Large non-factorizable contributions in B→ a 0 a 0 decays
aa r X i v : . [ h e p - ph ] J a n JSNU/PHY-TH-2015
Large non-factorizable contributions in B → a a decays Defa Dou, Xin Liu ab , and Jing-Wu Li c School of Physics and Electronic Engineering,Jiangsu Normal University, Xuzhou, Jiangsu 221116, People’s Republic of China
Zhen-Jun Xiao d Department of Physics and Institute of Theoretical Physics,Nanjing Normal University, Nanjing, Jiangsu 210023, People’s Republic of China (Dated: September 22, 2018)We investigate three tree-dominated B → a a decays for the first time in the perturbativeQCD(pQCD) approach at leading order in the standard model, with a standing for the lightscalar a (980) state, which is assumed as a meson based on the model of conventionaltwo-quark ( q ¯ q ) structure. All the topologies of the Feynman diagrams such as the non-factorizable spectator ones and the annihilation ones are calculated in the pQCD approach.It is of great interest to find that, contrary to the known B → ππ decays, the B → a a decays are governed by the large non-factorizable contributions, which give rise to the large B → a a decay rates in the order of − ∼ − , although the a meson has an extremelysmall vector decay constant f a . Also observed are large direct CP-violating asymmetriesaround and for the B → a a and a +0 a − modes. These sizable predictions couldbe easily examined at the running Large Hadron Collider and the near future Super-B/Belle-II experiments. The future precision measurements combined with these pQCD predictionsmight be helpful to explore the complicated QCD dynamics and the inner structure of thelight scalar a , as well as to complementarily constrain the unitary angle α . PACS numbers: 13.25.Hw, 12.38.Bx, 14.40.Nd a Corresponding author b Electronic address: [email protected] c Electronic address: [email protected] d Electronic address: [email protected]
I. INTRODUCTION
As we know, the nature of the light scalar states such as a (980) is not yet well understood atboth theoretical and experimental aspects. Also the identification and the classification of theselight scalars remain as a long-standing puzzle(for latest review, see, e.g. [1]) to be resolved. How-ever, it is fortunate for the people that the light scalars as products in the heavy flavor meson de-cays have been detected, for example, D → SP, SV , B → SP, SV , even B → SS modes [1, 2]with S, P, and V being the light scalar, pseudoscalar, and vector mesons, respectively, whichwill provide unique places and play very important roles on investigating the physical propertiesof light scalars. It is generally believed that the ongoing Large Hadron Collider(LHC) experi-ments can provide rich data on the B , B s , and B c meson decaying into light scalars. And morepromisingly, the forthcoming Super-B/Belle-II factory scheduled in 2018 with a high luminosity > ∼ cm − s − [3, 4] will produce much more events about the relevant decays. The studies onthe above mentioned decays can also provide more constraints complementarily on the parametersin the standard model(SM), hint the exotic new physics beyond the SM, etc.In this work, we will investigate the CP-averaged branching ratios and the CP-violating asym-metries of the B → a (980) a (980) decays by employing the perturbative QCD(pQCD) ap-proach [5–7] with the low energy effective Hamiltonian [8] in the SM. It should be noted thatthe a (980) state here will be assumed as a meson in the model of conventional two-quark ( q ¯ q ) structure. Moreover, hereafter, the a (980) will be abbreviated as a for the sake of simplicitythroughout the paper. To our knowledge, heretofore, no other B → SS processes have beenstudied explicitly in the factorization approaches based on the QCD dynamics, apart from the B u,d,s → K ∗ (1430) ¯ K ∗ (1430) decays [9] by two of our authors (X. Liu and Z.J. Xiao). Becausethe scalar meson has either tiny or vanishing vector decay constant [10, 11], the contributions aris-ing from the factorizable emission diagrams in the B → SS decays are usually highly suppressed,which is dramatically different from the known B → P P, P V, V V decays. In other words, for ex-ample, in contrast to the extensively investigated B → ππ decays, the large measured B → a a decay rates may indicate large non-factorizable spectator scattering and/or annihilation contribu-tions, which would hint some useful information on the B → ππ decays, the presently known puz-zle to be resolved, because they embrace the same components at the quark level. In the heavy B meson decays, the above mentioned large contributions from non-factorizable spectator and anni-hilation diagrams are often considered as the small and/or negligible higher order or higher powercorrections in the naive factorization approach [13]. Therefore, the channels involving an emittedscalar state in the heavy flavor meson decays are suggested to test the breaking effects of the fac-torization assumption, e.g. [14]. Though the QCD improved factorization approach [15, 16] goingbeyond the naive factorization, the end-point singularities make it less predictive because the non-factorizable spectator scattering contributions and the annihilation ones have to be parametrizedwith the tunable parameters, which are always determined by the experimental measurements. Asone of the popular factorization approaches based on the QCD dynamics, the pQCD approachinvolves no end-point singularities by retaining the parton transverse momentum k T . Based on k T factorization theorem, the double logarithms arising from the overlap of soft and collinear diver-gences generated in the radiative corrections are resummed into an important Sudakov factor tosuppress the long-distance contribution [17]. Armed with this pQCD approach, all the transitionform factors, the non-factorizable spectator diagrams, and the annihilation diagrams are perturba-tively calculable, besides the factorizable spectator diagrams. Note that, as far as the annihilationcontributions are concerned, soft-collinear effective theory [18] and pQCD approach have an ex-tremely different effect on the perturbative calculations [19, 20]. However, the predictions on thepure annihilation decays based on the pQCD approach can accommodate the experimental datawell, for example, see Refs. [21–24]. We will therefore put the controversies aside and adopt thepQCD approach in our analyses.The paper is organized as follows. Section II is devoted to the analytic expressions for thedecay amplitudes of B → a a modes in the pQCD approach. The numerical results and phe-nomenological analyses on the CP-averaged branching ratios and the CP-violating asymmetries ofthe considered decays are given in Sec. III. We summarize and conclude in Sec. IV. II. PERTURBATIVE CALCULATIONS
For the considered B → a a decays, the related weak effective Hamiltonian H eff [8] can bewritten as H eff = G F √ (cid:26) V ∗ ub V ud [ C ( µ ) O u ( µ ) + C ( µ ) O u ( µ )] − V ∗ tb V td [ X i =3 C i ( µ ) O i ( µ )] (cid:27) + H . c . , (1) In fact, the cancelation of the decay amplitudes indeed occurred between the two non-factorizable spectator dia-grams in the B → P P, P V, V V channels, for example, see Ref. [12]. with the Fermi constant G F = 1 . × − GeV − , the Cabibbo-Kobayashi-Maskawa(CKM)matrix elements V , and the Wilson coefficients C i ( µ ) at the renormalization scale µ . The localfour-quark operators O i ( i = 1 , · · · , are written as(1) current-current(tree) operators O u = ( ¯ d α u β ) V − A (¯ u β b α ) V − A , O u = ( ¯ d α u α ) V − A (¯ u β b β ) V − A ; (2)(2) QCD penguin operators O = ( ¯ d α b α ) V − A X q ′ (¯ q ′ β q ′ β ) V − A , O = ( ¯ d α b β ) V − A X q ′ (¯ q ′ β q ′ α ) V − A ,O = ( ¯ d α b α ) V − A X q ′ (¯ q ′ β q ′ β ) V + A , O = ( ¯ d α b β ) V − A X q ′ (¯ q ′ β q ′ α ) V + A ; (3)(3) electroweak penguin operators O = 32 ( ¯ d α b α ) V − A X q ′ e q ′ (¯ q ′ β q ′ β ) V + A , O = 32 ( ¯ d α b β ) V − A X q ′ e q ′ (¯ q ′ β q ′ α ) V + A ,O = 32 ( ¯ d α b α ) V − A X q ′ e q ′ (¯ q ′ β q ′ β ) V − A , O = 32 ( ¯ d α b β ) V − A X q ′ e q ′ (¯ q ′ β q ′ α ) V − A . (4)with the color indices α, β and the notations (¯ q ′ q ′ ) V ± A = ¯ q ′ γ µ (1 ± γ ) q ′ . The index q ′ in thesummation of the above operators runs through u, d, s , c , and b . The standard combinations a i ofWilson coefficients are defined as follows, a = C + C , a = C + C , a i = C i + C i ± i = 3 − . (5)where the upper(lower) sign applies, when i is odd(even).Similar to B → ππ decays [6, 25], there are eight types of diagrams contributing to B → a a modes at leading order(LO) in the pQCD approach, as illustrated in Fig. 1. They can be clas-sified into two types of topologies as emission and annihilation, respectively. And each kind oftopology contains factorizable diagrams such as Fig. 1(a) and 1(b), in which a hard gluon con-nects the quarks in the same meson, and non-factorizable diagrams such as Fig. 1(c) and 1(d), inwhich a hard gluon attaches the quarks in two different mesons. By evaluating all these Feynmandiagrams, one can obtain the decay amplitudes of B → a a decays. Because the above men-tioned diagrams are the same as those in B → K ∗ (1430) ¯ K ∗ (1430) modes [9], and also the light FIG. 1. (Color online) Typical Feynman diagrams for B → a a decays at leading order in the pQCDapproach. scalar mesons are considered, the formulas of B → a a decays are therefore same as those of B → K ∗ (1430) ¯ K ∗ (1430) ones just by replacing the wave functions and input parameters corre-spondingly. Hence the analytic formulas for the B → a a decays are not explicitly presented inthis paper.By taking various contributions from the relevant Feynman diagrams into consideration, thetotal decay amplitudes for three tree-dominated B → a a channels can then be read as,1. for B → a +0 a − decay mode, A ( B → a +0 a − ) = λ u (cid:20) C M nfs + C M nfa (cid:21) − λ t (cid:20) ( C + C ) M nfs + ( C + 2 C −
