Revealing Treacherous Points for Successful Light-Front
Phenomenological Applications
LC2005, Cairns, July 14, 2005
Motivation
• LFD Applications to Hadron Phenomenology
-GPD,SSA,…(JLAB,Hermes,…)
-B Physics (Babar,Belle,BTeV,LHCB,…)
-QGP,Quark R & F (RHIC,LHC ALICE,…)• Significance of Zero-Mode Contributions
-Even in J+ (G00 in Vector Anomaly)
-Angular Condition(Spin-1 Form Factors,…)
-Equivalence to Manifestly Covariant Formulation
How do we find where they are?
Outline• Common Belief of Equivalence - Exactly Solvable Model - Heuristic Regularization ~ Arc Contribution
• Vector Anomaly in W± Form Factors- Brief History- Manifestly Covariant Calculation
• Pinning Down Which Form Factors- Dependence on Formulations- Direct Power-Counting Method
• Conclusions
Common Belief of Equivalence
∫ 0dk
Manifestly Covariant Formulation
Equal t Formulation Equal = t + z/c Formulation
∫ −dk
(Time Ordered Amps)
However, the proof of equivalence is treacherous.B.Bakker and C.Ji, PRD62,074014 (2000)
Heuristic regularization to recover the equivalence.
B.Bakker, H.Choi and C.Ji, PRD63,074014 (2001)
Exactly Solvable Model of Bound-States
⎟⎟⎠
⎞⎜⎜⎝
⎛+=+=
Φ=Φ+−−+− ∫ dim11for2n
dim13for4n)(),()(})){(( 2222 llkKldkimkpimk p
npεε
S.Glazek and M.Sawicki, PRD41,2563 (1990)
...5int +ΨΨΦ+ΨΨΦ= sps gigL γ
Electromagnetic Form Factor
)()'(||' 2qFppipJp μμ +=
H.Choi and C.Ji, NPA679, 735 (2001)
Equivalent Result in LFD
)()'(||' 2qFppipJp ±±± +=
Valence Nonvalence
+
)()()()( 2222cov qFqFqFqF nvvaltot
+++ +==
)()'(||' 2qFppipJp ±±± +=
20
2 1),(
),(
)2()(
MRwhere
x
xRdx
NqFnv α
αααα
ααπ
α +=
−+= ∫−
However, the end-point singularity exists in F-(q2).
B.Bakker and C.Ji, PRD62, 074014 (2000)
Heuristic Regularizationto recover the equivalence
)()()(),(),( 222
cov
0
qFqFqFforx
RxRdx tottot
−+ ==−−
∫ ααααα
ε
γ μμ
ikkSwhere
pkSpkS
+Λ−
Λ=
−−=Γ
Λ
ΛΛ
22
2
)(
)'()(
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
211221
1111
DDDDDD
Arc Contribution in LF-Energy Contour
€
dk− (k−)2
(k− − k1−)(k− − k2
−)(k− − k3−)−∞
∞
∫ = −i dθ = −iπarc
∫
€
€
k1− k2
− k3−
€
dk− = dk−
−∞
+∞
∫ + dk−
arc
∫ = 0contour
∫
€
dk−
−∞
+∞
∫ = − dk−
arc
∫€
With the arc contribution, we find
€
Fnv− (q2) =
N
π (2 + α )dx
0
α
∫ R(x,α ) − R(α ,α )
α − x
Form Factor Results
( )MeVExptMeVf
MeVMeVmm du
25.04.92.5.92
900,250
±=
=Λ==
π
( )MeVExptMeVf
MeVMeVm
K
ss
1.14.113:5.112
910,480
±=
=Λ=
( )MeVExptMeVf
GeVGeVm
D
cc
9.154:6.108
79.1,78.1
≤=
=Λ=
Standard Model
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
μμ vv
e
vb
t
s
c
d
u
e
1
0
3/1
3/2
−
−
€
Q f = 0f
∑ (Anomaly − Free Condition)
• Utility of Light-Front Dynamics (LFD)• “Bottom-Up” Fitness Test of Model TheoriesB.Bakker and C.Ji, PRD71,053005(2005)
CP-Even Electromagnetic Form Factors of WGauge Bosons[ ]
⎭⎬⎫
⎩⎨⎧
+Δ
+−Δ+−++=Γ βαμ
αμββ
μα
μβα
μαβαβ
μμαβ κ qqpp
M
QqgqggqgqgppAie
W
)'(2
)())(()(2)'(
2
At tree level, for any q2,
0,0,1 =Δ=Δ= QA κBeyond tree level,
⎭⎬⎫
⎩⎨⎧
++−++−= )()'(2
)()()()'( 232
22
21 qFpp
M
qqqFqgqgqFgppJ
W
μβαα
μββ
μααβ
μμαβ
μαβ
μαβ
κ
Jie
qFQ
qFqF
qFA
−=Γ
=Δ−
+=Δ−
=
),()(
),(2)()(
),(
23
21
22
21
One-loop Contributions in S.M.
