#### Abstract

The significant divergence between the SM predictions and experimental measurements for the ratios, with , implies possible hint of new physics in the flavor sector. In this paper, motivated by the “ puzzle” and abundant data samples at high-luminosity heavy-flavor experiments in the future, we try to probe possible effects of new physics in the semileptonic decays induced by transitions in the model-independent vector and scalar scenarios. Using the spaces of NP parameters obtained by fitting to the data of and , the NP effects on the observables including branching fraction, ratio , lepton spin asymmetry, and lepton forward-backward asymmetry are studied in detail. We find that the vector type couplings have large effects on the branching fraction and ratio . Meanwhile, the scalar type couplings provide significant contributions to all of the observables. The future measurements of these observables in the decays at the LHCb and Belle-II could provide a way to crosscheck the various NP solutions to the “ puzzle”.

#### 1. Introduction

Thanks to the fruitful running of the factories and Large Hadron Collider (LHC) in the past years, most of the mesons decays with branching fractions have been measured. The rare -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 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 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 reported by the Heavy-Flavor Average Group (HFAG) are [7]which deviate from the SM predictions(see [8] in the first equation of (2) and [9] in the second equation of (2)) at the levels of and errors, respectively. Moreover, when the correlations between and are taken into account, the tension would reach up to level [7]. Besides, the ratio has recently been measured by the LHCb collaboration [10], which also shows an excess of about from the central value range of the corresponding SM predictions . In addition, another mild hint of NP in the induced decay has been observed by the BaBar and Belle collaborations [11–14]; the deviation is at the level of [15].

The large deviations in 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 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 mesons, the vector ground states of system, mesons, with quantum number of and [66–69], also can decay through the transitions at quark-level. Therefore, in principle, the corresponding NP effects might enter into the semileptonic decays as well. The decay occurs mainly through the electromagnetic process , and the weak decay modes are very rare. Fortunately, thanks to the rapid development of heavy-flavor experiments instruments and techniques, the 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 and the weak decays with branching fractions are hopeful to be observed [70, 73, 74]. Moreover, the LHC experiment also will provide a lot of experimental information for weak decays due to the much larger beauty production cross-section of collision relative to collision [75].

Recently, some interesting theoretical studies for the weak decays have been made within the SM in [73, 74, 76–82]. In this paper, motivated by the possible NP explanation for the puzzles, the corresponding NP effects on the semileptonic 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 () of semileptonic decays are explored by using the spaces of various NP couplings obtained through the measured .

Our paper is organized as follows. In Section 2, after a brief description of the effective Lagrangian for the transitions, the theoretical framework and calculations for the 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 decays in a model-independent scheme. The most general effective Lagrangian at for the () transition can be written as [19, 21, 40, 46]where is the Fermi coupling constant, denotes the CKM matrix elements, and is the negative/positive projection operator. Assuming the neutrinos are left-handed and neglecting the tensor couplings, the effective Lagrangian can be simplified aswhere and are the effective NP couplings (Wilson coefficients) defined at . 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 decay can be written as the product of leptonic () and hadronic () tensors,where the superscripts and 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, corresponds to the operator . For convenience in writing, these superscripts are omitted below. Inserting the completeness relationthe product of and can be further expressed asHere, is the polarization vector of the virtual intermediate states, which is boson in the SM and named as in this paper for convenience of expression. The quantities and are Lorentz invariant and therefore can be evaluated in different reference frames. In the following evaluation, and will be calculated in the -meson rest frame and the center-of-mass frame, respectively.

##### 2.2. Kinematics for Decays

In the -meson rest frame with daughter -meson moving in the positive -direction, the momenta of particles and areFor the four polarization vectors, , one can conveniently choose [83, 84]where and , with and being the momentum transfer squared, are the energy and momentum of the virtual . The polarization vectors of the initial -meson can be written as

In the center-of-mass frame, the four momenta of lepton and antineutrino pair are given aswhere , , and is the angle between the and three-momenta. In this frame, the polarization vector takes the form

##### 2.3. Hadronic Helicity Amplitudes

For the decay, the hadronic helicity amplitudes and are defined bywhich describe the decay of three helicity states of meson into a pseudoscalar meson and the four helicity states of virtual . It should be noted that in , (15) and (16), should always be equal to .

