New physics in b→ se+e−: A model independent analysis

versality (LFU) violating ratios RK and RK∗ by the LHCb collaboration. In 2014, the LHCb collaboration reported the first measurement of the ratio RK ≡ Γ(B+ → K+μ+μ−)/Γ(B+ → K+e+e−) in the di-lepton invariant mass-squared, q2, range 1.1 ≤ q2 ≤ 6.0 GeV2 [1]. The measured value 0.745 −0.074(stat.)±0.036(syst.) deviates from the SM prediction of ≈ 1 [2–4] by 2.6σ. Including the Run-II data and an update of the Run-I analysis, the value of RK was updated in Moriond-2019. The updated value 0.846 −0.054 (stat.) +0.016 −0.014(syst.) [5] is still ' 2.5σ away from the SM. The hint of LFU violation is further observed in another flavor ratio RK∗ . This ratio RK∗ ≡ Γ(B0 → K∗0μ+μ−)/Γ(B0 → K∗0e+e−) was measured in the low (0.045 ≤ q2 ≤ 1.1 GeV2) as well as in the central (1.1 ≤ q2 ≤ 6.0 GeV2) q2 bins by the LHCb collaboration [6]. The measured values are

The measured values suffer from large statistical uncertainties and hence consistent with the SM predictions. The ratios R K and R K * are essentially free from the hadronic uncertainties, making them extremely sensitive to new physics (NP) in b → se + e − or/and b → sµ + µ − transition(s).
Further, there are a few anomalous measurements which are related to possible NP in b → sµ + µ − transition only. These include measurements of angular observables, in particular P 5 , in [8][9][10] and the branching ratio of B s → φ µ + µ − [11]. By virtue of these measurements, it is natural to assume NP only in the muon sector to accommodate all b → s + − data. A large number of global analyses of b → s + − data have been performed under this assumption [12][13][14][15][16][17][18]. NP amplitude in b → sµ + µ − must have destructive interference with the SM amplitude to account for R K , R K * < 1. Hence the NP operators in this sector are constrained to be in vector/axial-vector form. The global analyses found three different combinations of such operators which can account for all the data. Possible methods to distinguish between these allowed NP solutions are investigated in refs. [19][20][21][22]. However, the predicted value of R low K * for the solutions with NP only in b → sµ + µ − still differs significantly from the measured value. This requires presence of NP in b → se + e − along with b → sµ + µ − , see for e.g, [23,24].
While the LFU ratios R K and R K * are theoretically clean, other observables in b → sµ + µ − sector which show discrepancy with SM, in particular the angular observables B → K * µ + µ − and B s → φµ + µ − , are subject to significant hadronic uncertainties dominated by undermined power corrections. So far, the power corrections can be estimated only in the inclusive decays. For exclusive decays, there are no theoretical description of power corrections within QCD factorization and SCET framework. The possible NP effects in these observables can be masked by such corrections.
The disagreement with the SM depends upon the guess value of power corrections. Under the assumption of ∼ 10% non-factorisable power corrections in the SM predictions, the measurements of these observables show deviations from the SM at the level of 3-4σ. However, if one assumes a sizable non-factorisable power corrections, the experimental data can be accommodated within the SM itself [25][26][27][28]. It is therefore expected that these tensions might stay unexplained until Belle-II can measure the corresponding observables in the inclusive b → sµ + µ − modes [26].
Therefore, if one considers the discrepancies in clean observables in b → s + − sector, which are R K and R K * , then NP only in b → se + e − is as natural solution as NP in b → sµ + µ − sector. In this work, we consider this possibility and perform a model independent analysis with NP restricted to b → se + e − sector. To the best of our knowledge, this is the first work where NP only in b → se + e − transition is considered in a model independent manner. In this scenario, we need the NP operators to increase the denominators of R K and R K * . Hence, the need for interference with SM amplitude is no longer operative. We consider NP in the form of vector/axial-vector (V/A), scalar/pseudoscalar (S/P) and tensor (T) operators. We show that solutions based on V/A operators predict values of R K /R K * , including R low K * , which are in good agreement with the measured values. The scalar NP operators can account for the reduction in R K but not in R K * and hence are ruled out. The coefficients of pseudoscalar operators are very severely constrained by the current bound on the branching ratio of B s → e + e − and these operators do not lead to a reduction of R K /R K * . It is not possible to get a solution to the R K /R K * problem using only tensor operators [29] but a solution is possible in the form of a combination of V/A and T operators, as shown in ref. [30]. In this work, we will limit ourselves to solutions involving either one NP operator or two similar NP operators at a time. We will not consider solutions with two or more dissimilar operators.
