Understanding the first measurement of B ( B → Kν ¯ ν )

Recently, Belle II reported on the first measurement of B ( B ± → K ± ν ¯ ν ) which appears to be almost 3 σ larger than predicted in the Standard Model. We point out the important correlation with B ( B → K ∗ ν ¯ ν ) so that the measurement of that decay mode could help restrain the possible options for building the model of New Physics. We interpret this new experimental result in terms of physics beyond the Standard Model by using SMEFT and find that a scenario with coupling only to τ can accommodate the current experimental constraints but fails in getting a desired R exp D ( ∗ ) /R SM D ( ∗ ) , unless one turns the other SMEFT operators that are not related to b → sℓℓ or/and b → sνν .


I. INTRODUCTION
Experimental studies of the angular distribution of B → K * (→ Kπ)µµ [1,2] and B → Kµµ [3,4] offered a number of observables, the study of which could help unveiling the effects of presence of New Physics (NP), i.e. physics beyond the Standard Model (BSM).It appeared, however, that interpreting several apparent deviations with respect to the Standard Model (SM) predictions required a very good control over the hadronic uncertainties, and in particular those related to the lowenergy operators coupling to cγ µ c.Those latter contributions are particularly problematic in the regions of q 2 = (p µ + + p µ − ) 2 populated by the cc resonances.Clearly, in order to resolve the effects of BSM physics from those related to the SM weak interaction in these regions, one needs to evaluate the relevant hadronic matrix element of a non-local operator, which cannot be done yet in lattice QCD.Instead, one resorts to using various assumptions such as the quark-hadron duality to treat the problem perturbatively (even though the energy window is narrow) [5][6][7], or a specific hadronic model [8][9][10][11].That problem is commonly circumvented when measuring R K ( * ) = B ′ (B → K ( * ) µµ)/B ′ (B → K ( * ) ee) [12], where B ′ is used to indicate that the partial branching fractions are measured in the interval q 2 ∈ [1.1, 6] GeV 2 , well below the first cc resonance, m 2 J/ψ = (3.097GeV) 2 .First measurements of R K and R K * , as well as of B(B s → µµ) [13][14][15][16] indicated an important departure from the SM predictions.Many models used to accommodate these deviations were used to constrain the BSM couplings, most of which also implied a significant deviation of B(B → K ( * ) ν ν) from its SM value.Very recently, however, LHCb reported R K = 0.998(90), R K * = 0.930(97) [17], thus fully consistent with R SM K ( * ) = 1.00(1) [18].Even though it is experimentally much more challenging, B → K ( * ) ν ν is theoretically cleaner than the equivalent mode with charged leptons instead of neutrinos in the final state.This is so because the coupling to the problematic operators involving cc resonances is absent.The remaining non-perturbative QCD obstacle, that both decay modes share, is a reliable theoretical estimate of the hadronic matrix elements of the local operators in the entire physical region, 0 ≲ q 2 ≤ (m B − m K ( * ) ) 2 .That task is also very challenging for lattice QCD because the available q 2 range is too large.This is why the lattice QCD results are used to constrain the parameters entering a model q 2 -dependence of the relevant form factor, necessary for the SM prediction of B(B → Kν ν).Interestingly, in the case of B → Kν ν the q 2 shape of the hadronic form factor can be checked experimentally by measuring the partial branching fractions B ′ (B → Kν ν), as discussed in Ref. [19].
In the following we will often use the ratio for which the experimental upper bounds exist [20]: Very recently, the first of these bounds was superseded by the first measurement of this decay mode by Belle II, , which appears to be 2.9σ larger than its SM estimate. 1This then leads to In this letter we discuss how this departure from the SM prediction can be interpreted in terms of generic BSM scenarios in the SMEFT framework, which then requires one to remain consistent with the stringent bounds on NP in b → s transitions arising from the measured R K ( * ) mentioned above, as well as from the measured B(B s → µµ) = 3.35(27) × 10 −8 [16], which is consistent with the SM estimate B(B s → µµ) SM = 3.66(4) × 10 −8 [23].

