Discrepancies in simultaneous explanation of Flavor Anomalies and IceCube PeV Events using Leptoquarks

Leptoquarks have been suggested to solve a variety of discrepancies between the expected and observed phenomenon. In this paper, we show that the scalar doublet Leptoquark with Hypercharge 7/6 can simultaneously explain the recent measurement of $R_{K}$, $R_{K^*}$, the excess in anomalous magnetic moment of muon, and the observed excess in IceCube HESE data. For appropriate choice of couplings, the flavor anomalies are generated at one-loop level and IceCube data is explained via resonant production of the Leptoquark. Several constraints from LHC searches are imposed on the model parameter space.


Introduction
Leptoquarks are solution to the problem of matter unification which appear naturally in many theories beyond the Standard Model (SM). For example, scalar quarks in R-Parity Violating Supersymmetry (RPV) have Leptoquark like Yukawa couplings [1] whereas vector Leptoquarks arise in Grand Unification Theories (GUT) based on SU (5) and SO(10) [2][3][4]. The unique feature of Leptoquarks is that they couple simultaneously to Standard Model (SM) quarks and leptons, thus providing ample testing grounds and applications to variety of discrepancies between theory and experiments.
In Section 2 we describe the model of Leptoquark and motivate the texture of the coupling matrices that has been used in this paper. In Section 3 we explain the excess in (g − 2) µ using this model. In Section 4 we explain the recent measurement of R K and R K * within our framework followed by the explanation for IceCube High Energy Starting Events (HESE) in Section 5. In Section 6 we discuss the results of this analysis and obtain the parameter space for simultaneous explanation. In the next section, we do the LHC analysis for the benchmark point and obtain the constraints. In the end, we conclude with some model-dependent and model-independent statements.

Model Description
In this paper, we consider the scalar Leptoquark ∆ = (3, 2, 7/6) whose interactions with the SM fields is given as [44], where y L(R) are the Yukawa-like couplings of the Leptoquark. For simplicity, we have assumed the couplings to be real. We have not shown the kinetic and Higgs interactions for brevity, however they are relevant for the discussion that follows. We refer the reader to reference [44] for a comprehensive analysis. We can rewrite (1) in terms of the mass eigenstates ∆ 5/3 and ∆ 2/3 , where the superscript denotes electric charge. In terms of these states, the Lagrangian (1) is written as, where V and U are the CKM and PMNS matrices respectively. In common literature [44], this model is also known as R 2 .
The observed negligible branching ratios of the flavor violating decays of leptons (for example, τ → µγ and µ → eγ) put stringent constraints on the inter-generation couplings of the Leptoquark. For all practical purposes, this implies that It has been argued in previous works that the this Leptoquark model results in R K ≈ 1, and R K * ≈ 1 because of the tree level contribution to b → sµµ [10]. This clearly contradicts the recent measurements by LHCb. It was pointed out in [9] that, if one assumes then the tree level contribution is negligible and the leading contribution of comes from a one-loop process. It will be shown in Section 4 that this results in R K < 1, and R K * < 1 which is in agreement with the latest experiments. We chose the former solution as it is also favoured by (g − 2) µ . As mentioned in [9], non-zero y cµ L results in tree level contribution to b → clν l which contradicts the observed R(D) and R(D ). Hence, we also assume that y cµ L = 0.
In order to avoid undesired contribution to other rare decays of the B meson, such as b → dl + l − , we assume that y dµ R ≈ 0.
With these constraints, the coupling matrices are, For brevity, we will use y uµ L = λ 1 , y tµ L = λ 2 , and y bµ R = λ 3 for the remainder of this paper. We will also use M 1 (M 2 ) to denote the mass of ∆ 5/3 (∆ 2/3 ).
In subsequent sections, it will be pointed out that the LHC constraints limit M 1 ≥ 1100 GeV. For our analysis, we take the lower limit and generate constraints on the remaining parameters. If future searches increase the lower limit considerably, the expressions will change accordingly. Having said that, there are only four free parameters in our model In the subsequent sections, we investigate various constraints on the model parameters coming from (g − 2) µ , flavor anomalies, IceCube data, and LHC.
The experimentally measured value of the anomalous magnetic moment of muon is slightly larger than the prediction from the Standard Model. This discrepancy has been attributed to a variety of new physics scenarios [31-33, 45, 46]. At present, the difference is [47], In this model, both of the mass eigenstates contribute to (g − 2) µ and one can estimate the contribution using expressions given in [44]. Keeping M 1 = 1100 GeV, the leptoquark contribution to (g − 2) µ is given as, where the approximation is obtained using the benchmark point M 2 = 1000 GeV. From the above expressions one can see that the leading contribution does not depend on M 2 . It is also clear that the product λ 2 λ 3 ≈ 10 −3 gives the correct estimate for (g − 2) µ . In Section 6 we use a ∆ µ = δa µ to constrain the parameter space of the model.

