- Email: [email protected]
B→Kη,Kη′ Decays
Nuclear Physics B (Proc. Suppl.) 186 (2009) 391–394 www.elsevierphysics.com
B → Kη, Kη Decays
∗
T. N. Phama a
Centre de Physique Th´eorique, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France
The nonet symmetry scheme seems to describe rather well the masses and η−η mixing angle of the ground state pseudo-scalar mesons and is thus expected to be also a good approximation for the matrix elements of the pseudoscalar density operators which play an important role in charmless two-body B decays with η or η in the final state. In this talk, I would like to report on a recent work on the B − → K − η, K − η decay using nonet symmetry for the matrix elements of pseudo-scalar density operators. We find that the branching ratio B → P P , with an η meson in the final state agrees well with data, while those with an η meson are underestimated by 20 − 30%. This could be considered as a more or less successful prediction for QCDF, considering the theoretical uncertainties involved. This could also indicate that an additional power-suppressed terms could bring the branching ratio close to experiment, as with the B → K ∗ π and B → K ∗ η decay for which the measured branching ratios are much bigger than the QCDF predictions.
1. Introduction The B → Kη, Kη decays have been analysed in recent papers [1–3] in QCD Factorization (QCDF), in perturbative QCD (pQCD) [4,5] and in soft collinear effective theory (SCET) [6] . In QCDF the B → Kπ branching ratio could be understood with a moderate contribution from annihilation terms [7,8]. Similarly, without fine tuning, the B → Kη branching ratio is predicted to be larger than that of B → Kπ in qualitative agreement with experiment, but is still underestimated by 20 − 30% compared with the measured value. Apart from the power-suppressed O(1/mb ) annihilation terms, the main theoretical uncertainties are the B → η transition form factor and the pseudo-scalar density matrix elements for η . Historically, there is an approximate SU (3) relation between the octet pseudo-scalar density [9] but there is no known explicit expression for the singlet pseudo-scalar density in the nonet symmetry scheme. In this talk I would like to discuss a recent work [10] in which we show that nonet symmetry for the quark mass term in η − η implies nonet symmetry for the pseudo-scalar density matrix elements. With the nonet symmetry ∗ Talk given at QCD08, the 14th International QCD Conference, 7-12th July 2008, Montpellier (France)
0920-5632/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2008.12.088
expression for the pseudo-scalar density matrix elements in η−η , we obtain in QCDF a B → Kη branching ratio, though sufficiently large, is still below the measured value by 20 − 30%, but a large B → η form factor or additional powersuppressed terms could bring the predicted value closer to experiment. 2. Nonet symmetry in the η − η system Since QCD interactions through the exchange of gluons are flavor-independent, the wave function for the pseudo-scalar meson nonet is also expected to be flavor-independent in the limit of vanishing current quark mass. The quark mass term is the leading term in the large Nc expansion while higher order terms in the chiral Lagrangian [11] is O(1/Nc ) and is thus suppressed in the large Nc limit. This justifies the nonet symmetry for the pseudo-scalar meson mass matrix, the off-diagonal quark mass term < η0 |HSB |η8 > then gives an η − η mixing angle θ = −18◦ in good agreement with the value determined from the two-photon decay width of η and η as mentioned in [10]. However, from the Gell-MannOkubo (GMO) mass formula, we would have m2η = m28 − tan θ2 (m2η − m28 ) ◦
(1)
which gives, for θ = −18 , mη = 483 MeV, about 60 MeV below experiment. This indicates that
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T.N. Pham / Nuclear Physics B (Proc. Suppl.) 186 (2009) 391–394
