Pion Electromagnetic and Transition Form Factors with the ...lc2013/downloads/talks/...Pion...

Pion Electromagnetic and Transition Form Factors withthe Light-Front Approach

Venturing off the Light-Cone - Local Versus Global Features20-24 May 2013, Skiathos, Greece

J. Pacheco B. C. de Melo

aLaboratório de F́ısica Téorica e Computacional-LFTC, UNICSUL (Brazil) andbInstituto de F́ısica Téorica, IFT, UNESP, (Brazil)

Collaborators: Bruno El-Bennich (LFTC) and T. Frederico (CTA-ITA)

May 24, 2013

J. Pacheco B. C. de Melo ( aLaboratório de F́ısica Téorica e Computacional-LFTC, UNICSUL (Brazil) and b Instituto de F́ısica TéoPion Electromagnetic and Transition Form Factors with the Light-Front ApproachMay 24, 2013 1 / 42

Outline

1 Motivation

2 Light-FrontOverview of the Light-Front

3 Electromagnetic Current: General

4 Triangle DiagramElastic Process: Electromagnetic Form Factors

5 General Matrix Elements

6 Pion Decay: Transition Form Factor

7 Remarks


Motivation

Motivations

• π0 → γγ Most Important Example of the Triangle Anomaly

• π0 Meson is the Lightest Meson cannot Decay to AnotherHadronic State

• π0 → γγ Is Conected to the Adler-Bell-Jackiw Anomaly

• Babar Experiment (2009)

• Belle Experiment (2012)


Motivation

0 10 20 30 40

Q2 [GeV/c]

2

0.00

0.05

0.10

0.15

0.20

0.25

0.30Q

2 F

πγγ(q

2 )

CLEO (1998)Belle (2012)Babar (2009)Perturbative QCD

Pion Form Factor λ=10 Γ= 0.3 znpi0=15 Gauss=80 m

q=0.384 GeV npion=10

• QCD: 2fπ =⇒ Brodsky & Lepage (1980)J. Pacheco B. C. de Melo ( aLaboratório de F́ısica Téorica e Computacional-LFTC, UNICSUL (Brazil) and b Instituto de F́ısica TéoPion Electromagnetic and Transition Form Factors with the Light-Front ApproachMay 24, 2013 4 / 42

Light-Front Overview of the Light-Front

Light-Front Coordinates

Four-Vector =⇒ xµ = (x0, x1, x2, x3) = (x+, x−, x⊥)

x+ = t + z x+ = x0 + x3 =⇒ Timex− = t − z x− = x0 − x3 =⇒ PositionMetric Tensor and Scalar product

x · y = xµyµ = x+y+ + x−y− + x1y1 + x2y2 =x+y− + x−y+

2− ~x⊥~y⊥

p+ = p0 + p3

p− = p0 − p3

p⊥ = (p1, p2)


Electromagnetic Current: General

Dirac Matrix and Electromagnetic Current

γ+ = γ0 + γ3 =⇒ Electr. Current J+ = J0 + J3γ− = γ0 − γ3 =⇒ Electr. Current J− = J0 − J3γ⊥ = (γ1, γ2) =⇒ Electr. Current J⊥ = (J1, J2)

pµxµ =p+x−+p−x+

2 − ~p⊥~x⊥x+, x−, ~x⊥ =⇒ p+, p−, ~p⊥p− =⇒ Light-Front Energyp2 = p+p− − (~p⊥)2 =⇒ p− = (~p⊥)

2+m2

p+On-shell

Bosons =⇒ SF (p) = 1p2−m2+ıǫFermions =⇒ SF (p) = /p+mp2−m2+ıǫ +

γ+

2p+

Phys. Rept. 301, (1998) 299-486, Brodsky, Pauli and Pinsky


Triangle Diagram Elastic Process: Electromagnetic Form Factors

Γ = (γµ, γµγ5)

k − P

P k

k − P ′

P ′Figure: P (π,K ,D,B) −→ P ′ (π,K ,D,B)