II. EFFECTIVE THEORY CONSIDERATIONS
The low energy effective theory relevant to the b → sνν decay is described by the effective Hamiltonian, with in a standard notation with λ t = V tb V * ts , the suitable combination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements.In the SM we know that [24][25][26][27][28].A detailed discussion of hadronic uncertainties and those arising from the choice of λ t can be found in Ref. [19].Here we will just mention that in the SM the values of decay rates of the charged B-meson read: and they do not comprise the triply Cabibbo-suppressed tree-level contribution, B + → τ + (→ K + ν) ν (also subtracted away in the experimental data analysis [21]).It is convenient to factor out the SM contribution to the branching fractions and write so that the NP piece, δB ν ν K ( * ) , can be expressed in terms of δC νiνj L,R , the BSM contribution to the left-or right-handed operator given in Eq. (5).It is then straightforward to , then we have [19,24]: The shaded gray area correspond to 1σ and 2σ of the recent Belle II result for B(B → Kνν).The red point corresponds to the SM predictions for these observables.We also show the region of experimentally excluded B(B → K * νν) values [20] (gray hatched area), as well as the region which is not accessible within the EFT approach (purple hatched area), cf.Eq. (9) .
where the sum over neutrino flavor indices is understood, i, j ∈ {1, 2, 3}.In the above expression, η K = 0, and η K * = 3.33 (7).In that way, one can easily check the response of B(B → K * νν) to the new experimentally established B(B → Kνν), which is shown in Fig. 1.Note, in particular, that we find the following relation between B → K * ν ν and B → Kν ν, which is depicted by the hatched-dark blue region labelled as "EFT" in Fig. 1 (see also [29]).By switching on either δC These solutions become strongly restricted after imposing the experimental bound B(B → K * νν) exp < 2.7 × 10 −5 [20].In particular, the large positive values of δC R are discarded, and the above ranges become: where we also give the resulting B(B → K * νν) compatible with B(B → Kνν) exp to 2σ and the experimental bound on B(B → K * νν) exp .These allowed ranges are highlighted in Fig. 1 with darker hues.At this stage we consider the BSM contributions to be neutrino flavor universal, which is a relevant information when interpreting the values for δC L,R .However, the correlation shown in Fig. 1 and the ranges of R K * νν given in Eq. ( 11) remain as such regardless of whether we assume the NP to be lepton flavor universal or not.
One can go a step further and predict the behavior of a fraction of the B → K * νν corresponding to a specific polarization state of the outgoing K * .For example, one can check how F L , the fraction of decay rate corresponding to the longitudinally polarized K * (cf.Refs.[24,30]), responds to a non-zero δC L or δC R . 2 to better emphasize the fact that F L remains insensitive to δC L ̸ = 0, whereas it becomes drastically depleted with respect to the SM value, F SM L = 0.49 (7), when δC R ̸ = 0 is chosen such that it is consistent with both B(B → Kνν) exp and the experimental bound on B(B → K * νν) exp , corresponding to negative values of δC R and the darker purple region in Fig. 2. In fact, we obtain that with δC R ̸ = 0, the value of F L gets more than 50% suppressed with respect to its SM value.From the plot in Fig. 2 we read off: This is a clear prediction that could be tested experimentally.Note, in particular, that F L has a much smaller theoretical uncertainty in the SM than the B(B → K * ν ν), since most of the form factor uncertainties cancel out in the ratio.

