Eur. Phys. J. C (2020) 80:204 https://doi.org/10.1140/epjc/s10052-020-7754-8

Regular Article - Theoretical Physics

A comparative study of the S1 and U1 leptoquark effects in the light regime

Ilja Doršner1, Svjetlana Fajfer2,3, Monalisa Patra2,a 1 University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture in Split (FESB), Rudera- Boškovi´ca 32, 21000 Split, Croatia 2 Jožef Stefan Institute, Jamova 39, P.O. Box 3000, 1001 Ljubljana, Slovenia 3 Department of Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia

Received: 17 December 2019 / Accepted: 19 February 2020 © The Author(s) 2020

Abstract We study the phenomenology of two lepto- Grand Unified Theories based on Pati–Salam model [2,3], , the SU(2) singlets S1 and U1, with SU(5) [4], SO(10) [5,6], with R-parity vio- regard to the latest experimental data from the low-energy fla- lation [7], and composite models [8,9]. Most recently LQs vor physics measurements, LHC, and the IceCube have been singled out as the most promising candidates experiment. We consider a scenario when scalar (vector) lep- for the explanations of anomalies in the low-energy fla- toquark S1 (U1) couples exclusively to the and vor physics experiments [10–13] concerning the B the (charged ) of all flavors, where the lep- semileptonic decays that hint at the flavor universality toquark in question couples to the SM lepton doublets. The violation. The anomalies in question, i.e., RD(∗) and RK (∗) , couplings of S1 (U1) to the up-type quarks and the charged usually require that the LQs couple strongly to the heavy leptons (neutrinos) are in turn uniquely determined via SU(2) quarks and leptons. This particular regime has thus been symmetry. We find that the most important constraints on studied extensively in the context of both scalar [14–19] and the leptoquark parameter space originate from flavor physics vector [17,18,20–25] LQs with few notable exceptions [26]. measurements, followed by the LHC search limits that take Weare interested, in this manuscript, in the scenarios when over the flavor physics ones in the large LQ mass regime. We LQs primarily couple to the quarks of the first generation furthermore show that S1 (U1) marginally improves (spoils) and neutrinos of all flavors and investigate the viability of the fit of the current IceCube data with respect to the SM the associated parameter space spanned by the LQ masses case within the region of parameter space that is otherwise and coupling strengths in view of the latest experimental consistent with various low-energy flavor physics measure- data from flavor physics, LHC, and the South Pole situated ments and the latest LHC input. Our study offers an up-to- IceCube detector. We accordingly study the implications of date analysis for these two leptoquarks in view of the latest the most recent and the most relevant experimental results experimental data. on the parameter space for two representative LQ scenar- ios. One scenario features scalar LQ S1 and the other uses vector LQ U1, where both fields are singlets under the SM 1 Introduction SU(2) group, allowing them to couple to both the left- and the right-handed quarks and leptons. We perform, in partic- The leptoquarks (LQs) are hypothetical that directly ular, a thorough analysis of the viability of the S1 (U1) sce- couple a Standard Model (SM) quark to a lepton. There are nario assuming non-zero couplings between S1 (U1), down 12 (10) types of multiplets [1] under the SM gauge group quark, and neutrinos (charged leptons) of all three gener- SU(3) × SU(2) × U(1) with this ability if one assumes ations, where the LQ in question couples to the SM lepton presence (absence) of the right-handed neutrinos. They can doublets. Consequentially, S1 (U1) couples up-type quarks to either be of scalar or vector nature but are, in all instances, charged leptons (neutrinos). We also entertain the possibility triplets under the SM SU(3) group. The LQs emerge in a nat- that S1 (U1) couples down quark () to the right- ural way in many New Physics (NP) proposals such as the handed neutrinos to investigate the sensitivity of the latest IceCube data to constrain the associated parameter space. a e-mail: [email protected] (corresponding author)

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The outline of the paper is as follows. We describe the We consider the scenario where U1 only couples to the down two LQ scenarios and the flavor ansatz considered in our quark and charged leptons of all three generations with work in Sect. 2. The constraints from the low-energy flavor physics experiments for these two LQ scenarios are presented χ L ≡ χ L ,χL ≡ χ L ,χL ≡ χ L . (2.3) in Sect. 3. The LHC constraints from the single LQ and the 11 de 12 dμ 13 dτ LQ pair productions are discussed in Sect. 4. We then perform the data analysis of the IceCube PeV events within these two χ R ≡ We also analyse the possibility when the couplings 11 frameworks in Sect. 5. The combined analysis using the low- χ R χ R ≡ χ R χ R ≡ χ R U uνe , 12 uνμ , and 13 uντ of 1 with the up quark energy flavor observables, along with the LHC results and and the right-handed neutrinos are switched on and equal to the latest IceCube data, for both S1 and U1, is presented in each other. All other U1 couplings are set to zero. Sect. 6. Finally we conclude in Sect. 7.