The paper is organized as follows. In Sec. II, we discuss the methodology adopted in this work. The fit results for NP in the form of V/A operators are shown in Sec. III. In Sec. III A, we discuss methods to discriminate between different V/A solutions and comment on the most effective angular observables which can achieve this discrimination. Finally, we present our conclusions in Sec. IV.

II. METHODOLOGY
We analyze the R K /R K * anomalies within the framework of effective field theory by assuming NP only in b → se + e − transition. We intend to identify the set of operators which can account for the measurements of R K /R K * . We consider NP in the form of V/A, S/P and T operators and analyze scenarios with either one NP operator (1D) at a time or two similar NP operators (2D) at a time.
In the SM, the effective Hamiltonian for b → s + − transition is where G F is the Fermi constant, V ts and V tb are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and P L,R = (1 ∓ γ 5 )/2 are the projection operators. The effect of the operators O i , i = 1−6, 8 can be embedded in the redefined effective Wilson coefficients (WCs) as C 7 (µ) → C eff 7 (µ, q 2 ) and C 9 (µ) → C eff 9 (µ, q 2 ). We now add following NP contributions to the SM effective Hamiltonian, where C NP, e 9,10 , C , e 9,10 , and C e SS,SP,P S,P P,T,T 5 are the NP WCs. Using simple symmetry arguments, we can argue that S/P operators can not provide a solution to R K /R K * discrepancy. The operators containing the quark bilinearsb can lead to B → Ke + e − transition but not B → K * e + e − transition. Such operators can account for R K but not R K * .
On the other hand, the operators containing the quark pseudoscalar bilinear sγ 5 b can not lead to B → Ke + e − . These operators can not account for R K . In addition, the contribution of these operators to B s → e + e − is not subject to helicity suppression. Hence, the coefficients C e P S and C e P P are constrained to be very small. The current upper limit on B(B s → e + e − ) < 9.4 × 10 −9 at 90% C.L., leads to the condition |C e P S | 2 + |C e P P | 2 0.01, whereas one needs 120 |C e P S | 2 + |C e P P | 2 345, 9 |C e P S | 2 + |C e P P | 2 29, to satisfy the experimental constraint on R low K * and R central K * respectively. Therefore, we will not consider S/P operators in our fit procedure.
We define the χ 2 function as Here uses the most precise form factor predictions obtained in the light cone sum rule (LCSR) [37,38] approach. The non-factorisable corrections are incorporated following the parameterization used in Ref. [36,37]. These are also compatible with the calculations in Ref. [39].
We obtain the values of NP WCs by minimizing the χ 2 using CERN minimization code Minuit [40,41]. We perform the minimization in two ways: (a) one NP operator at a time and (b) two NP operators at a time. Since we do the fit with fifteen data points, it is expected that an NP scenario with a value of χ 2 min ≈ 15 provides a good fit to the data. We also define pull = ∆χ 2 where ∆χ 2 = (χ 2 SM − χ 2 min ). Since χ 2 SM ≈ 27, any scenario with pull 3.0 can be considered to be a viable solution. In the next section, we present our fit results and discuss them in details.

III. VECTOR/AXIAL-VECTOR NEW PHYSICS
There are four cases for one operator fit and six cases for two operators fit. For all of these cases, we list the best fit values of WCs in Table I   and C NP,e 10 scenarios provide a good fit to the b → se + e − data. However, the other two 1D scenarios, C ,e 9 and C ,e 10 , fail to provide any improvement over the SM. Therefore, we reject them on the basis of ∆χ 2 or pull. In the case of 2D framework, all six combinations improve the global fit as compared to the SM.