W.A.Bardeen,R.Gastmans and B.Lautrup, NPB46,319(1972)G.Couture and J.N.Ng, Z.Phys.C35,65(1987)E.N.Argyres et al.,NPB391,23(1993)J.Papavassiliou and K.Philippidas,PRD48,4255(1993)
One-loop Contributions in S.M.
W.A.Bardeen,R.Gastmans and B.Lautrup, NPB46,319(1972)G.Couture and J.N.Ng, Z.Phys.C35,65(1987)E.N.Argyres et al.,NPB391,23(1993)J.Papavassiliou and K.Philippidas,PRD48,4255(1993)
One-loop Contributions in S.M.
W.A.Bardeen,R.Gastmans and B.Lautrup, NPB46,319(1972)G.Couture and J.N.Ng, Z.Phys.C35,65(1987)E.N.Argyres et al.,NPB391,23(1993)J.Papavassiliou and K.Philippidas,PRD48,4255(1993)
One-loop Contributions in S.M.
W.A.Bardeen,R.Gastmans and B.Lautrup, NPB46,319(1972)G.Couture and J.N.Ng, Z.Phys.C35,65(1987)E.N.Argyres et al.,NPB391,23(1993)J.Papavassiliou and K.Philippidas,PRD48,4255(1993)
Vector Anomaly in Fermion Triangle Loop
“Sidewise” channel “Direct” channel
""""
2
2
""""
)()(26
)()(
DirectSidewise
WFDirectSidewise
MG
Δ=Δ
+Δ=Δπ
κκ
L.DeRaad, K.Milton and W.Tsai, PRD9, 2847(1974); PRD12, 3972(1975)
Vector Anomaly RevisitedSmearing of charge (SMR)
Pauli-Villars Regulation (PV1, PV2)
Dimensional RegularizationDR4,DR2)
B.Bakker and C.Ji, PRD71,053005(2005)
Manifestly Covariant Calculation
[ ]313121
1
0
1
0321 )()(
12
1
yDDxDDDdydx
DDD
x
−+−+= ∫∫
−
kik == κκ ,00
∫ΓΓ
−−Γ+Γ=
+ −−)()
2(
)2
()2
(
)()(
)(
22
2
22
2
α
βαβπ
κ
κκ
βαα
β
n
nn
aa
dn
n
n
Manifestly Covariant Results
4323133 )()()()( DRPVPVSMR FFFF ===
2
3
1
4)2()2(
3
2
4)2()2(
6
1
4)2()2(
22
2
2
412212
2
2
412112
2
2
41212
WF
fDRPV
fDRPV
fDRSMR
MGg
QgFFFF
QgFFFF
QgFFFF
=
⎟⎠
⎞⎜⎝
⎛−++=+
⎟⎠
⎞⎜⎝
⎛++=+
⎟⎠
⎞⎜⎝
⎛++=+
π
π
π
LFD Results
)22(2,2),22(2),(2
),(4/0,||',
32
21003321031
2222''
FFFpGFpGFFFpGFFpG
qQMQwithframeqinphJphG Whh
ηηηηηη
η
−−=−=++=+=
−====++++
−+++
+++
++
+++
J+
LFD Results
)22(2,2),22(2),(2
),(4/0,||',
32
21003321031
2222''
FFFpGFpGFFFpGFFpG
qQMQwithframeqinphJphG Whh
ηηηηηη
η
−−=−=++=+=
−====++++
−+++
+++
++
+++
( ) ∫∫ ≠−++−−+
=⊥
⊥⊥
++ 0
)1(
)1(
2 221
2
221
22
1
023
2
..00 Qxxmk
Qxxmkkddx
M
pQgG
W
f
MZ π
J+
q+=0
LFD Results
)22(2,2),22(2),(2
),(4/0,||',
32
21003321031
2222''
FFFpGFpGFFFpGFFpG
qQMQwithframeqinphJphG Whh
ηηηηηη
η
−−=−=++=+=
−====++++
−+++
+++
++
+++
( ) ( ) [ ]+−++++++
+−+
++
++ ++−+=+
⎥⎥⎦
⎤
⎢⎢⎣
⎡+=+ GGG
pFFG
G
pFF )41()21(
4
12,
2
12 00
0012
0012 ηη
ηη
( ) ∫∫ ≠−++−−+
=⊥
⊥⊥
++ 0
)1(
)1(
2 221
2
221
22
1
023
2
..00 Qxxmk
Qxxmkkddx
M
