Deep learnig analysis of the inverse seesaw in a 3-3-1 model at the LHC

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(1)JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.1 (1-17). Physics Letters B ••• (••••) ••••••. 1. 1. Contents lists available at ScienceDirect. 2. 2 3. 3. Physics Letters B. 4. 4 5. 5. 6. 6. www.elsevier.com/locate/physletb. 7. 7. 8. 8. 9. 9. 10. 10. 11 12 13. Deep learning analysis of the inverse seesaw in a 3-3-1 model at the LHC. 11 12 13 14. 14 15 16 17 18 19 20. a. b. c. D. Cogollo , F.F. Freitas , C.A. de S. Pires , Yohan M. Oviedo-Torres. a, c. , P. Vasconcelos. d. 15 16. a. 17. Departamento de Física, Universidade Federal de Campina Grande, Caixa Postal 10071, 58429-900, Campina Grande, Paraíba, Brazil b Departamento de Física, Universidade de Aveiro and CIDMA, Campus de Santiago, 3810-183 Aveiro, Portugal c Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970, João Pessoa, PB, Brazil d Unidade Acadêmica de Engenharia de Produção - CDSA, Universidade Federal de Campina Grande, Caixa Postal 112, 58540-000, Sumé, PB, Brazil. 18 19 20 21. 21 22. a r t i c l e. i n f o. 22. a b s t r a c t. 23. 23 24 25 26 27 28. Article history: Received 14 August 2020 Received in revised form 28 October 2020 Accepted 2 November 2020 Available online xxxx Editor: A. Ringwald. Inverse seesaw is a genuine TeV scale seesaw mechanism. In it active neutrinos with masses at eV scale requires lepton number be explicitly violated at keV scale and the existence of new physics, in the form of heavy neutrinos, at TeV scale. Therefore it is a phenomenologically viable seesaw mechanism since its signature may be probed at the LHC. Moreover it is successfully embedded into gauge extensions of the standard model as the 3-3-1 model with the right-handed neutrinos. In this work we revisit the implementation of this mechanism into the 3-3-1 model and employ deep learning analysis to probe such setting at the LHC and, as main result, we have that if its signature is not detected in the next LHC running with energy of 14 TeVs, then, the vector boson Z of the 3-3-1 model must be heavier than 4 TeVs. © 2020 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .. 29 30 31 32 33. 24 25 26 27 28 29 30 31 32 33. 34. 34. 35. 35. 36. 36. 37 38 39 40 41 42 43 44 45 46 47. 37. 1. Introduction. 38. Seesaw mechanisms [1–4] are seem as the simplest proposals to solve the long-standing problem of the smallness of the neutrino masses. Recently researchers have focused their investigations on phenomenologically viable seesaw mechanisms, as inverse seesaw one [4], since their signatures may be probed at the LHC [5]. The distinguishable aspect of the inverse seesaw (ISS) mechanism is the fact that it is a genuine TeV scale seesaw mechanism and according to the original idea [4] its implementation requires the addition of six new neutrinos (N i R , S iL with i = 1, 2, 3) to the standard model particle content composing the following bilinear terms [6],. L = −ν¯ L m D N R − S¯ L M N R −. 48 49 50 51 52 53 54. 1 2. 0 Mν = ⎝ mD 0. m TD 0 M. (1). 57 58 59. ⎞. 0 MT ⎠ .. (2). 64 65 66. 43 44 45 47 49 50 51 53 54 55. Considering the hierarchy μ << m D << M, the diagonalization of this 9 × 9 mass matrix provides the following effective neutrino mass matrix for the standard neutrinos:. mν = m TD ( M T )−1 μ M −1 m D .. 56 57 58. (3). 59 60 61. 61 63. 42. 52. μ. 60 62. 41. 48. 55 56. 40. 46. S¯ L μ S L C + H.c.,. where m D , M and μ are generic 3 × 3 complex mass matrices. These terms can be arranged in the following 9 × 9 neutrino mass matrix in the basis (ν L , N LC , S L ),. ⎛. 39. E-mail addresses: diegocogollo@df.ufcg.edu.br (D. Cogollo), felipefreitas@ua.pt (F.F. Freitas), cpires@fisica.ufpb.br (C.A. de S. Pires), ymot@estudantes.ufpb.br (Y.M. Oviedo-Torres), pablo.wagner@professor.ufcg.edu.br (P. Vasconcelos). https://doi.org/10.1016/j.physletb.2020.135931 0370-2693/© 2020 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .. 62 63 64 65 66.

(2) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.2 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 1 2 3 4 5 6 7 8 9 10 11 12 13 14. The double suppression by the mass scale connected with M turns it possible to have such scale much below than that one involved in the canonical seesaw mechanism [1–3]. It happens that standard neutrinos with mass at sub-eV scale are obtained for m D at electroweak scale, M at TeV scale and μ at keV scale. In this case all the new six neutrinos may develop masses around TeV scale or less, and their mixing with the standard neutrinos is modulated by the ratio m D M −1 . The core of the ISS mechanism is that the smallness of the neutrino masses is guaranteed by assuming that the μ scale is small and, in order to bring heavy neutrino masses down to TeV scale, it has to be at the keV scale. In this regard it was showed in [7] that the SU (3)C × SU (3) L × U (1) N with right-handed neutrinos (331RHN) [8] has the main ingredients for realizing the ISS mechanism. However, a probe of the ISS mechanism in 331RHN at the LHC is missing. The proposal of this work is to complete this job and probe the ISS in 331RHN at the LHC. For this purpose we review the model, the mechanism, and employ deep learning to probe the signature of the mechanism at the LHC by means of the production of these new neutrinos and their detection in the form of leptons as final products. This work is organised as follows: in Sec. 2 we revised the implementation of the ISS into the 331RHN and present the charged and neutral currents of interest for our analysis. In Sec. 3 we perform our analysis by applying deep learning techniques to probe both the ISS and the 331RHN. In Sec. 4 we present our conclusions.. 19 20. 23. In order to implement the ISS mechanism into the 3311RHN we have to add three left-handed neutral fermions in the singlet form to the original leptonic content of model,. ⎛. 26 27 28 29. ν. Q iL. 2. u i R ∼ 3, 1,. 44 45 46 47 48 49 50 51 52 53 54 55. Q 3L. 64 65 66. 8 9 10 11 12 13 14. ⎛. 0. . . , di R ∼ 3, 1, −. 1. . . , di R ∼ 3, 1, −. 3. 1 3. 2. . 26 27 28 29. , d3 R ∼. 3. 38 40 41. 3, 1, −. ⎛. ∗ ( L bL )n G ab lmn ( L aL )lc m. . ρ. 35. 39. . ⎞. 1. . , T R ∼ 3, 1 , −. 3. 2. . 42. (7). .. 3. ρ+. 43 44 45. ⎞. ⎛. 0. 46. ⎞. 47 48 49 50 51 52 53 54 55. 1 + G ab L¯ aL χ ( N bL )C + ( N L )C μ N L + H.c., . 2. 56 57. (8). 1. 58 59. where a, b = 1, 2, 3, i , j = 1, 2 and l, m, n = 1, 2, 3. For the sake of simplicity, we consider charged leptons in a diagonal basis. Observe that the last line of this lagrangian includes the terms that trigger the ISS mechanism. As usual, we assume that only η0 , ρ 0 and χ 0 develop vacuum expectation values (VEVs) other than zero and we consider the following expansions around the VEVs: 0. 25. 37. + h3a Q¯ 3L ηua R + g 3a Q¯ 3L ρ da R + hia Q¯ i L ρ ∗ ua R + ya L¯ a L ρ ea R. 0. 23. 36. −LY = f i j Q¯ i L χ ∗ dj R + ( f 33 Q¯ 3L χ T R ) + g ia Q¯ i L η∗ da R. 0. 22. 34. (6). ,. ± , Z and the photon A plus five new ones U 0 , U , W ± and Z . The gauge sector is composed by the standard ones, W μ μ μ μ μ μ μ This particle content allows the following Yukawa interactions,. 2. 20. 33. . 0†. −. 19. 32. η χ 1 2 1 η = ⎝ η− ⎠ ∼ (1, 3, − ), ρ = ⎝ ρ 0 ⎠ ∼ (1, 3, ), χ = ⎝ χ − ⎠ ∼ (1, 3, − ). 3 3 3 + 0 0 ρ η χ. 1. 18. 31. The scalar sector keeps the original content,. 59. 63. (5). u3 1 ⎠ ⎝ , = d3 ∼ 3, 3, 3 T L. u 3 R ∼ 3, 1,. 58. 62. 7. 30. 3. . 57. 61. 6. 24. N aL ∼ (1, 1, 0) ,. ⎞. ⎛. 56. 60. (4). where i = 1, 2 while the third family will transform as triplet,. 42 43. ;. di = ⎝ −u i ⎠ ∼ (3, 3∗ , 0), di L. . 39 41. 3. ⎞. 36. 40. 5. 21. L. ⎛. 34. 38. 1. . where a = 1, 2, 3 which corresponds to three families of leptons. For completeness reasons, we present the quark content. As it is well known, in the quark sector, two families must transform as anti-triplet. This is so to cancel anomalies. Here we make the following choice:. 33. 37. . laR ∼ (1, 1, −1) ,. 31. 35. ⎞. C a. 30 32. νa. L aL = ⎝ la ⎠ ∼ 1, 3, −. 24 25. 4. 17. 21 22. 3. 16. 2. Some essential points of the model and of the mechanism. 17 18. 2. 15. 15 16. 1. 60 61 62 63 64 65. η , ρ , χ → √ ( v η,ρ ,χ + R η,ρ ,χ + i I η,ρ ,χ ).. (9). 2. 2. 66.

