electromagnetic meson form factor from a relativistic ...electromagnetic meson form factor from a...
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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
Meson Form Factors and Point-Form RQM
Pion Form Factor
Result
Summary
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
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
(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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
Comparison with Experiment
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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
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach
Meson Form Factors and Point-Form RQM Pion Form Factor Result Summary
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
1.000.10 10.000.01
1.00
0.20
0.10
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Q2@GeV2D
FHQ2L
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
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
Electromagnetic Meson Form Factor from a Relativistic Coupled-Channel Approach