pQgG
W
f
MZ π
LFD Results for Other Regularizations
⎟⎠
⎞⎜⎝
⎛++=+=+=+ +
6
1
4)2()2()2()2(
2
2
412cov
1200
120
12 πf
DRSMRSMRSMR
QgFFFFFFFF
0212 )2( ++ PVFF
⎟⎠
⎞⎜⎝
⎛++=+=+=+ +
3
2
4)2()2()2()2(
2
2
412cov
11200
1120
112 πf
DRPVPVPV
QgFFFFFFFF
00212 )2( PVFF + ⎟
⎠
⎞⎜⎝
⎛−++=+3
1
4)2()2(
2
2
412cov
212 πf
DRPV
QgFFFF
Pinning Down Which Form Factors• Jaus’s -dependent formulation yields
zero-mode contributions both in G00 and G01.
W.Jaus, PRD60,054026(1999);PRD67,094010(2003)
• However, we find only G00 gets zm-contribution.
B.Bakker,H.Choi and C.Ji,PRD67,113007(2003)
H.Choi and C.Ji,PRD70, 053015(2004)• Also,discrepancy exists in weak transition form
factor A1(q2)=f(q2)/(MP+MV).
Power Counting Method
H.Choi and C.Ji, PRD, in press.
Electroweak Transition Form Factors
€
< P2;1h | JV −Aμ | P1;00 >= ig(q2)εμναβεν
* Pα qβ
− f (q2)ε*μ − a+(q2)(ε* ⋅P)P μ − a−(q2)(ε* ⋅P)qμ
where
€
P = P1 + P2, q = P1 − P2
€
< JV −Aμ >h = i
d4k
(2π )4
SΛ1(P1 − k)Sh
μ SΛ 2(P2 − k)
Dm1DmDm2
∫
where
€
Dm = k 2 − m2 + iε,
SΛ i(Pi) = Λi
2 /(Pi2 − Λi
2 + iε),
Shμ = Tr ( / p 2 + m2)γ μ (1− γ 5)( / p 1 + m1)γ 5(−/ k + m)ε* ⋅Γ[ ],
Γ μ = γ μ −(P2 − 2k)μ
D,
and
€
(1) Dcov (MV ) = MV + m2 + m,
(2) Dcov (k ⋅P2) = 2k ⋅P2 + MV (m2 + m) − iε[ ] / MV ,
(3) DLF (M0) = M0 + m2 + m.
Power Counting Method
€
< JA+ >z.m.
h ∝ limα →1
dxα
1
∫ (1− x)2
(1−α )2Sh
+(km1
− ) ⋅⋅⋅[ ]
= limα →1
(1−α ) dz0
1
∫ (1− z)2 Sh+(km1
− ) ⋅⋅⋅[ ],
where
€
x = α + (1−α )z and ⋅⋅⋅[ ] is regular as α →1.
€
Sh= 0+ Power Counting :
(1) (1− x)−1 = (1−α )(1− z)[ ]−1
for Dcov (MV ),
(2) (1− x)0 for Dcov (k ⋅P2),
(3) (1− x)−1/ 2 = (1−α )(1− z)[ ]−1/ 2
for DLF (M0).
Conclusions• The common belief of equivalence between manifestly
covariant and LF Hamiltonian formulations is quite treacherous unless the amplitude is absolutely convergent.
• The equivalence can be restored by using regularizations with a cutoff parameter even for the point interactions taking
limit.• The vector anomaly in the fermion-triangle-loop is real and
shows non-zero zero-mode contribution to helicity zero-to zero amplitude for the good current.
• In LFD, the helicity dependence of vector anomaly is also seen as a violation of Lorentz symmetry.
• For the good phenomenology, it is significant to pin down which physical observables receive non-zero zero-mode contribution.
• Power counting method provides a good way to pin down this.