(3) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.3 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 1. With this set of VEVs, the last line of the Yukawa Lagrangian above provides the following mass terms for the neutrinos:. 2 3 4 5 6 7 8 9 10 11 12 13 14 15. Lνmass = ν¯ R m D ν L + ν¯ R M N L +. 18 19 20 21 22 23. 26 27 28 29 30. Lνmass =. 1 2. 8. (12). 41. 14. 44 45 46 47 48 49. . 18. 0 M. (14). μ. m D 3×3 03×3. . . . , M R 6×6 =. MTD 3×6 M R 6×6. 03×3. M D 6×3. M 3T×3. 03×3 M 3×3. 51 52 53 54 55 56 57 58 59. . μ3×3. (15). ,. W M ν9×9 W =.

(4) −1. 32. ⎞.

(5) −1 (M D )† (M R )† ⎟.

(6) −1 ⎠ 1 −1 † † 1 − 2 (M R ) M D (M D ) (M R ). MD. 33 34. (17). 66. 35 36 37. mlight 3×3 06×3. 03×6 mheav y. . 39. (18). ,. 6×6. (19). Observe that the matrix in Eq. (18) is not diagonal. It is a block diagonal matrix. The diagonalization of the mass matrix in Eq. (16) is done through the unitary matrix V = W U , such that V T M ν9×9 V = mdiag , with U defined as:. U P MN S 0. 0 UR. . 43 44 46 47 48 49 50. (20). ,. 51 52. with U P M N S being the PNMS matrix that diagonalizes mlight while U R diagonalizes mheav y , and mdiag is the diagonal mass matrix with nine eigenvalues. The explicit form of V is. . ⎛. ⎜ 1− V ⎜ ⎝. 41. 45. mlight = m TD ( M )−1 μ( M T )−1m D. . 40 42. 1 −1 where mlight = −M TD M− R M D and mheav y = M R . When we plug M D and M R in mlight we obtain the canonical inverse seesaw mass expression for the standard neutrinos:. 1 (M D )† 2.

(7) −1 M R (M R )† MD U P N M S −1. −(M R ). MD U P N M S. †. (M D ) (M R ). 1−. †.

(8) −1. 1 (M R )−1 M D (M D )† 2. ⎞. UR. (M R ). †.

(9) −1 . UR. ⎟ ⎟. ⎠. 53 54 55 56 57. (21). 58 59 60 61. In the end of the day we have. . 63 65. 30 31. 61. 64. 26. 38. . 60 62. 25. 29. −(M R )−1 M D. T. U=. 23. 28. (16). such that,. 22. 27. . 1 † † ⎜ 1 − 2 (M D ) M R (M R ). W ⎝. 20. 24. This last matrix can be block diagonalized. For this purpose let us definife the matrix W ,. ⎛. 19 21. so that we have the following block matrix where M R is supposed invertible,. 50. 15 17. 0 MT ⎠ .. 42 43. 13. ⎞. m TD. 39 40. 10 12. (13). 37 38. 9 11. This is the mass matrix that characterize the ISS mechanism. The hierarchy M m D μ provides a seesaw relation for the masses of the standard neutrinos. In order to see this it is useful to define the matrices,. M ν9×9 =. 7. 16. 0 Mν = ⎝ mD 0. 34 36. 6. (11). ( S L )c M ν S L + H.c.,. ⎛. 33 35. 5. with the mass matrix M ν having the texture,. M D 6×3 =. 3 4. with M ab and m Dab being Dirac mass matrices,. with this last one being antisymmetric. Considering the basis S L = ν L , ν C L , N L we can write Lνmass in the following form. 31 32. (10). vχ M ab = G ab √ 2 vρ m Dab = G ab √ 2. 24 25. 2. 2. ( N L )c μ N L + H.c.. where the 3 × 3 matrices are defined as. 16 17. 1. 1. U T W T Mν W U =. mν 0. 0 mR. 62. . 63. (22). ,. 64 65. with mν = diag (m1 , m2 , m3 ) and m R = diag (m4 , .... , m9 ).. 66. 3.

(10) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.4 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 1 2 3. The matrix V connects the flavor basis S L =. T. L. . 1. νaL = U P M N S − (M D )† M R (M R )†. 5. 2. 6. ζbL =. 7 8 10. . 1 T νL , ν C L , N L =(νL , ζL )T with the physical one which we call n L = (n0iL , nkL ) where n0i L. with i = 1, 2, 3 and nk1 with k = 1, 2, ..., 6. The relation between flavor and mass eigenstates, S L = V n L , is given explicitly by.. 4. 9. .

