arXiv:hep-ph/0504145v3 25 May 2005
Scalar interactions and the polarizations of
B → φK
∗Chuan-Hung Chen
a∗and Chao-Qiang Geng
b†aDepartment of Physics, National Cheng-Kung University, Tainan, 701 Taiwan bDepartment of Physics, National Tsing-Hua University, Hsin-Chu , 300 Taiwan
(Dated: February 2, 2008)
Abstract
We try to understand the polarization puzzle in B → φK∗ decays with a simple Higgs model associated with flavor changing neutral current at tree level. The new interactions can effectively reduce the longitudinal polarization |A0|2. In particular, we find that if the couplings of b-quark in
different chiralities to Higgs are the same, the transverse polarization |A⊥|2 can receive the largest
contribution and its value can be as large as 30%. On the other hand, with opposite sign in the couplings, the other transverse polarization |Ak|2 is enhanced.
∗ Email: [email protected] † Email: [email protected]
In terms of naive helicity analysis, it is well known that the transverse polarizations of vector bosons are associated with their masses. It was expected that the partitions of vector meson polarizations in B decays should have the same behavior. As a result, the ratio of various polarizations in two-body B meson decays can be estimated to be
|A0|2 : |A⊥|2 : |Ak|2 ∼ 1 : m2 V M2 B : m 2 V M2 B , (1)
where A0 and Ak belong to the mixtures of S and D-wave decay amplitudes while A⊥ the
P-wave one, which satisfy the identity X
λ=0,k,⊥
|Aλ|2 = 1 . (2)
According to Eq. (1), it is believed that in B decays with light vector mesons A⊥(k)are much
smaller than A0. The expectation is confirmed by BELLE [1] and BABAR [2] in B → ρρ
decays, in which the longitudinal parts occupy over 95%. Furthermore, when the final states include heavy vector mesons, transverse polarizations can be relatively large. The conjecture is verified in B → J/ΨK∗ decays [3, 4], in which the longitudinal contribution is only about
60%.
However, the rule in Eq. (1) seems to be broken in B → φK∗ decays. From the recent
measurements of BELLE [5] and BABAR [4, 6], summarized in the Table I, it is quite clear that the longitudinal polarizations of B → K∗φ are only around 50%. To solve the anomalous polarizations, the authors of Refs. [7, 8, 9, 10, 11] have proposed some solutions by introducing proper mechanisms such as large annihilation effect due to (S − P ) ⊗ (S + P ) interactions [7], the enhanced transversality from transverse gluon emitted by b → sg(∗)
[8], final state interactions [9, 10] and new sets of form factors [11]. All above proposals are related to the uncertainities of low energy QCD. The possible new physics effects are also studied in the literature [12]. In this paper, firstly we reexamine the branching and polarization fractions of B → φK∗ in the framework of perturbative QCD (PQCD) by
fixing the hard scale for the involving Wilson coefficients within the SM [13]. And then, we introduce a new type of scalar interactions, which allows flavor changing neutral current (FCNC) at tree level. We will display that the new interactions could explain the branching ratios (BRs) and various polarizations in B → φK∗.
TABLE I: The polarization fractions and relative phases for B → φK∗.
Model Polarization BELLE BABAR
K∗+φ |A0|2 0.52 ± 0.08 ± 0.03 0.46 ± 0.12 ± 0.03 |A⊥|2 0.19 ± 0.08 ± 0.02 φk(rad) 2.10 ± 0.28 ± 0.04 φ⊥(rad) 2.31 ± 0.20 ± 0.07 K∗0φ |A0|2 0.45 ± 0.05 ± 0.02 0.52 ± 0.05 ± 0.02 |A⊥|2 0.30 ± 0.06 ± 0.02 0.22 ± 0.05 ± 0.02 φk(rad) 2.39 ± 0.24 ± 0.04 2.34+0.23−0.20± 0.05 φ⊥(rad) 2.51 ± 0.23 ± 0.04 2.47 ± 0.25 ± 0.05 parametrized as [14] M(λ) = ǫ∗1µ(λ)ǫ∗2ν(λ)a gµν+ b Pµ 2P1ν + i c ǫµναβP1αP2β . (3)
Consequently, the helicity amplitudes are given by
H00 = −1
2m1m2
M2
B− m21− m22)a + 2MB2p2b ,
H±± = a ∓ MBp c,
where p is the magnitude of vector meson momenta. Note that we can define the polarization amplitudes to be A0 = H00 (P h|Hh|2)1/2 , Ak(⊥)= 1 √ 2(P h|Hh|2)1/2 (H++± H−−), (4)
so that Eq. (2) is satisfied. The relative phases between Aλ are described by φk(⊥) =
Arg(Ak(⊥)/A0). In order to merge the results calculated by PQCD [13], we rewrite Eq. (3)
as
M = MB2ML+ MB2MNǫ∗1T · ǫ2T∗ + iMTεαβγρǫ∗1Tǫ∗2TP1γP2ρ.