For transition, the matrix elements of the vector and axial-vector currents can be written in terms of form factors and aswith the sign convention . Furthermore, using the equations of motion,one can write the matrix elements of scalar and pseudoscalars currents asin which and are the running quark masses.

Then, by contracting above hadronic matrix elements with the polarization vectors in the -meson rest frame, we obtain five nonvanishing helicity amplitudesIt is obvious that only the amplitudes with survive.

##### 2.4. Leptonic Helicity Amplitudes

Expanding the leptonic tensor in terms of a complete set of Wigner’s -functions [9, 83, 87], can be rewritten as a compact formin which and run over 1 and 0, and run over their components, and massless right-handed antineutrinos with . In (27), are the leptonic helicity amplitudes defined asIn the center-of-mass frame, taking the exact forms of the spinors and polarization vectors, we finally obtain four nonvanishing contributions

##### 2.5. Observables of Decays

With the amplitudes obtained in above subsections, we then present the observables considered in our following evaluations. The double differential decay rate of decay is written aswhere the factor is caused by averaging over the spins of initial state . Using the standard convention for -function [88], we finally obtain the double differential decay rates with a given leptonic helicity state , which areUsing (35) and (36), one can get the explicit forms of various observables of decays as follows: (i)The differential decay rate is(ii)The dependent ratio is where denotes the light lepton.(iii)The lepton spin asymmetry is(iv)The forward-backward asymmetry is

The SM results can be obtained from above formulae by taking .

In the following evaluations, in order to fit the NP spaces, we also need the observables of 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]For the well-measured Fermi coupling constant , the masses of mesons and leptons, and the running masses of quarks at , we take their central values given by PDG [88]. The total decay widths (or lifetimes) of 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 dominates the decays of meson, we take the approximation ; the latter has been evaluated within different theoretical models [90–96]. In this paper, we adopt the most recent results [95, 96]

Then the residual inputs are the transition form factors, which are crucial for evaluating the observables of and decays. For the 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 decays; numerically, their values read [7]However, for the 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 and transitions. Using the inputs , , , , and , we obtain the results at ,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 read [98, 99]where is the state of with quantum number of ( and 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 ,* i.e.*, only considering the NP effects of couplings(ii)Scenario II: taking ,* i.e.*, only considering the NP effects of 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 processes and for the NP is absent. In the following discussion, the allowed spaces of NP couplings are obtained by fitting to and (1), with the data varying randomly within their 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 and Type Couplings

In this subsection, we vary couplings and while keeping all other NP couplings to zero. Under the constraints from the data of and , the allowed spaces of new physics parameters, and , are shown in Figure 1. In the fit, the 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].

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From Figure 1, we find that (i) the allowed spaces of 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 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 , and the coupling presents sizable positive (negative) NP contribution to fit data. The situation of solution B is similar, but only 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

Using the values of NP couplings given by (54), we then present our theoretical predictions for and -integrated in Table 1, in which the SM results are also listed for comparison. The -dependence of differential observables , , , and for and decays is shown in Figure 2; the case of and ( and ) is similar to the one of () decay, not shown here. The following are some discussions and comments:(1)From Table 1, it can be seen that the branching fractions of induced decays are at the level of , while the 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 (or ) transition is mainly attributed to the relation of total decay widths, , illustrated by (42), (43), and (44).(2)Comparing with the SM results, one can easily find from Table 1 that are enhanced about 20% by the NP contributions of and . 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 instead of decay rate due to the cancelation of nonperturbative errors; therefore 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 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 even though the theoretical errors are considered. So, future measurements on decays can make further test on the NP models which provide possible solutions to the and 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 and in the whole region, which can be understood from the following analyses. Because the NP contribution of solution A is dominated by the left-handed coupling , we can find that in the limit of . As a result, the NP contributions (solution A) to the numerator and denominator of and cancel each other out to a large extent. For , the cases of solutions B, C, and D are similar to solution A.

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##### 3.3. Scenario II: Effects of and Type Couplings

In this subsection, we only consider the effects of scalar interactions and and take the other NP couplings to be zero. Under the constraint from the data of and , the allowed spaces of and are shown in Figure 3. Similar to scenario I, four solutions for and 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 isUsing these values, we present in Table 1 our numerical predictions of scenario II for the observables, and -integrated . Moreover, the distributions of differential observables ,