Post on 24-Aug-2020
Light-front zero-mode issue for the transition
form factors between pseudoscalar and vector
mesons
(in collaboration with C-R. Ji)
Lightcone2011, SMU, Dallas, Texas, May 23~27
Based on:
NPA 856, 95 (2011) & PLB 696, 518 (2011)
Ho-Meoyng Choi
Kyungpook National University, Daegu, Korea
Outline
1. Motivation
2. LF covariant form factors for P V l νl and
P Vl+l - transitions in exactly solvable model- provide the method that pin-down the existence/absence of
the LF zero-mode contributions
3. Application to the LF quark model(LFQM)
4. Summary
1. Motivation
◦ Exclusive P V(P) l νl and P V(P) l+l - decays of mesons:
- useful testing ground of SM & beyond SM
- theoretically difficult to understand due to the nonperturbative hadronic
form factors
22
2||||)( factorFormVknown
dq
dCKM
Theoretical uncertainty!
1. Motivation
◦ Exclusive P V(P) l νl and P V(P) l+l - decays of mesons:
- useful testing ground of SM & beyond SM
- theoretically difficult to understand due to the nonperturbative hadronic
form factors
22
2||||)( factorFormVknown
dq
dCKM
▫ In our previous works , we analyzed semileptonic and rare P P decays [PLB 460,461(99), PRD 80, 054016(09) by Choi & Ji; PRD 65, 074032(02) by Choi, Ji, Kisslinger ;
PRD81, 054003 (10), JPG 37, 085005(10) by Choi]
- obtained LF covariant form factors (f+ , f-, and fT) in the q+=0 frame
[PRD 80, 054016(09) , JPG 37, 085005(10)]
Theoretical uncertainty!
1. Motivation
◦ Exclusive P V(P) l νl and P V(P) l+l - decays of mesons:
- useful testing ground of SM & beyond SM
- theoretically difficult to understand due to the nonperturbative hadronic
form factors
22
2||||)( factorFormVknown
dq
dCKM
▫ In our previous works , we analyzed semileptonic and rare P P decays [PLB 460,461(99), PRD 80, 054016(09) by Choi & Ji; PRD 65, 074032(02) by Choi, Ji, Kisslinger ;
PRD81, 054003 (10), JPG 37, 085005(10) by Choi]
- obtained LF covariant form factors (f+ , f-, and fT) in the q+=0 frame
[PRD 80, 054016(09) , JPG 37, 085005(10)]
Theoretical uncertainty!
▫ In this work, we extend our previous studies to semileptonic and rare P
V decays
])()()[()(
)()(||)1;(
22**2
*2
12
qqaPqaPqfJ
qPqigPPqqhPVJ
hA
hV
Semileptonic P →V transition
P1P2
qq )1( 5
])()()[()(
)()(||)1;(
22**2
*2
12
qqaPqaPqfJ
qPqigPPqqhPVJ
hA
hV
Rare P →V transitions
)()()(||
)(||
2
2
**
55
*2
10
qTPqqPPqqiqVJ
qPqiTPqqiqVJ
h
h
)()(
)( 2
3
2* qTP
qP
qqq
P1P2
,)1( 5 qq qq )1( 5
Nj=pj2-m2
j+iε(j=1,2) Nq=k2-m2+iεNΛj=pj
2-Λ2j+iε(j=1,2)
(Λi=momentum cutoffs)
2. Manifestly Covariant BS model (Semileptonic PV decay)
m
m1 m2
p1=P1-k p2=P2-k
kP1P2
q
l
21 21
4
4 )(
)2( NNNNN
SkdiNJ
q
hAV
hAV
21
1)1(
1)1( 55
NN
Bakker,Choi,Ji(03)
Nj=pj2-m2