chiral logarithms and chiral Lagrangian higher order terms [11,12] which are second order in SU (3) breaking as the (sin θ)2 term in Eq.(1) could contribute to m8 and shift mη upward by a similar amount with the result that the η mass is very close to the GMO value and a large η − η mixing angle is obtained, rather than the small value of −10◦ given by the GMO formula for m8 . We now use the nonet symmetry mass term to derive the pseudo-scalar density matrix element for η, η which allows a calculation of B → Kη as shown in the following sections. The usual way to derive the pseudo-scalar density matrix elements between the vacuum and the pseudo-scalar meson nonet is to consider the matrix elements of the divergence of the axial vector currents between the vacuum and pseudo-scalar meson nonet. For π, K meson, we have [10]: ¯ fπ B0 (mu + md ) = (mu + md )0|¯ u iγ5 d|ud, u iγ5 s|u¯ s. (2) fK B0 (mu + ms ) = (mu + ms )0|¯ and for π 0 fu B0 (mu + md ) = (mu + md )0|¯ u iγ5 u|u¯ u. (3) with the π and K meson masses the usual expressions in terms of B0 and the current quark mass [11,13]. To first order in SU (3) breaking quark mass term, the decay constants fq (q = u, d, s) is (putting fqq¯ = fq ), fπ = fud¯ ≈ fu , fK = fu¯s = (1 + ) fud¯, (4) fs = (1 + 2 ) fu ≈ (1 + ) fK . Consider now the divergence of the I = 0 axial vector current: u γμ γ5 u+d¯γμ γ5 d), An μ = (¯
As μ = s¯ γμ γ5 s. (5)
we have: ¯ 5 d) + 2 ¯iγ5 u + md diγ ∂An = 2(mu u ∂As = 2ms s¯iγ5 s +
αs ˜ G G. 4π
αs ˜ G G. (6) 4π (7)
Taking the matrix elements of ∂An and ∂As between the vacuum and η0,8 , we obtain: 1 2 1 ˆ =fu √ m20 − fu √ (m20 + B0 (ms + 2m)) 3 3 3
√ 1 2 2 1 (m ˆ − ms ) + 2 √ m0|¯ ˆ u iγ5 u|u¯ u,(8) f u √ B0 3 6 3 1 2 1 ˆ = fs √ m20 − fs √ (m20 + B0 (ms + 2m)) 3 3 3 √ 2 2 2 1 (m ˆ − ms ) + 2 √ ms 0|¯ s iγ5 s|s¯ s.(9) f s √ B0 3 6 3 Similarly, for η8 :
√ 1 2 1 2 2 (m ˆ − ms ) fu √ B0 (2ms + m) ˆ = −fu √ B0 3 6 3 3 1 ˆ u iγ5 u|u¯ u, (10) + 2 √ m0|¯ 6 √ 2 2 1 2 2 (m ˆ − ms ) ˆ = −fs √ B0 −fs √ B0 (2ms + m) 3 6 3 3 2 s iγ5 s|s¯ s. (11) − 2 √ ms 0|¯ 6 In deriving the above expressions, we have used the nonet symmetry mass formula, i.e m28 = B0 32 (2ms + m), ˆ m20 = m ¯ 20 + B0 32 (ms + 2m) ˆ and √ 2 2 m08 = B0 3 2(−ms +m) ˆ where m ˆ = (mu +md )/2 and m ¯ 0 is the anomaly contribution to m0 , the singlet η0 mass. The second term on the r.h.s of Eqs. (8-9) and the first term on the r.h.s of Eqs. (10-11) are the pole terms due to η − η mixing. In the limit mu = md =√0, the l.h.s and r.h.s of Eq. (10) become fu m28 / 6 in agreement with the divergence equation Eq. (6). Because of cancellation between the pole contribution and other quark mass terms, Eqs. (8-9) are reduced to the simplified form of Eq. (2) or Eq. (3). The pseudo-scalar density matrix elements in η0 are then given by: 0|¯ u iγ5 u|u¯ u = B0 fu , 0|¯ s iγ5 s|s¯ s = B0 fs . (12) The same expression for η8 is obtained similarly from Eqs. (10-11). Thus, 0|¯ u iγ5 d|π + , 0 + 0|¯ u iγ5 u|π and 0|¯ u iγ5 s|K , and the matrix element 0|¯ u iγ5 u|u¯ u and 0|¯ s iγ5 s|s¯ s in η0,8 are, apart from the decay constant fq , essentially the same, given by the parameter B0 and are consistent with nonet symmetry. Since experimentally, m208 = −(0.81±0.05) m2K is rather close to the nonet symmetry value of m208 −0.90 m2K [13], we expect nonet symmetry for the pseudo-scalar density matrix elements in
393
T.N. Pham / Nuclear Physics B (Proc. Suppl.) 186 (2009) 391–394
η − η would be valid to this accuracy. Since m28 gets about 15% increase from higher order terms L4 , L5 , L6 , L8 and chiral logarithms, Eqs. (10-11) s in η will be increased by a show that 0|¯ s iγ5 s|s¯ similar amount. A possible similar 15% increase for m20 would also increase 0|¯ s iγ5 s|s¯ s in η0 by a similar amount and would be additional source of enhancement for the B → Kη branching ratio.
