Analysis of B→a1(1260)(b1(1235))K* decays in the perturbative QCD approach

Within the framework of the perturbative quantum chromodynamics (PQCD) approach, we study the charmless two-body decays B→a1(1260)K*, b1(1235)K*. Using the decay constants and the light-cone distribution amplitudes for these mesons derived from the QCD sum rule method, we find the following results. (a) Our predictions for the branching ratios are consistent with the QCD factorization (QCDF) results within errors, but much larger than the naive factorization approach calculation values. (b) We predict that the anomalous polarizations occurring in the decays B→φK*, $rHK* also happen in B→a1K* decays, while they do not happen in B→b1K* decays. Here the contributions from the annihilation diagrams play an important role in explaining the larger transverse polarizations in the B→a1K* decays, while they are not sensitive to the polarizations for the B→b1K* decays. (c) Our predictions for the direct CP-asymmetries agree well with the QCDF results within errors. The decays B̄0 → b+1K*−, B− → b01K*− have larger direct CP-asymmetries, which could be measured by the present LHCb experiment and the forthcoming Super-B experiment.


Introduction
In general, mesons are classified in J P C multiplets. There are two types of orbitally excited axial-vector mesons, namely, 1 ++ and 1 +− . The former includes a 1 (1260), f 1 (1285), f 1 (1420), and K 1A , which compose the 3 P 1 nonet; the latter includes b 1 (1235), h 1 (1170), h 1 (1380), and K 1B , which compose the 1 P 1 nonet. With the exception of a 1 (1260) and b 1 (1235), these axialvector mesons have an important characteristic: each flavor state can mix with any other flavor state from the other nonet or the same nonet. There is no mixing between a 1 (1260) and b 1 (1235) because of their opposite C parities. They also do not mix with the other mesons. So, compared with other axial-vector mesons, these two mesons should have less uncertainty regarding their inner structures.
Like the decay modes B → VV, the charmless decays B→a 1 (1260)K * , b 1 (1235)K * also have three polarization states and so are expected to have rich physics. In many B → VV decays, the information on branching ratios and polarization fractions among various he-licity amplitudes have been studied by many authors [1][2][3][4]. Through polarization studies, some underlying helicity structure in the decay mechanism is suggested. It has been found that the polarization fractions follow the naive counting rule, that is , where f L,N,T denote the longitudinal, parallel, and perpendicular polarization fractions, respectively, and m B (m V ) is the B(V) meson mass. If the contributions from the factorizable emission amplitudes are suppressed for some decay modes, however, this counting rule might be modified dramatically by other contributions. For example, highly anomalous longitudinal polarization fractions of about 50% have been measured in the decays B → ρK * , φK * [5], apart from the decay B − →K * − ρ 0 , which has a large longitudinal polarization fraction of (96 +6 −16 )% [5] (the newer measurement is (90±20)% [6]). Whether similar results also occur in the decay modes B → a 1 (1260)K * , b 1 (1235)K * is worth researching. We know that a 1 (1260) has some similar behaviors to the vector meson, so one can expect the branching ratios and polarization fractions of the decays B→a 1 (1260)K * and B→ρK * , where a 1 (1260) and ρ are scalar partners of each other, to have similar characteristics. This is not the case for b 1 (1235) because of the different characteristics of its decay constant and lightcone distribution amplitudes (LCDAs) compared with those of a 1 (1260). For example, the longitudinal decay constant is very small for the charged b 1 (1235) states and vanishes under the SU (3) limit. It is zero for the neutral b 0 1 (1235) state. The transverse decay constant of a 1 (1260), on the other hand, vanishes under the SU (3) limit. In the isospin limit, the chiral-odd (-even) LCDAs of the b 1 (1235) meson are symmetric (antisymmetric) under the exchange of quark and antiquark momentum fractions, which is exactly opposite to the symmetric behavior for a 1 (1260). In view of these differences, one can expect that B→a 1 (1260)K * and B→b 1 (1235)K * should give very different results. On the theoretical side, the decays B → a 1 (1260)K * , b 1 (1235)K * have been studied by Cheng and Yang in [7], where the branching ratios are very different to those calculated by the naive factorization approach [8]. To clarify such large differences is another motivation of this work. On the experimental side, only the upper limits for some of the considered decays are available [9,10].
In the following, a 1 (1260) and b 1 (1235) are denoted as a 1 and b 1 in some places for convenience. The layout of this paper is as follows. In Section 2, we analyze these decay channels using the perturbative quantum chromodynamics (PQCD) approach. The numerical results and discussion are given in Section 3. The conclusions are presented in the final part.

