heavy mesons electromagnetic form factor with light-front … · 2007. 12. 10. · b meson elect....

OutlineLight-Front

Electromagnetic CurrentLight-Font Integration

Triangle DiagramPionkaon

B Meson Elect. Form FactorConclusions

Heavy Mesons Electromagnetic Form Factor with

Light-Front Approach

II LAWHEP - 2007- Sao Miguel das Missoes - RG

J. Pacheco B. C. de Meloa, B. El-Bennichb, B. Loiseaub,J.-P. Dedonderb and T. Fredericoc

aInstituto de Fısica Teorica, IFT, UNESP, aLaboratorio de Fısica Teorica eComputacional, LFTCC, UNICSUL (Brazil), bUniversite Pierre & Marie Curie,LPNHE (France) and c Instituto Tecnologico de Aeronautica, ITA,CTA (Brazil)

December 4, 2007

J. Pacheco B. C. de Meloa , B. El-Bennichb , B. Loiseaub ,J.-P. Dedonderb and T. FredericocHeavy Mesons Electromagnetic Form Factor with Light-Front App

OutlineLight-Front




Light-FrontOverview of the Light-Front

Electromagnetic Current

Light-Font Integration

Triangle Diagram

PionPion

kaonKaon

B Meson Elect. Form Factor

Conclusions


OutlineLight-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 =⇒ Time

x− = t − z x− = x0 − x3 =⇒ Position

Metric 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)


OutlineLight-Front




Dirac Matrix and Electromagnetic Current

γ+ = γ0 + γ3 =⇒ Electr. Current J+ = J0 + J3

γ− = γ0 − γ3 =⇒ Electr. Current J− = J0 − J3

γ⊥ = (γ1, γ2) =⇒ Electr. Current J⊥ = (J1, J2)

pµxµ = p+x−+p−x+

2 − ~p⊥~x⊥x+, x−, ~x⊥ =⇒ p+, p−, ~p⊥p− =⇒ Light-Front Energy

p2 = p+p− − (~p⊥)2 =⇒ p− = (~p⊥)2+m2

p+ On-shell

Bosons =⇒ SF (p) = 1p2−m2+ıǫ

Fermions =⇒ SF (p) = /p+m

p2−m2+ıǫ+ γ+

2p+

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


OutlineLight-Front




An Example

I cov1 =

∫

d4k1

(k2 − m2 + ıǫ)3=

ıπ3

2m26= 0

IFL1 =

∫

d2k⊥dk+dk− 1

(k+k− − k2⊥ − m2 + ıǫ)3

= 0

Double Pole =⇒ k− =k2⊥

+m2−ıǫ

k+


OutlineLight-Front




Pole Dislocation Method

p+ =⇒ p′+ = p+ + δ

Boson Eletromagnetic Current

Breit Frame =⇒ q− = 0, q+ =⇒ 0+, ~q⊥ 6= 0

J+ = J− + restoration covariance term

J⊥ ∝ q+ ⇒ 0J. de Melo, Sales and T.Frederico Nucl. Phys. B631,(1998) 574c-579c.

Ward-Takahashi Identity =⇒ Pair ContribuitionNaus, de Melo and Frederico

Few-Body Syst. 24, 1998, 99-107J. Pacheco B. C. de Meloa , B. El-Bennichb , B. Loiseaub ,J.-P. Dedonderb and T. FredericocHeavy Mesons Electromagnetic Form Factor with Light-Front App

OutlineLight-Front




Pµ P′µγ5 γ5

γ+ , γ−

kµ − P′µkµ − Pµ

kµ


OutlineLight-Front




Frame

q+ = −q− =√

−q2 sinα

qx =√

−q2 cos α, qy = 0

q2 = q+q− − (q⊥)2 .

Breit Frame (α = 0) =⇒ q+ → 0 , q− = 0 ; ~q 6= 0

J+π = J0 + J3 =⇒ No Pair Term Contribuition

J−π = J0 − J3 =⇒ Pair Term Contribuition

J.P.B.C. de Melo, T. Frederico and H.L. NausPhy.Rev. C59 (1999) 2278J.P.B.C. de Melo, T. Frederico, E. Pace and G. SalmeNucl. Phys. A 707 (2002) 399


OutlineLight-Front




Pion

Lagrangian to Vertex π → qq

LI = −ımfπ

~π · qγ5~τq

• Electromagnetic Current

Jµ = −ı2em2

f 2π

Nc

∫

d4k

(2π)4Tr [S(k)γ5

×S(k − P ′)γµS(k − P)γ5Λ(k,P ′)Λ(k,P)]