3 Low-energy constraints 2 Leptoquark scenarios The LQ interaction ansatz defined in the previous section We briefly review in this section the LQ scenarios we con- can lead to leptonic decays of pseudoscalar or fla- sider in our work. The two representative scenarios that are vor changing processes at both the tree and the one-loop addressed in our analyses are the scalar LQ S and the vector 1 levels. The LQ couplings to the first generation quarks and LQ U . 1 are strongly constrained by the atomic parity vio- lation (APV) experimental results. The experimental upper = (¯, , / )  2.1 Scalar leptoquark S1 3 1 1 3 bounds on the  →  γ decay branching ratios, with the LQ contribution coming in the loop, will also constrain the ( )× ( )× ( ) We study the signatures of S1, whose SU 3 SU 2 U 1 couplings of S with the quarks and leptons. These branch- (¯, , / ) 1 quantum numbers are 3 1 1 3 , on the flavor, LHC, and ing ratios receive contribution from both the left-handed as IceCube observables. In our normalisation the well as the right-handed couplings of the quarks to the lep- of S1 is 1/3 in the absolute units of the electron charge. The tons. The upper limits on the lepton flavor violating decays relevant Lagrangian terms, in the mass eigenstate basis, are of μ and τ leptons are obtained from various experiments of the form with BR(μ → eγ) < 4.2 × 10−13 [27], BR(τ → eγ) < . × −8 (τ → μγ ) < . × −8 L ⊃−( L ) ¯C 1 ν j + ( ∗ L ) ¯Ci j 3 3 10 , and BR 4 4 10 [28]@ y U 1 j dL S1 L V y ijuL S1eL 90% C.L.. S1 also contributes at the tree level to the rare + R ¯C 1 ν j + . ., 0 + − y1 j dR S1 R h c (2.1) flavor process D → μ μ . The most recent measurement of this branching ratio comes from LHCb [29] and reads , (= , , ) where the subscripts i j 1 2 3 denote the flavor of the BR(D0 → μ+μ−)<7.6 × 10−9. The same couplings con- quarks and leptons, V is the Cabibbo-Kobayashi-Maskawa tribute to the D0 − D¯ 0 oscillations. Following the study of the (CKM) mixing matrix and U is the Pontecorvo-Maki- LQ effects in the D0 − D¯ 0 oscillations explained in detail in Nakagawa-Sakata (PMNS) mixing matrix. We work under Refs. [1,30], we require that the LQ contributions are smaller the assumption that the only non-zero S1 couplings are 0 − ¯ 0 L L L L L L than the current bounds on the D D mixing amplitude. y ≡ y ν , y ≡ y ν , and y ≡ y ν . We also enter- 11 d e 12 d μ 13 d τ Since the LQs, in our case, yield new contributions to tain the possibility that S1 couples to the right-handed neu-  → γ R R R R , the APV measurements, the rare meson decays, and trinos and we set the couplings y ≡ y ν , y ≡ y ν , and 11 d e 12 d μ the ratio of the leptonic decays of the , R ≡ R y13 ydντ to be equal to each other, if and when switched we take into account all these constraints. on. All the other LQ Yukawa couplings are set to zero. Note Lepton flavor violation in the sector that the S1 couplings with the up-type quarks and charged The contribution of weak singlets S1 and U1 to the pion leptons are fixed by the CKM mixing matrix. muonic decays is different from the pion electron decays due to the different values of e and μ couplings with the first 2.2 Vector leptoquark U1 = (3, 1, 2/3) generation quarks as well as the dependence on me and mμ. The effects of this type can be exposed by the lepton flavor π π The relevant Lagrangian terms for the U1 LQ, in the mass universality ratios Re/μ and Rτ/μ, where eigenstate basis, are (π − → −ν)¯ (τ − → π −ν)¯ † L i μ j L ¯1 μ j π BR e π BR L ⊃ (V χ U) u¯ γ U ,μν + χ d γ U ,μe R /μ = , Rτ/μ = , ij L 1 L 1 j L 1 L e BR(π − → μ−ν)¯ BR(π − → μ−ν)¯ +χ R ¯1 γ μ ν j + . .. 1 j u R U1,μ R h c (2.2) (3.1) 123 Eur. Phys. J. C (2020) 80:204 Page 3 of 14 204

Table 1 Numerical values of parameters used in our calculation, taken from PDG [32]

−13 −8 −13 τD (s) 4.1 ×10 τπ (s) 2.603 ×10 ττ (s) 2.903 ×10 m D (GeV) 1.86 mπ (GeV) 0.140 mτ (GeV) 1.7768 −3 mc (GeV) 1.28 me (GeV) 0.51 ×10 mμ (GeV) 0.105

fD (MeV) 212 fπ (MeV) 130.41

Fig. 1 The allowed parameter space after taking into account the results from the most relevant low-energy flavor experiments

π | = ( . ± . )× with the experimental result Re/μ exp 1 2327 0 0023 decay amplitude, with S1 and the up-type quarks running in −4 π | = ( . ± . ) × the loop. The Z branching ratio to a pair of leptons has been 10 and the SM value Re/μ SM 1 2352 0 0001 −4 π precisely measured at LEP [32], thereby imposing constraints 10 [31](seeEq.(A.2). The measured ratio is Rτ/μ|exp = 0.1082 ± 0.0005 [32], while the SM value is found to be on the S1 parameter space. We have used formula for the π one-loop contribution of S computed in Ref. [33] and found Rτ/μ|SM = 0.1088 ± 0.0002, using Eq. (A.2). 1 We list the formulas for the branching ratios of the pion, that the bounds on the LQ couplings from the Z leptonic 0 branching ratio are not up to par with the other experimental , and the τ lepton for the S1 and U1 cases in Appendix A and specify numerical values of input param- constraints considered before. The loop level contribution to → ¯ eters relevant for our analysis in Table 1. We further- the Z decay amplitude in case of U1 is also negligible more summarize in Fig. 1 results of a randomized scan for the parameter space that survives the other low-energy ∈ ( , . ) flavor physics experiments. within the parameter space (mS1 300 GeV 1 5TeV , yL , yL , yL ∈ (0.0, 0.8)) and (m ∈ (500 GeV, We next discuss the LHC limits on the LQ masses and dνe dνμ dντ U1 . ) χ L ,χL ,χL ∈ ( . , . ) their couplings. 2 5TeV , de dμ dτ 0 0 0 8 ) that takes into account the constraints from the pion sector, APV, the rare radiative decays  → γ , and D0 → μ+μ− decays. The plots in Fig. 1 show currently allowed parameter spaces of the left- 4 Constraints from the Large handed couplings of the down quark and the first generation leptons as a function of the LQ mass as well as the correla- The LQ couplings to the quark-lepton pairs have been con- tions between different left-handed couplings. We find that strained through both direct and indirect searches in a num- yL and yL (χ L and χ L ) cannot be simultaneously large dνe dνμ de dμ ber of collider experiments. Prior to the LHC era, the LQs due to conflict with the current results from the low-energy were searched for at LEP [34], HERA [35,36] and Teva- sector in the S1 (U1) case. This can be clearly seen in the tron [37,38]. The LQs have been hunted for at the LHC panels of the second column of Fig. 1. We therefore mainly mainly through pair production [39–41] but there are also work, in what follows, in the presence of the left-handed cou- several searches/recasts that rely on the single LQ produc- pling of the down quark and the first generation leptons, with tion [42] as well as dilepton [43] and monolepton [26] Drell- the other couplings being set to zero. Yan processes to generate constraints. For a summary of cur- The flavor experiments constrain the parameter space of rently available bounds on the LQ masses and associated cou- the vector LQ more tightly than that of the scalar one. The plings from the LHC searches for various flavor final states ¯ S1 LQ also contributes at the loop level to the Z →  see, for example, Refs. [44,45]. 123 204 Page 4 of 14 Eur. Phys. J. C (2020) 80:204

