Anatomy of B_{s}\to VV decays and effects of next-to-leading order contributions in the perturbative QCD factorization approach

# Anatomy of Bs→VV decays and effects of next-to-leading order contributions in the perturbative QCD factorization approach

Da-Cheng Yan    Xin Liu    Zhen-Jun Xiao Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, Jiangsu 210023, China School of Physics and Electronic Engineering, Jiangsu Normal University, Xuzhou 221116, China Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing, Jiangsu 210023, China
July 25, 2019
###### Abstract

By employing the perturbative QCD (PQCD) factorization approach, we calculated the branching ratios, CP-violating asymmetries, the longitudinal and transverse polarization fractions and other physical observables of the thirteen charmless hadronic decays with the inclusion of all currently known next-to-leading order (NLO) contributions. We focused on the examination of the effects of all those currently known NLO contributions and found that: (a) for the measured decays and , the NLO contributions can provide to enhancements to the leading order (LO) PQCD predictions of their CP-averaged branching ratios, and consequently the agreement between the PQCD predictions and the measured values are improved effectively after the inclusion of the NLO contributions; (b) for the measured decays, the NLO corrections to the LO PQCD predictions for and are generally small in size, but the weak penguin annihilation contributions play an important role in understanding the data about their decay rates, and ; (c) the NLO PQCD predictions for above mentioned physical observables do agree with the measured ones and the theoretical predictions from the QCDF, SCET and FAT approaches; (d) for other considered decays, the NLO PQCD predictions for their decay rates and other physical observables are also basically consistent with the theoretical predictions from other popular approaches, future precision measurements could help us to test or examine these predictions.

###### pacs:
13.25.Hw, 12.38.Bx, 14.40.Nd

Key Words: meson decays; The PQCD factorization approach; Branching ratios; Polarization fractions; Relative phases

## I Introduction

During the past three decades, the two-body charmless hadronic decays, with being the light vector mesons and , have been studied by many authors based on rather different factorization approaches qcdf2 (); qcdf07 (); qcdf09 (); qcdfwa (); qcdfll (); gf99 (); ali07 (); jpg06 (); pqcd2 (); scet (); scetv (). Several such decay modes, such as decay, have been observed by CDF and LHCb experiments cdf (); lhcbks (); lhcbks1 (); lhcbphi (); lhcbphi2 (); lhcbphiks (); lhcbrhophi (); lhcb0 (); pdg2016 (); hfag2016 (). When compared with the similar (here , and ) decays, are indeed much more complicated due to the fact that more helicity amplitudes should be taken into account. The decays can offer, consequently, rich opportunities for us to test the Stand Model (SM) and to search for the exotic new physics beyond the SM.

Experimentally, a large transverse polarization fraction of was firstly observed in 2003 by BABAR and Belle Collaborations babe (). The new world averages of as given by HFAG-2016 hfag2016 () for decays, for example, are the following:

 fL(B+→VK∗+) = ⎧⎪⎨⎪⎩0.50±0.05,for  V=ϕ,0.78±0.12,for  V=ρ0,0.41±0.19,for  V=ω,, (1) fL(B0→VK∗0) = ⎧⎪⎨⎪⎩0.497±0.017,for  V=ϕ,0.40±0.14,for  V=ρ0,0.70±0.13,for  V=ω,, (2)

These measured values were in strong confliction with the general expectation in the naive factorization ansatz naive1 (), which is the so-called “polarization puzzle” 2004pft (); 2004pa (); lipa (); lipa1 (). The similar deviations also be observed later for and decays Aubert:2008bc (); hfag2016 ().

For the charmless decays studied in this paper, the similar puzzles have also been observed by CDF and LHCb Collaboration for , and decay modes cdf (); lhcbks (); lhcbks1 (); lhcbphi (); lhcbphi2 (); lhcbrhophi (). The new world averages of and as given by HFAG-2016 hfag2016 () for these three decay modes are the following:

 ¯B0s→ϕϕ : fL=0.361±0.022,f⊥=0.306±0.023, (3) ¯B0s→K∗ϕ : fL=0.51±0.17,f⊥=0.28±0.12, (4) ¯B0s→K∗0¯¯¯¯¯K∗0 : fL=0.201±0.070,f⊥=0.38±0.11. (5)

More measurements are expected in the near future.

