Decoding the B → Kνν excess at Belle II: kinematics, operators, and masses

An excess in the branching fraction for B + → K + νν recently measured at Belle II may be a hint of new physics. We perform thorough likelihood analyses for different new physics scenarios such as B → KX with a new invisible particle X , or B → Kχχ through a scalar, vector, or tensor current with χ being a new invisible particle or a neutrino. We find that vector-current 3-body decay with m X ≃ 0 . 6 GeV—which may be dark matter—is most favored, while 2-body decay with m X ≃ 2 GeV is also competitive. The best-fit branching fractions for the scalar and tensor cases are a few times larger than for the 2-body and vector cases. Past BaBar measurements provide further discrimination, although the best-fit parameters stay similar.

Assuming the excess is due to NP, there are two simple interpretations of the "νν", which is undetected experimentally.One (the "2-body scenario") is that "νν" is a single new particle, X, which is invisible or decays invisibly experimentally [21,22].Another (the "3body scenario") is that it is a pair of particles χ 1 χ 2 , where each χ i may be either a new particle or a SM (anti-)neutrino (e.g.[10,11,15,19]).The 3-body scenario can be further categorized into three sub-scenarios depending on whether the b → s transition happens via a scalar, vector, or tensor operator.The question, therefore, is which scenario seems to be most favored by data on statistical grounds.
In this work, we perform likelihood analyses mainly using the binned ITA data of Ref. [7], and identify the preferred ranges of the mass and branching fraction in each scenario.Then, comparing the best-fit points between different scenarios, we find that the vector-current 3-body scenario with m χ1,2 ≃ 0.6 GeV is most statistically favored, but the 2-body decay with m X ≃ 2 GeV is also comparable.We also find that in the scalarand tensor-current 3-body scenarios the best-fit branching fractions are a few times larger than the aforementioned value by Belle II.This is because the scalar and tensor event distributions have a very different shape from a SM-like (vector) event distribution assumed in the Belle II analysis.Finally, we also explore further discrimination between scenarios by including the past BaBar data.