The LQs are pair-produced through either gg or qq¯ fusion brown line in case of LQ pair production and the subsequent that is primarily dominated by the QCD interactions. There is decay to light quarks and leptons. The allowed region from also a Yukawa coupling contribution to the LQ pair produc- the single LQ production is the area below the green (for tion, corresponding to a t-channel process, with its amplitude qe+e−) and blue (for qμ+μ−) lines. In the single LQ pro- being proportional to the product of the two relevant Yukawa duction, the process pp → qe+e− is proportional to the couplings. This t-channel process is highly suppressed com- Yukawa coupling yL (χ L ), whereas pp → qμ+μ− is pro- dνe de pared to the QCD driven one, unless the Yukawa couplings L χ L portional to ydνμ ( dμ)fortheS1 (U1) case. Therefore no are rather large. The production cross section at the LHC limit is obtained from the process pp → qμ+μ−, in case of for a pair of vector LQs, when compared to the scalar ones, yL (χ L ) = yL (χ L ). dνe de dντ dτ additionally depends on the underlying theory for the origin The low-energy experiments, as discussed in the previous of vector LQs. The relevant trilinear and quartic couplings of section, do not allow simultaneous presence of large values vector LQs to a or a pair of is then completely for the couplings of the first generation quarks to fixed by the extended gauge invariance of the model wherein and . Therefore the choice yL (χ L ) = yL (χ L ) is dνe de dνμ dμ the vector LQs appear as fundamental objects. We here work strongly constrained by the flavor physics measurements. in the limit, where the vector LQ is some low energy mani- The single LQ production has more stringent limits than festation of a more fundamental theory at high energy scale. the pair production for ee final state at large LQ masses. The vector LQ-gluon interaction terms can then be obtained This is due to the final state phase space. The blue dotted ( ) from the most general SU 3 invariant effective Lagrangian line in Fig. 2c is incomplete as the experimental result from given by pp → qμ+μ− is provided up to the LQ mass of 1.8 TeV.The † μν most stringent constraint on the available parameter space L ⊃−ig κU G U ,ν, (4.1) V s 1,μ 1 currently comes from the low-energy flavor experiments denoted by the purple line. The LHC direct searches from where Gμν is the gluon field strength tensor and κ is a dimen- the LQ pair production currently exclude m < 700 GeV, sionless parameter which we consider to be 1 for our calcu- S1 and m < 1700 GeV,irrespective of the choice of couplings lations. U1 as long as LQ decays promptly. A study done in Ref. [26] Dedicated studies have been performed at the LHC, has shown that stringent limits can also be obtained on the assuming LQ pair production and a 100% branching ratio strength of the LQ coupling to the first generation quarks (BR) of LQ decaying into a charged lepton and jet ( jj+−, and left-handed electrons and muons through the monolep- where  = e,μ) or to jet and missing energy ( jjνν). Con- ton searches. sidering the model Lagrangian in Eq. (2.1)(Eq.(2.2)) in We study next in details the effects of S and U on the the S (U ) case, the final states relevant for our analysis 1 1 1 1 observed IceCube PeV events. are jjee, jjμμ, and jjνν, with jν having the dominant branching ratio. There being no distinction made in the LHC LQ searches in case of the light quark jets (u, d, c, s), we can consider the LHC limit directly. The upper limits on 5 PeV events in IceCube the LQ production cross section times BR2 for these final states are provided by the LHC collaborations [39,46,47]. The observation of the High Energy Starting Events above The LQs coupling to first-generation quarks and electrons 100 TeV at the South Pole situated IceCube detector [48–51], or muons are also sought in single production processes, i.e., consistent with a flux of high-energy astrophysical neutrinos + − pp →   j [42]. This process occurs via s- and t- channel from outside the galaxy, has motivated a large number of quark-gluon fusion and is directly proportional to the Yukawa studies that explore the IceCube potential to test various NP coupling of the u(d) quarks to the leptons. The single pro- models. Since some of the most studied NP sources are var- duction of LQs at 8 TeV LHC [42] is also considered in our ious LQ scenarios our intention is to investigate whether the analysis. latest IceCube data [51] might offer an independent way to We show in Fig. 2 the allowed parameter space for two constrain the LQ mass mLQ and the strength of its couplings different choices of values of relevant couplings of S1 (U1) to the quark-lepton pairs that would be on par with the flavor as a function of mS1 (mU1 ). In an effort to confront the LHC and collider physics constraints. constraints with the flavor physics measurements, we con- The high-energy neutrinos coming from outside the atmo- sider two particular scenarios: (i) yL (χ L ) = yL (χ L ) and dνe de dνμ dμ sphere are detected in the IceCube detector by observing the (ii) yL (χ L ) = yL (χ L ). The region below the purple line Cherenkov light emitted by the secondary charged particles dνe de dντ dτ in Fig. 2 is currently allowed by the low-energy experiments produced in the interaction of the neutrinos with the nucleus at the 2 σ level, as discussed in Sect. 3. The parameter space present in the ice. The charged current (CC) and the neutral allowed by the LHC, at 95% C.L., is the region below the current (NC) interactions have distinctive topologies depend- 123 Eur. Phys. J. C (2020) 80:204 Page 5 of 14 204