In terms of the well known effective Hamiltonian for the inclusive b → ss¯s process [15], the various transition matrix elements in B → φK∗ are written as
where H = L, N and T , fφ(B) is the decay constant of φ (B), and EH(N ) and A (N )
H denote
the factorization (nonfactorization) contributions of emission and annihilation topologies, respectively. We note that the Wilson coefficients of weak interactions have been included in {E} and {A} and their explicit expressions can be found in Ref. [13]. For simplicity, we will fix the scale, estimated by the energy of the exchanged hard gluon, to be around t =pΛM¯ B where ¯Λ ∼ MB−mb with mb being the b-quark mass. The various contributions
associated with different scales are shown in Table II. Here, we have neglected the values
TABLE II: Values (in units of 10−3) of transition matrix elements associated with different hard scales of t (GeV).
t EL ELN AL EN ENN
2.0 −13.90 0.37 + i0.37 1.37 − 8.05 −2.09 (−1.28 + i0.04)10−1
1.8 −14.91 0.40 + i0.39 1.49 − i8.69 −2.24 (−1.41 + i0.04)10−1 1.6 −16.13 0.43 + i0.42 1.62 − i9.50 −2.42 (−1.57 + i0.03)10−1
t AN ET ETN AT
2.0 −2.07 + i3.15 −4.08 (−2.65 − i0.25)10−1 −3.90 + i6.46
1.8 −2.23 + i3.40 −4.38 (−2.92 − i0.29)10−1 −4.22 + i6.98 1.6 −2.44 + i3.71 −4.73 (−3.26 − i0.33)10−1 −4.62 + i7.62
of AN since they are much smaller than the others. In our numerical estimations, we
have used fφ = 0.237 GeV, fφT = 0.22 GeV, fK∗ = 0.22 GeV, fKT∗ = 0.17 GeV, fB =
0.19 GeV, mφ = 1.02 GeV, mK∗ = 0.89 GeV and MB = 5.28 GeV. The wave functions
of φ and K∗ are refered to the results of light-cone sum rules (LCSRs) [16]. Using the
values of Table II, the BR and polarizations in Bd → φK∗0 can be easily obtained. To
illustrate the effects of nonfactorization and annihilation, we fix t = 1.6 GeV and we find that (BR, |A0|2, |A⊥|2) = (8.1 × 10−6, 0.93, 0.03)|EH, (10.93 × 10
−6, 0.73, 0.12)|
EH,AH and
(9.42×10−6, 0.62, 0.17)|
EH,AH,EHN for contributions with {EH}, {EH, AH} and {EH, AH, E
N H},
respectively. From the results, we find that the effects of annhilation and nonfactorization can enhance A⊥, but they are still not enough to explain the central values of data in
Table I. For completeness, we present the results with differnt hard scales in Table III. From Table III, we note that the polarizations are stable in different scales. It is difficult to further reduce the logitudinal polarization without introducing new mechinism. As a
TABLE III: Branching ratios (in units of 10−6), polarizations and relative phases with different
hard scales of t (GeV) for Bd→ φK∗0 in the SM.
t BR |A0|2 |Ak|2 |A⊥|2 φk(rad) φ⊥(rad) 2.0 6.93 0.628 0.206 0.166 2.16 2.15 1.8 8.02 0.625 0.207 0.167 2.15 2.14 1.6 9.46 0.622 0.209 0.169 2.15 2.13
comparsion, we also calculate the decay of B+ → ρ+K∗0 in the SM and, explicitly, we find
that (BR, |A0|2, |A⊥|2)|B+→ρ+K∗0 = (14.69×10−6, 0.72, 0.13)|E
H,AH,EHN. Note that the current
experimental data of BABAR and BELLE for (BR, |A0|2) are ((17.0+3.5−3.9) ×10−6, 0.79 ±0.09)
[17] and ((8.9 ± 1.7 ± 1.2) × 10−6, 0.43 ± 0.11+0.05
−0.02) [18], respectively, which are not consistent
with each other. Due to these inconclusive results in B → ρK∗, in this study we regard the
polarization anomaly happens only in the decays of B → φK∗.
