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Probing the Effects of New Physics in [[bar.B].sup.*] [right arrow] Pl[[bar.[nu]].sub.l] Decays.

1. Introduction

Thanks to the fruitful running of the B factories and Large Hadron Collider (LHC) in the past years, most of the [B.sub.u,d] mesons decays with branching fractions [??] O([10.sup.-7]) have been measured. The rare B-meson decays play an important role in testing the standard model (SM) and probing possible hints of new physics (NP). Although most of the experimental measurements are in good agreement with the SM predictions, several indirect hints for NP, the tensions or the so-called puzzles, have been observed in the flavor sector.

The semileptonic [bar.B] [right arrow] [D.sup.(*)] l[[bar.[nu]].sub.l] decays are induced by the CKM favored tree-level charged current, and therefore, their physical observables could be rather reliably predicted in the SM and the effects of NP are expected to be tiny. In particular, the ratios defined by [mathematical expression not reproducible] are independent of the CKM matrix elements, and the hadronic uncertainties canceled to a large extent; thus they could be predicted with a rather high accuracy. However, the BaBar [1, 2], Belle [3-5], and LHCb [6] collaborations have recently observed some anomalies in these ratios. The latest experimental average values for [mathematical expression not reproducible] reported by the Heavy-Flavor Average Group (HFAG) are [7]

[mathematical expression not reproducible], (1)

which deviate from the SM predictions

[mathematical expression not reproducible], (2)

(see [8] in the first equation of (2) and [9] in the second equation of (2)) at the levels of 2.2[sigma] and 3.4[sigma] errors, respectively. Moreover, when the correlations between RD and [R.sup.*.sub.D] are taken into account, the tension would reach up to 3.9[sigma] level [7]. Besides, the ratio [R.sub.J/[psi]] [equivalent to] B([B.sub.c] [right arrow] J/[psi][[tau].sup.-][[bar.[nu]].sub.[tau]])/B([B.sub.c] [right arrow] J/[psi][[mu].sup.-][??[].sub.[mu]]) has recently been measured by the LHCb collaboration [10], which also shows an excess of about 2[sigma] from the central value range of the corresponding SM predictions [0.25, 0.28]. In addition, another mild hint of NP in the b [right arrow] ul[bar.[nu]] induced B [right arrow] [tau][bar.[nu]] decay has been observed by the BaBar and Belle collaborations [11-14]; the deviation is at the level of 1.4[sigma] [15].

The large deviations in [mathematical expression not reproducible] and possible anomalies in the other decay channels mentioned above imply possible hints of NP relevant to the lepton flavor violation (LFV) [15]. The investigations for these anomalies have been made extensively both within model-independent frameworks [16-37] and in some specific NP models where the b [right arrow] c[tau][[bar.[nu]].sub.[tau]] transition is mediated by leptoquarks [16, 17, 38-46], charged Higgses [16, 47-59], charged vector bosons [16, 60, 61], and sparticles [62-65].

In addition to B mesons, the vector ground states of b[bar.q] system, [B.sup.*] mesons, with quantum number of [n.sup.2s+1][L.sub.J] = [1.sup.3][S.sub.1] and [J.sup.P] = [1.sup.-] [66-69], also can decay through the b [right arrow] (u, c)l[[bar.[nu]].sub.l] transitions at quark-level. Therefore, in principle, the corresponding NP effects might enter into the semileptonic [B.sup.*] decays as well. The [B.sup.*] decay occurs mainly through the electromagnetic process [[bar.B].sup.*] [right arrow] [bar.B][gamma], and the weak decay modes are very rare. Fortunately, thanks to the rapid development of heavy-flavor experiments instruments and techniques, the [B.sup.*] weak decays are hopeful to be observed by the running LHC and forthcoming SuperKEK/Belle-II experiments [70-72] in the near future. For instance, the annual integrated luminosity of Belle-II is expected to reach up to ~13 a[b.sup.-1] and the [B.sup.*] weak decays with branching fractions > O([10.sup.-9]) are hopeful to be observed [70, 73, 74]. Moreover, the LHC experiment also will provide a lot of experimental information for [B.sup.*] weak decays due to the much larger beauty production cross-section of pp collision relative to [e.sup.+][e.sup.-] collision [75].

Recently, some interesting theoretical studies for the [B.sup.*] weak decays have been made within the SM in [73, 74, 76-82]. In this paper, motivated by the possible NP explanation for the [mathematical expression not reproducible] puzzles, the corresponding NP effects on the semileptonic [B.sup.*] decays will be studied in a model-independent way. In the investigation, the scenarios of vector and scalar NP interactions are studied, respectively; their effects on the branching fraction, differential branching fraction, lepton spin asymmetry, forward-backward asymmetry, and ratio [R.sup.*.sub.P] (P = D, [pi], K) of semileptonic [B.sup.*] decays are explored by using the spaces of various NP couplings obtained through the measured [mathematical expression not reproducible].

Our paper is organized as follows. In Section 2, after a brief description of the effective Lagrangian for the b [right arrow] (u, c)l[[bar.[nu]].sub.l] transitions, the theoretical framework and calculations for the [[bar.B].sup.*] [right arrow] Pl[[bar.[nu]].sub.l] decays in the presence of various NP couplings are presented. Section 3 is devoted to the numerical results and discussions for the effects of various NP couplings. Finally, we give our conclusions in Section 4.