II. TESTING NEW PHYSICS WITH BELLE II
Given the latest ITA, Belle II reports the results in terms of a variable (q 2 rec ) that is only loosely related to q 2 ≡ (p B − p K ) 2 , we carefully analyze distortions in the reported signal shape compared to the true q 2 distributions expected in the various scenarios.We will also find it essential to take into account the effects of a very non-uniform signal efficiency.We further combine the results with past measurements to narrow down the possibilities.
Since the 4-momentum of the tagged B meson is not reconstructed in ITA, the missing mass-squared, q 2 = (p B − p K ) 2 , is approximated by q 2 rec : where E B,K and ⃗ p B,K are the energies and momenta of the observed B and K in the B B center-of-mass frame.The resolution in E K measurement is O(1)%, leading to negligible smearing in q 2 rec .Given the bin size of 1 GeV 2 chosen in Ref. [7], the smearing in q 2 rec almost entirely stems from the absence of the ⃗ p K (q 2 ) • ⃗ p B term in the inference of q 2 rec .By numerical simulation, we find that events with a given value of q 2 get distributed almost Number of B + → K + νν events at Belle II [7] (black dots, with error bars shown) after background subtraction, as a function q 2 rec , using inclusive tagging in the η(BDT2) > 0.98 signal region.In blue is shown the SM distribution [5,6].The red line shows the predicted distribution of events for a 3-body decay B + → K + χχ by a vector current, with mχ = 0.6 GeV and B(B + → K + χχ) = 3.2×10 −5 in addition to the SM.The green line shows B + → K + νLχ via a scalar current for mχ = 0 and B(B + → K + χχ) = 7.3 × 10 −5 .The yellow line shows the distribution for the 2-body decay B + → K + X with B(B + → K + X) = 0.7 × 10 −5 and mX = 2 GeV.uniformly in q 2 rec as follows: where ⃗ p * K (q 2 ) is the momentum of the K in the rest frame of the B meson, |⃗ p B | = 0.33 GeV, and E B = 5.29 GeV.For example, B → KX events with m X = 2 GeV would be spread over 2.5 GeV 2 < q 2 rec < 5.5 GeV 2 as shown in Fig. 1.Similarly, the SM distribution in Fig. 1 is obtained by applying the momentum spreading to a vector current for massless fermions with a total branching ratio as given in Eq. (1).The signal yield of B + → K + νν is extracted from the top-right panel in Fig. 18 of Ref. [7] by subtracting the B B and continuum backgrounds, which is shown as black dots in Fig. 1.
The signal efficiency, ϵ(q 2 ), is another essential input to process the various scenarios.Although two efficien-cies are reported in the latest Belle II analysis, they are not applicable to the samples with the highest "BDT 2 " cut, BDT 2 > 0.98, where the excess from the SM background is visible.Therefore we deduce the efficiencies to reproduce the shape of the reported B → Kνν distribution.We tune the q 2 -dependent efficiency such that the vector-current mediated distribution with the bestfit normalization of the ITA analysis, B = 2.7 × 10 −5 , is multiplied by q 2 -dependent efficiency and binned with the q 2 rec smearing, and then the resultant distribution is matched with the one reported in Fig. 18 of Ref. [7].The resulting efficiency, shown in Fig. 2 after interpolation in q 2 , is similar in shape to the one with BDT 2 > 0.95 reported in the earlier Belle II analysis [9].
For the statistical combination, we construct the binned likelihood, L = i f P (N obs i ; N ex i ), using Poisson statistics for the ith bin i includes the background and the SM B → Kνν with as well as the yield of new physics, where N B ± = 3.99 × 10 8 at Belle II.We obtain the loglikelihood, −2 ln L, which is denoted as χ 2 for simplicity.This method accounts for only the statistical uncertainty.As a check, we obtain B = (2.6±0.4(stat))×10−5 for B + → K + νν for ITA, which is reasonably consistent with B = (2.7 ± 0.5(stat)) × 10 −5 reported by Belle II.

III. TESTING 2-BODY SCENARIO
First, we calculate the log-likelihood χ 2 , based on the 2-body decay B + → K + X scenario.We find the minimum χ 2 min and obtain the preferred parameter space by evaluating χ 2 − χ 2 min .Since there is a significant mass preference, we perform the 2D fit in the m X -B space, and the contours with 1, 2, and 3σ are shown in the upper-left pane in Fig. 3.
Although our method correctly accounts for the statistical uncertainty, the systematic uncertainty can be comparable based on the Belle II report.We expect that including systematic uncertainty lowers the χ 2 by a factor of two, resulting in the allowed region being enlarged by ∼ 40%, but the tendency of the preferred parameters would not be affected.
To extract the preferred mass, we use the profile loglikelihood: finding the best χ 2 for each mass by profiling B, which is shown in the bottom left panel.We find a narrow range of mass, m X = 1.97 ± 0.05(stat) GeV.The example of distribution in the ITA is shown in Fig. 1.A similar method is applied for the branching fraction, leading to B = 0.79 +0. 16 −0.13(stat) × 10 −5 .This is in accordance with a recent study of the 2-body scenario [21], but it does not agree with the other [22].We wish to emphasize that the expected number of events in the For each plot in the upper panel, the likelihoods profiling one variable (along the B direction and along the mNP direction) are provided in the lower panel.The three columns correspond to different NP scenarios: on the left panel is the 2-body scenario B + → K + X, for which mNP = mX , the center panel is for the 3-body decay B + → + χχ, mNP = mχ mediated by a vector current, while the right column corresponds to the 3-body decay B + → K + χRνL, = mχ, where νL is a SM neutrino.FIG. 4. Normalized distributions for B + → K + χ1χ2 via scalar, vector, and tensor operators as functions of q 2 for massless χ1,2 (solid lines) as well as massive χ1,2 (dotted line) with an equal mass of 0.6 GeV.The distributions do not take experimental efficiency into account.
given q 2 rec binning is calculated following Eq.( 5) so that we appropriately fit the signal with the data of the ITA.