III. SMEFT CONSIDERATIONS
When discussing the BSM scenarios in which the new degrees of freedom enter the stage well above the electroweak scale, it is convenient to work in the SM effective field theory (SMEFT), invariant under the full [31].This allows one to relate the b → sν ν and b → sℓℓ via SU (2) L gauge symmetry [24,[32][33][34][35][36].Of all the d = 6 operators in FIG. 2. Similar to Fig. 1 we show how the fraction of B → K * νν with longitudinally polarized K * responds to the variation of δCL or δCR.Note that the yellow and purple curves correspond to those shown in Fig. 1, each associated with the allowed ranges of the Wilson coefficient given in Eq. 11.
we select those relevant to our study, namely, O O (1) where Q and L denote the quark and lepton SU (2) L doublet, respectively, while u, d, e stand for the quark and lepton weak singlets.In what follows, we will work in the flavor basis defined with diagonal down-type quark Yukawa matrix, i.e. with the CKM matrix element in the upper component of Since we are focusing on the b → s processes we will fix k = 2 and l = 3, and discuss the two subsets of operators (14,15) separately.

A. Quark bilinears and Higgs
Firstly, we consider the operators with quark-bilinears and the Higgs current, as defined in Eq. ( 14).These contribute to B → Kν ν via a tree-level induced Z-coupling to the (sγ µ b) current, as depicted in the right panel of lepton-flavor universal contributions [24], e.g. the precisely determined B s → µµ branching ratio.Moreover, a double insertion of these coefficients will also have an effect in B s − Bs mixing.Writing the Z-boson interaction Lagrangian with down-type quarks as where we find (1) with and θ W the Weinberg angle.These relations can be used to match onto the relevant low-energy effective four-fermion operators, after integration of the SM vector bosons [37].In particular, one finds, for example, that the contribution from NP to B s → µµ depends on the SMEFT operators as δC ℓiℓi 10 ∝ C (1) Hq 23 − C Hd 23 .Using the constraints from B(B s → µµ) [16] and ∆m Bs [38,39], we determine the 2σ confidence intervals for the coefficients [C Hq ̸ = 0 : C (3) We find that the B(B ± → K ± ν ν) value can only be enhanced by ≈ 20%, which is largely insufficient to accommodate the deviation shown in Belle-II data.

B. Four-fermion operators
We turn now our attention to the four-fermion operators defined in Eq. (15).Let us first explicitly rewrite L (6) SMEFT using the operators given in Eq. (15).We have: where we have not written the charged-current operators that are CKM-suppressed.By comparing the above Lagrangian with the one given in Eq. ( 4) it is easy to identify: As it was discussed in Ref. [19], several simple scenarios can be distinguished if we switch one of the above SMEFT operators at the time.For example, by allowing only C (1) lq ̸ = 0 we get the simplest scenario with a Z ′ boson coupled only to the left-handed fermions.If, instead, the Z ′ is considered to be member of the weak triplet then one has only C (3) lq corresponds to the BSM model with a triplet of scalar leptoquarks with hypercharge Y = 1/3 [41].

IV. PHENOMENOLOGY
So far we did not touch on the issue of lepton flavor.In this Section we discuss simple BSM scenarios and check whether or not one can build a simple scenario compatible with experimental data, and most importantly with B(B → Kνν) exp .In doing so we will separately treat the lepton flavor conserving case in which the couplings are diagonal (i = j), from the lepton flavor violating, i.e. with non-diagonal couplings (i ̸ = j) being non-zero.≡ C (1,3) lq lq and C ld lead to the same prediction (purple curve in the plot) which cannot be reconciled with B(B → Kνν) exp .Instead, by only allowing C (1) lq ̸ = 0 (orange curve) one can get its small fraction to be consistent with both B(B → Kνν) exp and B(Bs → µµ) exp to 2σ, which is also highlighted in orange in the plot.The red point corresponds to the SM values (i.e.C (1) the couplings to muon only, i.e. i = j = 2.The best suited quantity to check the validity of such a solution is B(B s → µµ) for which we need to include the modification of the Wilson coefficient C 10 = C SM 10 + δC 10 , that in terms of C (1,3) lq and C ld writes: In Fig. 4 we show how B(B s → µµ) varies when switching C (1,3) lq or C ld one at the time.Clearly, any C (3) lq or C ld cannot alone enhance B(B → Kνν) SM in order to be consistent with experiment.The situation improves when C (1) lq ̸ = 0 in which case one can reach the 2σ edge of B(B → Kνν) exp , while remaining consistent with B(B s → µµ) exp to 2σ as well.We find that the resulting allowed C (1) lq /Λ 2 ∈ [0.0129, 0.0131] TeV −2 , corresponding to the highlighted darker orange band in Fig. 5, and which translates into a very stringent bound: 2.16 ≤ B(B → K * νν) × 10 5 ≤ 2.19.This scenario, however, seems quite unlikely even though it would be consistent with the recent LHCb results regarding the lepton flavor universality R K ( * ) .In fact, consistency with R K ( * ) would require C