We now try to find out if there exists some kind of new interactions which can induce large transverse polarizations in B → φK∗, but not in the others, such as B → ρρ and B → ρK∗.
Naturally, one could try the scalar interactions in which the couplings between the scalar and fermions are proportional to the fermion masses (mf). In these models, the down-quark pair
production is expected to be one order of magnitude smaller than that of the strange-quark pair. However, as known, the couplings in one-Higgs-doublet and type I two-Higgs-doublet models are suppressed by mf/mW. Although there is an enhancement factor tan β in the
type II Higgs model, the effects of the b → s flavor change (FC) transition are one-loop suppressed. In order to get large transverse polarizations in B → φK∗, we consider a new
type of scalar interactions in which FCNC at tree level is allowed. Our another reason to try scalar interactions is that the new contributions on transverse polarizations should avoid the light meson mass dependence or the power suppression of mφ/MB, , unlike the SM
in which hφ|¯sγµs|0i = mφǫφµ arises. For an illustration, we consider the hadronic matrix
element hφK∗|¯bs ¯ss|Bi. To get the factorizable parts, we need do the Fierz transformation. Explicitly, we have
hφK∗|¯bs ¯ss|Bi ∝ 1
4Nchφ|¯sσ
µνs|0i hK∗|¯bσµνs|Bi + . . . , (6)
i[γµ, γν]/2 and {. . .} denotes contributions from other operators such as the vectors and
axial-vectors, which are suppressed by a factor of mφ/mB at the amplitude level. Since the
nonlocal structure of φ is related to the term /ǫφTP Φ/ Tφ with ΦTφ being the twist-2 φ meson
wave function, the factor hφ|¯sσµνs|0i ∝ ǫµφPν − ǫνφPµ which is clearly independent of mφ.
Hence, the hadronic suppression of scalar interactions can be only from color factor and Fierz coefficients. Next, we will demonstrate that scalar interactions have important influence on B → φK∗. Before introducing a specific model, we start from a general interaction with a
scalar boson S, given by
Lef f = Cbs¯bPRs + CsbsP¯ Rb + CsssP¯ Rs S + H.c. (7)
where PL(R) = 1 ∓ γ5. Since we are not dealing with the CP problem, the parameters Cij are
regarded as real numbers. From Eq. (7) the effective interaction for the process of b → ss¯s is derived to be Lef f = Css m2 S ¯b (CbsPR+ CsbPL) s ¯ss , (8)
where mS is the mass of the scalar.
The new contribution to B → φK∗ due to the scalar interaction are shown in Fig. 1,
where (a) and (b) stand for the factorizable and nonfactorizable effects, respectively. Since we have assumed that the couplings of the scalar interaction to fermions are proportional to mf, the annihilation topologies can be neglected due to the suppression of mu/v or md/v,
comparing to emission topologies (Fig. 1) associated with ms/v. Similar to the SM case,
b 1 − γ5 s s s q q or (a) or (b)
FIG. 1: Diagrams for hadronic transition matrix elements due to the scalar interaction of ¯b(1 − γ5)s ¯ss with (a) factorizable and (b) nonfactorizable contributions.
the decay amplitudes for various helicities could be written as MN PH =
CssCsb
2Ncm2S
where FH and NH are the factorizable and nonfactorizable effects, respectively. Here, for
simplicity, we have only presented the contributions of Cbs. The result of Csb can be obtained
by changing the sign in H = L and N. Csb and Cbs have the same contributions for H = T .
The explicit expressions of FH and NH are shown in Appendix I and their values are given
in Table II.
TABLE IV: Values (in units of 10−2) of transition matrix elements for scalar interactions.