IV. TESTING 3-BODY SCENARIOS
The same method is applied to the 3-body case B + → K + χ 1 χ 2 where we have multiple scenarios to consider.We assume that χ 1,2 are fermionic in this work.However, some operators that contribute to B → K decays also contribute to B → K * decays.Belle-II has recently obtained an upper limit on the branching fraction B(B 0 → K * νν) < 1.8 × 10 −5 [2], but a measurement of the branching fraction has not been obtained yet.We consider operators that do not have significant effects on B → K * : 1. Scalar operator: the interaction is given by where Λ S is a heavy scale.We strictly distinguish this case from the pseudo-scalar case with bγ 5 s as that would contribute only to B → K * and not to B → K, while the scalar case contributes only to B → K and not to B → K * .In our work, we consider the case where χ 1 = ν L , which is massless for our purpose, and χ 2 is a new fermion, possibly massive.Note that in Ref. [23] it was shown that a scalar current distribution for the related kaon decay mode K + → π + νν implies lepton number violation (LNV) for SM-invariant operators, since such a current can only be generated via ∆L = 2 odd mass-dimension operators.We consider the possibility of χ 2 being the right-handed neutrino N , for which case there exists non-LNV operators at dimension-6 which generate a scalar current, e.g. the operator 2. Vector operator: the interaction is given by where Λ V is a heavy scale.χ 1 and χ 2 may be a pair of SM ν L ν L or that of new (massive) neutral fermions.The quark part of this operator is a pure vector, which is crucially different from a SMlike V −A current.As shown in Fig. 2 of [15], an additional V −A contribution that fits the B → K excess would be severely excluded by the absence of a corresponding excess in B → K * , while the pure vector case has the least impact on B → K * and is still allowed by data.Therefore, for our interest in identifying the best scenarios, we do not consider any axial-vector component in the quark bilinear.
3. Tensor operator: the interaction is given by where Λ T is a heavy scale.As in the scalar case above, we consider χ 1 = ν L and χ 2 to be a new neutral fermion.
In each of these three cases, we compute the q 2 spectrum of the decay rate dΓ/dq 2 , and the distributions are given in Fig. 4. The relevant matrix elements and form factors for the B → K decay are given in [25,26] and also in Appendix A. In the ITA, the distributions are modified due to the efficiency and smeared by q 2 rec .Two representative cases are shown in Fig. 1.
We find the favored parameter space of the 3-body scenarios using the same likelihood analysis as we perform for the 2-body one, and the results for the cases of the vector and scalar operators are shown in Fig. 3.For the vector case, the fit is best at m χ = 0.62 +0.10 −0.09GeV and B = (3.5 +0.8 −0.6 (stat)) × 10 −5 (see the middle panel), which is close to the measurement made at Belle II.Since this massive χ be can be stable, it is a candidate of dark matter.
For the scalar operator, the best-fit mass is m χ = 0, while the branching fraction is preferred to be high, (7.9 +1.6 −1.5 (stat) × 10 −5 ) (see the right panel).This is due to that the q 2 -spectrum peaking at high q 2 ∼ 20 GeV 2 , as seen in Fig. 4, is multiplied by the efficiency which drops at high q 2 , see Fig. 2. Therefore, most of the signal events are discarded by the ITA, which statistically favors a higher branching fraction to explain the excess.
For the tensor case, since the q 2 -distribution is very similar to the scalar case, as seen in Fig. 4, the fitted results are also similar, and the preferred branching fraction is B = (6.4± 1.2(stat)) × 10 −5 .We do not show the tensor case in Fig. 3 for the sake of brevity, but we quote χ 2 min for the fit to the decay B → Kχν in Table I.For the best-fit points in Fig. 3, we find Λ V ≈ 6.5 TeV for the 3-body vector current and Λ S ≈ 4.1 TeV for the 3-body scalar current.The 2-body decay occurs at tree-level, for which we find an effective coupling λ X ≈ 3 × 10 −4 for L ⊃ −λ X bsX.
As already addressed, our analysis does not include systematic uncertainty.The systematic uncertainty would enlarge the allowed parameter space by about 40% unless the uncertainty has a strong q 2 dependence.