Flavor Anomalies
In the last two decades, loop-induced b → s transitions have been playing an active role in understanding the physics beyond the Standard Model. Starting from the first observation of B → K * γ, many decays involving b → s transitions have been observed. Two of the key observables for LFU violating decays of the B meson are R K and R K * , defined as It was shown in [48] that within the SM, the hadronic uncertainties in these expressions cancel which results in R K ,R K * ≈ 1. However recent measurement of R K * by LHCb has reported 2.1 − 2.3σ and 2.3 − 2.5σ deviations in the low-q 2 (0.045 -1.1 GeV 2 ) and centralq 2 (1.1 -6 GeV 2 ) regions, respectively [6]. A deviation of 2.6σ from SM has also been reported in R K [5]. We use the standard prescription of effective Hamiltonian to evaluate the contribution of the Leptoquark to R K and R K * .
The most general effective Hamiltonian for b → sl − l + is given as where O i are the operators and C i are the Wilson Coefficients (WCs) which can be written as where δC i represent the shifts due to new physics. Global analyses have been performed to fit δC i to the experimental results which yield interesting correlations between various WCs [49,50]. The operators relevant for the model are The expressions for all other operators can be found in [51]. As usual, the doubly CKM suppressed contributions from V ub V * us have been neglected.
For the model in consideration, the Leptoquark contributes to b → sµ + µ − at one-loop level ( Fig. 1) and results in non-zero δC 9 and δC 10 only.
we can write, where, , The contribution of up-quark is CKM suppressed. We have used Package-X [52] and the unitary gauge to evaluate the loop-functions F 1 and F 2 .
To evaluate R K and R K from the WCs, we use the simplified expressions from [53] and obtain, Immediately one can observe that the solution −1 < A 1 < 0 and A 2 = 0 is consistent with latest results. This was also the conclusion in [9].
Recent measurement B s → µ − µ + by LHCb is in close agreement with the SM and provides a constraint on the model [54]. In the operator basis (14), branching ratio of B s → µ − µ + can be written as [55] BR In general, this process gets contribution from C 10 , C ( ) S and C ( ) P as well. However, we are ignoring them as these WCs are zero in the SM as well as the model under consideration. In the SM, B(B s → µ + µ − ) is (3.65 ± 0.23) × 10 −9 [56] while LHCb has measured it to be 2.8 +0.7 −0.6 × 10 −9 [54]. For the model considered in this paper, (24) is using parameters given in [55]. Again, one can see that the solution −1 < A 1 < 0 and A 2 = 0 is consistent with the experiments. With these expressions, one can write the observables in terms of the couplings as, In passing, one can note that these expressions do not explicitly depend on λ 1 . This is due to the fact that the term proportional to λ 1 will enter the expression due to u-quark in the loop which is CKM suppressed. Henceforth, the term 'flavor anomalies' will be used to refer to R K and R K * with imposed constraints from BR(B s → µ − µ + ).

