊗kπ ⊗ k ′N

+

ke // k ′e

kγ

kN //⊗kπ ⊗ k ′N

. . .+

∆VV ′

4m3

gνλq2

∑α′′Nα

′′′N απ

∫Dkπ

1

ω′′N ω′′′N

(ω′′N − ωN + ω′′π

)−1×

(ω′′N − ωN + ω′′′π

)−1Jλe (e, e ′) Q5

π(N,N ′′) Jνπ(π′′, π′′′) Q5π(N ′′,N)



Hadronic diagrams

i.e., ⟨V ′N ′e ′

∣∣Vopt

∣∣VNe⟩ = . . .

ke //

kγ

k ′e

kN //⊗ k ′N

+

ke //

kγ

k ′e

kN //⊗

kπ

⊗ ⊗ k ′N+

ke //

kγk ′e

kN //⊗kπ⊗ k ′N



Contents


2 Hadronic level

3 Quark level

4 Outlook



Quark-level unity operators

Insert quark-level unity operators, e.g.

Iqqqe =∑∫

DV Dke Dkq2 Dkq3

(ωq1 + ωq2 + ωq3 + ωe)3

2ωq1

∣∣Vqqqe⟩⟨Vqqqe∣∣Photon coupling to bare nucleon now 6 diagrams:

⟨V ′N ′e ′

∣∣V 0opt

∣∣VNe⟩ =⟨V ′N ′e ′

∣∣KeγPNeγINeγIqqqeγK †q1γIqqqe∣∣VNe⟩+

+⟨V ′N ′e ′


+⟨V ′N ′e ′


+⟨V ′N ′e ′

∣∣IqqqeKq1γIqqqeγINeγPNeγK†eγ

∣∣VNe⟩++⟨V ′N ′e ′


∣∣VNe⟩++⟨V ′N ′e ′


∣∣VNe⟩D. Kupelwieser (Graz) Meson cloud effects 18 / 23


Spectator conditions & Quark current

Spectator condition for single “struck” quark :⟨V ′q′q′q′e ′γ′

∣∣K †q1γ

∣∣Vqqqe⟩ =

=∆VV ′∆ee′∆q2q′2∆q3q′3

(−1)√M ′3

qqqeγM3qqqe

⟨Vq′1γ

′∣∣K †q1γ

∣∣Vq1

⟩(etc.)

with electromagnetic interaction (pointlike quark!)⟨Vq′1γ

′∣∣K †q1γ

∣∣Vq1

⟩= e Qq1

[uµq1

(~kq1) γν uµ′q1(~k ′q1

)]ενµ′γ (~k ′γ)

The IqqqeγINeγ give rise to three-quark wave functions⟨V ′3q′e ′γ′

∣∣VNeγ⟩ = NNeγ3q′ ∆VV ′∆ee′∆γγ′⟨3q′∣∣N⟩



Quark-level diagrams

Have to treat following diagrams:⟨V ′N ′e ′

∣∣Vopt

∣∣VNe⟩ =

= 3×ke //

kγ

k ′e

kN I I // k ′N

+

ke //

kγ

k ′e

⊗

kN //⊗kπ

⊗ k ′N

+ 3×

ke //

kγ

k ′e

kN ⊗

kπ

I I //⊗ k ′N



Bare photon–nucleon vertex

Results for first diagram:

Hadron picture (as before):⟨V ′N ′e ′

∣∣Vopt

∣∣VNe⟩ = −∆VV ′

m3 q2Jeν(~ke , µe , ~k

′e , µ′e) JνN(~kN , µN , ~k

′N , µ

′N)

Quark picture:⟨V ′N ′e ′

∣∣V 0opt

∣∣VNe⟩ = −∆VV ′

m3 q2Jeν(~ke , µe , ~k

′e , µ′e)× . . .

. . . ×

2√ω′NωN

3∑i=1

∑(µq1 ,µq2 ,µq3 ,µ

′qi

)

∏j 6=i

(∫d3kqjωqj

)×

× 1

ω′qi ωqi

√ω′q1

ω′q2ω′q3

(∑ω′qk)√(∑

ω′qk)

√ωq1ωq2ωq3

(∑ωqk

)√(∑ωqk

) ×

×⟨N ′∣∣q′i {qj 6=i}

⟩ ⟨q1 q2 q3

∣∣N⟩ Jνqi (~kqi , µqi , ~k ′qi , µ′qi )D. Kupelwieser (Graz) Meson cloud effects 21 / 23


Contents


2 Hadronic level

3 Quark level

4 Outlook



Outlook

Finally, what’s left to do:

Insertion of three-quark wave functionsfrom sophisticated constituent quark model (e.g. XCQM)

Extract electromagnetic form factors(also for diagrams with pion loop)

Use these form factors in hadronic diagrams

Pion form factors from analog procedure (R. Kleinhappel)

Sum over all hadronic diagrams yields overall form factorsfor entire problem

Thank you!


Meson cloud e ects in the electromagnetic hadron structure · Constituent quark model, M N = M qqq...

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Transcript of Meson cloud e ects in the electromagnetic hadron structure · Constituent quark model, M N = M qqq...