1

Covariant Light Front Approach for s-wave and p-wave Mesons

ICHEP 2004, Beijing

Chun-Khiang Chua Academia Sinica (Taipei)

Based on PRD69, 074025 (2004)In collaboration with Hai-Yang Cheng and Chien-Wen Hwang

2

Introduction The interest in even-parity charmed mesons has been revived by recent discoveries: Two narrow resonances: Ds0* (2317) (BABAR 03): 2S+1LJ = 3P0, Ds1(2460) (CLEO 03): P1

1/2

Two broad resonances: D0* (Belle 03; FOCUS 03) and D1(2427) (Belle 03). The only systematic analysis for s- to p-wave transitions is the Isgur-Scora-Grinstein-Wise

(ISGW) QM (ISGW 89), which is non-relativistic (NR).

However, relativistic effect could be important in B and D decays.

Light-front QM, which is the only relativistic QM, has been employed to obtain decay constant and weak form factors (Jaus 90,91,96; Ji,Chung,Cotanch 92, Cheng,Cheung,Hwang 97)

- so far it has been applied only to s- to s-wave meson transitions - there exist some ambiguities in extracting the physical quantities (non-covariant).

Covariant LFQM have been constructed - in (Cheng,Cheung,Hwang,Zhang 98) within the framework of HQET - in (Jaus 99; Bakker Choi Ji 02) without using the HQ limit - both apply to s- to s-wave meson transitions only

We wish to extend the covariant LFQM in (Jaus 99) to even-parity, p-wave mesons and study the corresponding Isgur-Wise functions.

3

Decay constants Mesons can be annihilated by vector, axial vector currents:

Two classes of constraints for decay constants: (a) In SU(N) limit (b) In HQ limit

,)'(||0

,)'(||0'

'

PfPSV

PifPPA

S

P

.')','(||0

,')','(||0')1(3

'

)1(3

A

V

fMPAA

fMPVV

.0 ,0 1 AS ff

,03

1

3

2 ,

3

2

3

1 , 312/3312/1

AAASAAAPV fffffffff

)2 ,1( ),1 ,0( ),1 ,0( 2/32/32/12/12/12/1)(P

lightjJ

2S+1LJ = 3P1, 1P1

4

In the one-loop approximation, we obtain:

f(S,3A) ~ f(P,V) with m2→ – m2

It is easily seen that in the SU(N) limit (m’1=m2).

, )(

16

,)( )(

16

2''

2'1

2'0

2'21

''2

'3

2''

,

2'12'

2'1

'1

2'012'

02'

21

'

,'2'3,

1

1

1

3

3

3

pw

mm

MMxx

hpddx

M

Nf

pw

mmpmmmMx

MMxx

hpddx

M

Nf

A

ACA

AV

AVCAV

0 ,0 1 AS ff

),()(

4

16 122'12'

02'

21

','2

3, xmxmMMxx

hpddx

Nf SPC

SP

M0:kinetic mass h: vertex functions x1,2: momentum frac.of quark, antiquark.

5

fS(su)=21 MeV is close to finite energy sum-rule result. fP(cs)> fP(cu), fS(cs)<fS(cu) due to the different relative signs. Small fS(cs) is favorable from decay (Belle 03). Consistent with SU(N) and HQ expectations.

*0sDDB

Width of W-Fn, ~QCD

PVSAA

PVSAA

<

>

6

Form Factors Form factors are calculated in one-loop approximation:

For technical reason, form factors are obtained in spacelike region (q2<0). We fit them with

for B(D)M transition and then analytically continue them to timelike region (q2>0) (Jaus 96).

FF(P→S,3A) ~ FF(P→P,V) with m1”→ – m1” (mass of final state quark)

)/()/(1

)0()(

4)(

42)(

22

DBDB mqbmqa

FqF

7

Form Factors: numerical results (B …)

)/()/(1

)0()(

44222

BB mqbmqa

FqF

,1,0 BBa FF

8

Form Factors: numerical results (B K …)

)/()/(1

)0()(

44222

BB mqbmqa

FqF

Start to show deviation, due to ms→ – ms

9

Form Factors: numerical results (BD …)

It is non-trivial to have correct signs, relations that are consistent with HQS.