Theoretically, a number of strategies were proposed to resolve the above mentioned ”polarization puzzle” within and/or beyond the SM. For example, the weak penguin annihilation contributions in QCD factorization (QCDF) approach was proposed by Kagan Kann (), the final state interactions were considered in Refs. final (); fsi (); 2004pft (), the form-factor tuning in the perturbative QCD (PQCD) approach was suggested by Li lipa (), and even the exotic new physics effects have been studied by authors in Refs. np (); np1 (). Obviously, it is hard to get a good answer to this seemingly long-standing puzzle at present. However, according to the statement in Ref. lipa1 (), the complicated QCD dynamics involved in such decays should be fully explored before resorting to the possible new physics beyond the SM. Therefore, the QCDF approach  qcdf2 (); qcdf07 (); qcdf09 (); qcdfwa (); qcdfll (); qcdfv (), the soft-collinear effective theory (SCET)  scet (); scetv () and the PQCD approach  ali07 (); Li2002 (); jpg06 (); pqcd2 (); pqcdv (); pqcdv1 (), have been adopted to investigate these kinds of decays systematically.

The two-body charmless hadronic decays have been systematically studied in the PQCD approach at leading order (LO) in 2007 ali07 (). Recently, the authoers of Ref. pqcd2 () made improved estimations for the modes by keeping the terms with the higher power of the ratios in the PQCD approach, with and being the masses of the initial and final states. However, there still existed some issues to be clarified, e.g. the measured large decay rates for and decays, the latest measurement of a smaller for decay, etc.

Therefore, we would like to revisit those two-body charmless decays by taking into account all currently known next-to-leading order (NLO) contributions in the PQCD factorization approach. We will focus on the effects of the NLO contributions arising from various possible sources, such as the QCD vertex corrections (VC), the quark loops (QL), and the chromomagnetic penguins Li05 (); nlo05 () in the SM. As can be seen from Refs. fan2013 (); xiao08b (); nlo05 (); Li05 (); xiao2014 (), the NLO contributions do play an important role in understanding the known anomalies of physics such as the amazingly large decay rates fan2013 (); xiao08b (), the longitudinal-polarization dominated  Li05 () and the evidently nonzero , i.e., the famous “-puzzle” nlo05 (); xiao2014 (), and so forth. Very recently, we extend these calculations to the cases such as decays xiao14a (), decays xiao14b () and decays xiao17 (). We found that the currently known NLO contributions can interfere with the LO part constructively or destructively for those considered meson decay modes. Consequently, the agreement between the PQCD predictions and the experimental measurements of the CP-averaged branching ratios, the polarization fractions and CP-violating asymmetries was indeed improved effectively due to the inclusion of the NLO contributions.

This paper is organized as follows. In Sec. II, we shall present various decay amplitudes for the considered decay modes in the PQCD approach at the LO and NLO level. We show the PQCD predictions and several phenomenological analyses for the branching ratios, CP-violating asymmetries and polarization observables of thirteen decays in Sec III. A short summary is given in Sec. IV.

## Ii Decay amplitudes at LO and NLO level

We treat the meson as a heavy-light system and consider it at rest for simplicity. By employing the light-cone coordinates, we define the meson with momentum , the emitted meson with the momentum along the direction of , and the recoiled meson with the momentum in the direction of (Here, and are the light-like dimensionless vectors), respectively, as the following,

 P1 = mBs√2(1,1,0T),P2=MBs√2(1−r23,r22,0T),P3=MBs√2(r23,1−r22,0T), (6)

The polarization vectors of the final states can then be parametrized as:

 ϵL2 = 1√2r2(1−r23,−r22,0T),ϵL3=1√2r3(−r23,1−r22,0T), ϵT2 = (0,0,1T),ϵT3=(0,0,1T). (7)

with being the longitudinal(transverse) polarization vector.

