Constraining Flavor Changing Interactions from LHC Run-2 Dilepton Bounds with Vector Mediators
CConstraining Flavor Changing Interactions from LHC Run-2Dilepton Bounds with Vector Mediators
Farinaldo S. Queiroz , Clarissa Siqueira , and Jos´e W. F. Valle ∗ Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany Departamento de F´ısica, Universidade Federal da Para´ıba,Caixa Postal 5008, 58051-970, Jo˜ao Pessoa - PB, BrazilAHEP Group, Instituto de F´ısica Corpuscular C.S.I.C./Universitat de Valencia Edificio de Institutos de Paterna,C/Catedratico Jos´e Beltran, 2 E-46980 Paterna (Valencia) - SPAIN
Within the context of vector mediators, is a new signal observed in flavor changing interactions,particularly in the neutral mesons systems K − ¯ K , D − ¯ D and B − ¯ B , consistent with dileptonresonance searches at the LHC? In the attempt to address this very simple question, we discuss thecomplementarity between flavor changing neutral current (FCNC) and dilepton resonance searchesat the LHC run 2 at 13 TeV with 3 . − of integrated luminosity, in the context of vector mediatorsat tree level. Vector mediators, are often studied in the flavor changing framework, specially inthe light of the recent LHCb anomaly observed at the rare B decay. However, the existence ofstringent dilepton bound severely constrains flavor changing interactions, due to restrictive limitson the Z (cid:48) mass. We discuss this interplay explicitly in the well motivated framework of a 3-3-1scheme, where fermions and scalars are arranged in the fundamental representation of the weakSU(3) gauge group. Due to the paucity of relevant parameters, we conclude that dilepton data leavelittle room for a possible new physics signal stemming from these systems, unless a very peculiartexture parametrization is used in the diagonalization of the CKM matrix. In other words, if asignal is observed in such flavor changing interactions, it unlikely comes from a 3-3-1 model. I. INTRODUCTION
The Standard Model (SM) has passed all precisiontests thus far, and it is the best description of nature.Although, we need physics beyond the standard modelso as to account for neutrino masses and dark matter.Many models that address these puzzles are plagued byflavor changing neutral current (FCNC) processes, whichare, however, absent in the SM at tree-level, thanks to theGIM mechanism [4]. Therefore, precise measurement offlavor transition processes, such as those from neutral me-son oscillations, K − ¯ K , D − ¯ D and B d − ¯ B d , whichare forbidden in the SM at tree level, provide an excel-lent laboratory to test new physics models, due to lackof standard model background. Conversely, flavor chang-ing charged currents, are overwhelmed by numerous Wboson processes.That said, flavor changing neutral currents are oftenexamined in the context of neutral vector gauge bo-son, Z (cid:48) . A multitude of Abelian and non-Abelian mod-els predict the existence of extra neutral gauge bosons.Generally speaking they provide a straightforward cross-correlation among observables, such as FCNC and Z (cid:48) atthe LHC. Simplified models have become powerful toolsin this endeavor, since they capture the main features ofUV–complete models [5–8]. However, at the end of theday one needs a full theory to draw conclusive statements.In this attempt, we will address the complementarity be-tween flavor changing neutral currents and dilepton res- ∗ [email protected] The concept of minimal flavor violation has guided us at how tosuppress new physics interactions [1–3]. onance searches at the LHC, which refers to those withcharged lepton pairs in the final state [9], in the con-text of electroweak extensions of the SM, based on the SU (3) c ⊗ SU (3) L ⊗ U (1) N gauge group, shortly referredas 3-3-1 models.3-3-1 models are self-consistent if there exists onlythree generations due to the combined effect of trianglegauge anomalies cancellations and QCD asymptotic free-dom [10–14]. Moreover, the model furnishes a suitableenvironment for neutrino masses through see-saw mech-anisms [15–28], dark matter [29–45], explanation of thestrong CP problem in the quark sector [46, 47], first-orderphase transitions [48–50], lepton number violation pro-cesses [51–58], and several others [59–80]. 