10

Form Factors: comparison (s to s-wave)

Basically our results agree with others. Closer to Melikhov-Stech (MS) model predictions.

11

Form Factors: comparison (B to D**) Non-relativistic treatment should be OK in the b to c transition.

At low q2 , most FFs agree with ISGW2 calculation within 40%. q2-dependence is different in general.

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Form Factors: comparison (heavy to light) Using LCSR, Chernyak (01) obtained while we have 0.26. For B→ a1 FF:

is unlikely, since we expect However, recently BaBar gives

Our and QSR results are several times smaller

46.0)0()1450(1,0

0 BaF

61

0 10)1.42.46.42()( aBBr

)1.0(1,01,01 OAV BBa 11

0 BaV

)(

)(

01

011

1

BB

BaBa

AfFf

VfFf

13

Comparison with experiment From decays, we obta

in (following Cheng 03)

Our predictions,

are in agreement with data (recent BaBar results also seem to support this).

c

s

c

b

q

B

D

sJD

c

s

c

b

q

B

D

sJD

)( 0(*)ssJsJ DDDDB

MeV 190110

MeV 7347

)2460(

)2317(

1

*0

s

s

D

D

f

f

MeV 117 MeV, 71 )2460()2317( 1*0

ss

DDff

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Comparison with experiment Compare with B→ D** decays:

Our predictions are in agreement with data

CLFQM ISGW2 SCET Neubert Expt.

0.91 0.67 1 0.35 0.80±0.07 ±0.16 (BaBar)

0.77 ± 0.15 (Belle)

1.8 ±0.8 (CLEO)))2420('(

))2460((

1

0*2

DBr

DBr

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Heavy Quark Limit

We use both top-down and bottom up approach and we obtain consistent results.

- Top-down: we re-derive (Cheng,Cheung,Hwang,Zhang 98) Feynman rules. - Bottom-up: apply the “heavy quark mass infinity” limit to our analytic results.

Decay constant HQ relations checked.

In HQ limit FFs are related to some universal IW functions. - One IW function ( for P to P,V transitions. - Two IW functions (1/2, 3/2) for P to S, A transitions.

,03

1

3

2 ,

3

2

3

1 , 312/3312/1

AAASAAAPV fffffffff

16

Heavy Quark Limit: Our results are close to ISGW. -Relativistic effect is irrelevant at zero recoil (w=v·v’=1). -Model dependence between LF and ISGW are not sig

nificant in the HQ limit.

A recent Lattice QCD calculation (Becirevic et al. 04) gives 1/2(1)=0.38±0.05, 3/2(1)=0.58±0.08

ISGW

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Conclusion: A first RQM treatment of decay constants and FFs for p-wave mesons.

For heavy to heavy transition - BD** transition form factors agree with ISGW2 - predictions on decay constants (ISGW2 didn’t provide decay constants) and

BD**rates are in good agreement with data.

For the heavy to light transition (such as Ba0,1 transition) - the covariant LF model gives quite different results from NRQM - hard to understand the Ba1 data.

The heavy quark limit (and SU(N)F limit) of decay constants and form factors are examined

- the universal IW functions and are obtained - and close to the ISGW2 and a recent lattice results - the Bjorken and Uraltsev sum rules for the IW functions are fairly satisfied.

We apply the formalism to B→K*, K1,2 decays (obtaining T1K*(0)=0.24 in goo

d agreement with data) and pentaquark decays.

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Back Up Slides

19

Introduction

There are so many mesons beside s-wave mesons, pseudoscalar (P) and vector (V), (you may take a look up at PD

G...)

The interest in even-parity charmed mesons is revived by recent discoveries:

Two narrow resonances: D*s0(2317) (BABAR 03): 3P0, Ds1(2460) (CLEO 03): P0

1/2

and two broad resonances: D0*(2308) and D1(2427) (Belle 03). Three body decays of B mesons have been recently studied at the B factories: BaBar and Belle. The p-wave resonances observ

ed in three-body decays begin to emerge.