(11) −1. . MD U P M N S ai. . V =. 12. 2. V V NN. V Nν. . T. T −1. mlight = G (G ). −1. μ(G ) G. v2 ρ. 2. 32 33 34 35 36 37 38 39 40. 11. G =⎝ 0 . 0 and. 46 48 49. m 1 0,. 54. with. U P MN S. 59 60 61 62 63 64 65 66. 0 g 22 0. ⎛. 0 0.019 0 ⎠⎝ 0 g 33 0. g 12 0 − g 23. ⎞. 23 24 25 26 27 28 29 30 31. (28). 32 33 34 35. ⎞. 36 37 38 39 40 41. 0 0 0.07 0 ⎠, 0 0.04. ⎛. 0 g 13 g 23 ⎠ ⎝ −4.26 × 10−3 0 −4.97 × 10−3. m2 ≈ 8.7 × 10. −3. (29). 42 43 44. 4.26 × 10−3 0 −6.62 × 10−3. m3 ≈ 4.8 × 10. eV,. −2. eV,. ⎞ −3. 4.97 × 10 6.62 × 10−3 ⎠ . 0. 46 47. (30). μ presented above, the diagonalization of the mass matrix (31). 1.0 × 10−5 4.3 × 10−5 3.4 × 10−5. 51 53 54 55 56. 0.80 0.58 0.12 ⎝ −0.48 0.52 0.70 ⎠ . 0.34 −0.62 0.70. 9.6 × 10−6 ⎝ η = 1.0 × 10−5 3.0 × 10−6. 50 52. (32). 57 58 59. This U P M N S implies in the following mixing angles θ12 = 36o , θ23 = 45o and θ13 = 7o which recover the experimental values in Eq. (27). Let us check if the values for G and G above are in accordance with non-unitarity constraint [10]. On substituting the set of values of G and G in η yields,. ⎛. 48 49. ⎞. ⎛. 56 58. (27). With these set of values for G, G and for the values of the VEVs v, v χ and mlight in Eq. (26) furnishes. 53. 57. 22. 45. ⎛. 0 G = ⎝ − g 12 − g 13. 47. 55. 20. ⎞. 44. 52. 19 21. 2. ⎞. 17 18. To simplify our job we consider v η = v ρ = v. Thus, the constraint v 2η + v 2ρ = (246GeV)2 implies v = 174 GeV. It is supposed that v χ lies around TeV. Here we assume 5 TeV. We also consider μ = 0.3 I keV where I is the identity matrix. Regarding the Yukawa couplings G and G , we consider the scenario where G is diagonal but non-degenerate. Moreover, for the sake of simplicity, we neglect CP phases and consider the charged leptons in a diagonal basis. All this considered, as illustrative case we take,. 43. 51. (26). 2.0 × 10−3 3.5 × 10−5 8.0 × 10−3 |η| < ⎝ 3.5 × 10−5 8.0 × 10−4 5.1 × 10−3 ⎠ , 8.0 × 10−3 5.1 × 10−3 2.7 × 10−3.

(12) −1 where η = 12 (M D )† M R (M R )† MD .. 42. 50. 15 16. ⎛. g. 12 14. 2. .. 2. ⎛. 11 13. Moreover, the current status of neutrino physics allows that at least one of the three neutrinos may be massless. Returning to our model, in it the masses of the active neutrinos are obtained by diagonalizing mlight in Eq. (26) which involves many free parameters in the form of Yukawa couplings G and G . With such a large set of free parameters, there is a great deal of possible solutions that lead to the correct neutrino mass spectrum and mixing in Eq. (27). However due to the non-unitarity of the mixing matrix V νν any set of values for the entries in G and G that do the job must obey the following constraints [10],. 41. 45. 10. sin (2θ12 ) 0.86 , sin (2θ23 ) 0.92 , sin (2θ13 ) 0.092.. 24. 31. 9. m221 7.59 × 10−5 eV2 , m231 2.43 × 10−3 eV2 ,. 23. 30. vχ. 7 8. Remember that G is an anti-symmetric matrix implying that one eigenvalue of the neutrino mass matrix in Eq. (26) is null. Solar, reactor, accelerator and atmospheric neutrino experiments have determined [9],. 22. 29. 2. 5 6. (24). (25). 2. 21. 28. 1 nkL .. .. 18. 27. bk. ak. 4. (23). Returning to mlight , on substituting m D = √G v ρ and M = √ v χ , we obtain. 17. 26. . 1 nkL ;. G. 16. 25. UR. 2 3. . νN. νν V. 15. 20. n0iL + (M D )† (M R )†.

(13) −1.

(14) −1 1 −(M R )−1 M D U P M N S bi n0iL + U R − (M R )−1 M D (M D )† (M R )† UR. 13. 19. For simplicity, we will define the matrix V in the following form:. 11. 14. . 1. ⎞. 3.0 × 10−6 3.4 × 10−5 ⎠ , 4.5 × 10−5. 60 61 62 63 64. (33). 65 66. 4.

(15) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.5 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 1 2 3 4 5 6 7 8 9 10 11. which respect the bounds in Eq. (28). Regarding the six new neutrinos, by diagonalizing mheav y = M R in Eq. (15), our illustrative example yields (n11L , n16L ), with masses. 1. neutrinos is due to the small value of the lepton number violating parameter μ”. So we developed the basic aspects of the implementation of the ISS mechanism within the 331RHN and presented an illustrative example that recovers the current experimental results involving neutrino oscillation. Our wish now is to probe this scenario at the LHC. We do this by means of the production of pairs of heavy neutrinos, n1i , and L their subsequent detection in the form of leptons as main final products. The main contributions for the processes we study are those ± intermediated by the standard charged gauge boson W and Z . The neutral and charged currents of interest are presented below. For previous studies of the signature of the inverse seesaw mechanism in other scenarios, see Refs.: [11]. We present, first, the charged current with W ± which are composed by the following terms,. 4. ∼ 373.28 GeV, (n12L , n15L ) with masses ∼ 220.84 GeV and (n13L , n14L ) with masses around ∼ 96.32 GeV. “The mass degeneracy of sterile. 12 13 14. 3 3 g . Ln W = − √. 15 17. Lnn Z = − G. 23 24 25. +. 26. 29. 2 cos θ W. 6 3 . n1. 32. with G =. 2 1−2 sin θW. Lnn Z =F. 34. g 2 cos θ W. +. 35. 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54. 3−4 sin θ W. 61 62 63 64 65 66. n1. kL. 16. 22 23. μ 1 km γ nmL ] Z μ ,. 24 25. ( V ν N† V ν N ). 26 27 29. 3 .

(16) n¯0 iL ( V N ν )bi ( V N ν )bj γ μn0jL +. 6 3 . 30.

(17) 1 n¯0 iL ( V N ν )bi ( V N N )bm γ μnmL. 31 32. i ,b=1 m=1. i , j ,b=1.

(18) n1 kL γ μ ( V N N )bk ( V N ν )bj n0jL +. 6 3 . (36). 33 34.

(19) 1 n1 kL ( V N N )bk ( V N N )bm γ μnmL ]Zμ ,. 35. k,m=1 b=1. 36 37. .. 38.