The momenta carried by the light anti-quark in the initial and final mesons are chosen as follows:

 k1 = (x1,0,k1T),k2=(x2(1−r23),x2r22,k2T),k3=(x3r23,x3(1−r22),k3T), (8)

The integration over and will lead conceptually to the decay amplitudes in the PQCD approach,

 A(B0s→V2V3) ∼ ∫dx1dx2dx3b1db1b2db2b3db3 (9) ×Tr[C(t)ΦBs(x1,b1)ΦV2(x2,b2)ΦV3(x3,b3)H(xi,bi,t)St(xi)e−S(t)],

in which, is the conjugate space coordinate of transverse momentum , stands for the Wilson coefficients evaluated at the scale , and denotes the hadron wave functions, which are nonperturbative but universal inputs, of the initial and final states. The kernel describes the hard dynamics associated with the effective ”six-quark interaction” exchanged by a hard gluon. The Sudakov factors and together suppress the soft dynamics in the endpoint region effectively li2003 ().

### ii.1 Wave functions and decay amplitudes

Without the endpoint singularities in the evaluations, the hadron wave functions are the only input in the PQCD approach. These nonperturbative quantities are process independent and could be obtained with the techniques of QCD sum rule and/or Lattice QCD, or be fitted to the measurements with good precision.

For meson, its wave function could be adopted with the Lorentz structure ali07 (); pqcd2 ()

 ΦBs = 1√6(P/Bs+mBs)γ5ϕBs(k), (10)

in which the distribution amplitude is modeled as

 ϕBs(x,b) = NBsx2(1−x)2exp[−m2Bs x22ω2Bs−12(ωBsb)2], (11)

with being the shape parameter. We take GeV for the meson based on the studies of lattice QCD and light-cone sum rule wbs1 (); wbs2 (); wbs3 (). The normalization factor will be determined through the normalization condition: with the decay constant GeV.

For the vector meson, the longitudinally and transversely polarized wave functions up to twist-3 are given by wbs3 (); pball98 ()

 ΦLV=1√6[mV\makebox[0.0pt][l]/ϵLϕV(x)+\makebox[0.0pt][l]/ϵL\makebox[−1.5pt][l]/PϕtV(x)+mVϕsV(x)] Φ⊥V=1√6[mV\makebox[0.0pt][l]/ϵTϕvV(x)+\makebox[0.0pt][l]/ϵT\makebox[−1.5pt][l]/PϕTV(x)+mViϵμνρσγ5γμϵνTnρvσϕaV(x)], (12)

where and are the momentum and the mass of the light vector mesons, and is the corresponding longitudinal(transverse) polarization vector Li2002 (). Here is Levi-Civita tensor with the convention .

The twist-2 distribution amplitudes and can be written in the following form wbs3 (); pball98 ()

 ϕV(x)=3fV√6x(1−x)[1+a∥1VC3/21(t)+a∥2VC3/22(t)], (13) ϕTV(x)=3fTV√6x(1−x)[1+a⊥1VC3/21(t)+a⊥2VC3/22(t)], (14)

where , is the decay constants of the vector meson with longitudinal(transverse) polarization. The Gegenbauer moments here are the same as those in Refs. wbs3 (); pball07 (); pball98 ():

 a∥(⊥)1ρ=a∥(⊥)1ω=a∥(⊥)1ϕ=0,a∥(⊥)1K∗=0.03±0.02(0.04±0.03) , a∥(⊥)2ρ=a∥(⊥)2ω=0.15±0.07(0.14±0.06)a∥(⊥)2ϕ=0(0.20±0.07) , a∥(⊥)2K∗=0.11±0.09(0.10±0.08) . (15)

For the twist-3 distribution amplitudes , for simplicity, we adopt the asymptotic forms ali07 (); pqcd2 ()

 ϕtV(x)=3fTV2√6t2,ϕsV(x)=3fTV2√6(−t), ϕvV(x)=3fV8√6(1+t2),ϕaV(x)=3fV4√6(−t). (16)

The above choices of vector-meson distribution amplitudes can essentially explain the polarization fractions of the measured , and decayslipa (); lipa1 (); pqcdv (), together with the right branching ratios.

