A Decay amplitudes of B\to SP,SV

# Revisiting charmless hadronic B decays to scalar mesons

March, 2013

Revisiting charmless hadronic decays to scalar mesons

Hai-Yang Cheng, Chun-Khiang Chua, Kwei-Chou Yang, Zhi-Qing Zhang

Taipei, Taiwan 115, Republic of China

Department of Physics, Chung Yuan Christian University

Chung-Li, Taiwan 320, Republic of China

Department of Physics, Henan University of Technology

Zhengzhou, Henan 450052, P.R. China

Abstract

Hadronic charmless decays to scalar mesons are studied within the framework of QCD factorization (QCDF). Considering two different scenarios for scalar mesons above 1 GeV, we find that the data favor the scenario in which the scalars and are the lowest lying bound states. This in turn implies a preferred four-quark nature for light scalars below 1 GeV. Assuming being a lowest lying state, we show that the data of and can be accommodated in QCDF without introducing power corrections induced from penguin annihilation, while the predicted and are too small compared to experiment. In principle, the data of modes can be explained if penguin-annihilation induced power corrections are taken into account. However, this will destroy the agreement between theory and experiment for . Contrary to the pseudoscalar meson sector where has the largest rate in 2-body decays of the meson, we show that . The decay is found to have a rate much smaller than that of in QCDF, while it is the other way around in pQCD. Experimental measurements of these two modes will help discriminate between these two different approaches. Assuming 2-quark bound states for and , the observed large rates of and modes can be explained in QCDF with the mixing angle in the vicinity of . However, this does not necessarily imply that a 4-quark assignment for is ruled out because of extra diagrams contributing to . Irrespective of the mixing angle , the predicted branching fraction of is far below the Belle measurement and this needs to be clarified in the future.

## I Introduction

In the past few years there are some progresses in the study of charmless hadronic decays with scalar mesons in the final state both experimentally and theoretically. On the experimental side, measurements of decays to the scalar mesons such as , and have been reported by BaBar and Belle; see Tables 1 and 2 for a summary of the experimental results. It is well known that the identification of scalar mesons is difficult experimentally and the underlying structure of scalar mesons is not well established theoretically. The experimental measurements of and , where stand for scalar, vector and pseudoscalar mesons, respectively, will provide valuable information on the nature of the even-parity mesons. On the theoretical side, hadronic decays to scalar mesons have been studied in the QCD-inspired approaches: QCD factorization (QCDF) (6); (7); (8); (9); (10) and pQCD (11); (12); (13); (14); (15); (16); (17); (18); (19); (20).

In this work, we would like to revisit the study of the 2-body charmless decays and within the framework of QCDF for the following reasons: (i) In (6) we have missed some factorizable terms (more precisely, the and emission terms) in the expressions for the decay amplitudes of . (ii) Attention has not been paid to the relative sign difference of the vector decay constants between and and between and or and in our previous study. (iii) There were some errors in our previous computer code which may significantly affect some of the calculations done before. (iv) Progress has been made in the past in the study of transition form factors in various approaches (21); (22); (23); (24); (25); (26); (14). (v) Experimental data for some of decays such as and are now available. (vi) It is known that in order to account for the penguin-dominated decay modes within the framework of QCDF, it is necessary to include power corrections due to penguin annihilation (27); (28). In the present work, we wish to examine if the same effect holds in the scalar meson sector; that is, if the penguin-annihilation induced power corrections are also needed to explain the penguin dominated and decays.

This paper is organized as follows. We specify in Sec. 2 various input parameters for scalar mesons, such as decay constants, form factors and light-cone distribution amplitudes. The relevant decay amplitudes are briefly discussed in Sec. 3. Results and detailed discussions are presented in Sec. 4. Conclusions are given in Sec. 5. We lay out the explicit decay amplitudes of in Appendix A.

