Applications of Light-Front Dynamics in
Hadron Physics
1. Anomaly and Zero Modes
2. Application to DVCS and GPDs
Chueng-Ryong Ji
North Carolina State University
June 7, 2013
5th & 6th Lectures
Outline for AM
• Common Belief of Equivalence
• Unexpected Surprises and Treacherous Points
• Chiral Anomaly and Zero Modes in LFD
• Anomaly Free Condition in Standard Model
• Power Counting Method
ò0dk
Common Belief of Equivalence
Manifestly Covariant Formulation
Equal t Formulation Equal t = t + z/c Formulation
ò-dk
(Time Ordered Amps) S
However, the proof of equivalence is treacherous.
B.Bakker and C.Ji, PRD 62, 074014 (00)
Heuristic regularization to recover the equivalence.
B.Bakker, H.Choi and C.Ji, PRD63,074014 (01)
Arc-contribution removes the mystery.
B.Bakker, M.DeWitt, C.Ji, Y.Mishchenko, PRD72, 076005(05)
Electromagnetic Form Factor
)()'(||' 2qFppipJp
52222522 )'(
'
)()2(||'
impk
mpk
impk
mpk
imk
mkTr
kdNpJp
n
n
p ' | J± | p = i (p+ p ')± F±(q2 )
F+(q2 )=?
F-(q2 )
Equivalent Result in LFD
)()'(||' 2qFppipJp ±±± +=
Valence Nonvalence
+
2
0
2 1),(),(
)2()(
MRwhere
x
xRdx
NqFnv
a
aaa
a
a
ap
a+
=-+
= ò-
However, in bad(-) current, the end-point singularity exists without arc contribution.
B.Bakker and C.Ji, PRD62, 074014 (2000)
Fcov (q2 ) = Fval
+ (q2 )a=0 = Fval+ (q2 )a¹0 + Fnv
+ (q2 )a¹0
q2 = -q^2 < 0 q2 = q+q- - q^
2 > 0Alright in good(+) current:
Arc Contribution in LF-Energy
Contour
dk-(k-)2
(k- - k1-)(k- - k2
-)(k- - k3-)
-¥
¥
ò = -i dq = -iparc
ò
k1- k2
- k3-
dk- = dk-
-¥
+¥
ò + dk-arc
ò = 0contour
ò
dk-
-¥
+¥
ò = - dk-arc
ò
With the arc contribution, we find
Fnv- (q2) =
N
p (2 + a)dx
0
a
òR(x,a) - R(a,a)
a - xB.Bakker, M.DeWitt, C.Ji, Y.Mishchenko, PRD72, 076005(2005)
Heuristic Regularization
to recover the equivalence
)()()(),(),( 222
cov
0
qFqFqFforx
RxRdx tottot
ikkSwhere
pkSpkS
22
2
)(
)'()(
211221
1111
DDDDDD
u(p) /kg 51
/p - /k - Mg 5 /k u(p) = u(p)[ /k - /p + M ]g
5 1
/p - /k - Mg 5[ /k - /p + M ]u(p)
= 4M 2 u(p)g 51
/p - /k - Mg 5u(p)+ 2M u(p)u(p)+ u(p) /ku(p)
Ji, Melnitchouk, Thomas, PRL 110, 179191 (2013)
=
“treacherous” k+ = 0 (end-point) term
Note also the relation between PV and PS theories.
ˆ S PV = -i2gA
fp
æ
è ç
ö
ø ÷
2
t ×
t
d4k
(2p )4/ k g5( / p - / k + M)g5 / k
Dp DNò
SPV =1
2u (p,s) ˆ S PV u(p,s)
s
å
Dp = k2 - mp
2 + ie
DN = (p - k)2 - M 2 + ie
LFD
I =1
2dk+dk-ò
1
k+k- - m2 + ie=
1
2
dk +
k+dk-ò
1
k- -m2
k++ i
e
k+
ò
m2
k+- i
e
k+
x
x
k+ > 0
k+ < 0
m2
k+- i
e
k+
``Moving Pole”
X X
Capture the pole!
