Applications of Light-Front Dynamics in Hadron Physics 1. What is ...

Applications of Light-Front Dynamics in
Hadron Physics
Chueng-Ryong Ji
North Carolina State University
1. What is light-front dynamics (LFD)?
2. Why is LFD useful in hadron physics?
3. Any first principle QCD predictions?
4. Any use of effective field theory?
5. Any treacherous points?
6. Any relevance to JLab experiments?
June 5-7, 2013
Questions

Outline(tentative)
• Tutorials of LFD (5 th , 9-10am)
• Interpolating QFT (5 th , 1-2pm)
• Perturbative QCD Predictions (6 th ,9-10am)
• Chiral Effective Field Theory (6 th , 1-2pm)
• Anomaly and Zero Modes(7 th, 10:15-11:15am)
• Application to DVCS and GPDs (7 th ,1-2pm)

Interpolation between Instant and Front Forms
K. Hornbostel, PRD45, 3781 (1992)
C.Ji and C. Mitchell, PRD64,085013 (2001)
C.Ji and A. Suzuki, PRD87,065015 (2013)

Interpolating Hadronic Wavefunction
x 0 ! x ˆ+ " x +
Invariant under kinematic transformations
( P ! , J ! ) 0 ! ! " " / 4 (P + , P ! !
, J 3 , E ! !
, K 3 )

" = 0
p 0
= p 0
0 < " < # /4
pˆ +
= p 0 cos" $ p 3 sin"
= $ p0 sin" + p 3 cos"
pˆ
" = # /4
p +
= p $
p $
= p +
#p 3
= p 3 !
!

Σ(a)+Σ(b)=1/(s-m 2 ) ; s=2 GeV, m=1GeV
J-shape peak & valley :
P z
= "
s(1" C)
2C
; C = cos(2#)

g µ" =
$ 1 0 0 0 '
&
)
&
0 #1 0 0
)
& 0 0 #1 0 )
&
)
% 0 0 0 #1(
"
$
!
! d 3 x T 0µ = P µ =
$
$
$
# $
P 0
P 1
P 2
P 3
%
'
'
'
'
&'
#
%
!
! d 3 x (T 0µ x ! "T 0µ x ! ) = J µ! %
=
%
%
$ %
T µ" =
# L
#(# µ
$ k
) #" $ k
% g µ" L
!
& ;
k
"
$
$
P ˆµ = $
$
$
#
g ˆ µ " ˆ
=
0 K 1 K 2 K 3
"K 1 0 J 3 "J 2
"K 2 "J 3 0 J 1
"K 3 J 2 "J 1 0
" µ
T µ# = 0
% cos2# 0 0 sin2# (
'
*
'
0 $1 0 0
*
' 0 0 $1 0 *
'
*
& sin2# 0 0 $cos2# )
P ˆ+ = cos! P 0 + sin! P 3
P ˆ1 = P 1
P ˆ2 = P 2
P ˆ! = sin! P 0 ! cos! P 3
!
&
(
(
(
(
'(
J ˆ µ " ˆ
=
%
'
' !
' = ) dx ˆ! d 2 x( T
'
'
&
ˆ+ ˆµ
$ 0 E ˆ 1 E ˆ 2 #K 3 '
&
# E ˆ 1 0 J 3 # F ˆ )
&
1
# E ˆ 2 #J 3 0 # F ˆ
)
&
2 )
&
% K 3 F ˆ 1 F ˆ
)
2 0 (
E ˆ 1 = J 2 sin" + K 1 cos"
E ˆ 2 = K 2 cos" # J 1 sin"
F ˆ 1 = K 1 sin" # J 2 cos"
F ˆ 2 = J 1 cos" + K 2 sin"

Interpolating Poincaré Algebra
[P ˆµ , P ˆ! ] = 0 [P ˆµ , J ˆ" ˆ # ] = i(g ˆµ ˆ" P ˆ # ! g ˆµ ˆ # P ˆ" )
[J ˆµ ˆ! , J ˆ" ˆ # ] = i(g ˆµ ˆ # J ˆ! ˆ" + g ˆ! ˆ" J ˆµ ˆ # ! g ˆµ ˆ" J ˆ! ˆ # ! g ˆ! ˆ # J ˆµ ˆ" )
e.g.
[P ˆ+ , J ˆ+ˆ! ] = i(g ˆ+ˆ+ P ˆ! ! g ˆ+ˆ! P ˆ+ )
[P ˆ+ ,!K 3 ] = i(P ˆ! cos2! ! P ˆ+ sin2!)
Return of Prodigal Son
Exp "i# K 3
!
" #$ /4
[K 3 , P + ] = !iP +
( ) | x + > $ | x + >
One more kinematic generator appears only in the front form.
Maximum number (7) of members in the stability group.