General Matrix Elements

• Three-point Function ⇒ The General Matrix Element

〈

p′∣

∣Jµq(

q2)∣

∣ p〉

=Nc

(2π)4

∫

d4k Tr[

ΛM′(

k , p′)

SF(

k − p′)

Jµq S(k − p)ΛM (k , p)SF (k)]

• PS − qq̄ Vertex: γ5• Fermi Propagator =⇒ SF (p) = i/p−mi+iε


General Matrix Elements

Eletromagnetic Form Factor and Pseudoscalar Constant

〈

p′∣

∣Jµq(

q2)∣

∣ p〉

=(

p + p′)µ

F emPS (q2).

• Weak Decay Constant of the Pseudoscalar Mesons

〈0 |Aµ(0)| p〉 = ı√2fpsp

µ

=⇒ Aµ = q̄γµγ5 τ2q(x)

ıpµfπ =m

fπNc

∫

dk4

(2π)4Tr[

γµγ5S(k)γ5S(k− p)]

ΛM(k,p)

• Plus Component E.M. Current: γ+ = γ0 + γ3


Pion Decay: Transition Form Factor

Some QCD REMARKS

• R = σ(e+e−→qq̄)

σ(e+e−→µ−µ+) = Nc∑

Q2q = Nc[

(23)2 + (−13 )

2 + (−13 )2]

= 23Nc =2

For Ecm > 2 GeV ≫ 2mu , 2md , 2ms , for (u,d,s)

• R = σ(e+e−→qq̄)σ(e+e−→µ−µ+) = Nc[

(23)2 + (−13 )

2 + (−13 )2 + (23)

2]

= 109 Nc

For Ecm > 3 GeV ≫ 2mc , for (u,d,s,c)

• R = σ(e+e−→qq̄)σ(e+e−→µ−µ+) = Nc[

(23)2 + (−13 )

2 + (−13 )2 + (23)

2 + (−13 )2]

=119 Nc

For Ecm > 10 GeV ≫ 2mb, for (u,d,s,c,b)⋆ If NC = 3 !!!



10-1

1

10

10 2

10 3

1 10 102

ρ

ωφ

ρ

J/ψ ψ(2S)ZR

S GeV

Ref.: J. Beringer et al. (Particle Data Group), Phys. Rev. D86, 010001(2012)



• π0 → γγ =⇒ Γ =g2π0γγ

α2emm3π

16π3f 2π

• Exp.: ΓExp ≃ 7.80 ⇐⇒ Nc = 3, gπγγ = 0.5, fπ ≃ 92 MeV

• Effective Theory:

Γ = m3π

64π |Aγγ |2

Where: |Aγγ |2 = αNc3πfπAgain: =⇒ Nc = 3 and fπ ≃ 92 MeV

Ref.Dynamics of the Standard Model byJ. F. Donoghue, E. Golowich and B. R. Holtein



e

γ

q

q̄

γ⋆Q2

e

π0

Figure: Feynman Diagram for γγ⋆ → qq̄ → π0



.

π0

(a)

Figure: Pion Decay Diagram (a).



.

π0

(b)

Figure: Pion Decay Diagram (b).



Effective Interaction Lagrangian

Lintπq = −ıM

fπ~π · ~qγ5~τq ;

Where:→ M: Constituent Quark Mass→ fπ: Weak Decay Constant→ π: Pion Field→ q: Quark Field• ~=c=1



Tµν Tensor : Amplitude (a) and (b)

Tµν = tµν(k1, k2) + tµν(k2, k1)

• After Calculation of Trace in Spinor and Flavour Basis:

tµν =4

3

M2

fπe20Ncǫµναβk

α1 k

β2 I (k

21 ) ;

I (k21 ) =

∫

d4k

(2π)41

((k2 − k)2 −M2 + ıǫ)1

(k2 −M2 + ıǫ)1

((kπ − k)2 −M2 + ıǫ).



