Form factors of few-body systems - SMU PhysicsMotivationFramework and examplesHeavy-light form factors. Point form and heavy-quark symmetryConclusions Framework I Use point form of

Motivation Framework and examples Heavy-light form factors. Point form and heavy-quark symmetry Conclusions

Form factors of few-body systemsfrom point form to front form

Marıa Gomez Rocha, Elmar P. Biernat, Wolfgang Schweiger

Karl-Franzens-Universitat Graz

May 25, 2011

Light-Cone 2011 - SMU Dallas - Texas

Marıa Gomez Rocha, Elmar P. Biernat, Wolfgang Schweiger Karl-Franzens-Universitat Graz

Form factors of few-body systems


Contents

Motivation

Framework and examples

Heavy-light form factors. Point form and heavy-quark symmetry

Conclusions




Motivation

Structure of hadrons in terms of their constituents⇒ Form factors

−→ dσ

dΩ=(dσ

dΩ

)point

|F (Q2)|2

Aim: formalism to describe hadron currents in terms of the constituents’currents

1. Poincare invariance

2. Non-perturbative nature of strong interactions

Conditions for the current:

1. Lorentz covariance

2. Current conservation

3. Cluster separability




Framework

I Use point form of Relativistic Quantum Mechanics:

Pµ contains interactions, J , K interaction free

I B.T.-construction:

Pµ = MV µfree ⇒ V µ conserved during interactions

I Relativistic coupled channel approachI Strong interactions → Mconf

eqq = Meqq + UconfI Electromagnetic interactions → K

I Mass-eigenvalue equation(M confeqq K

K† M confeqqγ

)(|ψeqq〉|ψeqqγ〉

)= m

(|ψeqq〉|ψeqqγ〉

)K → vertex operator: emission/absorption of a photon by q, q or e

〈V ′; e′, q′, q′, γ′|K|V ; e, q, q, 〉 ∝ v0δ(V ′ − V )〈e′, q′, q′, γ′|Lint|e, q, q〉




Framework








K† M confeqqγ


)= m



〈V ′; e′, q′, q′, γ′|K|V ; e, q, q, 〉 ∝ v0δ(V ′ − V )〈e′, q′, q′, γ′|Lint|e, q, q〉




Framework








K† M confeqqγ


)= m



〈V ′; e′, q′, q′, γ′|K|V ; e, q, q, 〉 ∝ v0δ(V ′ − V )〈e′, q′, q′, γ′|Lint|e, q, q〉Marıa Gomez Rocha, Elmar P. Biernat, Wolfgang Schweiger Karl-Franzens-Universitat Graz



Optical potential derivation by Feshbach reduction

(M confeqq K

K† M confeqqγ


)= m


)−→ (Meqq −m)|ψeqq〉 = K†(Meqqγ −m)−1K︸︷︷︸

Vopt(m)

|ψeqq〉




Framework

Derivation of the current...

Compare transitions

Constituent level

〈v′;~k′e, µ′e;~k′C , n|Vconst

opt (m)|v;~ke, µe~kC , n〉 ∝ jµe Jmicroµ

Hadronic level

〈v′;~k′e, µ′e;~k′M |V hadronopt (m)|v;~ke, µe;~kM 〉 ∝ jµe J

pointµ F (Q2)




Pseudoscalar mesons: The pion π± form-factor[Elmar P.Biernat, W.Schweiger, K.Fuchsberger, W.Klink, Phys.Rev C 79, 055203 (2009)]

Jµ(~k′C ,~kC) =

∑µ′qµq

∫d3k′q . . . j

µµ′qµq

(~k′q ,~kq) S Ψ(k′q , µ

′q)Ψ(kq , µq)