(20) −1. U R ) = V ν N . Our illustrative example yields,. ⎞ 0 0 −2.8 × 10−3 1.4 × 10−3 −1.4 × 10−3 2.8 × 10−3 ⎠; V νN ⎝ 0 3.7 × 10−3 −5.5 × 10−3 5.5 × 10−3 −3.7 × 10−3 0 − 3 0 −6.3 × 10−3 6.3 × 10−3 0 −2.2 × 10−3 2.2 × 10. ⎜ −7.0 × 10−1 ⎜ ⎜ 0 ⎜ ⎜ −3.91 × 10−5 ⎜ ⎝ 7.0 × 10−1 −1.10 × 10−5. 0 0 7.0 × 10−1 −5.68 × 10−5 1.10 × 10−5 −7.0 × 10−1. 7.0 × 10−1 0 0 −7.0 × 10−1 3.91 × 10−5 −5.68 × 10−5. 7.0 × 10−1 0 0 7.0 × 10−1 −3.91 × 10−5 5.68 × 10−5. 0 0 7.0 × 10−1 5.68 × 10−5 −1.10 × 10−5 7.0 × 10−1. 0. 41. 44. (37). 45 46 47. ⎞. −7.0 × 10−1 ⎟ ⎟ ⎟ 0 ⎟, 3.91 × 10−5 ⎟ ⎟ −7.0 × 10−1 ⎠ 1.10 × 10−5. 48 49 50 51 52 53 54 55 56 57. (38). that along with Eq. (37) allows us to perform the analysis for this production. Before go into the analysis, with the charged and neutral currents at hand, first thing to do is to check if our illustrative example obeys the rare lepton flavor violation (LFV) process μ → e γ constraint. Such process is allowed by the second coupling in Eq. (34). The branching ratio for the process mediated by these six heavy neutrinos is given by [12], 5. 40. 43. Such pattern of mixing is due to the simple choice of the parameters G and μ. In the next section we are going to probe the signature of this mechanism by producing the lightest new neutrinos, n13L and n14L , at the LHC. Observe that as ( V ν N )13 and ( V ν N )14 are null, then these neutrinos do not form charged currents with the electrons. For this reason the analysis done in the next section is based on the production of these neutrinos and their final products in the form of muons. Concerning neutral currents, we also explore the direct production of Z and its subsequent decay into a pair of n13L or n14L . The interactions that generate these processes are the last terms of the Eqs. (35) and (36). Our illustrative example yields the following values for the mixing matrix V N N ,. 0. 39. 42. ⎛. V NN. 15. 21. (35). In the second line of Eq. (34) there appear the mixing matrix ((M D )† (M R )†. ⎛. 11. 20. 1 n¯0 iL ( V νν † V ν N )im γ μ nmL. 57. 60. 6 6 . 10. 19. set of interactions that matter for us here. In the first line of Eq. (34) we have the mixing matrix This is the −1 1 − 12 (M D )† M R (M R )† M D U P M N S = V νν . Due to the smallness of the second term, see values in Eq. (33), we take V νν U P M N S .. 56. 59. μ 0 kj γ n jL +. 9. 28. [. 6 3 . 2 with F = 2 cos θ2W. 55. 58. kL. ( V ν N† V νν ). 8. 18. i =1 m =1. i , j =1. k =1 b , j =1. 36 38. n¯0 iL ( V νν † V νν )i j γ μn0jL +. 7. 17. and,. 3−4 sin2 θ W. 33. 37. (34). k =1 m =1. 30 31. [. k =1 j =1. 27 28. g. 6 3 . 6. 14. ai. ak. 3 . 5. 13. The neutral current interactions with Z have two contributions. The first one is. 21 22. 2. 2 a =1 k =1. 18 20. ¯aL γ μ U P M N S. 3. 12. .

(21) −1 † † − − (M D ) M R (M R ) M D U P M N S n0iL W μ 1. 6 3.

(22) −1 g 1 −. ¯aL γ μ (M D )† (M R )† −√ UR nkL Wμ + H.c.. 16. 19. 2 a =1 i =1. . 2. 58 59 60 61 62 63 64 65 66.

(23) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.6 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 2. 6 3 m 1 αW sin2 (θ W )m5μ ν N ) ( V ν N ) I ( nkL )|2 , B R (μ → e γ ) ≈ × | ( V ek μ k 256π 2m4W μ m2W k =1. 1 2 3 4 5. 2 3 4. where. 6. I (x) = −. 7 8. 1. 2x3 + 5x2 − x 4(1 − x)3. 5. −. 3x3 ln x 2(1 − x)4. (39). . 2. 11. = 3.3 × 10−2 , sin2 (θW ) = 0.231, mμ = 105 Mev, m W = 80.385 Gev, μ = − 16 3 × 10 Mev. The present values of these parameters are found in [13]. Our illustrative example provides B R (μ → e γ ) ≈ 1.4 × 10−13 . This is very close to the current bound that is B R (μ → e γ ) < 4.2 × 10−13 [14]. So, this case may be confirmed or excluded at the next. 12. running of the MEG experiment.. 9 10. In the above branching ratio expression we use. αW =. g 4π. 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34. is through the direct production of the Z and its subsequent decay into a. 20. L. The second production mechanism for the neutrinos pair of. n1i . L. n1i L. The final state for this type of channel will appear as pair of high boosted muons, pair of leptons and missing transverse. energy (μ± μ∓ ∓ ν. ± ν ) or pair of high boosted muons and 4 light jets. We investigate both channels and explore the phenomenological features of this model and how the signatures of the n1i can appear at the listed final states at the LHC.1 L To do so, we generate an UFO [15] file using the FeynRules [16]. This UFO file is latter used by the MadGraph5 [17] package to produce the hard scattering processes we want to investigate. All the hard scattering processes are further pass to Pythia version 8.1 [18] and Delphes [23] in order to hadronize and include the detector effects to make the data from of Monte-Carlo pseudo-events be as close as possible to the data produced by the LHC at 14 TeV.. | < 3.0, |η | < 2.7,. 43. ZW. 52 54. Zt b¯ → μ. 55 56. − ±. (41). 39. 47 48 49 50 51 52 54. μ e νe bb¯. 55 56. For the event selection we impose the following criteria:. 57. one electron (positron) with p eT > 25 GeV, and / E T > 15 GeV ;. 58 59. μ. 60 62. (42). a pair of μ with p T > 25 GeV each, ;. (43). a pair of μ with p T > 25 GeV each and reconstructed object W ± .. (44). μ. 61. 58 59 60 61 62 63. 63. 64. 64 66. 34. 53. − +. 57. 65. 33. 46. → μ μ e νe +. 32. 45. Zt t¯ → μ+ μ− e + νe b(e − ν¯ e b¯ ). 53. 31. 44. from the fact that in our model the couplings between W ± , n13L or W ± , n14L and μ are relatively large, allowing a sizable cross section for the production at the LHC. As consequence for this choice we have as main irreducible background the channels: +. 28. 43. L. ±. 27. 42. This choice allows us to reconstruct, with a good accuracy, the full decay chain generated by the n1i . Another reason for this choice stems. 51. 26. 41. n13L (n14L ) → μ± W ∓ , W ± → e ± νe .. 46. 25. 40. with the decay chain for the neutrino. 45. 24. 38. (40). pp → W ± → μ± n13L (n14L ),. 42. 23. 37. We focus our investigation in the production of the lightest new neutrinos. Thus, we are going to analyze the channel. 41. 22. 36. R j j ,bb,. > 0.01 .. 40. 21. 35. j ,b. p T > 20 GeV, p T > 30 GeV,. |η. 19. 30. As mentioned earlier, this is one of the main production mechanisms for the production of n1i and is displayed in Fig. 1. To investigate L this channel we generate 450000 events with 14 TeV centre of mass energy. To stay safely away from infrared and colinear divergences, we apply the basic cuts of Eq. (40) at the generation level. j ,b. 18. 29. 3.1. pp → μ± μ∓ e ± νe channel. 39. 50. L. plus missing energy (μ± μ∓ ± ν ) or 2 muons and 2 jets (μ± μ∓ j j).. 38. 49. 16 17. 37. 48. 11. through the s-channel in a proton-proton collision. In the particular case of our illustrative example, the W ± can further decay into a μ lepton and the neutrinos n1i . On the other hand, the n1i can decay into μ and W ± . Then this channel can have as final product 3 leptons. 36. 47. 10. 15 L. 35. 44. 9. 14. There are two major production channels for the n1i neutrinos. The first one is via vector gauge boson W ± , which can be produced. 16 18. 8. 13. 3. Analysis of the production mechanism and main channels. 15 17. 7. 12. 13 14. 6. 1. We remark that it is also possible that the sterile neutrinos can also be produced via vector boson fusion. However such process is subleading because of the phase space suppression. In view of this such channel is not considered. 6. 65 66.