12 ( C − C )) M nfa + ( C − C ) M P nfa + (2 C + 12 C ) M P nfa + ( a − a ) × f B F P fa + ( a + a ) F P fs (cid:21) , (6)where λ u = V ∗ ub V ud and λ t = V ∗ tb V td . We adopt F and M to denote the contributions from ( V − A )( V − A ) operators in the factorizable and non-factorizable diagrams, respectively.Analogously, F P and M P are chosen to denote the contributions from ( V − A )( V + A ) operators, and F P and M P are taken to denote the contributions from ( S − P )( S + P ) operators which result from the Fierz transformation of the ( V − A )( V + A ) operators.The subscripts f s , nf s , f a , and nf a are the abbreviations for factorizable emission, non-factorizable emission, factorizable annihilation, and non-factorizable annihilation, respec-tively.2. for B + → a +0 a decay mode, √ A ( B + → a +0 a ) = λ u (cid:20) ( C + C ) M nfs (cid:21) − λ t (cid:20)
12 ( C + 3 C ) F P fs + 32 ( C + C ) M nfs + 32 C M P nfs (cid:21) , (7)3. for B → a a decay mode, √ A ( B → a a ) = λ u (cid:20) C ( M nfa − M nfs ) (cid:21) − λ t (cid:20) − ( a − a ) F P fs + ( C −
12 ( C +3 C )) M nfs − C M P nfs + ( C + 2 C −
12 ( C − C )) M nfa +( C − C ) M P nfa + (2 C + 12 C ) M P nfa + ( a − a ) f B F P fa (cid:21) . (8)It is worth mentioning that the highly suppressed F fs has been safely neglected in all of the abovedecay amplitudes for the considered B → a a decays due to the either extremely small or van-ishing vector decay constant. Furthermore, based on the discussions of F fa below Eq. (40) inRef. [9], the factorizable annihilation contributions induced by the V ± A currents are thereforenaturally absent because of the isospin symmetry between u and d quarks in the above analyticaldecay amplitudes. III. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we will make theoretical predictions on the CP-averaged branching ratios andthe CP-violating asymmetries for the B → a a decay modes considered. In numerical calcula-tions, central values of the input parameters will be used implicitly unless otherwise stated. Firstly,we shall make several essential discussions on the input quantities. A. Input quantities
For B meson, the distribution amplitude in the impact b space, with b being the conjugate spacecoordinate of transverse momentum k T , has been proposed [5–7], φ B ( x, b ) = N B x (1 − x ) exp " − (cid:18) xm B ω b (cid:19) − ω b b , (9)where the normalization factor N B is related to the decay constant f B through the following nor-malization condition, Z dxφ B ( x, b = 0) = f B √ N c . (10)with the color factor N c = 3 . The shape parameter ω b has been fixed at . GeV associated with N B = 91 . by using the rich experimental data on the B mesons with f B = 0 . GeV based onlots of calculations of form factors and other well-known decay modes of B meson in the pQCDapproach [5, 6, 26].For the light scalar a , its leading twist light-cone distribution amplitude φ a ( x, µ ) can be gen-erally expanded as the Gegenbauer polynomials [10, 27]: φ a ( x, µ ) = 3 √ N c x (1 − x ) (cid:26) f a ( µ ) + ¯ f a ( µ ) ∞ X m =1 B m ( µ ) C / m (2 x − (cid:27) , (11)where f a ( µ ) and ¯ f a ( µ ) , B m ( µ ) , and C / m ( t ) are the vector and scalar decay constants, Gegen-bauer moments, and Gegenbauer polynomials, respectively. For the vector and scalar decay con-stants, ¯ f a = µ a f a with µ a = m a m ( µ ) − m ( µ ) and m a = 0 . GeV, where m and m are therunning current quark masses in the scalar a . For neutral scalar a meson, which cannot be pro-duced by the vector current, the vector decay constant f a = 0 is guaranteed by charge conjugationinvariance. But the quantity ¯ f a = f a µ a remains finite. In fact, for the charged a ± meson, thevector decay constant f a ± also vanishes in the isospin limit. The reason is that f a ± is proportionalto the mass difference between the constituent d and u quarks, which will result in f a ± being of or-der m d − m u . Hence, the contribution from the first term in Eq. (11), namely, f a , can be neglectedsafely. In other words, the factorizable spectator diagrams could not contribute to B → a a de-cays through the vector currents. We shall use the same light-cone distribution amplitudes for bothneutral and charged a mesons for simplicity in this paper.The values for scalar decay constant and Gegenbauer moments in the a distribution amplitudeshave been investigated at scale µ = 1 GeV [10]: ¯ f a = 0 . ± .