Properties of Jµ

I Lorentz covariance

I Current conservation

I PROBLEM: Cluster separability




Problem: Cluster separability violationThe limit k →∞

Bakamjian-Thomas construction ⇒ cluster properties are violated

⇒ Jµ(~k′C ,~kC) = f1(Q2, s)(k′C + kC)µ + f2(Q

2, s)(k′e + ke)µ

E.P. Biernat & W. Schweiger

(((((((((((((hhhhhhhhhhhhhJµ(k′C , kC) = F (Q2)(k′C + kC)µ

For |~kC | → ∞,

I f1(Q2, s)→ F (Q2), and f2(Q2, s)→ 0 : Cluster properties are restored

I Usual structure recovered Jµ(k′C , kC) = F (Q2)(k′C + kC)µ

I Equivalence with standard front form result for J+ in q+ = 0 frame[Chung, Coester, Polyzou; PLB 205, 1988]




Problem: Cluster separability violationThe limit k →∞

Bakamjian-Thomas construction ⇒ cluster properties are violated

⇒ Jµ(~k′C ,~kC) = f1(Q2, s)(k′C + kC)µ + f2(Q

2, s)(k′e + ke)µ

E.P. Biernat & W. Schweiger

(((((((((((((hhhhhhhhhhhhhJµ(k′C , kC) = F (Q2)(k′C + kC)µ

For |~kC | → ∞,

I f1(Q2, s)→ F (Q2), and f2(Q2, s)→ 0 : Cluster properties are restored

I Usual structure recovered Jµ(k′C , kC) = F (Q2)(k′C + kC)µ

I Equivalence with standard front form result for J+ in q+ = 0 frame[Chung, Coester, Polyzou; PLB 205, 1988]




Vector meson form factors

I Extend the definition of the current to spin-1 mesons

I Consider the most general independent linear combination ofcovariants including Ke := k′e + ke → 11 covariants

Jµ(kC , µC ; k′C , µ′C ;Ke) =

=

[f1ε′∗ · ε+ f2

(ε′∗ · q)(ε∗ · q)2m2

C

]KµC + gM

[ε′∗µ(ε · q)− εµ(ε′∗ · q)

]+

m2C

2Ke · kC

[b1(ε′∗ · ε) + b2

(q · ε′∗)(q · ε∗)m2C

+ b3m2C

(Ke · ε′∗)(Ke · ε∗)(Ke · kC)2

+b4(q · ε′∗)(Ke · ε)− (q · ε)(Ke · ε′∗)

2(Ke · kC)

]Kµe

+

[b5m

2C

(Ke · ε′∗)(Ke · ε)(Ke · kC)2

+ b6(q · ε′∗)(Ke · ε)− (q · ε)(Ke · ε′∗)

2Ke · kC

]KµC

+b7m2C

ε′∗µ(ε ·Ke) + εµ(ε′∗ ·Ke)Ke · kC

+ b8qµ (q · ε′∗)(Ke · ε) + (q · ε)(Ke · ε′∗)

2Ke · kC




Vector mesons: ρ form factorsResults and remarks [E. P. Biernat, PhD Thesis]

I limit k →∞ does not remove spurious form factors B5(Q2),B6(Q2), B7(Q2) and B8(Q2)

I B7, B8: violation of current conservationI (B5 +B7): violation of the so called angular condition

(1 + 2η)J011 + J0

1−1 − 2√

2ηJ010 − J0

00 = −(B5 +B7) 6= 0I BUT: physical form factors F1(Q2), F2(Q2), GM (Q2) can be

uniquely extracted from good matrix elements J011, J

01−1, J

211

I Resemblance to covariant light-front approach of Karmanov et al.[Phys.Rept.300, 1998]




Heavy-light form factors




Heavy-light form factors

I Mesons in which mq 6= mq, in particular mQ mq

I Explore the heavy-quark limit −→ FORM FACTORS of heavy-lightsystems

1. Test heavy-quark symmetry conditions ⇒ universal form factor:Isgur-Wise function ξ(v · v′)

2. Point form is expected to be an appropriate framework for suchsystems [B. D. Keister, Phys.Rev. D 46 7 (1992)]

3. We present results for ξ(v · v′) in EM and EW processes




Heavy quark symmetry and PFRQM[M. Neubert/Physics Reports 245 (1994)]

I Heavy-light systems → additional symmetriesI Picture: Q surrounded by a cloud of light quarks and gluons.