),2( ),,,1( ),,0( ),,1( ),,0( 23

11

13

03

13

0112 PTPPAPSSVSLJP J

SP

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Feynman rules for vertices

M(2S+1LJ) iM (incoming meson)

Pseudoscalar (1S0) HP vector (3S1) iHv((p1p2)WV)

scalar (3P0) -i HS

axial (3P1) -iH3A ((p1p2)W3A) axial (1P1) -iH1A (p1p2)W1A

tensor (3P2) iHv((p1p2)WV)

(p1p2)

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Decay constants Mesons can be annihilated by vector, axial vector currents:

Two classes of constraints for decay constants: (a) In SU(N) limit (quark masses are identical) Under Charge conjugation (for neutral states): A A, VV; P(1S0):+, S(3P0):+,3P1:+ ,1P1:- [C=()L+S] Applies to charged states through SU(N) symmetry. (b) In HQ limit

,)'(||0

,)'(||0'

'

PfPSV

PifPPA

S

P

.')','(||0

,')','(||0')1(3

'

)1(3

A

V

fMPAA

fMPVV

.0 ,0 1 AS ff

,03

1

3

2 ,

3

2

3

1 , 312/3312/1

AAASAAAPV fffffffff

)2 ,1( ),1 ,0( ),1 ,0( 2/32/32/12/12/12/1)(P

lightjJ

2S+1LJ = 3P1, 1P1

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Decay Constants Step 1: Write down the Feynman amplitude (just like in the usual covariant calculation).

Step 2: Pass to LF formalism: perform the contour integration by closing the upper complex p’1-=(p0-p3)1. (Chang, Ma 69)

Step 3: Use the widely-used LF vertex functions. Extension. Step 4: Separation of spurious contribution by the inclusion

of zero mode contribution (Jaus 99, Chang et al. 73).

dppddppd 24

2

1

)1 ,0 ,0 ,1(~ 0'1 p

22

22at residue mp

23

We obtain:

It is easy to see that in the SU(N) limit (m’1=m2).

, )(

16

,)( )(

16

,)( )(

16

2''

2'1

2'0

2'21

''2

'3

2''

2'12'

2'1

'1

2'012'

02'

21

''2

'3

2''

2'12'

2'1

'1

2'012'

02'

21

''2

'3

1

1

1

3

3

3

pw

mm

MMxx

hpddx

M

Nf

pw

mmpmmmMx

MMxx

hpddx

M

Nf

pw

mmpmmmMx

MMxx

hpddx

M

Nf

A

A

ACA

ACA

V

VCV

0 ,0 1 AS ff

),()(

4

16

),()(

4

16

122'12'

02'

21

''2

3

122'12'

02'

21

''2

3

xmxmMMxx

hpddx

Nf

xmxmMMxx

hpddx

Nf

SCS

PCP

M0:kinetic mass h: vertex functions x1,2: momentum frac.of quark, antiquark.

24

Note that ,|3

1|

3

2| ,|

3

2|

3

1| 1

31

12/311

31

12/11 PPPPPP

25

Meson can be boosted to have any momentum without affecting internal momentum in its wave function.

Meson spin is taken care of by using the Melosh transformation (R).

Orbital angular momentum are incorporated through LLz.

zz

z

zz

z

LLLL

LLSS

zzz

C

JJLS

JJLS

zJS

Y

pxpxRJJLSSLLSN

pp

pqpqpp

ppPpdpdJLPM

,),( ),( ;|;1

),,,(

,),(),(| ),,,(

)~~~()2(2}}{{),,(|

212121

1111112121

2133

23

1312

26

Form Factors There are many

form factors in

P P,V,S,A,T

transitions.

For example, in

P P transition:

0212)(

)()()(||)(

Fq

qPqF

q

qPqP

fPPfPPPPVPP

V A

P P F0, F1

P V V A0,1,2

P S F0, F1

P A V0,1,2 A

P T h k,b+,b-

27

Heavy Quark Limit: IW functions

28

Heavy Quark Limit: numerical results

There are two sum-rules: Bjorken and Uraltsev. For example,

n

n

n

n

4

1|)1(||)1(| 2)(

2/12)(

2/3