020 GeV , B = − . ± . , B = 0 . ± . . (12)As for the twist-3 distribution amplitudes φ Sa and φ Ta , we here adopt the asymptotic forms inour numerical calculations for simplicity [10]: φ Sa = 12 √ N c ¯ f a , φ Ta = 12 √ N c ¯ f a (1 − x ) . (13)The QCD scale (GeV), masses (GeV), and B meson lifetime(ps) are [1, 5, 6] Λ ( f =4)MS = 0 . , m W = 80 . , m B = 5 . , m b = 4 . τ B + = 1 . , τ B = 1 . , m a = 0 . . (14)For the CKM matrix elements, we adopt the Wolfenstein parametrization and the updated pa-rameters A = 0 . , λ = 0 . , ¯ ρ = 0 . ± . , and ¯ η = 0 . ± . [1].Utilizing the above chosen distribution amplitudes and the relevant input parameters, we canget the numerical results in the pQCD approach for the form factor F B → a , at maximal recoil asfollows, F B → a , ( q = 0) = 0 . +0 . − . ( ω b ) +0 . − . ( ¯ f a ) +0 . − . ( B a i ) , (15)where the errors arise from the shape parameter ω b in B meson distribution amplitude, the scalardecay constant ¯ f a , and the Gegenbauer moments B a i ( i = 1 , in the light a distribution ampli-tude, respectively. This value agrees well with . +0 . − . as given in Ref. [27]. The tiny deviationis just from the zero vector decay constant f a assumed in this work. B. CP-averaged branching ratios and CP-violating asymmetries
In this subsection, we will analyze the CP-averaged B → a a branching ratios and the CP-violating asymmetries in the pQCD approach at LO level. For B → a a decays, the decay ratecan be written as Γ = G F m B π (1 − r a ) |A ( B → a a ) | , (16)where the decay amplitudes A can be referred correspondingly in Eqs. (6-8). Using the decayamplitudes obtained in last section, it is straightforward to numerically evaluate the CP-averagedbranching ratios with errors as collected in Eqs. (17)-(19), Br ( B → a a − ) = 1 . +0 . − . ( ω b ) +0 . − . ( ¯ f a ) +0 . − . ( B a i ) +0 . − . (CKM) × − , (17) Br ( B + → a a ) = 6 . +2 . − . ( ω b ) +1 . − . ( ¯ f a ) +3 . − . ( B a i ) +0 . − . (CKM) × − , (18) Br ( B → a a ) = 2 . +1 . − . ( ω b ) +0 . − . ( ¯ f a ) +1 . − . ( B a i ) +0 . − . (CKM) × − ; (19) The form factor F B → a , can be extracted directly from Eq. (29) in [9] with the state S being a . Of course, thereaders can also refer to Ref. [27] for more details. The dominant errors are induced by the uncertainties of the shape parameter ω b = 0 . ± . GeVfor B meson, the scalar decay constant ¯ f a , and the Gegenbauer moments B a i ( i = 1 , for thescalar a (see Eq. (12) for detail), respectively. It is worth stressing that the effective constraints onthe above mentioned non-perturbative parameters might be helpful to explore the QCD dynamicsinvolved in these decays and to reveal the inner structure of the light scalar a state.From Eqs. (17)-(19), one can obviously observe that the large B → a a decay rates are inthe order of − ∼ − calculated in the pQCD approach at LO level, which could be easilydetected through the dominant a to ηπ (or ππ ) final state [28] at the running LHC and the forth-coming Super-B/Belle-II experiments. As mentioned in the Introduction, some decays involvingscalar mesons were suggested as the ideal channels to test the validation of the factorization as-sumption [14]. It is therefore worth stressing that the B + → a +0 a mode would be the best choice,because it only contains a significantly suppressed factorizable emission contribution and a neg-ligible non-factorizable emission contribution as proposed in naive factorization, but has a largebranching ratio that could be easily tested in the near future experiments. Therefore, the obser-vation of this large B + → a +0 a decay rate, on one hand, could offer an effective test to thebreaking effects of the factorization assumption; on the other hand, might verify the q ¯ q compo-nents of the light scalar a evidently. Furthermore, it is surprising to find that the conventionallyso-called ”color-suppressed” B → a a mode has the largest branching ratio as . × − ,which is highly different from the known color-suppressed B → P P modes, such as the famous B → π π channel with very small branching ratio around O (10 − ) , although they embracethe same components at the quark level. Consequently, the hierarchy of the branching ratios ex-hibits theoretically as Br ( B → a a ) ∼ Br ( B → a +0 a − ) > Br ( B + → a +0 a ) in the pQCDapproach, which is also dramatically different from that in the B → ππ decays as Br ( B → π + π − ) & Br ( B + → π + π ) >> Br ( B → π π ) within theoretical errors [6, 12, 24, 25] and Br ( B + → π + π ) & Br ( B → π + π − ) > Br ( B → π π ) within experimental uncertain-ties [1, 2], respectively. In terms of the central values of the B → a a decay rates, the followingrelation can be easily found, Br ( B → a a ) > Br ( B → a +0 a − ) > Br ( B + → a +0 a ) , (20)which can be traced back to the factorization formulas as given in Eqs. (6)-(8). Specifically, thetree dominant contributions of these three decays are C ( M nfa − M nfs ) , C M nfs + C M nfa , and ( C + C ) M nfs , respectively, in which C is much larger than C in magnitude with C ∼ . C ∼ − . at the m b scale, and M nfs ( M nfa ) stands for the amplitude of the non-factorizableemission (annihilation) diagrams induced by the tree operators O , . The underlying reason isthat, as presented in Eq. (11), the asymmetric leading twist distribution amplitude φ a ( x ) turnsthe originally destructive interferences induced by the symmetric one φ AP ( x ) between the twonon-factorizable emission diagrams, namely, Fig. 1(c) and 1(d), in the B → P P decays into thepresently constructive ones in the B → a a modes. Meanwhile, the analogous phenomenon alsooccurs in the annihilation topologies. Note that the values of M nfa are usually a bit smaller thanthose of M nfs in modulus, because the former is always power /m B suppressed with m B beingthe B meson mass. It is interesting to note that the QCD behavior in light scalar a is greatlydifferent from that in the pseudoscalar pion, which can be seen apparently that the leading twist a (pion) distribution amplitude is governed by the odd(even) Gegenbauer polynomials [10, 29, 30].Therefore, large non-factorizable contributions are observed in the B → a a decays.In view of the surprisingly large Br ( B → a a ) and the amazingly small Br ( B → π π ) in the pQCD approach at LO level, respectively, we here present the numerical decay ampli-tudes (See Tables I and II for detail) arising from every topology to clarify the aforementionedpredictions explicitly. It can be clearly seen that the decay amplitudes in the B → a a decaysexhibit very different pattern from those in the B → ππ ones, although they embrace the samediagrams at the quark level: the former modes determined by the non-factorizable contributionswith a larger scalar decay constant ¯ f a ∼ . GeV, while the latter ones dominated by the factor-
TABLE I. The factorization decay amplitudes(in unit of − GeV ) of the charmless hadronic B → a a decays in the pQCD approach at leading order level, where only the central values are quoted forclarification.Decay modes A fs A nfs A nfa A fa B → a +0 a − . − i .
390 1 . − i . − . − i . − .
044 + i . B + → a +0 a − .
018 + i . − .
268 + i .
926 0 . . B → a a . − i .
284 2 . − i . − . − i . − .
035 + i . The topological amplitudes A fs , A nfs , A nfa , and A fa shown in the Tables I and II stand for the decay amplitudesof factorizable emission, non-factorizable emission, non-factorizable annihilation, and factorizable annihilationdiagrams, respectively. TABLE II. Same as Table I but for the charmless hadronic B → ππ decays.Channels A fs A nfs A nfa A fa B → π + π − − . − i .
957 0 .
095 + i . − .
047 + i .
159 0 .
038 + i . B + → π + π − . − i . − . − i .
082 0 . . B → π π − . − i .
104 0 .
153 + i . − .
033 + i .
113 0 .