### ii.2 Example of the LO decay amplitudes

In the SM, for the considered decays induced by the transition with , the weak effective Hamiltonian can be written asburas96 (),

 Heff = GF√2{VubV∗uq[C1(μ)Ou1(μ)+C2(μ)Ou2(μ)]−VtbV∗tq[10∑i=3Ci(μ)Oi(μ)]}+% h.c. (17)

where the Fermi constant GeV, and is the Cabbibo-Kobayashi-Maskawa(CKM) matrix element, are the Wilson coefficients and are the local four-quark operators. For convenience, the combinations of the Wilson coefficients are defined as usual ali07 (); pqcd2 ():

 a1=C2+C1/3,a2=C1+C2/3, ai=Ci+Ci±1/3,(i=3−10), (18)

where the upper(lower) sign applies, when is odd(even).

At leading order, as illustrated in Fig. 1, there are eight types of Feynman diagrams contributing to the decays, which can be classified into three types: the factorizable emission diagrams ( Fig. 1(a) and 1(b)); the nonfactorizable emission diagrams (Fig. 1(c) and 1(d)); and the annihilation diagrams (Fig. 1(e)-1(h)). As mentioned in the Introduction, the considered thirteen modes have been studied at LO in the PQCD approach ali07 (); pqcd2 (). The factorization formulas of decay amplitudes with various topologies have been presented explicitly in Ref. ali07 (). Therefore, after the confirmation by our independent recalculations, we shall not collect those analytic expressions here for simplicity. In this work, we aim to examine the effects of all currently known NLO contributions to the considered decay modes in the PQCD approach to see whether one can improve the consistency between the theory and the experiment in the SM or not, which would be help for us to judge the necessity of the exotic new physics beyond the SM.

For decays, both of the longitudinal and transverse polarizations will contribute. Then, the decay amplitudes can be decomposed into three partspqcd2 ():

 A(ϵ2,ϵ3)=iAL+i(ϵT2⋅ϵT3)AN+(ϵμναβnμvνϵTα2ϵTβ3)AT, (19)

where and correspond to the longitudinally, normally and transversely polarized amplitudes, respectively, whose detailed expressions can be inferred from Refs. ali07 (); pqcd2 ().

### ii.3 NLO contributions

In the framework of the PQCD approach, many two-body charmless decays have been investigated by including currently known NLO contributions,for example, in Refs. fan2013 (); nlo05 (); xiao14a (); xiao14b (); xiao17 (); xiao08b (); xiao2014 (); zhang09 (); zhou12 (). Of course, some NLO contributions are still not known at present, as discussed in Ref. fan2013 (). The currently known NLO corrections to the LO PQCD predictions of decays are the following:

• (a) The NLO Wilson coefficients (NLO-WC), the renormalization group running matrix at NLO level and the strong coupling constant at two-loop level as presented in Ref. buras96 ();

• (b) The NLO contributions from the vertex corrections (VC) Li05 (); nlo05 () as illustrated in Figs. 2(a)-2(d);

• (c) The NLO contributions from the quark-loops (QL)  nlo05 (); Li05 () as shown in Figs. 2(e)-2(f);

• (d) The NLO contributions from the chromo-magnetic penguin (MP) operator o8g2003 (); Li05 (); nlo05 () as illustrated in Figs. 2(g)-2(h).

In this paper, we adopt directly the formulas for all currently known NLO contributions from Refs. nlo05 (); Li05 (); o8g2003 (); xiao08b (); fan2013 (); xiao14a (); xiao14b (); xiao2014 () without further discussions about the details. Moreover, some essential comments should be given for those still unknown NLO corrections to the nonfactorizable emission amplitudes and the annihilation amplitudes as follows:

• (a) For the nonfactorizable emission diagrams as shown in Fig. 1, since the hard gluons are emitted from the upper quark line of Fig. 1(c) and the upper anti-quark line of Fig. 1(d) respectively, the contribution from these two figures will be strongly cancelled each other, the remaining contribution is therefore becoming rather small in magnitude. In NLO level, another suppression factor will appear, the resultant NLO contribution from the hard-spectators should become much smaller than the dominant contribution from the ¡±tree¡± emission diagrams (Fig. 1(a) and 1(b)).