• kµπ = kµ1 + kµ2 ⇐⇒ Pion Momentum

• kµ1 = qµ ⇐⇒ Momentum Transfer

I (k21 ) =

∫

d4k

(2π)41

((k2 − k)2 −M2 + ıǫ)

× 1(k2 −M2 + ıǫ)((kπ − k)2 −M2 + ıǫ)

.

• Ref. Frame: q+ = q− = 0 and q⊥ 6= 0



k− Integration:

I (k21 ) =1

2(2π)4

∫

dk+d2k+dk−

1

(k+(k+π − k+)(k+2 − k+)(k− −k2⊥+M2−ıǫ

k+)

× 1(k−2 − k− −

(~k2−~k)2⊥+M2−ıǫ

(k+2 −k+)

)(k−π − k− − (~kπ−~k)2⊥+M

2−ıǫ

(k+π −k+)),

• k+π = k+2 and k−π = k−2 .



I (q2) = − 12(2π)3

∫

dxd2k⊥

x(1− x)21

(k−2 k+2 −

k2⊥+M2

x− (

~k2−~k)2⊥+M2

1−x )

× 1(k−π k

+π − k

2⊥+M2

x− (

~kπ−~k)2⊥+M2

1−x )

Integral Contribuitioni) k+ < 0 =⇒ Not Contributeii) k+ > k+π =⇒ Not Contributeiii) 0 < k+ < kπ =⇒ Contribute ( On-Shell Quark )



• Integral Results

I (q2) = − 12(2π)3

∫

dxd2k⊥

x(1− x)21

(k−2 k+2 −

k2⊥+M2

x− (

~k2−~k)2⊥+M2

1−x )

× 1(k−π k

+π − k

2⊥+M2

x− (

~kπ−~k)2⊥+M2

1−x ),

• =⇒ 0 < k+ < k+π ⇐⇒ 0 < x < 1



• Relative momentum: ~K⊥ = (1− x)~k⊥ − x(~kπ − ~k)⊥

I (q2) =1

2(2π)3

∫

dxd2K⊥

(1− x)1

((~K + x~q)2⊥+M2)(m2π −M20 )

• Free Mass Operator

M20 =K 2⊥+M2

x(1− x)



Pion Eletromagnetic Transition Form Factor

jµ = e20ǫµναβǫ

νγq

αkβπFπ0(q2)

Fπ0(−q2) =Nc

6π3M2

fπ

∫

dxd2K⊥

(1− x)1

((~K + x~q)2⊥+M2)(M20 −m2π)

• ǫνγ ⇒ Polarization // Real Photon



Wave Function

• The Wave Function // Replaced

1

−m2π +M20→ π

32 fπ

M√M0Nc

Φπ(K2)

• Wave Function Φ(k2) Normalization∫

d3KΦ2π(k2) = 1 ,

Ref.

• T. Frederico and G. Miller, Phy. Rev. D45 (1992) 071901ibid. Phy. Rev. D50 (1994) 210• J.P.B.C. de Melo, T. Frederico and H.L. NausPhy.Rev. C59 (1999) 2278



• Final Pion Transition Form Factor

Fπ0(−q2) =√NcM

6π32

∫

dxd2K⊥

(1− x)√M0

Φπ(K2)

((~K − x~q)2⊥+M2)

• Soft Pion Limit ⋆:

Fγπ0(0) =1

4π2fπ

⋆ J. S. Schwinger, Phy. Rev.82, (1951) 664.S. L. Adler, Phys. Rev. Phy. Rev. 177, (1969) 2426.J. S. Bell, R. Jackiw, Nuovo Cimento, 60 , (1969) 47.



Wave Function

i) Gaussian Wave Function

Φπ =

(

8r2NR3π

)3/4

exp

[

−(

4

3

)

(rNRk)2

]

ii ) Hydrogen-Atom Wave Function

Φπ =1

2π

(√3

rNR

)5/2 [

1

(34 r−2NR + k

2)2

]



• Two Independents Parametersi) Quark Mass: Mii) Non-Relativistic Charge Radius: rNR

r2NR = −6d

dq2

∫

d3KΦ(|~K + ~q2)Φπ(K )



• Neutral Pion Radius

r2π0 =

√NcM

Fπ0(0)π32

∫

dxd2K⊥

(1− x)√M0

−~K 2⊥+M2

(−~K 2⊥+M2)3

Φπ(K2) .