3-3-1 modelsare burden with FCNC interactions and they naturallyarise at tree level in 331 model because one of the gen-erations has to transform differently from the other two,breaking the universality and leading to flavor changinginteractions involving the new neutral gauge boson Z (cid:48) .In principle, there are also other sources of FCNC in themodel involving the CP-even and -odd neutral scalar, butthose are suppressed [81].In summary, in this work, we will investigate the de-gree of complementarity among flavor changing interac-tions and dilepton resonance searches at the LHC at 13TeV with 3 . f b − of integrated luminosity using ATLASanalysis [9], which are linked to the Z (cid:48) boson. Due tothe paucity of relevant parameters dictating the resultsof both observables, and the fact that other 3-3-1 mod-els feature mild changes in the Z (cid:48) interactions with SMquarks, we are able to draw general conclusions whichare applicable to many 3-3-1 models.The paper is structured as follows: In Sec. II we brieflydiscuss the key aspects of the model relevant for our rea- a r X i v : . [ h e p - ph ] N ov soning; In Sec. III, we obtain LHC bounds in the modelusing dilepton ATLAS 13 TeV data. In Sec. IV, we ob-tain FCNC stemming from the 3-3-1 model with right-handed neutrinos and outline the region which a FCNCsignal can be seen in agreement with LHC data. II. THE MODEL
The SU (3) c ⊗ SU (3) L ⊗ U (1) N gauge symmetry meansthat the fermions can be placed in the fundamental re-presentation of SU (3) L , i.e triplets. In order to repro-duce the SM spectrum the SM doublet should be en-closed. The third component in the model is arbitraryand can vary from neutrinos, heavy neutrino fermionsand even exotic charged leptons, depending on the quan-tum number assignments. There are two ways to incorpo-rate right-handed neutrinos in the model. One can eitheradd three singlet right-handed neutrinos, or change thequantum numbers of the fermions in such way that right-handed neutrinos are embedded in the SU (3) L triplet.The latter scenario leads to an interesting and minimalmodel, which is the model we concentrate on, also shortlyrefereed as 331 r.h.n firstly presented in [82–84]. Thus thelepton sector is, f aL = ν al e al ( ν cR ) a ∼ (1 , , − / , e aR ∼ (1 , , − , (1)where a = 1 , , SU (3) L , whereas the second and third one as anti-triplet as follows, Q L = u d u (cid:48) L ∼ (3 , , / ,u R ∼ (3 , , / , d R ∼ (3 , , − / , u (cid:48) R ∼ (3 , , / ,Q iL = d i u i d (cid:48) i L ∼ (3 , ¯3 , ,u iR ∼ (3 , , / , d iR ∼ (3 , , − / , d (cid:48) iR ∼ (3 , , − / , (2)where i = 2 ,
3, with q (cid:48) being heavy exotic quarks withelectric charges Q ( u (cid:48) ) = 2 / Q ( d (cid:48) , ) = − / χ with, (cid:104) χ (cid:105) = v χ , (3) where v χ is the vacuum expectation value of the neu-tral scalar responsible for breaking SU (3) L ⊗ U (1) N into SU (2) L ⊗ U (1) Y , give rises to the exotic quark massesvia the Yukawa Lagrangian, L χyuk = λ ¯ Q L u (cid:48) R χ + λ ij ¯ Q iL d (cid:48) jR χ ∗ + H.c., (4)where χ ∼ (1 , , − / SU (2) ⊗ U (1) Y breaks into electromagnetismwhen two triplets ρ, η acquire a vev with, (cid:104) ρ (cid:105) = v ρ , (cid:104) η (cid:105) = v η , (5)giving rise to quark and charged lepton masses throughthe Yukawa lagrangian, L Y uk = λ a ¯ Q L d aR ρ + λ ia ¯ Q iL u aR ρ ∗ + G ab ¯ f aL ( f bL ) c ρ ∗ + G (cid:48) ab ¯ f aL e bR ρ + λ a ¯ Q L u aR η + λ ia ¯ Q iL d aR η ∗ + H.c. (6)with the scalar triplets transforming as ρ ∼ (1 , , / η ∼ (1 , , − / M W ± = 14 g v , M Z = M W ± /C W ,M Z (cid:48) = g − S W ) (cid:20) C W v χ + v C W + v (1 − S W ) C W (cid:21) ,M V ± = 14 g ( v χ + v ) , M U = 14 g ( v χ + v ) , (7)where Z (cid:48) , V ± and U , U † are new gauge bosons pre-dicted by the model, with v = v ρ + v η . We have nowhighlighted the key features of the model relevant to ourreasoning, thus it is a good timing to discuss the colliderphenomenology. III. DILEPTON RESONANCE SEARCHES ATTHE LHC
Heavy dilepton resonance searches at the LHC (seeFig.1) have proven to be an effective channel to probe
FIG. 1. Feynman diagram relevant for dilepton production atthe LHC. new physics models due to relatively good efficien-cies/acceptance and well controlled background whichcomes mostly from Drell-Yann processes [92–94] . Using8 TeV center-of-energy and 20 f b − of integrated lumi-nosity ATLAS collaboration has placed restrictive limitson the mass of gauge bosons arising in some new physicsmodels [96], but an assessment particularly devoted to3-3-1 models was performed in [97] ruling out Z (cid:48) massesbelow 2 .