(a) (b)

(c) (d)

Fig. 2 The allowed parameter space in case of S1 and U1 from differ- gle LQ) production, at the 95% C.L., at the 13TeV (8TeV) LHC. The ent experiments discussed in the text. The area below the brown (blue region enclosed by the purple line is allowed by the low-energy flavor and green) line is compatible with the LQ searches through pair (sin- experiments, at the 2σ level

ing on the flavor of the incoming neutrinos. The shower-like where T is the exposure time in seconds comprising 2635 events are induced by CC of νe and ντ interactions and NC days of data taking [51], NA is the Avogadro’s number interactions of neutrinos of all flavors. The tracks are pro- 6.022×1023, k is showers and tracks for each channel duced in the CC interactions of νμ and ντ (τ produced in ν ={e,μ,τ} induced by the charged and neutral current the final state decays to ντ νμμ, giving a distinctive double interactions (ch) for an incoming neutrino flux of type f cascade signature). The expected total number of events at (astrophysical (a), conventional atmospheric (ν)orprompt the IceCube from the NC or CC interactions in the deposited atmospheric flux (p)). The effective mass of the detector, i , f ( k,ch) energy interval [Edep Edep] can be written as Meff Etrue , is a function of the true electromagnetic equiv- alent energy and is defined as the mass of the target mate-   rial times the efficiency of converting the true deposited E f ∞ φ f k,ch, f dep f d ν energy of the event into an observed signal. The energy N = ν ( ν) , , ν TNA dEdep dE Attν E ( k ch, ,σ( k ch)) Ei 0 dEν resolution function is given by R Etrue Edep Etrue  dep 1 and is represented by a Gaussian distribution [52]. The × ( k,ch) ( k,ch, ,σ( k,ch)) dyMeff Etrue R Etrue Edep Etrue effect of the earth’s attenuation, in case of neutrino’s energy 0 above a few TeV, where the mean free path inside the σ ch( , ) d ν Eν y earth becomes comparable to the distance travelled by the , (5.1) dy 123 204 Page 6 of 14 Eur. Phys. J. C (2020) 80:204

Fig. 3 The relevant Feynman diagrams for the neutrino-quark interactions mediated by the scalar LQ S1

Fig. 4 The relevant Feynman diagrams for the neutrino-quark interactions mediated by the vector LQ U1

neutrino, is denoted by Attν (Eν). The incoming neutrino mass and the couplings considered here and is therefore f ν flux is given by dφν /dEν, where the incoming astro- neglected. The differential j N cross sections, in the pres- physical neutrino flux follows the isotropic single unbro- ence of the S1 interactions, are given by Eq. (B.8)in ken power-law spectrum. This spectrum is given by [51] Appendix B.1. The vector LQ U1 contributes to both the NC and CC   interactions. The relevant Feynman diagrams, in the pres- φastro −γ d ν Eν ence of U1, are shown in Fig. 4.Theui in Fig. 4 repre- = 3 0 f , (5.2) dEν 100 TeV sents the contributions from all three generations of up-type quarks. Note, however, that the charm contributions towards where f is the fraction of neutrinos of each flavor .The the NC processes, due to small PDFs, are of the order of fit is performed assuming a (1/3 : 1/3 : 1/3)⊕ flavor 0.001% and can be safely neglected. The differential ν j N ratio, which yields the best fit value for the spectral index cross sections in the presence of the U1 interaction are given γ = . +0.20 = . +1.46 × by Eq. (B.10) in Appendix (B.2). The U LQ compared to 2 89−0.19, with a normalization 0 6 45−0.46 1 − − − − 10 18 GeV cm 2s 1sr 1 at 1σ significance. the S1 case interferes with the SM leading to interesting The neutrino- differential cross section for differ- features. Since the U1 LQ interferes with the SM contri- ent channels in case of the CC and the NC interactions is bution, we show in Fig. 5 the ratio of the νe N total cross σ ch( , )/ sections for the SM + U and the SM for different values given by d ν Eν y dy. The SM differential cross section 1 is given by Eq. (B.1) in Appendix B. At the IceCube detec- of masses and couplings. The interference effect is clearly χ L tor the neutrinos interact with the present in the visible for low values of mass and large values of de.The ice. We assume that the natural ice nucleus can be treated seven years of IceCube data have fewer events when com- as an isoscalar with 10 and 8 . We calcu- pared to the SM in the 200–300 TeV energy range whereas late the event spectra of showers and tracks for each fla- for energies above 1000 TeV there are more events when vor in case of SM assuming an isotropic power-law spec- compared to the SM. It can be seen from Fig. 5 that there is trum. Since we find that the largest contribution to the event a crossover in the relevant energy range making it an inter- spectra comes from the νe showers, the esting feature for a more detailed study. We would like to χ L χ L should be sensitive to the NP effects if one is to have an point out that the inclusion of dμ and/or dτ will push the enhanced effect compared to the SM. We therefore study crossover away from the interesting energy range. The three next the effect of S1 and U1 on the IceCube spectrum when couplings then have to be adjusted so as to get the required these LQs couple the first generation quark to the elec- effect. tron. We study whether the SM + LQ scenarios result in The scalar LQ S1 mediates the NC interactions νd → νd a better or a worse fit of the IceCube data compared ¯ and νd → ν d and the CC interactions νu¯ → d, to the SM case by calculating parameter δ that corre-  ¯  2 νu¯ →  d, and dν → u, dν → u , where  = sponds to the percent change in χ [53]. We accordingly . The Feynman diagrams for the relevant processes are define shown in Fig. 3. The charm contribution towards the t- bins (N − N )2 channel CC process depicted in Fig. 3(iii), due to small χ 2 = modeli datai , . model N PDFs, is maximally around 0 001% for the choice of the i≥100 TeV datai 123 Eur. Phys. J. C (2020) 80:204 Page 7 of 14 204

