Electromagnetic Meson Form Factor from a Relativistic ...Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach Elmar P. Biernat, W. Schweiger (University of

Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary

Electromagnetic Meson Form Factor from a

Relativistic Coupled-Channel Approach

Elmar P. Biernat, W. Schweiger (University of Graz)W. H. Klink (University of Iowa), K. Fuchsberger (CERN)

July 2, 2009

E.P. Biernat

Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach


Meson Form Factors and Point-Form RQM

Pion Form Factor

Result

Summary

E.P. Biernat



Meson Form Factors

◮ electron-hadron scattering experiments ⇒ hadrons: bound states ofconstituents

◮ meson M: composite system of quark-antiquark,e.g. pion π+: u d+. . .⇒ M: non-pointlike interactions with the electromagnetic field

◮ bound state dynamics cannot be treated perturbatively⇒ no full understanding of electromagnetic structure of hadrons fromQCD

◮ phenomenological quantity: meson form factor

describes non-pointlike aspect of hadronic structure of M

E.P. Biernat



Point-Form Relativistic Quantum Mechanics

◮ Relativistic Quantum Mechanics (RQM): Poincare invariant treatment ofsystems with finite many degrees of freedom

◮ find a representation of generators{

Pµ, Jµν}

on H that satisfy Poincare

algebra[

Pµ, Pν]

= 0,[

Jµν , Pρ]

= i(

gνρPµ − gµρPν)

[

Jµν , Jλσ]

= −i(

gµλJνσ − gνλJµσ + gνσ Jµλ − gµσ Jνλ)

1. free theory: easy to achieve

2. interacting theory: non-linear constraints on interaction terms that

are hard to satisfy

◮ point form of dynamics: Pµ dynamical, Jµν kinematical⇒ manifest Lorentz-covariant formulationDirac; Rev.Mod.Phys., 1949

E.P. Biernat



Bakamjian-Thomas Construction

◮ Bakamjian-Thomas construction: interactions included in mass operatorBakamjian, Thomas; Phys.Rev.,1953

◮ point-form RQM: Pµ

free + Pµ

int =(

Mfree + Mint

)

Vµ

free

⇒{

M, Vfree, Jµν

free

}

satisfy certain commutation relations

M2 Casimir ⇒ linear constraints on interaction terms

◮ point-form QFT (canonical field quantization on a space-timehyperboloid):E.P.B., Klink, Schweiger, Zelzer; Ann.Phys., 2008

Vµ

free cannot be factored out of Pµ

E.P. Biernat



Velocity States

◮ multiparticle states, basis: velocity statesKlink; Phys.Rev.C,1998

(simultaneous eigenstates of Mfree and Vµfree )

|v ; k1, µ1; k2, µ2; . . . ; kn, µn〉 := U (Bc (v)) |k1, µ1; k2, µ2; . . . ; kn, µn〉

◮ orthogonality relation

〈v ′; k′1, µ

′1; k

′2, µ

′2; . . . ; k

′n, µ

′n| v ; k1, µ1; k2, µ2; . . . ; kn, µn〉

∝ v0 δ3(v′ − v)k0n

(∑

ni=1

k0i )

3

(∏n−1i=1 k0

i δ3(k′i − ki )

) (∏n

i=1 δµ′

iµi

)

◮ completeness relation

11,2,...,n ∝ ∑

{µi}

∫d3vv0

(∏n−1

i=1d3ki

k0i

)(∑ n

i=1 k0i )

3

k0n

×|v ; k1, µ1; k2, µ2; . . . ; kn, µn〉〈v ; k1, µ1; k2, µ2; . . . ; kn, µn|

E.P. Biernat



Meson Form Factor

◮ usual approaches: phenomenological ansatz for electromagnetic current

◮ our approach:

1. treat electron-meson scattering within Bakamjian-Thomas approach⇒ correct relativistic behaviour guaranteed