j+iε(j=1,2) Nq=k2-m2+iεNΛj=pj
2-Λ2j+iε(j=1,2)
(Λi=momentum cutoffs)
2. Manifestly Covariant BS model (Semileptonic PV decay)
m
m1 m2
p1=P1-k p2=P2-k
kP1P2
q
l
21 21
4
4 )(
)2( NNNNN
SkdiNJ
q
hAV
hAV
21
1)1(
1)1( 55
NN
Bakker,Choi,Ji(03)
])())(1()[()( *
511522 mkmpmpTrS hAV
D
kP
)2( 2
(Model dependent) D factors used in this work
m
m1 m2
p1=P1-k p2=P2-k
kP1P2
q
l
D
kP
)2( 2
mmMD
M
immMPkD
mmMD
LF
con
2
'
0
2
222cov
22
)3(
)(2)2(
)1(
x
mk
x
mkM
1
22'2
2
2'2'
0
Zero-mode issue in Light-Front Calculation
m
m1 m2
p1=P1-k p2=P2-k
kP1P2
q
Valence (0 < k+
< P2+):
(∆ < x < 1 )
Nonvalence (P2+
< k+
< P1+) :
(0 < x < ∆)
P1+
P2+
q+ q
+
P1+
P2+
= +on-mass-shell
on-mass-shellk2=m2
p12=m1
2
and Λ12
x=p1+/P1
+, ∆ = q+ / P1+
=γ+
m1
m
m2
P1+
P2+
q+
(i) Plus current(μ= +)
1-xx x
LF Valence contribution(∆ < x < 1 ): k-=k on-
(ii) Perpendicular current (μ=⊥) : a-
=γ⊥
+ +γ⊥
γ⊥
γ+γ⊥
γ+
γ+
: ( g, a+, f) and (T1, T2, T3)
p+m=(pon+m) + (1/2) γ+(p- -p-on)
(propagating) (instantaneous)
No instantaneous
for the J+
since (γ+)2=0
Nonvalence (0 < x < ∆ ) vs zero-mode contribution
q+=∆P1+
P1+
P2+
p12=m1
2
and Λ12
LF Zero-mode : Nonvanishing nonvalence contribution as q+
(∆ and x)0
2121
111111
2121NNNNNNNNNN qq
2
1
2
2
2
1
2
221
1
0ln
)(2lim
m
m
mm
xi
NNN
pdk
qnv
E.g.) p12=m1
2 (i.e. N10)
)( 222
iii pmm
021
1
NNN
px
q
n
(if n > 0)
Zero-mode depending on the D factors
D
kP
)2( 2 q+=∆P1
+
P1+
P2+
p12=m1
2
and Λ12
21
1
0
)/(lim
NNN
Dpdk
qnv
0 for D = Dcov (n=1)
= DLF (n=1/2)
δ(x) for D=Dcon (n=0)
D ~ (1/x)n
2/1
2
'
0
1
2
222cov
0
22
)/1()3(
)/1()(2
)2(
)/1()1(
xmmMD
xM
immMPkD
xmmMD
LF
con
Zero-mode contribution to <JV-Aμ>h & <J0(5)
μ>h
In q+ 0 frame:
D
pJ
D
pJ
MZ
h
MZ
hA
1..
05
1..
0
D
pJ
D
pJ
MZ
h
MZ
hA
1..
05
1..
0
D
pbapJ
MZ
hA
11
..
1
D
pJ
D
pJ
MZ
h
MZ
hA
1..
05
1..
0
D
pbapJ
MZ
hA
11
..
1
Only a- receives Z.M.
if D=Dcov or DLF is used!
Effective inclusion of zero-mode in valence region: ω-dependent LF covariant approach
Carbonell, Desplanques,Karmanov, Mathiot(98)
(i) On-shell amp. is independent of the orientation of LF plane ω∙x=0
(ii) Off-shell amp. depends on the orientation of LF plane ω∙x=0
→ <J⊥> acquires a spurious ω dependence!
Zero-mode associated with p1- ↔ spurious ω dependence in covariant LF dynamics
Jaus(99)
Effective inclusion of zero-mode in valence region: ω-dependent LF covariant approach
)1(
1
)1(
2
)1(
11 CP
AqAPp
)0,2,0(),,(
Decompose p1 in terms of (P =P1 + P2, q, ω) (Jaus 99)
2
22
1
22
1
2
0
2
12 ][)21()(q
qkPqqMxmmMMxZ
qNZC 2
)1(
1
Carbonell, Desplanques,Karmanov, Mathiot(98)
(i) On-shell amp. is independent of the orientation of LF plane ω∙x=0
(ii) Off-shell amp. depends on the orientation of LF plane ω∙x=0
→ <J⊥> acquires a spurious ω dependence!