029 + i . izable emission contributions with a smaller f π ∼ . GeV, apart from the special B → π π channel. As mentioned above, the underlying reason is that these considered modes include dra-matically different QCD dynamics. Notice that, for the B → a a decays, because of the vanishedvector decay constant f a ∼ , A fs come only from the penguin contributions induced by the ( S + P )( S − P ) operators, which are from the ( V + A )( V − A ) ones by Fierz transformation.However, the phenomenologies shown in B → a a decays indicate that the famous B → ππ puzzle could be resolved if a new QCD mechanism is resorted to enhance the non-factorizablecontributions. Of course, it is nontrivial to resolve the B → ππ puzzle just by including the largenon-factorizable contributions. This point has been clarified in the literatures, for example, seeRefs. [12, 31].Because of the large errors induced by the much less constrained hadronic parameters such asthe scalar decay constant ¯ f a , the Gegenbauer moments B and B in the a distribution ampli-tudes, we derive the ratios of the branching ratios, in which the parameter uncertainties may begreatly canceled and be more helpful for measurements in the relevant experiments, R ≡ Br ( B → a +0 a − ) Br ( B + → a +0 a ) ≈ . +0 . − . ( ω b ) +0 . − . ( ¯ f a ) +0 . − . ( B a i ) +0 . − . (CKM) , (21) R ≡ Br ( B → a +0 a − ) Br ( B → a a ) ≈ . +0 . − . ( ω b ) +0 . − . ( ¯ f a ) +0 . − . ( B a i ) +0 . − . (CKM) , (22) R +0 ≡ Br ( B + → a +0 a ) Br ( B → a a ) ≈ . +0 . − . ( ω b ) +0 . − . ( ¯ f a ) +0 . − . ( B a i ) +0 . − . (CKM) ; (23)It is well known that the B → ππ modes can provide important information to constrain theCKM unitary angle α . As they contain the same quark diagrams as the B → ππ decays, it isgenerally believed that the B → a a processes can also provide complementary constraints onthe angle α . Here, we show the α dependent branching ratios of the B → a a decays in thepQCD approach at the LO level. Based on Eqs. (6)-(8), the decay amplitudes of B → a a decays2can be rewritten as follows, A = V ∗ ub V ud T − V ∗ tb V td P = V ∗ ub V ud T (1 + ze i ( α + δ ) ) , (24)where the weak phase α = arg h − V ∗ tb V td V ∗ ub V ud i , the ratio z = | V ∗ tb V td /V ∗ ub V ud | · | P/T | , and δ is therelative strong phase between tree( T ) and penguin( P ) amplitudes. Correspondingly, the decayamplitudes of the ¯ B → a a decays can be read as, A = V ub V ∗ ud T − V tb V ∗ td P = V ub V ∗ ud T (1 + ze i ( − α + δ ) ) , (25)Therefore, the CP-averaged branching ratio of the B → a a decays shall be the following, Br ( B → a a ) = ( |A| + |A| ) / | V ∗ ub V ud T | (1 + 2 z cos α cos δ + z ) . (26)It is thus easy to see that the CP-averaged branching ratio is a function of cos α for the given ratio z and the strong phase δ , which can be perturbatively calculated in the pQCD approach. This givesa potential method to determine the CKM angle α by measuring the CP-averaged branching ratioswith precision. The dependence on the CKM weak phase α of the CP-averaged branching ratiosfor B → a +0 a − (Solid line), B + → a +0 a (Dashed line), and B → a a (Dash-dotted line) decays,respectively, are presented in Fig. 2, where the central values of the predictions in the pQCDapproach are simply quoted for clarification. Then we can directly observe that the central decayrates for the B → a a decays in the pQCD approach at LO level correspond to the value around ◦ of the CKM angle α , which agrees well with the constraints from various experiments [1]. B r ( a + a - )[ - ] (degree) B r ( a + a )[ - ] (degree) B r ( a a )[ - ] (degree) FIG. 2. (Color online) Dependence on the CKM angle α of the B → a +0 a − (Solid line), a +0 a (Dashed line),and a a (Dash-dotted line) decay rates at leading order in the pQCD approach, respectively. Now we turn to the evaluations of the CP-violating asymmetries of B → a a decays in thepQCD approach. For B + → a +0 a decay, the direct CP-violating asymmetry A CP can be defined3as: A dirCP = |A f | − |A f | |A f | + |A f | , (27)Using Eq. (27), it is easy to calculate the direct CP-violating asymmetry for the considered B + → a +0 a mode as listed in Eq. (28), A dirCP ( B + → a +0 a ) = − . +0 . − . ( ω b ) +0 . − . ( B a i ) +0 . − . (CKM)% , (28)This tiny direct CP-violating asymmetry would be hard to be measured because of the extremelysmall