The PQCD calculation
The PQCD approach has proven an effective theory to handle hadronic B decays in many works [2,3,11,12]. Because the transverse momentum of the valence quarks in the hadrons is taken into account, one will encounter double logarithm divergences when the soft and the collinear momenta overlap. Fortunately, these large double logarithms can be re-summed into the Sudakov factor [13]. There is also another type of double logarithm which arises from the loop corrections to the weak decay vertex. These double logarithms can also be re-summed and result in the threshold factor, which decreases faster than any other power of the momentum fraction in the threshold region, which removes the endpoint singularity. This factor is often parameterized into a simple form which is independent of channels, twists and flavors [14]. Certainly, when the higher order diagrams only suffer from soft or collinear infrared divergence, it is easy to cure using the eikonal approximation [15]. Controlling these kinds of divergences reasonably makes the PQCD approach more self-consistent.
In the standard model, the related weak effective Hamiltonian H eff mediating the b → s type transitions can be written as [16] Here the function Q i (i = 1,···,10) is the local fourquark operator and C i is the corresponding Wilson coefficient. V p(t)b , V p(t)s are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. The standard four-quark operators are defined as: where α and β are the SU (3) color indices; L and R are the left-and right-handed projection operators with L = (1−γ 5 ), R = (1+γ 5 ). The sum over q runs over the quark fields that are active at the scale μ = O(m b ), i.e. (q {u,d,s,c,b}). At leading order, there are eight types of single hard gluon exchange diagrams which contribute to the decays we are considering. These can be divided into emission type diagrams and annihilation type diagrams, each type including two factorizable and two nonfactorizable diagrams. Due to space limitations, these diagrams are not shown here. Combining the contributions from the different diagrams, the total decay amplitudes for these decays can be written as where F LL,j ea 1 denotes the amplitudes of the factorizable emission diagrams, where one can extract out the B→a 1 transition form factor. If we exchange the positions of a 1 andK * , we get the amplitudes F LL,j eK * and F SP,j eK * . As for the amplitudes of the non-factorizable emission diagrams, M LL,j ea 1 and M LR,j ea 1 are those relevant to the decays considered. The amplitudes M LL,j eK * and M SP,j eK * are obtained by exchanging a 1 andK * in the non-factorizable emission diagrams. In a similar way, for the annihilation diagram amplitudes F LL,j aa 1 and F SP,j aa 1 are from the factorizable annihilation diagrams, while M LL,j aa 1 and M LR,j aa 1 are from the non-factorizable annihilation diagrams. Note that the upper labels LL, LR, and SP denote the , and (S−P )(S+P ) currents, respectively, and j denotes three types of polarizations (one longitudinal and two transverse), denoted by L, N and T . Limitations of space prevent us from giving the analytical expressions for these amplitudes. The combinations of the Wilson coefficients are defined as usual: The amplitudes for those decays involving the b 1 meson can be found from Eq. (4)-Eq. (6) above by substituting the b 1 meson wave functions for a 1 ones.

Numerical results and discussions
For the wave function of the heavy B meson, we take [11] Here only the contribution of the Lorentz structure φ B (x,b) is taken into account, since the contribution of the second Lorentz structureφ B is numerically small [17] and has been neglected. For the distribution amplitude φ B (x,b) in Eq. (10), we adopt the following model: where ω b is a free parameter, taken to be ω b =0.4±0.04 GeV in numerical calculations, and N B = 91.745 is the normalization factor for ω b =0.4. This is the same wave function as in Ref. [11], which gives the best fit for most of the measured hadronic B decays. In these decays, both the longitudinal and the transverse polarizations are involved for the vector meson K * .

033102-3
Its distribution amplitudes are defined as where n(v) is the unit vector having the same (opposite) direction as the motion of the vector meson and x is the momentum fraction of quark q 2 . The upper (sub)leading twist wave functions can be parameterized as where The distribution amplitudes of the axial-vectors a 1 (b 1 ) have the same format as those of the K * meson except for the factor iγ 5 from the left-hand side: where A represents a 1 and b 1 . Their (sub) leading twist wave functions also have the same parameter formats as those of the K * , which can be obtained by replacing K * with A in Eq. (14). The corresponding functions φ(x), h(x), g(x) for the axial-vector are written as where the zeroth Gegenbauer moments a ⊥ 0 (a 1 )=a 0 (b 1 )= 0 and a 0 (a 1 )= a ⊥ 0 (b 1 )= 1. Here t = 2x−1, and the other decay constants and Gegenbauer moments are listed in Table 1.
