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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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
Light-FrontOverview of the Light-Front
Electromagnetic Current
Light-Font Integration
Triangle Diagram
PionPion
kaonKaon
B Meson Elect. Form Factor
Conclusions
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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)
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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+
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
Pµ P′µγ5 γ5
γ+ , γ−
kµ − P′µkµ − Pµ
kµ
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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.
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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−
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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)
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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)
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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)
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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 )
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
Kaon
Kaon Wave Function
Φ(x , ~k⊥) =N
(1 − x)21
(m2K+ − M2
0 )(m2K+ − M2
R)
Normalization: FK+(0) = F−K+(0) = 1
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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.
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
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
Electromagnetic CurrentLight-Font Integration
Triangle DiagramPionkaon
B Meson Elect. Form FactorConclusions
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
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