Revealing Treacherous Points for Successful Light-Front Phenomenological Applications

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)

Vector Anomaly in Fermion Triangle Loop

“Sidewise” channel “Direct” channel

""""

2

2

""""

)()(26

)()(

DirectSidewise

WFDirectSidewise

QQ

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


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


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.

Revealing Treacherous Points for Successful Light-Front Phenomenological Applications

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