2. Theoretical Framework and Calculation

2.1. Effective Lagrangian and Amplitudes. We employ the effective field theory approach to compute the amplitudes of [[bar.B].sup.*] [right arrow] Pl[[bar.[nu]].sub.l] decays in a model-independent scheme. The most general effective Lagrangian at [mu] = O([m.sub.b]) for the b [right arrow] p[l.sup.-][[bar.[nu]].sub.l] (p = u, c) transition can be written as [19, 21, 40, 46]

[mathematical expression not reproducible], (3)

where [G.sub.F] is the Fermi coupling constant, [V.sub.pb] denotes the CKM matrix elements, and [P.sub.L,R] = (1 [+ or -] [[gamma].sub.5])/2 is the negative/positive projection operator. Assuming the neutrinos are left-handed and neglecting the tensor couplings, the effective Lagrangian can be simplified as

[mathematical expression not reproducible], (4)

where [V.sub.L,R] and [S.sub.L,R] are the effective NP couplings (Wilson coefficients) defined at [mu] = O([m.sub.b]). In the SM, all the NP couplings will be zero.

We use the method of [83-87] to calculate the helicity amplitudes. The square of amplitudes for the [[bar.B].sup.*] [right arrow] P[l.sup.-][[bar.[nu]].sub.l] decay can be written as the product of leptonic ([L.sub.[mu][nu]]) and hadronic ([H.sub.[mu][nu]]) tensors,

[mathematical expression not reproducible], (5)

where the superscripts i and j refer to four operators in the effective Lagrangian given by (4) (the tensors related to the scalar and pseudoscalar operators can be understood through the relations given by (21) and (22)); in the SM, i = j corresponds to the operator [bar.p][[gamma].sub.[mu]] (1 - [[gamma].sub.5])b[bar.l][[gamma].sup.[mu]](1 - [[gamma].sub.5])[nu]. For convenience in writing, these superscripts are omitted below. Inserting the completeness relation

[summation over (m,n)] [[bar.[epsilon]].sub.[mu]] (m) [[bar.[epsilon]].sup.*.sub.[nu]] (n) [g.sub.mn] = [g.sub.[mu][nu]], (6)

the product of [L.sub.[mu][nu]] and [H.sup.[mu][nu]] can be further expressed as

[L.sub.[mu][nu]][H.sup.[mu][nu]] = [summation over (m,m',n,n')] L (m, n) H(m', n') [g.sub.mm'][g.sub.nn']. (7)

Here, [[bar.[epsilon]].sub.[mu]] is the polarization vector of the virtual intermediate states, which is [W.sup.*] boson in the SM and named as [omega] in this paper for convenience of expression. The quantities L(m, n) [equivalent to] [L.sup.[mu][nu]] [[bar.[epsilon]].sub.[mu]](m) [[bar.[epsilon]].sup.*.sub.[nu]](n) and H(m, n) [equivalent to] [H.sup.[mu][nu]] [[bar.[epsilon]].sup.*.sub.[mu]](m) [[bar.[epsilon]].sub.[nu]](n) are Lorentz invariant and therefore can be evaluated in different reference frames. In the following evaluation, H(m, n) and L(m, n) will be calculated in the [B.sup.*]-meson rest frame and the l - [[bar.[nu]].sub.l] center-of-mass frame, respectively.

2.2. Kinematics for [[bar.B].sup.*] [right arrow] P[l.sup.-][[bar.[nu]].sub.l] Decays. In the [B.sup.*]-meson rest frame with daughter P-meson moving in the positive z-direction, the momenta of particles [B.sup.*] and P are

[mathematical expression not reproducible]. (8)

For the four polarization vectors, [[bar.[epsilon]].sup.[mu]] ([[lambda].sub.[omega]] = t, 0, [+ or -]), one can conveniently choose [83, 84]

[mathematical expression not reproducible], (9)

where [mathematical expression not reproducible] being the momentum transfer squared, are the energy and momentum of the virtual [omega]. The polarization vectors of the initial [B.sup.*]-meson can be written as

[[epsilon].sup.[mu]] (0) = (0, 0, 0, 1), [[epsilon].sup.[mu]] ([+ or -]) = 1/[square root of 2] (0, [- or +] 1, -i, 0). (10)

In the l - [[bar.[nu]].sub.l] center-of-mass frame, the four momenta of lepton and antineutrino pair are given as

[mathematical expression not reproducible], (11)

where [E.sub.l] = ([q.sup.2] + [m.sup.2.sub.l])/2 [square root of ([q.sup.2])], [absolute value of ([[??].sub.l])] = ([q.sup.2] - [m.sup.2.sub.l])/2 [square root of ([q.sup.2])], and [theta] is the angle between the P and l three-momenta. In this frame, the polarization vector [[bar.[epsilon]].sup.[mu]] takes the form

[[bar.[epsilon]].sup.[mu]] (t) = (1, 0, 0, 0), [[bar.[epsilon]].sup.[mu]] (0) = (0, 0, 0, 1), [[bar.[epsilon]].sup.[mu]] ([+ or -]) = 1/[square root of 2] (0, [- or +]1, -i, 0). (12)

2.3. Hadronic Helicity Amplitudes. For the [[bar.B].sup.*] [right arrow] P[l.sup.-][[bar.[nu]].sub.l] decay, the hadronic helicity amplitudes [mathematical expression not reproducible] and [mathematical expression not reproducible] are defined by

[mathematical expression not reproducible], (13)

[mathematical expression not reproducible], (14)

[mathematical expression not reproducible], (15)

[mathematical expression not reproducible], (16)

which describe the decay of three helicity states of [B.sup.*] meson into a pseudoscalar P meson and the four helicity states of virtual [omega]. It should be noted that [[lambda].sub.[omega]] in [mathematical expression not reproducible], (15) and (16), should always be equal to t.