which gives γ = 66◦ ( |Vtb | = 1) and α = 91.8◦ , in good agreement with the value found in the current UT-fit value of (88 ± 16)◦ . For other pa= 80 MeV, fu = fπ , rameters, we use ms (2 GeV)
3. The B − → K − (η, η ) and B − → π − (η, η ) decays
F Bη,Bη from the u quark content in η and η :
The B → M1 M2 decay amplitude in QCD Factorization(QCDF) is given by[7,8]: GF ∗ Vpb Vps × A(B → M1 M2 ) = √ 2 p=u,c 10 10 p , ai M1 M2 |Oi |BH + fB fM1 fM2 bi (13) i=1
F0Bπ (0) = 0.258,
F0BK (0) = 0.33,
(17)
F Bη (0)= 0.58 F0Bπ (0), F Bη (0)=0.40 F0Bπ (0). (18) and the pseudo-scalar density matrix elements: 0|¯ s iγ5 s|η =Cη B0 fs , 0|¯ s iγ5 s|η =Cη B0 fs .(19) obtained with the s quark content Cη = −0.57, Cη = 0.82 and an η − η mixing angle (−22 ± 3)◦ [13]
i
where the QCD coefficients api contain vertex corrections, penguin corrections, and hard spectator scattering contributions. The hadronic matrix elements M1 M2 |Oi |BH of the tree and penguin operators Oi are given by factorization model [2,14] and bi are annihilation terms. The values for api , p = u, c and bi computed from the expressions in [7,8] at the renormalization scale μ = mb and with mb = 4.2 GeV are given in the published work [10]. We estimate the CKM matrix element Vub and the CKM angle γ from the (db) unitarity triangle [15]: ∗ |Vcb Vcd cos2 α | | sin β . (14) |Vub | = 1 + ∗ | |Vud sin2 α ◦ With α = (99+13 −9 ) [16] and |Vcb | = (41.78±0.30± −3 0.08) × 10 [17], we find
|Vub | = 3.60 × 10−3 .
fs = fπ 1 + 2( ffKπ − 1) , the current theoretical values [19]:
(15)
which is quite close to the exclusive data [17] |Vub | = (3.33 − 3.51) × 10−3 . Similarly, we use the current determination 0 ¯0 |Vtd /Vts | = (0.208+0.008 −0.006 ) from the Bs − Bs mixing measurements [18] to obtain the angle γ: ∗ cos2 α |Vcb Vcd | | sin γ . (16) 1 + |Vtd | = |Vtb∗ | sin2 α
Table 1 The branching ratio B(B → P η, P η ) in QCDF Decay Modes QCDF BR (×10−6 ) Exp. [20] B − → π− π0 5.05 5.7 ± 0.4 ¯ 0 → K −π+ 18.25 19.04 ± 0.6 B B − → π− η 3.39 4.4 ± 0.4 B − → π− η 1.91 2.6+0.6 −0.5 − − B →K η 0.43 2.2 ± 0.3 B − → K − η 48.26 69.7+2.8 −2.7
As shown in Table 1, with a moderate annihilation term (ρA = 0.6) and the current theoretical value for the F Bπ and F BK form factors [19], QCDF predictions are in reasonable agreement with experiment, except for the B → Kη branching ratio which is underestimated by 20 − 30%. One could increase the F Bη form factor to produce better agreement with experiment for B − → π − η for which the prediction in Table 1 is below the Babar value of (4.0 ± 0.8 ± 0.4) × 10−6 [20] and to bring the predicted B → Kη branching ratio closer to experiment, but so far there seems to be no evidence for a large B → η form factor compared with the nonet symmetry
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T.N. Pham / Nuclear Physics B (Proc. Suppl.) 186 (2009) 391–394
value as seen from the new Babar [21] upper limit B(B + → η + ν)/B(B + → η + ν) < 0.57 which is consistent with nonet symmetry for the B → η, η form factors given in Eq. (18). 4. Conclusion We have shown that nonet symmetry for the pseudo-scalar meson mass term implies nonet symmetry for the pseudo-scalar density matrix elements. With this approximate relation, we obtained an improved estimate for the B → P η (P = K, π ) branching ratios. With a moderate annihilation contribution consistent with the measured B → Kπ branching ratio, we find that a major part of the B → Kη branching ratio could be obtained by QCDF. Without fine tuning or a large F B→η form factor, we find that the B → Kη branching ratio is underestimated by 20 − 30%. This could be considered as a more or less successful prediction for QCDF, considering the theoretical uncertainties involved. This could also indicate that an additional powersuppressed terms could bring the branching ratio close to experiment, as with the B → K ∗ π [22] and B → K ∗ η decay for which the measured branching ratios are much bigger than the QCDF prediction.