The following input parameters are also used in our numerical calculations [18,19]: , which are usually the input parameters in other QCD approaches such as QCD factorization (QCDF). For comparison, we list both the results calculated by the PQCD approach and the light-cone sum rules (LCSR) in Table 2, the latter being the input parameters for QCDF calculations. Certainly, our calculations are consistent with previous PQCD calculation results [20].
The decay widths for these channels can be expressed as where r a 1 (b 1 ) = m a 1 (b 1 ) /M B . M σ is the decay amplitude and can be calculated in the PQCD approach; the subscript σ represents the helicity states of the two final mesons with the longitudinal (L), normal (N) and transverse (T) components. According to Lorentz structure 033102-4 analysis, the amplitudes are decomposed into We can define the longitudinal H 0 and transverse H ± helicity amplitudes as where κ= P 2 ·P 3 m a 1 (b 1 ) m K * . They satisfy the relation When we consider the polarization fractions, another equivalent set of definitions of helicity amplitudes is often used, If the normalization factor satisfying |A 0 | 2 +|A | 2 +|A ⊥ | 2 = 1, then A 0 , A , A ⊥ denote the longitudinal, parallel and perpendicular polarization fractions, respectively.
QCD , the threshold resummation parameter c, and the Gegenbauer moments in K * or a1(b1) meson. Using the input parameters as specified in this section, it is easy to get the branching ratios for the decays considered. The results are listed in Table 3, where the first error comes from the uncertainty in the B meson shape parameter ω b = 0.40±0.04 GeV, the second error is induced by the hard scale-dependent variation from Λ (4) QCD =0.25±0.05, the third is from the threshold resummation parameter c varying from 0.3 to 0.4, and the last error is from the Gegenbauer moments in K * and a 1 (b 1 ) mesons. Table 3. Branching ratios (in units of 10 −6 ) for the decays B → a1(1260)K * and B → b1(1235)K * . In our results, the errors for these entries correspond to the uncertainties from ωB, Λ (4) QCD , the threshold resummation parameter c, and the Gegenbauer moments in the final state mesons, respectively. For comparison, we also list the results predicted by the QCDF approach [7] and the naive factorization approach [8].
this work [7] [8] In our predictions, the branching ratio of the decaȳ B 0 →a 0 1K * 0 is larger than that of the decay B − →a 0 1 K * − . This is mainly due to the amplitudes of the factorizable emission diagrams, F ea 1 and F eK * , having opposite interference effects for these two decays: constructive for the decay a 0 1K * 0 , destructive for the decay a 0 1 K * − . The decaȳ B 0 → a 0 1K * 0 therefore receives a larger real part for the penguin amplitudes. Though the decay B − →a 0 1 K * − has much larger contributions from tree amplitudes, these are CKM suppressed and cannot change the branching ratio too much. In order to characterize the contributions from tree operators and the symmetry breaking effects between B − andB 0 mesons, it is useful to define two ratios: If one neglects the tree operators and the electro-weak penguins, the ratios obey the following limits Here our predictions for these two ratios are 1.02 and 0.55, respectively. The results predicted by the QCDF approach are 0.98 and 0.52, respectively. If the future data for R 1 show a large deviation from our value, the contributions from electro-weak penguin operators might have an important effect, as the contributions from tree operators cannot change the branching ratio ofB 0 →a + 1 K * − too much. If the future data for R 2 have a large deviation from our value, some mechanism beyond factorization, or even new physics, might have an important effect, because the factorizaton formulae for 033102-5 B 0 →b + 1 K * − and B − →b 0 1 K * − are exactly the same if the neutral b 0 1 meson decay constant vanishes. Table 3 shows that, when compared with other results, our predictions are consistent with the QCDF results within (large) theoretical errors, while in stark disagreement with the naive factorization approach, where the nonfactorizable effects are described by the effective number of colors N eff c . For some decays, where the contributions from the emission diagrams are dominant or the branching ratios have a strong dependence on the correlative form factors, the naive factorization approach can give a reasonable prediction, while for decays where the annihilation diagrams play an important role, this approach shows some disadvantages. On the experimental side, BarBar has searched for the decays B→a − 1K * 0 , b 1 K * and set upper limits on their branching ratios ranging from 3.3 to 8.0×10 −6 at the 90% confidence level [9,10]. These upper limits are obtained, however, by assuming that B(a ± 1 → π + π − π ± ) = B(a ± 1 → π 0 π 0 π ± ) and B(a ± 1 (b ± 1 )→ρ 0 (ω)π ± )=1. Furthermore, the background signals may have an important effect on these upper limits, such as the background decay channel B→a 2K * 0 in studies of the decay B→a 1K * 0 . In view of these disagreements, we strongly suggest that LHCb and the forthcoming Super-B experiments accurately measure these decay modes.