• (b) For the annihilation diagrams as shown in Fig. 1(e)-1(h), the corresponding NLO contributions are in fact doubly suppressed by the factors and , and consequently must become much smaller than those dominant LO contribution from Fig. 1(a) and 1(b).

Therefore, it is reasonable for us to expect that those still unknown NLO contributions in the PQCD approach are in fact the higher order corrections to the already small LO pieces, and should be much smaller than the dominant contribution for the considered decays, say less than 5% of the dominant ones.

According to Refs. Li05 (); nlo05 (), the vertex corrections can be absorbed into the redefinition of the Wilson coefficients by adding a vertex-function to them.

 a1,2(μ) → a1,2(μ)+αs(μ)9πC1,2(μ)V1,2(M), ai(μ) → ai(μ)+αs(μ)9πCi+1(μ)Vi(M),for  i=3,5,7,9, aj(μ) → aj(μ)+αs(μ)9πCj−1(μ)Vj(M),for  j=4,6,8,10, (20)

where denotes the vector meson emitted from the weak vertex ( i.e. the in Fig. 2(a)-2(d)). The expressions of the vertex-functions with both longitudinal and transverse components can be found easily in Refs. qcdfpppv (); nlovc ().

The NLO “Quark-Loop” and “Magnetic-Penguin” contributions are in fact a kind of penguin corrections with insertion of the four-quark operators and the chromo-magnetic operator , respectively, as shown in Figs. 2(e,f) and 2(g,h). For the transition, for example, the corresponding effective Hamiltonian and can be written in the following form:

 H(ql)eff = −∑q=u,c,t∑q′GF√2V∗qbVqsαs(μ)2πCq(μ,l2)[¯bγρ(1−γ5)Tas](¯q′γρTaq′), (21) Hmpeff = −GF√2gs8π2mbV∗tbVtsCeff8g¯siσμν(1+γ5)TaijGaμνbj, (22)

where is the invariant mass of the gluon which attaches the quark loops in Figs. 2(e,f), and the functions can be inferred from Refs. Li05 (); nlo05 (); fan2013 (); xiao08b (); xiao2014 (). The in Eq. (22) is an effective Wilson coefficient with the definition of  buras96 ().

With explicit evaluations, we find the following three points:

1. For the pure annihilation decays of , and , they do not receive the NLO contributions from the vertex corrections, the quark-loop and the magnetic-penguin diagrams. The NLO correction to these decay modes comes only from the NLO-WCs and the the strong coupling constant at the two-loop level.

2. For the and channels with only transition and no annihilation diagrams, the ”quark-loop” and ”magnetic-penguin” diagrams cannot contribute to these two decay modes. The related NLO contributions are mainly induced by the vertex corrections to the emitted or mesons.

3. For the remaining seven decay modes, besides the LO decay amplitudes, all of the currently known NLO contributions should be taken into account as follows:

 A(u),iρ0K∗0→A(u),iρ0K∗0+M(u,c),iρ0K∗0,A(t),iρ0K∗0→A(t),iρ0K∗0−M(t),iρ0K∗0−M(g),iρ0K∗0,A(u),iρ−K∗+→A(u),iρ−K∗++M(u,c),iρ−K∗+,A(t),iρ−K∗+→A(t),iρ−K∗+−M(t),iρ−K∗+−M(g),iρ−K∗+,A(u),iωK∗0→A(u),iωK∗0+M(u,c),iωK∗0,A(t),iωK∗0→A(t),iωK∗0−M(t),iωK∗0−M(g),iωK∗0,A(u),iϕK∗0→A(u),iϕK∗0+M(u,c),iϕK∗0,A(t),iϕK∗0→A(t),iϕK∗0−M(t),iϕK∗0−M(g),iϕK∗0,A(u),iK∗−K∗+→A(u),iK∗−K∗++M(u,c),iK∗−K∗+,A(t),iK∗−K∗+→A(t),iK∗−K∗+−M(t),iK∗−K∗+−M(g),iK∗−K∗+,A(u),iK∗0¯K∗0→A(u),iK∗0¯K∗0+M(u,c),iK∗0¯K∗0,A(t),iK∗0¯K∗0→A(t),iK∗0¯K∗0−M(t),iK∗0¯K∗0−M(g),iK∗0¯K∗0,A(u),iϕϕ→A(u),iϕϕ+M(u,c),iϕϕ,A(t),iϕϕ→A(t),iϕϕ−M(t),iϕϕ−M(g),iϕϕ, (23)

where and the terms stand for the LO amplitudes, while and are the NLO ones, which describe the NLO contributions arising from the up-loop, charm-loop, QCD-penguin-loop, and magnetic-penguin diagrams, respectively.