65 TeV in the 3-3-1 model with right-handedneutrinos.Here we take the dilepton results from LHC run IIdata at 13 TeV with L = 3 . − [9], which has givenrise to stringent limits on the Z (cid:48) mass of several modelsincluding the sequential standard model reading 3 . pp → Z (cid:48) → l + l − , where l = e, µ , was simulated using MadGraph5[98, 99] with the CTEQ6L parton distribution function[100] using efficiencies/acceptances described in [96].Similarly to previous analysis we selected the signalevents using the cuts, • E T ( e ) >
30 GeV , E T ( e ) >
30 GeV , | η e | < . • p T ( µ ) >
30 GeV , p T ( µ ) >
30 GeV , | η µ | < . •
500 GeV < M ll < M ll being the dilepton invariant mass.These signals are peaked at the Z (cid:48) mass, thus one canuse cuts the dilepton invariant mass to discriminate sig-nal from background. In summary, since no excess ofevents has been observed we can re-interpret ATLAS re-sults to derive a limit on the Z (cid:48) mass. Re-analyzing theATLAS dilepton results we found M Z (cid:48) > IV. FCNC IN THE 3-3-1
All mesons are unstable, with the longest-lived lastingfor only a few hundredths of a microsecond. Although See [95] for an excellent review about LEP-II limits
FIG. 2. Diagram contributing to K − ¯ K mass difference inthe 3-3-1 model with right-handed neutrinos.FIG. 3. Diagram contributing to D − ¯ D mass difference inthe 3-3-1 model with right-handed neutrinos.FIG. 4. Diagram contributing to B d − ¯ B d mass difference inthe 3-3-1 model with right-handed neutrinos. no meson is stable, those of lower mass are nonethelessmore stable than the most massive mesons, and are easierto observe in colliders. In particular the K meson is abound state composed of d ¯ s , implying that kaons cannotbe their own antiparticles. There must be then two differ-ent neutral kaons, differing by two units of strangeness,i.e. K and ¯ K (see Fig. 2). The eigenstates which areobtained after mass diagonalization are known as Kaonlong ( K L ) and Kaon short ( K S ) which yield opposite CPvalue, with K L decaying into three pions, and K S intotwo pions. Since K L is slightly heavier than three pionmasses, its lifetime is much longer than the K S . Thephysics of Kaon mixing is a explicit example of the im-portance of the CP symmetry in weak interactions. Themass difference of these mesons is precisely measured tobe (∆ m K ) = 3 . × − MeV. In a similar vein, themesons D made of c ¯ u and B d composed of d ¯ b have massdifference (∆ m D ) = 4 . × − MeV, m D = 1865 MeVand (∆ m B d ) = 3 . × − MeV [101–103] (see Figs.3-4and Table I). Hence, new physics FCNC processes whichmight yield sizeable contributions to the mass differencesabove can be probed using these meson systems . In the3-3-1 model these FCNC processes that contribute to themass difference of these meson systems surface throughthe neutral current mediated by Z (cid:48) gauge boson (scalarcontributions are dwindled). That said, in order to de-rive the 3-3-1 corrections to these mass differences in apedagogic way, we need first to derive the neutral currentin the 3-3-1 model. As in the SM the Z bosons does notmediated FCNC, only the Z (cid:48) does through, L Z (cid:48) u = g C W (cid:32) (3 − S W )3 (cid:112) − S W (cid:33) [¯ u aL γ µ u aL ] Z (cid:48) µ − g C W (cid:32) − S W )3 (cid:112) − S W (cid:33) [¯ u L γ µ u L ] Z (cid:48) µ , (8) L Z (cid:48) d = g C W (cid:32) (3 − S W )3 (cid:112) − S W (cid:33) (cid:2) ¯ d aL γ µ d aL (cid:3) Z (cid:48) µ − g C W (cid:32) − S W )3 (cid:112) − S W (cid:33) (cid:2) ¯ d L γ µ d L (cid:3) Z (cid:48) µ , (9)with a = 1 , ,
3, i.e. running through the three gener-ations. Notice that Eqs. (8) and (9) are in the mass-eigenstate basis, but we need to move to the flavor basisin order to connect to meson observables using the trans-formations, uct L,R = U L,R u (cid:48) c (cid:48) t (cid:48) L,R , dsb L,R = V L,R d (cid:48) s (cid:48) b (cid:48) , (10)where the matrices U L,R and V L,R are 3 × V CKM = ( U L ) † ( V L ) [107–109]. Using thistransformations one can find [110–112], L K − ¯ K Z (cid:48) eff = 4 √ G F C W − s W M Z M Z (cid:48) | ( V L ) ∗ ( V L ) | | ¯ d (cid:48) L γ µ d (cid:48) L | , L D − ¯ D Z (cid:48) eff = 4 √ G F C W − s W M Z M Z (cid:48) | ( U L ) ∗ ( U L ) | | ¯ u (cid:48) L γ µ u (cid:48) L | , L B d − ¯ B d Z (cid:48) eff = 4 √ G F C W − s W M Z M Z (cid:48) | ( V L ) ∗ ( V L ) | | ¯ d (cid:48) L γ µ d (cid:48) L | , (11)and consequently, See [104–106] for relevant reviews. (∆ m K ) Z (cid:48) = 4 √ G F C W − S W M Z M Z (cid:48) | ( V L ) ∗ ( V L ) | f K B K η K m K , (∆ m D ) Z (cid:48) = 4 √ G F C W − S W M Z M Z (cid:48) | ( U L ) ∗ ( U L ) | f D B D η D m D , (∆ m B d ) Z (cid:48) = 4 √ G F C W − S W M Z M Z (cid:48) | ( V L ) ∗ ( V L ) | f B B B η B m B , (12)with G F being the Fermi constant, S W ( C W ) the sine(cossine) of the Weinberg angle, and B K , B D , B B thebag parameters, f K , f D , f B the decay constants, and η K , η D , η B the QCD leading order correction obtained in[104–106], and m K , m D , m B the masses of the mesons.In table I we summarize the values of these parameters.We emphasize that the Z (cid:48) does mediate FCNC in the3-3-1 model because the hadronic generations do nottransform identically under SU (3) L . In Eqs. (8)-(12) u a = u, d, t and d a = d, s, b for a = 1 , , q (cid:48) representing the flavor eigenstate of a given quark. Input parameters∆ m K = 3 . × − MeV m K = 497 .