(24) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.7 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. 10. 10. 11. 11. 12. 12. 13. 13. 14. 14. 15. 15. 16. 16. 17. 17. 18. 18 19. 19 20. 20. Fig. 1. Production of n1(3,4) at the LHC via W channel.. 21. 21 22. 22. Table 1 Cross sections, in fb, for signal and background processes after successive selection criteria of Eqs. (40) –(44).. 23. 23 24. 24 25 26 27. Process cross section (fb). Basic selection Eq. (40). Selection 1 Eq. (42). Selection 2 Eq. (43). Selection 3 Eq. (44). W±. 2.28 × 10−2. 1.54 × 10−2. 2.7 × 10−3. 1.7 × 10−3. 104.32 0.3 3.98 × 10−2. 71.82 2.845 × 10−1 2.92 × 10−2. 68.79 2.24 × 10−1 2.88 × 10−2. 47.51 2.125 × 10−1 2.13 × 10−2. → μ± n 1. 1 3 L (n4 L ). W ± Z , ( W ± → e ± νe b, Z → μ+ μ− ) t t¯ Z , (t ( W b) → e ± νe b, Z → μ+ μ− ) Ztb, (t ( W b) → e + νe b, Z → μ+ μ− ). 28 29. 25 26 27 28 29 30. 30. 31. 31 32 33 34. After we impose the selection criteria described in Eqs. (40)–(44), we are able to analyze the kinematics (dimension-full) and angular (dimension-less) observables from the final state particles produced by this channel. This analysis has the purpose of increase the significance of detecting n13L (n14L ) at the next LHC run. We choose the following observables:. 38. Table 2 Kinematic (Dimension-full) and angular (Dimension-less) observables selected to study the channel pp → μ± μ∓ e ± νe . We include dimensionless observables in two different referential frames: Center of Mass frame (top row) and n1i rest frame (bottom row), where θi , j is the angle between the respective particles from. 39. either the final state or reconstructed objects, W , n1i , and R (i , j ) is the separation in the. 37. L. L. 40. Dimension-full. 41. M (μ+ , μ− ), M (e , μ+ , μ− ) M (n1 (n1 )), M (e ± , ν ),. 42. Laboratory referential frame. 3L. 43. e. 46. T. 4L. μ. μ. n13L (n14L ). η × φ plane defined by. (φ)2 + (η)2 .. n1i. referential. e. cos(θ ), cos(θμ+ ,μ− ), cos(θμ1 , W ), e ,/ E T. cos(θμ2 , W ), cos(θμ1 ,n1 ), cos(θμ2 ,n1 ), iL. cos(θW ,n1 ), cos(θ. μ1 , / E T. iL. T. 51 52 53 54. iL. ), cos(θμ ,/E ) 1 T. L. L. cos(θμ1 ,e )n1 , cos(θμ2 ,e )n1 , cos(θ iL. iL. L. )n1 W ,/ E iL T. R (μ+ , μ− ). n1i. L. , R (μ1 , W )n1. iL. 43 44. )1, , R (μ1 , / E T n. ) 1 , R (e, / )1 R (μ2 , W )n1 , R (μ2 , / E E T n T n iL. iL. 45. iL. 46 47. iL. 48. In Table 2 we present the distributions for the observables of our analysis, and in Figs. 2–4 we display the respective distributions. One naive approach is a simple cut and count analysis using the reconstructed n13L (n14L ) from the final state muon and reconstructed W boson. However, due to the number of events for the background remained after the selection, even when we impose a cut window around the mass predicted for the n13L (n14L ), buries completely our signal. To overcome this problem we make use of a Deep learning algorithm trained to distinguish the signal over the main irreducible background using the observables described before. We present the details of the architecture and training methodology in the section 3.3.. 59 60 61. 64 65 66. 50 51 52 53 54 56 57. Another production mechanism for the n1i is through the production and subsequent decay of Z , see Fig. 5. L To investigate this channel we apply the same workflow where we generate 450000 events with 14 TeV and the same basic generation cuts described in Eq. (40). We then pass the hard scattering events through Pythia and Delphes to finally select the events based on the following selection criteria:. 58 59 60 61 62. 62 63. 49. 55. 3.2. Z channel. 57 58. 38. 42. 55 56. 37. 41. 48 50. 36. 40. ), R (μ1 , W ) R (μ1 , / E T. R (μ2 , W ), R (μ2 , / E T ), R (e , / E T ), ) R (μ1 , n1i L ), R (μ2 , n1i L ), R ( W , / E T. R (μ+ , μ− ),. cos(θ )n1 , cos(θμ1 , W )n1 , cos(θμ2 , W )n1 , e ,/ E i i i. frame. 47 49. 34. 39. Dimensionless. p T1 , p T 1 , p T 2 , p T L , p W T. 44 45. 33 35. 35 36. 32. a pair of electron and positron with p eT > 25 GeV, and / E T > 15 GeV μ. (45). a pair of μ with p T > 25 GeV each,. (46). a pair of μ with p T > 25 GeV each and two reconstructed W ± .. (47). μ. 7. 63 64 65 66.

(25) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.8 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. 10. 10. 11. 11. 12. 12. 13. 13. 14. 14. 15. 15. 16. 16. 17. 17. 18. 18. 19. 19. 20. 20. 21. 21. 22. 22. 23. 23 24. 24 25 26 27. Fig. 2. Kinematic (dimension-full) observables for the pp → μ± μ∓ e ± νe channel. The blue region represents the kinematic distribution for the signal events, while the orange, green and red lines are the Zt b¯ , W Z and t t¯ + Z respective backgrounds. The met variable corresponds to total missing transverse energy. We highlight the observables were the signal plays a dominant role on the distributions.. 25 26 27. 28. 28. 29. 29. 30. 30. 31. 31. 32. 32. 33. 33. 34. 34. 35. 35. 36. 36. 37. 37. 38. 38. 39. 39. 40. 40. 41. 41. 42. 42. 43. 43. 44. 44. 45. 45. 46. 46. 47. 47. 48. 48. 49. 49. 50. 50. 51. 51 52. 52 53. Fig. 3. Angular (dimensionless) observables for the pp → μ± μ∓ e ± νe channel. Cosine of the angle between selected particles at the Center of mass and n1i , L particle frames.. 54. The subscript indicate the observable is be taking in the n13L (n14L ) reference frame. The blue region represents the kinematic distribution for the signal events, while the. 55 56. vector direction. orange, green and red lines are the Zt b¯ , W Z and t t¯ + Z respective backgrounds. The met variable corresponds to / E T. 59 60 61 62 63 64 65 66. 54 55 56 57. 57 58. 53. The W ± bosons are reconstructed from the final state electrons and the / E T . In our simulations we set the value for the Z mass to 4 TeV and n13L (n14L ) to 96.31 GeV which are consistent with the current estimate limits [21,22] for the expected Z mass. In Figs. 6 we √ display the cross section for a given range of Z mass against the n1i ones for s = 14, 20 and 100 TeV. The region explored in this paper L. offers a sizeable cross section for the production of a Z and its subsequent decay into n1i . L For the main irreducible background we have:. 58 59 60 61 62 63 64. • Zt t¯ → μ+ μ− e + νe be − ν¯ e b¯ • Z W + W − → μ+ μ− e + νe e − ν¯ e. 65 66. 8.