2. try to extract hadronic current from 1-photon-exchange opticalpotential

3. look for current conservation and cluster properties

E.P. Biernat



Elastic Electron-Meson Scatteringe(pe , σe) + M (pM) → e(p′

e , σ′e) + M (p′

M)

photon

electron

quark levelhadronic level antiquark

quarkmeson

form factor

calculate optical potential on hadronic and quark level

⇒ extract form factor from comparison

E.P. Biernat



2-Channel SystemHadronic Level

◮ mass eigenstate |Ψ〉 =

(|ΨeM〉|ΨeMγ〉

)

◮ coupled-channel mass operator M =

(MeM K†

K MeMγ

)

◮ mass eigenvalue equationM|Ψ〉 = m|Ψ〉 ⇒ system of coupled equations:

MeM |ΨeM〉 + K†|ΨeMγ〉 = m|ΨeM〉MeMγ |ΨeMγ〉 + K |ΨeM〉 = m|ΨeMγ〉

◮ solve for |ΨeMγ〉 ⇒ non-linear eigenvalue equation for m

K†(

MeMγ + m)−1

K︸︷︷︸

=Vopt(m)

|ΨeM〉 =(

MeM + m)

|ΨeM〉

E.P. Biernat



Vertex InteractionHadronic Level

◮ matrix element of optical potential in velocity states basis⟨

v ′; k′e , µ

′e ; k

′M

∣∣∣Vopt (m)

∣∣∣ v ; ke , µe ; kM

⟩

=

⟨

v ′; k′e , µ

′e ; k

′M

∣∣∣∣K†

(

MeMγ + m)−1

1eMγK

∣∣∣∣v ; ke , µe ; kM

⟩

◮ electromagnetic vertex interactionKlink; Nucl.Phys.A, 2003⟨

v ′; k′e , µ

′e ; k

′M ; kγ , µγ

∣∣∣K

∣∣∣ v ; ke , µe ; kM

⟩

∝ v0δ3 (v − v′)

×⟨

v ; k′e , µ

′e ; k

′M ; kγ , µγ

∣∣∣

(

F (∆m, . . . )LMγ

int (0) + Leγ

int (0))∣∣∣ v ; ke , µe ; kM

⟩

assumption: total-velocity conservation at electromagnetic vertices

(approximation)

E.P. Biernat



Optical PotentialHadronic Level

◮ matrix element of optical potential⟨

v ′; k′e , µ

′e ; k

′M

∣∣∣Vopt (m)

∣∣∣ v ; ke , µe ; kM

⟩

∝ v0δ3 (v − v′) F(Q2, . . .

)jµ (k′

M ; kM)gµν

q2 jν (k′e , µ

′e ; ke , µe)

where

jµ(kM ; k′M) := e(kM + k ′

M)µ and

jµ(ke , µe , k′e , µ

′e) := −euµ′

e(k′

e) γµuµe (ke)

are the (conserved) transition currents

gµν

q2 photon propagator where q = kM − k ′M

⇒ 2 currents connected by photon exchange

E.P. Biernat



Properties of the CurrentsHadronic Level

◮ jµ (k′M ; kM) , jν (k′

e , µ′e ; ke , µe) do not transform like four vectors

◮ jµ (p′M ; pM) = Bc (v)µ

ν jν (k′M ; kM)

jµ (p′e , σ

′e ; pe , σe)

= Bc(v)µν jν (k′

e , µ′e ; ke , µe)D

12∗

µ′

eσ′

e(R−1

W (Bc(v), k ′e))D

12µeσe (R

−1W (Bc(v), ke))

transform like four vectors

E.P. Biernat



Optical PotentialQuark Level

◮ confinement: instantaneous potential

MeM → MeM = Meqq + Mconf

MeMγ → MeMγ = Meqqγ + Mconf

◮ mass eigenstate |Ψ〉 =

(|ΨeM〉|ΨeMγ〉

)

◮ coupled-channel mass operator M =

(MeM K†

qγ

Kqγ MeMγ

)

◮ K†qγ

(

MeMγ + m)−1

Kqγ

︸︷︷︸

=Vqopt(m)

|ΨeM〉 =(

MeM + m)

|ΨeM〉

E.P. Biernat



Optical PotentialQuark Level

◮ matrix element of optical potential on quark level⟨

v ′; k′e , µ

′

e; k′

M , n′, m′j , j

′∣∣∣V

qopt (m)

∣∣∣ v ; ke , µe

; kM , n, mj , j⟩

=

⟨

v ′; k′e ; k

′M . . .