• S(p) = 1

/p−m+ıǫ


OutlineLight-Front




Pion

Vertex Function

• Nonsymmetric Vertex Function

Λ(k,P) =N

((P − k)2 − m2R + ıǫ)

• Symmetric Vertex Function

Λ(k,P) =N

(k2 − m2R + ıǫ)

+N

((P − k)2 − m2R + ıǫ)

• Ref. Nucl. Phys. A 707 (2002) 399-424


OutlineLight-Front




Pion

Electromagnetic Form Factor

• jµ = e(Pµ + P ‘µ)Fπ(q2)• ”J+” and J− Components of the Electromagnetic Current• Integration Intervals in k−

I) 0 < k+ < P+

II) P+ < k+ < P′+

Form Factor

Fπ(q2) = F (I )π (q2, α) + F (II )

π (q2, α)

where• F

(I )π (q2, α) → Valence Contribuition (qq).

• F(II )π (q2, α) → Non-Valence Contribuition.


OutlineLight-Front




Pion

Valence and Non-Valence

k − P

P k

k − P ′

P ′

k − P

P k

k − P ′

P ′J. Pacheco B. C. de Meloa , B. El-Bennichb , B. Loiseaub ,J.-P. Dedonderb and T. FredericocHeavy Mesons Electromagnetic Form Factor with Light-Front App

OutlineLight-Front




Pion

Pion Wave Function

• Dirac Propagator in the Light-Front:/k+m

k2−m2+ıǫ=

/kon+m

k+(k−−k−on+

ıǫk+ )

+ γ+

2k+

• Bethe-Salpeter Amplitude:

Ψ(k,P) = mfπ

/k+m

k2−m2+ıǫγ5Λ(k,P)

/k−/P+m

(k−P)2−m2+ıǫ

• The Wave Function Appear after Elimination• Instantaneous Terms• Factors with Gamma Matrix in the Numerador• Factors k+ and (P+ − k+) in the Denominator and Integrationk−


OutlineLight-Front




Pion

Final Pion Wave Function

Φ(k+, ~k⊥;P+, ~P⊥) =P+

m2π − M2

0

[ N(1 − x)(m2

π −M2(m2,m2R))

+N

x(m2π −M2(m2

R ,m2))

]

• here: N =√

Nc C mfπ

and

M2(m2a,m

2b) =

k2⊥

+m2a

x+

(P−k)2⊥

+m2b

1−x− P2

⊥

• Free Mass Operator M20 = M2(m2,m2)


OutlineLight-Front




Pion

Dispersion Relations

• Bound State Mass M: s = (k1 + k2)2, k2

1 = m21, k2

2 = m22 .

• Pole Amplitude: s = M2

• Wave Function: φ(s) = Gv (s)s−M2

• Vertex Function Gv (s)

• Normalization Condition:∫ ∞(m1+m2)2

G 2v (s) ρ(s,m1,m2)

π(s−M2)2ds = 1


OutlineLight-Front




Pion

• Spectral Density (Pseudoscalar)

ρP(s,m1,m2) =λ1/2(s,m2

1,m22)

8πs[s − (m2 − m1)

2] θ(

s − (m1 + m2)2)

• λ(s,m21,m

22) ≡ (s + m2

1 − m22)

2 − 4sm21 ,

• Wave Function: φP(s) = NPπ√2

√s2−(m2

1−m22)

2√s−(m2−m1)2

1s3/4 w(k) ,

• Gaussian: w(k) = exp(

−4νk2/m212

)

where m12 = m1m2/(m1 + m2)


OutlineLight-Front




Pion

Parameters and Numerical Results

• NonSymmetric Vertex• Quark Mass mq: m = 0.220 GeV• Regulator Mass mR : m = 0.946 GeV• Fit for fπ = 101.0 MeV• Pion Mass: 0.140 GeV• Charge Radius 〈r2〉 = 6 ∂

∂q2 Fπ = 0.67fm


OutlineLight-Front




Pion

• Symmetric Vertex• Quark Mass mq = 0.220 GeV• Regulator Mass mR = 0.6 GeV• Fit for f

expπ = 92.4 MeV

• Pion Mass: 0.140 GeV• Charge Radius: 〈r2〉 = 6 ∂

∂q2 Fπ = 0.74fm

10% Bigger than Experimental Value (rexp = 0.67 ± 0.02fm)


OutlineLight-Front




Pion

0 2 4 6 8 10

Q2=−q

2[GeV/c]

