Light-front zero-mode issue for the transition form ... · (in collaboration with C-R. Ji)...

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




form factors

22


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)]


1. Motivation




form factors

22


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)]


▫ 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)


)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






Jaus(99)


)1(

1

)1(

2

)1(

11 CP

AqAPp

)0,2,0(),,(


2

22

1

22

1

2

0

2

12 ][)21()(q

qkPqqMxmmMMxZ

depends on p1-

qNZC 2

)1(

1

ω free terms






Jaus(99)


)1(

1

)1(

2

)1(

11 CP

AqAPp

)0,2,0(),,(


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






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.


Our method: No Z.M.!


Jaus method: T2 & T3 receive Z.M.


)]()()()([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

QQ

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




hypQQ

QQQQQQ

VVV

VkmkmH

0

2222

coul

QQ

QQs Vmm

SS

rrbra 22

3

2

3

4)(


Variational Principle

0|)(| 00

VH

)2/exp(~ 22 k




hypQQ

QQQQQQ

VVV

VkmkmH

0

2222

coul

QQ

QQs Vmm

SS

rrbra 22

3

2

3

4)(


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

QQ

QQ

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

Light-front zero-mode issue for the transition form ... · (in collaboration with C-R. Ji)...

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