For [B.sup.*] [right arrow] P transition, the matrix elements of the vector and axial-vector currents can be written in terms of form factors V([q.sup.2]) and [A.sub.0,1,2]([q.sup.2]) as

[mathematical expression not reproducible], (17)

[mathematical expression not reproducible], (18)

with the sign convention [[epsilon].sub.0123] = -1. Furthermore, using the equations of motion,

i[[partial derivative].sub.[mu]] ([bar.p][[gamma].sup.[mu]]b) = [[m.sub.b] ([mu]) - [m.sub.p] ([mu])] [bar.p]b, (19)

i[[partial derivative].sub.[mu]] ([bar.p][[gamma].sub.[mu]][[gamma].sub.5]b) = - [[m.sub.b] ([mu]) + [m.sub.p] ([mu])] [bar.p][[gamma].sub.5]b, (20)

one can write the matrix elements of scalar and pseudoscalars currents as

[mathematical expression not reproducible], (21)

[mathematical expression not reproducible], (22)

in which [m.sub.b]([mu]) and [m.sub.p]([mu]) are the running quark masses.

Then, by contracting above hadronic matrix elements with the polarization vectors in the [B.sup.*]-meson rest frame, we obtain five nonvanishing helicity amplitudes

[mathematical expression not reproducible], (23)

[mathematical expression not reproducible], (24)

[mathematical expression not reproducible], (25)

[mathematical expression not reproducible]. (26)

It is obvious that only the amplitudes with [mathematical expression not reproducible] survive.

2.4. Leptonic Helicity Amplitudes. Expanding the leptonic tensor in terms of a complete set of Wigner's [d.sup.J]-functions [9, 83, 87], [L.sub.[mu][nu]][H.sup.[mu][nu]] can be rewritten as a compact form

[mathematical expression not reproducible], (27)

in which J and J' run over 1 and 0, [[lambda].sup.(').sub.[omega]] and [[lambda].sup.l] run over their components, and massless right-handed antineutrinos with [mathematical expression not reproducible]. In (27), [mathematical expression not reproducible] are the leptonic helicity amplitudes defined as

[mathematical expression not reproducible], (28)

[mathematical expression not reproducible]. (29)

In the l - [[bar.[nu]].sub.l] center-of-mass frame, taking the exact forms of the spinors and polarization vectors, we finally obtain four nonvanishing contributions

[mathematical expression not reproducible], (30)

[mathematical expression not reproducible], (31)

[mathematical expression not reproducible], (32)

[mathematical expression not reproducible]. (33)

2.5. Observables of [[bar.B].sup.*] [right arrow] P[l.sup.-][[bar.[nu]].sub.l] Decays. With the amplitudes obtained in above subsections, we then present the observables considered in our following evaluations. The double differential decay rate of [[bar.B].sup.*] [right arrow] P[l.sup.-][[bar.[nu]].sub.l] decay is written as

[mathematical expression not reproducible], (34)

where the factor 1/3 is caused by averaging over the spins of initial state [[bar.B].sup.*]. Using the standard convention for [d.sup.J]-function [88], we finally obtain the double differential decay rates with a given leptonic helicity state ([[lambda].sub.l] = [+ or -]1/2), which are

[mathematical expression not reproducible], (35)

[mathematical expression not reproducible]. (36)

Using (35) and (36), one can get the explicit forms of various observables of [[bar.B].sup.*] [right arrow] P[l.sup.-][[bar.[nu]].sub.l] decays as follows:

(i) The differential decay rate is

[mathematical expression not reproducible]. (37)

(ii) The [q.sup.2] dependent ratio is

[R.sup.*.sub.P] ([q.sup.2]) [equivalent to] [d[GAMMA] ([[bar.B].sup.*] [right arrow] P[[tau].sup.-][[bar.[nu]].sub.[tau]])/d[q.sup.2]]/ [d[GAMMA] ([[bar.B].sup.*] [right arrow] P[l'.sup.-][[bar.[nu]].sub.l'])/d[q.sup.2]], (38)

where l' denotes the light lepton.