614 MeV √ B K f K = 135 MeV η K = 0 . m D = 4 . × − MeV m D = 1865 MeV √ B D f D = 187 MeV η D = 0 . m B d = 3 . × − MeV m B = 5279 . √ B B f B = 208 MeV η B = 0 . V CKM = (13) . ± . . ± . . ± . . ± . . ± . . ± . . +0 . − . . +0 . − . . ± . . Now to compute the theoretical prediction from the3-3-1 model to the mass difference systems under studyas a function of the Z (cid:48) mass, we simply need to pluginto Eq.12 the parameters summarized in Table I, know-ing the entries of the quark mixing matrices V uL and V dL .These entries are bound by the CKM matrix (see Eq. 13),which is reasonably well measured but the constraintson the individual entries of the matrices ( V uL and V dL )are loose [109]. Therefore, one can work on two pos-sible regimes which we name as parametrization 1 and parametrization 2 , which yield the strongest and weakest3-3-1 contributions to FCNC processes respectively, whilekeeping the CKM matrix intact. In the parametrization1 , we found, V L = V R = .
97 0 .
23 0 . .
23 0 .
97 0 . .
043 0 .
089 0 . and, U L = U R = . − .
45 0 . − . − .
89 0 . . .
054 0 . , whereas for the parametrization 2 we found, V L = V R = . − . . − . − . . . . . and, U L = U R = . − . . − . − . . . . . . Δ m B d [ G e V ] −15 −14 −13 −12 −11 −10 M Z' [GeV] Δm Bd ExclusionLHC Exclusion parametrization 1parametrization 2
FIG. 5. ∆ m B d × Z (cid:48) mass for two different parametrizationsof the quark mixing matrices. The pink region is ruled out byconstraints on ∆ m B d , wheres the shaded blue region indicatethe exclusion limit on the Z (cid:48) mass from LHC. We have now collected all information needed topresent the degree of complementarity between FCNCand dilepton searches at the LHC in the context of thevector mediator, Z (cid:48) taking into account the uncertaintiesin which such constraints are subject to.In Fig. 5 we show the 3-3-1 contribution to ∆ m B d for parametrizations 1-2 as a function of the Z (cid:48) mass andwe overlay in pink and blue the existing limits on theon the B d mass difference, and on the Z (cid:48) mass comingfrom dilepton resonance searches at the LHC. Only us-ing parametrization 1 meson physics gives rise to a limitstronger than LHC one on the Z (cid:48) mass. In other words, Δ m D [ G e V ] −19 −18 −17 −16 −15 −14 −13 −12 M Z' [GeV] Δm D ExclusionLHC Exclusion parametrization 1parametrization 2
FIG. 6. ∆ m D × Z (cid:48) mass for two different parametrizationsof the quark mixing matrices. The pink region is excluded byconstraints on ∆ m D and the blue region is ruled out by theLHC limit on the Z (cid:48) mass. Δ m K [ G e V ] −19 −18 −17 −16 −15 −14 −13 M Z' [GeV] Δm K ExclusionLHC Exclusion parametrization 1parametrization 2
FIG. 7. ∆ m K × Z (cid:48) mass for two different parametrizationsof the quark mixing matrices. The pink region is excluded byconstraints on ∆ m K and the blue region is ruled out by theLHC limit on the Z (cid:48) mass. if in the near future a signal is observed in the B d systembelow the current limit, that would be consistent withLHC searches for a neutral vector boson. The 3-3-1 con-tribution to FCNC processes using parametrization 2 israther small, with LHC bound driving the limit on the Z (cid:48) mass.Moreover, in Figs.6-7 we see that the 3-3-1 correctionsto the mass difference of the K and D mesons is quitedwindled. Thus LHC rules out any possibility for a possi-ble signal in the foreseeable future coming from the 3-3-1model, since the LHC limits on the Z (cid:48) mass is very strin-gent and robust, which reads M Z (cid:48) > parametrization 1 for the B d meson system. V. CONCLUSION