FL NL FN NN FT FN
9.74 −2.85 + i0.26 −6.91 0.44 + i0.31 −20.27 −0.038 + i0.29
For a specific model, we concentrate on the generalized two-Higgs-doublet model (Model III) and the corresponding Yukawa Lagrangian for down-type quarks is described by [19]
L(III)Y = ηijDQ¯iLΦ1DjR+ ξijDQ¯iLΦ2DjR+ H.c. , (10)
where the indices i(j) represent the possible quark flavors and ξD
ij denote the allowed FC
effects. The vacuum expectation values (VEVs) of neutral Higgs fields are denoted by hΦ0
1(2)i = v1(2). For convenience, we can choose a proper basis such that only one scalar
field possesses the VEV. Hence, the new scalar fields could be chosen to be φ0
1 = cos βΦ01+ sin βΦ0 2 = (v + H0 + iχ0)/ √ 2 and φ0 2 = − sin βΦ01 + cos βΦ02 = (H1 + iH2)/ √ 2, where v = pv2
1 + v22 is the vacuum expectation value (VEV) of φ01, cos β(sin β) = v1(2)/v, H0(1)
and H2 are CP-even and CP-odd Higgs bosons, and χ0 is Goldstone boson, respectively.
Since H0 and H1 are not physical eigenstates, the mass eigenstates could be parametrized by a mixing angle α as h0
SM = H0cos α + H1sin α and h0 = −H0sin α + H1cos α. When α
goes to zero, h0
SM becomes the SM Higgs. It is known that to get naturally small FCNC at
tree level, one can use the ansatz [19, 20] ξijD = λij
√m
imj
v . (11)
It has been analyzed phenomenologically that the coupling λsb for the transition of b → s
may not be small and it could be as large as O(10) [19]. Besides the coupling λsb, for
b → ss¯s, we also need the information on λss, which is flavor conserved. To understand the
order of magnitude on λss, we refer to the case of type II model, in which the corresponding
coupling ¯ssH1is m
tan β = v2/v1, λss could be order of tan β ∼ mt/mb. We note that the large enhancement
of λss is natural only for the type II model. In our considered type III model, we shall set
λss = O(100), which implies that the Higgs coupling to the strange quark ξDss = λssms/v
is O(10−2). Since we only concern the non-leptonic decays, it is clear that the value of
λµµ = O(1) for the muonic coupling given in Ref. [19] can be relaxed and there are no
stringent limits for λij.
To estimate the influence of scalar interactions, we set λss = 90, λsb = 5 and mH = 150
GeV and take ζ = λbs/λsb as a variable. By using the results of Tables II and IV, we present
the BR and polarizations of Bd → φK∗0 for different values of ζ in Table V. In Fig. 2, we
show BR and |Ak(⊥)|2 as functions of (a) λss with ζ = 0.2 and mH = 150 GeV, (b) ζ with
λss = 90 and mH = 150 GeV and (c) mH with ζ = 0.2 and λss = 90, respectively. From
TABLE V: Branching ratios (in units of 10−6), polarizations and relative phases of Bd → φK∗0
by combining the results of Tables II and IV with t = 1.6 GeV, mH = 150 GeV, λss = 90 and
λsb= 5. ζ BR |A0|2 |Ak|2 |A⊥|2 φk(rad) φ⊥(rad) −1.0 11.28 0.56 0.30 0.14 2.16 2.18 −0.6 11.25 0.56 0.27 0.17 2.24 2.23 −0.2 11.28 0.55 0.24 0.21 2.22 2.26 0.0 11.32 0.54 0.23 0.23 2.21 2.27 0.2 11.37 0.53 0.22 0.25 2.20 2.28 0.6 11.53 0.52 0.19 0.29 2.17 2.30 1.0 11.76 0.50 0.17 0.33 2.15 2.31
the results of Table V and Fig. 2, we find that |A⊥|2 increases (decreases) if ζ > 0 (< 0). In
particular, when ζ > 0.6, |A⊥|2 can be as large as 30%. We remark that the contributions
of λsb and λbs to ML,N are opposite in sign but to MT the same sign. Therefore, if we
take λsb = λbs, the scalar interactions can only contribute to |A⊥|2. On the other hand, if
λsb = −λbs, only |Ak|2 gets affected as shown in Table V.