B. New Physics coupling to two or three lepton species
With respect to the previous case, the situation considerably improves if we turn on the couplings to both electrons and muons, in such a way that C ≡ C, which is what we plot in Fig. 5. Like before, R K ( * ) remains unchanged with respect to its SM value, and overall consistency with the b → s data is achieved.However, as a result of the severe constraint arising from B(B s → µµ) exp , in both cases we obtain C/Λ 2 ∈ [0.012, 0.013] TeV −2 , which then results in either R K * νν ∈ (3.6, 3.9) e,µ or R K * νν ∈ (4.8, 5. 3) e,µ,τ .Since both these intervals are larger than R K * (exp) νν < 2.7, all three scenarios discussed so far cannot meet the experimental constraints.
Another important shortcoming of the two scenarios discussed here, is that the contribution to ratios R ( * ) , where l ∈ (e, µ), is absent, since this needs to go through the triplet operator C (3) lq (cf.Eq. ( 20)).We remind the reader that the most recent averages of experimental values for R D ( * ) [42], R D = 0.257 (29), R D * = 0.284( 12) (23) are larger than predicted in the SM [43][44][45], leading to R D /R SM D = 1.19 (10) and R D * /R SM D * = 1.15 (5), which can be averaged for our purpose to: From Figure 4 one can clearly see that the constraints from B s → µµ cannot be respected in the case C One concludes that modification of the couplings to τ 's is what is desired in order to be consistent with experimental data.More specifically one needs C  ̸ = 0 such that B(B → Kνν) and R D ( * ) are simultaneously consistent with their experimental values is possible as we avoid the constraint coming from B(B s → µµ) exp .We note that in this situation from which we see that C lq ̸ = 0 is significantly enhanced by V cs /V cb = 24 (1).The solution is shown in Fig. 6.We find that the acceptable C (3) lq falls in: As can be seen from ( 25), and in view of Eq. ( 24), only negative C lq values will lead to R exp D ( * ) /R SM D * > 1, and we obtain R D ( * ) /R SM D * ∈ [1.06, 1.12], also highlighted in Fig. 6.
However, the C lq range allowed by R lequ , e.g.Ref. [60].As it is well known, accommodating R (exp) D ( * ) results in an increase of decay rates to a pair of τ -leptons in the final state [46].Indeed we find that the scenarios compatible with R (exp) D ( * ) and B(B → Kνν) (exp) , and consistent with the C (3) lq values given in Eq. ( 26) yield:

B(B →
If we drop the requirement of compatibility with R  [9,11] .(28) Note, in particular, that we only consider O (3) lq in the predictions shown in Eq. ( 27)- (28).However, the connection between the b → sν τ ντ and b → sτ τ transitions is more general, as the operators O (1) lq and O ld also contribute to both transitions.Therefore, if B(B → Kνν) is confirmed to be considerably larger than the SM value, one should expect a sizable deviation from the SM in B(B s → τ τ ) and B(B → K ( * ) τ τ ) as well.The only exception to this conclusion is the scenario where C

D. Lepton flavor violating case
We can now assume that the difference between B(B → Kνν) exp and its SM prediction can be described by turning on the off-diagonal couplings to lepton flavors (i ̸ = j).This can be done with a number of various assumptions.Here, as an example, we turn either C As a consequence the decay rates of the corresponding lepton flavor violating modes will be significant.By using the expressions derived in Ref. [47] we obtain: not far from but consistent with the experimental bound [48,49], B (B → Kµτ ) exp < 4.8 × 10 −5 at 90% C.L.