(26) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.9 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. 10. 10. 11. 11. 12. 12. 13. 13. 14. 14. 15. 15. 16. 16. 17. 17. 18. 18. 19. 19. 20. 20. 21. 21. 22. 22. 23. 23. 24 25 26 27. Fig. 4. Angular (dimensionless) observables for the pp → μ± μ∓ e ± νe channel. Separation in the frames. The subscript script indicate the observable is be taking in the. n1i L. η × φ plane between selected particles at the Center of mass and n1i ,L particle. reference frame. The blue region represents the kinematic distribution for the signal events, while. vector direction. the orange, green and red lines are the Zt b¯ , W Z and t t¯ + Z respective backgrounds. The met variable corresponds to / E T. 24 25 26 27. 28. 28. 29. 29. 30. 30. 31. 31. 32. 32. 33. 33. 34. 34. 35. 35. 36. 36. 37. 37. 38. 38. 39. 39. 40. 40. 41. 41. 42. 42. 43. 43. 44. 44. 45. 45. 46. 46. 47. 47. 48. 48. 49. 49. 50. 50. 51. 51. 52. 52. 53. 53 54. 54 55. 55. Fig. 5. Production of n1(3,4) at the LHC via Z channel.. 56. 56. 57. 57. 58. 58 59. 59 60 61 62 63 64 65 66. Table 3 Cross sections, in fb, for signal and background processes after successive selection criteria of Eqs. (40) –(47).. 60 61. Process cross section (fb). Basic selection Eq. (40). Selection 1 Eq. (45). Selection 2 Eq. (46). Selection 3 Eq. (47). Z → n13L n¯ 13L W + W − Z , ( W ± → e ± νe b, Z → μ+ μ− ) t t¯ Z , (t ( W b) → e ± νe b, Z → μ+ μ− ). 4.32 × 10−2 4.0 × 10−2 3.0 × 10−1. 3.88 × 10−2 2.7 × 10−2 2.234 × 10−1. 3.8 × 10−2 2.52 × 10−2 1.81 × 10−1. 3.36 × 10−2 1.42 × 10−2 1.24 × 10−1. 9. 62 63 64 65 66.

(27) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.10 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. 10. 10. 11. 11. 12. 12. 13. 13. 14. 14. 15. 15. 16. 16. 17. 17. 18. 18. 19. 19. 20. 20. 21. 21. 22. 22. 23. 23. 24. 24. 25. 25. 26. 26. 27. 27. 28. 28. 29. 29. 30. 30. 31. 31. 32. 32. 33. 33. 34. 34. 35. 35. 36. 36. 37. 37. 38. 38. 39. 39. 40. 40. 41. 41. 42. 42. 43. 43. 44. 44. 45. 45. 46. 46. 47. 47. 48. 48. 49. 49. 50. 50. 51. 51. 52. 52. 53. 53. 54. 54 55. 55 56 57. Fig. 6. Cross section times branching ratio dependency of the processes pp → Z → n13L n¯ 13L (left) and pp → Z → n14L n¯ 14L (right) for the masses of the Z’ (y-axis) and n13L (n14L ) for 14 TeV (top row), 28 TeV (middle row) and 100 TeV (bottom row) centre of mass energy.. 60 61. This channel contains six leptons as final state particles, 4 visible (μ+ , μ− , e + , e − ) and 2 invisible (νe , ν¯ e ), which opens up the number of observables we can use to distinguish the signal over background. We choose the following dimension-full and dimensionless variables, see Table 4, and in Figs. 7–9 we display the respective distributions. 3.3. Deep learning analysis: methods and results. 66. 60 61 63 64. 64 65. 59. 62. 62 63. 57 58. 58 59. 56. After we select the events and gather the kinematic and angular information we can feed this information into a Neural Network (NN) designed to proper separate signal over background. Due to the simplicity of the data-set of our events, which store the information 10. 65 66.

(28) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.11 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 1 3. Table 4 Kinematic (Dimension-full) and angular (Dimension-less) observables selected to study the channel. We include dimensionless observables in three different referential frames: Center of Mass frame (top row), n1i rest frame (middle row) and n¯ 1i rest frame (bottom row), where θi , j is the angle. 4. between the respective particles from either the final state or reconstructed objects, W , n1i , and R (i , j ) is the separation in the. 5. by. 2. L. . L. 7. M T (e − , E T ), M T (e + , E T ), p (μ− ), p (μ+ ), p (e − ),. Laboratory referential frame. 9 10 12 13. /. T. /. T. T. p T (e + ), p T ( W + ), p T ( W − ), p T (n1iL ), p T (¯n1iL ), / ET ,. n1i referential frame L. cos(θ. e − ,/ E T. cos(θ. T. ), cos(θ. μ− ,e+. 4. T. cos(θμ+ , W − ), cos(θμ+ ,e+ ), cos(θ + ), μ ,/ ET cos(θn1 , W + ), cos(θn1 ,μ− ), cos(θn1 ,e+ ), iL. iL. iL. cos(θn¯ 1 , W − ), cos(θn¯ 1 ,μ+ ), cos(θn¯ 1 ,e− ), iL iL iL cos(θW + , W − ), cos(θμ+ ,μ− ), cos(θe+ ,e− ) cos(θ. )n1 , e + ,/ E iL T. cos(θμ− , W + )n1 , iL. 16. cos(θ. iL. iL. iL. iL. )n1 , cos(θμ− ,e + )n1 μ− , / E iL iL T. n¯ 1i referential frame L. cos(θ. )n¯ 1 , e − ,/ E iL T. cos(θμ+ , W − )n¯ 1 , iL. cos(θn¯ 1 , W − )n¯ 1 , cos(θn¯ 1 ,μ+ )n¯ 1 , iL iL iL iL cos(θ + )n¯ 1 , cos(θμ+ ,e− )n¯ 1. 19. μ ,/ ET. 20. iL. iL. + − + T ), R (e , E T ), R (e , e ), − + − + R (μ , W ), R (μ , e ), R (μ− , E. 7. R (n1iL , W + ), R (n1iL , μ− ), R (n1iL , e+ ), R (¯n1iL , W − ), R (¯n1iL , μ+ ), R (¯n1iL , e− ), R ( W + , W − ), R (μ+ , μ− ), R (e+ , e− ). 10. /. ), cos(θμ− ,/E ),. cos(θn1 , W + )n1 , cos(θn1 ,μ− )n1 ,. 18. R (e− , E. ), cos(θe+ ,/E ), cos(θe− ,e+ ),. μ− , W +. 15 17. 3. 6. Dimensionless. M (n1iL ), M (¯n1iL ), M (μ+ , μ− , e + ), M (μ+ , μ− ), M (e + , e − ). 11. 2. 5. Dimension-full. 14. η × φ plane defined. (φ)2 + (η)2 .. 6 8. L. 1. /. /T ), ), R (μ+ , W − ), R (μ+ , e+ ), R (μ+ , / E T. ) 1 , R (μ− , W + ) 1 , R (e+ , / E T n n iL. iL. R (n1iL , W + )n1 , R (n1iL , μ− )n1 , iL iL ) 1 , R (μ− , e+ ) 1 R (μ− , / E T niL. niL. ) 1 , R (μ+ , W − ) 1 , R (e− , / E T n¯ n¯ iL. iL. R (¯n1iL , W − )n¯ 1 , R (¯n1iL , μ+ )n¯ 1 , iL iL ) 1 , R (μ+ , e− ) 1 R (μ+ , / E T n¯ iL. n¯ iL. 8 9 11 12 13 14 15 16 17 18 19 20. 21. 21. 22. 22. 23. 23. 24. 24. 25. 25. 26. 26. 27. 27. 28. 28. 29. 29. 30. 30. 31. 31. 32. 32. 33. 33. 34. 34. 35. 35. 36. 36. 37. 37. 38. 38. 39. 39. 40. 40. 41. 41. 42. 42. 43. 43. 44. 44 45. 45 46 47. Fig. 7. Kinematic (dimension-full) observables for the pp → μ± μ∓ e ± νe e ∓ νe channel. The blue region represents the kinematic distribution for the signal events, while the orange and green lines are the t t¯ Z and W W Z backgrounds.. 49. 49 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66. 47 48. 48 50. 46. from the events as tables where each row corresponds to an event entry and the columns are the observables, we decide to work with a fully connected NN. However, we still have to choose some important parameters for the NN: number of layers, number of neurons, kernel initializes, etc. The decision of choose the correct parameters directly reflect the efficiency of our NN, which can be translated into significance of discovery, or not, of the particles predicted by the model. This selection is often refereed as hyperparameter optimization. A first approach is to use “brute force” to tune the hyperparameters by using a grid search, but the number of combinations and the computational time to test each one of them increases exponentially. More efficient ways beyond grid search are random sampling or using gaussian process algorithms to learn the best hyperparameters. Another way to tackle this problem is to use genetic/evolutionary algorithms, as in Ref. [19]. To test the different architectures, as well the modifications and fining tuning of the parameters, we set up an evolutionary algorithm to test the different combinations of parameters by creating a set of populations. In our case we restrict the population to 25 models, and keep the top 5 models with highest accuracy, after 5 rounds (generations) we obtain the top 3 architectures sorted by accuracy and we select the best one to continue our analysis. This full process takes around 2 hours in a NVIDIA GTX 1070 GPU. We use Tensorflow 2.0 [24] to build, train and evaluate our models. The best architecture and hyperparameters found by our genetic algorithm consist of a 5 layers NN each one with 512 neurons with a Rectified Linear Unit (a.k.a. ReLU) activation function with the exception of the top layers which consist of a layer with 4 neurons, one for each channel analysed (μ n13L (μ n14L ), Zt b¯ , W Z , t t¯ Z ), and a sigmoid as activation function. We also found that initial random weights for the layers sampled from normal distribution and L2 regularization with a value of 10−7 gives the best significance. We also found a 11. 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66.