penguin contributions in magnitude, although the large strong phase can be obtained dueto the constructive interferences between the two non-factorizable emission diagrams with theasymmetric a leading twist distribution amplitude, which is very different from that in the B + → π + π mode with the small non-factorizable emission contributions, relative to the purely realamplitudes from the factorizable emission diagrams in the pQCD approach at LO level.As to the CP-violating asymmetries for the neutral decays B → a a , the effects of B − B mixing should be considered. The CP-violating asymmetries of B ( B ) → a a − and a a decays are time dependent and can be defined as A CP ≡ Γ (cid:16) B (∆ t ) → f CP (cid:17) − Γ ( B (∆ t ) → f CP )Γ (cid:16) B (∆ t ) → f CP (cid:17) + Γ ( B (∆ t ) → f CP )= A dirCP cos(∆ m ∆ t ) + A mixCP sin(∆ m ∆ t ) , (29)where ∆ m is the mass difference between the two B d mass eigenstates, ∆ t = t CP − t tag isthe time difference between the tagged B ( B ) and the accompanying B ( B ) with opposite b flavor decaying to the final CP-eigenstate f CP at the time t CP . The direct- and mixing-inducedCP-violating asymmetries A dirCP and A mixCP can be written as A dirCP = | λ CP | −
11 + | λ CP | , A mixCP = 2Im( λ CP )1 + | λ CP | , (30)with the CP-violating parameter λ CP λ CP ≡ η f V ∗ tb V td V tb V ∗ td · h f CP | H eff | B ih f CP | H eff | B i . (31)where η f is the CP-eigenvalue of the final states. Then the direct- and mixing-induced CP-violatingasymmetries for the B → a +0 a − and a a decays in the pQCD approach at LO level can becalculated as, A dirCP ( B → a +0 a − ) = 31 . +3 . − . ( ω b ) +10 . − . ( B a i ) +1 . − . (CKM)% , (32) A mixCP ( B → a +0 a − ) = 0 . +9 . − . ( ω b ) +7 . − . ( B a i ) +9 . − . (CKM)% , (33)4 A dirCP ( B → a a ) = 16 . +1 . − . ( ω b ) +5 . − . ( B a i ) +0 . − . (CKM)% , (34) A mixCP ( B → a a ) = 4 . +4 . − . ( ω b ) +4 . − . ( B a i ) +9 . − . (CKM)% , (35)where we have neglected the vanishing theoretical errors for the CP-violations in B → a a de-cays arising from the scalar decay constant ¯ f a of a meson. It is interesting to see that thesetwo channels, namely, B → a +0 a − and B → a a , have large branching ratios and large directCP asymmetries simultaneously, which could be easier to be measured at the running LHC ex-periments and the forthcoming Super-B/Belle-II factory, and have the potential to reveal the QCDdynamics and the inner structure involved in the light scalar a meson.Similarly, based on Eqs. (24), (25), and (28), the direct CP-violating asymmetry can also beexpressed as the function of the CKM angle α , A dirCP = 2 z sin α sin δ z cos α cos δ + z . (36)Then the precise measurements on these large direct CP violations can also provide the constraintson the CKM angle α potentially. The variation of the direct CP-violating asymmetries with theCKM angle α for the B → a +0 a − (Solid line) and a a (Dashed line) decays is shown in Fig. 3.Again, the central value about ◦ of the CKM angle α can be utilized to produce the abovementioned large direct CP violations. A d i r C P ( a + a - ) (degree) A d i r C P ( a a ) (degree) FIG. 3. (Color online) Dependence on the CKM angle α of the B → a +0 a − (Solid line) and a a (Dashedline) direct CP violations at leading order in the pQCD approach, respectively. IV. SUMMARY
In summary, we studied the two-body charmless hadronic B → a a decays, which have thesame Feynman diagrams as the B → ππ modes at the quark level, by employing the pQCD factor-5ization approach based on the k T factorization theorem. Based on the assumption of two-quark( q ¯ q )structure of the light scalar a state, we make theoretical predictions on the CP-averaged branch-ing ratios and the CP-violating asymmetries of the considered B → a a channels in the SM. Dueto the large non-factorizable contributions induced by the asymmetric leading twist distributionamplitude of a meson, large branching ratios in the order of − ∼ − have been predictedin the pQCD approach at LO level. At the same time, large direct CP violations around and in the B → a a and a +0 a − decays have also been observed. It is therefore expected thatthe large branching ratios plus the large CP asymmetries would be easier to be measured at therunning LHC experiments and the forthcoming Super-B/Belle-II factory, if a is indeed the q ¯ q bound