Zero-mode associated with p1- ↔ spurious ω dependence in covariant LF dynamics
Jaus(99)
Effective inclusion of zero-mode in valence region: ω-dependent LF covariant approach
)1(
1
)1(
2
)1(
11 CP
AqAPp
)0,2,0(),,(
Decompose p1 in terms of (P =P1 + P2, q, ω) (Jaus 99)
2
22
1
22
1
2
0
2
12 ][)21()(q
qkPqqMxmmMMxZ
depends on p1-
qNZC 2
)1(
1
ω free terms
Carbonell, Desplanques,Karmanov, Mathiot(98)
(i) On-shell amp. is independent of the orientation of LF plane ω∙x=0
(ii) Off-shell amp. depends on the orientation of LF plane ω∙x=0
→ <J⊥> acquires a spurious ω dependence!
Zero-mode associated with p1- ↔ spurious ω dependence in covariant LF dynamics
Jaus(99)
Effective inclusion of zero-mode in valence region: ω-dependent LF covariant approach
)1(
1
)1(
2
)1(
11 CP
AqAPp
)0,2,0(),,(
Decompose p1 in terms of (P =P1 + P2, q, ω) (Jaus 99)
2
22
1
22
1
2
0
2
12 ][)21()(q
qkPqqMxmmMMxZ
depends on p1-
qNZC 2
)1(
1
Removing ω-dependence[C1 (1)=0 or Nq→ Z2] ↔ Effectively include Z.M.
in the valence region! (i.e. p1- → -Z2)
ω free terms
Carbonell, Desplanques,Karmanov, Mathiot(98)
(i) On-shell amp. is independent of the orientation of LF plane ω∙x=0
(ii) Off-shell amp. depends on the orientation of LF plane ω∙x=0
→ <J⊥> acquires a spurious ω dependence!
Zero-mode associated with p1- ↔ spurious ω dependence in covariant LF dynamics
Jaus(99)
What is p1- → -Z2 prescription?
2
22
1
22
1
2
0
2
12 ][)21()(q
qkPqqMxmmMMxZ
]][[
1),(
2)('2
)2(1
2)('
0
2
)2(1
2
)('
)2(1
)2(1
MMMMx
kx
1
0
221
2
3)',(])[,(
)1(16
1kxZkxkd
x
dx
MZ q NNNNN
pkdi
. 21
1
4
4
21)2(
222112122
,q
PqxZ
q
qkxqppZp etc.
or simply,
Difference between Jaus’s and Our Methods
Jaus:
222
1121
22,
q
PqxZ
q
qkx
D
q
D
pp
D
Z
D
p
regardless of D factors (Dcon, Dcov, DLF)
Our:
222
1121
22,
q
PqxZ
q
qkx
D
q
D
pp
D
Z
D
p
conconconcon
0,0)cov(
11
)cov(
1
LFLF D
pp
D
p
Existence(O(source element)) or absence (X) of the zero-mode contribution
to (g, a+, a-, f) depending on and h
AVJ DkPV /)2( 2
Our method: only a- receives Z.M.
when D=Dcov(LF) is used!
Jaus method: f & a- receive Z.M.
when D=Dcov(LF) is used!
Our method: No Z.M.!
when D=Dcov(LF) is used!
Jaus method: T2 & T3 receive Z.M.
when D=Dcov(LF) is used!
)]()()()([2
1)( 2222
2
2
1
2
2
2
0 qaqqaMMqfM
qA
)()2/1()( 2..2
2
2..
0 qaqMqA MZMZ
1
0
22211
2
2
3
2..
0
])[()1(16
)(
Zmmkdx
dx
M
N
qA MZ
hypQQ
QQQQQQ
VVV
VkmkmH
0
2222
coul
QQs Vmm
SS
rrbra 22
3
2
3
4)(
),,(),(),,( iiiiiiiiQQkxRkxkx
Key idea of our LFQM: Using the variational principle to the QCD-motivated
effective Hamiltonian, we fix the model parameters!