Based on the values χ 2 min of different scenarios, shown in Table I I.The likelihood minima χ 2 min for 2-body scenario (2b), and several 3-body scenarios: two vector cases with mχ = 0 (V) and mχ ≃ 0.6 GeV (V ′ ); the scalar (S) and tensor (T) cases.The rows correspond to the choice of analyzed datasets: first only the Belle II ITA data [7], then adding the BaBar data [4] within the signal region (SR), and finally including outside the SR, q 2 < 0.8m 2 B .The bold ones indicate the best-fit or competitive scenarios with α(H0 = V ′ ) > 10%.The χ 2 values for the SM are also shown.Only statistical uncertainties are considered.
hypothesis H 0 can be evaluated by [27] Within the Belle II ITA, we find that the best scenario is the 3-body decay with vector-operator and m χ = 0.62 GeV, and the best 2-body scenario is also competitive.However, the scalar-current 3-body scenario is disfavored, and adding mass to χ makes it even worse.A SM-like shape, i.e. the vector-current 3-body scenario with massless χ, is also disfavored.Even if unaccounted uncertainties, such as the systematics, lower χ 2 , say by a factor of two, the scalar and vector operators with massless χ would still have α < 10% against the best-fit scenario.a. Combining with other measurements.-Having considered the Belle II ITA, we now combine the analysis with past BaBar measurements of B → hνν with h = K 0 , K + based on the conventional hadronic tagging [4].The shape with the BaBar data (after the background is subtracted) with several new physics scenarios in Fig. 5.A major difference from the latest Belle II ITA is that the analysis uses q 2 rather than q 2 rec , and the resolution of q 2 is significantly better than the adopted bin size.Therefore, we can ignore smearing effects.
For their signal region s B ≡ q 2 /m 2 B < 0.3, we calculate the log-likelihood χ 2 for this data, which we then combine with the result of Belle II ITA.The combined fits are shown as magenta dashed lines in Fig. 3, and the values of χ 2 min are in Table I.While the preferred parameter regions barely change in all scenarios, the difference of χ 2 min between the 2-body scenario and the best 3-body scenario increases from 2 to 5 because the signal of 2-body decay should be localized in the second bin is seen.
Although s B > 0.3 (q 2 ≳ 8.4 GeV 2 ) is outside the  [4] with background subtracted, as a function of q 2 .The colored lines show the same benchmark scenarios as Fig. 1.
signal region and requires more careful treatment of the uncertainties, it is interesting to examine this region for further discrimination.For some 3-body scenarios with the scalar and tensor operators, we expect an excess in high q 2 because the efficiency in this BaBar study stays flat until q 2 ∼ 20 GeV 2 while the efficiency of the Belle II ITA is very small for q 2 > 10 GeV 2 .Thus, increasing the branching fraction to fit the Belle II ITA would be in tension with B + → K + νν result at BaBar, as seen in Fig. 5.For this reason, the χ 2 min values of these scenarios are significantly worsened, see Table I.The best scenario is still the 3-body decay with massive χ via the vector operator, but the 2-body one is comparable.
Other data sets that may be relevant for us, but are not included, are the following: 1.In [7], Belle II also reported the q 2 result in the HTA, see Fig. 6.Since the data has a small overlap with the ITA dataset, we are unable to combine them.However, we see the trend that the third bin seems to have a tension with the 2-body scenario with m X = 2 GeV.
2. Belle results via hadronic [1] and semileptonic [2] tagging: the number of events is measured in bins of E ECL , which is the residual energy deposited in the electromagnetic calorimeter (ECL).Since E ECL is not related to q 2 , we are unable to include this data for testing the various scenarios.
3. BaBar results via semileptonic tagging [3]: The binning is done in bins of |p K |, the magnitude of the momentum of the K + in the center-of-mass frame.While |p K | has a one-to-one correspondence to q 2 rec , the final results are reported as an average of 20 different BDT analyses.Due to this unconventional statistical treatment, we cannot safely add this data to our analysis.