In summary, we have studied how scalar interactions effectively affect the polarizations in B → φK∗ decays. We have illustrated that for the scalar interactions with FCNC
9 10 11 12 10 6 BR 5 5.2 5.5 10|Α 0 | 2 80 90 100 λ110 ss 2.1 2.4 2.7 10|Α ⊥ | 2 (a) 8 10 12 10 6 BR 4.8 5.1 5.4 5.7 10|Α 0 | 2 -1 -0.5 0 0.5 1 ζ 1.2 2.4 3.6 10|Α ⊥ | 2 (c) 8 10 12 10 6 BR 5.1 5.4 5.7 10|A 0 | 2 150 200 250 mH 2 2.4 2.8 10|Α ⊥ | 2 (b)
FIG. 2: Branching ratio and |Ak(⊥)|2 in B
d → φK∗0 as functions of (a) λss with ζ = 0.2 and
mH = 150 GeV, (b) ζ with λss = 90 and mH = 150 GeV and (c) mH with ζ = 0.2 and λss = 90,
respectively. The solid, dashed and dotted curves stand for t = 2.0, 1.8 and 1.6 GeV, respectively.
polarization |A0|2 can be 50% and the transverse polarization |A⊥|2 30%. We have also
found that the sign of ζ = λsb/λbs = Csb/Cbs controls the relative magnitudes of Ak and
A⊥.
Acknowledgments
This work is supported in part by the National Science Council of R.O.C. under Grant #s: NSC-93-2112-M-006-010 and NSC-93-2112-M-007-014.
I. APPENDIX: DECAY AMPLITUDES FOR NEW SCALAR INTERACTIONS
The transition matrix elements of factorizable and nonfactorizable effects for the effec-tive interaction ¯b(1 − γ5)s ¯ss are given as follows: the factorizable amplitudes with various
helicities are FL = 2πCFMB2 Z 1 0 dx1dx3 Z ∞ 0 b1db1b3db3ΦB(x1){[(1 + x3− rφ)ΦK∗(x3) +rK∗(1 − 2x3+ rφx3) Φt(x3) + ΦsK∗(x3)]Ee(t (1) e )he(x1, x3, b1, b3) +[2rK∗(1 − rφ)ΦsK∗(x3)]Ee(t (2) e )he(x3, x1, b3, b1)}, (12)
FN = −2πCFMB2 Z 1 0 dx1dx3 Z ∞ 0 b1db1b3db3ΦB(x1){[(1 + x3− rφ)ΦTK∗(x3) +rK∗(1 − 2x3) (Φv(x3) + ΦaK∗(x3)) − rφrK∗((2 + x3)ΦvK∗(x3) − x3Φ a K∗(x3))] ×Ee(t(1)e )he(x1, x3, b1, b3) + [rK∗(1 − rφ) (ΦvK∗(x3) + Φ a K∗(x3))] ×Ee(t(2)e )he(x3, x1, b3, b1)}, (13) FT = −4πCFMB2 Z 1 0 dx1dx3 Z ∞ 0 b1db1b3db3ΦB(x1){[(1 + x3+ rφ)ΦTK∗(x3) +rK∗(1 − 2x3) (Φv(x3) + ΦaK∗(x3)) − rφrK∗((2 + x3)ΦaK∗(x3) − x3Φ v K∗(x3))] ×Ee(t(1)e )he(x1, x3, b1, b3) + [rK∗(1 − rφ) (ΦvK∗(x3) + Φ a K∗(x3))] ×Ee(t(2)e )he(x3, x1, b3, b1)}, (14)
where {Φ} denote the distribution amplitudes of φ and K∗ mesons. We consider the effects
up to twist-3. The hard function he and Ee factor are
he(x1, x3, b1, b3) = K0(√x1x3Mbb1)St(x3)[θ(b1− b3)K0(√x3MBb1)I0(√x3MBb3)
+θ(b3− b1)K0(√x3MBb3)I0(√x3MBb1)],
Ee(t) = αs(t)SB(t)SK∗(t).