E. Expectations from concrete models
So far we have been agnostic about the possible origin of these effective operators, not focusing on any concrete model [41,50,51].It is worth mentioning that a BSM model with a weak triplet of scalar leptoquarks S 3 cannot be made consistent with all the data discussed here.The reason is well known, namely it cannot give a significant increase to R D ( * ) [50,51].The U 1 vector leptoquark, in a simplified model approach (involving couplings to the left-handed operators only), cannot be made consistent with data.This is so because at tree-level C (1) lq , or δC L = 0, which is excluded by the new Belle II result for B(B → Kνν) exp .It is important to stress, however, that the U 1 being a massive vector necessarily calls for a UV completion, which typically requires also the inclusion of additional fermionic degrees of freedom.The full theory of the U 1 has been extensively studied, and in particular it has been shown that the relation C lq is broken at one-loop, leading to an increase of B(B → Kνν) of up to ≈ 50% with respect to the Standard Model, given the current values of R D ( * ) [52,53].Another possibility is to take the couplings to τ 's verifying C (1) lq , which is valid in a scenario with the so called S 1 leptoquark.In that case one gets R D ( * ) /R SM D * ≈ 1.02, thus much smaller than in the case discussed in Sec.IV C. The way out would then be to turn on the right handed couplings [50].Similarly, in the case of the so called R 2 leptoquarks one needs to turn on the right-handed couplings to quarks (and left-handed to τ ), which in the SMEFT language means that C ld ̸ = 0 [54]

F. Summary
In this letter, we discuss the possible consequences on B(B → K * νν) due to the recent Belle II measurement of B(B → Kνν) which was found to be nearly 3σ larger than predicted.We find that the values B(B → K * νν) ≲ 1.8 × B(B → K * νν) SM could only occur if the coupling to the right-handed operator bR γ µ s R is non-zero, while larger branching fractions would be consistent with couplings to either left-handed or right-handed operators (or a combination of both).
By relying on SMEFT we show that the increase in B(B → Kνν) cannot be described by switching on the couplings to one or more lepton flavors if that choice comprises the coupling to muons.The reason is that B(B s → µµ) exp provides a very stringent constraint on the values of the Wilson coefficient C (1) lq , which is the only one that can provide the desired enhancement to B(B → K ( * ) νν).In the SMEFT Lagrangian the Wilson coefficients relevant to b → sℓℓ and b → sνν decay modes are also related to the semileptonic b → cℓν, so that one should make sure that the resulting R D ( * ) > R SM D * , as suggested by experiment.For that it is necessary that C (3) lq ̸ = 0 or to have the scalar and tensor operators that are not related to b → sℓℓ and b → sνν [50].We find that a significant increase in R D ( * ) /R SM D * can be achieved if we allow only the coupling to τ and not to other species.In this way one can find the agreement with all of the abovementioned experimental constraints, including R K ( * ) , except that R D ( * ) /R SM D * ≈ 1.05.We also considered a possibility with the off-diagonal couplings to lepton species.Accommodating B(B → Kνν) exp in this setting implies a large B(B → Kµτ ), not too far from the current experimental bound.
We need to emphasize that all the couplings discussed in this paper are well within the ranges allowed by the experimental data collected in the region of high-p T tails of the differential distribution of pp → ℓℓ (+ soft jets), cf.Ref. [56].
One final comment regards the experimental value used in our analysis, which is the new measurement of B(B → Kνν) only.In [21], also an average with the previous bound from Belle-II was provided [20], giving B(B → Kνν) exp = 1.4(4) × 10 −5 .How considering this average affects our conclusions can be mostly read off the same plots we presented, the main effect being a higher compatibility with the bound on R K * ν in the left-handed scenario, and an increased (although still below the SM) range for F L .Similarly, the τ -only coupling scenario would call for a larger right-handed contribution in order to enhance the contributions to R

NOTE ADDED
While this paper was in writing the results of studies similar to our's were released in Refs.[29,57].In particular the authors of Ref. [29] interpret the new Belle II results in terms of the SMEFT operators reaching the conclusions which agree with ours for the most part.