IceCube PeV Events
During the first four years of its operation, the IceCube neutrino observatory at the South pole has observed more number of PeV events than expected. This has resulted in a lot of interesting studies in various fields [57][58][59]. Resonant production of Leptoquark by interactions of astrophysical neutrinos with partons has been proposed as a possible explanation of the excess in PeV events at IceCube [38][39][40][41][42][43]. In the model considered in this paper, the following neutrino interactions are possible: Neutral Current (NC) Like:ν i u It is important to distinguish between the CC and NC interactions due the difference in their deposited energy signature [60,61]. Ideally speaking, one should also distinguish between shower and track events as the observed PeV events are only shower type. However, one can attribute this to the smallness of statistics and hence we do not consider this difference.
The number of events due to Leptoquark contribution in the deposited energy interval where T = 1347 days is the total exposure time, N A = 6.023×10 23 cm −3 water equivalent is the Avogadro's Number, and ch denotes the interaction channel (NC or CC). Other terms in the expression are discussed in [60]. For each neutrino or anti-neutrino flavor, an isotropic, power-law flux parametrized as is assumed. The best fit values from IceCube [62] (32) are obtained using likelihood analysis of the data from 10 TeV -10 PeV. We use the central values in our analysis.
It is evident from the structure of coupling matrices (8) that the model only admits interactions between incoming antineutrino (neutrino) with u-and t-(anti-u-and anti-t-) quarks. It is seen that the Parton Distribution Function (PDF) of t-quark is negligible as compared to that of u-quark. Hence, we only consider interaction with u-quark in our analysis. The differential cross-section for this process is given as [41] dσ dy where s = 2M N E ν , and U(x, Q 2 ) is the PDF of u-quark in an isoscalar proton evaluated at energy Q 2 . In terms of the valence and sea quark distributions, one can write [38] We have used the Mathematica package MSTW [63] to obtain these PDFs.
The dependence of event rate on couplings is captured by Given the mass of the Leptoquark (M 2 ) and the couplings, we are now in a position to estimate the contribution of Leptoquark to the IceCube HESE events. We use the standard χ 2 analysis to estimate the couplings that provide the best fit to the data. In order to estimate whether adding Leptoquark contribution results in a better or worse fit to data, we use the statistic which represents the percentage change in χ 2 . We only use the data for which non-zero number of events are observed at IceCube.

A simultaneous explanation
In this model, we have four free parameters as was pointed out before. However, the Leptoquarks state ∆ 2/3 does not feature in any explanation of the flavor anomalies and hence these do not depend on M 2 . It is also seen that for M 2 ∈ (600 − 1400) GeV, the dependence of (g − 2) µ on M 2 is very weak. Hence, the flavor anomalies and (g − 2) µ effectively depend only on the three free couplings in the model. In Fig. 2, we have shown the parameter space that explains the flavor anomalies a (g − 2) µ for M 1 = 1100 GeV and M 2 = 1000 GeV. It can be seen from Fig. 2 that the resolution to flavor anomalies requires λ 2 ∼ O(1) whereas (g − 2) µ constrains λ 3 ∼ O(10 −3 ) for λ 1 < 6. Using this, and equations (32)- (36), one sees that the number of events at IceCube only depends on the coupling λ 1 . Since, ∆ 5/3 does not feature in the explanation for IceCube, these predictions are independent of M 1 and only depend on M 2 . In Fig. 3, we show the variation of the statistic δ with M 2 for various choice of coupling λ 1 . It can be seen that a Leptoquark of mass 800 -1400 GeV can give 20-35% improvement to the fit. In Fig. 4, we show the contribution of Leptoquark for the benchmark point M LQ =1 TeV, λ 1 ≈ 6. which gives δ 35.  It is evident that for the aforementioned choices of Leptoquark parameters, one can satisfactorily explain the observed excess in the IceCube HESE Data. However, such an explanation requires large couplings and TeV scale Leptoquarks. Such a scenario should be testable at LHC and is the subject of study in the next section.