Fig. 5 The ratio of νe N total cross sections between the SM + U1 and the SM for different masses and couplings of U1

Fig. 6 The contour lines of constant δ in the m ( )-yL (χ L ) plane in case of U results in a fit comparable or better than the SM one. The S1 U1 dνe de 1 showing the percent change of χ 2 compared to the SM case. The region region above the blue line is excluded at the 2 σ level by the limits from to the right of the dotted curve in case of S1 and above the dotted curve the APV experiments

2 2 χ − χ + The hatched region above the blue line in Fig. 6a is currently δ = 100% × SM SM LQ , (5.3) χ 2 excluded at the 2 σ level by the APV results. The region to SM the right of the black dotted curve in Fig. 6a results in a fit to where the observed number of events Ndatai in each bin the IceCube data that is marginally better than the SM one. i is compared with the LQ scenario prediction and, in The contours of constant value of δ in the SM + U1 sce- = , χ L our case, LQ S1 U1. We consider the events in the nario are shown in Fig. 6bforthemU1 - de parameter space. neutrino deposited energy range [60 TeV, 10 PeV] that is The region above the black dotted curve results in a fit of divided in 20 logarithmic energy bins. We initially use the IceCube data that is better than the SM one. It can be only the data for the bins with the non-zero number of seen, through comparison of Fig. 6a, b, that the U1 sce- events. nario has much larger region of parameter space that results The SM value with the best fit value of γ and C0 from the in a better fit compared to the SM case than the S1 sce- 2 IceCube data results in a χ value of 0.15. This shows that the nario due to the fact that U1 signatures interfere with the σ χ L current IceCube data is quite compatible with the SM. The SM. The 2 limit on de from the APV experiment, i.e., χ L ≤ . × /( NP contribution to the number of events in each bin depends de 0 34 mU1 1 TeV), as a function of mU1 is shown on the values of yL , y R , and m (χ L , χ R , by the blue line, with the region above the blue line being dνe,dνμ,dντ dν S1 de,dμ,dτ uν and mU1 )intheS1 (U1) case. We present in Fig. 6a contours excluded. of constant δ for the SM + S scenario in the m -yL plane. The couplings of S and U with the right-handed neu- 1 S1 dνe 1 1 Since the S1 contribution simply adds to the SM one, a small trinos are first fixed to zero, for simplicity. We find that the mass and a large value for the LQ-neutrino-quark coupling best fit to the recent IceCube data in the U1 case is obtained = χ L = . will lead to an enhanced number of events in each bin. This for mU1 710 GeV and de 1 25 and results in a 9.5% is beneficial for (detrimental to) the bins where there is an improvement over the SM fit. This is in contrast to the S1 observed excess (lack) of events compared to the SM case. case, which for most of the parameter space considered in 123 204 Page 8 of 14 Eur. Phys. J. C (2020) 80:204

Fig. 7 The total event rate, with the LQ contribution for m = 710 GeV (χ L = 1.25, χ L ,χL = 0) (left panel), and m = 710 GeV (yL = 1.2, U1 de dμ dτ S1 dνe L = L = ydνμ ydντ 0) (right panel). The gray shaded region and the bin with zero events are not included in the fit

(a) (b)

Fig. 8 The m -yL (left panel) and m -χ L (right panel) parameter left of the brown line is currently disallowed at 95% C.L. by the 13TeV S1 dνe U1 de space where the region above the blue line is disallowed at 2 σ level LHC data on the LQ direct searches via the jjνν, jjee,and jjμμ final from 2635 days of IceCube data. We also show the low energy disal- states. The single LQ production is also included in the S1 case lowed region (space above purple line) at the 2σ level. The region to the

our work either results in a fit worse or comparable to the 6 Combined analysis of the low-energy flavor physics, SM. Therefore we show in the left (right) plot of Fig. 7 the LHC, and IceCube constraints χ L = . contribution of U1 (S1) for mass of 710 GeV and de 1 25 (yL = 1.25), which gives δ = 9.5% (δ =−2.9%). All Our goal is to combine the low-energy flavor physics, LHC, dνe other couplings are set to zero. and IceCube constraints on the parameter spaces associated We see from the above analysis that no significant devia- with the S1 and U1 scenarios. The summary of our analysis tion from the SM prediction is seen in the current IceCube of these constraints on the m (m )-yL (χ L ) parameter S1 U1 dνe de data. We use this information to put an upper bound on the space is shown in Fig. 8. The LHC constraints from the LQ yL (χ L ) coupling as a function of m (m ) through a pair production with the dijet + MET, jjee, and jjμμ final dνe de S1 U1 binned likelihood analysis with the Poisson likelihood func- states are considered and the currently allowed space, at 95% tion [54]. This constraint obtained on the S1 and U1 parameter C.L., is shown by the area below (right of) the brown line space is then compared with the results from the low-energy in case of S1 (U1). The region below 850GeV (1.6TeV) in flavor experiments and the LHC in the next section. the S1 (U1) case is completely excluded by the LHC data. The region below the purple line in Fig. 8 is allowed by

123 Eur. Phys. J. C (2020) 80:204 Page 9 of 14 204

(a) (b)

