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Covariant Light Front Approach for s-wave and p-wave Mesons ICHEP 2004, Beijing

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. Introduction.

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Covariant Light Front Approach for s-wave and p-wave Mesons ICHEP 2004, Beijing

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  1. Covariant Light Front Approach for s-wave and p-wave MesonsICHEP 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): P11/2 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 (Jaus99; 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 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). M0:kinetic mass h: vertex functions x1,2: momentum frac. of quark, antiquark.

  5. Width of W-Fn, ~LQCD P V S A A • 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. P V S A A < >

  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)

  7. Form Factors: numerical results (B p…)

  8. Form Factors: numerical results (B K …) Start to show deviation, due to ms→– ms

  9. Form Factors: numerical results (BD …) • 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.

  12. 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

  13. Comparison with experiment • From decays, we obtain (following Cheng 03) • Our predictions, are in agreement with data (recent BaBar results also seem to support this).

  14. Comparison with experiment • Compare with B→ D**p decays: • Our predictions are in agreement with data

  15. 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 (x) for P to P,V transitions. - Two IW functions (t1/2, t3/2) for P to S, A transitions.

  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 significant in the HQ limit. • A recent Lattice QCD calculation (Becirevic et al. 04) gives t1/2(1)=0.38±0.05, t3/2(1)=0.58±0.08 ISGW

  17. Conclusion: • A first RQM treatment of decay constants and FFs for p-wave mesons. • For heavy to heavy transition - BD** transition form factors agree with ISGW2 - predictions on decay constants (ISGW2 didn’t provide decay constants) and BD**p rates are in good agreement with data. • For the heavy to light transition (such as Ba0,1 transition) - the covariant LF model gives quite different results from NRQM - hard to understand the Ba1pdata. • The heavy quark limit (and SU(N)F limit) of decay constants and form factors are examined - the universal IW functions x(w), t1/2(w) and t3/2(w) are obtained - t1/2(w) and t3/2(w) 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*g, K1,2gdecays (obtaining T1K*g(0)=0.24 in good agreement with data) and pentaquark decays.

  18. 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 PDG...) • 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): P01/2 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 observed in three-body decays begin to emerge.

  20. Feynman rules for vertices

  21. 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): Am Am, Vm-Vm; P(1S0):+, S(3P0):+,3P1:+ ,1P1:- [C=(-)L+S] Applies to charged states through SU(N) symmetry. (b) In HQ limit 2S+1LJ = 3P1, 1P1

  22. 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).

  23. We obtain: • It is easy to see that in the SU(N) limit (m’1=m2). M0:kinetic mass h: vertex functions x1,2: momentum frac. of quark, antiquark.

  24. Note that

  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 jLLz.

  26. Form Factors • There are many form factors in P  P,V,S,A,T transitions. • For example, in P  P transition:

  27. Heavy Quark Limit: IW functions

  28. Heavy Quark Limit: numerical results • There are two sum-rules: Bjorken and Uraltsev. For example,

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