5. Acknowledgments I would like to thank S. Narison and the organizers of QCD 2008 for the warm hospitality extended to me at Montpellier. This work was supported in part by the EU contract No. MRTNCT-2006-035482, ”FLAVIAnet”. REFERENCES 1. M. Beneke and M. Neubert, Nucl. Phys. B 651 (2003) 225. 2. B. Dutta, C. S. Kim, S. Oh, and G. H. Zhu, Eur. Phys. J. C 37 (2004) 273. 3. J. M. Gerard and E. Kou, Phys. Rev. Lett. 97 (2006) 261804. 4. E. Kou and A. I. Sanda, Phys. Lett. B 525 (2002) 240.
5. Y. Y. Charng, T. Kurimoto, and H.-n. Li, Phys. Rev. D 74 (2006) 074024. 6. A. R. Williamson and J. Zupan, Phys. Rev. D 74 (2006) 014003. 7. M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B 606 (2001) 245. 8. M. Beneke and M. Neubert, Nucl. Phys. B 675 (2003) 333. 9. M. Gell-Mann, R. J. Oakes and B. Renner, Phys. Rev. 175 (1968) 2195. 10. T. N. Pham, Phys. Rev. D 77 (2008) 014024. 11. J. Gasser and H. Leutwyler, Nucl. Phys. B 250 (1985) 465. 12. J. M. Gerard and E. Kou, Phys. Lett. B 616 (2005) 85. 13. J. F. Donoghue, E. Golowich, and B. R. Holstein, Dynamics of the Standard Model, Cambridge University Press, Cambridge, (1992). 14. A. Ali, G. Kramer, and C. L¨ u, Phys. Rev. D 58 (1998) 094009. 15. T.N. Pham, invited talk at 2nd Workshop on the CKM Unitarity Triangle, Durham, April 2003, hep-ph/0306271. 16. W.-M. Yao et al. (Particle Data Group) J. Phys. G 33 (2006) 1. 17. E. Barberio, Talk at the XXIII International Symposium on Lepton and Photon Interactions at High Energy, Aug 13-18 2007, Daegu, Korea. 18. A. Abulencia et al, [CDF Collaboration], hepex/0606027, Phys. Rev. Lett. 97, 062003 (2006). 19. P. Ball and R. Zwicky, Phys. Rev. D 71, (2005) 014015. 20. Heavy Flavor Averaging Group, E. Barberio et al., arXiv:0704.3575 [hep-ex]. 21. B. Aubert et al. (BABAR Collaboration), arXiv:0805.2408 [hep-ex]. 22. T. N. Pham and G. H. Zhu, Phys. Rev. D 69 (2004) 114016.
B → Kη, Kη Decays
∗
T. N. Phama a
Centre de Physique Th´eorique, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France
The nonet symmetry scheme seems to describe rather well the masses and η−η mixing angle of the ground state pseudo-scalar mesons and is thus expected to be also a good approximation for the matrix elements of the pseudoscalar density operators which play an important role in charmless two-body B decays with η or η in the final state. In this talk, I would like to report on a recent work on the B − → K − η, K − η decay using nonet symmetry for the matrix elements of pseudo-scalar density operators. We find that the branching ratio B → P P , with an η meson in the final state agrees well with data, while those with an η meson are underestimated by 20 − 30%. This could be considered as a more or less successful prediction for QCDF, considering the theoretical uncertainties involved. This could also indicate that an additional power-suppressed terms could bring the branching ratio close to experiment, as with the B → K ∗ π and B → K ∗ η decay for which the measured branching ratios are much bigger than the QCDF predictions.