k+ = rcosf k- = rsinf
I =dr
r0
¥
ò dz2
z - (ia + 1-a 2 +e)éë
ùû
z - (ia - 1-a 2 +e)éë
ùû
ò Þ ip logm2
z = e2if
en)
0
0
0
>
v
¶m Jm = 0 ; qmu( ¢p )g
mu(p)
= u( ¢p )[ / ¢p - /p]u(p)
= u( ¢p )[m - m]u(p)
= 0
¶m J5m = 0 ; qmu( ¢p )g
mg5u(p)
= u( ¢p )[ / ¢p - /p]g5u(p)
= u( ¢p )[ / ¢p g5 +g5 /p]u(p)
= 2mu( ¢p )g5u(p)
= 0 if m = 0
Tree Level
Loop Level ¶m J
m = 0
¶m J5m =
e2
16p 2eabgdFab Fgd
Classical
symmetry
is broken
due to
infinite
degrees
of freedom
in quantum
fields.
C.Ji & S.Rey, PRD53,5815(1996)
Standard Model
t
t vv
e
v
b
t
s
c
d
u
e
1
0
3/1
3/2
)(0 ConditionFreeAnomalyQf
f
CP-Even Electromagnetic Form
Factors of W Gauge Bosons
qqpp
M
QqgqggqgqgppAie
W
)'(2
)())(()(2)'(
2
At tree level, for any q2, 0,0,1 QA
Beyond tree level,
)()'(2
)()()()'( 2322
2
2
1 qFppM
qqqFqgqgqFgppJ
W
Jie
qFQ
qFqF
qFA
),()(
),(2)()(
),(
2
3
2
1
2
2
2
1
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
WF
DirectSidewise
MG
L.DeRaad, K.Milton and W.Tsai, PRD9, 2847(1974);
PRD12, 3972(1975)
Vector Anomaly Revisited Smearing of charge (SMR)
Pauli-Villars Regulation (PV1, PV2)
Dimensional Regularization (DR4,DR2)
B.Bakker and Ji, PRD71, 053005 (2005)
Manifestly Covariant Results
4323133 )()()()( DRPVPVSMR FFFF
2
3
1
4)2()2(
3
2
4)2()2(
6
1
4)2()2(
2
2
2
2
412212
2
2
412112
2
2
41212
WF
f
DRPV
f
DRPV
f
DRSMR
MGg
QgFFFF
QgFFFF
QgFFFF
LFD Results
)22(2,2),22(2),(2
),(4/0,||',
3
2
21003321031
2222''
FFFpGFpGFFFpGFFpG
qQMQwithframeqinphJphG Whh
J+
LFD Results
)22(2,2),22(2),(2
),(4/0,||',
3
2
21003321031
2222''
FFFpGFpGFFFpGFFpG
qQMQwithframeqinphJphG Whh
0)1(
)1(
2 2212
22
1
22
1
0
23
2
..00Qxxmk
Qxxmkkddx
M
pQgG
W
f
MZ
J+ q+=0
LFD Results
)22(2,2),22(2),(2
),(4/0,||',
3
2
21003321031
2222''
FFFpGFpGFFFpGFFpG
qQMQwithframeqinphJphG Whh
GGG
pFFG
G
pFF )41()21(
4
12,
2
12 00
00
1200
12
0)1(
)1(
2 2212
22
1
22
1
0
23
2
..00Qxxmk
Qxxmkkddx
M
pQgG
W
f
MZ
9
2
3
1
2
1
4)2()2(
6
1
4)2()2(
2
2
412
00
212
2
2
412
0
212
f
DRDR
f
DRDR
QgFFFF
QgFFFF
LFD Results for Other
Regularizations
6
1
4)2()2()2()2(
2
2
412
cov
12
00
12
0
12
f
DRSMRSMRSMR
QgFFFFFFFF
0
212 )2( PVFF
3
2
4)2()2()2()2(
2
2
412
cov
112
00
112
0
112
f
DRPVPVPV
QgFFFFFFFF
00
212 )2( PVFF
3
1
4)2()2(
2
2
412
cov
212
f
DRPV
QgFFFF= ?