(iii) The lepton spin asymmetry is

[A.sup.P.sub.[lambda]] ([q.sup.2]) = [d[GAMMA] [[[lambda].sub.l] = -1/2]/d[q.sup.2] - d[GAMMA][[[lambda].sub.l] = 1/2]/d[q.sup.2]]/ [d[GAMMA] [[[lambda].sub.l] = -1/2]/d[q.sup.2] + d[GAMMA][[[lambda].sub.l] = 1/2]/d[q.sup.2]]. (39)

(iv) The forward-backward asymmetry is

[A.sup.P.sub.[theta]] ([q.sup.2]) = [[[integral].sup.0.sub.-1] d cos [theta] ([d.sup.2][GAMMA]/d[q.sup.2]d cos [theta]) - [[integral].sup.1.sub.0] d cos [theta] ([d.sup.2][GAMMA]/d[q.sup.2]d cos [theta])]/d[GAMMA]/d[q.sup.2]. (40)

The SM results can be obtained from above formulae by taking [V.sub.L] = [V.sub.R] = [S.sub.L] = [S.sub.R] = 0.

In the following evaluations, in order to fit the NP spaces, we also need the observables of [bar.B] [right arrow] [D.sup.(*)][l.sup.-][[bar.[nu]].sub.l] decays, which have been fully calculated in the past years. In this paper, we adopt the relevant theoretical formulae given in [46].

3. Numerical Results and Discussions

3.1. Input Parameters. Before presenting our numerical results and analyses, we would like to clarify the values of input parameters used in the calculation. For the CKM matrix elements, we use [89]

[absolute value of ([V.sub.cb])] = [4.181.sup.+0.028.sub.-0.060] x [10.sup.-2], [absolute value of ([V.sub.ub])] = [3.715.sup.+0.060.sub.-0.060] x [10.sup.-3]. (41)

For the well-measured Fermi coupling constant [G.sub.F], the masses of mesons and leptons, and the running masses of quarks at [mu] = [m.sub.b], we take their central values given by PDG [88]. The total decay widths (or lifetimes) of [B.sup.*] mesons are essential for estimating the branching fraction; however there is no available experimental data until now. According to the fact that the electromagnetic process [B.sup.*] [right arrow] B[gamma] dominates the decays of [B.sup.*] meson, we take the approximation [[GAMMA].sub.tot]([B.sup.*]) [equivalent] [GAMMA]([B.sup.*] [right arrow] B[gamma]); the latter has been evaluated within different theoretical models [90-96]. In this paper, we adopt the most recent results [95, 96]

[[GAMMA].sub.tot] ([B.sup.*+]) [equivalent] [GAMMA]([B.sup.*+] [right arrow] [B.sup.+][gamma]) = ([468.sup.+73.sub.-75]) eV, (42)

[[GAMMA].sub.tot] ([B.sup.*0]) [equivalent] [GAMMA]([B.sup.*0] [right arrow] [B.sup.0][gamma]) = (148 [+ or -] 20) eV, (43)

[[GAMMA].sub.tot] ([B.sup.*0.sub.s]) [equivalent] [GAMMA]([B.sup.*0.sub.s] [right arrow] [B.sup.0.sub.s] [gamma]) = (68 [+ or -] 17) eV. (44)

Then the residual inputs are the transition form factors, which are crucial for evaluating the observables of [[bar.B].sup.*] [right arrow] P[l.sup.-][[bar.[nu]].sub.l] and [bar.B] [right arrow] [D.sup.(*)][l.sup.-][[bar.[nu]].sub.l] decays. For the B [right arrow] [D.sup.(*)] transitions, the scheme of Caprini, Lellouch, and Neubert (CLN) parametrization [97] is widely used, and the CLN parameters can be precisely extracted from the well-measured [bar.B] [right arrow] [D.sup.(*)][l.sup.-][[bar.[nu]].sub.l] decays; numerically, their values read [7]

[mathematical expression not reproducible]; (45)

[mathematical expression not reproducible]. (46)

However, for the [[bar.B].sup.*.sub.u,d,s] [right arrow] [P.sub.u,d,s] transition, there is no experimental data and ready-made theoretical results to use at present. Here, we employ the Bauer-Stech-Wirbel (BSW) model [98, 99] to evaluate the form factors for both [[bar.B].sup.*] [right arrow] P and [bar.B] [right arrow] [D.sup.(*)] transitions. Using the inputs [m.sub.u] = [m.sub.d] = 0.35 GeV, [m.sub.s] = 0.55 GeV, [m.sub.c] = 1.7 GeV, [m.sub.b] = 4.9 GeV, and [omega] = [square root of (<[[??].sup.2.sub.[perpendicular to]>)] = 0.4 GeV, we obtain the results at [q.sup.2] = 0,

[mathematical expression not reproducible]; (47)

[mathematical expression not reproducible]; (48)

[mathematical expression not reproducible]; (49)

[mathematical expression not reproducible]; (50)

[mathematical expression not reproducible]; (51)

[mathematical expression not reproducible]. (52)

To be conservative, 15% uncertainties are assigned to these values in our following evaluation. Moreover, with the assumption of nearest pole dominance, the dependences of form factors on [q.sup.2] read [98, 99]

[mathematical expression not reproducible], (53)

where [B.sub.q]([J.sup.P]) is the state of [B.sub.q] with quantum number of [J.sup.P] (J and P are the quantum numbers of total angular momenta and parity, respectively).