We have investigated the degree of complementaritybetween FCNC in the neutral mesons systems K − ¯ K , D − ¯ D and B d − ¯ B d in the context of vector media-tors, using the 3-3-1 model with right-handed neutrinosas framework. Our goal was to assess the possibility ofexplaining a possible FCNC signal in these systems hav-ing in mind the stringent limits stemming from dileptonresonance searches at the LHC. After briefly presentingthe model we derived the 13 TeV LHC 3 . f b − limit onthe Z (cid:48) mass which reads 3 TeV. Then we proceeded tothe 3-3-1 corrections to the mass differences of the threemesons above. We found that the 3-3-1 contributes ap-preciably only the B d mass difference. Using two differ-ent parametrizations, one that enhances, parametrization1 and other that suppresses parametrization 2 the 3-3-1 contribution to the latter, we concluded that bounds on the Z (cid:48) rising from dilepton resonance searches gen-erally impose much stronger limits than FCNC ones.Conversely, a small window for a signal in the B d sys-tem exists if parametrization 1 is used. Therefore, if aFCNC signal is seen in these mesons systems in the fore-seeable future, unless a parametrization very similar to parametrization 1 is advocated, the 3-3-1 model cannotnot offer a feasible solution. ACKNOWLEDGMENTS
The authors thank the anonymous referee for sev-eral suggestions. This work is supported by theSpanish grants FPA2014-58183-P, Multidark CSD2009-00064, SEV-2014-0398 (MINECO) and PROME-TEOII/2014/084 (GVA). CS acknowledges support fromCNPq, Brazil. JV thanks Carlos Pires for hospitality atthe Departamento de F´ısica, Universidade Federal daPara´ıba in Jo˜ao Pessoa, Brazil. FSQ thanks UNICAMPand UFABC for the hospitality during final stages ofthis project. FSQ is specially grateful to Alex Dias andVanidia Dias for discussions and hospitality. [1] A. J. Buras, P. Gambino, M. Gorbahn, S. Jager, andL. Silvestrini, Phys. Lett.
B500 , 161 (2001), arXiv:hep-ph/0007085 [hep-ph].[2] A. J. Buras,
Theoretical physics. Proceedings, 43rdCracow School, Zakopane, Poland, May 30-June 8,2003 , Acta Phys. Polon.
B34 , 5615 (2003), arXiv:hep-ph/0310208 [hep-ph].[3] G. D’Ambrosio, G. F. Giudice, G. Isidori, andA. Strumia, Nucl. Phys.
B645 , 155 (2002), arXiv:hep-ph/0207036 [hep-ph].[4] S. L. Glashow, J. Iliopoulos, and L. Maiani,
Meetingof the Italian School of Physics and Weak InteractionsBologna, Italy, April 26-28, 1984 , Phys. Rev. D2 , 1285(1970).[5] B. Allanach, F. S. Queiroz, A. Strumia, and S. Sun,(2015), arXiv:1511.07447 [hep-ph].[6] A. Celis, W.-Z. Feng, and D. Lst, JHEP , 007 (2016),arXiv:1512.02218 [hep-ph].[7] G. Blanger and C. Delaunay, (2016), arXiv:1603.03333[hep-ph].[8] W. Altmannshofer, C.-Y. Chen, P. S. B. Dev, andA. Soni, (2016), arXiv:1607.06832 [hep-ph].[9] G. Aad et al. (ATLAS), ATLAS-CONF-2015-70 (2015).[10] M. Singer, J. W. F. Valle, and J. Schechter, Phys. Rev. D22 , 738 (1980).[11] F. Pisano and V. Pleitez, Phys. Rev.
D46 , 410 (1992),arXiv:hep-ph/9206242 [hep-ph].[12] R. Foot, O. F. Hernandez, F. Pisano, and V. Pleitez,Phys. Rev.
D47 , 4158 (1993), arXiv:hep-ph/9207264[hep-ph].[13] J. C. Montero, F. Pisano, and V. Pleitez, Phys. Rev.
D47 , 2918 (1993), arXiv:hep-ph/9212271 [hep-ph].[14] A. G. Dias and V. Pleitez, Phys. Rev.
D80 , 056007(2009), arXiv:0908.2472 [hep-ph]. [15] R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. , 912 (1980).[16] P. Minkowski, Phys. Lett. B67 , 421 (1977).[17] J. Schechter and J. W. F. Valle, Phys. Rev.
D22 , 2227(1980).[18] R. N. Mohapatra and G. Senjanovic, Phys. Rev.
D23 ,165 (1981).[19] G. Lazarides, Q. Shafi, and C. Wetterich, Nucl. Phys.
B181 , 287 (1981).[20] J. Schechter and J. W. F. Valle, Phys. Rev.
D25 , 774(1982).[21] W.-Y. Keung and G. Senjanovic, Phys. Rev. Lett. ,1427 (1983).[22] G. Senjanovic, Int. J. Mod. Phys. A26 , 1469 (2011),arXiv:1012.4104 [hep-ph].[23] D. V. Forero, S. Morisi, M. Tortola, and J. W. F. Valle,JHEP , 142 (2011), arXiv:1107.6009 [hep-ph].[24] L. Dorame, S. Morisi, E. Peinado, J. W. F. Valle,and A. D. Rojas, Phys. Rev. D86 , 056001 (2012),arXiv:1203.0155 [hep-ph].[25] C.-Y. Chen, P. S. B. Dev, and R. N. Mohapatra, Phys.Rev.