## Ii Physical properties of scalar mesons

In order to study the hadronic charmless decays containing a scalar meson in the final state, it is necessary to specify the quark content of the scalar meson. For scalar mesons above 1 GeV we have explored in (6) two possible scenarios in the QCD sum rule method, depending on whether the light scalars and are treated as the lowest lying states or four-quark particles: (i) In scenario 1, we treat as the lowest lying states, and as the corresponding first excited states, respectively, and (ii) we assume in scenario 2 that are the lowest lying resonances and the corresponding first excited states lie between  GeV. Scenario 2 corresponds to the case that light scalar mesons are four-quark bound states, while all scalar mesons are made of two quarks in scenario 1. Phenomenological studies in (6); (7) imply that scenario 2 is preferable, which will be also reinforced in this work. Indeed, lattice calculations have confirmed that and are lowest-lying -wave mesons (29), and indicated that (or ) and (or ) are -wave tetraquark mesonia (30). 9

### ii.1 Decay constants and form factors

Decay constants of scalar, pseudoscalar and vector mesons are defined as

 ⟨S(p)|¯q2γμq1|0⟩=fSpμ,⟨S|¯q2q1|0⟩=mS¯fS, ⟨P(p)|¯q2γμγ5q1|0⟩=−ifPpμ,⟨V(p)|¯q2γμq1|0⟩=fVmVε∗μ, ⟨V(p,ε∗)|¯q2σμνq1|0⟩=f⊥V(pμε∗ν−pνε∗μ). (1)

For scalar mesons, the vector decay constant and the scale-dependent scalar decay constant are related by equations of motion

 μSfS=¯fS,with  μS=mSm2(μ)−m1(μ), (2)

where and are the running current quark masses and is the scalar meson mass. For the neutral scalar mesons , and , vanishes owing to charge conjugation invariance or conservation of vector current, but the quantity remains finite. It is straightforward to show from Eq. (II.1) that the decay constants of the scalar meson and its antiparticle are related by

 ¯f¯S=¯fS,f¯S=−fS. (3)

Indeed, from Eq. (2) we have, for example,

 fa−0(μ)=¯fa0md(μ)−mu(μ)ma0,fa+0(μ)=¯fa0mu(μ)−md(μ)ma0. (4)

Therefore, the vector decay constants of and are of opposite sign.

In (6) we have applied the QCD sum rule method to estimate the decay constant for various scalar mesons as summarized in Table 3. Note that a recent sum rule calculation in (25) yields a smaller in scenario 2 for and . In this work we shall use the values of and taken from (33). For the decay constants and of the and mesons defined by

 ⟨0|¯qγμγ5q|η(′)⟩=i1√2fqη(′)pμ,⟨0|¯sγμγ5s|η(′)⟩=ifsη(′)pμ, (5)

we shall follow the results of (34).

For the and transition form factors defined in the conventional way (35), we will use the results obtained using the QCD sum rule method (36). Form factors for transitions are defined by (32)

 ⟨S(p′)|Aμ|B(p)⟩ = −i[(Pμ−m2B−m2Sq2qμ)FBS1(q2)+m2B−m2Sq2qμFBS0(q2)], (6)

where and . The momentum dependence of the form factor is usually parameterized in a 3-parameter form

 F(q2)=F(0)1−a(q2/m2B)+b(q4/m4B). (7)

The parameters , and for transitions are summarized in Table 4 obtained using the covariant light-front quark model (32). Form factors are also available in other approaches, such as light-cone sum rule (21); (22); (23); (24); (25) and pQCD (26); (14). In general, form factors calculated by sum rule and pQCD methods are larger than that obtained using the quark model. For example, is of order 0.26 in the covariant light-front quark model (32), while it is found to be 0.45 (25), 0.49 (22); (24) in the sum rule method and 0.60 (26) and 0.76 (14) in pQCD (all evaluated in scenario 2). We will come to this point later.