Fig. 9 The allowed parameter space at 95% C.L. from the 13TeV (right panel). The contours of constant value of δ,asdefinedinEq.(5.3), LHC direct LQ searches is shown in brown in the yL -y R plane, with show the percent change in χ 2. The purple points are allowed at the 2σ dνe dν L = L = = = level by the low-energy flavor constraints discussed in Sect. 3 ydνμ ydντ 0formS1 800 GeV (left panel) and mS1 1TeV

R the flavor observables and the radiative decays of leptons. of interest. This is expected since the coupling ydν only The region above the blue line is currently disallowed by appears quadratically in the final state of the νN cross sec- σ R IceCube data at 2 level. For our statistical analysis of the tion. Note that ydν is also tightly constrained by the flavor IceCube constraints, the LQ mass and the couplings are kept observables and we present in Fig. 9 the allowed region by as free parameters, with γ and C0 fixed to the IceCube best fit purple points. data. We have used, for our numerical calculation, the (1/3 : The inclusion of the U1 couplings to the right-handed 1/3 : 1/3) flavor ratio for the incoming flux. Even though the neutrinos follows the same pattern as in the S1 case. Pro- IceCube data alone favors U1 over S1 the actual parameter vided that the right-handed neutrinos do not contribute to space allowed by the low-energy flavor experiments and LHC the initial state, the current IceCube data are not sensi- corresponds to the region where S1 (U1) marginally improves tive to these couplings. The consideration of the right- (spoils) the IceCube data fit when compared to the SM case. handed neutrinos in the initial state will lead to a change Overall we find that the limits obtained from the most recent in the initial flux at the source. This flux will depend on R IceCube data are considerably weaker when compared to the the mass of ν and also on the possible decay channels R constraints from the low-energy observables and direct LQ in case of heavy ν . This is beyond the scope of present searches at the LHC. This is mostly due to the current lack analysis. The resulting final high-energy cosmic neutrino of statistics in the high-energy bins of the IceCube spectrum. flux ratios on earth by the possible mixing between the R We next discuss the effects of inclusion of ydν on our anal- three active neutrinos and the fourth have ysis. The leading processes for the LQ pair production at the been studied in Ref. [55]. The explanation of the PeV neu- LHC will be via the initial state of gg, uu¯, and dd¯. The cross trinos at IceCube, with the consideration of heavy right- sectionintheS1 case will be particularly enhanced through handed neutrino, acting as a dark has been studied ¯ R the dd initial state for large values of the right-handed ydν in Ref. [56]. couplings. That regime will generate large branching ratio of S1 to jν and will, therefore, be strongly constrained by the jjνν final state searches at the LHC. We show in Fig. 9 the allowed parameter space from the LHC in brown in the 7 Conclusions yL -y R plane with yL = yL = 0form = 800 GeV dνe dν dνμ dντ S1 (left panel) and 1TeV (right panel) and contours of constant We analyse the impact of the latest low-energy flavor physics values of δ in green. Since the IceCube data are more sen- measurements, LHC search limits, and IceCube data on the parameter space of the electroweak SU(2) singlet scalar sitive to the νe coupling we consider this particular choice to check the effect of y R on yL . Clearly, the inclusion of (vector) LQ S1 (U1). We perform, in particular, a thorough dν dνe R analysis of the viability of the S1 (U1) scenario assuming ydν slightly spoils the fit of the IceCube data in the region non-zero couplings between S1 (U1), down quark, and neu- 123 204 Page 10 of 14 Eur. Phys. J. C (2020) 80:204 trinos (charged leptons) of all three generations, where the A Formulas LQ in question couples to the SM lepton doublets. Con- sequentially, the SU(2) symmetry of the SM requires that We list here the different branching ratios used for our anal- S1 (U1) couples up-type quarks to charged leptons (neutri- ysis in Sect. 3. nos). + − We