D88 , 033014 (2013), arXiv:1306.2342 [hep-ph].[26] S. M. Boucenna, S. Morisi, Q. Shafi, and J. W. F.Valle, Phys. Rev.
D90 , 055023 (2014), arXiv:1404.3198[hep-ph].[27] C. Bonilla, R. M. Fonseca, and J. W. F. Valle, Phys.Rev.
D92 , 075028 (2015), arXiv:1508.02323 [hep-ph].[28] J. W. F. Valle and C. A. Vaquera-Araujo, Phys. Lett.
B755 , 363 (2016), arXiv:1601.05237 [hep-ph].[29] C. A. de S. Pires and P. S. Rodrigues da Silva, JCAP , 012 (2007), arXiv:0710.2104 [hep-ph].[30] J. K. Mizukoshi, C. A. de S. Pires, F. S. Queiroz,and P. S. Rodrigues da Silva, Phys. Rev.
D83 , 065024(2011), arXiv:1010.4097 [hep-ph]. [31] J. D. Ruiz-Alvarez, C. A. de S. Pires, F. S. Queiroz,D. Restrepo, and P. S. Rodrigues da Silva, Phys. Rev.
D86 , 075011 (2012), arXiv:1206.5779 [hep-ph].[32] F. S. Queiroz,
Proceedings, Workshop on Dark Mat-ter, Neutrino Physics and Astrophysics CETUP* 2013:7th International Conference on Interconnection be-tween Particle Physics and Cosmology (PPC 2013):Lead/Deadwood, South Dakota, USA, July, 8-13, 2013 ,AIP Conf. Proc. , 83 (2014), arXiv:1310.3026[astro-ph.CO].[33] C. Kelso, C. A. de S. Pires, S. Profumo, F. S. Queiroz,and P. S. Rodrigues da Silva, Eur. Phys. J.
C74 , 2797(2014), arXiv:1308.6630 [hep-ph].[34] S. Profumo and F. S. Queiroz, Eur. Phys. J.
C74 , 2960(2014), arXiv:1307.7802 [hep-ph].[35] C. Kelso, H. N. Long, R. Martinez, and F. S. Queiroz,Phys. Rev.
D90 , 113011 (2014), arXiv:1408.6203 [hep-ph].[36] P. V. Dong, N. T. K. Ngan, and D. V. Soa, Phys. Rev.
D90 , 075019 (2014), arXiv:1407.3839 [hep-ph].[37] P. V. Dong, D. T. Huong, F. S. Queiroz, and N. T.Thuy, Phys. Rev.
D90 , 075021 (2014), arXiv:1405.2591[hep-ph].[38] D. Cogollo, A. X. Gonzalez-Morales, F. S. Queiroz, andP. R. Teles, JCAP , 002 (2014), arXiv:1402.3271[hep-ph].[39] R. Martnez, J. Nisperuza, F. Ochoa, and J. P. Rubio,Phys. Rev.
D90 , 095004 (2014), arXiv:1408.5153 [hep-ph].[40] R. Martinez, J. Nisperuza, F. Ochoa, J. P. Rubio,and C. F. Sierra, Phys. Rev.
D92 , 035016 (2015),arXiv:1411.1641 [hep-ph].[41] P. S. Rodrigues da Silva, (2014), arXiv:1412.8633 [hep-ph].[42] P. V. Dong, C. S. Kim, D. V. Soa, and N. T. Thuy,Phys. Rev.
D91 , 115019 (2015), arXiv:1501.04385 [hep-ph].[43] R. Martinez and F. Ochoa, JHEP , 113 (2016),arXiv:1512.04128 [hep-ph].[44] D. T. Huong and P. V. Dong, (2016), arXiv:1605.01216[hep-ph].[45] C. A. de S. Pires, P. S. Rodrigues da Silva, A. C. O.Santos, and C. Siqueira, (2016), arXiv:1606.01853 [hep-ph].[46] A. G. Dias, V. Pleitez, and M. D. Tonasse, Phys. Rev. D67 , 095008 (2003), arXiv:hep-ph/0211107 [hep-ph].[47] A. G. Dias and V. Pleitez, Phys. Rev.
D69 , 077702(2004), arXiv:hep-ph/0308037 [hep-ph].[48] V. Q. Phong, H. N. Long, V. T. Van, and N. C. Thanh,Phys. Rev.
D90 , 085019 (2014), arXiv:1408.5657 [hep-ph].[49] V. Q. Phong, H. N. Long, V. T. Van, and L. H. Minh,Eur. Phys. J.
C75 , 342 (2015), arXiv:1409.0750 [hep-ph].[50] H. N. Long, (2015), 10.3844/pisp.2016.1.14,arXiv:1501.01852 [hep-ph].[51] A. Palcu, Mod. Phys. Lett.
A22 , 939 (2007), arXiv:hep-ph/0701066 [hep-ph].[52] P. V. Dong and H. N. Long, Phys. Rev.