3. Light-Front Quark Model PRD59, 074015(99); PLB460, 461(99) by Choi and Ji
Key idea of our LFQM: Using the variational principle to the QCD-motivated
effective Hamiltonian, we fix the model parameters!
3. Light-Front Quark Model PRD59, 074015(99); PLB460, 461(99) by Choi and Ji
hypQQ
QQQQQQ
VVV
VkmkmH
0
2222
coul
QQs Vmm
SS
rrbra 22
3
2
3
4)(
),,(),(),,( iiiiiiiiQQkxRkxkx
Variational Principle
0|)(| 00
VH
)2/exp(~ 22 k
Key idea of our LFQM: Using the variational principle to the QCD-motivated
effective Hamiltonian, we fix the model parameters!
3. Light-Front Quark Model PRD59, 074015(99); PLB460, 461(99) by Choi and Ji
hypQQ
QQQQQQ
VVV
VkmkmH
0
2222
coul
QQs Vmm
SS
rrbra 22
3
2
3
4)(
),,(),(),,( iiiiiiiiQQkxRkxkx
Variational Principle
0|)(| 00
VH
)2/exp(~ 22 k
1
1,
9
32
3
8
2
3
2
2
3
,
2
2
2
1
2/222
IImm
SSbaI
mKem
IM
s
QQi
im
iQQi
PRD80,054016(09)
Model mq ms mc mb qc sc cc qb sb cb bb
Linear 0.22 0.45 1.8 5.2 0.4679 0.5016 0.6509 0.5266 0.5712 0.8068 1.1452
HO 0.25 0.48 1.8 5.2 0.4216 0.4686 0.6998 0.4960 0.5740 1.0350 1.8025
Optimized model parameters(in unit of GeV) and meson mass spectra
Experiment
Linear potential
Harmonic oscillator(HO)potential
Input masses
BR(Our) BR(Our) BR(Exp)
D0 ρeν HO
Linear
0.0269 |Vcd|2
0.0282 |Vcd|2
(1.42 ± 0.14) x 10-3
(1.49 ± 0.14) x 10-3
(1.9 ± 0.4) x 10-3
D0 K*eν HO
Linear
0.0246 |Vcs|2
0.0247 |Vcs|2
(2.36 ± 0.50) %
(2.37 ± 0.50) %
(2.17 ± 0.16) %
Ds φeν HO
Linear
0.0249 |Vcs|2
0.0257 |Vcs|2
(2.39 ± 0.51) %
(2.47 ± 0.51) %
(2.49 ± 0.14) %
B0 ρlν HO
Linear
21.38 |Vub|2
26.98 |Vub|2
(2.44 ± 0.49) x 10-4
(3.09 ± 0.62) x 10-4
(2.77 ± 0.34) x 10-4
B0 D*lν HO
Linear
34.31 |Vbc|2
35.29 |Vbc|2
(5.14 ± 0.29) %
(5.29 ± 0.30) %
(5.05± 0.12) %
Used CKM: |Vcd| = 0.230 ± 0.011 |Vcs| = 0.98 ± 0.11
|Vub| = (3.38 ± 0.36) x 10-3 |Vcb| = (38.7 ±1.1) x 10-3
2*
22/123
*3
2*0 ||)()()1(
48)( cbDD
F VwFwPwmG
DBdw
d
)(0011.00387.0
|| exp
exclusive
Vcb
*
22
*
2
*
*
2 DB
DB
DB
DB
mm
qmm
mm
PPw
World average(HFAG2011):
BR= (2.77 ±0.18±0.16) x 10-4
|Vub|=(3.05 – 3.73) x 10-3
(extracted from B π)
LFQM:
BR= (1.95 – 2.44) x 10-4
for |Vub|=(3.02 – 3.38) x 10-3
4. Summary
1. Study exclusive semileptonic and rare P → V transitions:
- obtain LF covariant form factors (g, a+, a-, f, T1, T2, T3) in
q+=0 frame
(in comparison with manifestly covariant calculation)
3. Hadron phenomenology:
- Extend the present work to more realistic LFQM
- Comparision with experiment in B-factory and LHCb etc.
2. Zero-mode issue:
- For D=Dcov or DLF, only a- form factor receives zero-mode!
- Effective inclusion of zero-mode in the valence region