Now, we can calculate the decay amplitudes and in the PQCD approach. As mentioned in Eq. (19), for decays, there are three individual polarization amplitudes . For the longitudinal components, the NLO decay amplitudes and can be written as:

 M(ql),LV2V3 = (24) −2r2ϕs2(x2)ϕ3(x3)+r3(1−2x3)ϕ2(x2)(ϕs3(x3)+ϕt3(x3))−2r2r3ϕs2(x2)((2+x3)ϕs3(x3) −x3ϕt3(x3))]⋅α2s(ta)⋅he(x1,x3,b1,b3)⋅exp[−Sab(ta)]C(q)(ta,l2)+[2r3ϕ2(x2)ϕs3(x3) −4r2r3ϕs2(x2)]ϕs3(x3)⋅α2s(tb)⋅he(x3,x1,b3,b1)⋅exp[−Sab(tb)]C(q)(tb,l′2)},
 M(mp),LV2V3 = 8m6BsC2F√6∫10dx1dx2dx3∫∞0b1db1b2db2b3db3ϕBs(x1,b1) (25) ×{[(−1+x3)[2ϕ3(x3)−r3(x3−1)ϕt3(x3)+r3(x3+3)ϕs3(x3)]ϕ2(x2) +r2x2(3ϕs2(x2)−ϕt2(x2))[(1+x3)ϕ3(x3)−r3(2x3−1)(ϕs3(x3)+ϕt3(x3))] +r2r3(x3−1)(3ϕs2(x2)+ϕt2(x2))(ϕt3(x3)−ϕs3(x3))] ⋅α2s(ta)hg(xi,bi)⋅exp[−Scd(ta)]Ceff8g(ta) −[4r3ϕ2(x2)−2r2r3x2(3ϕs2(x2)−ϕt2(x2))]ϕs3(x3) ⋅α2s(tb)⋅h′g(xi,bi)⋅exp[−Scd(tb)]⋅Ceff8g(tb)}.

with .

The transverse components and of the corresponding decay amplitudes can be written in the form of

 M(ql),NV2V3 = −8m4Bsr2C2F√6∫10dx1dx2dx3∫∞0b1db1b3db3ϕBs(x1,b1){[ϕT3(x3)(ϕa2(x2)+ϕv2(x2)) (26) +r3(x3+2)(ϕa2(x2)ϕa3(x3)+ϕv2(x2)ϕv3(x3))−r3x3(ϕa2(x2)ϕv3(x3)+ϕv2(x2)ϕa3(x3))] ⋅α2s(ta)⋅he(x1,x3,b1,b3)⋅exp[−Sab(ta)]C(q)(ta,l2) +[r2r3(ϕa2(x2)ϕa3(x3)+ϕa2(x2)ϕv3(x3)+ϕv2(x2)ϕa3(x3)+ϕv2(x2)ϕv3(x3))]ϕs3(x3) ⋅α2s(tb)⋅he(x3,x1,b3,b1)⋅exp[−Sab(tb)]C(q)(tb,l′2)},
 M(ql),TV2V3 = −8m4Bsr2C2F√6∫10dx1dx2dx3∫∞0b1db1b3db3ϕBs(x1,b1){[ϕT3(x3)(ϕa2(x2)+ϕv2(x2)) (27) +r3(x3+2)(ϕa2(x2)ϕv3(x3)+ϕv2(x2)ϕa3(x3))−r3x3(ϕa2(x2)ϕa3(x3)+ϕv2(x2)ϕv3(x3))] ⋅α2s(ta)⋅he(x1,x3,