I Dynamics of the heavy hadron controlled by the dynamics of theheavy constituent quark

I Matrix elements do not depend onI heavy quark mass ⇒ flavor symmetryI heavy quark spin ⇒ spin symmetry

I The heavy-quark limit eliminates the heavy quark mass from thedescription:

mQ 'MQmq

mQ→ 0

I State vector in terms of velocities becomes natural: |~p µ〉 −→ |~v µ〉









mQ 'MQmq

mQ→ 0










mQ 'MQmq

mQ→ 0










mQ 'MQmq

mQ→ 0

I State vector in terms of velocities becomes natural: |~p µ〉 −→ |~v µ〉Marıa Gomez Rocha, Elmar P. Biernat, Wolfgang Schweiger Karl-Franzens-Universitat Graz



The limit mQ →∞

I(((((((((hhhhhhhhhNon-relativistic limit: the momentum transfer Q2 is allowed to go to∞ too, light degrees of freedom behave relativistically

I Reexpress Q2 in terms of v · v′

Q2 = (p− p′)2 = m2Q(v − v′)2 ⇒ Q2 = m2

Q(1− v · v′)

I The infinite-mass limits have to be taken in such a way that v · v′stays constant




Electromagnetic Form Factors




The form factor and its limit mQ →∞

Jµ = Jµq + Jµq

I Let the system be qq −→ Qq

I Take now a model wave function: h.o. ψ(~kq) = 1√4π

2

π14 a

32

exp

− ~k2q

2a2

I Use v · v′ variables: replace Q2 = 2mQ(1− v · v′)

mQ→∞




The form factor and its limit mQ →∞

Jµ = Jµq + Jµq

I Let the system be qq −→ Qq

I Take now a model wave function: h.o. ψ(~kq) = 1√4π

2

π14 a

32

exp

− ~k2q

2a2

I Use v · v′ variables: replace Q2 = 2mQ(1− v · v′)

mQ→∞Marıa Gomez Rocha, Elmar P. Biernat, Wolfgang Schweiger Karl-Franzens-Universitat Graz



The limit mQ →∞

I Jµq → 0I JµQ simplifies to −→ (J0, 0, 0, 0)

I (kM + k′M )µ simplifies to −→ (mQ

√2(1 + v · v′), 0, 0, 0)

We can write the current in the usual form

JµQ = F (v · v′)(k′M + kM )µ

I no k−dependence




⇒ The Isgur-Wise function ξ(v · v′)

ξ(v · v′) =∑µµ′

∫d3k′q

√ωq

ω′q

√2

1− v · v′×

×1

2D

1/2µµ′

[R−1W

(kq

mq, B(v)

)RW

(k′q

mq, B(v′)

)]

×ψout(~k′q)ψin(~kq)

I We end up with a simpleanalytical expression for EMform factors in the heavy quarklimit

I Independence of mQ:Universal

2 4 6 8 10v.v’0.0

0.2

0.4

0.6

0.8

1.0

ΞHv.v’L

a = 0.55GeVMarıa Gomez Rocha, Elmar P. Biernat, Wolfgang Schweiger Karl-Franzens-Universitat Graz



Weak Form Factors




Weak decays

Properties of heavy-quark symmetry can be checked:

I Flavor symmetry relates EM and weak form factors

I Example: Let (Qq) −→ B−(bu)

〈B(v′)|bγµb|B(v)〉 ←→ 〈D(v′)|cγµb|B(v)〉

[M. Neubert/Physics Reports 245 (1994)]

I ∃ a relation between weak and EM form factors, such that they coincidein the mQ →∞ limit

I Compute weak form factors and compare the mQ →∞ limits




Weak decays

Compare transitions

I Kem ∝ LQEDint −→ Kw ∝ Lw

int

I M couples three channels → |ψB−〉, |ψW−D0〉, |ψD0νee−〉I The four-momentum transfer is time-like ⇒ v · v′ = m2

B+m2D−Q

2

2mBmD

I No k-dependence: k is fixed (initial state is at rest)