### ii.2 Distribution amplitudes

In general, the twist-2 light-cone distribution amplitude (LCDA) of the scalar meson has the form

 ΦS(x,μ)=fS6x(1−x)[1+μS∞∑m=1Bm(μ)C3/2m(2x−1)], (8)

where are Gegenbauer moments and are Gegenbauer polynomials. The general twist-3 LCDAs are given by

 ΦsS(x) = ¯fS[1+∞∑m=1am(μ)C1/2m(2u−1)], ΦσS(x) = ¯fS6x(1−x)[1+∞∑m=1bm(μ)C1/2m(2u−1)]. (9)

Since and even Gegenbauer coefficients are suppressed, it is clear that the twist-2 LCDA of the scalar meson is dominated by the odd Gegenabuer moments. In contrast, the odd Gegenbauer moments vanish for the and mesons. The Gegenbauer moments and in scenarios 1 and 2 obtained using the QCD sum rule method (6) are listed in Table 3. The Gegenbauer moments and for twist-3 LCDAs have been computed in (37); (25).

Since the decay constants vanish for the neutral scalar mesons and , it follows from Eq. (8) that

 ΦS(x,μ)=¯fS6x(1−x)∞∑m=1Bm(μ)C3/2m(2x−1) (10)

for these neutral scalar mesons.

As stressed in (6), it is most suitable to define the LCDAs of scalar mesons including decay constants. However, it is more convenient in practical calculations to factor out the decay constants in the LCDAs and put them back in the appropriate places. In the ensuing discussions, we will use the LCDAs with the decay constants being factored out.

### ii.3 Mixing angle between f0(980) and f0(500) and between η and η′

In the naive 2-quark model with ideal mixing for and , is purely an state, while is a state with . However, there also exist some experimental evidences indicating that is not purely an state. For example, the observation of (38) clearly shows the existence of the non-strange and strange quark content in . Therefore, isoscalars and must have a mixing

 (|f0(980)⟩|f0(500)⟩)=(cosθsinθ−sinθcosθ)(|n¯n⟩|s¯s⟩). (11)

Various mixing angle measurements have been discussed in the literature and summarized in (6); (39). A recent measurement of the upper limit on the branching fraction product by LHCb leads to (40).

For the and mesons, it is more convenient to consider the flavor states , and labeled by the , and , respectively. Neglecting the small mixing with , we write

 (|η⟩|η′⟩)=(cosϕ−sinϕsinϕcosϕ)(|ηq⟩|ηs⟩), (12)

where (41) is the mixing angle in the and flavor basis.

## Iii Decay amplitudes in QCD factorization

We shall use the QCD factorization approach (27); (28) to study the short-distance contributions to the decays with . In QCD factorization, the factorizable amplitudes of above-mentioned decays can be found in (6) and (7). However, the expressions for the decay amplitudes of involving a neutral or given in (6) are corrected in Appendix A as some factorizable contributions were missed before. The effective parameters with appearing in Eq. (A) can be calculated in the QCD factorization approach (27). In general, they have the expressions

 api(M1M2)=(ci+ci±1Nc)Ni(M2)+ci±1NcCFαs4π[Vi(M2)+4π2NcHi(M1M2)]+Ppi(M2), (13)

where , the upper (lower) signs apply when is odd (even), are the Wilson coefficients, with , is the emitted meson and shares the same spectator quark with the meson. The quantities account for vertex corrections, for hard spectator interactions with a hard gluon exchange between the emitted meson and the spectator quark of the meson and for penguin contractions. The expression of the quantities reads

 Ni(M2)={0,i=6,8~{}and~{}M2=V,1,else. (14)

The explicit expressions of , , and weak annihilation contributions described by the terms and are given in (6) and (7) for and , respectively. 10

Power corrections in QCDF always involve troublesome endpoint divergences. We shall follow (27) to model the endpoint divergence in the annihilation and hard spectator scattering diagrams as

 XA=ln(mBΛh)(1+ρAeiϕA),XH=ln(mBΛh)(1+ρHeiϕH), (15)

with being a typical scale of order 500 MeV, and