LHC constraints
Since Leptoquarks carry color charge, they can by singly or pair produced in pp collisions. Subsequent decays of these Leptoquarks in the detector will give rise to jets, leptons, and neutrinos. This gives very interesting final states of the form jjll, jjlν, jlν, jjν, jνν, etc and has been the subject of various studies [64][65][66][67][68][69][70][71][72][73][74]. As these neutrinos are not seen by the detector, they appear as a Missing Transverse Energy (MET). For the LHC analysis, we have implemented the model using FeynRules (v2) [75] and simulate the above processes using MadGraph (v5) [76] which uses Pythia (v8) [77] for parton showering. We then use CheckMATE (v2) [78] to find the value of statistical parameter, r defined as for several points in the parameter space. Here, S and ∆S represents signal and its uncertainty. The numerator represents 95% confidence limit on number of events obtained using CheckMATE and the denominator represents 95% experimental limits on the number of events. The approximate functional form is obtained using linear interpolation. Parameter space with r ≥ 1 is excluded and the results are summarized in Fig. 5.
Constraints from jjll: When the Leptoquarks are pair produced in pp collisions, each Leptoquark can decay into a charged lepton and a quark. Recently, ATLAS collaboration performed a search for new physics signature of lepton-jet resonances based on √ s = 13 TeV data [79] wherein pair production of Leptoquarks was studied based on events like eejj and µµjj. The analysis gives an upper limit on branching ratio of first and second generation Leptoquark to ej and µj respectively. Although, our model has inter-generation couplings, we use these limits to constrain the free parameters in our model. We find that, as it couples to only second generation of leptons. This puts a lower limit on mass of Leptoquark as, M 1 ≥ 1100 TeV.
We use the lower limit to generate other constraints and for flavor analysis. For ∆ 2/3 state, which does not provide any constraints from this analysis.
Constraints from jjνν: When the Leptoquark state ∆ 2/3 is pair produced, each can decay into a neutrino and a quark giving rise to a peculiar Dijet + MET signature. The parameters M 1 and λ 2 are fixed from flavor observables and this process only depends on M 2 and λ 1 . We use the 13 TeV ATLAS search [80] to find constraints on this parameter space.
Constraints from jνν: If the Leptoquark ∆ 2/3 is singly produced, it can decay into a quark and a neutrino giving rise to Monojet signal at the LHC. Again, this process only depends on the parameters M 2 and λ 1 . We use the 8 TeV ATLAS search [81] to find constraints on this parameter space.
Other Constraints: We find that the Monojet constraints are strong enough to rule out the entire parameter space that explains IceCube PeV events and we do not provide results for other processes. However, in passing, we note that the constraints from jlν final state are much stronger. This maybe relevant for future tests of Leptoquark models.

Conclusion
The discrepancy in anomalous magnetic moment of muon, the observed excess in PeV events at IceCube, and the lepton flavor universality violation in B decays are some of the biggest challenges facing the Standard Model. A simultaneous explanation for these problems is desirable. An ad-hoc solution such as Leptoquarks, if it can successfully address these issues, will shed more light on the unification scenarios that contain them. One such attempt was made in this paper using a scalar doublet Leptoquark. The peculiar feature of this model is that the flavor anomalies are explained at one-loop level. Because of the loop suppression, one does not require either very small couplings or very heavy Leptoquarks. We find that one can explain the B-anomalies R K and R K * with O(1) coupling and TeV scale Leptoquark. In the past, similar parameters have been invoked to explain IceCube events and a unified explanation seemed possible. However, we find that in order to explain IceCube data, one needs Leptoquark coupling to first generation quarks and neutrinos. This coupling will give rise to Monojet and Dijet signals at LHC, both of which are severely constrained. Because of this, any attempt to explain IceCube events using such Leptoquarks would be in conflict with LHC data. This conclusion was also reached for a Scalar Triplet in [38], and for Scalar Singlet in [71]. Any unification scenario that has Leptoquark like states, IceCube explanation in such theories (e.g. R-Parity Violating MSSM [43]) should also be in conflict.