(29) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.12 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. 10. 10. 11. 11. 12. 12. 13. 13. 14. 14. 15. 15. 16. 16. 17. 17. 18. 18. 19. 19. 20. 20. 21. 21. 22. 22. 23. 23. 24. 24. 25. 25. 26. 26. 27. 27. 28. 28. 29. 29. 30. 30. 31. 31. 32. 32. 33. 33. 34. 34. 35. 35. 36. 36. 37. 37. 38. 38. 39. 39. 40. 40. 41. 41. 42. 42. 43. 43. 44 45 46. Fig. 8. Angular (dimensionless) observables for the pp → μ± μ∓ e ± νe e ∓ νe channel. The blue region represents the angular distribution of our signal, while the orange and green lines are the t t¯ Z and W W Z backgrounds. The subscript script indicate the observable is be taking in the n1i object reconstructed reference frame. L. 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66. 46 48. 48 50. 45 47. 47 49. 44. similar architecture for the channel n13L n¯ 13L (n14L n¯ 14L ), with the only difference that at the top layer we have 3 neurons, one for each channel. (n13L n¯ 13L (n14L n¯ 14L ), t t¯ Z , W W Z ). Our data sets consist of tables where each row corresponds to an event entry and the columns are the kinematics and angular distributions we described in the sections. Due to the selection criteria 1 and 3 we impose into the signal and backgrounds events, we ended up with an imbalanced number of events for each channel, this can lead the DNN model to over-fit towards the majority class, which turns the model unable to make correct predictions for the classes we are interested. To overcome this problem we balance the original data set using Synthetic Minority Over-sampling Technique (SMOTE) [25], we first dived the original data set into 80% to generate the balance data set and 20% to use our validation set. (See Table 5.) We can evaluate the performance of our NN by look into the signal efficiency over the background rejection. The left panel of Fig. 10 show the signal efficiency and the background rejection for both channels analysed while the right panel gives us the normalized number of entries for a given NN prediction score. A simple figure to evaluate how good is the signal-background separation is the area under the ROC curve, AUC. The closer AUC is to one, the better we should expect the backgrounds can be cleaned up for a giving signal efficiency. We are interested in obtaining not only the acceptance and rejection factors, but mainly the statistical significance of the signal. To do so we can use the predictions made by our NN to estimate the number of events expected and from the number of events for each of the analysed channels get the estimate Asimov significance, which depends on the integrated luminosity and systematic uncertainties which are often disregarded in machine learning studies. The Asimov estimate of significance [20], a well-established approach to evaluate likelihood-based tests of new physics taking into account the systematic uncertainty on the background normalization, can then be used 12. 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66.

(30) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.13 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. 10. 10. 11. 11. 12. 12. 13. 13. 14. 14. 15. 15. 16. 16. 17. 17. 18. 18. 19. 19. 20. 20. 21. 21. 22. 22. 23. 23. 24. 24. 25. 25. 26. 26. 27. 27. 28. 28. 29. 29. 30. 30. 31. 31. 32. 32. 33. 33. 34. 34. 35. 35. 36. 36. 37. 37. 38. 38. 39. 39. 40. 40. 41. 41. 42. 42. 43. 43. 44 45 46. Fig. 9. Angular (dimensionless) observables for the pp → μ± μ∓ e ± νe e ∓ νe channel. The subscript script indicate the observable is be taking in the n1i object reconstructed L reference frame.. 44 45 46. 47. 47. 48. 48. 49. 49. 50. 50. 51 52 53 54 55 56. Table 5 Our data set. The first dimension corresponds to the number of events entries of each channel and the second dimension is the number of features (i.e. kinematic and angular variables). The first column shows the number of events survived after we apply the selection cuts Eq. (45) – Eq. (47), the Training (SMOTE) shows the balanced data sets after we apply the SMOTE algorithm to 80% of the original events. The Test/Validation sets are the remain 20% of the original selected events.. 51 52 53 54 55 56. 57. Process. Original. Training (SMOTE). Test/Validation. 57. 58. W ± → μ± n13L (n14L ) W ± Z , ( W ± → e ± νe b, Z → μ+ μ− ) t t¯ Z , (t ( W b) → e ± νe b, Z → μ+ μ− ) Ztb, (t ( W b) → e + νe b, Z → μ+ μ− ). (32060, 41) (205162, 41) (318443, 41) (240033, 41). (254528, (254528, (254528, (254528,. (6243, 41) (41019, 41) (63915, 41) (47963, 41). 58. Total (channel 1) Z → n13L n¯ 13L (n14L n¯ 14L ) W + W − Z , ( W ± → e ± νe b, Z → μ+ μ− ) t t¯ Z , (t ( W b) → e ± νe b, Z → μ+ μ− ). (890011, 41) (350140, 77) (159303, 77) (185562, 77). (1018112, 41) (279963, 77) (279963, 77) (279963, 77). (159140, 41) (70177, 77) (32023, 77) (36801, 77). 62. Total (channel 2). (695005, 77). (839889, 77). (139001, 77). 59 60 61 62 63 64 65 66. 13. 41) 41) 41) 41). 59 60 61 63 64 65 66.

(31) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.14 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. 10. 10. 11. 11. 12. 12. 13. 13. 14. 14. 15. 15. 16. 16. 17. 17. 18. 18. 19. 19. 20. 20. 21. 21. 22. 22. 23. 23. 24. 24. 25. 25. 26. 26. 27. 27. 28. 28. 29. 29. 30. 30. 31. 31. 32. 32. 33. 33. 34. 34. 35. 35. 36. 36. 37. 37. 38. 38. 39. 39. 40. 40. 41. 41. 42. 42. 43. 43. 44. 44. 45. 45. 46. 46. 47. 47. 48. 48. 49. 49. 50. 50. 51. 51. 52 53 54. Fig. 10. Signal efficiency over background rejections and prediction scores assigned by the Neural Network for the signal channel pp → μ± μ∓ e± νe e∓ νe (b) and their respective backgrounds.. μ± μ∓ e± νe (a) and pp →. 57 58 59 60 61 62 63 64 65 66. 53 54 55. 55 56. 52. for a more careful estimate of the signal significance at the training and testing phases of construction of the classifier. The formula of the Asimov signal significance is given by. Z A = 2 (s + b) ln. . (s + b)(b + σb2 ) b2 + (s + b)σb2. . . σb2 s − 2 ln 1 + σb b(b + σb2 ) b2. 1/2. 57 58. ,. (48). where, for a given integrated luminosity, s is the number of signal events, b is the number of background events, and the uncertainty associated with the number of background events is given by σb . In Fig. 11 we plot the estimate Asimov significance dependency over the classification score assigned by the NN. Despite the relative higher cross-section for the process pp → W → μn1iL and the 99% accuracy achieved by the NN, the overwhelm irreducible background we have for this channel dominates the uncertainties for the Asimov significance. This imposes a bigger challenge 14. 56. 59 60 61 62 63 64 65 66.