Limit to q+=0 or q2=q+q--q2 0 or q+0
and Analytic Continuation
from q20 (timelike)
= å +
Absent in the limit q+ 0 ?
q+
Issue in Hadron Phenomenology
• Zero-Mode
• Even if q+→0, the off-diagonal elements do not
go away in some cases.
0(....)lim0
qp
pq
dk
For example, G00 has the zero-mode contribution
in the calculation of spin-1 form factors.
B.Bakker,H.M.Choi and C.Ji,Phys.Rev.D65,116001(2002)
+
n n+2
n n
n n
q+
Typical Electroweak Transition Form Factors
< P2 | JV -Am | P1 >=< P2 |V
m | P1 >
= f+(q2)(P1 + P2)
m + f-(q2)(P1 - P2)
m
and
JV -Am = V m - Amwhere
< P2 | Am | P1 >= 0
for 0- to 0
- transition.
LF-covariant -dependent formulation:W.Jaus,PRD60,054026(99)
Power counting method:H.-M.Choi & C.Ji, PRD80, 054016 (09)
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, PRD72, 013004(2005)
Electroweak Transition Form Factors
< P2;1h | JV -Am | P1;00 >= ig(q
2)emnaben* Paqb
- f (q2)e*m - a+(q2)(e* × P)Pm - a-(q
2)(e* × P)qm
where
P = P1 + P2, q = P1 - P2
< JV -Am >h = i
d4k
(2p)4SL1 (P1 - k)Sh
mSL 2 (P2 - k)
Dm1 DmDm2ò
where
Dm = k2 - m2 + ie,
SL i (Pi) = L i2 /(Pi
2 - L i2 + ie),
Shm = Tr ( / p 2 + m2)g
m (1- g5)( / p 1 + m1)g5(-/ k + m)e* × G[ ],
Gm = g m -(P2 - 2k)
m
D,
and
(1) Dcov (MV ) = MV + m2 + m,
(2) Dcov (k × P2) = 2k × P2 + MV (m2 + m) - ie[ ] / MV ,
(3) DLF (M0) = M0 + m2 + m.
Power Counting Method
where
Sh= 0+ Power Counting :
(1) (1- x)-1 = (1-a)(1- z)[ ]-1
for Dcov (MV ),
(2) (1- x)0 for Dcov (k × P2),
(3) (1- x)-1/ 2 = (1-a)(1- z)[ ]-1/ 2
for DLF (M0).
g a+ f a-
x x x O( )
x
x O( ) O( , )
x
x x x
x
x x x
1
VJ1
AJ JA+
0, JA
+
1, JA
^
1
JA+
0, JA
+
1
(P2 - 2k)m
Dcon
(P2 - 2k)m
Dcov
(P2 - 2k)m
DLF
JA+
0
Z .M .
JA^
1
Z .M .
JA+
0
Z .M .
JA^
1
Z .M .
Existence(O(source element)) or absence (X) of the zero-mode
contribution to (g, a+, a-, f) depending on and JV-Am
hGV
m =g m - (P2 - 2k)m / D
Dcon = M2 + m2 + m ~ (1/ x)0as x® 0, D cov=
2k ×P2 + M2 (m2 + m)- ie
M2~ (1/ x)1 as x® 0
DLF = M0 + m2 + m ~ (1/ x)1/2as x® 0
T1 T2 T3
x x x
x
O( )
O( )
x
x x
x
x x
10
J J5+
0, J5
+
1
(P2 - 2k)m
Dcon
(P2 - 2k)m
Dcov
(P2 - 2k)m
DLF
J5+
0
Z.M .
Tensor form factors Ti (i=1,2,3) for the rare P Vl+l- decays
J5+
0, J5
+
1
J0m
hº V(P2,eh
*) | qis mnqnq | P(P1) = iemnaben
*PaqbT1
J5m
hº V | qis mnqng5q | P(P1) = [e
*m (P ×q)- (e* × q)Pm ]T2 + (e* × q) qm -
q2
(P × q)Pm
é
ëê
ù
ûúT3
J5+
0
Z.M .