• Limit: −q2 → ∞ =⇒ ∼ q2

Λπ0 = lim−q2→∞

[−q2Fπ0(−q2) =√NcM

6π32

∫

dxd2K⊥

(1− x)√M0

Φπ(K2)



Some Results

Table-I: fπ : 92.4 MeV Fixed

model mu,d [GeV ] rnr [fm] < r2 > [fm2] < r2π0 > [fm

2]

Gaussian 0.220 0.345 0.637 0.6830.330 0.472 0.655 0.552

Hydrogen 0.220 0.593 0.795 0.7820.330 0.708 0.807 0.582

Exp.[PDG] 0.672±0.008



Table-II: Quark Mass fixed : mu,d = 0.220 GeV

model fπ [MeV ] rnr [fm] < r2 > [fm2] < r2π0 > [fm

2]

Gaussian 92.4 0.345 0.637 0.68397.0 0.303 0.589 0.657110.0 0.172 0.406 0.664

Hydrogen 92.4 0.593 0.795 0.78297.0 0.543 0.750 0.767110.0 0.410 0.626 0.720

Exp.[PDG] 92.2 ± 0.021 0.672 ± 0.008



Exp. References:

• Pion Electromagnetic Form Factor⋆ R. Baldini, et al., Eur. Phys. J. C 11 (1999), 709Nucl. Phys. A666-667 (2000), 3⋆ J. Volmer et al., (The Jefferson Lab Fπ Collaboration),Phy. Rev. Lett. 86 (2001), 1713⋆ T. Horn at al., Phys. Rev. Lett. 97, (2006), 192001⋆ V. Tadevosyan et al., Phys. Rev. C 75, (2007), 055205

• Pion Transition Form Factor⋆ (Belle-2012) S. Uehara et al., Phys. Rev. 86, (2012) 092007⋆ (Babar-2009) B.Aubert et al., Phy. Rev. D 80, (2009) 052002⋆ (CLEO-1998) J. Gronberg et al., Phy. Rev. D 57, (1998) 33



0 2 4 6 8 10

Q2 = -q

2 [GeV/c]

2

0.00

0.20

0.40

0.60

0.80

1.00|F

π(q2

)|Volmer et al. (Exp.)Horn et al. (Exp.) Frascati (Exp.)Tadevosyan et al. (Exp.)Gaussian: mq=0.220 GeV ; rπ = 0.637 fmGaussian: mq=0.330 GeV ; rπ = 0.655 fmHydrogen: mq=0.220 GeV; rπ = 0.795 fmHydrogen: mq=0.330 GeV; rπ = 0.807 fm





0 2 4 6 8 10

Q2 = -q

2[GeV/c]

2

0.00

0.20

0.40

0.60

0.80

1.00

Q2 F

π(q2

)Baldini et Al.Volmer et al. (Exp.)Horn et al (Exp.)Tadevosyan (Exp.)Gaussian: mq=0.220 GeV; rπ = 0.637 fmGaussian: mq=0.330 GeV; rπ = 0655 fmHydrogen: mq=0.220 GeV; rπ = 0.795 fm Hydrogen: mq=0.330 GeV; rπ = 0.807 fm





0 10 20 30 40

Q2[GeV/c]

2

0.00

0.01

0.10

1.00|F

πγγ(q

2 )|

CLEO (1998)BaBar (2009)Belle (2012)Gaussian: mq=0.220 GeV; rπ=0.637 fm ; fπ = 92.4 MeVGaussian: mq=0.330 GeV ; rπ=0.655 fm; fπ=92.4 MeVHydrogen: mq=0.220 GeV; rπ=0.795 fm; fπ=92.4 MeVHydrogen: mq=0.330 GeV ; rπ=0.807 fm; fπ=92.4 MeV





0 10 20 30 40

Q2 [GeV/c]