V. CONCLUSION
In this work, we perform a binned-likelihood analysis of different NP scenarios mainly based on the recent Belle II ITA data on the decay B + → K + νν.The NP scenarios we consider are: (a) 2-body scenario B → KX, and (b) the 3-body scenario B → Kχ 1 χ 2 .For (b) we consider several operators (scalar, vector, and tensor) as well as several masses for the new particles.
We find it crucial to account for the fact that the Belle II ITA data is binned not in the momentum transfer q 2 but in q 2 rec , and for the non-uniform efficiency.We augment our analysis with past BaBar data.Our results are listed in Table I and in Fig. 3.
As seen in Table I, the best fit NP scenario is a 3-body decay of the B + via a vector current into K + and a χχ pair, where m χ ≃ 0.6 GeV.This χ can be stable and hence a possible dark matter candidate.Table I also tells us that the 2-body scenario is not much worse, and so it still remains competitive.
The right panel of Fig. 3, shows the preferred parameter region in the scalar-current 3-body scenario.Massless χ fits best, for a branching fraction of B(B + → Kχ R ν L ) ∼ 8 × 10 −5 .This is about a factor of 3 higher than the value in Eq. ( 2).This is an especially clear demonstration of the importance of considering the nonuniform signal efficiency in inferring the branching fraction of NP from the reported excess at Belle II.
FIG.1.Number of B + → K + νν events at Belle II[7] (black dots, with error bars shown) after background subtraction, as a function q 2 rec , using inclusive tagging in the η(BDT2) > 0.98 signal region.In blue is shown the SM distribution[5,6].The red line shows the predicted distribution of events for a 3-body decay B + → K + χχ by a vector current, with mχ = 0.6 GeV and B(B + → K + χχ) = 3.2×10 −5 in addition to the SM.The green line shows B + → K + νLχ via a scalar current for mχ = 0 and B(B + → K + χχ) = 7.3 × 10 −5 .The yellow line shows the distribution for the 2-body decay B + → K + X with B(B + → K + X) = 0.7 × 10 −5 and mX = 2 GeV.

5 BK 1 5 FIG. 3 .
FIG.3.The upper panel shows 2D plots of χ 2 − χ 2 min in the B-mNP plane for different scenarios.For each plot in the upper panel, the likelihoods profiling one variable (along the B direction and along the mNP direction) are provided in the lower panel.The three columns correspond to different NP scenarios: on the left panel is the 2-body scenario B + → K + X, for which mNP = mX , the center panel is for the 3-body decay B + → + χχ, mNP = mχ mediated by a vector current, while the right column corresponds to the 3-body decay B + → K + χRνL, = mχ, where νL is a SM neutrino.

6 FIG. 5 .
FIG. 5.Number of B → Kνν events (black, error bars shown) for 471×10 6 B B pairs at BaBar[4] with background subtracted, as a function of q 2 .The colored lines show the same benchmark scenarios as Fig.1.

FIG. 6 .
FIG.6.Number of B + → K + νν events at Belle II[7] (black dots, with error bars shown) after background subtraction using hadronic tagging, as a function of q 2 .The colored lines show the same benchmark scenarios as in Fig.1.
, we can distinguish the preferred ones out of them.Testing one hypothesis H 1 against the null