1. Introduction The B → Kη, Kη decays have been analysed in recent papers [1–3] in QCD Factorization (QCDF), in perturbative QCD (pQCD) [4,5] and in soft collinear effective theory (SCET) [6] . In QCDF the B → Kπ branching ratio could be understood with a moderate contribution from annihilation terms [7,8]. Similarly, without fine tuning, the B → Kη branching ratio is predicted to be larger than that of B → Kπ in qualitative agreement with experiment, but is still underestimated by 20 − 30% compared with the measured value. Apart from the power-suppressed O(1/mb ) annihilation terms, the main theoretical uncertainties are the B → η transition form factor and the pseudo-scalar density matrix elements for η . Historically, there is an approximate SU (3) relation between the octet pseudo-scalar density [9] but there is no known explicit expression for the singlet pseudo-scalar density in the nonet symmetry scheme. In this talk I would like to discuss a recent work [10] in which we show that nonet symmetry for the quark mass term in η − η implies nonet symmetry for the pseudo-scalar density matrix elements. With the nonet symmetry ∗ Talk given at QCD08, the 14th International QCD Conference, 7-12th July 2008, Montpellier (France)
0920-5632/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2008.12.088
expression for the pseudo-scalar density matrix elements in η−η , we obtain in QCDF a B → Kη branching ratio, though sufficiently large, is still below the measured value by 20 − 30%, but a large B → η form factor or additional powersuppressed terms could bring the predicted value closer to experiment. 2. Nonet symmetry in the η − η system Since QCD interactions through the exchange of gluons are flavor-independent, the wave function for the pseudo-scalar meson nonet is also expected to be flavor-independent in the limit of vanishing current quark mass. The quark mass term is the leading term in the large Nc expansion while higher order terms in the chiral Lagrangian [11] is O(1/Nc ) and is thus suppressed in the large Nc limit. This justifies the nonet symmetry for the pseudo-scalar meson mass matrix, the off-diagonal quark mass term < η0 |HSB |η8 > then gives an η − η mixing angle θ = −18◦ in good agreement with the value determined from the two-photon decay width of η and η as mentioned in [10]. However, from the Gell-MannOkubo (GMO) mass formula, we would have m2η = m28 − tan θ2 (m2η − m28 ) ◦
(1)
which gives, for θ = −18 , mη = 483 MeV, about 60 MeV below experiment. This indicates that
392
T.N. Pham / Nuclear Physics B (Proc. Suppl.) 186 (2009) 391–394
chiral logarithms and chiral Lagrangian higher order terms [11,12] which are second order in SU (3) breaking as the (sin θ)2 term in Eq.(1) could contribute to m8 and shift mη upward by a similar amount with the result that the η mass is very close to the GMO value and a large η − η mixing angle is obtained, rather than the small value of −10◦ given by the GMO formula for m8 . We now use the nonet symmetry mass term to derive the pseudo-scalar density matrix element for η, η which allows a calculation of B → Kη as shown in the following sections. The usual way to derive the pseudo-scalar density matrix elements