∣∣∣∣1eqqK

†qγ1eqqγ

(

MeMγ + m)−1

1eMγ1eqqγKqγ1eqq

∣∣∣∣v ; ke ; kM . . .

⟩

◮ electromagnetic vertex interaction⟨

v ′; k′e , µ

′

e; k′

q , µ′q ; kq′ , µq′ ; kγ , µ

γ

∣∣∣Kqγ

∣∣∣ v ; ke , µe

; kq, µq ; kq, µq

⟩

∝ v0δ3 (v − v′)

×⟨

v ; k′e , µ

′

e; k′

q, µ′q; k

′q , µ

′q; kγ , µ

γ

∣∣∣

(

Lqγ

int (0) + Leγ

int (0))∣∣∣ v ; ke , µe

; kq , µq ; kq , µq

⟩

E.P. Biernat



Microscopic Current

◮

⟨

v ′; k′e , µ

′

e; k′

M , n′, m′j , j

′∣∣∣V

qopt (m)

∣∣∣ v ; ke , µe

; kM , n, mj , j⟩

∝ v0δ3 (v − v′) Jµ (k′M ; kM)

gµν

q2 jν(

k′e , µ

′

e; ke , µe

)

◮ (conserved) microscopic meson current

Jµ (k′M ; kM)

∝ ∑ ∫d3k ′

q · · ·√

mqq

m′

qqΨ∗

n′j′m′

jµ′

q µ′

q

(

k′q

)

D12µ′

qµ′

q

(RW

(k ′

q, B−1

(v ′qq

)))

× (Qq + Qq) jµ(k′

q, µ′q ; k

′′′q , µ′′′

q

)D

12µ′′′

q µ′′′

q

(

RW

(

k ′′′q , B

(v ′′′qq

)))

×D12

µ′

qµ′′′

q

(

RW

(

k ′′′q , B−1

(v ′qq

)B

(v ′′′qq

)))

Ψnjmj µ′′′

q µ′′′

q

(

k′′′q

)

E.P. Biernat



Current Properties

◮ Jµ (k′M ; kM) does not transform as a four vector under Lorentz

transformations

◮ Jµ(

p′

M; p

M

)

:= Bc(v)µν Jν (k′

M ; kM) transforms like a four vector,

conserved

◮ momentum transfer: k′M − kM = k′

q − kq but k ′0M − k0

M 6= k ′0q − k0

q

⇒ 4-momentum transfer to cluster and to quark are in general not the

same

E.P. Biernat



Wave Function

◮ cluster wave function Ψnjmj µq µq

(

kq

)

= Cjmj

l ml smsC s ms

12µq

12µq

unl(|kq|)Yl ml(ˆkq)

defined by

Krassnigg, Schweiger, Klink; Phys.Rev.C, 2003⟨

v ; ke , µe ; kq, µq ; kq, µq|v ; ke , µe; kM , n, mj , j

⟩

∝ v0δ3(v − v)δ3(ke − ke)δµeµe

√k0e k0

qq

(k0e +k0

qq)3

√

k0ek0

M

(k0e+k0

M)3×

√k0q k0

q

(k0q+k0

q )3

∑Ψnjmj µq µq

(

kq

)

D12µq µq

(RW (kq, Bc(vqq)))D12µqµq

(RW (kq, Bc(vqq)))

this definition of Ψnjmj µq µq

(

kq

)

is not independent of additional

spectators (Bakamjian-Thomas construction)⇒ cluster separability violation

E.P. Biernat



Form Factor◮ identify form factor by comparing matrix elements of optical potential on

hadronic and quark level

F(Q2, ...