FIG. 1 .
FIG. 1.The correlation between B(B → K * νν) and B(B → Kνν) decays with respect to the variation of δCL or δCR.The shaded gray area correspond to 1σ and 2σ of the recent Belle II result for B(B → Kνν).The red point corresponds to the SM predictions for these observables.We also show the region of experimentally excluded B(B → K * νν) values[20] (gray hatched area), as well as the region which is not accessible within the EFT approach (purple hatched area), cf.Eq.(9) .

( 1 )
Hq ] 23 , [C (3) Hq ] 23 , and [C Hd ] 23 , switching on one at a time at Λ = 1 TeV and accounting for the renormalization group evolution from Λ down to µ = m b [40].The allowed intervals for the coefficients can then be translated onto intervals for the B → Kν ν and B → K * ν ν branching fractions, C

FIG. 5 .
FIG. 5. Similar to Fig. 4, except that in this case we turn on all of the lepton species with [C] 11 = [C] 22 = [C] 33 where C stands for either C (1) lq , C (3) lq or C ld .
Quite obviously, in that case R K ( * ) remains at its SM value and thus consistent with recent experimental analyses at LHCb.Compatibility with B(B → Kνν) exp is so improved that even a small 1σ overlap with B(B → Kνν) exp can be reached.To go deep into the 1σ region of both B(B → Kνν) exp and B(B s → µµ) one can take the flavor universal situation, and assume C

lq 33 .
Moreover, this lepton-flavor universal scenario predicts R D ( * ) = R SM D ( * ) which disagrees with Eq. (24).Also the case C with data, since it predicts R D ( * ) < R SM D ( * ) in correlation with an enhanced B(B → Kνν).
C. New Physics coupling to taus onlyFinding C incompatible with Eq.(26).Nonetheless, including other operators contributing to b → cτ ν, but not to b → sν ν, one can find scenarios allowing us to simultaneously explain the deviations in R νν K and R D ( * ) .Example of such scenarios are C

FIG. 6 .
FIG. 6.The curve shows R D ( * ) /R SM D * as a function of B(B → Kνν) varying C (3) lq 33 (purple) or C ld 33 (blue).The highlighted purple region correspond to C (3) lq 33 allowed by B(B → Kνν) exp which becomes restricted to the hatched region once the condition R K * (exp) νν < 2.7 is imposed.That range is smaller than R exp D ( * ) /R SM D * to 2σ (horizontal gray area).The red point corresponds to the SM.

ACKNOWLEDGMENTS
This project has received funding /support from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 860881-HIDDeN and No 101086085-ASYMMETRY.The work of LA is supported by the Swiss National Science Foundation (SNF) under contract 200020-204428.LA is also grateful to the Mainz Institute for Theoretical Physics (MITP) of the Cluster of Excel-lence PRISMA+ (Project ID 39083149), for its hospitality and its partial support during the completion of this work.
. It thus can explain the new experimental result B(B → Kνν) exp .Moreover, by combining the constraints B(B → Kνν) exp and [∆m s ] exp /[∆m s ] SM we get the bound m R2 ≲ 3 TeV and m S1 ≲ 4.5 TeV.This scenario, however, fails to explain R exp D ′ boson coupled to left-handed τ and sγ µ P R b, again giving rise to C ld ̸ = 0 [55].While it can accommodate all of the b → s constraints, it does not contribute to b → cτ ν.
( * ) /R SM D * .Another similar model is the one with a Z