We now impose the stringent condition that a NP solution must predict the values of R K , R low K * and R central K * to be within 1σ of their measured values. In order to identify solutions satisfying this condition, we calculate the predictions of R K /R K * for all good fit scenarios. The predicted values of these quantities are listed in Table II  0.91 ± 0.    Table III.

A. Discriminating V/A solutions
The differential distribution of the four-body decay B → K * (→ Kπ)e + e − can be parametrized as the function of one kinematic and three angular variables. The kinematic variable is q 2 = (p B − p K * ) 2 , where p B and p K * are respective four-momenta of B and K * mesons. The angular variables are defined in the K * rest frame. They are (a) θ K the angle between B and K mesons where K meson comes from K * decay, (b) θ e the angle between momenta of e − and B meson and (c) φ the angle between K * decay plane and the plane defined by the e + − e − momenta. The full decay distribution can be expressed as [42,43] where I(q 2 , θ e , θ K , φ) = I s 1 sin 2 θ K + I c 1 cos 2 θ K + (I s 2 sin 2 θ K + I c 2 cos 2 θ K ) cos 2θ e +I 3 sin 2 θ K sin 2 θ e cos 2φ + I 4 sin 2θ K sin 2θ e cos φ +I 5 sin 2θ K sin θ e cos φ +(I s 6 sin 2 θ K + I c 6 cos 2 θ K ) cos θ e + I 7 sin 2θ K sin θ e sin φ +I 8 sin 2θ K sin 2θ e sin φ + I 9 sin 2 θ K sin 2 θ e sin 2φ.
The twelve angular coefficients I i . Therefore, there could be twelve CP averaged angular observables which can be defined as [42,43] The longitudinal polarization fraction of K * , F L , depends on the distribution of the events in the angle θ K (after integrating over θ e and φ) and the forward-backward asymmetry, A F B , is defined in terms of θ e (after integrating over θ K and φ). We can write these two quantities in terms of S (a) i as follows In addition to the S i observables, one can also investigate the NP effects on a set of optimized observables P i . In fact, the observables P i are theoretically cleaner in comparison to the form factors dependent observables S i . These two sets of observables are related to each other through the following relations [44] [15,19] 0.341 ± 0.020 0.338 ± 0.022 0.325 ± 0.020 0.349 ± 0.020 Table IV: Average values of B → K * e + e − angular observables A F B and F L in SM as well as for the allowed NP V/A solutions listed in Table. III.  Table IV and the q 2 plots are shown in Fig. 2. From the predictions, we observe the following features: • In low q 2 region, the SM prediction of A F B (q 2 ) has a zero crossing at ∼ 3.5 GeV 2 . For the NP solutions, the predictions are negative throughout the low q 2 range. However, the A F B (q 2 ) curve is almost the same for S-I and S-II whereas for S-III, it is markedly different.
Therefore an accurate measurement of q 2 distribution of A F B can discriminate between S-III and the remaining two NP solutions.
• In high q 2 region, the SM prediction of A F B is 0.368 ± 0.018 whereas the predictions for the three solutions are almost zero. If A F B in high q 2 region is measured to be small, it provides additional confirmation for the existence of NP, which is indicated by the reduced values of R K and R K * . All the three NP solutions induce a large deviation in A F B , but the discriminating capability of A F B is extremely limited.
• The S-I and S-II scenarios can marginally suppress the value of F L in low q 2 region compared to the SM whereas for S-III, the predicted value is consistent with the SM. In high q 2 region, F L for all three scenarios are close to the SM value. Hence F L cannot discriminate between the allowed V/A solutions.
Hence we see that neither A F B nor F L have the power to discriminate between all the three allowed V/A NP solutions. Therefore, we now study optimized observables P i in B → K * e + e − decay. In particular, we investigate the distinguishing ability of P 1,2,3 and P 4,5,6,8 . We compute the average values of these seven observables for the SM along with three NP scenarios in two different q 2 bins, q 2 ⊂ [1.1, 6.0] and [15,19] GeV 2 . These are listed in Tab V. We also plot these observables as a function of q 2 for the SM and the three solutions. The q 2 plots for P 1,2,3 and • The observable P 1 in the low q 2 region can discriminate between all three NP solutions, particularly S-I and S-II. The sign of P 1 is opposite for these scenarios. Hence an accurate measurement of P 1 can distinguish between S-I and S-II solutions. In fact, measurement of P 1 with an absolute uncertainty of 0.05 can confirm or rule out S-I and S-II solutions by more than 4σ. In the high-q 2 region, the predictions for all allowed solutions are consistent with the SM.