state. Furthermore, the large non-factorizable contributions in the B → a a decays canhint some important information on resolving the famous B → ππ puzzle, although this is non-trivial work as clarified in the literatures [12, 31]. The detection of these considered decays mightbe helpful to investigate the QCD dynamics in the channels and to explore the inner structure ofthe light scalar a state. The investigation of the B → a a decays could also provide more com-plementary constraints on the CKM weak phase α , since the same components as the B → ππ modes exist in the considered B → a a ones at the quark level. Frankly speaking, the predic-tions in the present work suffered from large uncertainties induced by the much less constrainedhadronic parameters such as the Gegenbauer moments B a and B a , which need further studies inthe non-perturbative QCD(such as QCD sum rule and/or Lattice QCD) calculations and the rele-vant experimental measurements(e.g., at BESIII, LHC, Super-B/Belle-II, etc.) on the productionsand/or decays involving the a state. ACKNOWLEDGMENTS
This work is supported by the National Natural Science Foundation of China under GrantsNo. 11205072, No. 11235005, and No. 11047014, and by a project funded by the Priority Aca-demic Program Development of Jiangsu Higher Education Institutions (PAPD). [1] K. A. Olive et al. [Particle Data Group Collaboration], Chin. Phys. C , 090001 (2014).[2] Y. Amhis et al. [3] T. Gershon and A. Soni, J. Phys. G , 479 (2007).[4] M. Bona et al. [SuperB Collaboration], arXiv:0709.0451 [hep-ex]; T. Aushev et al. [Belle-II Collabo-ration], arXiv:1002.5012 [hep-ex].[5] Y. Y. Keum, H.-n. Li, and A. I. Sanda, Phys. Lett. B , 6 (2001); Phys. Rev. D , 054008 (2001).[6] C. D. L ¨u, K. Ukai, and M. Z. Yang, Phys. Rev. D , 074009 (2001).[7] H.-n. Li, Prog. Part. Nucl. Phys. , 85 (2003).[8] G. Buchalla, A.J. Buras, and M.E. Lautenbacher, Rev. Mod. Phys. , 1125 (1996).[9] X. Liu, Z. J. Xiao and Z. T. Zou, J. Phys. G , 025002 (2013).[10] H. Y. Cheng, C. K. Chua, and K. C. Yang, Phys. Rev. D , 014017 (2006); ibid. , 014034 (2008).[11] H. Y. Cheng and J. G. Smith, Ann. Rev. Nucl. Part. Sci. , 215 (2009).[12] X. Liu, H.-n. Li and Z. J. Xiao, Phys. Rev. D , 114019 (2015).[13] M. Bauer, B. Stech and M. Wirbel, Z. Phys. C , 103 (1987); M. Wirbel, B. Stech and M. Bauer, Z.Phys. C , 637 (1985).[14] M. Diehl and G. Hiller, J. High Energy Phys. , 067 (2001); S. Laplace and V. Shelkov, Eur. Phys.J. C , 431 (2001).[15] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. , 1914 (1999); Nucl.Phys. B , 313 (2000).[16] D. s. Du, D. s. Yang and G. h. Zhu, Phys. Lett. B , 46 (2000).[17] H.-n. Li, Phys. Rev. D , 105 (1997); Phys. Lett. B , 347 (1997).[18] C. W. Bauer, D. Pirjol, I. Z. Rothstein and I. W. Stewart, Phys. Rev. D , 054015 (2004).[19] C.M. Arnesen, Z. Ligeti, I.Z. Rothstein, and I.W. Stewart, Phys. Rev. D , 054006 (2008).[20] J. Chay, H.-n. Li, and S. Mishima, Phys. Rev. D , 034037 (2008).[21] C.D. L ¨u and K. Ukai, Eur. Phys. J. C , 305 (2003).[22] Y. Li, C.D. L ¨u, Z.J. Xiao, and X.Q. Yu, Phys. Rev. D , 034009 (2004).[23] A. Ali, G. Kramer, Y. Li, C.D. L ¨u, Y.L. Shen, W. Wang, and Y.M. Wang, Phys. Rev. D , 074018(2007).[24] Z.J. Xiao, W.F. Wang, and Y.Y. Fan, Phys. Rev. D , 094003 (2012); Y.L. Zhang, X.Y. Liu, Y.Y. Fan,S. Cheng, and Z.J. Xiao, Phys. Rev. D , 014029 (2014).[25] H.-n. Li, S. Mishima, and A. I. Sanda, Phys. Rev. D , 114005 (2005).[26] C. D. L ¨u and M. Z. Yang, Eur. Phys. J. C , 515 (2003).[27] R. H. Li, C. D. L ¨u, W. Wang and X. X. Wang, Phys. Rev. D , 014013 (2009). [28] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D , 111102 (2004); ibid. , 111102 (2007);S. Uehara et al. [Belle Collaboration], Phys. Rev. D , 032001 (2009); R. Aaij et al. [LHCb Collab-oration], Phys. Rev. D , 072005 (2013).[29] V. L. Chernyak and A. R. Zhitnitsky, Phys. Rept. , 173 (1984); A. R. Zhitnitsky, I. R. Zhitnitskyand V. L. Chernyak, Sov. J. Nucl. Phys. , 284 (1985) [Yad. Fiz. , 445 (1985)]; V. M. Braunand I. E. Filyanov, Z. Phys. C , 157 (1989) [Sov. J. Nucl. Phys. , 511 (1989)] [Yad. Fiz. , 818(1989)]; V. M. Braun and I. E. Filyanov, Z. Phys. C , 239 (1990) [Sov. J. Nucl. Phys. , 126 (1990)][Yad. Fiz. , 199 (1990)].[30] P. Ball, J. High Energy Phys. , 005 (1998); P. Ball, J. High Energy Phys. , 010 (1999).[31] H.-n. Li and S. Mishima, Phys. Rev. D , 034023 (2011); ibid.90