D77 , 057302(2008), arXiv:0801.4196 [hep-ph].[53] P. T. Giang, L. T. Hue, D. T. Huong, and H. N. Long,Nucl. Phys.
B864 , 85 (2012), arXiv:1204.2902 [hep-ph].[54] L. T. Hue, D. T. Huong, and H. N. Long, Nucl. Phys.
B873 , 207 (2013), arXiv:1301.4652 [hep-ph]. [55] L. T. Hue, H. N. Long, T. T. Thuc, andT. Phong Nguyen, Nucl. Phys.
B907 , 37 (2016),arXiv:1512.03266 [hep-ph].[56] V. Van Vien, Braz. J. Phys. , 467 (2015), [Erratum:Braz. J. Phys.45,no.6,807(2015)], arXiv:1506.07388[hep-ph].[57] A. C. B. Machado, J. Montao, and V. Pleitez, (2016),arXiv:1604.08539 [hep-ph].[58] R. M. Fonseca and M. Hirsch, JHEP , 003 (2016),arXiv:1606.01109 [hep-ph].[59] J. Schechter and J. W. F. Valle, Phys. Rev. D25 , 2951(1982).[60] D. Cogollo, H. Diniz, C. A. de S. Pires, and P. S. Ro-drigues da Silva, Mod. Phys. Lett.
A23 , 3405 (2009),arXiv:0709.2913 [hep-ph].[61] D. Cogollo, H. Diniz, C. A. de S. Pires, and P. S.Rodrigues da Silva, Eur. Phys. J.
C58 , 455 (2008),arXiv:0806.3087 [hep-ph].[62] C. A. de S. Pires, F. S. Queiroz, and P. S. Rodrigues daSilva, Phys. Rev.
D82 , 105014 (2010), arXiv:1002.4601[hep-ph].[63] A. Alves, E. Ramirez Barreto, A. G. Dias, C. A.de S. Pires, F. S. Queiroz, and P. S. Rodrigues da Silva,Phys. Rev.
D84 , 115004 (2011), arXiv:1109.0238 [hep-ph].[64] A. Alves, E. Ramirez Barreto, A. G. Dias, C. A.de S. Pires, F. S. Queiroz, and P. S. Rodrigues da Silva,Eur. Phys. J.
C73 , 2288 (2013), arXiv:1207.3699 [hep-ph].[65] W. Caetano, C. A. de S. Pires, P. S. Rodrigues da Silva,D. Cogollo, and F. S. Queiroz, Eur. Phys. J.
C73 , 2607(2013), arXiv:1305.7246 [hep-ph].[66] A. E. Crcamo Hernndez, R. Martnez, and F. Ochoa,(2013), arXiv:1309.6567 [hep-ph].[67] C. Kelso, P. R. D. Pinheiro, F. S. Queiroz, and W. Shep-herd, Eur. Phys. J.
C74 , 2808 (2014), arXiv:1312.0051[hep-ph].[68] D. Cogollo, F. S. Queiroz, and P. Vasconcelos, Mod.Phys. Lett.
A29 , 1450173 (2014), arXiv:1312.0304 [hep-ph].[69] A. Doff and A. A. Natale, Phys. Rev.
D87 , 095004(2013), arXiv:1303.3974 [hep-ph].[70] A. C. B. Machado, J. C. Montero, and V. Pleitez, Phys.Rev.
D88 , 113002 (2013), arXiv:1305.1921 [hep-ph].[71] F. S. Queiroz and W. Shepherd, Phys. Rev.
D89 ,095024 (2014), arXiv:1403.2309 [hep-ph].[72] A. Alves, A. Berlin, S. Profumo, and F. S. Queiroz,Phys. Rev.
D92 , 083004 (2015), arXiv:1501.03490 [hep-ph].[73] A. J. Buras and F. De Fazio, JHEP , 010 (2016),arXiv:1512.02869 [hep-ph].[74] A. Doff, Eur. Phys. J. C76 , 33 (2016), arXiv:1512.05493[hep-ph].[75] A. Doff and C. Siqueira, Phys. Lett.
B754 , 294 (2016),arXiv:1512.03256 [hep-ph].[76] C. A. de S. Pires, J. G. Rodrigues, and P. S.Rodrigues da Silva, Phys. Lett.
B759 , 322 (2016),arXiv:1602.08126 [hep-ph].[77] J. G. Ferreira, C. A. de S. Pires, P. S. Rodrigues daSilva, and A. Sampieri, Phys. Rev.
D88 , 105013 (2013),arXiv:1308.0575 [hep-ph].[78] G. De Conto and V. Pleitez, (2016), arXiv:1606.01747[hep-ph]. [79] A. E. Crcamo Hernndez, H. N. Long, and V. V. Vien,Eur. Phys. J.