With the theoretical formulae and inputs given above, we then proceed to present our numerical results and discussion, which are divided into two scenarios with different simplification for our attention to the types of NP couplings as follows:

(i) Scenario I: taking [S.sub.L] = [S.sub.R] = 0, i.e., only considering the NP effects of [V.sub.L,R] couplings

(ii) Scenario II: taking [V.sub.L] = [V.sub.R] = 0, i.e., only considering the NP effects of [S.sub.L,R] couplings

In these two scenarios, we consider all the NP parameters to be real for our analysis. In addition, we assume that only the third generation leptons get corrections from the NP in the b [right arrow] (u,c)l[[bar.[nu]].sub.l] processes and for l = e,[mu] the NP is absent. In the following discussion, the allowed spaces of NP couplings are obtained by fitting to [R.sub.D] and [mathematical expression not reproducible] (1), with the data varying randomly within their 1[sigma] error, while the theoretical uncertainties are also considered and obtained by varying the inputs randomly within their ranges specified above.

3.2. Scenario I: Effects of [V.sub.L] and [V.sub.R] Type Couplings. In this subsection, we vary couplings [V.sub.L] and [V.sub.R] while keeping all other NP couplings to zero. Under the constraints from the data of [R.sub.D] and [R.sup.*.sub.D], the allowed spaces of new physics parameters, [V.sub.L] and [V.sub.R], are shown in Figure 1. In the fit, the B [right arrow] [D.sup.(*)] form factors based on CLN parametrization and BSW model are used, respectively; it can be seen from Figure 1 that their corresponding fitting results are consistent with each other, but the constraint with the former is much stronger due to the relatively small theoretical error. Therefore, in the following evaluations and discussions, the results obtained by using CLN parametrization are used. In addition, our fitting result in Figure 1 agrees well with the ones obtained in the previous works, for instance, [26, 35].

From Figure 1, we find that (i) the allowed spaces of ([V.sub.L], [V.sub.R]) are bounded into four separate regions, namely, solutions A-D. (ii) Except for solution A, the other solutions are all far from the zero point (0, 0) and result in very large NP contributions. Taking solution C (D) as an example, the SM contribution is completely canceled out by the NP contribution related to [V.sub.L], and the [V.sub.R] coupling presents sizable positive (negative) NP contribution to fit data. The situation of solution B is similar, but only [V.sub.L] coupling presents sizable NP contribution. Numerically, one can easily conclude that the NP contributions of solutions B-D are about two times larger than the SM, which seriously exceeds our general expectation that the amplitudes should be dominated by the SM and the NP only presents minor corrections. In this point of view, the minimal solution (solution A) is much favored than solutions B-D. So, in our following discussions, we pay attention only to solution A, which is replotted in Figure 1(b) and numerical result is

[V.sub.L] = [0.14.sup.+0.06.sub.-0.06], [V.sub.R] = [0.05.sup.+0.06.sub.-0.07]. solution A (54)

Using the values of NP couplings given by (54), we then present our theoretical predictions for B([[bar.B].sup.*] [right arrow] P[[tau].sup.-][[bar.[nu]].sub.[tau]]) and [q.sup.2]-integrated [R.sup.*.sub.P] in Table 1, in which the SM results are also listed for comparison. The [q.sup.2]-dependence of differential observables d[GAMMA]/d[q.sup.2], [R.sup.*.sub.P], [A.sup.P.sub.[lambda]], and [A.sup.P.sub.[theta]] for [B.sup.*-] [right arrow] [D.sup.0][[tau].sup.-][[bar.[nu]].sub.[tau]] and [[pi].sup.0][[tau].sup.-][[bar.[nu]].sub.[tau]] decays is shown in Figure 2; the case of [[bar.B].sup.*0] [right arrow] [D.sup.+][[tau].sup.-][[bar.[nu]].sub.[tau]] and [[bar.B].sup.*0.sub.s] [right arrow] [D.sup.+.sub.s] [[tau].sup.-][[bar.[nu]].sub.[tau]] ([[bar.B].sup.*0] [right arrow] [[pi].sup.+][[tau].sup.-][[bar.[nu]].sub.[tau]] and [[bar.B].sup.*0.sub.s] [right arrow] [K.sup.+][[tau].sup.-][[bar.[nu]].sub.[tau]]) is similar to the one of [B.sup.*-] [right arrow] [D.sup.0][[tau].sup.-][[bar.[nu]].sub.[tau]] ([B.sup.*-] [right arrow] [[pi].sup.0][[tau].sup.-][[bar.[nu]].sub.[tau]]) decay, not shown here. The following are some discussions and comments:

(1) From Table 1, it can be seen that the branching fractions of b [right arrow] c[tau][[bar.[nu].sub.[tau]] induced [[bar.B].sup.*.sub.u,d,s] decays are at the level of O([10.sup.-8]-[10.sup.-7]), while the b [right arrow] u[tau][[bar.[nu].sub.[tau]] induced decays are relatively rare due to the suppression caused by the CKM factor. In addition, the difference between the branching fractions of three decay modes induced by b [right arrow] c[tau][[bar.[nu]].sub.[tau]] (or b [right arrow] u[tau][[bar.[nu]].sub.[tau]]) transition is mainly attributed to the relation of total decay widths, [[GAMMA].sub.tot]([B.sup.*-]) : [[GAMMA].sub.tot]([[bar.B].sup.*0]) : [[GAMMA].sub.tot]([B.sup.*0.sub.s]) ~ 1 : 2 : 6, illustrated by (42), (43), and (44).