(32) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.15 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. 10. 10. 11. 11. 12. 12. 13. 13. 14. 14. 15. 15 16. 16 17 18. Fig. 11. Asimov significance versus NN classifier score for W → μ band represents the systematic uncertainties for the background.. n1i L. (left panel) and. n13L n¯ 13L (n14L n¯ 14L ). (right panel) channels for 3000 fb−1 with 1% systematic error. The blue. 18 19. 19. 20. 20 21 22 23 24 25 26. Table 6 Projected Asimov significance of Eq. (48) for integrated luminosities of 100, 300, 1000 and 3000 fb−1 at the 14 TeV LHC for the given systematic uncertainty. In the last column we show the naive combination of both LHC experiments for an integrated luminosity of 3000 fb−1 .. 31. 22 23 24. 100 fb−1. 300 fb−1. 1000 fb−1. 3000 fb−1. ATLAS+CMS combined (3 ab−1 ). 25. 1%. 0.17 ± 0.352 6.07 ± 0.608. 0.30 ± 0.610 10.51 ± 1.053. 0.54 ± 1.113 19.18 ± 1.923. 0.94 ± 1.928 33.22 ± 3.331. 1.95 ± 4.0 ( W → μn1iL ) 68.97 ± 6.915 ( Z → n1iL n¯ 1iL ). 26. → n1iL ) ( Z → n1iL n¯ 1iL ) → n1iL ) ( Z → n1iL n¯ 1iL ). 28. 5%. 29 30. 21. Systematics. 27 28. 17. 10%. 0.17 ± 0.352 6.07 ± 0.608. 0.30 ± 0.610 10.50 ± 1.054. 0.54 ± 1.113 19.17 ± 1.925. 0.94 ± 1.928 33.14 ± 3.340. 1.95 ± 4.0 ( W 68.88 ± 6.922. 0.17 ± 0.352 6.06 ± 0.608. 0.30 ± 0.610 10.50 ± 1.054. 0.54 ± 1.114 19.11 ± 1.930. 0.94 ± 1.928 32.90 ± 3.364. 1.95 ± 4.0 ( W 68.57 ± 6.956. μ. μ. 27 29 30 31. 32. 32. 33. 33. 34. 34. 35. 35. 36. 36. 37. 37. 38. 38. 39. 39. 40. 40. 41. 41. 42. 42. 43. 43. 44. 44. 45. 45. 46. 46. 47. 47. 48. 48. 49. 49 50. 50 51. Fig. 12. Luminosity (fb−1 ) versus Asimov significance for. 52. bands correspond to 2σ confidence level. The dashed lines show the luminosity milestones of 60 fb−1 (RUN 1), 150 fb−1 (RUN 2) and 3000 fb−1 .. μ n13L (μ n14L ) (left panel) and n13L n¯ 13L (n14L n¯ 14L ) (right panel) channels with 1% of background systematic error. The. 54. 54 56 57 58 59 60 61 62 63 64 65 66. 52 53. 53 55. 51. Z. n1iL n¯ 1iL. to one who intend to probe such particle using this channel alone. Meanwhile, the process pp → → offers a new window to probe not only the n1iL but the aforementioned Z boson. The smaller backgrounds cross-section and the 100% accuracy achieved by the NN allow us to safely probe this channel and estimate higher significance using current LHC luminosity. Combining all these factors if the Z is not discovery in this channel, we can exclude this model with a Z mass below 4 TeV using current LHC luminosity. However, from Fig. 6 we still have a wide range of mass to explore and use the analysis we developed so far as main guideline to constrain the parameters of the 331RHN. (See Table 6.) We can project the Asimov significance for a range of luminosity values. In Fig. 12 we have the projected significance with 1% systematic error versus the expected luminosity. The bands correspond to the projected systematic uncertainties. Due to the systematic dominance over the W → μn1iL channel, we can only achieve 3σ significance at 3000 fb−1 ; yet, the projected significance for the Z → n1iL n¯ 1iL shows a better perspective with 10.5 σ of significance using the RUN-2 luminosity and around 33 σ at 3000 fb−1 showing the sensitivity power not only of the analysis we developed, but the channel Z → n1iL n¯ 1iL as well. 15. 55 56 57 58 59 60 61 62 63 64 65 66.

(33) JID:PLB AID:135931 /SCO Doctopic: Phenomenology. [m5G; v1.297] P.16 (1-17). Physics Letters B ••• (••••) ••••••. D. Cogollo, F.F. Freitas, C.A. de S. Pires et al.. 1. 4. Conclusions. 1 2. 2 3 4 5 6 7 8 9 10. In this work we revisited, in details, the implementation of the inverse seesaw mechanism into the 3-3-1 model with right-handed neutrinos and, then, probed their signatures, in the form of heavy neutrinos, at the LHC by means of deep learning techniques. The spectrum of mass for these new neutrinos may vary from some hundreds of GeVs up to TeV scale. Our analysis considered the production of such neutrinos by means of the processes pp → W ± → μ± n1(3,4) → μ± μ∓ e ± νe and pp → Z → n1(3,4) n1(3,4) → μ+ μ− e + e − νe ν¯ e . We L L L applied deep learning techniques in conjunction with evolutionary algorithms in our analysis and concluded that the second process is much more efficient than the first one. As main result we have that the second process allows we probe not only the signal of the ISS mechanism, but also the model in question, i.e., the 331RHN. According to our analysis if the Z is not discovered in this channel, we can exclude within 6 σ at 95% of confidence level this model with a Z mass below 4 TeV using current LHC luminosity. Declaration of competing interest. 15. 20 21 22 23 24. 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66. 9 10. 14 15. 18. D. Cogollo is partly supported by the Brazilian National Council for Scientific and Technological Development (CNPq), under grants 436692/2018-0. Y.M. Oviedo-Torres acknowledges the financial support from CAPES under grants 88887.485509/2020-00. C. Pires is partly supported by the Brazilian National Council for Scientific and Technological Development (CNPq), under grants No. 304423/2017-3. F.F. Freitas is supported by the project From Higgs Phenomenology to the Unification of Fundamental Interactions PTDC/FIS-PAR/31000/2017 grant BPD-32 (19661/2019), and P. Vasconcelos was partly supported by the Brazilian National Council for Scientific and Technological Development (CNPq).. 19 20 21 22 23 24 25. References. 26 27. 27 28. 8. 17. 25 26. 7. 16. Acknowledgements. 18 19. 6. 13. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.. 16 17. 5. 12. 13 14. 4. 11. 11 12. 3. [1] M. Gell-Mann, P. Ramond, R. Slansky, in: P. van Nieuwenhuizen, D.Z. Freedman (Eds.), Supergravity, North-Holland, Amsterdam, 1979; T. Yanagida, in: O. Sawada, A. Sugamoto (Eds.), Proceedings of the Workshop on the Unified Theory and the Baryon Number in the Universe, KEK Report No. 79-18, Tsukuba, Japan, 1979; R.N. Mohapatra, G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912. [2] M. Magg, C. 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