between the vacuum and the pseudo-scalar meson nonet is to consider the matrix elements of the divergence of the axial vector currents between the vacuum and pseudo-scalar meson nonet. For π, K meson, we have [10]: ¯ fπ B0 (mu + md ) = (mu + md )0|¯ u iγ5 d|ud, u iγ5 s|u¯ s. (2) fK B0 (mu + ms ) = (mu + ms )0|¯ and for π 0 fu B0 (mu + md ) = (mu + md )0|¯ u iγ5 u|u¯ u. (3) with the π and K meson masses the usual expressions in terms of B0 and the current quark mass [11,13]. To first order in SU (3) breaking quark mass term, the decay constants fq (q = u, d, s) is (putting fqq¯ = fq ), fπ = fud¯ ≈ fu , fK = fu¯s = (1 + ) fud¯, (4) fs = (1 + 2 ) fu ≈ (1 + ) fK . Consider now the divergence of the I = 0 axial vector current: u γμ γ5 u+d¯γμ γ5 d), An μ = (¯
As μ = s¯ γμ γ5 s. (5)
we have: ¯ 5 d) + 2 ¯iγ5 u + md diγ ∂An = 2(mu u ∂As = 2ms s¯iγ5 s +
αs ˜ G G. 4π
αs ˜ G G. (6) 4π (7)
Taking the matrix elements of ∂An and ∂As between the vacuum and η0,8 , we obtain: 1 2 1 ˆ =fu √ m20 − fu √ (m20 + B0 (ms + 2m)) 3 3 3
√ 1 2 2 1 (m ˆ − ms ) + 2 √ m0|¯ ˆ u iγ5 u|u¯ u,(8) f u √ B0 3 6 3 1 2 1 ˆ = fs √ m20 − fs √ (m20 + B0 (ms + 2m)) 3 3 3 √ 2 2 2 1 (m ˆ − ms ) + 2 √ ms 0|¯ s iγ5 s|s¯ s.(9) f s √ B0 3 6 3 Similarly, for η8 :
√ 1 2 1 2 2 (m ˆ − ms ) fu √ B0 (2ms + m) ˆ = −fu √ B0 3 6 3 3 1 ˆ u iγ5 u|u¯ u, (10) + 2 √ m0|¯ 6 √ 2 2 1 2 2 (m ˆ − ms ) ˆ = −fs √ B0 −fs √ B0 (2ms + m) 3 6 3 3 2 s iγ5 s|s¯ s. (11) − 2 √ ms 0|¯ 6 In deriving the above expressions, we have used the nonet symmetry mass formula, i.e m28 = B0 32 (2ms + m), ˆ m20 = m ¯ 20 + B0 32 (ms + 2m) ˆ and √ 2 2 m08 = B0 3 2(−ms +m) ˆ where m ˆ = (mu +md )/2 and m ¯ 0 is the anomaly contribution to m0 , the singlet η0 mass. The second term on the r.h.s of Eqs. (8-9) and the first term on the r.h.s of Eqs. (10-11) are the pole terms due to η − η mixing. In the limit mu = md =√0, the l.h.s and r.h.s of Eq. (10) become fu m28 / 6 in agreement with the divergence equation Eq. (6). Because of cancellation between the pole contribution and other quark mass terms, Eqs. (8-9) are reduced to the simplified form of Eq. (2) or Eq. (3). The pseudo-scalar density matrix elements in η0 are then given by: 0|¯ u iγ5 u|u¯ u = B0 fu , 0|¯ s iγ5 s|s¯ s = B0 fs . (12) The same expression for η8 is obtained similarly from Eqs. (10-11). Thus, 0|¯ u iγ5 d|π + , 0 + 0|¯ u iγ5 u|π and 0|¯ u iγ5 s|K , and the matrix element 0|¯ u iγ5 u|u¯ u and 0|¯ s iγ5 s|s¯ s in η0,8 are, apart from the decay constant fq , essentially the same, given by the parameter B0 and are consistent with nonet symmetry. Since experimentally, m208 = −(0.81±0.05) m2K is rather close to the nonet symmetry value of m208 −0.90 m2K [13], we expect nonet symmetry for the pseudo-scalar density matrix elements in
393
T.N. Pham / Nuclear Physics B (Proc. Suppl.) 186 (2009) 391–394
η − η would be valid to this accuracy. Since m28 gets about 15% increase from higher order terms L4 , L5 , L6 , L8 and chiral logarithms, Eqs. (10-11) s in η will be increased by a show that 0|¯ s iγ5 s|s¯ similar amount. A possible similar 15% increase for m20 would also increase 0|¯ s iγ5 s|s¯ s in η0 by a similar amount and would be additional source of enhancement for the B → Kη branching ratio.