(2) Comparing with the SM results, one can easily find from Table 1 that B([[bar.B].sup.*] [right arrow] P[[tau].sup.-][[bar.[nu]].sub.[tau]]) are enhanced about 20% by the NP contributions of [V.sub.L] and [V.sub.R]. It can also be clearly seen from Figures 2(a) and 2(b). However, as shown in Figures 2(a) and 2(b), due to the large theoretical uncertainties caused by the form factors, the NP hints are hard to be totally distinguished from the SM results.

(3) The theoretical uncertainties can be well controlled by using the ratio [R.sup.*.sub.P] instead of decay rate due to the cancelation of nonperturbative errors; therefore [R.sup.*.sub.P] is much suitable for probing the NP hints. From the last three rows of Table 1, it can be found that the NP prediction for [R.sub.*.sub.P] significantly deviates from the SM result. In particular, as Figures 2(c) and 2(d) show, the NP effects can be totally distinguished from the SM at [q.sup.2] [greater than or equal to] 7 [GeV.sup.2] even though the theoretical errors are considered. So, future measurements on [[bar.B].sup.*] [right arrow] P[[tau].sup.-][[bar.[nu]].sub.[tau]] decays can make further test on the NP models which provide possible solutions to the [R.sub.D] and [mathematical expression not reproducible] problems.

(4) From Figures 2(e)-2(h) it can be found that the NP contribution of solution A has little effect on the observables [A.sup.P.sub.[lambda]] and [A.sup.P.sub.[theta]] in the whole [q.sup.2] region, which can be understood from the following analyses. Because the NP contribution of solution A is dominated by the left-handed coupling [V.sub.L], we can find that [absolute value of (M([[bar.B].sup.*] [right arrow] P[l.sup.-][[bar.[nu]].sub.l]))] [varies] [[absolute value of ((1 + [V.sub.L]))].sup.2] in the limit of (1 + [V.sub.L]) [much greater than] [V.sub.R]. As a result, the NP contributions (solution A) to the numerator and denominator of [A.sup.P.sub.[lambda]] and [A.sup.P.sub.[theta]] cancel each other out to a large extent. For [A.sub.P.sub.[lambda]], the cases of solutions B, C, and D are similar to solution A.

3.3. Scenario II: Effects of SL and SR Type Couplings. In this subsection, we only consider the effects of scalar interactions [S.sub.L] and [S.sub.R] and take the other NP couplings to be zero. Under the 1[sigma] constraint from the data of RD and [R.sup.*.sub.D], the allowed spaces of [S.sub.L] and [S.sub.R] are shown in Figure 3. Similar to scenario I, four solutions for [S.sub.L] and [S.sub.R] are found in scenario II, which can be seen from Figure 3(a); and the fitting results obtained by using form factors in CLN parametrization and BSW model are consistent with each other. Solutions B-D result in so large NP contributions; therefore, in the following discussion, we pay our attention to solution A, which are replotted in Figure 3(b). The numerical result of solution A is

[S.sub.L] = -[0.46.sup.+0.24.sub.-0.24], [S.sub.R] = [0.70.sup.+0.23.sub.-0.24]. (55)

Using these values, we present in Table 1 our numerical predictions of scenario II for the observables, B([[bar.B].sup.*] [right arrow] P[[tau].sup.-][[bar.[nu]].sub.[tau]]) and [q.sup.2]-integrated [R.sup.*.sub.P]. Moreover, the [q.sup.2] distributions of differential observables d[GAMMA]/d[q.sup.2], [R.sup.*.sub.P], [A.sup.P.sub.[lambda]], and [A.sup.P.sub.[theta]] are shown in Figure 4. The following are some discussions for these results:

(1) From Table 1 and Figures 4(a) and 4(b), it can be found that the B([B.sup.*] [right arrow] P[[tau].sup.-][[bar.[nu]].sub.[tau]]) and [R.sup.*.sub.P] can be enhanced about 15% compared with the SM results by the NP contributions. Similar to the situation of scenario I, the NP effect of [S.sub.L] and [S.sub.R] on [R.sup.*.sub.P] is much more significant than the one on branching fraction due to the theoretical uncertainties of [R.sup.*.sub.P] which can be well controlled. In particular, as Figures 4(a) and 4(b) show, the spectra of the SM and NP for [R.sup.*.sub.P] can be clearly distinguished at middle [q.sup.2] region.

(2) The main difference between the effects of scalar and vector couplings on the [[bar.B].sup.*] [right arrow] P[[tau].sup.-][[bar.[nu]].sub.[tau]] decays is that the former only contributes to the longitudinal amplitude, which can be found from (37). As a result, their effects on B([[bar.B].sup.*] [right arrow] P[[tau].sup.-][[bar.[nu]].sub.[tau]]) and [R.sup.*.sub.P] are a little different, which can be seen by comparing Figures 2(a)-2(d) with Figures 4(a)-4(d).