Kinematic Operators
(Members of Stability Group)
Exp ("i# $ ˆ i
) | x
+ ˆ
> % | x
+ ˆ
>
[ ˆ! i , P ˆ+ ] = 0
!
!
" = 0
#J 2
J 1
!
" ˆ i = F ˆ i cos2# $ E ˆ i sin2#
" ˆ 1 = #J 2 cos$ # K 1 sin$
ˆ " 2 = J 1 cos$ # K 2 sin$
(J 3 , P 1 , P 2 , Pˆ!)
!
" = # /4
$E 1 = $(J 2 + K 1 ) / 2
E 2 = (J 1 $ K 2 ) / 2

!
same
!
!
p 0
remain at rest
P 0 = M ; p 3 = 0
particle at rest
p 0 = M , p 1 = p 2 = p 3 = 0
(pˆ +
= M cos" , pˆ
= M sin")
#
!
Under
ˆ " i transformation
P 0 = M +
can move
!
!
p "
2
p 0 + p 3
2M ; p3 = #
!
p "
2
2M
same
!
(p 0 ) 2 " (p 3 ) 2 = (M +
!
p #
2
2M )2 " ("
!
p #
2
2M )2 = M 2 + ! p # 2 = 2p + p " > 0
Rational Energy-Momentum Dispersion Relation
Vacuum gets simpler in LFD.

!
Angular Momentum
[J i ,J j ] = i" ijk
J k ,[J i ,M] = 0
{ }
T = Exp "i(# ˆ
1
$
1 + # ˆ
2
$ 2 )
[" ˆ i ," ˆ j ] = i# " ˆ k , [" ˆ i , M] = 0
ijk
!
T | n >=| p,n >
ˆ " i | p,n >= TJ i | n >
!
" ˆ i = TJ i T +

" ˆ ( 3 = J 3 pˆ
+ ˆ
#
z $ ( p ! !
%
&' ˆ +
)
%
),
*
- / M sin.
!
" ˆ
%
= J ! (
%
+ p ! z
%
cos. J 3 + ˆ $ ( p ! !
5 /
%
&' ˆ 2
%
)
1
0
pˆ #
+ M sin. 4 # (ˆ z & p ! /
)
%
)sin. 1
K 3 +
* 5
3
0
" #$ /4
pˆ
!
!
p %
$ E ˆ 2 +
%
5
,
#
+ M sin. 4
3 - 5
" 3 = J 3 + z ˆ # ( !
! '
" $
= ( z ˆ % ( p ! & E $ !& p ! + F $
+ p !
$
K 3 ) &
)
" 3 = W +
E $
% p !
$
) / p +
!
p $
p +
( p ) $
p + J 3 + z ˆ # E !
$
% !
*
+
, / M
p + W ˆµ = 1 2 ! ˆµ ˆ" ˆ# ˆ$ p ˆ" J ˆ# ˆ$
[" 3 , Stability Group Members] = 0

Interpolating Spinors
ˆ! 3 û (1) (1)
CR
= (+1)û CR

Interpolating Spinors
ˆ! 3 û (2) (2)
CR
= ("1)û CR

Interpolating Helicity Amplitude
ˆM(! 1
, ! 2
, ! 3
, ! 4
) = ˆ"( ˆp 2
, ! 2
)# ˆµû( ˆp 1
, ! 1
)û( ˆp 3
, ! 3
)# ˆµ ˆ"( ˆp 4
, ! 4
)

Jacob-Wick Helicity vs. Light-Front Helicity
Invariant under kinematic transformations
Related by a rotation