find that the limits obtained from the most recent Ice- BR(D0 → μ μ )   Cube data are considerably weaker when compared to the 2 2 G 4mμ  mμ v2 = τ f 2 m3 F 1 −  constraints from the low-energy observables and direct LQ D D D π 2  2 64 m D m D mS searches at the LHC and we quantify this inferiority. We  1 attribute this disparity in constraining power to the current 2 ( ∗ L ) ( ∗ L )  lack of statistics in the high-energy bins of the IceCube spec- V y 12 V y 22 (A.1) trum. Even though the IceCube data alone favors U1 over S1, the viable parameter space allowed by the low-energy fla- The π → ν¯ and the τ − → π −ν branching ratios at the vor physics and the LHC data analyses singles out the region leading order in SM are given by: where S1 (U1) marginally improves (spoils) the IceCube data   2 2 2 fit when compared to the SM case. In this region, in the S1 G F 2 2 2 m (π → ν)¯ | = τπ f mπ m |V | − , ) ≥ ≥ . BR SM π  11 1 2 (U1 case, mS1 900 GeV (mU1 1 6 TeV) and the relevant 8π mπ S U   1-neutrino-down quark ( 1-neutrino-up quark) coupling is 2 2 2 − − G F 2 3 2 mπ small. We have also verified that the couplings of both S1 and BR(τ → π ν)| = ττ fπ mτ |V | 1 − . SM 16π 11 m2 U1 to the right-handed neutrinos are not being sensitive to τ the current IceCube data provided that the right-handed neu- (A.2) trinos only contribute to the final state. Since we investigate (π → ν)¯ scenarios where LQs primarily couple to the first genera- The electroweak corrections to BR were calcu- (τ − → π −ν) tion quarks, the most important constraints originate from lated in Ref. [31] and for BR in Ref. [57]. The flavor physics measurements, followed by the LHC search relevant branching ratios in the LQ models is given by, limits that take over flavor physics limits in the large LQ   mass regime. 2 2 2 G F 2 3 m BR(π → ν)¯ = τπ fπ mπ 1 − 8π mπ2 Acknowledgements We are grateful to Olcyr Sumensari, Boris Panes, ⎡ ⎛ ⎞ and Damir Beˇcirevi´c for helpful discussions. M.P. would like to thank 2 2 v2 3 ⎣ m 2 m 2 ⎝ ∗ L ∗ L ⎠ Anushree Ghosh for discussions on IceCube. S.F. and M.P. acknowl- |V11| + Re V (y )1 j U (y ν)1i m2 m2 2 11 q ji q edge support of the Slovenian Research Agency through research core π π CmLQ = ⎛ i 1 funding No. P1-0035. v2 3 ⎝ m L L 2 + (y )1 j |(y ν)1i | Data Availability Statement This manuscript has no associated data 2 mπ q q CmLQ = or the data will not be deposited. [Authors’ comment: The experimental i 1 ⎞⎤ data used in the present study has been published by the IceCube col- 3  L 2   (y )1 j  laboration [51]. The LHC results used throughout the manuscript are +  | R |2 2  q  ⎠⎦ , C y1i mπ   published by the ATLAS [39]andCMS[46,47] collaborations.] mu + m i=1 d   Open Access This article is licensed under a Creative Commons Attri- G2 2 2 (τ − → π−ν) = τ F 2 2 − mπ bution 4.0 International License, which permits use, sharing, adaptation, BR τ fπ mπ mτ 1 16π mτ2 distribution and reproduction in any medium or format, as long as you ⎡ ⎛ ⎞ give appropriate credit to the original author(s) and the source, pro- 2 2 v2 3 ⎣ mτ | |2 + mτ ⎝ ∗ ( L ) ∗ ( L ) ⎠ vide a link to the Creative Commons licence, and indicate if changes V11 Re V11 yq 13 U3i yqν 1i mπ2 mπ2 Cm2 were made. The images or other third party material in this article LQ i= ⎛ 1 are included in the article’s Creative Commons licence, unless indi- v2 3 cated otherwise in a credit line to the material. If material is not ⎝ mτ L L 2 + (y )13 |(y ν)1i | included in the article’s Creative Commons licence and your intended 2 mπ q q CmLQ = use is not permitted by statutory regulation or exceeds the permit- i 1 ⎞⎤ ted use, you will need to obtain permission directly from the copy- 3  L 2   (y )13  right holder. To view a copy of this licence, visit http://creativecomm +  | R |2 2  q  ⎠⎦ , C y1i mπ   (A.3) mu + m ons.org/licenses/by/4.0/. i=1 d Funded by SCOAP3. L = L ( †χ L ) L = ∗ L (χ L ) = ( ) with yqν y U V U , yq V y , C 4 2 ,  and C = 1(2) in the S1(U1) case. The light quark masses are determined at the LQ mass scale. The subscript j in Eq. (A.3) takes on the values of 1 and 2 for e− and μ− respectively. 123 Eur. Phys. J. C (2020) 80:204 Page 11 of 14 204