2

0.00

0.20

0.40

0.60

0.80

1.00

|Fπ(

q2 )|Pion Form Factor

λ=10 Γ= 0.3 znpi0=15 Gauss=80 mq=0.384 GeV npion=10

Jefferson Lab. (Exp.) Frascati (Exp.)Symmetric VertexNonsymmetric vertex whitout pair termsNonsymmetric vertex with pair termsQCD Sum RulesVector Meson Dominance


OutlineLight-Front




Pion

0 2 4 6 8 10

Q2=-q

2[GeV/c]

2

0.00

0.20

0.40

0.60

0.80

1.00

|Fπ(q

2 )|

Jefferson Lab. (Exp.)Tadevosyan et al. (Exp.)Horn et al. (Exp.) Frascati (Exp.)mq=0.220 GeV (Dispersion Relation)mq=0.220 GeV MR=0.946 GeV (NS Model)mq=0.350 GeV MR=0.946 GeV (NS Model)mq=0.350 GeV MR=0.6946 GeV (NS Model)mq=0.350 GeV mR=0.946 GeV (Symmetric)mq=0.220 GeV mR=0.547 GeV (Symmetric)Disperion with Pol.mq=0.350 GeV mR=0.6946 GeV (Symmetric)

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

q=0.384 GeV npion=10


OutlineLight-Front




Kaon

Kaon

• |K+ > → |us >

• Breit Frame =⇒ q+ = 0 , q− = 0 ; ~q 6= 0• Pµ = (P0,P0,−qx/2, 0) =⇒ Initial State• P ′µ = (P0,P0, qx/2, 0) =⇒ Final State, P0 =

√

m2π + q2

x/4


OutlineLight-Front




Kaon

Elect. Form Factor

F+q (q2) = −eq

N2g2Nc

4π3P+

∫

d2k⊥dx

xN+

q θ(x)θ(1 − x)

×Φ∗fq (x , k⊥)Φi

q(x , k⊥) ,

F+q (q2) = [ q ↔ q in F+

q (q2) ]

F+K+(q2) = F+

q (q2) + F+q (q2)

F−K+(q2) =

[

F−q (q2) + F−

q (q2)]

(I )+

[

F−(q2)]

(II )


OutlineLight-Front




Kaon

Kaon Wave Function

Φ(x , ~k⊥) =N

(1 − x)21

(m2K+ − M2

0 )(m2K+ − M2

R)

Normalization: FK+(0) = F−K+(0) = 1


OutlineLight-Front




Kaon

kaon Parameters

• Quark Masss : mq = 0.220 GeV mq = 0.419 GeV• Regulator Mass: mR : m = 0.946 GeV• Decay Constant: f

expK = 113 MeV

• Kaon Mass: 0.496 GeV• Charge Radius: 〈r2〉 = 6 ∂

∂q2 FK = 0.354 fm2

• Exp.: 〈r2〉expK = 0.340 ± 0.05 fm2

• Ref.: Nucl. Phys. A 610 (2007) 610.


OutlineLight-Front




Kaon

0.01 0.10 1.00

-q2 [(GeV/c)

2]

0

0.2

0.4

0.6

0.8

1

|FK

+ (q

2 )|2


OutlineLight-Front




Kaon

0.0 1.0 2.0 3.0 4.0

-q2 [(GeV/c)

2]

0

0.2

0.4

0.6

0.8

1|F

K+ (

q2 )|

2


OutlineLight-Front




0 2 4 6 8 10

Q2=-q

2[GeV/c]

2

0.00

0.20

0.40

0.60

0.80

1.00

|FΒ(q

2 )|2

Dipersion Relation Approach (Gaussian wave function)m=0.384 GeV MR=7.95 GeVmu=0.350 GeV MR=7.95 GeV mu=0.340 GeV mb=4.90 GeV mR=7.95 GeV mu=0.335 GeV mb=4.90 GeV mR=7.95 GeVmu=0.333 GeV mb=4.9 GeV mR=7.95 GeV


OutlineLight-Front




Final Remarks

• Light-Front =⇒{

Bound States

Covariance

• Rotational Invariance Broken =⇒ k− Problematic

• Terms

{

− Good

− Bad

• Electromagnetic Current: “ + ” , “- ” and “ ⊥ ”

• Particles

− Bosons

− Pseudoscalar

− Vector

• Pairs Terms Contribuition =⇒ Full Covariance Restorate• New Informations about Bound States qq =⇒ B Meson• Next =⇒ D Mesons and Vector Mesons


heavy mesons electromagnetic form factor with light-front … · 2007. 12. 10. · b meson elect....

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