Canonical Quantization
Simple 1+1 dim QFT:
L = 1 2 ! µ! ! µ ! " 1 2 m2 ! 2
L = C #
$
2 (! ˆ+!) 2 " (! ˆ"!) 2
%
& + S(! ˆ+!)(! ˆ"!)" 1 2 m2 ! 2
C = cos(2!), S = sin(2!)
! (x) = !L
!(! 0
") = !0 "
! (x) = C! ˆ+" + S! ˆ""
Conjugate field:
Equal-time quantization:
[! (x),"(y)] x 0 =y 0 = !i#(x 3 ! y 3 )
[! (x),"(y)] x ˆ+ =y ˆ+ = !i#(x ˆ! ! y ˆ! )

T µ" =
&
k
# L
#(# µ
$ k
) #" $ k
% g µ" L
Pˆ+
Energy
= "
ˆ!
dx (! # ˆ+" ! L)
= 1 2
Periodic Boundary Condition:
!(x ˆ+ = 0, x ˆ! ) =
$
%
P 0 = # dx 3 (! ! 0
" L)
&
'
!(x ˆ+ , x ˆ! ! !) = !(x ˆ+ , x ˆ! + !)
" dx ˆ! C{(# ˆ+") 2 + (# ˆ!") 2 }+ m 2 " 2
.
/
n=!.
!
! n
= n 2 + C m! $
# &
" " %
Pˆ+ = !
#
! " !
2
1
(
a n
e !i "
$
n" # !
*
4"# n )*
%
'x ˆ!
&
$ !" & n
' S n $
(#
& a + n
a n
% " C %
n
"
$
+ a + n
e i n" # !
[a m
, a n + ] = ! mn
%
'x ˆ!
&
+
-
,-

Symmetry Breaking
4
4
3
3
2
2
1
1
2 1 1 2
! !"
= ! +"
2 1 1
L ! L " = L # m 2 ! " # 1 2 m2 ! 2
( ) 1/2 !
Pˆ+ ! Pˆ+ " = Pˆ+ + m3 !
(a
C 1/4 0
+ a + 0
)

Nontrivial Vacuum State
| 0 > ! | " >
! !"
= ! +"
$ +! '
| ! > = exp&
i # dx ˆ" ! " (x ˆ" )) | 0 >
%
(
Translation in scalar field:
! (x ˆ+ = 0, x ˆ! ) = !i
"!
.
/
n=!.
" ! %
$ '
# ! &
"
(
n
4! a ne !i "
$
n! # !
*
)*
%
'x ˆ!
&
"
$
! a + n
e i n! # !
%
'x ˆ!
&
+
-
,-
#
| ! >= exp "(C 1/2 m!) ! 2 &
%
(exp#"(C 1/2 m!) 1/2 +
$ !a 0
&
'
$ 2 '
| 0 >
Condensation of Zero-Modes

Vacuum Energy
Pˆ+ | ! >= E !
| ! >
a e !a+ | 0 >= ! e !a+ | 0 >
Pˆ+
#
m!
| ! >"
C a ( +
1/2 0a 0
+ m3 !) 1/2 !
&
%
(a
C 1/4 0
+ a + 0
)(
%
( exp # )(C1/2 m!) 1/2 +
$ !a 0
&
' | 0 >
$
'
= ()m 2 ! 2 !) exp #)(C 1/2 m!) 1/2 +
$ !a 0
&
' | 0 >
+!
#
"!
E !
= "m 2 ! 2 ! = (" 1 2 m2 ! 2 )dx ˆ"
Independent of interpolation angle!

Recovery of Trivial Vacuum in LFD
#
| ! >= exp "(C 1/2 m!) ! 2
%
$ 2
&
(exp#"(C 1/2 m!) 1/2 +
$ !a 0
'
&
' | 0 >
| ! >"| 0 > as C " 0
However,
E !
and
< ! |!(x) | ! >= ""
are still independent of interpolation angle!

What is going on?
< ! |!(x) | ! >
'
+
=< 0 | exp #
$ (C 1/2 m!) 1/2 "(a + 0
" a 0
)%
a 0
+ a 0
&)
( 2(C 1/2 m!) 1/2
= ""
*
,exp#
$ "(C 1/2 m!) 1/2 "(a + 0
" a 0
)%
&
+
| 0 >
Complication is transferred from vacuum to operator.

Summary
• LFD is not just formal but consequential
in the analysis of physical observables.
• Longitudinal boost joins stability group in
LFD.
• LF helicity amplitudes are independent of
all references frames that are related by
front-form boosts.
• Energy-momentum dispersion relation
becomes rational and vacuum gets
simpler in LFD.