The Weak Isgur-Wise function

Compare and go to the limit mQ →∞ξW (v · v′) =

=∑µ′µ

∫d3k′u

√ωuω′u

√2

1 + v · v′1

2D

1/2

µ′µ

[RW

(~k′umu

, B(v′)

)]×

×ψout(~k′u)ψin(~ku)

⇒ Check universality

I Independence on theheavy-quark mass

I ξEM (v · v′) = ξW (v · v′)in the limit

⇒ Numerical equality withfront-form results[H.Y.Cheng et al., Phys.Rev.D55 (1997) 1559]

2 4 6 8 10v.v’0.0

0.2

0.4

0.6

0.8

1.0

1.2

ΞHv.v’LEM

ΞHv.v’LW




Comparison of the limits mQ →∞Analytical comparison

I Obtain 2 different expressions for ξ(v · v′) =⇒ Equivalent?

Electromagnetic form factor

ξEM (v · v′) =∑µ′µ

∫d3k′q

√√√√ ωq

ω′q

√2

1 + v · v′1

2D

1/2µµ′

R−1W

kq

mq

, B(v)

RW k′qmq

, B(v′)

××ψout(

~k′q)ψin(~kq)

Weak form factor

ξW (v · v′) =∑µ′µ

∫d3k′q

√√√√ ωq

ω′q

√2

1 + v · v′1

2D

1/2µµ′

RW ~k′qmq

, B(v′)

ψout(~k′q)ψin(~kq)

I ξEM and ξW only functions of v · v′ → ξW special case of ξEM withv = (1, 0, 0, 0)




Check of HQ (flavor) symmetry for mQ <∞b→ c

2 4 6 8 10v.v’0.0

0.2

0.4

0.6

0.8

1.0

1.2

mc=1.27 GeV, mb=3.3mc

RH1-Q2HmB+mDL2L-1F0

R F1

ΞHv.v’L

R = 2√mBmD

mB+mD

Jµ = F1(Q2)

((kB + k′D)µ − m2

B −m2D

Q2Qµ)

+ F0(Q2)m2B −m2

D

Q2Qµ




Check of HQ (flavor) symmetry for mQ <∞b→ c

2 4 6 8 10v.v’0.0

0.2

0.4

0.6

0.8

1.0

1.2

mc=10 GeV, mb=3.3mc

RH1-Q2HmB+mDL2L-1F0

R F1

ΞHv.v’L

R = 2√mBmD

mB+mD

Jµ = F1(Q2)

((kB + k′D)µ − m2

B −m2D

Q2Qµ)

+ F0(Q2)m2B −m2

D

Q2Qµ




Check of HQ (flavor) symmetry for mQ <∞t→ b

2 4 6 8 10v.v’0.0

0.2

0.4

0.6

0.8

1.0

1.2

mb=4.2 GeV, mt=40.6mb

RH1-Q2HmB+mDL2L-1F0

R F1

ΞHv.v’L

R = 2√mBmD

mB+mD

Jµ = F1(Q2)

((kB + k′D)µ − m2

B −m2D

Q2Qµ)

+ F0(Q2)m2B −m2

D

Q2Qµ




Vector meson form factors




P→V transition form factors

I Use heavy-quark spin symmetry to relate PS→PS to PS→VtransitionsConsider the process B → D∗lν

I Spins should rearrange in intermediate states to give S = 1 in thefinal state




P→V transition form factors

I Five covariants

〈D∗(p′, ε)|cγµ(1− γ5)b|B(p)〉 =2iεµναβ

MB +MD∗ε∗νp′αpβV (Q2)−

−(

(MB +MD∗ )ε∗µA1(Q2)−ε∗ ·Q

MB +MD∗(p+ p′)µA2(Q2)−

−2mD∗ε∗ ·QQ2

QµA3(Q2))− 2mD∗

ε∗ ·QQ2

QµA0(Q2)

I HQ spin symmetry can relate ξ(v · v′) and V (Q2), A0(Q2), A1(Q

2),A2(Q

2)