From Table 4, we find that the polarization charactistics for the decays B→a 1 K * and B→b 1 K * are very different: the transverse polarization amplitudes have almost equal values with (even a little stronger than) the longitudinal polarization amplitudes for the former, while the longitudinal polarization states are dominant for the latter. It seems that the anomalous polarizations which occur in B→φK * , ρK * decays also happen in B→a 1 K * decays, while they do not occur in B → b 1 K * decays. We also find that the contributions from the annihilation diagrams are very important for the final polarization fractions for B → a 1 K * decays: if these contributions are neglected, the longitudinal polarization fraction of the decay B − → a 0 1 K * − becomes 98.8%, those ofB 0 → a + 1 K * − , a 0 1K * 0 increase to about 90%, and that of the decay B − → a − 1K * 0 changes from 50.3% to 70.0%. The longitudinal polarizations of B → b 1 K * decays decrease very little if the annihilation type contributions are neglected, except for the decay B − → b − 1K * 0 , which has a larger reduction, changing from 96.2% to 86%. In short, compared to B → a 1 K * decays, the longitudinal polarizations of B → b 1 K * decays are not very sensitive to the annihilation type contributions. Now we turn to evaluations of the CP -violating asymmetries in the PQCD approach. Here we only research the decays B → b 1 K * , where the transverse polarization fractions are very small and range from 3.8 to 5.2%. It is easy to see that for these b 1 K * decay modes, the contributions from the transverse polarizations are very small, so we neglect them in our calculations. Consider these two matrix elements where γ is the CKM weak phase angle, defined via ; δ L is the relative strong phase between the tree and the penguin amplitudes, which are denoted as 'T L ' and 'P L ', respectively; and the term z L describes the ratio of penguin to tree contributions and is defined as Using Eq. (38) and Eq. (39), one can get the expression for the direct CP -violating asymmetry: .
(41) Table 4. Longitudinal polarization fraction (fL) and two transverse polarization fractions (f , f ⊥ ) for decays B→ a1(1260)K * and B→b1(1235)K * . In our results, the uncertainties are the same as those in Table 3. For comparison, the results of fL as predicted by the QCDF approach are displayed in parentheses.  Note that the contributions from the transverse polarizations have been neglected in the derivation of the direct CP -violating asymmetry.
Using the input parameters and the wave functions as specified in this section, one can find the PQCD predictions (in units of 10 −2 ) for the direct CP -violating asymmetries of the decays considered: QCD =0.25±0.05, the threshold resummation parameter c varying from 0.3 to 0.4, and the Gegenbauer moments in the final state mesons. In Fig. 1, we show the CKM angle γ dependence of the direct CP -violating asymmetries for the four decays above. It is particularly noteworthy that our predictions about the direct CP asymmetries of these decays are consistent with the QCDF results [23]: where the error comes from the parameters ρ A,H and arbitrary phases φ A,H . These are phenomenological parameters to cure the endpoint divergences in the amplitudes for the annihilation and hard spectator scattering diagrams.

Conclusion
In this paper, by using the decay constants and lightcone distribution amplitudes derived from the QCD sumrule method, we studied the B → a 1 K * , b 1 K * decays in the PQCD factorization approach and found that: 1) Our predictions for the branching ratios are consistent with the QCDF results within errors, but larger than the values calculated by the naive factorization approach. On the experimental side, some primary upper limit values are inexplicable. In view of these disagreements, we strongly suggest that LHCb and the forthcoming Super-B experiments perform further studies to accurately measure these decay modes.
2) The anomalous polarizations which occur in B → φK * , ρK * decays also happen in B → a 1 K * decays, but not in B→b 1 K * decays. Here, the contributions from the annihilation diagrams play an important role in explaining the larger transverse polarizations in the B → a 1 K * decays, while they are not sensitive to the polarizations in B→b 1 K * decays.
3) Our predictions for the direct CP -asymmetries agree well with the QCDF results, within errors. The decaysB 0 →b + 1 K * − , B − →b 0 1 K * − have larger direct CPasymmetries, which could be measured by the present LHCb and the forthcoming Super-B experiments.