T.Altomari & L. Wolfenstein, PRD37, 681(88)
Summary
• The common belief of equivalence between manifestly covariant and LF Hamiltonian formulations is not always realized unless nothing is missing.
• Arc, moving pole, zero-mode, etc. must be taken into
account.
• LFD contributes to fundamental understanding of anomaly intrinsic to QFT.
• Gauge symmetry appears to be intimately related to Lorentz symmetry in realization of anomaly free condition.
Outline for PM • Hadron Phenomenology at JLab
• JLab Kinematics in DVCS
(t < -|tmin|
• Original Formulation of DVCS with GPDs
(Valid only at a limited t region)
• Comparison between
Exact Tree-Level Result and the corresponding
Results from Original Formulation
• Hadronic Tensors in DVCS
Toward Generalization: Two Approaches
• Conclusion and Outlook
Hadron Physics at JLab
Wakamatsu (2010)
X.Ji (1997) Jaffe-Manohar (1990)
Chen et al. (2008)
Sq
Sg Lg
Lq
Sq
Sg Lg
Lq Sq
Sg Lg
Lq
Sq
Jg
Lq
Gauge-invariant extension (GIE)
Generalized Parton Distributions
1
21
( ) ( , , ) ( , , )4
q qE
q
tG t dx H x t E x tM
x xì üï ïí ýï ïî þ
+
-
- = +åò (for example),
• and the GPDs unify the description of inclusive and exclusive processes, connecting directly to the “normal” parton distributions:
1 1
2 2
1
1( , , 0) ( , , 0)
quarkq qqJ dxx H x t tL E xx x
ì üï ïí ýï ïî þ
+
-= DS+ = = + =ò
• GPDs provide access to fundamental quantities such as the quark orbital angular momentum that have not been accessible
e
e'
x+x x-x
H, E, H, E
g*
x = xB
2- xBg, p, h, r, w, K
H, E - unpolarized, H, E - polarized GPD
H, E, H, E
(1+ x)P (1- x)P
N N', D, L
The GPDs Define Nucleon Structure
JLab Kinematics t < -|tmin|
; ;
thesis
Coincidence Experiment
Ranges of Kinematic Variables
0 < Q2 <4E2M
2E + M
Q2
2ME< xBj =
Q2
2p × q
Table III in E12 - 06 - 114, Julie Roche et al.
Jlab 12 GeV Exclusive Kinematics
qg =14
Nucleon GPDs in DVCS Amplitude X.Ji,PRL78,610(1997): Eqs.(14) and (15)
Just above Eq.(14),
``To calculate the scattering amplitude, it is convenient to define
a special system of coordinates.”
Note here that = 0 .
Nucleon GPDs in DVCS Amplitude A.V.Radyushkin, PRD56, 5524 (1997): Eq.(7.1)
At the beginning of Section 2E (Nonforward distributions),
``Writing the momentum of the virtual photon as q=q’-ζp is equivalent to using the Sudakov decomposition in the
light-cone `plus’(p) and `minus’(q’) components in a situation when there is no transverse momentum .”
q = ¢ q -z p ,
z =Q2
2p × ¢ q ,
r = p - ¢ p
T mn ( p,q, ¢ q ) =1
2(p × ¢ q )ea
2
a
å [ -gmn +1
p × ¢ q (pm ¢ q n + pn ¢ q m )
æ
è ç
ö
ø ÷
´ u ( ¢ p ) ¢ / q u(p)TFa (z ) +
1
2Mu ( ¢ p )( ¢ / q / r - / r ¢ / q )u(p)TK
a (z )ì í î
ü ý þ
+ iemnabpa ¢ q b
p × ¢ q u ( ¢ p ) ¢ / q g5u(p)TG
a (z ) +¢ q × r
2Mu ( ¢ p )g5u(p)TP
a (z )ì í î
ü ý þ ]
Note here that ,i.e. only consistent at t=0,
neglecting nucleon mass.