I If HQ symmetry is satisfied, the IW function should be reached for largemasses

ξ(v · v′) = limmQ→∞

R∗V (Q2) = limmQ→∞

R∗A0(Q2) = limmQ→∞

R∗A2(Q2) =

= limmQ→∞

R∗

(1−

Q2

(mB +m2D∗ )

)−1

A1(Q2) (1)




Check HQ (spin) symmetryb→ c

2 4 6 8 10v.v’0.0

0.2

0.4

0.6

0.8

1.0

1.2

mc=1.27 GeV, mb=3.3mc

ΞA0Hv.v’L

ΞA2Hv.v’L

ΞA1Hv.v’L

ΞVHv.v’L

ΞHv.v’L





2 4 6 8 10v.v’0.0

0.2

0.4

0.6

0.8

1.0

1.2

mc=10 GeV, mb=3.3mc

ΞA0Hv.v’L

ΞA2Hv.v’L

ΞA1Hv.v’L

ΞVHv.v’L

ΞHv.v’L





2 4 6 8 10v.v’0.0

0.2

0.4

0.6

0.8

1.0

1.2

mc=50 GeV, mb=3.3mc

ΞA0Hv.v’L

ΞA2Hv.v’L

ΞA1Hv.v’L

ΞVHv.v’L

ΞHv.v’L





2 4 6 8 10v.v’0.0

0.2

0.4

0.6

0.8

1.0

1.2

mc=50 GeV, mb=3.3mc

ΞA0Hv.v’L

ΞA2Hv.v’L

ΞA1Hv.v’L

ΞVHv.v’L

ΞHv.v’L




Cluster properties in point-form heavy-light systems




The spurious form factor in heavy-light systemsElectromagnetic scattering

⇒ Jµ(k′M , kM ) = f(Q2, s)(k′M + kM )µ + g(Q2, s)(k′e + ke)µ

B 2 4 6 8 10ÈkÈ HGeVL0.0

0.2

0.4

0.6

0.8

1.0FHQ2,ÈkÈL

Q2=1 GeV2

Q2=0.5 GeV2

Q2=0.1 GeV2

Q2=0 GeV2

2 4 6 8 10ÈkÈ HGeVL0.0

0.2

0.4

0.6

0.8

1.0GHQ2,ÈkÈL

Q2=1 GeV2

Q2=0.5 GeV2

Q2=0.1 GeV2

Q2=0 GeV2

π+




The spurious form factor in heavy-light systemsElectromagnetic scattering

⇒ Jµ(k′M , kM ) = f(Q2, s)(k′M + kM )µ + g(Q2, s)(k′e + ke)µ

B 2 4 6 8 10ÈkÈ HGeVL0.0

0.2

0.4

0.6

0.8

1.0FHQ2,ÈkÈL

Q2=1 GeV2

Q2=0.5 GeV2

Q2=0.1 GeV2

Q2=0 GeV2

2 4 6 8 10ÈkÈ HGeVL0.0

0.2

0.4

0.6

0.8

1.0GHQ2,ÈkÈL

Q2=1 GeV2

Q2=0.5 GeV2

Q2=0.1 GeV2

Q2=0 GeV2

π+




Conclusions




Conclusions

I Relativistic formalism to derive current and form factors of boundfew-body systems consistent with the binding forces

I PFRQM convenient framework for treatment of heavy-light systems

I Bakamjian-Thomas construction provides sensible results if one ofthe quarks is heavy

I We are able to compute form factors of systems of mq 6= mq in themost general case. Heavy-quark symmetry emerges correctly in thelimit mQ →∞. Physical masses of heavy quark → considerableviolation of heavy-quark symmetry

I Analytical structure of IW function resembles those of other CQMapproaches (numerical equality with Front-form )

I Problems with cluster separability seem to be less serious forheavy-light systems, in particular for decay form factors, and vanishin the heavy-quark limit




Conclusions










Conclusions










Conclusions










Conclusions










Conclusions










Thank you!



Documents

Form factors of few-body systems - SMU PhysicsMotivationFramework and examplesHeavy-light form factors. Point form and heavy-quark symmetryConclusions Framework I Use point form of