(3) Another significant difference between the scalar and vector couplings is that only the leptonic helicity amplitudes of scalar type with [[lambda].sub.l] = 1/2 survive, which can be easily found from (35) and (36). Therefore, as Figures 4(e) and 4(f) show, the scalar couplings lead to significant NP effects on [A.sup.P.sub.[lambda]], which is obviously different from predictions of vector couplings in scenario I (Figures 2(e) and 2(f)). Besides, as Figures 4(e) and 4(f) show, [S.sub.L] and [S.sub.R] couplings also have large contributions to the [A.sup.P.sub.[theta]] at all [q.sup.2] region, which is another difference with the vector couplings (Figures 2(g) and 2(h)). Therefore, the future measurements on these observables will provide strict tests on the SM and various NP models.

4. Summary

In this paper, motivated by the observed "[mathematical expression not reproducible] and [R.sub.D] puzzles" and its implication of NP, we have studied the NP effects on the b [right arrow] (c,u)[l.sup.-][[bar.[nu]].sub.l] induced semileptonic [[bar.B].sup.*.sub.u,d,s] [right arrow] P[l.sup.-][[bar.[nu]].sub.l] (P = D, [D.sub.s], [pi], K) decays in a model-independent scheme. Using the allowed spaces of vector and scalar couplings obtained by fitting to the data of [mathematical expression not reproducible] and [R.sub.D], the NP effects on the decay rate, ratio [R.sub.*.sub.P], lepton spin asymmetry, and forward-backward asymmetry are studied in vector and scalar scenarios, respectively. It is found that the vector couplings present large contributions to the decay rate and [R.sup.*.sub.P], but their effects on [A.sup.P.sub.[lambda]] and [A.sup.P.sub.[theta]] are very tiny. Different from the vector couplings, the scalar couplings present significant effects not only on the decay rate and [R.sup.*.sub.P] but also on [A.sup.P.sub.[lambda]] and [A.sup.P.sub.[theta]]. The future measurements on the [[bar.B].sup.*.sub.u,d,s] [right arrow] P[l.sup.-][[bar.[nu]].sub.l] decays will further test the predictions of the SM and NP and confirm or refute possible NP solutions to [mathematical expression not reproducible] and [R.sub.D].

https://doi.org/10.1155/2018/7231354

Data Availability

The experimental data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11475055), the Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No. 201317), and Program for Innovative Research Team in University of Henan Province (Grant No. 19IRTSTHN018).

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Qin Chang (iD), (1) Jie Zhu, (1, 2) Na Wang (iD), (1) and Ru-Min Wang (2)

(1) Institute of Particle and Nuclear Physics, Henan Normal University, Henan 453007, China

(2) Institute of Theoretical Physics, Xinyang Normal University, Henan 464000, China

Correspondence should be addressed to Qin Chang; changqin@htu.edu.cn

Received 4 June 2018; Accepted 2 August 2018; Published 27 September 2018

Academic Editor: Enrico Lunghi

Caption: Figure 1: The allowed spaces of [V.sub.L] and [V.sub.R] obtained by fitting to [R.sub.D] and [mathematical expression not reproducible]. The red and green regions are obtained by using the form factors of CLN parametrization and BSW model, respectively. (b) shows the minimal result (solution A) of the four solutions shown in (a).

Caption: Figure 2: The [q.sup.2]-dependence of the differential observables d[GAMMA]/d[q.sup.2], [R.sup.*.sub.P], [A.sup.P.sub.[lambda]], and [A.sup.P.sub.[theta]] for [B.sup.*-] [right arrow] [D.sup.0][[tau].sup.-][[bar.[nu]].sub.[tau]] and [[pi].sup.0][[tau].sup.-][[bar.[nu]].sub.[tau]] decays within the SM and scenario I.

Caption: Figure 3: The allowed spaces of [S.sub.L] and [S.sub.R] obtained by fitting to the data of [R.sub.D] and [mathematical expression not reproducible]. The other captions are the same as in Figure 1.

Caption: Figure 4: The [q.sup.2]-dependence of the differential observables d[GAMMA]/d[q.sup.2], [R.sup.*.sub.P], [A.sup.P.sub.[lambda]], and [A.sup.P.sub.[theta]] for [B.sup.*-] [right arrow] [D.sup.0][[tau].sup.-][[bar.[nu]].sub.[tau]] and [[pi].sup.0][[tau].sup.-][[bar.[nu]].sub.[tau]] decays within the SM and scenario II.
Table 1: The theoretical predictions for the branching fractions of
[[bar.B].sup.*] [right arrow] P[[tau].sup.-][[bar.[nu]].sub.[tau]]
decays and [R.sup.*.sub.P] within the SM and the two scenarios. The
first error is caused by the uncertainties of form factors, CKM
factors, and [[GAMMA].sub.tot]([B.sup.*]); and the second error given
in the last two columns is caused by the NP couplings.