JLab Kinematics t < 0
• In JLab, the final hadron and final photon
move off the z-axis.
• To see the effect of taking t
Bare Bone Structure
“Bare Bone” VCS Amplitude at Tree Level
H(hq,h ¢ q ,sk,s ¢ k ) = em* ( ¢ q ,h ¢ q ) en (q,hq ) TS
mn + TUmn( )
TSmn =
ka + qa
Su ( ¢ k ,s ¢ k )g
mgagn u(k,sk )
TUmn =
ka - ¢ q aU
u ( ¢ k ,s ¢ k )gn gag mu(k,sk )
S = (k + q)2
U = (k - ¢ q )2
= ¢ k
Neglecting masses,
Identity:
g mgagn = gmagn + gang m - gmn ga + iemanbgbg5
Hadron Helicity Amplitude:
Keeping no transverse momentum in DVCS, we agree on
equivalent to the expression given by X. Ji and A.V. Radyushkin.
Using Sudakov vectors
we find
Full Amp vs. Reduced Amp
S-channel:
U-channel:
Checking Amplitudes • Gauge invariance of each and every polarized amplitude
including the longitudinal polarization for the virtual photon.
• Klein-Nishina Formula in RCS.
• Angular Momentum Conservation.
ll = l ¢ l = +1
2, sk = s ¢ k = +
1
2, h ¢ q = +1 ;
Allowed !
Checking Amplitudes • Gauge invariance of each and every polarized amplitude
including the longitudinal polarization for the virtual photon.
• Klein-Nishina Formula in RCS.
• Angular Momentum Conservation.
ll = l ¢ l = +1
2, sk = s ¢ k = +
1
2, h ¢ q = -1 ;
Prohibited !
Comparison
Swap
C.Carlson and C.Ji, Phys.Rev.D67,116002 (2003);
B.Bakker and C.Ji, Phys.Rev.D83,091502(R) (2011).
For any orders in Q
Exact Reduced
Number of Independent Amplitudes in VCS
Nucleon Target
3 ´ 2 ´ 2 ´ 22
= 12
12 independent tensor structures
M.Perrottet, Lett. Nuovo Cim. 7, 915 (1973);
R.Tarrach, Nuovo Cim. 28A, 409 (1975);
D.Drechsel et al.,PRC57,941(1998);
A.V.Belitsky, D.Mueller and A.Kirchner, NPB629, 323(2002);
A.V.Belitsky and D.Mueller, PRD82, 074010(2010)
A.V.Belitsky and D. Mueller,arXiv:1005.5209v1[hep-ph]
Biproducts of P,V,A,T, but not S
D.Drechsel,G.Knoechlein,A.Yu.Korchin,A.Metz and S.Scherer,
PRC 57, 941 (1998)
…
Biproducts of S,V,A,T, but not P
Gordon Decomposition and Extension
Possible Reconciliation
Jm = g mF1 + is mn qn2M
F2
= g m (F1 + F2) +(p + ¢ p )
m
2MF2
=(p + ¢ p )
m
2M
4M 2F1 + q2F2
4M 2 - q2- ie mnabg 5gn pa ¢ p b
2(F1 + F2)
4M 2 - q2
=(p + ¢ p )
m
2MF1 + i
s mn qn2M
(F1 + F2)
= g m (F1 +q2
4M 2F2) - ie
mnabg 5gn pa ¢ p bF2
2M 2
= is mn qn2M
(4M 2
q2F1 + F2) + ie
mnabg 5gn pa ¢ p b2F1
q2
All are equivalent!
V T
V S
S A
S T
V A
T A
Conclusion and Outlook
• We find that the XJ and AVR amplitudes for DVCS in terms of GPDs for t < 0 are not satisfactory.
• The determination of all independent structures is important for the discussion of GPDs and maintaining
EM gauge invariance is a crucial constraint.
• If all invariant structures are identified, the question whether one can measure GPDs in experiments where Q2 does not go to infinity may become more focused.
• LFD appears to be a promising tool for hadron phenomenology provided treacherous points are well taken care of.