which gives γ = 66◦ ( |Vtb | = 1) and α = 91.8◦ , in good agreement with the value found in the current UT-fit value of (88 ± 16)◦ . For other pa= 80 MeV, fu = fπ , rameters, we use ms (2 GeV)
3. The B − → K − (η, η ) and B − → π − (η, η ) decays
F Bη,Bη from the u quark content in η and η :
The B → M1 M2 decay amplitude in QCD Factorization(QCDF) is given by[7,8]: GF ∗ Vpb Vps × A(B → M1 M2 ) = √ 2 p=u,c 10 10 p , ai M1 M2 |Oi |BH + fB fM1 fM2 bi (13) i=1
F0Bπ (0) = 0.258,
F0BK (0) = 0.33,
(17)
F Bη (0)= 0.58 F0Bπ (0), F Bη (0)=0.40 F0Bπ (0). (18) and the pseudo-scalar density matrix elements: 0|¯ s iγ5 s|η =Cη B0 fs , 0|¯ s iγ5 s|η =Cη B0 fs .(19) obtained with the s quark content Cη = −0.57, Cη = 0.82 and an η − η mixing angle (−22 ± 3)◦ [13]
i
where the QCD coefficients api contain vertex corrections, penguin corrections, and hard spectator scattering contributions. The hadronic matrix elements M1 M2 |Oi |BH of the tree and penguin operators Oi are given by factorization model [2,14] and bi are annihilation terms. The values for api , p = u, c and bi computed from the expressions in [7,8] at the renormalization scale μ = mb and with mb = 4.2 GeV are given in the published work [10]. We estimate the CKM matrix element Vub and the CKM angle γ from the (db) unitarity triangle [15]: ∗ |Vcb Vcd cos2 α | | sin β . (14) |Vub | = 1 + ∗ | |Vud sin2 α ◦ With α = (99+13 −9 ) [16] and |Vcb | = (41.78±0.30± −3 0.08) × 10 [17], we find
|Vub | = 3.60 × 10−3 .
fs = fπ 1 + 2( ffKπ − 1) , the current theoretical values [19]:
(15)
which is quite close to the exclusive data [17] |Vub | = (3.33 − 3.51) × 10−3 . Similarly, we use the current determination 0 ¯0 |Vtd /Vts | = (0.208+0.008 −0.006 ) from the Bs − Bs mixing measurements [18] to obtain the angle γ: ∗ cos2 α |Vcb Vcd | | sin γ . (16) 1 + |Vtd | = |Vtb∗ | sin2 α
Table 1 The branching ratio B(B → P η, P η ) in QCDF Decay Modes QCDF BR (×10−6 ) Exp. [20] B − → π− π0 5.05 5.7 ± 0.4 ¯ 0 → K −π+ 18.25 19.04 ± 0.6 B B − → π− η 3.39 4.4 ± 0.4 B − → π− η 1.91 2.6+0.6 −0.5 − − B →K η 0.43 2.2 ± 0.3 B − → K − η 48.26 69.7+2.8 −2.7
As shown in Table 1, with a moderate annihilation term (ρA = 0.6) and the current theoretical value for the F Bπ and F BK form factors [19], QCDF predictions are in reasonable agreement with experiment, except for the B → Kη branching ratio which is underestimated by 20 − 30%. One could increase the F Bη form factor to produce better agreement with experiment for B − → π − η for which the prediction in Table 1 is below the Babar value of (4.0 ± 0.8 ± 0.4) × 10−6 [20] and to bring the predicted B → Kη branching ratio closer to experiment, but so far there seems to be no evidence for a large B → η form factor compared with the nonet symmetry
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T.N. Pham / Nuclear Physics B (Proc. Suppl.) 186 (2009) 391–394
value as seen from the new Babar [21] upper limit B(B + → η + ν)/B(B + → η + ν) < 0.57 which is consistent with nonet symmetry for the B → η, η form factors given in Eq. (18). 4. Conclusion We have shown that nonet symmetry for the pseudo-scalar meson mass term implies nonet symmetry for the pseudo-scalar density matrix elements. With this approximate relation, we obtained an improved estimate for the B → P η (P = K, π ) branching ratios. With a moderate annihilation contribution consistent with the measured B → Kπ branching ratio, we find that a major part of the B → Kη branching ratio could be obtained by QCDF. Without fine tuning or a large F B→η form factor, we find that the B → Kη branching ratio is underestimated by 20 − 30%. This could be considered as a more or less successful prediction for QCDF, considering the theoretical uncertainties involved. This could also indicate that an additional powersuppressed terms could bring the branching ratio close to experiment, as with the B → K ∗ π [22] and B → K ∗ η decay for which the measured branching ratios are much bigger than the QCDF prediction.
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