• The observable P 2 can be a good discriminant of S-III provided we have handle over its q 2 distribution in [1.1, 6.0] GeV 2 bin. In this bin, P 2 (q 2 ) has a zero crossing at ∼ 3.5 GeV 2 for the SM prediction whereas there is no zero crossing for any of the allowed solutions. Scenarios S-I and S-II predict large negative values for P 2 , around −0.3 whereas the S-III predicts  [15,19] 0.001 ± 0.000 0.036 ± 0.002 −0.036 ± 0.003 0.000 ± 0.000  and R K * anomalies.
• The P 3 observable in the low-q 2 region cannot discriminate between the allowed solutions.
However, in the high q 2 region, P 3 can uniquely discriminate the three solutions. In particular, the prediction of P 3 for S-III in the high q 2 is the same as the SM whereas the predictions for S-I and S-II are exactly equal and opposite.
• The P 4 in low-q 2 region can only distinguish S-II solution from the other two NP solutions and the SM. In high-q 2 region, it has a poor discrimination capability.
• In the low q 2 bin, P 5 has a zero crossing at ∼ 2 GeV 2 and has an average negative value in the SM. For all three NP solutions, there is no zero crossing in P 5 . Further, these scenarios scenarios which fail to explain R K and R K * simultaneously. The depletion in pull for these allowed solutions is due to inconsistency between the measured and predicted values of P 5 .
• In the both low and high-q 2 regions, the NP predictions for P 6 for all three scenarios are consistent with the SM.
• The P 8 in the low-q 2 region does not have any discrimination capability. The predicted values for all solutions are consistent with the SM. In the high-q 2 region, both S-III and SM predict P 8 values close to zero whereas S-I and S-II predict large positive and negative values, respectively.
From this detailed study of the behavior of the optimized observables P i , we find that both P 1 and P 4 at low-q 2 have the best capability to discriminate between all the three V/A solutions. The predicted values of P 1 are equal and opposite for S-I and S-II and of a much smaller magnitude for S-III. Moreover, each of the predicted values is appreciably different from the SM prediction.
Their magnitudes are quite large ∼ (0.3 − 0.6) with a relative theoretical uncertainty of about 10%.
Hence, a measurement of the variable P 1 , with an experimental uncertainty of about 0.05, will not not only confirm the presence of new physics in the b → se + e − amplitude but also can determine the correct WC of the NP operators. In the case P 4 , the predictions of all the three solutions have the same sign but their magnitudes are quite different. The theoretical uncertainty in the predictions is quite low too. So, P 4 observable also has a good capability to distinguish between the three V/A solutions.

IV. CONCLUSIONS
In this work, we intend to analyze R K ( * ) anomalies by assuming NP only in b → se + e − decay.
The effects of possible NP are encoded in the WCs of effective operators with different Lorentz structures. These WCs are constrained using all measurements in the b → se + e − sector along with lepton-universality-violating ratios R K /R K * . We show that scalar/pseudoscalar NP operators and tensor NP operators can not explain the data in b → se + e − sector. We consider NP in the form of V/A operators, either one operator at a time or two similar operators at a time. We find that there are several scenarios which can provide a good fit to the data. However, there are only three solutions whose predictions of R K /R K * , including R K * in the in the low-q 2 bin (0.045 GeV 2 ≤ q 2 ≤ 1.1GeV 2 ), match the data well. In order to discriminate between the three allowed V/A solutions, we consider several angular observables in the B → K * e + e − decay. The three solutions predict very different values for the optimized observables P 1 and P 4 in the low-q 2 bin. Both these observables also have the additional advantage that the theoretical uncertainties in their predictions are less than 10%. Hence a measurement of either of these observables, to an absolute uncertainty of 0.05, can lead to a unique identification of one of the solutions..