)jµ (k′

M ; kM) jµ (k′e , µ

′e ; ke , µe) = Jµ (k′

M ; kM) jµ (k′e , µ

′e ; ke , µe)

⇒ can depend also on other invariants, like e.g. total invariant mass of

system

◮ simple model for a pion: harmonic oscillator wave function

Ψ000µq−µq ∝ C 0012µq

12−µq

exp

(

− k2q

2a2

)

/a32

Chung, Coester, Polyzou; Phys.Lett.B, 1988

Coester, Polyzou; Phys.Rev.C, 2005

⇒ 2 parameters: mq, a

◮ kinematics

kM =

Q2

0√

k2M − Q2

4

, q =

Q

00

E.P. Biernat



Form Factor|kM |-Dependence

k@GeVD¥

0.7512

0.50.3

0 0.25 0.50

0.5

1

Q2@GeV2D

FHQ2L

Q2=0 GeV2

Q2=0.2 GeV2

Q2=0.5 GeV2

0 3.5 70

0.5

1

È kÓÖ

M È @GeVDfHD

m,ÈkÓÖ

MÈL

⇒ |kM |-dependence (⇐ approximation: velocity conservation at vertex) ⇒vanishes fast for increasing |kM | (> 2 GeV)

E.P. Biernat



Limit |kM | → ∞

◮ extract form factor where |kM |-dependence vanishes:|kM | → ∞ (simple analytic expression)

◮ Jµ (k′M ; kM) −→ F

(Q2

)jµ (k′

M ; kM)E.P.B., Schweiger, Fuchsberger, Klink; Few Body Syst., 2008

◮ electromagnetic pion form factor: overlap integral

F(Q2

)=

∫d3k ′

q

√mqq

m′

qqSΨ∗

n00

(

k′q

)

Ψn00

(

kq

)

E.P.B., Fuchsberger, Klink, Schweiger; Phys.Rev.C 2009

E.P. Biernat



Equivalence with Front Form

◮ momentum transfer k ′M − kM

|kM |→∞−→ k ′q − kq

◮ |kM | → ∞: Jµ (p′M ; pM) has correct continuity, covariance and

cluster-separability properties

◮ |kM | → ∞ means that subprocess γ∗M → M is considered in infinitemomentum frame⇒ equivalence with front form result:Chung, Coester, Polyzou; Phys.Lett.B, 1988

overlap integrals connected by variable transformation{

k′q

}

→ {k⊥, x}

E.P. Biernat



Form FactorInfluence of Quark Spins

S=1S¹1

0 0.15 0.30

0.5

1.

Q2@GeV2D

FHQ2L

0 5 100

0.3

0.6

Q2@GeV2D

Q2FHQ2L@GeV2D

E.P. Biernat



Comparison with Experiment

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E.P. Biernat



Comparison with Point-Form Spectator Model

◮ Point-Form Spectator Model (PFSM)Allen, Klink; Phys.Rev.C, 1998

Wagenbrunn, Boffi, Klink, Plessas, Radici; Phys.Lett.B, 2001

Boffi, Glozman, Klink, Plessas, Radici, Wagenbrunn; Eur.Phys.J.A, 2002

Melde, Canton, Plessas, Wagenbrunn; Eur.Phys.J.A, 2005

◮ Bakamjian-Thomas framework only for calculation of hadron wavefunctions

◮ spectator current ansatz for hadronic current with all required properties

◮ vqq = vM 6= v ′qq = v ′

M

(our approach: veqq = veM = v ′eqq = v ′

eM )

◮ form factor affected by shift of the whole spectrum

◮ for comparison: fix parameters that vector meson spectrum is reproducedas well as possible

E.P. Biernat



Comparison with Point-Form Spectator Model

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mΠ=0.77 GeVmq=0.34 GeV, a=0.31 GeV

CCPFSMHuber 2008Bebek 1978Bebek 1976Bebek 1974Brown 1973Amendolia 1986

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E.P. Biernat



Summary

◮ Poincare invariant coupled-channel approach to electron-meson scatteringin point form

◮ assumption: total velocity conservation at interaction vertices(approximation which satisfies Poincare invariance)⇒ form factor: dependence on Q2 and total invariant mass

◮ dependence on total invariant mass vanishes rather fast:for

√s → ∞ ⇒ equivalence with front form calculations can be

established

◮ outlook: extension to heavy-light systems, baryons, exchange currents ...

E.P. Biernat


Documents

Electromagnetic Meson Form Factor from a Relativistic ...Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach Elmar P. Biernat, W. Schweiger (University of