2 2 us(x, Q ) + ds(x, Q ) B Neutrino-nucleon differential cross sections + (R2 + R2) 2 u d +(s (x, Q2) + b (x, Q2))(L2 + R2) The neutrino-nucleon scattering in the SM gives rise to s s d d  the charged current (ν N → X) and the neutral current +( ( , 2) + ( , 2))( 2 + 2) , (ν N → ν X) interactions mediated by W and Z , cs x Q ts x Q Lu Ru (B.6) respectively. The target nucleon N is an isoscalar nucleon   2 with N = (n + p)/2, X is the hadronic final state, and m2 q¯0(x, Q2) = Z  = e,μ,τ. The SM differential cross sections in terms of Q2 + m2  Z the scaling variables are given as uv(x, Q2) + dv(x, Q2)   2 2σ CC 2 2 2 d νN 2G F m N Eν mW  = 2 2 2 us(x, Q ) + ds(x, Q ) dxdy π Q2 + m + (R2 + R2)  W  2 u d 2 2 2 xq(x, Q ) + xq¯(x, Q )(1 − y) , (B.1) 2 2 us(x, Q ) + ds(x, Q ) + (L2 + L2 ) 2σ NC 2 2 u d d νN G F m N Eν = 2 2 2 2 π +(ss(x, Q ) + bs(x, Q ))(L + R ) dxdy  2  d d  xq0(x, Q2) + xq¯0(x, Q2)( − y)2 , +( ( , 2) + ( , 2))( 2 + 2) , 1 (B.2) cs x Q ts x Q Lu Ru (B.7) where −Q2 is the invariant momentum-square transfer to with the chiral couplings given by L = 1−(4/3)x , L = the exchanged vector , m and m ( ) are the nucleon u W d N W Z − + ( / ) , =−( / ) = ( / ) and intermediate W(Z) boson masses, respectively, and G 1 2 3 xW Ru 4 3 xW , and Rd 2 3 xW , F = 2 θ θ = 1.166378 ×10−5 GeV−2 is the Fermi coupling constant. where xW sin W and W is the weak mixing angle. For ν¯ The differential distributions in Eqs. (B.1) and (B.2) are with the N cross sections Eqs. (B.1) and (B.2) are the same but respect to the Bjorken scaling variable x and the inelasticity with each quark distribution function replaced by the cor- parameter y, where responding anti-quark distribution function, and vice-versa, i.e., q(x, Q2) ↔¯q(x, Q2), q0(x, Q2) ↔¯q0(x, Q2).The 2 − parton distribution functions (PDFs) of the quarks are evalu- = Q = Eν E . x and y (B.3) ated at energy Q2, and the Mathematica package MSTW [58] 2m N yEν Eν is used throughout this work. E denotes the energy carried away by the outgoing lepton There are also neutrino-electron interactions, but they can or the neutrino in the laboratory frame while x is the fraction be generally neglected with respect to the neutrino-nucleon of the initial nucleon momentum taken by the struck quark. cross section because of the smallness of electron’s mass, 2 2 0 2 0 2 Here, q(x, Q ) and q¯(x, Q ) (q (x, Q ) and q¯ (x, Q ))are except for the resonant formation of the intermediate W − the quark and anti-quark density distributions in a , ν¯ = 2 /( ) = boson in the ee interactions at around Eν mW 2me respectively, summed over valence and sea quarks of all fla- 6.3 × 106 GeV, known as the Glashow resonance. The dif- vors relevant for CC (NC) interactions: ferential cross sections for all the neutrino electron reactions are listed in Ref. [59]. uv(x, Q2) + dv(x, Q2) q(x, Q2) = 2 B.1 ν j N cross sections in the presence of the S1 ( , 2) + ( , 2) us x Q ds x Q 2 2 + + ss(x, Q ) + bs(x, Q ), 2 The differential ν j N cross sections in the presence of the S1 (B.4) interaction are given by ( , 2) + ( , 2) 2 us x Q ds x Q 2 2 2 ch q¯(x, Q ) = + c (x, Q ) + t (x, Q ), d σν s s j N m N Eν L 2 ch 2 = |(y U) j | N (B.5) dxdy 16π 1      2 + + m2 1 uv dv + us ds q0(x, Q2) = Z 2 | − 2 +  |2 Q2 + m 2xmN Eν mS i S1 mS1 2 2  Z 1  ( , 2) + ( , 2) 1 u + d uv x Q dv x Q + (1 − y)2 s s , (B.8) ( 2 − − 2 )2 2 Q 2xmN Eν mS 2  1 ( , 2) + ( , 2) , + us x Q ds x Q ( 2 + 2 ) where j = 1,2,3 with ch =CC NC. The coefficients are Lu Ld N CC = 3 |( ∗ L ) |2 N NC = 2 given by k=1 V y 1k , 123 204 Page 12 of 14 Eur. Phys. J. C (2020) 80:204   3 |( L ) |2 + 3 | R |2  k=1 y U 1k k=1 y1k . The decay width S1 of nos or will have a neutrino of different flavor from the initial S1 given by one and are given by  3 3 mS ∗ 2 NC  = 1 |( L ) |2 + |( L ) |2 d σν S1 y U 1i V y ij j N Eν † L 2 16π = |(V χ U)1 j | i=1 i, j=1 dxdy 8G2  ⎛ F 3 + |y R |2 . (B.9) ⎜ 1i ⎜ 1 i=1 ⎝ (Q2 − 2xm Eν − m2 )2 N U1 Note that the effect of the right-handed couplings to the neu- ⎛ ⎞ trinos is only visible in NC interactions. ⎜ 3 3 ⎟ ⎜ 2 |χ R |2 + |( †χ L ) |2⎟ ⎝y 1k V U 1k ⎠ B.2 ν j N cross sections in the presence of the U1 k=1 k=1 k = j   0 2 0 2 + + The modified q (x, Q ) and q¯ (x, Q ) listed in Eqs. (B.6) uv dv + us ds and (B.7), in the presence of U , are given below. 2 2 1      2 2   1  2 ( , 2) + ( , 2) +(1 − y)2   0 2 m Z uv x Q dv x Q  2  q (x, Q ) = 2xmN Eν − m + iU mU Q2 + m2 2 U1 ⎞ 1 1 Z  2 2 3 us(x, Q ) + ds(x, Q )  + ⎟ + L2 † L 2 us ds ⎟ d |(V χ U)1k| ⎠ . 2 2  k=1 u (x, Q2) + d (x, Q2) k = j + s s (R2 + R2) 2 u d  uv(x, Q2) + dv(x, Q2) 2 /( 2 + 2 ) + In case of CC interactions, the coefficient mW Q mW in Eq. (B.1) is modified to 2    ( , 2) + ( , 2)  m2 +us x Q ds x Q Z     Lu 2  2  Q2 + m2 m2  m2 Z  W q(x, Q2) ⇒  W 2 (Q2 + m2 ) (Q2 + m2 ) |(V †χ LU) |2 1  W W + √ 1i  (B.10)  2 − − 2  χ L ( †χ L ) 2 2 2G F Q 2xMN Eν mU dj V U 1 j 1    1 + √  q(x, Q2) 2  2 2  2 2 2 2 2G Q − 2xm Eν − m m v( , ) + v( , ) F N U1 ¯0( , 2) = Z u x Q d x Q    q x Q 2  Q2 + m2 2 m2  m2 Z  W q¯(x, Q2) ⇒  W u (x, Q2) + d (x, Q2) (Q2 + m2 ) (Q2 + m2 ) + s s (R2 + R2) W W 2 u d χ L ( †χ L )  dj V U 1 j 2 2 + √ us(x, Q ) + ds(x, Q ) + L2 2 2G F 2 d   2  1  ( , 2) + ( , 2)  m2  q¯(x, Q2), (B.12) +us x Q ds x Q Z − 2 +    Lu 2xmN Eν mU i U1 mU1 2  Q2 + m2 1 Z  2 |(V †χ LU) |2 1  + √ 1 j  taking into account the interference terms only when the final G 2xm Eν − m2 + i m  state is similar to the SM. The decay width of U1 is given by 2 2 F N U1 U1 U1 (B.11)  m 3 3 where j = 1, 2, 3. Eqs. (B.10) and (B.11) correspond to the  = U1 |χ L |2 + |(V †χ LU) |2 U1 π 1i 1i 24 = = case when the same flavor neutrino is in the initial and the i 1  i 1 final states and there is an interference with the SM contri- 3 + |χ R|2 . bution. There will be additional contributions from the cases 1i (B.13) where the final state will consist of the right-handed neutri- i=1 123 Eur. Phys. J. C (2020) 80:204 Page 13 of 14 204

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