C76 , 242 (2016), arXiv:1601.05062 [hep-ph].[80] A. J. Buras and F. De Fazio, JHEP , 115 (2016),arXiv:1604.02344 [hep-ph].[81] D. Cogollo, A. V. de Andrade, F. S. Queiroz, andP. Rebello Teles, Eur. Phys. J. C72 , 2029 (2012),arXiv:1201.1268 [hep-ph].[82] R. Foot, H. N. Long, and T. A. Tran, Phys. Rev.
D50 ,R34 (1994), arXiv:hep-ph/9402243 [hep-ph].[83] H. N. Long, Phys. Rev.
D54 , 4691 (1996), arXiv:hep-ph/9607439 [hep-ph].[84] H. N. Long, Phys. Rev.
D53 , 437 (1996), arXiv:hep-ph/9504274 [hep-ph].[85] J. Kopp, P. A. N. Machado, M. Maltoni, andT. Schwetz, JHEP , 050 (2013), arXiv:1303.3011 [hep-ph].[86] M. C. Gonzalez-Garcia, M. Maltoni, and T. Schwetz,Nucl. Phys. B908 , 199 (2016), arXiv:1512.06856 [hep-ph].[87] J. Bergstrom, M. C. Gonzalez-Garcia, M. Maltoni, andT. Schwetz, JHEP , 200 (2015), arXiv:1507.04366[hep-ph].[88] F. Queiroz, C. A. de S. Pires, and P. S. R. da Silva,Phys. Rev. D82 , 065018 (2010), arXiv:1003.1270 [hep-ph].[89] C. A. d. S. Pires, (2014), 10.3844/pisp.2015.33.41,arXiv:1412.1002 [hep-ph].[90] A. G. Dias, C. A. de S. Pires, P. S. Rodrigues daSilva, and A. Sampieri, Phys. Rev.
D86 , 035007 (2012),arXiv:1206.2590.[91] S. M. Boucenna, J. W. F. Valle, and A. Vicente, Phys.Rev.
D92 , 053001 (2015), arXiv:1502.07546 [hep-ph].[92] T. Jezo, M. Klasen, D. R. Lamprea, F. Lyonnet, andI. Schienbein, JHEP , 092 (2014), arXiv:1410.4692[hep-ph].[93] T. Jeo, M. Klasen, D. R. Lamprea, F. Lyonnet, andI. Schienbein (2015) arXiv:1508.03539 [hep-ph].[94] M. Klasen, F. Lyonnet, and F. S. Queiroz, (2016),arXiv:1607.06468 [hep-ph].[95] F. del Aguila, J. de Blas, and M. Perez-Victoria, JHEP , 033 (2010), arXiv:1005.3998 [hep-ph].[96] G. Aad et al. (ATLAS), Phys. Rev. D90 , 052005 (2014),arXiv:1405.4123 [hep-ex]. [97] C. Salazar, R. H. Benavides, W. A. Ponce, and E. Ro-jas, JHEP , 096 (2015), arXiv:1503.03519 [hep-ph].[98] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Mal-toni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli,and M. Zaro, JHEP , 079 (2014), arXiv:1405.0301[hep-ph].[99] J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, andT. Stelzer, JHEP , 128 (2011), arXiv:1106.0522 [hep-ph].[100] H.-L. Lai, J. Huston, S. Mrenna, P. Nadolsky, D. Stump,W.-K. Tung, and C. P. Yuan, JHEP , 035 (2010),arXiv:0910.4183 [hep-ph].[101] N. Garron, in CKM unitarity triangle. Proceedings, 6thInternational Workshop, CKM 2010, Warwick, UK,September 6-10, 2010 (2011) arXiv:1102.1671 [hep-lat].[102] S. Durr et al. , Phys. Lett.
B705 , 477 (2011),arXiv:1106.3230 [hep-lat].[103] J. Beringer et al. (Particle Data Group), Phys. Rev.
D86 , 010001 (2012).[104] M. Misiak, S. Pokorski, and J. Rosiek, Adv. Ser. Di-rect. High Energy Phys. , 795 (1998), arXiv:hep-ph/9703442 [hep-ph].[105] G. Barenboim, F. J. Botella, and O. Vives, Phys. Rev. D64 , 015007 (2001), arXiv:hep-ph/0012197 [hep-ph].[106] A. Datta, Phys. Rev.
D78 , 095004 (2008),arXiv:0807.0795 [hep-ph].[107] N. Cabibbo,
Meeting of the Italian School of Physics andWeak Interactions Bologna, Italy, April 26-28, 1984 ,Phys. Rev. Lett. , 531 (1963), [,648(1963)].[108] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. ,652 (1973).[109] K. A. Olive et al. (Particle Data Group), Chin. Phys. C38 , 090001 (2014).[110] F. Pisano, D. Gomez Dumm, F. Pisano, andV. Pleitez, Mod. Phys. Lett. A9 , 1609 (1994),arXiv:hep-ph/9307265 [hep-ph].[111] H. N. Long and V. T. Van, J. Phys. G25 , 2319 (1999),arXiv:hep-ph/9909302 [hep-ph].[112] R. H. Benavides, Y. Giraldo, and W. A. Ponce, Phys.Rev.