The Sudakov factors for K∗ and B mesons and threshold resummation factor are given by
SB = exp −s(x1PB+, b1) − 2 Z t 1/b1 dµ µ γ(αs(µ)) , SK∗ = exp −s(x3P3+, b3) − s((1 − x3)P2+, b3) − 2 Z t 1/b2 dµ µ γ(αs(µ)) , St(x) = 21+2cΓ(3/2 + c) √ πΓ(1 + c)[x(1 − x)]c,
where γ = −αs/π which is the quark anomalous dimension, the variables (b1, b2, b3) are
conjugate to the parton transverse momenta (k1T, k2T, k3T), c = 0.4 for B → φK∗ decays,
and the explicit expression for s(x, b) can be found in Ref. [21]. The scale t(1)e and t(2)e are
chosen by
t(1)e = max(√x3MB, 1/b1, 1/b3),
The nonfactorizable amplitudes with various helicities are given as NL = −4πCFMB2p2Nc Z 1 0 d[x] Z ∞ 0 b1db1b2db2ΦB(x1){[x2Φφ(x2)ΦK∗(x3) +rK∗x3 Φt(x3) − ΦsK∗(x3) − rφrK∗Φtφ(x2) 0.5(x2+ 3x3)ΦtK∗(x3) + x2Φ s K∗(x3) +2rφx2Φtφ(x2)ΦK∗(x3) − rφrK∗ (x2 − x3)ΦtK∗(x3) + (x2+ x3)Φ s K∗(x3)] ×Ed(t(1)d )h (1) d (x1, x2, x3, b1, b2) − [2(1 − x2+ x3)Φφ(x2)ΦK∗(x3) +rK∗x3Φφ(x2) Φt(x3) + ΦsK∗(x3) − rφ(1 − x2) Φ t φ(x2) − Φsφ(x2) ΦK∗(x3) +2rφrK∗(1 − x2+ x3) Φtφ(x2)ΦtK∗(x3) − Φ s φ(x2)ΦsK∗(x3)] ×Ed(t(2)d )h (1) d (x1, x2, x3, b1, b2)}, (15) NN = −4πCFMB2p2Nc Z 1 0 d[x] Z ∞ 0 b1db1b2db2ΦB(x1){[−x2ΦTφ(x2)ΦTK∗(x3) +rφx2 Φvφ(x2) − Φφa(x2) ΦTK∗(x3)]Ed(t (1) d )h (1) d (x1, x2, x3, b1, b2) +[(1 − x2+ x3)ΦTφ(x2)ΦTK∗(x3) − rK∗x3ΦTφ(x2) (ΦvK∗(x3) + Φ a K∗(x3)) +rφ(1 − x2) Φvφ(x2) − Φaφ(x2) ΦTK∗(x3) − 2rφrK∗(1 − x2+ x3) × Φvφ(x2)ΦvK∗(x3) − Φ a φ(x2)ΦaK∗(x3)]Ed(t (2) d )h (2) d (x1, x2, x3, b1, b2), (16) NT = 8πCFMB2p2Nc Z 1 0 d[x] Z ∞ 0 b1db1b2db2ΦB(x1){[x2ΦTφ(x2)ΦTK∗(x3) +rφx2 Φvφ(x2) − Φφa(x2) ΦTK∗(x3)]Ed(t (1) d )h (1) d (x1, x2, x3, b1, b2) −[(1 − x2+ x3)ΦTφ(x2)ΦTK∗(x3) − rK∗x3ΦTφ(x2) (ΦvK∗(x3) + Φ a K∗(x3)) −rφ(1 − x2) Φvφ(x2) − Φaφ(x2) ΦTK∗(x3) + 2rφrK∗(1 − x2+ x3) × Φvφ(x2)ΦaK∗(x3) − Φ a φ(x2)ΦvK∗(x3)]Ed(t (2) d )h (2) d (x1, x2, x3, b1, b2), (17)
where the Sudakov factor for the φ meson is given as Sφ = exp −s(x2P2+, b2) − s((1 − x2)P2+, b2) − 2 Z t 1/b2 dµ µ γ(αs(µ)) , and the hard functions h(j)d are
h(j)d = [θ(b1− b2)K0(DMBb1)I0(DMBb2) + θ(b2 − b1)K0(DMBb2)I0(DMBb1)] × K0(DjMBb2) for D2j ≥ 0, iπ 2H (1) 0 ( q |D2 j|MBb2) for D2j ≤ 0
with D2 = x
1x3, D12 = (x1− x2)x3 and D22 = −(1 − x1− x2)x3. The scales t(j)d are chosen by
t(1)d = max(DMB, q |D2 1|MB, 1/b1, 1/b2), t(2)d = max(DMB, q |D2 2|MB, 1/b1, 1/b2).
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