revealing treacherous points for successful light-front phenomenological applications
DESCRIPTION
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,…) - PowerPoint PPT PresentationTRANSCRIPT
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Revealing Treacherous Points for Successful Light-Front
Phenomenological Applications
LC2005, Cairns, July 14, 2005
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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?
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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
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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)
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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
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Electromagnetic Form Factor
)()'(||' 2qFppipJp μμ +=
H.Choi and C.Ji, NPA679, 735 (2001)
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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)
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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
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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
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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=
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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)
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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
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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)
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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)
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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)
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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)
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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)
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Vector Anomaly RevisitedSmearing of charge (SMR)
Pauli-Villars Regulation (PV1, PV2)
Dimensional RegularizationDR4,DR2)
B.Bakker and C.Ji, PRD71,053005(2005)
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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
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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
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LFD Results
)22(2,2),22(2),(2
),(4/0,||',
32
21003321031
2222''
FFFpGFpGFFFpGFFpG
qQMQwithframeqinphJphG Whh
ηηηηηη
η
−−=−=++=+=
−====++++
−+++
+++
++
+++
J+
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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
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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
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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
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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.
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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
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€
< 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.
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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).
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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.