2

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Q2

Fπ

γγ(q

2 )

CLEO (1998)Belle (2012)Babar (2009)Perturbative QCDGaussian mq=0.220 GeV Gaussian: mq=0.330 GeVHydrogen mq=0.220 GeV Hydrogen: mq=0.330 GeV





0 10 20 30 40

Q2 [GeV/c]

2

0.00

0.10

0.20

0.30

0.40Q

2 F

πγγ(q

2 )

CLEO (1998)Belle (2012)Babar (2009)Perturbative QCDGaussian mq=0.220 GeV Gaussian: mq=0.330 GeVHydrogen mq=0.220 GeV Hydrogen: mq=0.330 GeVAdS/QCD Free Current (Brodsky et al.)





Fit Curves:

• Babar: Q2|F (Q2)| = A ( Q210 GeV 2

)β =⇒{

A = 0.182 GeVβ = 0.250

• Belle: Q2|F (Q2)| = A1 ( Q2

10 GeV 2)β1 =⇒

{

A1 = 0.167 GeVβ1 = 0.204

• Belle: Q2|F (Q2)| = B Q2Q2+C

=⇒{

B = 0.209 GeVC = 2.2 GeV 2



0 10 20 30

Q2 [GeV/c]

2

0.00

0.10

0.20

0.30

0.40Q

2 F

πγγ(q

2 )

CLEO (1998)Belle (2012)Babar (2009)Perturbative QCDGaussian mq=0.220 GeV Hydrogen mq=0.220 GeV ADSQCDBabar FitBelle Fit (with Babar expression)Belle Fit





Some Coments

• Theoretical Analyses: Explain Babar Data and Not ExplainModels try reproduce Babar:• Alteration of the asymptotic pion wave function or distribuitionamplitude• Dressing γ − qq̄-vertex with Phenomelogical Interactions, ie., likeVMD



Some References:

• M. V. Polyakov, JETP, 90 (2009) 228• S. Noguera and V. Vento, Eur. Phys. J. A48 (2012) 143• S. S. Agaev, V. M. Braun, N. Offen and F. A. Porkert,Phys. Rev. D86 (2012) 077504• I. Balakireva, W. Lucha, D. Melikhov, PoS (2012),arXiv:1212.5898v1, (Local-duality QCDSRL)• A. P. Bakulev, V. V. Mikhailov and A. V. Pimikov andN. G. Stefanis, Phys. Rev. 86 (2012) 031501, (QCDSRL)• S. J. Brodsky, Fu-Guang Cao andG. F. Teramond, Phys.Rev. D84 (2011) 075012 (ADS/QCD)• H.L.L. Roberts, C. D. Roberts, A. Bashir, L.X.Gutírrez andP.C. Tandy, Phy. Rev. C82 (2010) 065202 (Dyson-Schwinger.)• B. El-Bennich, de Melo, T. Frederico,Few-Body Syst. 52 (2013) 403. (Light-Front QCM)


Remarks

Some Remarks

Ligth-Front Approach:

• =⇒ Computation of Form-Factors and Decay Constants• =⇒ Easy to Test Different Analytical Models• =⇒ Correct Asymptotic Form-Factors• =⇒ Agreement with Experiments (CLEO and Belle) and OthersModels (but, also with Babar for small Q2)• =⇒ The Large Babar data is Not Consistent with QCD


Remarks

Thanks to Organizers Greece/Skiathos-LC2013

Obrigado, Thanks, Merci, Gracias, GrazieDanke, Efcharisto !!!

Support: Brazilian Agency FAPESP andLCTC-UNICSUL


MotivationLight-Front Overview of the Light-Front

Electromagnetic Current: General Triangle DiagramElastic Process: Electromagnetic Form Factors

General Matrix ElementsPion Decay: Transition Form FactorRemarks

Pion Electromagnetic and Transition Form Factors with the ...lc2013/downloads/talks/...Pion...

Documents

Transcript of Pion Electromagnetic and Transition Form Factors with the ...lc2013/downloads/talks/...Pion...