Obs.                                        SM Prediction

B([B.sup.*-] [right arrow]    [0.87.sup.+0.46.sub.-0.32] x [10.sup.-8]
[D.sup.0][[tau].sup.-]
[[bar.[nu]].sub.[tau]])

B([[bar.B].sup.*0]            [2.74.sup.+1.29.sub.-0.94] x [10.sup.-8]
[right arrow]
[D.sup.+][[tau].sup.-]
[[bar.[nu]].sub.[tau]])

B([[bar.B].sup.*0.sub.s]      [5.13.sup.+3.67.sub.-2.13] x [10.sup.-7]
[right arrow]
[D.sup.+.sub.s]
[[tau].sup.-]
[[bar.[nu]].sub.[tau]])

B([[bar.B].sup.*-]           [1.42.sup.+0.79.sub.-0.50] x [10.sup.-10]
[right arrow] [[pi].sup.0]
[[tau].sup.-]
[[bar.[nu]].sub.[tau]])

B([[bar.B].sup.*0]            [0.99.sup.+0.38.sub.-0.41] x [10.sup.-9]
[right arrow]
[[pi].sup.+][[tau].sup.-]
[[bar.[nu]].sub.[tau]])

B([B.sup.*0.sub.s]            [0.95.sup.+0.65.sub.-0.40] x [10.sup.-9]
[right arrow]
[K.sup.+][[tau].sup.-]
[[bar.[nu]].sub.[tau]])

[R.sup.*.sub.D]                    [0.298.sup.+0.012.sub.-0.010]

[R.sup.* .sub.[pi]]                 [0.677.sup.+0.013.sub.-0.014]

[R.sup.*.sub.K]                     [0.638.sup.+0.017.sub.-0.015]

Obs.                                         Scenario I

B([B.sup.*-] [right arrow]      [1.04.sup.+0.54+0.06.sub.-0.38-0.05]
[D.sup.0][[tau].sup.-]                     x [10.sup.-8]
[[bar.[nu]].sub.[tau]])

B([[bar.B].sup.*0]              [3.27.sup.+1.66+0.19.sub.-1.14-0.15]
[right arrow]                              x [10.sup.-8]
[D.sup.+][[tau].sup.-]
[[bar.[nu]].sub.[tau]])

B([[bar.B].sup.*0.sub.s]        [6.13.sup.+4.51+0.35.sub.-2.48-0.28]
[right arrow]                              x [10.sup.-7]
[D.sup.+.sub.s]
[[tau].sup.-]
[[bar.[nu]].sub.[tau]])

B([[bar.B].sup.*-]              [1.71.sup.+0.91+0.09.sub.-0.63-0.07]
[right arrow] [[pi].sup.0]                 x [10.sup.-10]
[[tau].sup.-]
[[bar.[nu]].sub.[tau]])

B([[bar.B].sup.*0]              [1.08.sup.+0.55+0.06.sub.-0.37-0.05]
[right arrow]                              x [10.sup.-9]
[[pi].sup.+][[tau].sup.-]
[[bar.[nu]].sub.[tau]])

B([B.sup.*0.sub.s]              [1.14.sup.+0.78+0.06.sub.-0.46-0.05]
[right arrow]                              x [10.sup.-9]
[K.sup.+][[tau].sup.-]
[[bar.[nu]].sub.[tau]])

[R.sup.*.sub.D]               [0.355.sup.+0.015+0.020.sub.-0.011-0.016]

[R.sup.* .sub.[pi]]           [0.816.sup.+0.017+0.044.sub.-0.012-0.035]

[R.sup.*.sub.K]               [0.770.sup.+0.021+0.042.sub.-0.015-0.034]

Obs.                                         Scenario II

B([B.sup.*-] [right arrow]      [1.00.sup.+0.51+0.03.sub.-0.36-0.04]
[D.sup.0][[tau].sup.-]                      x [10.sup.-8]
[[bar.[nu]].sub.[tau]])

B([[bar.B].sup.*0]              [3.13.sup.+1.52+0.10.sub.-1.13-0.11]
[right arrow]                               x [10.sup.-8]
[D.sup.+][[tau].sup.-]
[[bar.[nu]].sub.[tau]])

B([[bar.B].sup.*0.sub.s]        [5.89.sup.+3.93+0.20.sub.-2.39-0.22]
[right arrow]                               x [10.sup.-7]
[D.sup.+.sub.s]
[[tau].sup.-]
[[bar.[nu]].sub.[tau]])

B([[bar.B].sup.*-]              [1.74.sup.+0.94+0.10.sub.-0.62-0.10]
[right arrow] [[pi].sup.0]                 x [10.sup.-10]
[[tau].sup.-]
[[bar.[nu]].sub.[tau]])

B([[bar.B].sup.*0]              [1.09.sup.+0.52+0.06.sub.-0.39-0.06]
[right arrow]                               x [10.sup.-9]
[[pi].sup.+][[tau].sup.-]
[[bar.[nu]].sub.[tau]])

B([B.sup.*0.sub.s]              [1.20.sup.+0.87+0.08.sub.-0.47-0.08]
[right arrow]                               x [10.sup.-9]
[K.sup.+][[tau].sup.-]
[[bar.[nu]].sub.[tau]])

[R.sup.*.sub.D]               [0.341.sup.+0.048+0.011.sub.-0.026-0.012]

[R.sup.* .sub.[pi]]           [0.827.sup.+0.126+0.046.sub.-0.073-0.048]

[R.sup.*.sub.K]               [0.810.sup.+0.144+0.052.sub.-0.084-0.054]
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Title Annotation:Research Article
Author:Chang, Qin; Zhu, Jie; Wang, Na; Wang, Ru-Min
Publication:Advances in High Energy Physics
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2018
Words:9868
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